warren siegel fields book

arXiv:hep-th/9912205 21 Dec 1999YITP-99-67 FIELDSFIELDS Warren Siegel C. N. Yang Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook, New York 11794-3840 USA mailto:[email protected] http://insti.physics.sunysb.edu/~siegel/plan.html i CONTENTS Preface::::::::::::::::::::::::::::: ivSome eld theory texts ::::::::: xviii :::::::::::::::::: :::::::::::::::::: :::::::::::::::::: PART ONE: SYMMETRY :::::::::::::::::: I. Global A.Coordinates 1.Nonrelativity :::::::::::::: 3 2.Fermions:::::::::::::::::: 7 3.Lie algebra ::::::::::::::: 11 4.Relativity:::::::::::::::: 15 5.Discrete: C, P, T ::::::::: 20 6.Conformal::::::::::::::: 23 B.Indices 1.Matrices::::::::::::::::: 28 2.Representations :::::::::: 30 3.Determinants :::::::::::: 35 4.Classical groups :::::::::: 38 5.Tensor notation :::::::::: 40 C.Representations 1.More coordinates ::::::::: 45 2.Coordinate tensors ::::::: 47 3.Young tableaux :::::::::: 51 4.Color and avor :::::::::: 53 5.Covering groups :::::::::: 58 II. Spin A.Two components 1.3-vectors::::::::::::::::: 61 2.Rotations:::::::::::::::: 64 3.Spinors:::::::::::::::::: 66 4.Indices::::::::::::::::::: 67 5.Lorentz:::::::::::::::::: 70 6.Dirac:::::::::::::::::::: 76 7.Chirality/duality ::::::::: 78 B.Poincar e 1.Field equations ::::::::::: 80 2.Examples:::::::::::::::: 83 3.Solution:::::::::::::::::: 84 4.Mass::::::::::::::::::::: 89 5.Foldy-Wouthuysen ::::::: 92 6.Twistors::::::::::::::::: 96 7.Helicity:::::::::::::::::: 98 C.Supersymmetry 1.Algebra::::::::::::::::: 103 2.Supercoordinates :::::::: 104 3.Supergroups :::::::::::: 106 4.Superconformal ::::::::: 109 5.Supertwistors ::::::::::: 110III. Local A.Actions 1.General::::::::::::::::: 115 2.Fermions:::::::::::::::: 119 3.Fields::::::::::::::::::: 120 4.Relativity::::::::::::::: 122 5.Constrained systems ::::128 B.Particles 1.Free:::::::::::::::::::: 132 2.Gauges::::::::::::::::: 136 3.Coupling:::::::::::::::: 137 4.Conservation :::::::::::: 138 5.Pair creation :::::::::::: 141 C.Yang-Mills 1.Nonabelian :::::::::::::: 144 2.Lightcone::::::::::::::: 148 3.Plane waves ::::::::::::: 152 4.Self-duality ::::::::::::: 153 5.Twistors:::::::::::::::: 156 6.Instantons:::::::::::::: 159 7.ADHM::::::::::::::::: 163 8.Monopoles:::::::::::::: 165 IV. Mixed A.Hidden symmetry 1.Spontaneous breakdown :171 2.Sigma models ::::::::::: 173 3.Coset space ::::::::::::: 176 4.Chiral symmetry :::::::: 177 5.Stuckelberg::::::::::::: 180 6.Higgs::::::::::::::::::: 182 B.Standard model 1.Chromodynamics :::::::: 185 2.Electroweak ::::::::::::: 189 3.Families::::::::::::::::: 193 4.Grand Unied Theories ::195 C.Supersymmetry 1.Chiral:::::::::::::::::: 200 2.Actions::::::::::::::::: 202 3.Covariant derivatives ::::204 4.Prepotential ::::::::::::: 207 5.Gauge actions ::::::::::: 209 6.Breaking:::::::::::::::: 211 7.Extended::::::::::::::: 214 ii :::::::::::::::::::: :::::::::::::::::::: :::::::::::::::::::: PART TWO: QUANTA :::::::::::::::::::: V. Quantization A.General 1.Path integrals ::::::::::: 221 2.Semiclassical expansion ::225 3.Propagators ::::::::::::: 229 4.S-matrices:::::::::::::: 231 5.Wick rotation ::::::::::: 235 B.Propagators 1.Particles:::::::::::::::: 239 2.Properties::::::::::::::: 242 3.Generalizations :::::::::: 245 4.Wick rotation ::::::::::: 248 C.S-matrix 1.Path integrals ::::::::::: 253 2.Graphs::::::::::::::::: 257 3.Semiclassical expansion ::262 4.Feynman rules :::::::::: 266 5.Semiclassical unitarity :::272 6.Cutting rules :::::::::::: 274 7.Cross sections ::::::::::: 277 8.Singularities ::::::::::::: 280 9.Group theory ::::::::::: 282 VI. Quantum gauge theory A.Becchi-Rouet-Stora-Tyutin 1.Hamiltonian :::::::::::: 288 2.Lagrangian :::::::::::::: 292 3.Particles:::::::::::::::: 295 4.Fields::::::::::::::::::: 296 B.Gauges 1.Radial:::::::::::::::::: 300 2.Lorentz::::::::::::::::: 303 3.Massive::::::::::::::::: 305 4.Gervais-Neveu ::::::::::: 307 5.Super Gervais-Neveu ::::310 6.Spacecone::::::::::::::: 313 7.Superspacecone ::::::::: 317 8.Background-eld :::::::: 319 9.Nielsen-Kallosh ::::::::: 325 10.Super background-eld ::327 C.Scattering 1.Yang-Mills :::::::::::::: 331 2.Recursion::::::::::::::: 335 3.Fermions:::::::::::::::: 337 4.Masses:::::::::::::::::: 339 5.Supergraphs :::::::::::: 345VII. Loops A.General 1.Dimensional renormaliz'n350 2.Momentum integration ::353 3.Modied subtractions :::357 4.Optical theorem ::::::::: 361 5.Power counting :::::::::: 363 6.Infrared divergences ::::: 367 B.Examples 1.Tadpoles:::::::::::::::: 371 2.Eective potential ::::::: 374 3.Dimensional transmut'n :377 4.Massless propagators ::::378 5.Massive propagators ::::: 381 6.Renormalization group ::385 7.Overlapping divergences :388 C.Resummation 1.Improved perturbation ::395 2.Renormalons :::::::::::: 400 3.Borel::::::::::::::::::: 403 4.1/N expansion :::::::::: 406 VIII. Gauge loops A.Propagators 1.Fermion::::::::::::::::: 412 2.Photon::::::::::::::::: 415 3.Gluon::::::::::::::::::: 416 4.Grand Unied Theories ::422 5.Supermatter :::::::::::: 425 6.Supergluon :::::::::::::: 427 7.Bosonization :::::::::::: 432 8.Schwinger model :::::::: 435 B.Low energy 1.JWKB:::::::::::::::::: 441 2.Axial anomaly :::::::::: 444 3.Anomaly cancelation ::::447 4.0!2 :::::::::::::::: 450 5.Vertex:::::::::::::::::: 452 6.Nonrelativistic JWKB :::455 C.High energy 1.Conformal anomaly ::::: 460 2.e+e!hadrons:::::::: 463 3.Parton model ::::::::::: 465 iii :::::::::::::: :::::::::::::: :::::::::::::: PART THREE: HIGHER SPIN :::::::::::::: IX. General relativity A.Actions 1.Gauge invariance :::::::: 473 2.Covariant derivatives ::::476 3.Conditions :::::::::::::: 481 4.Integration :::::::::::::: 484 5.Gravity::::::::::::::::: 488 6.Energy-momentum :::::: 491 7.Weyl scale:::::::::::::: 494 B.Gauges 1.Lorentz::::::::::::::::: 500 2.Geodesics::::::::::::::: 502 3.Axial::::::::::::::::::: 505 4.Radial:::::::::::::::::: 507 5.Weyl scale:::::::::::::: 512 C.Curved spaces 1.Self-duality ::::::::::::: 517 2.De Sitter:::::::::::::::: 518 3.Cosmology :::::::::::::: 521 4.Red shift:::::::::::::::: 523 5.Schwarzschild ::::::::::: 525 6.Experiments :::::::::::: 531 7.Black holes :::::::::::::: 534 X. Supergravity A.Superspace 1.Covariant derivatives ::::538 2.Field strengths :::::::::: 543 3.Compensators ::::::::::: 547 4.Scale gauges :::::::::::: 549 B.Actions 1.Integration :::::::::::::: 555 2.Ectoplasm:::::::::::::: 558 3.Component transform'ns 561 4.Component approach ::::562 5.Duality::::::::::::::::: 565 6.Superhiggs :::::::::::::: 569 7.No-scale:::::::::::::::: 571 C.Higher dimensions 1.Dirac spinors :::::::::::: 574 2.Wick rotation ::::::::::: 577 3.Other spins ::::::::::::: 580 4.Supersymmetry ::::::::: 582 5.Theories:::::::::::::::: 585 6.Reduction to D=4 ::::::: 588XI. Strings A.Scattering 1.Regge theory :::::::::::: 595 2.Classical mechanics ::::: 598 3.Gauges::::::::::::::::: 601 4.Quantum mechanics ::::: 606 5.Anomaly:::::::::::::::: 609 6.Tree amplitudes ::::::::: 611 B.Symmetries 1.Massless spectrum ::::::: 618 2.Reality and orientation ::620 3.Supergravity :::::::::::: 621 4.T-duality::::::::::::::: 622 5.Dilaton::::::::::::::::: 624 6.Superdilaton :::::::::::: 626 7.Conformal eld theory ::628 8.Triality::::::::::::::::: 633 C.Lattices 1.Spacetime lattice :::::::: 637 2.Worldsheet lattice ::::::: 641 3.QCD strings :::::::::::: 643 XII. Mechanics A.OSp(1,1j2) 1.Lightcone::::::::::::::: 649 2.Algebra::::::::::::::::: 652 3.Action:::::::::::::::::: 655 4.Spinors::::::::::::::::: 657 5.Examples::::::::::::::: 659 B.IGL(1) 1.Algebra::::::::::::::::: 664 2.Inner product ::::::::::: 665 3.Action:::::::::::::::::: 667 4.Solution::::::::::::::::: 670 5.Spinors::::::::::::::::: 673 6.Masses:::::::::::::::::: 674 7.Background elds ::::::: 675 8.Strings:::::::::::::::::: 677 9.Relation to OSp(1,1 j2)::682 C.Gauge xing 1.Antibracket ::::::::::::: 685 2.ZJBV::::::::::::::::::: 688 3.BRST:::::::::::::::::: 692 AfterMath :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 696 iv ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: PREFACE ::::::::::::::::::::::::::::::: Scientic method Although there are many ne textbooks on quantum eld theory, they all have various shortcomings. Instinct is claimed as a basis for most discussions of quantum eld theory, though clearly this topic is too recent to aect evolution. Their subjectiv- ity more accurately identies this as fashion :( 1 )T h e old-fashioned approach justies itself with the instinct of intuition . However, anyone who remembers when they rst learned quantum mechanics or special relativity knows they are counter-intuitive;quantum eld theory is the synthesis of those two topics. Thus, the intuition in thiscase is probably just habit : Such an approach is actually historical ortraditional , recounting the chronological development of the subject. Generally the rst half (orvolume) is devoted to quantum electrodynamics, treated in the way it was viewed in the 1950's, while the second half tells the story of quantum chromodynamics, as it was understood in the 1970's. Such a \dualistic" approach is n ecessarily redundant, e.g., using canonical quantization for QED but path-integral quantization for QCD,contrary to scientic principles, which advocate applying the same \unied" methodsto all theories. While some teachers may feel more comfortable by beginning a topicthe way they rst learned it, students may wonder why the course didn't begin with the approach that they will wind up using in the end. Topics that are unfamiliar to the author's intuition are often labeled as \formal" (lacking substance) or even\mathematical" (devoid of physics). Recent topics are usually treated there as ad-vanced: The opposite is often true, since explanations simplify with time, as the topicis better understood. On the positive side, this approach generally presents topicswith better experimental verication. (2) In contrast, the fashionable approach is described as being based on the in- stinct of beauty . But this subjective beauty of artis not the instinctive beauty of nature, and in science it is merely a consolation. Treatments based on this approachare usually found in review articles rather than textbooks, due to the shorter life ex-pectancy of the latest fashion. On the other hand, this approach has more imagination than the traditional one, and attempts to capture the future of the subject. A related issue in the treatment of eld theory is the relative importance of con- cepts vs.calculations : (1) Some texts emphasize the concepts, including those which have not proven of practical value, but were considered motivational historically (in the traditional approach) or currently (in the artistic approach). However, many ap-proaches that were once considered at the forefront of research have faded into oblivionnot because they were proven wrong by experimental evidence or lacked conceptual v attractiveness, but because they were too complex for calculation, or so vague they lacked predicitive ability. Some methods claimed total generality, which they used to prove theorems (though sometimes without examples); but ultimately the only usefulproofs of theorems are by construction. Often a dualistic, two-volume approach isagain advocated (and frequently the author writes only one of the two volumes): Likethe traditional approach of QED volume + QCD volume, some prefer concept volume+ calculation volume. Generally, this means that gauge theory S-matrix calculationsare omitted from the conceptual eld theory course, and left for a \particle physics"course, or perhaps an \advanced eld theory" course. Unfortunately, the particlephysics course will nd the specialized techniques of gauge theory too technical tocover, while the advanced eld theory course will frighten away many students by itstitle alone. (2) On the other hand, some authors express a desire to introduce Feynman graphs as quickly as possible: This suggests a lack of appreciation of eld theory outside ofdiagrammatics. Many essential aspects of eld theory (such as symmetry breakingand the Higgs eect) can be seen only from the action, and its analysis also leads to better methods of applying perturbation theory than those obtained from a xed set of rules. Also, functional equations are often simpler than pictorial ones, especiallywhen they are nonlinear in the elds. The result of over-emphasizing the calculationsis a cookbook, of the kind familiar from some lower-division undergraduate courses intended for physics majors but designed for engineers. The best explanation of a theory is the one that ts the principles of scientic method : simplicity, generality, and experimental verication. In this text we thus take a more economical orpragmatic approach, with methods based on eciency and power. Unattractiveness or counter-intuitiveness of such methods become ad-vantages, because they force one to accept new and better ways of thinking aboutthe subject: The eciency of the method directs one to the underlying idea. Forexample, although some consider Einstein's original explanation of special relativityin terms of relativistic trains and Lorentz transformations with square roots as be-ing more physical, the concept of Minkowski space gave a much simpler explanationand deeper understanding that proved more useful and led to generalization. Manytheories have \miraculous cancelations" when traditional methods are used, which led to new methods (background eld gauge, supergraphs, spacecone, etc.) that not only incorporate the cancelations automatically (so that the \zeros" need not be cal-culated), but are built on the principles that explain them. We place an emphasison such new concepts, as well as the calculational methods that allow them to becompared with nature. It is important not to neglect one for the sake of the other,articial and misleading to try to separate them. vi As a result, many of our explanations of the standard topics are new to textbooks, and some are completely new: For example, (1) we derive the Foldy-Wouthuysen transformation by dimensional reduction from an analogous one for the massless case(subsections IIB3,5). (2) We derive the Feynman rules in terms of background eldsrather than sources (subsection VC1); this avoids the need for amputation of exter-nal lines for S-matrices or eective actions, and is more useful for background-eldgauges. (3) We obtain the nonrelativistic QED eective action, used in modern treat-ments of the Lamb shift (because it makes perturbation easier than the older Bethe-Salpeter methods), by eld redenition of the relativistic eective action (subsectionVIIIB6), rather than tting parameters by comparing Feynman diagrams from therelativistic and nonrelativistic actions. (In general, manipulations in the action areeasier than in diagrams.) (4) We present a somewhat new method for solving for the curvature in general relativity that is slightly easier than all previous methods (subsections IXA2,C5). There are also some completely new topics, like: (1) the anti-Gervais-Neveu gauge, where spin in U(N) Yang-Mills is treated in almost the sameway as internal symmetry | with Chan-Paton factors (subsection VIB4); (2) thesuperspacecone gauge, the simplest gauge for QCD (subsection VIB7); and (3) a new\(almost-)rst-order" superspace action for supergravity, analogous to the one forsuper Yang-Mills (subsection XB1). We try to give the simplest possible calculational tools, not only for the above reasons, but also so group theory (internal and spacetime) and integrals can be per-formed with the least eort and memory. (Some traditionalists may claim that theold methods are easy enough, but their arguments are less convincing when the orderof perturbation is increased. Even computer calculations are more ecient when leftas a last resort.) We give examples of (and excercises on) these methods, but notexhaustively. We also include more recent topics (or those more recently appreciatedin the particle physics community) that might be deemed non-introductory, but are commonly used, and are simple and important enough to include at the earliest level. For example, the related topics of (unitary) lightcone gauge, twistors, and spinorhelicity are absent from all eld theory texts, and as a result no such text performsthe calculation of as basic a diagram as the 4-gluon tree amplitude. Another missingtopic is the relation of QCD to strings through the random worldsheet lattice andlarge-color (1/N) expansion, which is the only known method that might quantita-tively describe its high-energy nonperturbative behavior (bound states of arbitrarilylarge mass). This text is meant to cover all the eld theory every high energy theorist should know, but not all that any particular theorist might need to know. It is not meant asan introduction to research, but as a preliminary to such courses: We try to ll in the vii cracks that often lie between standard eld theory courses and advanced specialized courses. For example, we have some discussion of string theory, but it is more oriented toward the strong interactions, where it has some experimental justication, ratherthan quantum gravity and unication, where its usefulness is still under investigation.We do not mention statistical mechanics, although many of the eld theory methodswe discuss are useful there. Also, we do not discuss any experimental results in detail;phenomenology and analysis of experiments deserve their own text. We give and applythe methods of calculation and discuss the qualitative features of the results, but donot make a numerical comparison to nature: We concentrate more on the \forest"than the \trees". Unfortunately, our discussions of the (somewhat related) topics of infrared-diver- gence cancelation, Lamb shift, and the parton model are sketchy, due to our inabilityto give fully satisfying treatments | but maybe in a Second Edition? Unlike all previous texts on quantum eld theory, this one is available for free over the Internet (as usual, from xxx.lanl.gov and its mirrors), and may be periodically updated. Errata, additions, and other changes will be posted on my web page at http://insti.physics.sunysb.edu/~siegel/plan.html until enough are accumulated fora new edition. Electronic distribution not only makes it more available, but easierto update to new editions, and you don't have to bother to lug your copy with youwhen you go anywhere (like home) that has a computer (so you can access it froma cartridge/diskette or the internet). It also oers the option of reading it on thecomputer, which saves space and trees (and the gures are nicer in color). ThePDF version allows searches that are more general than the Index, and includes an\outline" window with clickable \bookmarks" that is more convenient than the Tableof Contents, as well as the usual web links to xxx.lanl.gov (and a couple of otherplaces). Highlights This text also diers from others in most of the following ways: (1) We place a greater emphasis on mechanics in introducing some of the more elementary physical concepts of eld theory: (a) Some basic ideas, such as antiparticles, can be more sim-ply understood already with classical mechanics. (b) Some interactions can also betreated through rst-quantization: This is sucient for evaluating certain tree andone-loop graphs as particles in external elds. Also, Schwinger parameters can beunderstood from rst-quantization: They are useful for performing momentum inte-grals (reducing them to Gaussians), studying the high-energy behavior of Feynmangraphs, and nding their singularities in a way that exposes their classical mechanics viii interpretation. (c) Quantum mechanics is very similar to free classical eld the- ory, by the usual \semiclassical" correspondence between particles (mechanics) and waves (elds). They use the same wave equations, since the mechanics Hamiltonianor Becchi-Rouet-Stora-Tyutin operator is the kinetic operator of the correspondingclassical eld theory, so the free theories are equivalent. In particular, (relativistic)quantum mechanical BRST provides a simple explanation of the o-shell degreesof freedom of general gauge theories, and introduces concepts useful in string theory.As in the nonrelativistic case, this treatment starts directly with quantum mechanics,rather than by (rst-)quantization of a classical mechanical system. Since supersym-metry and strings are so important in present theoretical research, it is useful to havea text that includes the eld theory concepts that are prerequisites to a course onthese topics. (For the same reason, and because it can be treated so similarly to Yang-Mills, we also discuss general relativity.) (2) We also emphasize conformal invariance . Although a badly broken sym- metry, the fact that it is larger than Poincar e invariance makes it useful in many ways: (a) General classical theories can be described most simply by rst analyzing conformal theories, and then introducing mass scales by various techniques. This is particularly useful for the general analysis of free theories, and for constructingactions for supergravity theories. (b) Quantum theories that are well-dened withinperturbation theory are conformal (\scaling") at high energies. (A possible excep-tion is string theories, but the supposedly well understood string theories that arenite perturbatively have been discovered to be hard-to-quantize membranes in dis-guise nonperturbatively.) This makes methods based on conformal invariance usefulfor nding classical solutions, as well as studying the high-energy behavior of thequantum theory, and simplifying the calculation of amplitudes. (c) Theories whoseconformal invariance is not (further) broken by quantum corrections avoid certainproblems at the nonperturbative level. Thus conformal theories ultimately may be required for an unambiguous description of high-energy physics. ( 3 )W em a k ee x t e n s i v eu s eo f two-component (chiral) spinors , which are ubiqui- tous in particle physics: (a) The method of twistors (more recently dubbed \spinorhelicity") greatly simplies the Lorentz algebra in Feynman diagrams for massless(or high-energy) particles with spin, and it's now a standard in QCD. (Twistors are also related to conformal invariance and self-duality.) On the other hand, most texts still struggle with 4-component Dirac (rather than 2-component Weyl) spinornotation, which requires gamma-matrix and Fierz identities, when discussing QCDcalculations. (b) Chirality and duality are important concepts in all the interactions:Two-component spinors were rst found useful for weak interactions in the days of4-fermion interactions. Chiral symmetry in strong interactions has been important ix since the early days of pion physics; the related topic of instantons (self-dual solutions) is simplied by two-component notation, and general self-dual solutions are expressed in terms of twistors. Duality is simplest in two-component spinor notation, even whenapplied to just the electromagnetic eld. (c) Supersymmetry still has no convincingexperimental verication (at least not at the moment I'm typing this), but its the-oretical properties promise to solve many of the fundamental problems of quantumeld theory. It is an element of most of the proposed generalizations of the StandardModel. Chiral symmetry is built into supersymmetry, making two-component spinorsunavoidable. (4) The topics are ordered in a more pedagogical manner: (a) Abelian and non- abelian gauge theories are treated together using modern techniques. (Classical grav-ity is treated with the same methods.) (b) Classical Yang-Mills theory is discussed be-fore any quantum eld theory. This allows much of the physics, such as the StandardModel (which may appeal to a wider audience), of which Yang-Mills is an essentialpart, to be introduced earlier. In particular, symmetries and mass generation in theStandard Model appear already at the classical level, and can be seen more easily from the action (classically) or eective action (quantum) than from diagrams. (c) Only the method of path integrals is used for second-quantization. Canonical quantizationis more cumbersome and hides Lorentz invariance, as has been emphasized even byFeynman when he introduced his diagrams. We thus avoid such spurious concepts asthe \Dirac sea", which supposedly explains positrons while being totally inapplica-ble to bosons. However, for quantum physics of general systems or single particles,operator methods are more powerful than any type of rst-quantization of a classicalsystem, and path integrals are mainly of pedagogical interest. We therefore \review"quantum physics rst, discussing various properties (path integrals, S-matrices, uni-tarity, BRST, etc.) in a general (but simpler) framework, so that these properties neednot be rederived for the special case of quantum eld theory, for which path-integral methods are then sucient as well as preferable. (5)Gauge xing is discussed in a way more general and ecient than older meth- ods: (a) The best gauge for studying unitarity is the (unitary) lightcone gauge. Thisrarely appears in eld theory texts, or is treated only half way, missing the importantexplicit elimination of all unphysical degrees of freedom. (b) Ghosts are introduced by BRST symmetry, which proves unitarity by showing equivalence of convenient and manifestly covariant gauges to the manifestly unitary lightcone gauge. It can beapplied directly to the classical action, avoiding the explicit use of functional determi-nants of the older Faddeev-Popov method. It also allows direct introduction of moregeneral gauges (again at the classical level) through the use of Nakanishi-Lautrupelds (which are omitted in older treatments of BRST), rather than the functional x averaging over Landau gauges required by the Faddeev-Popov method. (c) For non- abelian gauge theories the background eld gauge is a must. It makes the eective action gauge invariant, so Slavnov-Taylor identities need not be applied to it. Beta functions can be found from just propagator corrections. (6)Dimensional regularization is used exclusively (with the exception of one-loop axial anomaly calculations): (a) It is the only one that preserves all possible sym-metries, as well as being the only one practical enough for higher-loop calculations. (b) We also use it exclusively for infrared regularization, allowing all divergences to be regularized with a single regulator (in contrast, e.g., to the three regulators used for the standard treatment of Lamb shift). (c) It is good not only for regularization, but renormalization (\dimensional renormalization"). For example, the renormaliza- tion group is most simply described using dimensional regularization methods. More importantly, renormalization itself is performed most simply by a minimal prescrip- tion implied by dimensional regularization. Unfortunately, many books, even amongthose that use dimensional regularization, apply more complicated renormalization procedures that require additional, nite renormalizations as prescribed by Slavnov- Taylor identities. This is a needless duplication of eort that ignores the manifestgauge invariance whose preservation led to the choice of dimensional regularization in the rst place. By using dimensional renormalization, gauge theories are as easy to treat as scalar theories: BRST does not have to be applied to amplitudes explicitly, since the dimensional regularization and renormalization procedure preserves it. (7) Perhaps the most fundamental omission in most eld theory texts is the expansion of QCD in the inverse of the number of colors : (a) It provides a gauge- invariant organization of graphs into subsets, allowing simplications of calculations at intermediate stages, and is commonly used in QCD today. (b) It is useful as a perturbation expansion, whose experimental basis is the Okubo-Zweig-Iizuka rule.(c) At the nonperturbative level, it leads to a resummation of diagrams in a way that can be associated with strings, suggesting an explanation of connement. Notes for instructors This text is intended for reference and as the basis for a full-year course on rela- tivistic quantum eld theory for second-year graduate students. A preliminary version of the rst two parts was used for a one-year course I taught at Stony Brook. Thechapter on gravity and pieces of early chapters cover a one-semester graduate rela- tivity course I gave several times here | I used most of the following: IA, IB3, IC2, IIA, IIIA-C5, VIB1, IX, XIA2-4, XIB4-5. The prerequisites (for the quantum eld xi theory course) are the usual rst-year courses in classical mechanics, classical elec- trodynamics, and quantum mechanics. For example, the student should be familiarwith Hamiltonians and Lagrangians, Lorentz transformations for particles and elec- tromagnetism, Green functions for wave equations, SU(2) and spin, and Hilbert space. Unfortunately, I nd that many second-year graduate students (especially many who got their undergraduate training in the USA) still have only an undergraduate level of understanding of the prerequisite topics, lacking a working knowledge of actionprinciples, commutators, creation and annihilation operators, etc. While most such topics are brie y reviewed here, they should be learned elsewhere. There is far more material here than can be covered comfortably in one year, mostly because of included material that should be covered earlier, but rarely is. Ideally, a modern curriculum for eld theory students should include: (1) courses on classical mechanics, nonrelativistic quantum mechanics, and classical electrodynamicsin the rst semester of graduate study, without overly reviewing aspects that should have been covered in undergraduate study (and in particular avoiding the enormous overlap of the last two subjects due to both covering primarily the solution of wave equations); (2) in the second semester, statistical mechanics as the sequel to clas- sical, relativistic quantum mechanics as the sequel to nonrelativistic, and classicalnonabelian eld theory (Yang-Mills and gravity) as the sequel to classical electrody- namics; (3) in the second year, one year of quantum eld theory, and at least one semester on \phenomenology" (model-building and direct comparison with observa- tions, including those for general relativity and cosmology); and (4) in the third year, more specialized courses, such as a semester on supersymmetry and strings. Unfor- tunately, in practice little of relativistic quantum mechanics and classical eld theory (other than electromagnetism) will have been covered previously, which means theywill comprise half of the \quantum" eld theory course, while the true quantum eld theory will be squeezed into the last half. One way to cut the material to t a one-year course is to omit Part Three, which can be left for a third semester on \advanced quantum eld theory"; then the rstsemester (Part One) is classical while the second (Part Two) is quantum. Further- more, the ordering of the chapters is somewhat exible: The \ ow" is indicated by the following \3D" plot: 8 < :lower spin &. higher spinclassical ! quantum symmetry elds quantize loop Bose I III V VII # IX XI XX I I Fermi II IV VI VIII xii where the 3 dimensions are spin (\ j"), quantization (\ h"), and statistics (\ s"): The three independent ows are down the page, to the right, and into the page. (The thirddimension has been represented as perpendicular to the page, with \higher spin" insmaller type to indicate perspective, for legibility.) To present these chapters in the1 dimension of time we have classied them as jhs, but other orderings are possible: jhs: I II III IV V VI VII VIII IX X XI XII jsh: I III V VII II IV VI VIII IX XI X XII hjs: I II III IV IX X V VI XI XII VII VIII hsj: I II III IX IV X V XI VI XII VII VIII sjh: I III V VII IX XI II IV VI VIII X XII shj: I III IX V XI VII II IV X VI XII VIII (However, the spinor notation of II is used for discussing instantons in III, so some rearrangement would be required, except in the jhs,hjs,a n dhsjcases.) For exam- ple, the rst half of the course can cover all of the classical, and the second quantum,dividing Part Three between them ( hjsor hsj). Another alternative ( jsh)i sao n e - semester course on quantum eld theory, followed by a semester on the Standard Model, and nishing with supergravity and strings. Although some of these (espe- cially the rst two) allow division of the course into one-semester courses, this shouldnot be used as an excuse to treat such courses as complete: Any particle physicsstudent who was content to sit through another entire year of quantum mechanics in graduate school should be prepared to take at least a year of eld theory. Notes for students Field theory is a hard course. (If you don't think so, name me a harder one at this level.) But you knew as an undergraduate that physics was a hard major. Students who plan to do research in eld theory will nd the topic challenging; those with less enthusiasm for the topic may nd it overwhelming. The main dierence between eldtheory and lower courses is that it is not set in stone: There is much more variation instyle and content among eld theory courses than, e.g., quantum mechanics courses, since quantum mechanics (to the extent taught in courses) was pretty much nished in the 1920's, while eld theory is still an active research topic, even though it has hadmany experimentally conrmed results since the 1940's. As a result, a eld theorycourse has the avor of research: There is no set of mathematically rigorous rules to solve any problem. Answers are not nal, and should be treated as questions: One should not be satised with the solution of a problem, but consider it as a rst steptoward generalization. The student should not expect to capture all the details of xiii eld theory the rst time through, since many of them are not yet fully understood by people who work in the area. (It is far more likely that instead you will discover details that you missed in earlier courses.) And one reminder: The only reason for lectures (including seminars and conferences) is for the attendees to ask questions (and not just in private), and there are no stupid questions (except for the infamous \How many questions are on the exam?"). Only half of teaching is the responsibility of the instructor. Outline The preceding Table of Contents lists the three parts of the text: Symmetry, Quanta, and Higher Spin. Each part is divided into four chapters, each of which has three sections, divided further into subsections. Each section is followed by references to reviews and original papers. Excercises appear throughout the text, immediately following the items they test: This purposely disrupts the ow of the text, forcing the reader to stop and think about what he has just learned. These excercises areinteresting in their own right, and not just examples or memory tests. This is not a crime for homeworks and exams, which at least by graduate school should be about more than just grades. The rst part of the text focuses on symmetry: The Poincar e group is special relativity, and is sucient to nd all free equations of motion for particles and elds. Internal symmetries include both global ones, used for classifying particles, and local ones, which describe the interactions of elds. The rst chapter discusses global symmetry, both spacetime and internal. Space- time symmetries covered include not only Poincar e but also Galilean (i.e., nonrel- ativistic, used as an introduction) and conformal (broken in nature, but still very useful). Parity, time reversal, and even charge conjugation are described simply in terms of classical mechanics. Lightcone bases are introduced. Some general proper- ties of Lie algebras are summarized, including fermions and anticommuting numbers. Classical groups are described using tensor methods and index notation, including Young tableaux. Dirac gamma matrices appear as coordinates for orthogonal groups. The color and avor symmetries of the particles of the Standard Model, and observed light hadrons, are given as examples. The second chapter extends the rst chapter's treatment of spacetime symmetry to include spin. The methods introduced in this chapter are the most ecient ones for handling Lorentz indices in QCD (or even pure Yang-Mills theory). Two-component spinor notation is introduced by the rotation group in three (space) dimensions: Ten- sor notation avoids Clebsch-Gordan-Wigner coecients. The study of the simple xiv algebraic properties of 2 2 matrices is extended straightforwardly from three dimen- sions to four, and applied to simple examples in free eld theory. The conformal group gives an easy and unied way to nd massless free eld equations in general,and dimensional reduction does the same for masses. As an application, we discussthe Foldy-Wouthuysen transformation (and its massless analog) for arbitrary spin,with minimal electromagnetic coupling to spin 1/2 as an example. (The case of non-minimal coupling will be useful in chapter VIII for the Lamb shift.) Twistors, relatedto conformal invariance and self-duality, yield a convenient and covariant method tosolve the massless equations, and also explain helicity. The chapter concludes with adiscussion of the general properties of supersymmetry and its representations, usingsuperspace and supertwistors. Local symmetries are covered in the third chapter. It begins with a discussion of the action principle, including fermions and constrained systems, applied to thesimple free examples given earlier (spins 1/2 and 1). The concepts of gauge invarianceand gauge xing are introduced through the simple case of the (spinless) relativisticparticle, whose classical mechanics will prove useful later in understanding severalfeatures of Feynman diagrams. As an example, classical pair creation and annihilation is examined. Finally, pure Yang-Mills theory is analyzed, including some solutions to the classical eld equations. Twistors are used to study self-duality and instantons.The lightcone gauge is used as a unitary gauge, and in combination with self-duality. Gauge symmetry is coupled to lower spins in chapter four. Following an intro- duction to spontaneous symmetry breakdown of global symmetries, nonlinear sigma models are considered as low-energy theories, particularly in the study of chiral sym-metry, and gauge invariance is used in their general construction. The use of scalarsto generate mass for vectors is illustrated rst by the free case of the St uckelberg model, generalized to the Higgs model, and applied to the Standard Model. Familiesand Grand Unied Theories are also described, as well as the basics for construct-ing actions for supersymmetric theories in superspace (including a brief discussion ofextended supersymmetry). The second part of the text covers the quantum aspects of eld theory, as revealed through perturbation theory. Although some have conjectured that nonperturbativeapproaches might solve the renormalization diculties found in perturbation, all ev-idence indicates these features survive in the complete theory. Chapter ve focuses on the method of quantization of classical theories based on path integrals. The chapter begins by considering various properties of quantumphysics in a general context | relation to canonical quantization, Wick rotation,unitarity, and causality | so that these items need not be repeated in the more xv specialized and complicated cases of eld theory. Path integrals are then applied to classical mechanics to explain the St uckelberg-Feynman propagator. Path integra- tion of eld theory produces a generating functional of background elds for Feynmandiagrams, as well as its connected and one-particle-irreducible parts. We use back-grounds elds instead of sources exclusively: All uses of Feynman diagrams involveeither the S-matrix or the eective action, both of which require the removal of ex-ternal propagators, which is equivalent to replacing sources with elds. The classical(tree) graphs are shown to give the perturbative solution to the classical eld equa-tions. Also described are the properties of the classical action needed for unitarity, thediagrammatic translation of unitarity and causality, the denition of cross sections,the relation of Landau singularities to classical mechanics, and the use of the quarkline rules for dealing with group theory in graphs easily. Throughout the chapter simple examples are given from scalar theories. Complications that arise from quantization of gauge theories are described in the sixth chapter. BRST symmetry is the easiest way to gauge x, and makes unitarityclear by relating general gauges to unitary gauges. Again a general discussion is given in the framework of quantum physics and canonical quantization, including the relation of Hamiltonian and Lagrangian approaches, so that later eld theorycan be addressed covariantly with path integrals. Various gauges are considered forYang-Mills elds: radial, Lorentz, Landau, Fermi-Feynman, unitary, renormalizable,Gervais-Neveu, anti-Gervais-Neveu, super Gervais-Neveu, spacecone, superspacecone,background-eld, and Nielsen-Kallosh. The spacecone gauge is used as the simplestmethod to calculate graphs in massless theories, with examples given from (massless)QCD, including the 4-gluon and 5-gluon tree amplitudes. The Fermi-Feynman gaugeis used to calculate all the 4-point tree amplitudes of QED, and their dierentialcross sections. The supergraph rules are derived for supersymmetric theories, andthe locality of the eective action in the anticommuting coodinates is shown to imply nonrenormalization theorems. General features of higher orders in perturbation theory due to momentum inte- gration are examined in chapter seven. Renormalization is explained (but not proven),and dimensional regularization is applied. Tadpole integrals are used to explain di-mensional transmutation through the example of the eective potential. The running of couplings with energy is shown through the evaluation of one-loop massless and massive propagator corrections. Some simple overlapping two-loop divergences areused to illustrate renormalization of subdivergences. The renormalization group equa-tions are introduced via dimensional regularization. The problems solved perturba-tively by renormalization are shown to reappear upon resummation of the expansion.Instantons and IR and UV renormalons are analyzed through a Borel transform in xvi the coupling, and the resultant ambiguities are related to nonperturbative vacuum values of composite elds. The expansion in the inverse of the number of colors is an approach to this problem, related to string theory, that also has uses at nite ordersof perturbation. Chapter eight applies these methods to gauge theories. Propagator corrections in QED and QCD are used to analyze the one-loop conformal anomaly and its relation to asymptotic freedom. Finite N=1 supersymmetric theories are considered as a solutionto the renormalon problem. The Schwinger model is given as another example fromtwo dimensions, illustrating interesting features at one loop such as bound states,bosonization, and the axial anomaly. The axial anomaly is then evaluated moregenerally, and considered in four dimensions in relation to constraints on electroweakmodels and electromagnetic pion decay. The nonrelativistic form of the eectiveaction useful for nding the Lamb shift (including the anomalous magnetic moment) isgiven as an example of vertex corrections. Finally, the production of hadrons throughelectron-positron annihilation, deep inelastic scattering, and Drell-Yan scattering arebrie y described as applications of perturbative QCD. Part Three treats general spins, particularly spin 2, which ultimately must be included in any complete theory of nature. Such spins are observed experimentallyfor bound states, but may be required also as fundamental elds. Gravity is described through the theory of general relativity in chapter nine. The classic experimental tests are described, including cosmology. The treatmentused is closely related to that applied to Yang-Mills theory, and diers from thatof most texts on gravity: (1) We emphasize the action for deriving eld equationsfor gravity (and matter), rather than treating it as an afterthought. (2) We makeuse of local (Weyl) scale invariance for cosmological solutions, gauge xing, eldredenitions, and studying conformal properties. In particular, other texts neglect the (unphysical) dilaton, which is crucial in such treatments (especially for generalization to supergravity and strings). (3) While most gravity texts leave spinors till the end,and treat them brie y, our discussion of gravity is based on methods that can beapplied directly to spinors, and therefore to supergravity and superstrings. (4) Ourmethod of calculating curvatures for purposes of solving the classical eld equationsis somewhat new, but probably the simplest, and is directly related to the simplestmethods for super Yang-Mills theory and supergravity. The approach of the previous chapter is generalized straightforwardly to super- gravity in chapter ten. Actions with matter are constructed, and are analyzed insuperspace, and in terms of component elds using component expansion and sep-aration of superconformal breaking (\compensator") terms. The spin-3/2 particle xvii is given mass by the superhiggs eect, and no-scale supergravity provides a model whereby the would-be resultant cosmological constant vanishes naturally. Extendedsupergravity is analyzed through general properties of extended supersymmetry andby reduction from higher dimensions. Strings are proposed in chapter eleven as an approach to studying the most im- portant yet least understood property of QCD: connement. Other methods havebeen proposed to study this phenomenon, but none have achieved explicit results formore than low hadron energy, which no more exhibits connement than chemistrydisproves the existence of free nuclei. Known string theories are not suitable for de-scribing hadrons quantitatively, but are useful models of observed properties, suchas Regge behavior. The classical and quantum theory of the simplest such model is analyzed, and qualitative features expected of general theories are described. The dis- cretization of the worldsheet of the string into a sum of Feynman diagrams is shownto exhibit features relevant to a string theory of hadrons. The nal chapter gives a general derivation of free actions for any gauge theory, based on adding equal numbers of commuting and anticommuting ghost dimensionsto the lightcone formulation of the Poincar e group. The usual ghost elds appear as components of the gauge elds in anticommuting directions, as do necessary auxiliaryelds like the determinant of the metric tensor in gravity. Gauge xing to the Fermi-Feynman gauge is automatic. The \antields" and \antibracket" of the Zinn-Justin-Batalin-Vilkovisky method appear naturally from the anticommuting coordinate that is the rst-quantized ghost of the Klein-Gordon equation. Following the body of the text (and preceding the Index) is the AfterMath, con- taining conventions and some of the more important equations. Acknowledgments I thank everyone with whom I have discussed eld theory, especially Gordon Chalmers, Marc Grisaru, Marcelo Leite, Martin Ro cek, Jack Smith, George Sterman, and Peter van Nieuwenhuizen. More generally, I thank the human race, without whom this work would have been neither possible nor necessary. December 20, 1999 xviii ::::::::::::: ::::::::::::: ::::::::::::: SOME FIELD THEORY TEXTS ::::::::::::: Comprehensive, traditional Complete texts; use canonical quantization for QED, then path integrals for QCD 1S. Weinberg, The quantum theory of elds , 3 v. (Cambridge University, 1995,6,9?) 609+489+c.500 pp.:First volume just QED; second volume contains many interesting topics; thirdvolume supersymmetry. By one of the developers of the Standard Model. 2M.E. Peskin and D.V. Schroeder, An introduction to quantum eld theory (Addison-Wesley, 1995) 842 pp.: Many applications; style similar to Bjorken and Drell. 3M. Kaku, Quantum eld theory: a modern introduction (Oxford University, 1993) 785 pp.:Includes introduction to supergravity and superstrings. 4C. Itzykson and J.-B. Zuber, Quantum eld theory (McGraw-Hill, 1980) 705 pp. (but with lots of small print ): Emphasis on QED. Somewhat specialized Basics, plus thorough treatment of an advanced topic 5J. Zinn-Justin, Quantum eld theory and critical phenomena , 3rd ed. (Clarendon, 1996) 1008 pp.: First 1/2 is basic text, with interesting treatments of many topics, but no S-matrix examples or discussion of cross sections; second 1/2 is statistical mechanics. 6G. Sterman, An introduction to quantum eld theory (Cambridge University, 1993) 572 pp.:First 3/4 can be used as basic text, including S-matrix examples; last 1/4 hasextensive treatment of perturbative QCD, emphasizing factorization. Basic; few S-matrix examples Should be supplemented with a \QED/particle physics text" 7L.H. Ryder, Quantum eld theory , 2nd ed. (Cambridge University, 1996) 487 pp.: Includes introduction to supersymmetry. 8D. Bailin and A. Love, Introduction to gauge eld theory ,2 n de d .( I n s t i t u t eo f Physics, 1993) 364 pp.: All the fundamentals. 9P. Ramond, Field theory: a modern primer , 2nd ed. (Addison-Wesley, 1989) 329 pp.:Short text on QCD: no weak interactions or Higgs. xix Classics Older but unconventional treatments from their originators; no Yang-Mills or Higgs 10N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized elds , 3rd ed. (Wiley, 1980) 620 pp.:Ahead of its time (1st English ed. 1959); early treatments of path integrals, causal-ity, background elds, and renormalization of all eld theories (not just QED). 11R.P. Feynman, Quantum electrodynamics: a lecture note and reprint volume (Ben- jamin, 1961) 198 pp.: Original treatment of quantum eld theory as we know it today, but from me- chanics; includes reprints of original articles (1949). QED/particle physics Numerous Feynman diagram calculations; no Yang-Mills or Higgs 12B. de Wit and J. Smith, Field theory in particle physics , v. 1 (Elsevier Science, 1986) 490 pp.:Oriented toward experimentalists (but wait till v. 2, due any day now...). 13A.I. Akhiezer and V.B. Berestetskii, Quantum electrodynamics (Wiley, 1965) 868 pp.: Extensive examples of lower-order QED diagrams. Advanced topics For further reading; including brief reviews of some standard topics 14Theoretical Advanced Study Institute in Elementary Particle Physics (TASI) pro- ceedings, University of Colorado, Boulder, CO (World Scientic):Annual collection of summer school lectures on recent research topics. 15W. Siegel, Introduction to string eld theory (World Scientic, 1988) 244 pp.: Reviews lightcone, BRST, gravity, rst-quantization, spinors, twistors, strings; besides, I like the author. 16S.J. Gates, Jr., M.T. Grisaru, M. Ro cek, and W. Siegel, Superspace: or one thou- sand and one lessons in supersymmetry (Benjamin/Cummings, 1983) 548 pp.: Covers supersymmetry, spinor notation, lightcone, St uckelberg elds, gravity, Weyl scale, gauge xing, background-eld method, regularization, and anoma-lies; same author as previous, plus three other guys whose names sound familiar. May soon be available for free where you found this book. 1 PART ONE: SYMMETRY The rst four chapters present a one-semester course on \classical eld theory". Perhaps a more accurate description would be \everything you should know before learning quantum eld theory". This is basically a study of global and local symme- tries: Classical dynamics represents only a certain limit of quantum dynamics, and not the one usually emphasized, but most of the symmetries of classical physics survive quantization. The phenomenon of symmetry breaking, and the related mechanisms ofmass generation, can also be seen at the classical level. In perturbative quantum eld theory, classical eld theory is simply the leading term in the perturbation expansion. Continuous symmetry is one of the most fundamental and important concepts of physics. In the framework of an action principle (which is required in quantumphysics), it is equivalent to conservation laws, which have been a cornerstone of physics since Newton. From a practical viewpoint, it simplies calculations by relating dier- ent solutions to equations of motion, and allowing these equations to be written moreconcisely by treating independent degrees of freedom as a single entity. In particular, local (\gauge") symmetries, which allow independent transformations at each coordi- nate point, are basic to all the fundamental interactions: All the fundamental forcesare mediated by particles described by Yang-Mills theory and its generalizations. Symmetries are the result of a redundant, but useful, description of a theory. (Note that here we refer to symmetries of a theory, not of a solution to the theory.) For example, translation invariance says that only dierences in position are measurable,not absolute position: We can't measure the position of the \origin". There are two ways to deal with this: (1) Choose an origin; i.e., make a \choice of coordinates". For example, place an object at the origin; i.e., choose the position of an object ata certain time to be the origin. (2) Work only in terms of dierences of coordinates, which are \translationally invariant". Although the latter choice is more physical, the former is usually more convenient: The use of redundant variables, together with symmetry, often gives a simpler description of a theory. Another example is quantum mechanics, where the arbitrariness of the phase of the wave function can be considereda symmetry: Although quantum mechanics can be reformulated in terms of phase- invariant probabilities, currents, or density matrices instead of wave functions, and this can be useful for some purposes of exposing physical properties, formulating andsolving the Schr odinger equation is simpler in terms of the wave function. The same applies to \local" symmetries, where there is an independent symmetry at each point of space and time: For example, quarks and gluons have a local \color" symmetry, 2 and are not (yet) observed independently in nature, but are simpler objects in terms of which to describe strong interactions than the observed hadrons (protons, neutrons, etc.), which are described by color-invariant products of quark/gluon wave functions, in the same way that probabilities are phase-invariant products of wave functions. (Note that in quantum mechanics there is a subtle distinction between observed andobserver that can obscure this symmetry if the observer is not invariant under it. This can always be avoided by choosing to dene the observer as invariant: For example, the detection apparatus can be included as part of the quantum mechanical system,while the observer can be dened as some \remote" recorder, who may be abstracted as even being translationally invariant. In practice we are less precise, and abstract even the detection apparatus to be invariant: For example, we describe the scatteringof particles in terms of the coordinates of only the particles, and deal with the origin problem as above in terms of just those coordinates.) Note that \global" (time-, and usually space-independent) symmetries can elim- inate a variable, but not its time derivative. For example, translation invarianceallows us to x (i.e., eliminate) the position of the center of mass of a system at some initial time, but not its time derivative, which is just the total momentum, whose conservation is a consequence of that same symmetry. A local symmetry, being timedependent, may allow the elimination of a variable at all times: The existence of this possibility depends on the dynamics, and will be discussed later. Of particular intrerest are ways in which symmetries can be made manifest. Fre- quently in the literature \manifest" is used vacuously; a \manifest symmetry" is anobvious one: If you know the group, the representation under consideration doesn't need to be stated, but can be seen from just the notation. (In fact, one of the main uses of index notation is just to manifest the symmetry.) Formulations where globaland local symmetries are manifest simplify calculations and their results, as well as clarifying their meaning. One of the main uses of manifest symmetry is rarely needing to explicitly perform a specic symmetry transformation. For example, one might need to examine a rela-tivistic problem in dierent Lorentz frames. Rather than starting with a description of the problem in one frame, and then explicitly transforming to another, it is much simpler to start with a manifestly covariant description, make one choice of frame,then make another choice of frame. One then never uses the messy square roots of the familiar Lorentz contraction factors (although they may appear at the end from kine- matic constraints). A more extreme example is the corresponding situation for local A. COORDINATES 3 symmetries, where such transformations are intractable in general, and one always starts with the manifestly covariant form. I. GLOBAL In the rst chapter we study symmetry in general, concentrating primarily on spacetime symmetries, but also discussing general properties that will have other applications in the following chapter. ::::::::::::::::::::::: ::::::::::::::::::::::: ::::::::::::::::::::::: A. COORDINATES ::::::::::::::::::::::: In this section we discuss the Poincar e (and conformal) group as coordinate trans- formations. This is the simplest way to represent it on the physical world. In later sections we nd general representations by adding spin. 1. Nonrelativity We begin by reviewing some general properties of symmetries, including as an example the symmetry group of nonrelativstic physics. In the Hamiltonian approach to mechanics, both symmetries and dynamics can be expressed conveniently in terms of a \bracket": the Poisson bracket for classical mechanics, the commutator for quan-tum mechanics. In this formulation, the fundamental variables (operators) are some set of coordinates and their canonically conjugate momenta, as functions of time. The (Heisenberg) operator approach to quantum mechanics then is related to classi-cal mechanics by identifying the semiclassical limit of the commutator as the Poisson bracket: For any functions AandBofpandq, the quantum mechanical commutator is ABBA=ih@A @pm@B @qm@B @pm@A @qm +O(h2) In other words, the true classical limit of ABBAis zero, since classically functions commute; thus the semiclassical limit is dened by lim h!01 h(ABBA) (which is really a derivative with respect to  h). We therefore dene the bracket for the two cases by [A;B]8 >< >:i@A @pm@B @qm@B @pm@A @qm semiclassically ABBA quantum mechanically 4I . G L O B A L The semiclassical denition of the bracket then can be applied to classical physics (where it was originally discovered). Classically AandBare two arbitrary functions of the coordinates qand momenta p; in quantum mechanics they can be arbitrary operators. We have included an \ i" in the classical normalization so the two agree in the semiclassical limit. We generally use (natural/Planck) units  h=1 ,s om a s si s measured as inverse length, etc.; when we do use an explicit  h, it is a dimensionless parameter, and appears only for dening Jeries-Wentzel-Kramers-Brillouin (JWKB) expansions or (semi)classical limits. Our indices may appear either as subscripts or superscripts, with preferences to be explained later: For nonrelativistic purposes we treat them the same. We also use the Einstein summation convention, that any repeated index in a product is summed over (\contracted"); usually we contract a superscript with a subscript: AmBmX mAmBm The denition of the bracket is equivalent to using [pm;qn]=in m (wheren mis the \Kronecker delta function": 1 if m=n,0i fm6=n) together with the general properties of the bracket [A;B]=[B;A]; [A;B]y=[Ay;By] [[A;B];C]+[ [B;C];A]+[ [C;A];B]=0 [A;BC ]=[A;B]C+B[A;C] The rst set of identities exhibit the antisymmetry of the bracket; next are the \Ja- cobi identities". In the last identity the ordering is important only in the quantummechanical case: In general, the dierence between classical and quantum mechanics comes from the fact that in the quantum case operator reordering after taking the commutator results in multiple commutators. Innitesimal symmetry transformations are then written as A=i[G;A];A 0=A+A whereGis the \generator" of the transformation. More explicitly, innitesimal gen- erators will contain innitesimal parameters: For example, for translations we have G=ipi)xi=i[G;xi]=i A. COORDINATES 5 whereiare innitesimal numbers. The most evident physical symmetries are those involving spacetime. For nonrel- ativistic particles, these symmetries form the \Galilean group": For the free particle,those innitesimal transformations are linear combinations of M=m; P i=pi;Jij=x[ipj]xipjxjpi;E =H=p2 i 2m;Vi=mxipit in terms of the position xi(i=1;2;3), momenta pi, and (nonvanishing) mass m, where [ij] means to antisymmetrize in those indices, by summing over all permuta- tions (just two in this case), with plus signs for even permutations and minus for odd.(In three spatial dimensions, one often writes J i=1 2ijkJjkto makeJi n t oav e c t o r . This is a peculiarity of three dimensions, and will lose its utility once we consider rela-tivity in four spacetime dimensions.) These transformations are the space translations(momentum) P, rotations (angular momentum | just orbital for the spinless case) J, time translations (energy) E, and velocity transformations (\Galilean boosts") V.( T h e m a s s Mis not normally associated with a symmetry, and is not conserved relativistically.) Excercise IA1.1 Let's examine the Galilean group more closely. Using just the relations for[x;p]a n d[A;BC ] (and the antisymmetry of the bracket): aFind the action on x iof each kind of inntesimal Galilean transformation. bShow that the nonvanishing commutation relations for the generators are [Jij;Pk]=ik[iPj]; [Jij;Vk]=ik[iVj]; [Jij;Jkl]=i[k [iJj]l] [Pi;Vj]=iijM; [H;Vi]=iPi For more than one free particle, we introduce an m,xi,a n dpifor each particle (but the same t), and the generators are the sums over all particles of the above ex- pressions. If the particles interact with each other the expression for His modied, in such a way as to preserve the commutation relations. If the particles also interact withdynamical elds, eld-dependent terms must be added to the generators. (External,nondynamical elds break the invariance. For example, a particle in a Coulomb po-tential is not translation invariant since the potential is centered about some point.)Note that for N particles there are 3N coordinates describing the particles, but stillonly 3 translations: The particles interact in the same 3-dimensional space. We canuse translational invariance to x the position of any one particle at a given time, butnot the rest: The dierences in position are translationally invariant. On the other 6I . G L O B A L hand, it is often useful not to x the position of any particle, since keeping this invari- ance (and the corresponding redundant variables) allows all particles to be treated equally. We might also consider using the dierences of positions themselves as thevariables, allowing a symmetric treatment of the particles in terms of translationally invariant variables: However, this would require applying constraints on the variables, since there are 3N(N-1)/2 dierences, of which only 3(N-1) are independent. We willnd similar features later for \local" invariances: In general, the most convenientdescription of a theory is with the invariance; the invariance can then be xed, or invariant combinations of variables used, appropriately for the particular application. The rotations (or at least their \orbital" parts) and space translations are exam- ples of coordinate transformations. In general, generators of coordinate transforma-tions are of the form G= i(x)pi)(x)=i[G;]=i@i where@i=@=@xiand(x) is a \scalar eld" (or \spin-0 wave function"), a function of only the coordinates. In classical mechanics, or quantum mechanics in the Heisenberg picture, time development also can be expressed in terms of the Hamiltonian using the bracket: d dtA=@ @t+iH;A =@ @tA+i[H;A] (The middle expression with the commutator of @=@t makes sense only in the quantum case, and is not dened for the Poisson bracket.) Again, this general relation is equivalent to the special cases, which in the classical limit are Hamilton's equationsof motion:dq m dt=i[H;qm]=@H @pm;dpm dt=i[H;pm]=@H @qm The Hamiltonian has no explicit time dependence in the absence of time-dependent nondyamical elds (external potentials whose time dependence is xed by hand, rather than by introducing the elds and their conjugate variables into the Hamilto-nian). Consequently, time development is itself a symmetry: Time translations are generated by the Hamiltonian; the @=@t term ind=dt term can be dropped when acting on operators without explicit time dependence. Invariance of the theory under a symmetry means that the equations of motion are unchanged under the transformation: dA dt0 =dA0 dt A. COORDINATES 7 To apply our above translation of innitesimal transformations into bracket language, we dene(d=dt)b y d dtA =d dt A+d dtA In the quantum case we can write d dt = iG;@ @t+iH ; which follows from the Jacobi identity using B=iGandC=@=@t +iH, and inserting Ainto the blank spaces of the commutators above. (The classical case can be treated similarly, except that the time derivatives are not written as brackets.) We then ndthat the generator of a symmetry transformation is conserved (constant), since 0=d dt = i@G @t[G;H ]; =idG dt; Excercise IA1.2 Show that the generators of the Galilean group are conserved, using the rela-tiond=dt =@=@t +i[H;] for the hamiltonian Hof a free particle. Solve the equations of motion for x(t)a n dp(t) in terms of initial conditions, and substi- tute into the expression for the generators to give an independent derivationof their time independence. In the cases where time dependence is not involved, symmetries can be treated in almost exactly the same way either classically or quantum mechanically using thecorresponding bracket (Poisson or commutator), by using the properties that theyhave in common. In particular, the fact that a symmetry generator G= m(x)pm is conserved means that we can solve for a component of pin terms of the constant G, and substitute the result into the remaining equations of motion. For example, translation invariance of a potential in a particular direction means that componentof the momentum is a constant ( dp 1=dt=@H=@q1= 0), rotational invariance about some axis means that component of angular momentum is a constant ( dJ=dt = @H=@ = 0), etc. 2. Fermions As we know experimentally, and we will see follows from relativistic eld theory, particles with half-integral spins obey Fermi-Dirac statistics. Let's therefore considerthe classical limit of fermions: This will lead to generalizations of the concepts ofbrackets and coordinates. Bosons obey commutation relations, such as [ x;p]=ih; 8I . G L O B A L in the classical limit they just commute. Fermions obey anticommutation relations, such asf;yg=hfor a single fermionic harmonic oscillator, where fA;Bg=AB+BA is the anticommutator. So, in the truly classical (not semiclassical) limit they an- ticommute, y+y= 0. Actually, the simplest case is a single real (hermitian) fermion: Quantum mechanically, or semiclassically, we have h=f;g=22 while classically 2= 0. There is no analog for a single boson: [ x;x]=x2x2=0 . This means that classical fermionic elds must be \anticommuting": Two such objects get a minus sign when pushed past each other. As a result, the product of two fermionic quantities is bosonic, while fermionic times bosonic gives fermionic. Excercise IA2.1 Show [B;C]=[A;D]=0) [AB;CD ]=1 2fA;Cg[B;D ]+1 2[A;C]fB;Dg To work with wave functions that are functions of anticommuting numbers, we rst must understand how to dene general properties of functions of anticommuting variables. For instance, given a single anticommuting variable , we need to be able to Taylor expand functions in , e.g., to nd a basis for the states. We therefore have an anticommuting derivative @=@ , satisfying @ @ 2 =0 from either anticommutativity or the fact functions of terminate at rst order in . We also need a integral to dene the inner product; indenite integration turns out to be enough. The most important property of the integral is integration by parts; then, when acting on any function of , Z d @ @ =0)Z d =@ @ where the normalization is xed for convenience. This also implies ( )= Excercise IA2.2 Prove this is the most general possibility for anticommuting integration by A. COORDINATES 9 considering action of integration and dierentiation on the most general func- tion of (which has only two terms). In general, when Taylor expanding a function of anticommuting variables we must preserve the statistics: If we Taylor expand a quantity that is dened to be commuting(bosonic), then the coecients of even powers of anticommuting variables will also be commuting, while the coecients of odd powers will be anticommuting (fermionic), to maintain the commuting nature of that term (the product of the variables and coecient). Similarly, when expanding an anticommuting quantity the coecients of even powers will also be anticommuting, while for odd powers it will be commuting. We can now consider operators that depend on both commuting (  m)a n da n t i - commuting ( ) classical variables, M=(m; ) Classically they satisfy the \graded" commutation relations (anticommutation if both elements are fermionic, commutation otherwise), not to be confused with the Poisson bracket, classically [M;Ng=0 :mnnm=m  m=  +  =0 This relation is then generalized to the the graded quantum mechanical commutator or Poisson bracket by [M;Ng=h MN; MN PN=M P where is constant, hermitian, and \graded antisymmetric": (MN]=0: (mn)= []= m+ m=0 For the standard normalization of canonically conjugate pairs of bosons m=i=(qi;pi) and self-conjugate fermions, we choose =; i;j=ijC;C=0 ii 0
Because of signs resulting from ordering anticommuting quantities, we dene derivatives unambiguously by their action from the left: @ @MN=N M 10 I. GLOBAL The general Poisson bracket then can be written as semiclassically [A;BgA @ @M NM@ @NB Since derivatives are normally dened to act from the left, there is a minus sign from pushing the rst derivative to the left if Aand that particular component of @=@M are both fermionic. Excercise IA2.3 Let's examine some properties of fermionic oscillators: aFor a single set of harmonic oscillators we have fa;ayg=1;fa;ag=fay;ayg=0 Show that the \number operator" ayahas the property fa;eiayag=0 (Hint: Since this system has only 2 states, the easiest way is to check the action on those states.) bDene eigenstates of the annihilation operator (\coherent states") by aji=ji whereis anticommuting. Show that this implies ayji=@ @ji;ji=eayj0i;e0ayji=j+0i;xayaji=jxi; hj0i=e*0;1=Z d*d e*jihj Dene wave functions in this space, ( )=hj i.T a y l o r e x p a n d t h e m i n , and compare this to the usual two-component representation using j0iand ayj0ias a basis. cDene the \supertrace" by str(A)=Z d*d e*hjAji Find the relation between any operator in this space and a 2 2 matrix, and nd the expression for the supertrace in terms of this matrix. dRepeat part bfor the bosonic oscillator ([ a;ay] = 1), where the Hilbert space is innite-dimensional, paying attention to signs, etc. Show that the analog of part cdenes the ordinary trace. A. COORDINATES 11 eFortwosets of fermionic oscillators, we dene fa1;ay 1g=fa2;ay 2g=1;o t h e rf;g=0 Show that the new operators ~a1=a1; ~a2=eiay1a1a2 (and their Hermitian conjugates) are equivalent to the original ones except that one set of the new oscillators commutes (not anticommutes) with the other ([~a1;~ay2] = 0, etc.), even though each set satises the same anticom- mutation relations with itself ( f~a1;~ay1g= 1, etc.). Thus, choice of statistics is relevant only for particles in the same state: at most one fermion, but unlimited bosons. (This change of oscillator basis is called a \Klein trans-formation". It can be useful for discrete sets of oscillators, but not for thoselabeled by a continuous parameter, because of the discontinuity in the com-mutation relations when the two labels are equal.) 3. Lie algebra Since the same symmetries can be expressed in terms of dierent kinds of brackets for classical and quantum theories, it can be useful to work with just those propertiesthat the Poisson bracket and commutator have in common, i.e., those that involveonly the bracket of two operators, not just their ordinary product: [A+B;C ]=[A;C]+[B;C]f o r n u m b e r s ; (distributivity) [A;B]=[B;A] (antisymmetry) [A;[B;C]] + [B;[C;A]] + [C;[A;B]] = 0 (Jacobi identity) with similar expressions (diering only by signs) for anticommutators or mixed com- mutators and anticommutators. Excercise IA3.1 Find the generalizations of the Jacobi identity using also anticommutators,corresponding to the cases where 2 or 3 of the objects involved are consideredas fermionic instead of bosonic. These properties also give an abstract denition of a form of multiplication, the \Lie bracket", which denes a \Lie algebra". (The rst property is true of algebrasin general.) Other Lie brackets include those dened by another, associative, form of 12 I. GLOBAL multiplication, such as matrix multiplication, or operator (innite matrix) multipli- cation as in quantum mechanics: In those cases we can write [ A;B]=ABBA,a n d use the usual properties of multiplication (distributivity and associativity) to derive the properties of the Lie bracket. (Another familiar example in physics is the \cross" product for three-vectors; however, this can also be expressed in terms of matrixmultiplication.) The most important use of Lie algebras for physics is for describing (continuous) innitesimal transformations, especially those describing symmetries. Excercise IA3.2 Using only the commutation relations of the generators of the Galilean group(excercise IA1.1), check all the Jacobi identities. For describing transformations, we can also think of the bracket as a derivative: The \Lie derivative" of Bwith respect to Ais dened as L AB=[A;B] As a consequence of the properties of the Lie bracket, this derivative satises the usual properties of a derivative, including the Leibniz rule. (In fact, for coordinatetransformations the Lie derivative is really a derivative with respect to the coordi- nates.) We can now dene nite transformations by exponentiating innitesimal ones: A 0(1 +iLG)A)A0= lim !0(1 +iLG)1=A=eiLGA In cases where we have [ A;B]=ABBA, we can also write eiLGA=eiGAeiG This follows from replacing Gon both sides with Gand taking the derivative with respect to, to see that both satisfy the same dierential equation with the same initial condition. We then can recognize this as the way transformations are performed in quantum mechanics: A linear transformation that preserves the Hilbert-space innerproduct must be unitary, which means it can be written as the exponential of an antihermitian operator. Just as innitesimal transformations dene a Lie algebra with elements A, nite ones dene a \Lie group" with elements g=e iG The multiplication law of two group elements follows from the fact the product of two exponentials can be expressed in terms of multiple commutators: eAeB=eA+B+1 2[A;B]+::: A. COORDINATES 13 We now have the mathematical properties that dene a group, namely: (1) a product, so that for two group elements g1andg2, we can dene g1g2, which is another element of the group (closure), (2) an identity element, so gI=Ig=g, (3) an inverse, where gg1=g1g=I, and (4) associativity, g1(g2g3)=(g1g2)g3. In this case the identity is 1 =e0, while the inverse is ( eA)1=eA. Since the elements of a Lie algebra form a vector space (we can add them and multiply by numbers), it's useful to dene a basis: G=iGi)g=eiiGi The parameters ithen also give a set of (redundant) coordinates for the Lie group. (Previously they were required to be inntesimal, for inntesimal transformations; now they are nite, but may be periodic, as determined by topological considerations that we will mostly ignore.) Now the multiplication rules for both the algebra andthe group are given by those of the basis: [G i;Gj]=ifijkGk for the (\structure") constants fijk=fjik, which dene the algebra/group (but are ambiguous up to a change of basis). They satisfy the Jacobi identity [[G[i;Gj];Gk]]=0)f[ijlfk]lm=0 A familiar example is SO(3) (SU(2)), 3D rotations, where fijk=ijkif we useGi= 1 2ijkJjk. Another useful concept is a \subgroup": If some subset of the elements of a group also form a group, that is called a \subgroup" of the original group. In particular, for a Lie group the basis of that subgroup will be a subset of some basis for the original group. For example, for the Galilean group Jijgenerate the rotation subgroup. Excercise IA3.3 Let's examine the subgroup of the Galilean group describing (spatial) coor- dinate transformations | rotations and spatial translations: aShow that the innitesimal transformations are given by xi=xjji+^i;ij=ji where the's are constants. bExponentiate to nd the nite transformations x0i=xjji+^i 14 I. GLOBAL cShow that  ijmust satisfy ikjlkl=ij both to preserve the scalar product, and as a consequence of exponentiating. (Hint: Use matrix notation, and nd the equivalent relation between  and 1.) dShow that the last equation implies det=1, while exponentiating can give onlydet = 1 (since +1 can't change continuously to 1). What is the physical interpretation of a transformation with det=1? (Hint: Consider a simple example.) These results can be generalized to include anticommutators: When some of the basis elements Giare fermionic, the corresponding parameters iare anticommuting numbers, the structure constants are dened by [ Gi;Gjg,e t c . . T h e n G=iGiis bosonic term by term, as is g, so bosons transform into bosons and fermions into fermions, but Taylor expansion in the 's will have both bosonic and fermionic co- ecients. (For example, for A=B,i fAis bosonic, then so is B, but if also is fermionic, then Bwill also be fermionic.) For some purposes it is more convenient to absorb the \ i" in the innitesimal transformation into the denition of the generator: G!iG)A=[G;A]=LGA; g =eG;[Gi;Gj]=fijkGk This aects the reality properties of G: In particular, if gis unitary ( ggy=I), as usually required in quantum mechanics, g=eiGmakesGhermitian ( G=Gy), while g=eGmakesGantihermitian ( G=Gy). In some cases anithermiticity can be an advantage: For example, for translations we would then have Pi=@iand for rotationsJij=x[i@j], which is more convenient since we know the i's in these (and any) coordinate transformations must cancel anyway. On the other hand, the U(1)transformations of electrodynamics (on the wave function for a charged particle) arejust phase transformations g=e i(whereis a real number), so clearly we want the expliciti; then the only generator has the representation Gi= 1. In general we'll nd that for our purposes absorbing the i's into the generators is more convenient for just spacetime symmetries, while explicit i's are more convenient for internal symmetries. A. COORDINATES 15 4. Relativity The Hamiltonian approach singles out the time coordinate. In relativistic theories time can be treated on equal footing with space, and it is useful to take advantage of this fact, so that the full Poincar e invariance is manifest. So, we treat the time tand spatial position xitogether as a four-vector (or D-vector in D 1s p a c ea n d1t i m e dimension) xm=(x0;xi)=(t;xi) wherem=0;1;:::;3( o rD1),i=1;2;3. Since the energy Eand three-momentum piare canonically conjugate to them, [pi;xj]=iij; [E;t]=+i we dene the 4-momentum as pm=(E;pi)=mnpn;pm=mnpn;[pm;xn]=imn; [pm;xn]=in m where we raise and lower indices with the \Minkowski metric", in an \orthonormal basis", mn=0 BBBB@01 2 3 01000 101 0 0200 1 0 300 0 11 CCCCA)p 0=p0=E in four spacetime dimensions, with obvious generalizations to higher dimensions. (Sometimes the metric with signs + is used; we prefer + ++ because it is more convenient for quantum calculations.) Therefore, we now distinguish upperand lower indices in general: At least for position and momentum, the upper-indexed x mandpmhave the usual physical interpretation (so xmandpmhave extra signs). This is consistent with our previous nonrelativistic notation, since 3-vector indices donot change sign upon raising or lowering. Of course, we could have done that much nonrelativistically. Relativity is a symmetry of kinematics and dynamics: In particular, a free, spinless, relativistic particle is completely described by the constraint p 2+m2=0 where we dene the covariant square p2=pmpm=pmpnmn=(p0)2+(p1)2+(p2)2+(p3)2 16 I. GLOBAL Our relativistic symmetry must leave this constraint invariant: Thus the metric de- nes the norm of a vector (and an invariant inner product). Therefore, to preserve Lorentz invariance it is important that we contract only an upper index with a lower index. For similar reasons, we have @m=@ @xm;@mxn=n m so quantum mechanically pm=i@m. Unlike the positive-denite nonrelativistic norm of a 3-vector Vi, for an arbitrary 4-vectorVmwe can have V28 < :< = >9 = ;0:8 < :timelike lightlike=null spacelike In particular, the 4-momentum is timelike for massive particles ( m2>0) and lightlike for massless ones (while \tachyons", with spacelike momenta and m2<0, do not exist, for reasons that are most clear from quantum eld theory). The quantum mechanics will be described later, but the result is that this con- straint can be used as the wave equation. The main qualitative distinction from the nonrelativistic case in the constraint nonrelativistic :2mE+~p2=0 relativistic :E2+m2+~p2=0 is that the equation for the energy Ep0is now quadratic, and thus has two solutions: p0=!; ! =p (pi)2+m2 Later we'll see how the second solution is interpreted as an \antiparticle". We also use (natural/Planck) units c= 1, so length and duration are measured in the same units;cthen appears only as a parameter for dening nonrelativistic expansions and limits. The translations and Lorentz transformations make up the Poincar e group, the symmetry that denes special relativity. (The Lorentz group in D 1s p a c ea n d1 time dimension is the \orthogonal" group \O(D 1,1)". The \proper" Lorentz group \SO(D1,1)", where the \S" is for \special", transforms the coordinates by a matrix whose determinant is 1. The Poincar eg r o u pi sI S O ( D 1,1), where the \I" stands for \inhomogeneous".) For the spinless particle they are generated by coordinate transformations GI=(Pa;Jab): Pa=pa;Jab=x[apb] A. COORDINATES 17 (where also a;b=0;:::;3). Then the fact that the physics of the free particle is invariant under Poincar e transformations is expressed as [Pa;p2+m2]=[Jab;p2+m2]=0 Writing an arbitrary innitesimal transformation as a linear combination of the gen- erators, we nd xm=xnnm+^m;mn=nm where the's are constants. Note that antisymmetry of mndoes not imply antisym- metry ofmn=mppn, because of additional signs. (Similar remarks apply to Jab.) Exponentiating to nd the nite transformations, we have x0m=xnnm+^m; mpnqpq=mn The same Lorentz transformations apply to pm, but the translations do not aect it. The condition on  follows from preservation of the Minkowski norm (or inner product), but it is equivalent to the antisymmetry of mnby exponentiating  = e (compare excercise IA3.3). Sincedxapais invariant under the coordinate transformations dened by the Pois- son bracket (the chain rule, since eectively pa@a), it follows that the Poincar e invariance of p2is equivalent to the invariance of the line element ds2=dxmdxnmn which denes the \proper time" s. Spacetime with this indenite metric is called \Minkowski space", in contrast to the \Euclidean space" with positive denite metric used to describe nonrelativistic length measured in just the three spatial dimensions. For the massive case, we also have pa=mdxa ds For the massless case ds= 0: Massless particles travel along lightlike lines. However, we can dene a new parameter such that pa=dxa d is well-dened in the massless case. In general, we then have s=m While this xes =s=m in the massive case, in the massless case it instead restricts s= 0. Thus, proper time does not provide a useful parametrization of the world 18 I. GLOBAL line of a classical massless particle, while does: For any piece of such a line, dis given in terms of (any component of) paanddxa. Later we'll see how this parameter appears in relativistic classical mechanics, and is useful for quantum mechanics andeld theory. Excercise IA4.1 The relation between xandpis closely related to the Poincar e conservation laws: aShow that dP a=dJab=0)p[adxb]=0 and use this to prove that conservation of PandJimply the existence of a parametersuch thatpa=dxa=d. bConsider a multiparticle system (but still without spin) where some of the particles can interact only when at the same point (i.e., by collision; theyact as free particles otherwise). Dene P a=PpaandJab=Px[apb]as the sum of the individual momenta and angular momenta. Show that momentum conservation implies angular momentum conservation,
Pa=0)
Jab=0 where \
" refers to the change from before to after the collision(s). Special relativity can also be stated as the fact that the only physically observable quantities are those that are Poincar e invariant. (Other objects, such as vectors, depend on the choice of reference frame.) For example, consider two spinless particles that interact by collision, producing two spinless particles (which may dier from theoriginals). Without loss of generality, we can describe this process in terms of justthe momenta. (Quantum mechanically, this is automatically a complete description;classically, the position is found by p=dx=d .) All invariants can be expressed in terms of the masses and the \Mandelstam variables" (not to be confused with time and proper time) s=(p 1+p2)2;t =(p1p3)2;u =(p1p4)2 where we have used momentum conservation, which shows that even these three quantities are not independent: p2 I=m2 I;p 1+p2=p3+p4)s+t+u=4X I=1m2 I A. COORDINATES 19 (The explicit index now labels the particle, for the process 1+2 !3+4.) The simplest reference frame to describe this interaction is the center-of-mass frame (actually the center of momentum, where the two 3-momenta cancel). In that Lorentz frame, using also rotational invariance, momentum conservation, and the mass-shell conditions, the momenta can be written in terms of these invariants as p1=1ps(1 2(s+m2 1m2 2);12;0;0) p2=1ps(1 2(s+m2 2m2 1);12;0;0) p3=1ps(1 2(s+m2 3m2 4);34cos; 34sin; 0) p4=1ps(1 2(s+m2 4m2 3);34cos;34sin; 0) cos  =s2+2st(Pm2 I)s+(m2 1m2 2)(m2 3m2 4) 41234 2 IJ=1 4[s(mI+mJ)2][s(mImJ)2] The \physical region" of momentum space is then given by s(m1+m2)2and (m3+m4)2,a n djcos j1. Excercise IA4.2 Derive the above expressions for the momenta in terms of invariants in the center-of-mass frame. Excercise IA4.3 Find the conditions on s;tanduthat dene the physical region in the case where all masses are equal. For some purposes it will prove more convenient to use a \lightcone basis" p=1p 2(p0p1))mn=0 BBBB@+23 +0100 100 0 200 1 0 300 0 11 CCCCA;p 2=2p+p+(p2)2+(p3)2 and similarly for the \lightcone coordinates" ( x;x2;x3). (\Lightcone" is an unfor- tunate but common misnomer, having nothing to do with cones in most usages.) In this basis the solution to the mass-shell condition p2+m2= 0 can be written as p=p=(pi)2+m2 2p (where now i=2;3), which more closely resembles the nonrelativistic expression. (Note the change on indices + $ upon raising and lowering.) A special lightcone basis is the \null basis", p=1p 2(p0p1);pt=1p 2(p2ip3);pt=1p 2(p2+ip3) 20 I. GLOBAL )mn=0 BBBB@+tt +0100 100 0 t 00 0 1 t 00 1 01 CCCCA;p 2=2p+p+2ptpt where the square of a vector is linear in each component. (We often use \ "t o indicate complex conjugation.) Excercise IA4.4 Show that for p2+m2=0(m20,pa6= 0), the signs of p+andpare always the same as the sign of the canonical energy p0. Excercise IA4.5 Consider the Poincar e group in 1 extra space dimension (D space, 1 time) for a massless particle. Interpret p+as the mass, and pas the energy. Show that the constraint p2= 0 gives the usual nonrelativistic expression for the energy. Show that the subgroup of the Poincar e group generated by all generators that commute with p+is the Galilean group (in D 1s p a c e and 1 time dimensions). Now nonrelativistic mass conservation is part ofmomentum conservation, and all the Galilean transformations are coordinate transformations. Also, positivity of the mass is related to positivity of the energy (see excercise IA4.4). 5 .D i s c r e t e :C ,P ,T By considering only symmetries than can be obtained continuously from the iden- tity (Lie groups), we have missed some important symmetries: those that re ect some of the coordinates. It's sucient to consider a single re ection of a spacelike axis,and one of a timelike axis; all other re ections can be obtained by combining these with the continuous (\proper, orthochronous") Lorentz transformations. (Spacelike and timelike vectors can't be Lorentz transformed into each other, and re ection of a lightlike axis won't preserve p 2+m2.) Also, the re ection of one spatial axis can be combined with a rotation about that axis, resulting in re ection of all three spatial coordinates. (Similar generalizations hold for higher dimensions. Note that the product of an even number of re ections about dierent axes is a proper rotation; thus, for even numbers of spatial dimensions re ections of all spatial coordinates areproper rotations, even though the re ection of a single axis is not.) The reversal of the spatial coordinates is called \parity (P)", while that of the time coordinate is called \time reversal" (\T"; actually, for historical reasons, to be explained shortly, this is A. COORDINATES 21 usually labeled \CT".) These transformations have the same eect on the momen- tum, so that the denition of the Poisson bracket is also preserved. These \discrete" transformations, unlike the proper ones, are not symmetries of nature (except in cer-tain approximations): The only exception is the transformation that re ects all axes(\CPT"). While the metric  mnis invariant under all Lorentz transformations (by deni- tion), the \Levi-Civita tensor" mnpqtotallyantisymmetric;  0123=0123=1 is invariant under only proper Lorentz transformations: It has an odd number of space indices and of time indices, so it changes sign under parity or time reversal. Consequently, we can use it to dene \pseudotensors": Given \polar vectors", whose signs change as position or momentum under improper Lorentz transformations, andscalars, which are invariant, we can dene \axial vectors" and \pseudoscalars" as V a=abcdBbCcDd; =abcdAaBbCcDd which get an extra sign change under such transformations (P or CT, but not CPT). There is another such \discrete" transformation that is dened on phase space, but which does not aect spacetime. It changes the sign of all components of themomentum, while leaving the spacetime coordinates unchanged. This transformationis called \charge conjugation (C)", and is also only an approximate symmetry innature. (Quantum mechanically, complex conjugation of the position-space wave function changes the sign of the momentum.) Furthermore, it does not preserve the Poisson bracket, but changes it by an overall sign. (The misnomer \CT" for timereversal follows historically from the fact that the combination of reversing the timeaxis and charge conjugation preserves the sign of the energy.) The physical meaningof this transformation is clear from the spacetime-momentum relation of relativisticclassical mechanics p=md x = d s : It is proper-time reversal, changing the sign of s. The relation to charge follows from \minimal coupling": The \covariant momentum" md x = d s =p+qA(for charge q) appears in the constraint ( p+qA) 2+m2=0i na n electromagnetic background; p!pthen has the same eect as q!q. In the previous subsection, we mentioned how negative energies were associated with \antiparticles". Now we can better see the relation in terms of charge conjuga- tion. Note that charge conjugation, since it only changes the sign of but does not eect the coordinates, does not change the path of the particle, but only how it isparametrized. This is also true in terms of momentum, since the velocity is given by 22 I. GLOBAL pi=p0. Thus, the only observable property that is changed is charge; spacetime prop- erties (path, velocity, mass; also spin, as we'll see later) remain the same. Another way to say this is that charge conjugation commutes with the Poincar e group. One way to identify an antiparticle is that it has all the same kinematical properties (mass, spin) as the corresponding particle, but opposite sign for internal quantum numbers (like charge). (Another way is pair creation and annihilation: See subsection IIIB5 below.) Quantum mechanically, we can identify a particle with its antiparticle by requiring the wave function or eld to be invariant under charge conjugation: For example, for a scalar eld (spinless particle), we have the reality condition (x)=*(x) or in momentum space, by Fourier transformation, ~(p)=[ ~(p)]* which implies the particle has charge zero (neutral). All these transformations are summarized in the table: CCT PTC PPT CPT s++ + t+++ ~x+++ E+++ ~p+++ (The upper-left 33 matrix contains the denitions, the rest is implied.) However, from the point of view of the \particle" there issome kind of kinematic change, since the proper time has changed sign: If we think of the mechanics of a particle as a one-dimensional theory in space (the worldline), where x()( a sw e l l as any such variables describing spin or internal symmetry) is a wave function or eld on that space, then !is T on that one-dimensional space. (The fact we don't get CT can be seen when we add additional variables: For example, if we describe internal U(N) symmetry in terms of creation and annihilation operators ayiandai, then C mixes them on both the worldline and spacetime. So, on the worldline we have the \pure" worldline geometric symmetry CT times C = T.) Thus, in terms of \zeroth quantization", worldlineT$spacetimeC A. COORDINATES 23 On the other hand, spacetimePandCTare simply internal symmetries with respect to the worldline (as are proper, orthochronous Poincar e transformations). 6. Conformal Although Poincar e transformations are the most general coordinate transforma- tions that preserve the mass condition p2+m2= 0, there is a larger group, the \conformal group", that preserves this constraint in the massless case. Transforma- tionsthat satisfy [a(x)pa;p2]=(x)p2 for somealso preserve p2= 0, although they don't leave p2invariant. Equivalently, we can look for coordinate transformations that scale dx02=(x)dx2 Excercise IA6.1 Find the conformal group explicitly in two dimensions, and show it's innite dimensional (not just the SO(2,2) described below). (Hint: Use lightconecoordinates.) This symmetry can be made manifest by starting with a space with one extra space and time dimension: y A=(ya;y+;y))y2=yAyBAB=(ya)22y+y where (ya)2=yaybabuses the usual D-dimensional Minkowski-space metric ab, and the two additional dimensions have been written in a lightcone basis (not to be confused for the similar basis that can be used for the Minkowski metric itself). With respect to this metric, the original SO(D 1,1) Lorentz symmetry has been enlarged to SO(D,2). This is the conformal group in D dimensions. However, rather than also preserving (D+2)-dimensional translation invariance, we instead impose the constraint and invariance y2=0; yA=yA This reduces the original space to the \projective" (invariant under the scaling) lightcone (which in this case really is a cone). These two conditions can be solved by yA=ewA;wA=(xa;1;1 2xaxa) 24 I. GLOBAL Projective invariance then means independence from e(y+), while the lightcone con- dition has determined y.y2= 0 implies y
dy= 0, so the simplest conformal invariant is dy2=(edw+wde)2=e2dw2=e2dx2 w h e r ew eh a v eu s e d w2=0)w
dw= 0. This means any SO(D,2) transformation onyAwill simply scale dx2,a n ds c a l e e2in the opposite way: dx02=e2 e02 dx2 in agreement with the previous denition of the conformal group. The explicit form of conformal transformations on xa=ya=y+now follows from their linear form on yA, using the generators GAB=y[ArB]; [rA;yB]=iB A of SO(D,2) in terms of the momentum rAconjugate to yA. (These are dened the same way as the Lorentz generators Jab=x[apb].) For example, G+just scalesxa. (Scale transformations are also known as \dilatations".) We can also recognize G+aas generating translations on xa. The only complicated transformations are generated byGa, known as \conformal boosts" (acceleration transformations). Since they commute with each other (like translations), it's easy to exponentiate to nd thenite transformations: y 0=eGy; G =vay[@a] for some constant D-vector va(where@A@=@yA). Since the conformal boosts act as \lowering operators" for scale weight (+ !a! ), only the rst three terms in the exponential survive: Gy=0;G ya=vay;G y+=vaya) y0=y;y0a=ya+vay;y0+=y++vaya+1 2v2y) x0a=xa+1 2vax2 1+v
x+1 4v2x2 usingxa=ya=y+,y=y+=1 2x2. Excercise IA6.2 Make the change of variables to xa=ya=y+,e=y+,z=1 2y2.E x p r e s s rAin terms of the momenta ( pa;n;s) conjugate to ( xa;e;z). Show that the conditionsy2=yArA=r2= 0 become z=en=p2= 0 in terms of the new variables. A. COORDINATES 25 Excercise IA6.3 Find the generator of innitesimal conformal boosts in terms of xaandpa. We actually have the full O(D,2) symmetry: Besides the continuous symmetries, and the discrete ones of SO(D 1,1), we have a second \time" reversal (from our second time dimension): y+$y)xa$xa 1 2x2 This transformation is called an \inversion". Excercise IA6.4 Show that a nite conformal boost can be obtained by performing a transla-tion sandwiched between two inversions. Excercise IA6.5 The conformal group for Euclidean space (or any spacetime signature) can beobtained by the same construction. Consider the special case of D=2 for theseSO(D+1,1) transformations. (This is a subgroup of the 2D superconformalgroup: See excercise IA6.1.) Use complex coordinates for the two \physical"dimensions: z= 1p 2(x1+ix2) aShow that the inversion is z$1 z* bShow that the conformal boost is (using a complex number also for the boost vector) z!z 1+v*z Excercise IA6.6 Any parity transformation (re ection in a spatial axis) can be obtained from any other by a rotation of the spatial coordinates. Similarly, when there ismore than one time dimension, any time reversal can be obtained from another(but time reversal can't be rotated into parity, since a timelike vector can't berotated into a spacelike one). Thus, the complete orthogonal group O(m,n)can be obtained from those transformations that are continuous from theidentity by combining them with 1 parity transformation and 1 time reversaltransformation (for mn 6=0). For the conformal group, nd the rotation (in terms of an angle) that rotates between the two time directions, and expressits action on x a. Show that for angle it produces a transformation that is 26 I. GLOBAL the product of time reversal and inversion. Use this to show that inversion is related to time reversal by nding the continuum of conformal transformationsthat connect them. Although conformal symmetry is not observed in nature, it is important in all approaches to eld theory: (1) First of all, it is useful in the construction of free the-ories (see subsections IIB1-4 below). All massive elds can be described consistentlyin quantum eld theory in terms of coupling massless elds. Massless theories are asubset of conformal theories, and some conditions on massless theories can be found more easily by nding the appropriate subset of those on conformal theories. This is related to the fact that the conformal group, unlike the Poincar e group, is \simple": It has no nontrivial subgroup that transforms into itself under the rest of the group(like the way translations transform into themselves under Lorentz transformations).(2) In interacting theories at the classical level, conformal symmetry is also important in nding and classifying solutions, since at least some parts of the action are confor- mally invariant, so corresponding solutions are related by conformal transformations(see subsections IIIC5-7). Furthermore, it is often convenient to treat arbitrary theo-ries as broken conformal theories, introducing elds with which the breaking is asso-ciated, and analyze the conformal and conformal-breaking elds separately. This is particularly true for the case of gravity (see subsections IXA7,B5,C2-3,XA3-4,B5-7). (3) Within quantum eld theory at the perturbative level, the only physical quantumeld theories are ones that are conformal at high energies (see subsection VIIIC1).The quantum corrections to conformal invariance at high energy are relatively sim- ple. (4) Beyond perturbation theory, the only quantum theories that are well dened may be just the ones whose breaking of conformal invariance at low energy is onlyclassical (see subsections VIIC2-3,VIIIA5-6). Furthermore, the largest possible sym-metry of a nontrivial S-matrix is conformal symmetry (or superconformal symmetryif we include fermionic generators). (5) Self-duality (a generalization of a condition that equates electric and magnetism elds) is useful for nding solutions to classical eld equations as well as simplifying perturbation theory, and is closely related to\twistors" (see subsections IIB6-7,C5,IIIC4-7). In general, self-duality is related toconformal invariance: For example, it can be shown that the free conformal theoriesin arbitrary even dimensions are just those with (on-mass-shell) eld strengths on which self-duality can be imposed. (In arbitrary odd dimensions the free conformal theories are just the scalar and spinor.) REFERENCES 1 F.A. Berezin, The method of second quantization (Academic, 1966): A. COORDINATES 27 calculus with anticommuting numbers. 2P.A.M. Dirac, P r o c .R o y .S o c . A126 (1930) 360: antiparticles. 3E.C.G. St uckelberg, Helv. Phys. Acta 14(1941) 588, 15(1942) 23; J.A. Wheeler, 1940, unpublished: the relation of antiparticles to proper time. 4S. Mandelstam, Phys. Rev. 112(1958) 1344. 5P.A.M. Dirac, Ann. Math. 37(1936) 429; H.A. Kastrup, Phys. Rev. 150(1966) 1186; G. Mack and A. Salam, Ann. Phys. 53(1969) 174; S. Adler, Phys. Rev. D6(1972) 3445; R. Marnelius and B. Nilsson, Phys. Rev. D22 (1980) 830: conformal symmetry. 6S. Coleman and J. Mandula, Phys. Rev. 159(1967) 1251: conformal symmetry as the largest (bosonic) symmetry of the S-matrix. 7W. Siegel, Int. J. Mod. Phys. A 4(1989) 2015: equivalence between conformal invariance and self-duality in all dimensions. 28 I. GLOBAL ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: B. INDICES ::::::::::::::::::::::::::::: In the previous section we saw various spacetime groups (Galilean, Poincar e, conformal) in terms of how they acted on coordinates. This not only gave them a simple physical interpretation, but also allowed a direct relation between classical and quantum theories. However, as we know from studying rotations in quantum theory in terms of spin, we will often need to study symmetries of quantum theories for which the classical analog is not so useful or perhaps even nonexistent. We therefore now consider some general results of group theory, mostly for con- tinuous groups. We use tensor methods, rather than the slightly more powerful but greatly less convenient Cartan-Weyl-Dynkin methods. Much of this section should be review, but is included here for completeness; it is not intended as a substitute for a group theory course, but as a summary of those results commonly useful in eld theory. 1. Matrices Matrices are dened by the way they act on some vector space; an n n matrix takes one n-component vector to another. Given some group, and its multiplication table (which denes the group completely), there is more than one way to represent it by matrices. Any set of matrices we nd that has the same multiplication table as the group elements is called a \representation" of that group, and the vector space onwhich those matrices act is called the \representation space." The representation of the algebra or group in terms of explicit matrices is given by choosing a basis for the vector space. If we include innite-dimensional representations, then a representation of a group is simply a way to write its transformations that is linear: 0=M is linear in . More generally, we can also have a \realization" of a group, where the transformations can be nonlinear. These tend to be more cumbersome, so we usually try to make redenitions of the variables that make the realization linear. A precise denition of \manifest symmetry" is that all the realizations used are linear. (One possible exception is \ane" or \inhomogeneous" transformations 0=M1 +M2, such as the usual coordinate representation of Poincar e transformations, since these transformations are still very simple, because they are really still linear, though not homogeneous.) For convenience, we write matrices with a Hilbert-space-like notation, but unlike Hilbert space we don't necessarily associate bras directly with kets by Hermitian conjugation, or even transposition. In general, the two spaces can even be dierent B. INDICES 29 sizes, to describe matrices that are not square; however, for group theory we are interested only in matrices that take us from some vector space into itself, so they are square. Bras have an inner product with kets, but neither necessarily has a norm (inner product with itself): In general, if we start with some vector space, written as kets, we can always dene the \dual" space, written as bras, by dening such aninner product. In our case, we may start with some representation of a group, in terms of some vector space, and that will give us directly the dual representation. (If the representation is in terms of unitary matrices, we have a Hilbert space, and thedual representation is just the complex conjugate.) So, we dene column vectors j iwith a basisj Ii, and row vectors h jwith a basishIj,w h e r eI=1;:::;n to describe nn matrices. The two bases have a relative normalization dened so that the inner product gives the usual component sum: j i=jIi I;hj=IhIj;hIjJi=J I)hj i=I I;hIj i= I;hjIi=I These bases then dene not only the components of vectors, but also matrices: M=jIiMIJhJj;hIjMjJi=MIJ where as usual the Ion the component (matrix element) MIJlabels the row of the matrixM,a n dJthe column. This implies the usual matrix multiplication rules, inserting the identity in terms of the basis, I=jKihKj) (MN)IJ=hIjMjKihKjNjJi=MIKNKJ Closely related is the denition of the trace, tr M =hIjMjIi=MII)tr(MN)=tr(NM) (We'll discuss the determinant later. The bra-ket notation is really just matrix nota- tion written in a way to clearly distinguish column vectors, row vectors, and matrices.) Thus, for example, we can easily translate transformation laws from matrix no- tation into index notation just by using a basis for the representation space: gjIi=jJigJI;GjIi=jJiGJI G=iGi;j i=iGj i=jIiii(Gi)IJ J) I=ii(Gi)IJ J The dual space isn't needed for this purpose. However, for any representation of a group, the transpose (MT)I J=MJI 30 I. GLOBAL of the inverse of those matrices also gives a representation of the group, since g1g2=g3) (g1)T1(g2)T1=(g3)T1 [G1;G2]=G3) [GT 1;GT 2]=GT 3 This is the dual representation, which follows from dening the above inner product to be invariant under the group: h ji=0) I=i Ji(Gi)JI The complex conjugate of a complex representation is also a representation, since g1g2=g3)g1*g2*=g3* [G1;G2]=G3) [G1*;G2*] =G3* From any given representation, we can thus nd three others from taking the dual and the conjugate: In matrix and index notation, 0=g : 0 I=gIJ J 0=(g1)T : 0I=g1 JI J 0=g* : 0. I=g*. I. J . J 0=(g1)y : 0. I=g*1. J. I . J since (g1)T,g*, and (g1)y(but notgT, etc.) satisfy the same multiplication algebra asg, including ordering. We use up/down and dotted/undotted indices to denote the transformation law of each type of index; contracting undotted up indices with undotted down indices preserves the transformation law as indicated by the remaining indices, and similarly for dotted indices. These four representations are not necessarily independent: Imposing relations among them is how the classical groups are dened (see subsections IB4-5 below). 2. Representations For example, we always have the \adjoint" representation of a Lie group/algebra, which is how the algebra acts on its own generators: G=iGi;A =iGi)A=i[G;A]=jifijkGk )i=ikj(Gj)ki;(Gi)jk=ifijk B. INDICES 31 This gives us two ways to represent the adjoint representation space: as either the usual vector space, or in terms of the generators. Thus, we either use the matrix A=iGi(for arbitrary representation of the matrices Gi, or treating Gias just abstract generators), and write A=i[G;A], or we can write Aas a row vector, hAj=ihij)hAj=ihAjG)ihij=ikj(Gj)kihij The adjoint representation also provides a convenient way to dene a (symmetric) group metric invariant under the group, the \Cartan metric": ij=trA(GiGj)=fiklfjlk For \Abelian" groups the structure constants vanish, and thus so does this metric. \Semisimple" groups are those where the metric is invertible (no vanishing eigenval- ues). A \simple" group has no nontrivial subgroup that transforms into itself under the rest of the group: Semisimple groups can be written as \products" of simple groups. \Compact" groups are those where it is positive denite (all eigenvalues pos- itive); they are also those for which the invariant volume of the group space is nite.For simple, compact groups it's convenient to choose a basis where  ij=cAij for some constant cA(the \Dynkin index" for the adjoint representation). For some general irreducible representation Rof such a group the normalization of the trace is trR(GiGj)=cRij=cR cAij Now the proportionality constant cR=cAis xed by the choice of R(only), since we have already xed the normalization of our basis. In general, the cyclicity property of the trace implies, for any representation, that 0=tr([Gi;Gj]) =ifijktr(Gk) sotr(Gi) = 0 for semisimple groups. Similarly, we nd fijkfijllk=it rA([Gi;Gj]Gk) is totally antisymmetric: For semisimple groups, this implies the total antisymmetry of the structure constants fijk, up to factors (which are absent for compact groups in ab a s i sw h e r e ijij). This also means the adjoint representation is its own dual. 32 I. GLOBAL (For example, for the compact group SO(3), we have ij=ikljlk=2ij.) Thus, we can write Ain a third way, as a column vector jAi=jiiijiijji We can also do this for Abelian groups, by dening an invertible metric unrelated to the Cartan metric: This is trivial for Abelian groups, since the generators themselves are invariant, and thus so is anymetric on them. An identity related to the trace one is the normalization of the value kRof the \Casimir operator" for any particular representation, ijGiGj=kRI Its proportionality to the identity follows from the fact that it commutes with each generator: [jkGjGk;Gi]=ifj ikfGj;Gkg=0 using the antisymmetry of the structure constants. (Thus it takes the same value on any component of an irreducible representation, since they are all related by grouptransformations.) By tracing this identity, and contracting the trace identity, c R cAdA=trR(ijGiGj)=kRdR )kR=cRdA cAdR wheredRtrR(I) is the dimension of that representation. Although quantum mechanics is dened on Hilbert space, which is a kind of com- plex vector space, more generally we want to consider real objects, like spacetime vectors. This restricts the form of linear transformations: Specically, if we absorbi's asg=e G, then in such representations Gitself must be real. These represen- tations are then called \real representations", while a \complex representation" is one whose representation isn't real in any basis. A complex representation space can have a real representation, but a real representation space can't have a complex rep- resentation. In particular, coordinate transformations (of real coordinates) have onlyreal representations, which is why absorbing the i's into the generators is a useful convention there. For semisimple unitary groups, hermiticity of the generators of the adjoint representation implies (using total antisymmetry of the structure constantsand reality of the Cartan metric) that the structure constants are real, and thus the adjoint representation is a real representation. More generally, any real unitary rep- resentation will have antisymmetric generators ( G=G*=G y)G=GT). If B. INDICES 33 the complex conjugate representation is the same as the original (same matrices up to a similarity transformation g*=MgM1), but the representation is not real, then it is called \pseudoreal". (An example is the spinor of SU(2), to be described in thenext section.) For any representation gof the group, a transformation g!g 0gg1 0on every group element gfor some particular group element g0clearly maps the algebra to itself, and preserves the multiplication rules. (Similar remarks apply to applying thetransformation to the generators.) However, the same is true for complex conjugation, g!g*: Not only are the multiplication rules preserved, but for any element g of that representation of the group, g* is also an element. (This can be shown, e.g., by dening representations in terms of the values of all the Casimir operators,contructed from various powers of the generators.) In quantum mechanics (wherethe representations are unitary), the latter is called an \antiunitary transformation".Although this is a symmetry of the group, it cannot be reproduced by a unitarytransformation, except when the representation is (pseudo)real. A very simple way to build a representation from others is by \direct sum". If we have two representations of a group, on two dierent spaces, then we can take theirdirect sum by just putting one column vector on top of the other, creating a biggervector whose size (\dimension") is the sum of that of the original two. Explicitly, ifwe start with the basis j ifor the rst representation and j0ifor the second, then the union (ji;j0i) is the basis for the direct sum. (We can also write jIi=(ji;j0i), where=1;:::;m ;0=1;:::;n ;I=1;:::;m;m +1;:::;m +n.) The group then acts on each part of the new vector in the obvious way: =ji ; =j0i0;gji=jig;gj0i=j0ig00 )j i=ji j0i0=j ijior( ) =  gj i=jig j0ig000or(g)=g0 0g00 (We can replace the with an ordinary + if we understand the basis vectors to be now in a bigger space, where the elements of the rst basis have zeros for the newcomponents on the bottom while those of the second have zeros for the new compo-nents on top.) The important point is that no group element mixes the two spaces:The group representation is block diagonal. Any representation that can be writtenas a direct sum (after an appropriate choice of basis) is called \reducible". For exam-ple, we can build a reducible real representation from an irreducible complex one by 34 I. GLOBAL just taking the direct sum of this complex representation with the complex conjugate representation. Similarly, we can take direct sums of more than two representations. A more useful way to build representations is by \direct product". The idea there is to take a colummn vector and a row vector and use them to construct a matrix, where the group element acts simultaneously on rows according to one representation and columns according to the other. If the two original bases are again jiandj0i, the new basis can also be written as jIi=j0i(I=1;:::;mn ). Explicitly, j i=ji j0i 0;g(ji j0i)=ji j0igg00)g00=gg00 or in terms of the algebra G00=G00+G00 A familar example from quantum mechanics is rotations (or Lorentz transformations), where the rst space is position space (so is the continuous index x), acted on by the orbital part of the generators, while the second space is nite-dimensional, and isacted on by the spin part of the generators. Direct product representations are usuallyreducible: They then can be written also as direct sums, in a way that depends onthe particulars of the group and the representations. Consider a representation constructed by direct product: In matrix notation ^G i=Gi I0+I G0 i Usingtr(A B)=tr(A)tr(B), and assuming tr(Gi)=tr(G0 i)=0 ,w eh a v e tr(^Gi^Gj)=tr(I0)tr(GiGj)+tr(I)tr(G0 iG0j) For example, for SU(N) (see subsection IB4 below) we can construct the adjoint rep- resentation from the direct product of the N-dimensional, \dening" representation and its complex conjugate. (We also get a singlet, but it will not aect the result forthe adjoint.) In that case we nd tr A(GiGj)=2NtrD(GiGj))cD cA=1 2N For most purposes, we use trD(GiGj)=ij(cD= 1) for SU(N), so cA=2N. B. INDICES 35 3. Determinants We now \review" some properties of determinants that will prove useful for the group analysis of the following subsections. Determinants can be dened in terms of the Levi-Civita tensor . As a consequence of its antisymmetry,  totally antisymmetric;  12:::n=12:::n=1)J1:::JnI1:::In=I1 [J1


In Jn] since each possible numerical index value appears once in each ,s ot h e yc a nb e matched up with 's. By similar reasoning, 1 m!K1:::KmJ1:::JnmK1:::KmI1:::Inm=I1 [J1


Inm Jnm] where the normalization compensates for the number of terms in the summation. This tensor is used to dene the determinant: det MIJ=1 n!J1:::JnI1:::InMI1J1


MInJn)J1:::JnMI1J1


MInJn=I1:::IndetM since anything totally antisymmetric in nindices must be proportional to the tensor. This yields an explicit expression for the inverse: (M1)J1I1=1 (n1)!J1:::JnI1:::InMI2J2


MInJn(detM )1 From this follows a useful expression for the variation of the determinant: @ @MIJdet M =(M1)JIdet M which is equivalent to lnd e tM =tr(M1M) ReplacingMwitheMgives the often-used identity l nd e teM=tr(eMeM)=tr M)det eM=etrM where we have used the boundary condition for M= 0. Finally, replacing Min the last identity with ln(1 +L) and expanding both sides to order Lngives general expressions for determinants of nnmatrices in terms of traces: det(1 +L)=etrln(1+L))det L =1 n!(tr L)n1 2(n2)!(tr L2)(tr L)n2+


Excercise IB3.1 Use the denition of the determinant (and not its relation to the trace) to show det(AB)=det(A)det(B) 36 I. GLOBAL These identities can also be derived by dening the determinant in terms of a Gaussian integral. We rst collect some general properties of (indenite) Gaussian integrals. The simplest such integral is Zd2x 2ex2=2=Z2 0d 2Z1 0dr rer2=2=Z1 0du eu=1 )ZdDx (2)D=2ex2=2=Zdxp 2ex2=2D =Zd2x 2ex2=2D=2 =1 The complex form of this integral is ZdDz*dDz (2i)Dejzj2=1 by reducing to real parameters as z=(x+iy)=p 2. These generalize to integrals involving a real, symmetric matrix Sor a Hermitian matrix Has ZdDx (2)D=2exTSx=2=(detS )1=2;ZdDz*dDz (2i)DezyHz=(det H )1 by diagonalizing the matrices, making appropriate redenitions of the integration variables, and identifying the determinant of a diagonal matrix. Alternatively, wecan use these integrals to dene the determinant, and derive the previous denition. The relation for the symmetric matrix follows from that for the Hermitian one by separatingzinto its real and imaginary parts for the special case H=S. If we treat zandz* as independent variables, the determinant can also be understood as the Jacobian for the (dummy) variable change z!H 1z,z*!z*. More generally, if we dene the integral by an appropriate limiting procedure or analytic continuation (for convergence), we can choose zandz* to be unrelated (or even separate real variables), and SandHto be complex. Excercise IB3.2 Other properties of determinants can also be derived directly from the integral denition: aFind an integral expression for the inverse of a (complex) matrix Mby using the identity 0=Z@ @zI(zJezyMz) bDerive the identity l nd e tM =tr(M1M) by varying the Gaussian de- nition of the (complex) determinant with respect to M. B. INDICES 37 An even better denition of the determinant is in terms of an anticommuting integral (see subsection IA2), since anticommutativity automatically gives the anti-symmetry of the Levi-Civita tensor, and we don't have to worry about convergence.We then have, for anymatrixM, Z d DydDeyM=det M whereycan be chosen as the Hermitian conjugate of or as an independent variable, whichever is convenient. From the denition of anticommuting integration, the onlyterms in the Taylor expansion of the exponential that contribute are those with theproduct of one of each anticommuting variable. Total antisymmetry in and in y then yields the determinant; we dene \ dDydD" to give the correct normalization. (The normalization is ambiguous anyway because of the signs in ordering the d's.) This determinant can also be considered a Jacobian, but the inverse of the commutingresult follows from the fact that the integrals are now really derivatives. Excercise IB3.3 Divide up the range of a square matrix into two (not necessarily equal) parts: In block form, M=AB CD and do the same for the (commuting or anticommuting) variables used in dening its determinant. Show that detAB CD =det D
det(ABD 1C)=det A
det(DCA1B) aby integrating over one part of the variables rst (this requires o-diagonal changes of variables of the form y!y+Ox, which have unit Jacobian), or bby rst proving the identity AB CD =IB D1 0IABD1C0 0DI 0 D1CI We then have, for any antisymmetric (even-dimensional) matrix A, Z d2DeTA=2=PfA; (PfA)2=det A by the same method as the commuting case (again with appropriate denition of the normalization of d2D; the determinant of an odd-dimensional antisymmetric matrix vanishes, since detM =detMT). However, there is now an important dierence: The 38 I. GLOBAL \Pfaan" is not merely the square root of the determinant, but itself a polynomial, since we can evaluate it also by Taylor expansion: PfAIJ=1 D!2DI1:::I2DAI1I2


AI2D1I2D which can be used as an alternate denition. (Normalization can be checked by examining a special case; the overall sign is part of the normalization convention.) 4. Classical groups The rotation group in three dimensions can be expressed most simply in terms of 22 matrices. This description is the most convenient for not only spin 1/2, but all spins. This result can be extended to orthogonal groups (such as the rotation, Lorentz, and conformal groups) in other low dimensions, including all those relevantto spacetime symmetries in four dimensions. There are an innite number of Lie groups. Of the compact ones, all but a nite number are among the \classical" Lie groups. These classical groups can be dened easily in terms of (real or complex) matrices satisfying a few simple constraints. (Theremaining \exceptional" compact groups can be dened in a similar way with a little extra eort, but they are of rather specialized interest, so we won't cover them here.) These matrices are thus called the \dening" representation of the group. (Sometimesthis representation is also called the \fundamental" representation; however, this term has been used in slightly dierent ways in the literature, so we will avoid it.) These constraints are a subset of: volume: Special:det(g)=1 metric:8 < :hermitian: Unitary: (anti)symmetric:Orthogonal: Symplectic:gg y= ggT= g gT= (y= ) (T=) ( T= ) reality:Real: pseudoreal (*):g*=g1 g*= g 1 wheregis any matrix in the dening representation of the group, while  ;; a r e group \metrics", dening inner products (while the determinant denes the volume, as in the Jacobian). For the compact cases  and can be chosen to be the identity, but we will also consider some noncompact cases. (There are also some uninteresting variations of \Special" for complex matrices, setting the determinant to be real or its magnitude to be 1.) B. INDICES 39 Excercise IB4.1 Write all the dening constraints of the classical groups (S, U, O, Sp, R, pseudoreal) in terms of the algebra rather than the group. Note the modied denition of unitarity, etc. Such things are also encountered in quantum mechanics with ghosts, since the resulting Hilbert space can have an indenite metric. For example, if we have a nite-dimensional Hilbert space where the inner product is represented in terms of matrices as h ji= y then \observables" satisfy a \pseudohermiticity" condition h jHi=hH ji) H=Hy and unitarity generalizes to hU jUi=h ji)UyU= Similar remarks apply when replacing the Hilbert-space \sesquilinear" (vector times complex conjugate of vector) inner product with a symmetric (orthogonal) or anti-symmetric (symplectic) bilinear inner product. An important example is when the wave function carries a Lorentz vector index, as expected for a relativistic description of spin 1; then clearly the time component is unphysical. The groups of matrices that can be constructed from these conditions are then: GL(n,C) [SL(n,C)] U: [S]U(n +,n) O: [S]O(n,C) Sp: Sp(2n,C)R: GL(n) [SL(n)] *: [S]U*(2n)U R * O [S]O(n +,n) SO*(2n) Sp Sp(2n) USp(2n +,2n) Of the non-determinant constraints, in the rst column we applied none (\GL" means \general linear", and \C" refers to the complex numbers; the real numbers \R" areimplicit); in the second column we applied one; in the third column we applied three, since two of the three types (unitarity, symmetry, reality) imply the third. (The corresponding groups with unit determinant, when distinct, are given in brackets.)T h e s es q u a r em a t r i c e sa r eo fs i z en ,n ++n, 2n, or 2n ++2n, as indicated. n +and nrefer to the number of positive and negative eigenvalues of the metric  or . O(n) diers from SO(n) by including \parity"-type transformations, which can't be 40 I. GLOBAL obtained continuously from the identity. (SSp(2n) is the same as Sp(2n).) For this reason, and also for studying \topological" properties, for nite transformations itis sometimes more useful to work directly with the group elements g, rather than parametrizing them in terms of algebra elements as g=e iG. U(n) diers from SU(n) (and similarly for GL(n) vs. SL(n)) only by including a U(1) group that commutes with the SU(n): Although U(1) is noncompact (it consists of just phase transforma-tions), a compact form of it can be used by requiring that all \charges" are integers (i.e., all representations transform as 0=eiq for group parameter ,w h e r eqis an integer dening the representation). Of these groups, the compact ones are just SU(n), SO(n) (and O(n)), and USp(2n) (all with n=0). The compact groups have an interesting interpretation in terms of various number systems: SO(n) is the unitary group of n n matrices over the real numbers, SU(n) is the same for the complex numbers, and USp(2n) is the same for the quaternions. (Similar interpretations can be made for some of the noncompactgroups.) The remaining compact Lie groups that we didn't discuss, the \exceptional" groups, can be interpreted as unitary groups over the octonions. (Unlike the classical groups, which form innite series, there are only ve exceptional compact groups,because of the restrictions following from the nonassociativity of octonions.) 5. Tensor notation Although historically group representations have usually been taught in the no- tation where an m-component representation of a group dened by n n matrices is represented by an m-component vector, carrying a single index with values 1 to m, a much more convenient and transparent method is \tensor notation", where a gen- eral representation carries many indices ranging from 1 to n, with certain symmetries(and perhaps tracelessness) imposed on them. (Tensor notation for a covering group is generally known as \spinor notation" for the corresponding orthogonal group: See subsection IC5.) This notation takes advantage of the property described above forexpressing arbitrary representations in terms of direct products of vectors. In termsof transformation laws, it means we need to know only the dening representation, since the transformation of this representation is applied to each index. There are at most four vector representations, by taking the dual and complex conjugate; weuse the corresponding index notation. Then the group constraints simply state the invariance of the group metrics (and their complex conjugates and inverses), which thus can be used to raise, lower, and contract indices: B. INDICES 41 volume: Special:I1:::In metric:8 < :hermitian: Unitary: (anti)symmetric:Orthogonal: Symplectic:. IJ IJ IJ reality:Real: pseudoreal (*):. IJ . IJ As a result, we have relations such as hIjJi=IJor IJ;h. IjJi=. IJ We also dene inverse metrics satisfying KIKJ= KI KJ=. KI. KJ=I J (and similarly for contracting the second index of each pair). Therefore, with uni- tarity/(pseudo)reality we can ignore complex conjugate representations (and dottedindices), converting them into unconjugated ones with the metric, while for orthogo-nality/symplecticity we can do the same with respect to raising/lowering indices: Unitary: . I=. IJ J Orthogonal: I=IJ J Symplectic: I= IJ J Real: . I=. IJ J pseudoreal (*): . I= . IJ J For the real groups there is also the constraint of reality on the dening representation:  . I( I)* = . I. IJ J Excercise IB5.1 As an example of the advantages of index notation, show that SSp is the sameas Sp. (Hint: Write one in the denition of the determinant in terms of 's by total antisymmetrization, which then can be dropped because it is enforcedby the other . One can ignore normalization by just showing detM =detI .) For SO(n +,n), there is a slight modication of a sign convention: Since then indices can be raised and lowered with the metric, I:::is usually dened to be the result of raising indices on I:::,w h i c hm e a n s 12:::n=1)12:::n=det  =(1)n 42 I. GLOBAL ThenI:::should be replaced with ( 1)nI:::in the equations of subsection IB3: For example, J1:::JnI1:::In=(1)nI1 [J1


In Jn] We now give the simplest explicit forms for the dening representations of the classical groups. The most convenient notation is to label the generators by a pair of fundamental indices, since the adjoint representation is obtained from the directproduct of the fundamental representation and its dual (i.e., as a matrix labeled by row and column). The simplest example is GL(n), since the generators are arbitrary matrices. We therefore choose as a basis matrices with a 1 as one entry and 0'severywhere else, and label that generator by the row and column where the 1 appears. Explicitly, GL(n): (G IJ)KL=L IJ K)GIJ=jJihIj This basis applies for GL(n,C) as well, the only dierence being that the coecients inG=IJGJIare complex instead of real. The next simplest case is U(n): We can again use this basis, although the matrices GIJare not all hermitian, by requiring thatIJbe a hermitian matrix. This turns out to be more convenient in practice than using a hermitian basis for the generators. A well known example is SU(2),where the two generators with the 1 as an o-diagonal element (and 0's elsewhere) are known as the \raising and lowering operators" J , and are more convenient than their hermitian parts for purposes of contructing representations. (This generalizesto other unitary groups, where all the generators on one side of the diagonal are raising, all those on the other side are lowering, and those along the diagonal give the maximal Abelian subalgebra, or \Cartan subalgebra".) Representations for the other classical groups follow from applying their deni- tions to the GL(n) basis. We thus nd SL(n): (G IJ)KL=L IJ K1 nJ IL K)GIJ=jJihIj1 nJ IjKihKj SO(n): (GIJ)KL=K [IL J])GIJ=j[IihJ]j Sp(n): (GIJ)KL=K (IL J))GIJ=j(IihJ)j As before, SL(n,C) and SU(n) use the same basis as SL(n), etc. For SO(n) and Sp(n) we have raised and lowered indices with the appropriate metric (so SO(n) includes SO(n +,n)). For some purposes (especially for SL(n)), it's more convenient to impose tracelessness or (anti)symmetry on the matrix , and use the simpler GL(n) basis. Excercise IB5.2 Our normalization for the generators of the classical groups is the simplest, and independent of n (except for subtracting out traces): B. INDICES 43 aFind the commutation relations of the generators (structure constants) for the dening representation of GL(n) as given in the text. Note that the values of all the structure constants are 0, i. Show that cD=1 (see subsection IB2). bConsider the GL(m) subgroup of GL(n) (m <n) found by restricting the range of the index of the above dening representation. Show the structure con- stants are the same as those given by starting with the above representationof GL(m). cFind the structure constants for SO(n) and Sp(n). dDirectly evaluate k DcA(=ijGiGj) for SL(n), SO(n), and Sp(n), and compare withcDdA=dD. Excercise IB5.3 At e n s o rt h a tp o p su pi nv a r i o u sc o n t e x t si s dijk=tr(GifGj;Gkg) It takes a very simple form in terms of dening indices: aShow that for SU(n) this tensor is determined to be, up to an overall normal- ization (that depends on the representation), tr GI1J1 GI2J2;GI3J3
[(231)+(312)]2 n[(132)+(213)+(321)]+4 n2(123) (ijk)J1 IiJ2 IjJ3 Ik from just the total symmetry of dijk(andGII= 0), since the only invariant tensor available is J I.( I fIJ:::were used, IJ:::would also be required, to balance the number of subscripts and superscripts; but their product can be expressed in terms of just 's also.) bCheck this result by using the explicit G's for the dening representation, and determine the proportionality constant for that representation. With the exception of the \spinor" representations of SO(n) (to be discussed in subsection IC5, section IIA, and subsection XC1), general representations can be obtained by reducing direct products of the dening representations. This means theycan be described by objects with multiple indices (up/down, dotted/undotted), where each index is that of a dening representation, and satisfying various (anti)symmetry and tracelessness conditions on the indices. 44 I. GLOBAL Excercise IB5.4 Consider the representations of SU(n) obtained from the symmetric and an- tisymmetric part of the direct product of two dening representations. For simplicity, one can work with the U(n) generators, since the U(1) pieces will appear in a simple way. aUsing tensor notation for the generators ( GIJ)KLMN, nd their explicit rep- resentation for these two representations. bBy evaluating the trace, show that the Dynkin index for the two cases is ca=n2;cs=n+2 cShow the sum of these two is consistent with the argument at the end of subsection IB2. Show each case is consistent with n=2, and the antisymmetriccase with n=3, by relating those cases to the singlet, dening, and adjoint representations. REFERENCES 1 H. Georgi, Lie algebras in particle physics: from isospin to unied theories (Benjamin/Cummings, 1982):best book on Lie groups; unlike other texts, covers not only more powerful Cartan-Weyl methods, but also more useful tensor methods; also has useful applications tononrelativistic quark model. 2S. Helgason, Dierential geometry and symmetric spaces (Academic, 1962); R. Gilmore, Lie groups, Lie algebras, and some of their applications (Wiley, 1974): noncompact classical Lie groups. C. REPRESENTATIONS 45 ::::::::::::::::::: ::::::::::::::::::: ::::::::::::::::::: C. REPRESENTATIONS ::::::::::::::::::: We now consider some of the more useful representations, as explicit examples of the results of the previous section. In particular, we consider symmetries of the quark model. 1. More coordinates We began our \review" of group theory by looking at how symmetries were rep- resented on coordinates. We now return to coordinates as a special case (particular representation) of the general results of the previous section. The idea is that the coordinates themselves are already a representation of the group, and the wave func-tions are functions of these coordinates. For example, for ordinary rotations we use wave functions that depend on position or momentum, which transforms as a vec- tor. (This is not always the case: For example, in our description of the conformal group the usual space and time coordinates transformed nonlinearly, and not just by multiplication by constant matrices unless the extra two coordinates were intro-duced.) This is the basic distinction between classical mechanics and classical eld theory: Mechanics uses the coordinates themselves as the basic variables, while eld theory uses functions of the coordinates. (Similarly, in quantum mechanics the wavefunctions are functions of the coordinates, while in quantum eld theory the wave functions are \functionals" of functions of the coordinates.) In general, the construction of such a \coordinate representation" starts with a given matrix representation (usually nite dimensional) ( G i)IJand then denes a new representation ^Gi=qI(Gi)IJpJ;[pI;qJg=J I;[q;qg=[p;pg=0 for some objects qandp, which are interpreted as either coordinates and their con- jugate momenta (up to a factor of i), or as creation and annihilation operators: The latter nomenclature is used when the boundary conditions allow the existence of a statej0icalled the \vacuum", satisfying pj0i= 0, so we can dene the other states as functions of qacting onj0i. (If the coordinates are fermionic, the distinction is moot, since by the usual Taylor expansion the Hilbert space is nite dimensional. See excercise IA2.3.) It is easy to check that ^Gisatisfy the same commutation relations asGi. In particular, if the matrices are in the adjoint representation, qican be inter- preted as the group coordinates themselves: This follows from considering the action of an innitesimal transformation on the group element g(q)=eiqiGi(or just the Lie algebra element G(q)=qiGi). 46 I. GLOBAL If we write these results in bra/ket notation, since [^Gi;qI]=qJ(Gi)JI;[^Gi;pI]=(Gi)IJpJ it is more natural to look at the action on bras: hqj=qIhIj;jpi=jIipI) ^Gihqj=hqjGi;^Gijpi=Gijpi Note that this vector space is coordinate space itself, not the space of functions of the coordinates; it is the same space on which Giis dened. (Of course, ^Giis dened on arbitrary functions of the coordinates; it has a reducible representation bigger than (Gi)IJ. Eectively, ( Gi)IJis represented on the space of functions linear in the coordinates.) Then, for example ^G1^G2hqj=^G1hqjG2=hqjG1G2 is obviously equivalent, while (ignoring any extra signs for fermions) ^G1^G2jpi=^G1G2jpi=G2^G1jpi=G2G1jpi at least gives an equivalent result for the commutator algebra [ ^G1;^G2]. This is the expected result for the dual representation Gi!GT i. Interesting examples are given by using the dening representation for G.F o r example, the commonly used oscillator representation for U(n) is U(n):GIJ=ayJaI;[aI;ayJg=J I where the oscillators can be bosonic or fermionic. For the SO and Sp cases, because we can raise and lower indices, and because of the (anti)symmetry on the indices, the interesting possibility arises to identify the coordinates with their momenta, with the statistics appropriate to the symmetry: Sp(n):GIJ=1 2z(IzJ);[zI;zJ]= IJ SO(n):GIJ=1 2 [I J];f I; Jg=IJ For SO(n) the representation is nite dimensional because of the Fermi-Dirac statis- tics, and is called a \Dirac spinor" (and the \Dirac matrices"). If the opposite statistics are chosen, the coordinates and momenta can't be identied: For example, bosonic coordinates for SO(n) give the usual spatial rotation generators GIJ=x[I@J]. Excercise IC1.1 Use this bosonic oscillator representation for U(2)=SU(2) U(1), and use the C. REPRESENTATIONS 47 SU(2) subgroup to describe spin. Show that the spin s(the integer or half- integer number that denes the representation) itself has a very simple ex- pression in terms of the U(1) generator. Show this holds in the quantummechanical case (by interpreting the bracket as the quantum commutator),giving the usual s(s+1) for the sum of the squares of the generators (with ap- propriate normalization). Use this result to show that these oscillators, actingon the vacuum state, can be used to construct the usual states of arbitraryspins. Excercise IC1.2 Considering SO(2n), divide up Iinto pairs of canonical (and complex) con- jugatesa1=( 1+i 2)=p 2, etc., sofa;ayg= 1. Write the SO(2n) generators in terms of aa,ayay,a n daya. Show that the aya's by themselves generate a U(n) subgroup. Decompose the Dirac spinor into U(n) representations. Show that the product of all the 's is related to the U(1) generator, and commutes with all the SO(2n) generators. Show that the states created by even or oddnumbers of a y's on the vacuum don't mix with each other under SO(2n), so the Dirac spinor is reducible into two \Weyl spinors". 2. Coordinate tensors Many groups can be represented on coordinates. Depending on the choice of coordinates, the coordinates may transform nonlinearly (i.e., as a realization, not arepresentation), as for the D-dimensional conformal group in terms of D (not D+2)coordinates. However, given the nonlinear transformation of the coordinates, thereare always representations other than the dening one (scalar eld) that we can im-mediately write down (such as the adjoint). We now consider such representations:These are useful not only for the spacetime symmetries we have already considered, but also for general relativity, where the symmetry group consists of arbitrary coor- dinate transformations. Furthermore, these considerations are useful for describingcoordinate transformations that are not symmetries, such as the change from Carte-sian to polar coordinates in nonrelativistic theories. When applied to quantum mechanics, we write the action of a symmetry on a state as =iG (or 0=eiG ), but on an operator as A=i[G;A]( o rA0= eiGAeiG). In classical mechanics, we always write A=i[G;A] (since classical objects are identied with quantum operators, not states). However, if G=m@mis a coordinate transformation (e.g., a rotation) and is a scalar eld, then in quantum 48 I. GLOBAL notation we can write (x)=[G;]=G=m@m (0=eGeG=eG) since the derivatives in Gjust dierentiate . (For this discussion of coordinate transformations we switch to absorbing the i's into the generators.) The coordinate transformation Ghas the usual properties of a derivative: [G;f(x)] =Gf)Gf1f2=[G;f 1f2]=(Gf1)f2+f1Gf2 eGf1f2=eGf1f2eG=(eGf1eG)(eGf2eG)=(eGf1)(eGf2) and similarly for products of more functions. The adjoint representation of coordinate transformations is a \vector eld" (in the sense of a spatial vector), a function that has general dependence on the coordinates (like a scalar eld) but is also linear in the momenta (as are the Poincar e generators): G=m(x)@m;V =Vm(x)@m)V=[G;V]=(m@mVnVm@mn)@n )Vm=n@nVmVn@nm The same result follows if we use the Poisson bracket instead of the quantum me- chanical commutator, replacing @mwithipmin bothGandV. Finite transformations can also be expressed in terms of transformed coordinates themselves, instead of the transformation parameter: (x)=em@m0(x)=0(em@mx) as seen, for example, from a Taylor expansion of 0,u s i n geG0=eG0eG.W e t h e n dene 0(x0)=(x))x0=em@mx This is essentially the statement that the active and passive transformations cancel. However, in general this method of dening coordinate transformations is not con- venient for applications: When we make a coordinate transformation, we want toknow 0(x). Working with the \inverse" transformation on the coordinates, i.e., our originale+G, ~xe+m@mx)0(x)=eG(x)=(~x(x)) So, for nite transformations, we work directly in terms of ~ x(x), and simply plug this intoin place ofx(x!~x(x)) to nd0as a function of x. C. REPRESENTATIONS 49 Similar remarks apply for the vector, and for derivatives in general. We then use x0=eGx)@0=eG@eG where@0=@=@x0,s i n c e@0x0=@x=. This tells us Vm(x)@m=eGV0m(x)@meG=V0m(x0)@0 m orV0(x0)=V(x). Acting with both sides on x0m, V0m(x0)=Vn(x)@x0m @xn On the other hand, working in terms of ~ xis again more convenient: Since for ~@=@=@~x ~@m=eG@meG="@~x(x) @x1# mn @n we have, for example, (Vm@m)0(x)=eG(Vm@m)(x)=Vm(~x(x))"@~x(x) @x1# mn @n(~x(x)) which yields an explicit expression for the transformed elds. A \dierential form" is dened as an innitesimal W=dxmWm(x). Its transfor- mation law under coordinate transformations, like that of scalar and vector elds, is dened byW0(x0)=W(x). For any vector eld V=Vm(x)@m,VmWmtransforms as a scalar, as follows from the \chain rule" d=dx0m@0 m=dxm@m. Explicitly, W0 m(x0)=Wn(x)@xn @x0m or in innitesimal form Wm=n@nWm+Wn@mn Thus a dierential form is dual to a vector, at least as far as the matrix part of coor- dinate transformations is concerned. They transform the same way under rotations,because rotations are orthogonal; however, more generally they transform dierently, and in the absence of a metric there is not even a way to relate the two by raising or lowering indices. Higher-rank dierential forms can be dened by antisymmetric products of the above \one-forms". These are useful for integration: Just as the line integralRW=Rdx mWmis invariant under coordinate transformations by denition (as long as we choose the curve along which the integral is performed in a coordinate-independent 50 I. GLOBAL way), so is a totally antisymmetric Nth-rank tensor (\ N-form")Wm1


mNintegrated on anN-dimensional subspace as Z dxm1


dxmNWm1


mN:W0 m1


mN(x0)=Wp1


pN(x)@xp1 @x0m1


@xpN @x0mN w h e r et h es u r f a c ee l e m e n t dxm1


dxmNis interpreted as antisymmetric. (The signs come from switching initial and nal limits of integration, as prescribed by the \ori-entation" of the hypersurface.) This is clear if we rewrite the integral more explicitlyin terms of coordinates  ifor the subspace: Then Z dxm1


dxmNWm1


mN(x)=Z di1


diNcWi1


iN()=Z dNi1


iNcWi1


iN() where cWi1


iN()=@xm1 @i1


@xmN @iNWm1


mN(x) is the result of a coordinate transformation that converts Nof thex's to's, an interpretation of the functions x() that dene the surface. Then any coordinate transformation on x!x0(not on) will leavecW() invariant. In particular, if the subspace is the full space, so we can look directly atR dNxm1


mNWm1


mN,w es e e that a coordinate transformation generates from WanN-dimensional determinant exactly canceling the Jacobian resulting from changing the integration measure dNx. Excercise IC2.1 For all of the following, use the exponential form of the nite coordinate trans-formation: Show that any (local) function of a scalar eld (without explicit x dependence additional to that in the eld) is also a scalar eld (i.e., satisesthe same coordinate transformation law). Show that the transformation lawof a vector eld or dierential form remains the same when multiplied by ascalar eld (at the same x). Show that V=V m@mis a scalar eld for any scalar eld and vector eld V. Show that [ V;W ] is a vector eld for any vector elds VandW. Excercise IC2.2 Examine nite coordinate transformations for integrals of dierential formsin terms of ~ xrather than x 0. Find the explicit expression for W0(x)i nt e r m s ofW(~x(x)), etc., and use this to show invariance: Z dxm1


dxmNW0 m1


mN(x)=Z d~xm1


d~xmNWm1


mN(~x) =Z dxm1


dxmNWm1


mN(x) C. REPRESENTATIONS 51 where in the last step we have simply substituted ~ x!xas a change of integra- tion variables. Note that, using the ~ xform of the transformation rather than x0, the transformation generates the needed Jacobain, rather than canceling one. From the above transformation law, we see that the curl of a dierential form is also a dierential form: @0 [m1W0 m2


mN](x0)=@0 [m1(@0 m2xp2)


(@0 mN]xpN)Wp2


pN(x) =[@[p1Wp2


pN](x)](@0 m1xp1)


(@0 mNxpN) because the curl kills @0@0xterms that would appear if there were no antisymmetriza- tion. Objects that transform \covariantly" under coordinate transformations, without such higher derivatives of x(orin the other notation), like scalars, vectors, dieren- tial forms and their products, are called (coordinate) \tensors". Getting derivatives of tensors to come out covariant in general requires special elds, and will be discussed in chapter IX. An important application of the covariance of the curl of dierentialforms is the generalized Stokes' theorem (which includes the usual Stokes' theorem and Gauss' law as special cases): Z dx m1


dxmN+1 1 (N+1)!@[m1Wm2


mN+1]=I dxm1


dxmNWm1


mN where the second integral is over the boundary of the space over which the rst is integrated. (We use the symbol \H " to refer to boundary integrals, including those over contours, which are closed boundaries of 2D surfaces.) 3. Young tableaux We now return to our discussion of nite-dimensional representations. In the previous section we gave the machinery for describing them using index notation,but examined only the dening representation in detail. Now we analyze general irreducible representations. All the irreducible nite-dimensional representations of the groups SU(N) can be described by tensors with lower N-valued indices with various (anti)symmetrizations.(An upper index can be replaced with N 1 lower indices by using the Levi-Civita tensor.) Although detailed calculations require explicit use of these indices, three properties can be more conveniently discussed pictorially: (1) the (anti)symmetriesof the indices, (2) the dimension (number of independent components) of the repre- sentation, and (3) the reduction of the direct product of two representations (which irreducible representations result, and how many of each). 52 I. GLOBAL A \Young tableau" is a picture representing an irreducible representation in terms of boxes arranged in a regular grid into rows and columns, such that the columns are aligned at the top, and their depths are nonincreasing to the right: for example, Each box represents an index, with antisymmetry among indices in any column, and symmetry among indices in any row. More precisely, since one can't simultaneously have these symmetries and antisymmetries, it corresponds to the result of taking anyarbitrary tensor with that many indices, rst symmetrizing the indices in each row, and then antisymmetrizing the indices in each column (or vice versa; symmetrizing and then antisymmetrizing and then symmetrizing again gives the same result asskipping the rst symmetrization, etc.). This gives a simple way to classify and symbolize each representation. (We can denote the singlet representation, which has no boxes, by a dot.) Note that the deepest column should have no more than N 1 boxes for SU(N) because of the antisymmetry. To calculate the dimension of the representation for a given tableau, we use the \factors over hooks" rule: (1) Write an \N" in the box in the upper-left corner, and ll the rest of the boxes with numbers that decrease by 1 for each step down and increase by 1 for each step to the right. (2) Draw (or picture in your mind) a \hook" for each box | a \" with its corner in the box and lines extending right and downout of the tableau. (3) The dimension is then given by the formula dimension =Y each boxinteger written there # boxes intersected by its hook For the previous example, we nd (listing boxes rst down and then to the right) N 8
N1 6
N2 3
N3 1
N+1 6
N 4
N1 1
N+2 4
N+1 2
N+3 3
N+2 1
N+4 1 The direct product of two Young tableaux A B is analyzed by the following rules: First, label all the boxes in B by putting an \a" in each box in the top row, \b" in the second row, etc. Then, take the following steps in all possible ways to nd theYoung tableaux resulting from the direct product: (1) Add all the \a" boxes from B to the right side and bottom of A, then \b" to the right and bottom of that, etc., to make a new Young tableaux. Any two tableaux constructed in this way with the samearrangement of boxes but dierent assignment of letters are considered distinct, i.e., multiple occurences of the same representation in the direct product. (2) No more than 1 \a" can be in any column, and similarly for the other letters. (3) Reading from C. REPRESENTATIONS 53 right to left, and then from top to bottom (i.e., like Hebrew/Arabic), the number of a's read should always be the number of b's, b's c's, etc. For example, a ba=baaaa ba ba Note that A B always gives the same result as B A, but one way may be simpler than the other. For a given value of N, a column of N boxes is equivalent to none(again by antisymmetry), while more than N boxes in a column gives a vanishing tableau. Excercise IC3.1 Calculate Check the result by nding the dimensions of all the representations and adding them up. These SU(N) tableaux also apply to SL(N): Only the reality properties are dif- ferent. Similar methods can be applied to USp(2N) (or Sp(2N)), but tracelessness(with respect to the symplectic metric) must be imposed in antisymmetrized indices,so these trace pieces must be separated out when considering the above rules. (I.e.,consider USp(2N) SU(2N).) Similar remarks apply to SO(N), which has a symmet- ric metric, but there are also \spinor" representations (see below). The additional irreducible representations then can be constructed from taking direct products ofthe above with the smallest spinors, and removing the \gamma-matrix" traces. 4. Color and avor We now consider the application of these methods to \internal symmetries" (those that don't act on the coordinates) in particle physics. The symmetries with experi- mental conrmation involve only the unitary groups (U and SU) of small dimension.However, we will nd later that larger unitary groups can be useful for approxima-tion schemes. (Also, larger unitary and other groups continue to be investigated forunication and other purposes, which we consider in later chapters.) The \Standard Model" describes all of particle physics that is well conrmed experimentally (except gravity, which is not understood at the quantum level). Itincludes as its \fundamental" particles: (1) the spin-1/2 quarks that make up the observed strongly interacting particles, but do not exist as asymptotic states, (2) the weakly interacting spin-1/2 leptons, (3) the spin-1 gluons that bind the quarks to-gether, which couple to the charges associated with SU(3) \color" symmetry, but also 54 I. GLOBAL are not asymptotic, (4) the spin-1 particles that mediate the weak and electromag- netic interactions, which couple to SU(2) U(1) \ avor", and (5) the yet unobserved spin-0 Higgs particles that are responsible for all the masses of these weakly interact-ing particles. These particles, along with their masses (in GeV) and (electromagnetic)charges (Q=Q+
Q), are: s=1 2 color:! quark (3) lepton (1) avor (
Q)(Q=1 6) (Q=1 2) +1 2u(.005)e(<108) 1 2d(.009)e(.0005109991) +1 2c(1.4)(<.00017) 1 2s(.18)(.10565839) +1 2t(174)(<.0182) 1 2b(4.7)(1.7770)s=1 color:! gluon electroweak avor (Q) (8) (1) 0g(0) (<2
1025) 0 Z(91.187) 1 W(80.4) s=0 (Q=0 )H (>77.5) The quark masses we have listed are the \current quark masses", the eective masses when the quarks are relativistic with respect to their hadron, and act as almost free.Nonrelativistic quark models use instead the \constituent quark masses", which in-clude potential energy from the gluons. This extra potential energy is about .30GeV per quark in the lightest mesons, .35 GeV in the lightest baryons; there is alsoa contribution to the binding energy from spin-spin interaction. Unlike electrody-namics, where the potential energy is negative because the electrons are free at largedistances, where the potential levels o (the top of the \well"), in chromodynamicsthe potential energy is positive because the quarks are free at high energies (shortdistances, the bottom of the well), and the potential is innitely rising. Masslessnessof the gluons is implied by the fact that no colorful asymptotic states have ever beenobserved. We have divided the spin-1/2 particles into 3 \families" with the same quantum numbers (but dierent masses). Within each family, the quarks are similar to the leptons, except that: (1) the masses and average charges ( Q) are dierent, (2) the quarks come in 3 colors, while the leptons are colorless, and (3) the neutrinos,to within experimental error, are massless, so they have half as many componentsas the massive fermions (1 helicity state each, instead of 2 spin states each). Thismeans that each lepton family has 1 SU(2) doublet and 1 SU(2) singlet. For symme-try (and better, quantum mechanical, reasons to be explained later), we also assumethe quarks have 1 SU(2) doublet, but therefore 2 SU(2) singlets. (Some experimentshave indicated small masses for neutrinos: This would require generalization of the C. REPRESENTATIONS 55 Standard Model, such as models with parity broken by interactions. Some examples of such theories will be discussed in subsection IVB4.) We rst look at the color group theory of the physical states, which are color singlets. The fundamental unobserved particles are the spin-1 \gluons", described by the Yang-Mills gauge elds, and the spin-1/2 quarks. Suppressing all but color indices, we denote the quark states by qi, and the antiquarks by qyi, where the indices are those of the dening representation of SU(n), and its complex conjugate. The quarks alsocarry a representation of a \ avor" group, unlike the gluons. The simplest avorfulstates are those made up of only (anti)quarks, with indices completely contracted byone factor of an SU(n) group metric: From the \U" of SU(n), we can contract dening indices with their complex conjugates, giving the \mesons", described by q yiqi(quark- antiquark), which are their own antiparticles. From the \S" of SU(n), we have the\baryons", described by  i1:::inqi1:::qin(n-quark), and the antibaryons, described by the complex conjugate elds. All other colorless states made of just (anti)quarkscan be written as products of these elds, and therefore considered as describingcomposites of them. Thus, we can approximate the ground states of the mesons by q yi(x)qi(x), which describe spins 0 and 1 because of the various combinations of spins (from1 2 1 2=01). The rst excited level will then be described by qy$ @q(where A$ @BA@B(@A)B and picks out the relative momentum of the two quarks): It includes spins 0, 1, and 2, etc., where each derivative introduces orbital angular momentum 1. (Similar remarksapply to baryons.) We can also have avorless states made from just gluons, called \glueballs": The ground states can be described by F ijFji,w h e r ee a c h Fis a gluon state (in the adjoint representation of SU(n)), and includes spins 0 and 2 (from thesymmetric part of 1 1). Because of their avor multiplets and (electroweak) inter- actions, many mesons and baryons corresponding to such ground and excited stateshave been experimentally identied, while the glueballs' existence is still uncertain.Actually, quarks and gluons can almost be observed independently at high energies,where the \strong" interaction is weak: The energetic particle appears as a \jet" | a particle of high energy accompanied by particles of much lower energy (perhaps too small to detect) in color-singlet combinations. (Depending on the available decaymodes, the jet might not be observed until after decaying, but still within a smallangle of spread.) We now look at the avor group theory of the physical hadronic states. In contrast to the previous paragraph, we now suppress all but the avor indices. Mesons M ij= 56 I. GLOBAL qyiqjare thus in the adjoint representation of avor U(m) ( m m,w h e r emis the dening representation and  mits complex conjugate), for both the spin-0 and the spin-1 ground states. The baryons are more complicated: For simplicity we considerSU(3) color, which accurately describes physics at observed energies. Then the colorstructure described above results in total symmetry in combined avor and Lorentzindices (from the antisymmetry in the color indices, and the overall antisymmetry forFermi-Dirac statistics). Thus, for the 3-quark baryons, the Young tableaux  for SU(m) avor are accompanied by the same Young tableaux for spin indices: Innonrelativistic notation, the rst tableau, being totally antisymmetric in avor in-dices, is also totally antisymmetric in the three two-valued spinor indices, and thusvanishes. Similarly, the last tableau describes spin 3/2 (total symmetry in both types of indices), while the middle one describes spin 1/2. Since only 3 avors of quarks have small masses compared to the hadronic mass scale, hadrons can be most conve-niently grouped into avor multiplets for SU(3) avor: The ground states are then,in terms of SU(3) avor multiplets, 8 1 for the pseudoscalars, 8 1 for the vectors, 8 for spin 1/2, and 10 for spin 3/2. Excercise IC4.1 What SU( avor) Young tableaux, corresponding to what spins, would we havefor mesons and baryons if there were 2 colors? 4 colors? However, the diering masses of the dierent avors of quarks break the SU(3) avor symmetry (as does the weak interaction). In particular, the mass eigenstatestend to be pure states of the various combinations of the dierent avors of quarks,rather than the linear combinations expected from the avor symmetry. Specically,the linear combinations predicted by an 8 1 separation for mesons (trace and traceless pieces of a 3 3 matrix) are replaced with particles that are more accurately described by a particular avor of quark bound to a particular avor of antiquark. (This isknown as \ideal mixing".) The one exception is the lighest mesons (pseudoscalars),which are more accurately described by the 8 1 split, for this restriction to the 3 lighter avors of quarks, but the mass of the singlet diers from that naively expected from group theory or nonrelativistic quark models. (This is known as the \U(1) problem".) The solution is probably that the singlet mixes strongly with the lightestpsuedoscalar glueball (described by tr  abcdFabFcd); the mass eigenstates are linear combinations of these two elds with the same quantum numbers. In any case, themost convenient notation for labeling the entries of the matrix M ijrepresenting the C. REPRESENTATIONS 57 various meson states for any particular spin and angular momentum of the quark- antiquark combination is that corresponding to the choice we gave earlier for the generators of U(n): Label each entry by a separate name, where the complex conjugateappears re ected across the diagonal. These directly correspond to the combinationof a particular quark with a particular antiquark, and to the mass eigenstates, withthe possible exception of the entries along the diagonal for the 3 lightest avors,where the mass eigenstates are various linear combinations. (However, the SU(2) ofthe 2 lightest avors is only slightly broken by the quark masses, so in that case thecombinations are very close to the 3 1 split of SU(2).) For example, for the lightest multiplet of mesons (spin 0, and relative angular mo- mentum 0 for the quark and antiquark, but not all of which have yet been observed),we can write the U(6) matrix (for the 6 avors of the 3 known families) M ij=0 BBBBBBBB@uu ud ucus utub du dd dc ds dt db cu cd cccsctcb su sd scssstsb tu td tctstttb bu bdbcbsbtbb1 CCCCCCCCA =0 BBBBBBBB@ud c s t b u u(:1395700)D0(1:8646)K(:49368)T0B(5:279) d+(")d D+(1:8693) K0(:49767)T+B0(5:279) c D0(")D(")c(2:980)D s(1:9685)T0 cB c sK+(")K0(")D+ s(")s T+ sB0 s t T0T T0 c T s tT b bB+(")B0(")B+ c B0 s T+ bb1 CCCCCCCCA where (approximately)  u=1p 20(:1349764) +1 2[0(:9578) +(:5473)] d=1p 20+1 2(0+);s=1p 2(0) in terms of the mass eigenstates (observed particles), with masses again in GeV, and ditto marks refer to the transposed entry. (We have neglected the important 58 I. GLOBAL contribution from the glueball.) For the corresponding spin-1 multiplet, fMij=0 BBBBBBBB@ud c s t b u! u(:7700)D*0(2:0067)K*(:8917)T*0B*(5:325) d+(")!dD*+(2:0100) K*0(:8961)T*+B*0(5:325) c D*0(")D*(")J= (3:09688)D* sT*0 cB* c sK *+(")K*0(")D*+ s (1:019413)T*+ sB*0 s t T*0T* T*0 c T* s T * b bB *+(")B*0(")B*+ c B*0 sT*+ b(9:4604)1 CCCCCCCCA where ! u=1p 2[!(:7819) +0(:7700)];!d=1p 2(!0) (with ss=, ideal mixing, also approximate). 5. Covering groups The orthogonal groups O(n +,n) are of obvious interest for describing Lorentz symmetry in spacetimes with n +space and ntime dimensions, or conformal sym- metry in spacetimes with n +1s p a c ea n dn 1 time dimensions. This means we should be interested in O(n) for n 6, and their \Wick rotations": transformations that put in extra factors of ito change some signs on the metric. Coincidentally, these are just the cases where the Lie algebras of the orthogonal groups are equivalent to those of some algebras for smaller matrices. The smaller representation then can be identied as the \spinor" representation of that orthogonal group. Since the \vec-tor", or dening representation space of the orthogonal group, itself is represented asa matrix with respect to the other group (i.e., the state carries two spinor indices),the other group may include certain phase transformations (such as 1) that cancel in the transformation of the vector. The other group is then called the \covering"group for that orthogonal group, since it includes those missing transformations inits dening representation. (As a result, its group space also has a more interesting topology, which we won't discuss here.) One way to discover these covering groups is to rst count generators, then try to construct explicitly the orthogonal metric on matrices. SO(n) has n(n 1)/2 gen- erators (antisymmetric matrices), Sp(n) has n(n+1)/2 (symmetric), and SU(n) has n 21 (traceless). (These are hermitian generators, since we applied reality or her- miticity.) So, for some group SO(n), we look for another group that has the samenumber of generators. Then, if the new group is dened on m m matrices, we look for conditions to impose on an m m matrix (not necessarily the adjoint) to get an C. REPRESENTATIONS 59 n-component representation. This is easy to do by inspection for small n; for large n it's easy to see that it can't work, since m will be of the order of n, and the simpleconstraints will give of the order of n 2components instead of n. We then construct the norm of this matrix Mastr(MyM), which is just the sum of the absolute value squared of the components, for SO(n), and the other orthogonal groups by Wick rotation. (Wick rotation aects mainly the reality conditions on M.) The identications for the Lie algebras are then: SO(2) = U(1), SO(1,1) = GL(1) SO(3) = SU(2) = SU*(2) = USp(2), SO(2,1) = SU(1,1) = SL(2) = Sp(2)SO(4) = SU(2) SU(2), SO(3,1) = SL(2,C) = Sp(2,C), SO(2,2) = SL(2) SL(2) SO(5) = USp(4), SO(4,1) = USp(2,2), SO(3,2) = Sp(4) SO(6) = SU(4), SO(5,1) = SU*(4), SO(4,2) = SU(2,2), SO(3,3) = SL(4) Note that the Euclidean cases are all unitary, while the ones with (almost) equal numbers of space and time dimensions are all real. There are also some similarrelations for the pseudoreal orthogonal groups: SO*(2) = U(1), SO*(4) = SU(2) SL(2), SO*(6) = SU(3,1), SO*(8) = SO(6,2) The norm and conditions for an m-spinor of SO(n +,n)a r e : n) 0 1 2 3 mnnorm symmetry :zT= reality :z*= 12z0z z0z(z0*=z0) 23zz   zzz 4z0z 0 00 zzTz 45zz  z(z =0 )1 2z1 2(z)z 6zz  z1 2z z 1 2(z)z Note that in all but the 2D cases the norms are associated with determinants: For D=3 and 4 the norm is given by the determinant, while for D=5 and 6 we use thefact that the determinant of an antisymmetric matrix is the square of the Pfaan. Excercise IC5.1 Show that for D=5 zzandzz give the same norm. (Hint: Consider [ ].) Unfortunately, for SO(n) for larger n, the spinor is as least as large as, and usu- ally larger than, the vector. In general, the spinor is like the \square root" of the vector, in that the vector can be found by taking the direct product of two spinors. 60 I. GLOBAL It is impossible to nd the spinor representation by taking direct products of vec- tors. This situation occurs only for orthogonal groups: In all other classical groups, all (nite-dimensional) representations are among those obtained from multiple di- rect products of vectors. Furthermore, in those cases the \irreducible" representa- tions (those that can't be divided into smaller representations) can be picked out by(anti)symmetrization, and by separating trace and traceless pieces (where traces are taken with the group metrics). Fortunately, for the above cases of orthogonal groups, we can perform the same construction starting with the spinor representations, sincethose are the \vectors" of non-orthogonal groups. REFERENCES 1 J. Schwinger, On angular momentum, Quantum theory of angular momentum: a collec- tion of reprints and original papers , eds. L.C. Biedenharn and H. Van Dam (Academic, 1965) p. 229:spin using spinor oscillators. 2Georgi, loc. cit. (IB). 3P.A.M. Dirac, P r o c .R o y .S o c . A117 (1928) 610. 4H. Weyl, Z. Phys. 56(1929) 330. 5M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley, 1962):detailed discussion of Young tableaux. 6M. Gell-Mann, Phys. Lett. 8(1964) 214; G. Zweig, preprints CERN-TH-401 and 412 (1964); Fractionally charged particles andSU 6,i nSymmetries and elementary particle physics , proc. Int. School of Physics \Ettore Majorana", Erice, Italy, Aug.-Sept., 1964, ed. A. Zichichi (Academic, 1965) p. 192:quarks. 7Particle Data Group (in 1998, C. Caso, G. Conforto, A. Gurtu, M. Aguilar-Benitez, C.Amsler, R.M. Barnett, P.R. Burchat, C.D. Carone, O. Dahl, M. Doser, S. Eidelman, J.L.F e n g ,M .G o o d m a n ,C .G r a b ,D . E .G r o o m ,K .H a g i w a r a ,K . G .H a y e s ,J . J .H e r n  andez, K. Hikasa, K. Honscheid, F. James, M.L. Mangano, A.V. Manoha, K. M onig, H. Mu- rayama, K. Nakamura, K.A. Olive, A. Piepke, M. Roos, R.H. Schindler, R.E. Shrock,M. Tanabashi, N.A. T ornqvist, T.G. Trippe, P. Vogel, C.G. Wohl, R.L. Workman, and W.-M. Yao), http://pdg.lbl.gov/pdg.html (published biannually, including European Phys. J. C3(1998) 1): Review of Particle Physics, a.k.a. Review of Particle Properties, a.k.a. Rosenfeld tables;tables of masses, decay rates, etc., of all known particles, plus useful brief reviews. A. TWO COMPONENTS 61 II. SPIN Special relativity is simply the statement that the laws of nature are symmetric under the Poincar e group. Free relativistic quantum mechanics or eld theory is then equivalent to a study of the representations of the Poincar e group. Since the con- formal group is a classical group, while its subgroup the Poincar e group is not, it is easier to rst study the conformal group, which is sucient for nding the masslessrepresentations of the Poincar e group. The massive ones then can be found by di- mensional reduction, which gives them in the same form as occurs in interacting eldtheories. In four spacetime dimensions we use the covering group of the conformalgroup, which is the easiest way to include spinors. These methods extend straight-forwardly to supersymmetry, a symmetry between fermions and bosons that includesthe Poincar e group. :::::::::::::::::: :::::::::::::::::: :::::::::::::::::: A. TWO COMPONENTS :::::::::::::::::: Although we have already specialized to spacetime symmetries, we have consid- ered arbitrary spacetime dimensions. We have also noted that many of the lower-dimensional Lie groups have special properties, especially with regard to coveringgroups. In this section we will take advantage of those features; specically, we ex-amine the physical case D=4, where the rotation group is SO(3)=SU(2), the Lorentzgroup is SO(3,1)=SL(2,C), and the conformal group is SO(4,2)=SU(2,2). 1. 3-vectors The most important nontrivial Lie group in physics is the rotations in three dimensions. It is also the simplest nontrivial example of a Lie group. This makesit the ideal example to illustrate the properties discussed in the previous chapter,as well as lay the groundwork for later discussions. We have already mentioned theorbital part of rotations, i.e., the representation of rotations on spatial coordinates.In this chapter we discuss the spin part; this is really the same as nding all (nitedimensional, unitary) representations. A useful way to understand spin, or general representations of the rotation group in three dimensions of space, is to consider properties of 2 2 matrices. (This way also generalizes in a very simple way to relativity, in three space and one time dimen-sions.) Consider such matrices to be hermitian, which is natural from the quantummechanical point of view. Then they have four real components, one too many for a 62 II. SPIN three-vector (but just right for a relativistic four-vector), so we restrict them to also be traceless: V=Vy;t r V =0 The simplest way to get a single number out of a matrix, besides taking the trace, is to take the determinant. By expanding a general matrix identity to quadratic order we nd an identity for 2 2 matrices det(I+M)=etrln(I+M)) 2detM =tr(M2)(tr M )2 It is then clear that in our case det V is positive denite, as well as quadratic, so we can dene the norm of this 3-vector as jVj2=2detV =tr(V2) This can be compared easily with conventional notation by picking a basis: V=1p 2V1V2iV3 V2+iV3V1 =~V
~)det V =1 2(Vi)2 where~are the Pauli matrices, up to normalization. As usual, the inner product follows from the norm: jV+Wj2=jVj2+jWj2+2V
W )V
W=detV +det Wdet(V+W)=tr(VW) Applying our previous identities for determinants to 2 2 matrices, we have MCMTC=Id e tM ; M1=CMTC(detM )1 where we now use the imaginary, hermitian matrix C=0 ii 0
If we make the replacement M!eMand expand to linear order in M, we nd M+CMTC=It rM This implies tr V =0, (VC)T=VC i.e., the tracelessness of Vis equivalent to the symmetry of VC. Furthermore, the combination of the trace and determinant identities tell us M2=Mt rMId e tM)V2=Id e tV =I1 2jVj2 A. TWO COMPONENTS 63 Here by \V2" we mean the square of the matrix, while \ jVj2"= (Vi)2is the square of the norm (neither of which should be confused with the component V2=Vi2 i.) Again expressing the inner product in terms of the norm, we then nd fV;Wg=(V
W)I Furthermore, since the trace of a commutator of two nite matrices is traceless, and picks up a minus sign under hermitian conjugation, we can dene an outer product (vectorvector = vector) by [V;W ]=p 2iVW Combining these two results, VW =1 2(V
W)I+1p 2iVW In other words, the product of two traceless hermitian 2 2 matrices gives a real trace piece, symmetric in the two matrices, plus an antihermitian traceless piece, antisym- metric in the two. Thus, we have a simple relation between the matrix product, the inner (\dot") product and the outer (\cross") product. Therefore, the cross product is a special case of the Lie bracket, or commutator. This way of treating vectors (except for factors of \ i") is basically Hamilton's \quaternions". Excercise IIA1.1 Check this result in two ways: aShow the normalization agrees with the usual outer product. Using only the above denition of VW,a l o n gw i t hfV;Wg=(V
W)I,s h o w IjVWj2=( [V;W ])2=I[jVj2jWj2(V
W)2] bUse components, with the above basis. Excercise IIA1.2 Write an arbitrary two-dimensional vector in terms of a complex number as V=1p 2(vxivy). aShow that the phase (U(1)) transformation V0=Veigenerates the usual rotation. Show that for any two vectors V1andV2,V1*V2is invariant, and identify its real and imaginary parts in terms of well known vector prod- ucts. What kind of transformation is V!V*, and how does it aect these products? 64 II. SPIN bConsider two-dimensional functions in terms of z=1p 2(x+iy)a n dz*= 1p 2(xiy). Show by the chain rule that @z=1p 2(@xi@y). Write the real and imaginary parts of the equation @z*V= 0 in terms of the divergence and curl. (Then Vis a function of just z.) cConsider the complex integral Idz 2iV where \H " is a \contour integral": an integral over a closed path in the complex plane dened by parametrizing dz=du(dz=du ) in terms of some real parameter u.T h i s i s u s e f u l i f Vcan be Laurent expanded as V(z)=P1 n=1cn(zz0)ninside the contour about a point z0there, since by con- sidering circles z=z0+reiwe nd only the 1 =(zz0) term contributes. Show that this integral contains as its real and imaginary parts the usual lineintegral and \surface" integral. (In two dimensions a surface element diersfrom a line element only by its direction.) Use this fact to solve Gauss' lawin two dimensions for a unit point charge as E=1=z. Excercise IIA1.3 Consider electromagnetism in 2 2 matrix notation: Dene the eld strength as a complex vectorF=p 2(E+iB). Write partial derivatives as the sum of a (rotational) scalar plus a (3-)vector as @=1p 2I@t+r,w h e r e@t=@=@t is the time derivative and ris the partial space derivatives written as a traceless matrix. Do the same for the charge density and (3-)current jas J=1p 2I+j. Using the denition of dot and cross products in terms of matrix multiplication as discussed in this section, show that the simple matrixequation@F=J, when separated into its trace and traceless pieces, and its hermitian and antihermitian pieces, gives the usual Maxwell equations r
B=0;r
E=;rE+@ tB=0;rB@tE=j (Note: Avoid the Pauli -matrices and explicit components.) 2. Rotations One convenience of representing three-vectors as 2 2 instead of 31 is that rota- tions are easier to write. Since vectors are hermitian, we expect their transformationsto be unitary: V 0=UVUy;Uy=U1 A. TWO COMPONENTS 65 It is easily checked that this preserves the properties of these matrices: (V0)y=(UVUy)y=V0;t r (V0)=tr(UVU1)=tr(U1UV)=tr(V)=0 Furthermore, it also preserves the norm (and thus the inner product): det(V0)=det(UVU1)=det(U)det(V)(det U )1=det V Unitary 22 matrices have 4 parameters; however, we can elimimate one by the condition det U =1 This eliminates only the phase factor in U, which cancels out in the transformation law anyway. Taking the product of two rotations now involves multiplying only 2 2 matrices, and not 3 3 matrices. We can also write Uin exponential notation, which is useful for going to the innitesimal limit: U=eiG)Gy=G; tr G =0 This means that Gitself can be considered a vector. Rotations can be parametrized by a vector whose direction is the axis of rotation, and whose magnitude is (1 =p 2) the angle of rotation: V0=eiGVeiG)V=i[G;V]=p 2GV We also now see that the Lie bracket we previously identied as the cross product is the bracket for the rotation group. Excercise IIA2.1 Evaluate the elements of the matrix eiGin closed form for a diagonal generator G. Generalize this result to arbitrary G. (Hint: Use rotational invariance.) The hermiticity condition on Vcan also be expressed as a reality condition: V=Vyand tr V =0)V*=CVC; (VC)* =C(VC)C where \ * " is the usual complex conjugate. A similar condition for Uis Uy=U1and det U =1)U*=CUC which is also a consequence of the fact that we can write Uin terms of a vector as U=eiV. As a result, the transformation law for the vector can be written in terms ofVCin a simple way, which manifestly preserves its symmetry: (VC)0=UVU1C=U(VC)UT 66 II. SPIN Excercise IIA2.2 Write an arbitrary rotation in two dimensions in terms of the slope (dy=dx )o f the rotation (the slope to which the x-axis is rotated) rather than the angle. (This is actually more convenient to measure if you happen to have a ruler, which you need to measure lengths anyway, but not a protractor.) This avoids trigonometry, but introduces ugly square roots. Show that the square roots can be eliminated by using the slope of half the angle of transformation as the variable. Show the relation to the variables used in writing 3D rotations in terms of 22m a t r i c e s .( H i n t :C o n s i d e r UandVCdiagonal.) 3. Spinors Note that the mapping of SU(2) to SO(3) is two-to-one: This follows from the factV0=VwhenUis a phase factor. We eliminated continuous phase factors from Uby the condition det U = 1, which restricts U(2) to SU(2). However, det(Iei)=e2i=1)ei=1 for 22 matrices. More generally, for any SU(2) element U,Uis also an element of SU(2), but acts the same way on a vector; i.e., these two SU(2) transformations give the same SO(3) transformation. Thus SU(2) is called a \double covering" of SO(3). However, this second transformation is not redundant, because it acts dierently on half-integral spins, which we discuss in the following subsections. The other convenience of using 2 2 matrices is that it makes obvious how to introduce spinors | Since a vector already transforms with two factors of U,w e dene a \square root" of a vector that transforms with just one U: 0=U ) y0= yU1 where is a two-component \vector", i.e., a 2 1 matrix. The complex conjugate of a spinor then transforms in essentially the same way: (C *)0=CU* *=U(C *) Note that the antisymmetry of Cimplies that must be complex: We might think that, sinceC * transforms in the same way as , we can identify the two consistently with the transformation law. But then we would have =C *=C(C *)* =CC* = A. TWO COMPONENTS 67 Thus the representation is pseudoreal. The fact that C * transforms the same way under rotations as leads us to consider the transformation 0=C * Since a vector transforms the same way under rotations as y, under this transfor- mation we have V0=CV*C=V which identies it as a re ection. Another useful way to write rotations on (like looking at VCinstead ofV)i s ( TC)0=( TC)U1 This tells us how to take an invariant inner product of spinors: 0=U ; 0=U) ( TC)0=( TC) In other words, Cis the \metric" in the space of spinors. An important dierence of this inner product from the familiar one for three-vectors is that it is antisymmetric. Thus, if andare anticommuting spinors, TC=TCT =TC where one minus sign comes from anticommutativity and another is from the anti- symmetry of C. Thus, it makes sense to take the norm of an anticommuting spinor as TC , which would vanish if were commuting. Of course, since rotations are unitary, we also have the usual y as an invariant, positive denite, inner product. 4. Indices The best way to discuss general spins is to use index notation, rather than matrix notation. Then a spinor rotates as 0 =U  with two-valued indices =; . The inner product is dened by = C=  where we have dened raising and lowering of indices by = C; =C  68 II. SPIN C=C=C=C=0 ii 0
paying careful attention to signs. (In general, we x signs by using a convention of contracting indices from upper-left to lower-right.) Then objects with many indicestransform as the product of spinors: A 0 ::: =UU:::U A::: An innitesimal transformation is then a sum: iA::: =GA::: +GA::: +:::+G A::: This is also true for C, even though it is an invariant constant: C0 =U UC =Cdet U =C A more interesting case is the vector: The transformation law is V0 =U UV  whereVis the symmetric VCconsidered earlier (in contrast to the antisymmetric C). There is basically only one identity in index notation, namely 0=1 2C[C ]=CC +C C+C C The expression vanishes because it is antisymmetric in those indices, and thus the indices must all have dierent values, but there are three two-valued indices. Anotherway to write this identity is to use the denition of C as the inverse of C: C C = )CC = [ ]     This tells us that antisymmetrizing in any pair of indices automatically contracts (sums over) them: Contracting this identity with an arbitrary tensor A , A[]=CC A =CA That means that we need to consider only objects that are totally symmetric in their free indices. This gives all spins: Such a eld with 2s indices describes spin s; we have already seen spins 0, 1/2, and 1. We have dened the transformation law of all elds with lower indices by con- sidering the direct product of spinors. Transformations for upper indices follow from multiplication with C: They all follow from 0= (U1) A. TWO COMPONENTS 69 Since the vertical position of the index indicates the form of the transformation law, we dene  =( )* where the \ " indicates complex conjugation. Thus, a hermitian matrix is written as M=(M)* =M )M=M So, for a vector we have V=V=V Spin s is usually formulated in terms of a (2s+1)-component \vector". Then one needs to calculate Clebsch-Gordan-Wigner coecients to construct Hamiltonians relating dierent spins. For example, to couple two spin-1/2 objects to a spin-1 object, one might write something like ~V
y~. The matrix elements of the Pauli matrices ~are the CGW coecients for the spin-1 piece of1 2 1 2=10. This method gets progressively messier for higher spins. On the other hand, in spinor notation such a term would be simply V ; no special coecients are necessary, only contraction of indices. Similarly the decomposition of products of spins involves only the picking out of the various symmetric and antisymmetric pieces: For example, for1 2 1 2, =1 2( ()+ [])=1 2 ()C  =V+CS where () means to symmetrize in those indices, by adding all permutations with plus signs. We have thus explicitly separated out the spin-1 and spin-0 parts Vand Sof the product. The square roots of various integers that appear in the CGW coecients come from permutation factors that appear in the normalizations of the various elds/wave functions that appear in the products: For example, A A =jAj2+3jA j2+3jA j2+jA j2 In the spinor index method, the square roots never appear explicitly, only their squares appear in normalizations: For example, in calculating a probability for A B!C, we evaluate hA BjCihCjA Bi hAjAihBjBihCjCi whereA,B,a n dCeach have 2s indices for spin s, and hjimeans contracting all indices (with the usual complex conjugation). 70 II. SPIN 5. Lorentz Consider now the general 2 2 hermitian matrix (V) . =Vy=V+Vt* VtV =1p 2V0+V1V2iV3 V2+iV3V0V1 =Va(a) .  where we distinguish the right spinor index by a dot because it will be chosen to transform dierently from the left one. For comparison, raising both spinor indices with the matrix Cas for SU(2), and the vector indices with the Minkowski metric (in either the orthonormal or null basis, as appropriate | see subsection IA4), we ndanother hermitian matrix (V) . =V+Vt* VtV =1p 2V0+V1V2+iV3 V2iV3V0V1 =Va(a).  In the orthonormal basis, aare the Pauli matrices and the identity, up to normal- ization. They are also the Clebsch-Gordan-Wigner coecients for spinor spinor = vector. In the null basis, they are completely trivial: 1 for one element, 0 for the rest, the usual basis for matrices. In other words, they are simply an arbitrary way (ac- cording to choice of basis) to translate a 2 2 (hermitian) matrix into a 4-component vector. Examining the determinant of (either version of) V, we nd the correct Minkowski norms: 2detV =2V+V+2VtVt*=(V0)2+(V1)2+(V2)2+(V3)2=V2 Thus Lorentz transformations will be those that preserve the hermiticity of this matrix and leave its determinant invariant: V0=gVgy;d e t g =1 (detg could also have a phase, but that would cancel in the transformation.) Thus g is an element of SL(2,C). In terms of the representation of the Lie algebra, g=eG;t r G =0 Thus the group space is 6-dimensional ( Ghas three independent complex compo- nents), the same as SO(3,1) (where ggT=)(G)T=G). Excercise IIA5.1 SL(2,C) also can be seen (less conveniently) from vector notation: aConsider the generators J() ab=1 2(Jabi1 2abcdJcd) A. TWO COMPONENTS 71 of SO(3,1). Find their commutation relations, and in particular show [J(+);J()]=0 . E x p r e s s J() 0iin terms of J() ij. ShowJ() ijhave the same commutation relations as Jij. Finally, take a general innitesimal Lorentz transformation in terms of Jaband rewrite it in terms of J() ij, paying special attention to the reality properties of the coecients. This demonstrates thatthe algebra of SO(3,1) is the same as that of SU(2) SU(2), but Wick rotated to SL(2,C). bApply the same procedure to SO(4) and SO(2,2) to derive their covering groups. Excercise IIA5.2 Consider relativity in two dimensions (one space, one time): aShow that SO(1,1) is represented in lightcone coordinates by x 0+=x+;x0=1x for some (nonvanishing) real number , and therefore SO(1,1) = GL(1). Write this one Lorentz transformation, in analogy to excercise IIA1.2a on rotationsin two space dimensions, in terms of an analog of the angle (\rapidity") forthose transformations that can be obtained continuously from the identity.Do the relativistic analog of excercise IIA2.2. bStill using lightcone coordinates, nd the parity and time reversal transfor- mations. Note that writing  as an exponential, so it can be obtained con-tinuously from the identity, restricts it to be positive, yielding a subgroup ofGL(1). Explicitly, what are the transformations of O(1,1) missing from thissubgroup? Which of P, (C)T, and (C)PT are missing from these transforma-tions, and which are missing from GL(1) itself? In index notation, we write for this vector V 0 . =g g*. . V .  while for a (\Weyl") spinor we have 0 =g  The metric of the group SL(2,C) is the two-index antisymmetric symbol, which is also the metric for Sp(2,C): In our conventions, C=C=C=C.. =0 ii 0
72 II. SPIN We also have the identities det L=1 2CC L L=1 2(tr L)21 2tr(L2);(L1)=CC L (det L )1 A[]=CC A ;A [ ]=0 discussed earlier in this section. As there, we use the metric to raise, lower, and contract indices: = C; .= . C. . V
W=V. W .  These results for SO(3,1) = SL(2,C) generalize to SO(4) = SU(2) SU(2) and SO(2,2) = SL(2) SL(2). As described earlier, the reality conditions change, so now SO(4) : (V0)* =V 0C C00;S O (2;2) : (V0)* =V 0 consistent with the (pseudo)reality properties of spinors for SU(2) and SL(2), where we now use unprimed and primed indices for the two independent group factors(V!gVg 0). Excercise IIA5.3 Take the explicit 2 2 representation for a vector given above, change the factors ofito satisfy the new reality conditions for SO(4) and SO(2,2), and show the determinant gives the right signatures for the metrics. A common example of index manipulation is to use antisymmetry whenever pos- sible to give vector products. For example, from the fact that V. V . C. . is antisym- metric in we have that V. V . =1 2 V2 where the normalization follows from tracing both sides. Similarly, V. W . +W. V . = V
W It then follows that V. W . V . =( V
WW. V . )V . =V
WV. 1 2V2W.  Antisymmetry in vector indices also implies some antisymmetry in spinor indices. For example, the antisymmetric Maxwell eld strength Fab=Fba, after translating vector indices into spinor, can be separated into its parts symmetric and antisymmet-ric in undotted indices; antisymmetry in vector indices (now spinor index pairs) then implies the opposite symmetry in dotted indices: F . ;. =F . ;. =1 4(F ()[. . ]+F [](. . ))=C. . f+Cf. . ;f=1 2F. ;. A. TWO COMPONENTS 73 Thus, an antisymmetric tensor also can be written in terms of a (complex) 2 2 matrix. We also need to dene complex (hermitian) conjugates carefully because Cis imaginary, and uses indices consistent with transformation properties:  .=( )*)  .=( )*;(  )y= . . V. =(V.)*) x. =x.  where we have used the spacetime coordinates as an example of a real vector (her- mitian 22 matrix). In general, hermitian conjugation properties for any Lorentz representation are dened by the corresponding product of spinors: For example, ( ())y=(. . )= (.. )) f.. (f)* More generally, we nd (T(1:::j)(. 1:::. k))y(1)j(j1)=2+k(k1)=2T(1:::k)(.1:::.j) As we'll see later, most spinor algebra involves, besides spinors, just vectors and antisymmetric tensors, which carry only two spinor indices, so matrix algebra is often useful. When using bra-ket notation for 2-component spinors, it is often convenientto distinguish undotted and dotted spinors. Furthermore, since spinor indices can be raised and lowered, we can always choose the bras to carry upper indices and the kets lower, consistent with our index-contraction conventions, to avoid extra signsand factors of C. We therefore dene (see subsection IB1) h j= hj;j i=ji ;[ j= .[.j;j ]=j.] . V=jiV. [. j)V*=j.]V.hj;f =jifhj)f*=j.]f.. [. j As a result, we also have h i=h i= ;[ ]= ..;h iy=[ ] h jVj]= V. . ;h jfji= f VW*+WV*=(V
W)I where we have used the anticommutativity of the spinor elds. From now on, we use this notation for the matrix representing a vector V(V. ), rather than the one with which we started ( V. ). 74 II. SPIN Excercise IIA5.4 Consider the generators G=x. @. +jihj and their Hermitian conjugates, where @ . =@=@x. . Show their algebra closes. What group do they generate? Find a subset of these generators thatcan be identied with (a representation of) the Lorentz group. Since we have exhausted all possible linear transformations on spinors (except for scale, which relates to conformal transformations), the only way to represent discrete Lorentz transformations is as antilinear ones: 0 =p 2n.  . ( 0=p 2n *) From its index structure we see that nis a vector, representing the direction of the re ection. The product of two identical re ections is then, in matrix notation 00=2n(n *)* =n2 )n2=1 where we have required closure on an SL(2,C) transformation ( 1). Thusnis a unit vector, either spacelike or timelike. Applying the same transformation to a vector, whereV. transforms like . , we write in matrix notation V0=2nV*n=n2V2(n
V)n (The overall sign is ambiguous, and depends on whether it is a polar or axial vector.) This transformation thus describes parity (actually CP, because of the complex con-jugation). In particular, to describe purely CP without any additional rotation (i.e., exactly re ection of the 3 spatial axes), in our basis we must choose a unit vector in the time direction, p 2n. =. ) 0= .( 0.= ) )V0. =V. which corresponds to the usual in vector notation, since in our basis .  a=a . To describe time reversal, we need a transformation that does not preserve the com- plex conjugation properties of spinors: For example, CPT is 0= ; 0. = . )V0=V A. TWO COMPONENTS 75 (The overall sign on Vis unambiguous.) In principle, whenever we work on a problem with both spinors and vectors we could use a mixed vector-spinor notation, converting between the usual basis for vectors and the spinor-index basis with identities such as a . .  b=b a;a .  .  a= . .  However, in practice it's much simpler to use spinor indices exclusively, since then one needs no -matrix identities at all, but only the trivial identities for the matrix Cthat follow from its antisymmetry. For example, converting the vector index on thematrices themselves into spinor indices ( a!. ), they become trivial: (. ) . = . .  (This is the same as saying an orthonormal basis of vectors has the components (Va)b=a bwhen the components are dened with respect to the same basis.) Thus, the most general irreducible (nite-dimensional) representation of SL(2,C) (and thus SO(3,1)) has an arbitrary number of dotted and undotted indices, and istotally symmetric in each: A (1:::m)(. 1:::. n). Treating a vector index directly as a dotted-undotted pair of indices (e.g., a=., which is just a funny way of labeling a 4-valued index), we can translate into spinor notation the two constant tensors ofSO(3,1): Since the only constant tensor of SL(2,C) is the antisymmetric symbol, theyc a nb ee x p r e s s e di nt e r m so fi t :  .;. =CC.. ; .;. ; . ;. =i(CC C.. C. . CC C.. C. . ) When we work with just vectors, these can be expressed in matrix language: V
W=tr(VW*) abcdVaWbXcYd(V;W;X;Y )=it r(VW*XY*Y*XW*V) Excercise IIA5.5 Prove this expression for the tensor (in either index or matrix version) by (1) showing total antisymmetry, (2) explicitly evaluating a nonvanishing component. 76 II. SPIN 6. Dirac The Dirac spinor we encountered earlier is a 4-component reducible representation in D=4: in terms of two (\left" and \right") two-component spinors, = L  R. The Hermitian metric  that denes the (Lorentz-invariant) Dirac spinor inner prod- uct  = y =  L R+h:c:;   y=(  R . L) takes the simple form =0 C..  C0 =p 2 0 The Dirac matrices are given by V=
V=0V.  V.0 =0V V*0 where the indices have been chosen to insure that the matrices always take a Dirac spinor to the same type of spinor. Since fV=;W=g=V
W,t h e matrices satisfy f a; bg=ab The extra sign is the result of normalizing the 's to be pseudohermitian with respect to the metric:  y1=+ . This Dirac spinor can be made irreducible by imposing a reality condition that relates Land R: The resulting \Majorana spinor" is then =1p 2   . The product of all the 's is a pseudoscalar: 1=2p 2 4!abcd a b c d=1p 2 i 0 0i. .! (This is usually called \ 5" in the literature for D=4 ,o r\ D"f o rD6=4 . W eh a v e renamed it for consistency with dimensional reduction.) It can be used to project aDirac or Majorana spinor onto its two two-component spinors:  =1 2(Ip 2i 1)=I 00 0
;0 00 I
Various identities for these matrices can be derived directly from the anticommu- tation relations: For example, a a=2; aa= a=a=; aa=b= a=a
b; aa=b=c= a=c=b=a= A. TWO COMPONENTS 77 tr(I)=4;t r (a=b=)=2a
b; tr (a=b=c=d=)=a
bc
d+a
db
ca
cb
d The trace identities follow from the fact that the only way to get a nonvanishing trace out of a product of matrices is when there are terms proportional to the identity; sincef a; bg=ab, this only happens when the indices are pairwise identical. The above results then follow from examination of relevant special cases. (Traces of odd numbers of matrices vanish. An exception is 1, until it is rewritten in terms of its denition as the product of the other -matrices.) Although use of the anticommutation relations is convenient for generalization of such identities to arbitrary dimensions, 2-spinor bra-ket notation is easier for deriving 4D identities. Since a Dirac spinor is the direct sum of a Weyl spinor and its complexconjugate, we write =j i L+j.] R.;  =  Rhj+ . L[.j In this notation, there is no need to use a spinor metric , just as in Minkowski 4-vector bra-ket notation there is no need for an explicit matrix to represent theMinkowski metric: It is included implicitly in the denition of the inner product for the basis elements ( h ajbi=aborhji=C). Thus hermitian conjugation is automatically pseudohermitian conjugation, etc.:  is y, from the eect of hermitian conjugating the basis vectors along with the components of the spinors they multiply. (See subsections IB4-5.) We then have simply . =ji[. jj. ]hj; +=jihj; =j.ih.j where we have replaced the vector index a!. on a. Excercise IIA6.1 Use this representation for the matrices and projection operators  for all of the following: aDerive . a=1


a=2n+1 . =a=2n+1


a=1 . a=1


a=2n . =1 2It r(a=1


a=2n) 1tr( 1a=1


a=2n) bRederive the trace identities above. (Hint: For the last identity, use the identityC[C ]= 0 repeatedly.) cShow that tr[(+) . .  . . ]=i .;. ; . ;.  by comparison with the expression of the previous subsection for . 78 II. SPIN 7. Chirality/duality are often called \chiral projectors"; 2-component spinors (not paired into Dirac spinors) are often called \chiral spinors", and appear in \chiral theories"; the two 2-component spinors of a Dirac spinor are often labeled as having left and right \chirality"; etc. When these two halves decouple, a theory can have a \chiral sym- metry" 0 =ei  Since chirality is closely related to parity (chiral spinors can represent CP, but need to be doubled to allow C, and thus P), Dirac spinors are often used to describe theories where parity is preserved, or softly broken, or to analyze parity violation specically, using 1to identify it. A similar feature appears in electrodynamics. We rst translate the theory into spinor notation: The Maxwell eld strength Fabis expressed in terms of the vector po- tential (\gauge eld") Aa, with a \gauge invariance" in terms of a \gauge parameter" with spacetime dependence. The gauge transformation Aa=@abecomes A0 . =A . @ .  where@ . =@=@x. . It leaves invariant the eld strength Fab=@[aAb]: F . ;. =@. A . @ . A. =1 4(F ()[. . ]+F [](. . )) =C. . f+Cf. . ;f=1 2@(. A). Maxwell's equations are @. fJ. They include both the eld equations (the hermitian part) and the \Bianchi identi- ties" (the antihermitian part). Excercise IIA7.1 We already saw VW*+WV* gave the dot product; show how VW*WV* is related to the \cross product" V[aWb]. Excercise IIA7.2 Write Maxwell's equations, and the expression for the eld strength in termsof the gauge vector, in 2 2 matrix notation, without using C's. Combine them to derive the wave equation for A. Maxwell's equations now can be easily generalized to include magnetic charge by allowing the current Jto be complex. (However, the expression for Fin terms of Ais A. TWO COMPONENTS 79 no longer valid.) This is because the \duality transformation" that switches electric and magnetic elds is much simpler in spinor notation: Using the expression given above for the 4D Levi-Civita tensor using spinor indices, F0 ab=1 2abcdFcd)f0 =if More generally, Maxwell's equations in free space (but not the expression for Fin terms ofA) are invariant under the continuous duality transformation f0 =eif (andJ0 . =eiJ . in the presence of both electric and magnetic charges). Excercise IIA7.3 Prove the relation between duality in vector and spinor notation. Show that Fab+i1 2abcdFcdcontains only fand notf.. . Excercise IIA7.4 How does complexifying J . modify Maxwell's equations in vector notation? In even time dimensions, Wick rotation kills the i(ori) in the spinor-index expression for 0; 0;0;0. Since the (discrete and continuous) duality transformation now contains no i, we can impose self-duality or anti-self-duality; i.e., that forf00 vanishes, since they are now independent and real instead of complex conjugates. These continuous chirality and duality symmetries on the eld strengths generalize to the free eld equations for arbitrary massless elds in four dimensions. For reasons to be explained in the following section, they distinguish the two polarizations of the waves described by such elds. They are closely related to conformal invariance: In higher dimensions, where not all free, massless theories are conformal (even on the mass shell), these symmetries exist exactly for those that are conformal. REFERENCES 1L.D. Landau and E.M. Lifshitz, Quantum mechanics, non-relativstic theory ,2 n de d . (Pergamon, 1965) ch. VIII:review of SU(2) spinor notation. 2B. van der Waerden, Gottinger Nachrichten (1929) 100: SL(2,C) spinor notation. 3E. Majorana, Nuo. Cim. 14(1937) 171. 4A. Salam, Nuo. Cim. 5(1957) 299; L. Landau, JETP 32(1957) 407, Nucl. Phys. 3(1957) 27; T.D. Lee and C.N. Yang, Phys. Rev. 105(1957) 1671: identication of neutrino with Weyl spinor. 80 II. SPIN :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: B. POINCAR E:::::::::::::::::::::::::::: The general procedure for nding arbitrary representations of the Poincar eg r o u p relevant to physics is to: (1) Describe spin 0. As we have seen, this means startingwith the coordinate representation, which is reducible, and apply the constraint p 2+ m2= 0 to get an irreducible one. (2) Find arbitrary, nite-dimensional, irreducible representations of the Lorentz group. This we have done in the previous section.(3) Take the direct product of these two representations of the Poincar e group, which give the orbital and spin parts of the generators. (The spin part of translationsvanishes.) We then need a further constraint to pick out an irreducible piece of thisproduct, which is the subject of this section. 1. Field equations We rst show how the derivation above of the massless particle from the confor- mal particle for spin 0 can be generalized to all \spins", i.e., all representations ofthe Poincar e group in arbitrary dimensions. There is a way to do this in terms of classical mechanics for all representations of the conformal group, by generalizing the description of the classical spinning particle. However, by analyzing the conformalparticle quantum mechanically instead, applying a set of constraints, it will be clearhow to generalize from conformal particles to general massless particles by weakeningthe constraints. The general idea is that the symmetry group for massive particles isthe Poincar e group, while that for massless particles includes also scale transforma- tions, and nally conformal particles have also conformal boosts. So, starting withthe conformal group and dropping anything to do with conformal boosts will givemassless particles. Massive particles then follow from dimensional reduction: addinga further spatial dimension and xing its component of momentum to a constant,the mass, so p 2!p2+m2. The equation p2+m2= 0, as an operator equation acting on a eld or wave function is the \Klein-Gordon (or relativistic Schr odinger) equation". States or elds that satisfy this (and the other) eld equation are called\on-(mass-)shell", while those that don't (or for which the equations haven't beenimposed) are \o-shell". We begin with a general representation of the conformal group SO(D,2) in terms of generators G AB,w h e r eA;B are D+2-component vector indices. We then impose constraints that are the conformally covariant form of p2= 0: Identifying (G+a;Gab;G+;Ga)=(Pa;Jab;
;Ka) B. POINCAR E8 1 (whereA=(;a)) as the generators for translations, Lorentz transformations, di- latations, and conformal boosts, we see that GAB=1 2GC(AGCB)1 D+2ABGCDGCD=0 is an irreducible piece of the product GG(symmetric and traceless) and includes: (G++;G+a;Gab;G+;Ga;G)=(P2;1 2fJab;Pbg+1 2f
;Pag;:::) where \:::" all have terms containing Ka. Excercise IIB1.1 Work out all theG's in terms of P,J,
,a n dK. In general theories, even massless ones, it is not always possible to have invariance under conformal boosts. (We'll see examples of this insubsection IXA7.) However, all massless theories are scale invariant, at least at the free level. (In D=4, free masslesstheories can always be made conformal on shell. However, the fact that even these theories can have actions that are not invariant under conformal boosts proves that it is sucient to add just dilatations to the Poincar e group. Furthermore, the fact that conformal boosts are not always an invariance in D >4 means that dropping them will give results in a dimension-independent form.) Therefore, only G ++andG+acan be dened in general massless theories, but we'll see that these are sucient to dene the kinematics. The former is just the masslessness condition, which we used to pick the constraints in the rst place. As we saw earlier,
just scales xa: We can therefore write the relevant generators as Pa=@a;Jab=x[a@b]+Sab;
=1 2fxa;@ag+w1=xa@a+w+D2 2 (We have used the antihermitian form of the generators.) The \scale weight" w+D2 2 is the real \spin" part of
, just as Sabis the spin part of the angular momentum Jab. To preserve the algebra it must commute with everything, and thus we can set it equal to a constant on an irreducible representation. We'll see shortly that its value is actually determined by the spin Sab. It is the engineering dimension of the corresponding eld. It has been normalized for later convenience; the value ofwdepends on the representation of S ab, but is independent of D. The dilatation generator
is not exactly antihermitian because the integration measure dDxisn't invariant under scaling. This is another reason wis determined, by the free action. The form we have given preserves reality of elds. The commutation relations for the spin parts, and the total generators, are the same as those for the orbital parts; e.g., [Sab;Scd]=[c [aSb]d] 82 II. SPIN (A convenient mnemonic for evaluating this commutator in general is to use Sab! x[a@b]instead.) Excercise IIB1.2 Find an expression for Kain terms of x,@,S,a n dwthat preserves the commutation relations. Evaluate all the constraints G, and express the inde- pendent ones in terms of just @,S,a n dw(nox). Substituting the explicit representation of the generators into the constraint G+a, and using the former constraint P2= 0 (when acting on wave functions on the right), we nd that all xdependence drops out, leaving for G+athe condition Sab@b+w@a=0 (paying careful attention to quantum mechanical ordering). This equation is the general eld equation for all spins (acting on the eld strength), in addition to theKlein-Gordon equation (which is redundant except for spin 0). Excercise IIB1.3 Dene spin for the conformal group by starting in D+2 dimensions: In terms of the (D+2)-dimensional coordinates y Aand their derivatives @A, GAB=y[A@B]+SAB Besides the previous conditions y2=@2=fyA;@Ag=0 impose the constraints, in analogy to the D-dimensional eld equations, and taking into account the symmetry between yand@, SAByB+wyA=SAB@B+w@A=0 Show that the algebra of constraints closes, if we include the additional con- straint 1 2S(ACSB)C+w(w+D 2)AB=0 Solve all the constraints with explicit y's for everything with an upper \ " index, reducing the manifest symmetry to SO(D 1,1), in analogy to the way y2= 0 was solved to nd y. Write all the conformal generators in terms of xa,@a,Sab,a n dw. B. POINCAR E8 3 2. Examples We now examine the constraints Sab@b+w@a= 0 in more detail. We begin by looking at some simple (but useful) examples. The simplest case is spin 0: Sab=0)w=0 The next simplest case (for arbitrary dimension) is the Dirac spinor: Sab=1 2[ a; b])Sab@b+w@a= a b@b+(w1 2)@a ) a@a=0;w =1 2 where we have separated out the pieces of the constraint that are irreducible with respect to the Lorentz group (e.g., by multiplying on the left with a). This gives the (massless) \Dirac equation" @= = 0. The next case is the vector: In terms of the basisjVi=Vajai, the spin is Sab=j[aihb]j However, the vector yields just another description of the scalar: Excercise IIB2.1 Apply the eld equations for general eld strengths to the case of a vectoreld strength. aFind the independent eld equations (assuming the eld strength is not just a constant) @ [aFb]=0;@aFa=0;w =1 Note that solving the rst equation determines the vector in terms of a scalar, while the second then gives the Klein-Gordon equation for that scalar, and the third xes the weight of the scalar to be the same as that found by startingwith a scalar eld strength. bSolve the second equation rst to nd a gauge eld that is not a scalar. All other representations can be built up from the spinor and vector. As our nal example, we consider the case where the eld is a 2nd-rank antisymmetric tensor, (S abF)cd=[c [aFb]d]) (Sab@b+w@a)Fcd=1 2@[aFcd]a[c@bFd]b+(w1)@aFcd )@[aFbc]=@bFab=0;w =1 which are Maxwell's equations, again separating out irreducible pi eces (e.g., by tracing and antisymmetrizing). 84 II. SPIN Excercise IIB2.2 Verify the representation of Lorentz spin given above for Fabby nding the commutation relations implied by this representation. Excercise IIB2.3 Consider the eld equations in 4D spinor notation for a general eld strength, totally symmetric in its mundotted indices and ndotted indices, S@. m@. =S.. @ . n@ .=0;w =1 2(m+n) where the spin acts on spinors as (S ) =C ( ); (S..  ). =C. (. . )((S ). =(S..  ) =0 ) aShow this implies @. :::. :::=@ .  :::. :::=0 bTranslate the eld equations into vector notation (in terms of Sab), nding Sab@b+w@a= 0 and an axial vector equation. cShow that the two equations are equivalent by deriving the equations of part afromSab@b+w@a= 0 alone, and from the axial equation alone. In each case, choosing the wrong scale weight wwould imply the eld was con- stant. Note that we chose the eld strengthFabto describe electromagnetism: The arguments we used to derive eld equations were based on physical degrees of free-dom, and did not take gauge invariance into account. In chapter XII we use more powerful methods to nd the gauge covariant eld equations for the gauge elds, and their actions. 3. Solution Free eld equations can be solved easily in momentum space. Then the simplest way to do the algebra is in the \lightcone frame". This is a reference frame, obtained by a Lorentz transformation, where a massless momentum takes the simple form pa=a +p+ (using only rotations), or the even simpler form pa=a +(using also a Lorentz boost), where again is the sign of the energy. In that frame the general eld equationSab@b+w@a= 0 reduces to Si=0;w =S+ B. POINCAR E8 5 The constraint Si= 0 determines S+to take its maximum possible value within that irreducible representation, since Siis the raising operators for S+: For any eigenstate of S+, S+jhi=hjhi)S+(Sijhi)=(SiS++[S+;Si])jhi=(h+1 ) (Sijhi) The remaining constraint then determines w: It is the maximum value of S+for that representation. By parity (+ $ ),wis the minimum, so w0;w=0,Sab=0 since ifS+= 0 for all states then Sab= 0 by Lorentz transformation. As we have seen by other methods (but can easily be derived by this method), w=1 2for the Dirac spinor and w= 1 for the vector; since general representations can be built from reducing direct products of these, we see that wis an integer for bosons and half- integer for fermions. If we describe a general irreducible representation by a Young tableau for SO(D 1,1) (with tracelessness imposed), or a Young tableau times a spinor (with also -tracelessness a a:::b= 0), then it is easy to see from the results for the spinor and vector, and antisymmetry in rows, that wis simply the number of columns of the tableau (its \width"), counting a spinor index as half a column: S+just counts the maximum number of \ " indices that can be stuck in the boxes describing the basis elements. (In fact, Dirac spinor Dirac spinor gives just all possible 1-column representations.) This leaves undetermined only SijandS+i.H o w e v e r , S+i(\creation operator") is canonically conjugate to Si(\annihilation operator"), so its action has also been xed: [Si;S+j]=ijS++Sij (Sijvanishes for i=j,s oS+iandSiare conjugate, though not \orthonormal". The constantS+was xed above to be nonvanishing, except for the trivial case of spin 0.) Thus only the \little group" SO(D 2) spinSijremains nontrivial: The original irreducible representation of SO(D 1,1) Lorentz spin Sabwas a reducible representa- tion of SO(D2) spinSij; the irreducible SO(D 2) representation with the highest value ofS+is picked out of this SO(D 1,1) representation. This solution also gives the eld strength in terms of the gauge eld: Working with just the highest- S+- weight states is equivalent to working with the gauge eld, up to factors of @+. 86 II. SPIN As an explicit example, for spin 1/2 we have simply = 0, which kills half the components, leaving the half given by + . For spin 1, we nd pbFab=0)Fa=0 p[aFbc]=0)only F+a6=0 )only F+i6=0 In the \lightcone gauge" A+=0 ,w eh a v e F+i=@+Ai, so the highest-weight part of Fabis the transverse part of the gauge eld. The general pattern, in terms of eld strengths, is then to keep only pieces with as many as possible upper + indices andno upperindices (and thus highest S +weight). In terms of the vector potential, we have Fabp[aAb])only Ai6=0 The general rule for the gauge eld is to drop indices, so the eld becomes an irreducible representation of SO(D 2). All + indices on the eld strength are picked up by the momenta, which also account for the scale weight of the eld strength: Allgauge elds have w= 0 for bosons and w= 1 2for fermions. Excercise IIB3.1 Using only the anticommutation relations f a; bg=ab, construct projec- tion operators from : These are operators  Ithat satisfy IJ=IJI noX ;X I=1 Because of time reversal symmetry +$ (or parity +$ ), these project onto two subspaces equal in size. A method equivalent to using the lightcone frame is to perform a unitary trans- formationUon the spin that is the inverse of the transformation on the coordi- nates/momentum that would take us to the lightcone frame: We want a Lorentztransformation  abon the eld equations, which are of the form Oabpb=0;Oab=Sab+wb a that has the eect UOabU1=acOcdb d; b apb=p0 a;p0a=a +p+) 0=UOabpbU1=acOcdb dpb=acOcdp0 d)Oabp0 b=0 Ifj isatises the original constraint, then Uj iwill satisfy the new one. If we like, we can always transform back at the end. This is equivalent to a gauge transformationin the eld theory. B. POINCAR E8 7 It is easy to check that the appropriate operator is U=eS+ipi=p+ Any operator Vathat transforms as a vector under Sab, [Sab;Vc]=V[ab]c but commutes with p, is transformed by UintoUVU1=V0as V0+=V+;V0i=Vi+V+pi p+;V0=V+Vipi p++V+(pi)2 2(p+)2 as follows from explicit Taylor expansion, which terminates because S+iact as low- ering operators (as for conformal boosts in subsection IA6). This yields the desiredresult V 0apa=Vap0 a+V+ 2p+p2 when we impose the eld equation p2=0 . Excercise IIB3.2 Check this result by performing the transformation explicitly on the con- straint. Before the transformation, the lightcone decomposition of the con- straint is (S++w)p++S+ipi=0 Sip++Sijpj+wpiSi+p=0 Sipi+(S++w)p=0 Show that after this transformation, the constraint becomes (S++w)p+=0 Sip++(S++w)pi1 2S+ip2 p+=0 Sipi+(S++w)pS+p2 p+1 2S+ipip2 p+2=0 Clearly these imply w=S+;Si=0 withp2=0 . On the other hand, if instead of using the lightcone identication of x+as \time", we choose to use the usual x0for purposes of nding the evolution of the system, then we want to consider transformations that do not involve p0, instead of not involving 88 II. SPIN the \energy" p. Thus, by p0-independent rotations alone, the best we can do is to choose pi=0;p1=! i.e., we can x the value of the spatial momentum, but not in a way that relates to the sign of the energy. The result is then p0>0:pa=a +p+ p0<0:pa=a p The result is similar to before, but now the positive and negative energy solutions are separated: In this frame the eld equations reduce to p0>0:Si=0;S+=w p0<0:S+i=0;S+=w Thus, while wtakes the same value as before, now the positive-energy states are associated with the highest weight of S+, while the negative-energy ones go with the lowest weight (and nothing between). The unitary transformation that achieves this result is a spin rotation that rotates Sabin the eld equations with the same eect as an orbital transformation that would rotate ( p1;pi)!(!;0). By looking at the special case D= 3 (where there is only one rotation generator), we easily nd the explicit transformation U=exp tan1jpij p1 S1ipi jpij Excercise IIB3.3 Perform this transformation: aFind the action of the above transformation on an arbitrary vector Va.( H i n t : Look atD= 3 to get the transformation on the \longitudinal" part of the vector.) In particular, show that V0apa=Vap0 a;p0a=a 0p0+a 1! bShow the eld equations are transformed as S0apa+wp0)!S10+wp0=p0(wp0 !S10)1 !S10p2 S1apa+wp1)1 !pi(!S1ip0S0i)+p1(wp0 !S10) Siapa+wpi) [ij1 !(!+p1)pipj](!S1jp0S0j)+pi(wp0 !S10) B. POINCAR E8 9 Note that the rst equation gives the time-dependent Schr odinger equation, with Hamiltonian H=1 w(S10p1S0ipi)!1 wS10! This diagonalizes the Hamiltonian H(in a representation where S10is diag- onal). Thus the only independent equations are p2=0;S10=(p0)w; S1i(p0)S0i=0 leading to the advertised result. cFind the transformation that rotates to the pidirection instead of the 1 di- rection, so H!1 wS0ipi jpjj! 4. Mass So far we have considered only massless theories. We now introduce masses by \dimensional reduction", identifying mass with the component of momentum inan extra dimension. As with the extra dimensions used for describing conformal symmetry, this extra dimension is just a mathematical construct used to give a simple derivation. (Theories have been postulated with extra, unseen dimensions that arehidden by \compactication": Space curls up in those directions to a size too small to detect with present experiments. However, no compelling reason has been given for why the extra dimensions should want to compactify.) The method is to: (1) extend the range of vector indices by one additional spatial direction, which we call \ 1"; (2) set the corresponding component of momentum to equal the mass, p 1=m and (3) introduce extra factors of ito restore reality, since @1=ip1=im,b y a unitary transformation. Since all representations can be constructed by directproducts of the vector and spinor, it's sucient to dene this last step on them. For the scalar this method is trivial, since then simply p 2!p2+m2. For the spinor, since any transformation on the spinor index can be written in terms of the gamma matrices, and the transformation must aect only the 1 direc- tion, we can use only 1. (For even dimensions, we can identify the 1of dimen- sional reduction with the one coming from the product of all the other 's, since in odd dimensions the product of all the 's is proportional to the identity.) We nd U=exp( 1=2p 2) : 1! 1; a!p 2 1 a 90 II. SPIN We perform this transformation directly on the spin operators appearing in the con- straints, or the inverse transformation on the states. Dimensional reduction, followed by this transformation, then modies the massless equation of motion as i@=!i@=m 1!p 2 1(i@=+mp 2) soi@= =0!(i@=+mp 2) = 0. The prescription for the vector is U=exp(1 2ij1ih1j):j1i!ij1i;h1j!ih1j (h1j1i=1 ) with the other basis states unchanged. This has the eect of giving each eld a i for each (1)-index. For example, for Maxwell's equations @[aFbc]!@[aFbc] @[aFb]1+imFab!@[aFbc] (redundant) i(@[aFb]1mFab) @bFab!@bFab+imFa1 @aF1a!@bFab+mFa1 i@aF1a (redundant) Note that only the mass-independent equations are redundant. Also, Fa1appears explicitly as the potential for Fab, but without gauge invariance. Alternatively, we can keep the gauge potential: Fab=@[aAb]!Fab=@[aAb] Fa1=@aA1imAa!Fab=@[aAb] iFa1=i(@aA1+mAa) This is known as the \St uckelberg formalism" for a massive vector, which maintains gauge invariance by having a scalar A1in addition to the vector: The gauge trans- formations are now Aa=@a!Aa=@a A1=im!Aa=@a iA1=im Excercise IIB4.1 Consider the general massive eld equations that follow from the general massless ones by dimensional reduction. One of these is S1a@a+wim =0 (before restoring reality). This scalar equation alone gives the complete eld equations for w=1/2 and 1 (antisymmetric tensors), 0 being trivial. aShow that for w=1/2 it gives the (massive) Dirac equation. bExpanding the state over explicit elds, nd the covariant eld equations it implies for w=1. Show these are sucient to describe spins 0 (vector eld B. POINCAR E9 1 strength: see excercise IIB2.1) and 1 ( FabandFa1). Note that S1aact as generalized matrices (the Dirac matrices for spin 1/2, the \Dun-Kemmer matrices" for w=1), where Sab=[S1a;S1b] cShow that these covariant eld equations imply the Klein-Gordon equation for arbitrary antisymmetric tensors. Show that in D=4 all antisymmetric tensors (coming from 0-5 indices in D=5) are equivalent to either spin 0 orspin 1, or trivial. (Hint: Use  abcd.) dConsider the reducible representation coming from the direct product of two Dirac spinors, and represent the wave function itself as a matrix: Sij = ~Sij + ~Sij wherei=(1;a)a n d ~Sijis the usual Dirac-spinor representation. Using the fact that any 44 (in D=4) matrix can be written as a linear combination of products of -matrices (antisymmetric products, since symmetrization yields anticommutators), nd the irreducible representations of SO(4,1) in , andrelate to part c. Excercise IIB4.2 Solve the eld equations for massive spins 1/2 and 1 in momentum space bygoing to the rest frame. The solution to the general massive eld equations can also be found by going to the rest frame: The combination of that and dimensional reduction is, in terms ofthe massive analog of lightcone components, p +=1p 2(p0+p1)=p 2m; p=1p 2(p0p1)=0;pi=0 wherepiare now the other D 1 (spatial) components. This xing of the momentum is the same as the lightcone frame except that p1has been replaced by p1,a n d thuspinow has D1 components instead of D 2. The solution to the constraints is thus also the same, except that we are left with an irreducible representation of the \little group" SO(D 1) as found in the rest frame for the massive particle, vs. one of SO(D2) found in the lightcone frame for the massless case. 92 II. SPIN 5. Foldy-Wouthuysen The other frame we used for the massless analysis, which involved only energy- independent rotations, can also be applied to the massive case by dimensional reduc- tion. The result is known as the \Foldy-Wouthuysen transformation", and is useful for analyzing interacting massive eld equations in the nonrelativistic limit. Replacing p1!p1=min our previous result, we have for the free case U=exp tan1j~pj m S1ipi j~pj ;U H U1=1 wS10! For purposes of generalization to interactions, it was important that in the free trans- formation (1) we used only the spin part of a rotation, since the orbital part could introduce explicit xdependence, and (2) we used only rotations, since a Lorentz boost would introduce p0dependence in the \parameters" of the transformation, which could generate additional p0(time derivative) terms in the eld equation. Excercise IIB5.1 Perform this transformation for the Dirac spinor, and then apply the reality-restoring transformation to obtain H!p 2 0! We then can use the diagonal representation 0=I 00 I
=p 2. (We can de- ne this representation, up to phases, by switching 0and 1of the usual representation.) In general the reality-restoring transformation will be unnec-essary for any spin, since applying the eld equation S 10=wpicks out a representation of the \little group" SO(D 1). In the interacting case the result generally can't be obtained in closed form, so it is derived perturbatively in 1 =m. The goal is again a Hamiltonian diagonal with respect toS10, to preserve the separation of positive and negative energies; we then can set S10=wto describe just positive energies. We thus choose the transformation to cancel any terms in Hthat are o-diagonal, which come from odd total numbers of \1" and \0" indices from the spin factors in any term: i.e., odd numbers of S0i andS1i(e.g., theS0ipiterm in the original H). For example, for coupling to an electromagnetic eld, the exponent of Uis generalized by covariantizing derivatives (minimal coupling @!r =@+iA), but also requires eld-strength ( EandB)t e r m s to cancel certain ones of those generated from commutators of these derivatives in the transformation: ra=@a+iAa) [ra;rb]=iFab B. POINCAR E9 3 Before performing this transformation explicitly for the rst few orders, we con- sider some general properties that will allow us to collect similar terms in advance.(Few duplicate terms would appear to the order we consider, but they breed like rabbits at higher orders.) We start with a eld equation Fthat can be separated into \even" termsEand \odd" onesO,e a c ho fw h i c hc a nb ee x p a n d e di np o w e r so f1 =m: F=E+O:E=1X n=1mnEn;O=1X n=0mnOn Note that the leading ( m+1) term is even; thus we choose only odd generators to transform away the odd terms in F, perturbatively from this leading term: F0=eGFeG;G =1X n=1mnGn SinceF0is even while Gis odd, we can separate this equation into its even and odd parts as F0=cosh(LG)E+sinh(LG)O 0=sinh(LG)E+cosh(LG)O (withLG=[G;] as in subsection IA3). Since we can perturbatively invert any Taylor-expandable function of LGthat begins with 1, we can use the second equation to give a recursion relation for Gn: Separating the leading term of F, E=mE1+
E;m[G;E1]=[G;
E]+LGcoth(LG)O which we can expand in 1 =m[after Taylor expanding LGcoth(LG)] to give an expres- sion for [Gn;E1]t os o l v ef o r Gn. We can also use the implicit solution for [ G;E] directly to simplify the expression for F0: F0=E+tanh(1 2LG)O For example, to order 1 =m2we have forF0 F0 1=E1;F0 0=E0;F0 1=E1+1 2[G1;O0] F0 2=E2+1 2[G2;O0]+1 2[G1;O1] To this order we therefore need to solve [G1;E1]=O0;[G2;E1]=O1+[G1;E0] 94 II. SPIN For our applications we will always have E1=1 wS10 unchanged by interactions. We have oversimplied things a bit in the above deriva- tion: For general spin we need to consider more than just even and odd terms; weneed to consider all eigenvalues of S 10: [S10;Fs]=sFs and nd the transformation that makes F0commute with it ( s= 0). The procedure is to rst divide into even and odd values of s, as above, then to divide the remaining even terms inF0into twice even values of s(multiples of 4) as the new E0and twice odd as the newO0, which are transformed away with the new twice odd G0,a n ds o on. This very rapidly removes the lower nonzero values of jsj(1!2!4!:::), which has a maximum value of 2 w(from the operators that mix the maximum value S10=wwith the minimum S10=w). For example, for the case of most interest, the Dirac spinor, the only eigenvalues (for operators) are 0 and 1, so the original even part does commute with S10, and the procedure need be applied only once. Furthermore, terms in Fof eigenvalue scan be generated only at order m1sor higher; so at any given order the procedure rapidly removes all undesired terms forany spin. Since the terms we want to cancel are exactly the ones with nonvanishing eigen- values ofS 10, they can always be written as [ G;S10]f o rs o m eG, so we can always nd a transformation to eliminate them: [S10;Gsn]=sGsn)Gsn=w sf[G;
E]+LGcoth(LG)Ogsn (This is just diagonalization of a Hermitian matrix in operator language.) In partic- ular for the Dirac spinor, since E1has only1 eigenvalues, it's easy to see that not only do all even operators commute with it, but all odd operators anticommute with it. (Consider the diagonal representation of E1:f1 00 1
;0 ab 0
g=0 . ) W et h e nh a v e simply w=1 2) (E1)2=1) [E1;
E]=fE1;Og=fE1;Gg=0 )mG=1 2f[G;
E]+LGcoth(LG)OgE1 As a nal step, we can apply the usual transformation U0=eimtS10=w B. POINCAR E9 5 which commutes with all but the p0term inE0to have the sole eect of canceling E1, eliminating the rest-mass term from the nonrelativistic-style expression for the energy. For the minimal electromagnetic coupling described above, we have besides E1 E0=0;O0=1 wS0ii w h e r ew eh a v ew r i t t e n a=pa+Aa(instead of a=ira,t os a v es o m e i's). There are no additional terms in Ffor minimal coupling for spin 1/2, but later we'll need to include nonminimal eective couplings coming from quantum (eld theoretic) eects. There are also extra terms for spins 0 and 1 because the eld strength is not the sameas the fundamental eld, so we'll treat only spin 1/2 here, but we'll continue to use the general notation to illustrate the procedure. Using the above results, we nd to order 1=m 2forF0 G1=S1ii;G 2=wS0iiF0i in agreement with with the free case up to eld strength terms. The diagonalized Schrodinger equation is then to this order, including the eect of U0, F0 1=0;F0 0=0;F0 1=1 2w[1 2fS1i;S0jgiFij+S10(i)2] F0 2=1 4[fS0i;S0jg(@iF0j)SijfiF0i;jg] For spin 1/2 we are done, but for other spins we would need a further transformation (beforeU0)t op i c ko u tt h ep a r to f F0 2that commutes with S10(by eliminating the twice odd part); the nal result is F0 2=1 4[1 2(fS1i;S1jgfS0i;S0jg)(@iF0j)+SijfiF0i;jg] It can also be convenient to translate into notation (as for the massless case, but with index 1! 1): We then write F0 1=0;F0 0=0;F0 1=1 2w[1 2fS+i;SjgiFij+S+(i)2] F0 2=1 4[1 2fS+(i;Sj)g(@iF0j)SijfiF0i;jg] In this notation the eigenvalue of S+=S10for any combination of spin operators can be simply read o as the number of indices minus the number of +. Excercise IIB5.2 Find the Hamiltonian for spin 1/2 in background electromagnetism, expanded nonrelativistically to this order, by substituting the appropriate expressions for the spin operators in terms of matrices, and applying S+=won 96 II. SPIN the right for positive/negative energy. (Ignore the reality-restoring transfor- mation.) -matrix algebra can be performed directly with the spin operators: For the Dirac spinor we have the identities S(a(bSc)d)=1 2b (ad c)acbd)fS+i;Sjg=1 2ij2SijS+ 6. Twistors Besides describing spin 1/2, spinors provide a convenient way to solve the condi- tionp2= 0 covariantly: Any hermitian matrix with vanishing determinant must have a zero eigenvalue (consider the diagonalized matrix), and so such a 2 2 matrix can be simply expressed in terms of its other eigenvector. Absorbing all but the sign of the nontrivial eigenvalue into the normalization of the eigenvector, we have p2=0)p. =pp.  for some spinor p.S i n c ep0is the (canonical) energy, the is the sign of the energy. This explains why time reversal (actually CT in the usual terminology) is not a lineartransformation. Note that p is a commuting object, while most spinors are fermionic, and thus anticommuting (at least in quantum theory). Such commuting spinors are called \twistors". Excercise IIB6.1 Show that, in terms of its energy Eand the angular direction ( ;)o fi t s 3-momentum, a massless particle is described by the twistor p=21=4p jEj(cos 2ei=2;sin 2ei=2) One useful way to think of twistors is in terms of the lightcone frame. In spinor notation, the momentum is p. =1 00 0
If we write an arbitrary massless momentum as a Lorentz transformation from this lightcone frame, then the twistor is just the part of the SL(2,C) matrix that con- tributes: p0. =p . g g. . = . . g g. . =gg. . =pp.  For this reason, the twistor formalism can be understood as a Lorentz covariant form of the lightcone formalism. The twistor construction thus gives a covariant way of constructing wave functions satisfying the mass-shell condition (Klein-Gordon equation) for the massless case, B. POINCAR E9 7 =0 ,w h e r e =@2=p2. We simply Fourier transform, and use the twistor expression for the momentum, writing the momentum-space wave function in terms of twistor variables (\Penrose transform"): (x)=Z d2pd2p.[exp(ix. pp. )+(p;p.)+exp(ix. pp. )(p;p.)] wheredescribe the positive- and negative-energy states, respectively. (The integral over p.can be performed also, eectively taking the Fourier transform with respect to that variable only, treating x. pas the conjugate.) We can extend the matrix notation of subsection IIA5-6 to twistors: hpj=p;jpi=p;[pj=p.;jp]=p. P=jpi[pj;P*=jp]hpj As a result, we also have for twistors hpqi=hqpi;[pq]=[qp];hpqihrsi+hqrihpsi+hrpihqsi=0 ;hpqi*=[qp] These properties do not apply to physical, anticommuting spinors, where h i= +h i,a n dh i6=0 . Another natural way to understand twistors is through the conformal group. We have already seen that the conformal group in D dimensions is SO(D,2). Since thisgroup in four dimensions is the same as SU(2,2), it's simpler to describe its general representations (and in particular spinors) in SU(2,2) spinor notation. Then the simplest way to generate representations of this group is to use spinor coordinates: We therefore write the generators as (see subsection IC1) G AB=BA1 4B ACC where we have subtracted out the trace piece to reduce U(2,2) to SU(2,2) and, consis- tently with the group transformation properties under complex conjugation, we havechosen the complex conjugate of the spinor to also be the canonical conjugate: The Poisson bracket is dened by [ A;B]=B A To compare with four-dimensional notation, we reduce this four-component spinor by recognizing it as a particular use of the Dirac spinor. Using the same representation as in subsection IIA6, we write A=(p;!.); A=(!;p.); . AB=0 C..  C0 98 II. SPIN Now the Poisson brackets are [!;p]= ;[!.;p. ]=. . The group generators themselves reduce to pp. ;!!. ;p (!);p(.!. );p!+p.!.2 which are translations, conformal boosts, SL(2,C) generators and their complex con- jugates, and dilatations. Another kind of twistor, related to position space instead of momentum space, follows from this (D+2)-coordinate description of conformal symmetry for D=4 (see subsection IA6). In practice, it's more convenient to work with invariances than con- straints. In this case, we can solve the lightcone constraint on Wick-rotated D=3+3 or 5+1 space, replacing 6-component conformal vector indices with 4-component con- formal spinor indices, with a position-space twistor: y2=1 4ABCDyAByCD=0)yAB=zAzB  whereAis an SL(4) (or SU*(4)) index and is an SL(2) (or SU(2)) index, and zA is real (with either two real or two pseudoreal indices). (Here the SL groups apply to 3+3 dimensions, the SU groups to 5+1.) Whereas yhad 61=5c o m p o n e n t s due to the constraint, zhas 4
23 = 5 components due to the SL(2) (SU(2)) gauge invariance of the above relation to y. These coordinates reduce to the usual by an SL(2) transformation: A=(;.);zA =( ;x.)) SL(2) gauge =  wheree=2. Excercise IIB6.2 Substitute this spinor-notation z(;x)i n t oyz2and compare with the vector-notation y(e;x) of subsection IA6. 7. Helicity A sometimes-useful way to treat the transverse spin operators Sijis in terms of Wabc=1 2P[aJbc]=1 2P[aSbc] which reduces to Sijin the rest frame, and (like the eld equations) can be written in terms of just the Poincar e generators. This is the part of Sabwhose commutator B. POINCAR E9 9 with the eld equations is proportional to the eld equations (i.e., it preserves the constraints). In D=4 this is the \Pauli-Luba nski (axial) vector" Wa=1 6Wbcdbcda We can choose our states to be eigenstates of a component of it: For example, for massless states W0=P0is called the \helicity". For massive states the helicity is dened asW0=j~Pj, but is less useful, especially since it is undened (0/0) in the rest frame. In that case one instead chooses a component in terms of a (momentum-dependent axial) vector s aassaWa,w h e r esaPa=0a n ds2=1=m2. Excercise IIB7.1 Show in both the massless and massive cases that Wabcreduces to the little group generators on shell by going to the appropriate reference frame. The twistor representation of the conformal group does not give the most general representation, but it does give all the (free) massless ones. The reason it givesmassless ones is that this representation satisies the constraint (see subsection IIB1) G [AB][CD]=G[A[CGB]D]traces =0 which includes p2= 0 as well as all the equations that follow from p2=0b yc o n f o r m a l transformations. As a consequence, this representation also satises GACGCBtrace =hGAB wherehis the helicity. This equation may be more recognizable in SO(4,2) notation, as 1 8ABCDEFGCDGEF=ihGAB This equation includes, as its lowest mass-dimension part (as dened by dilatations), the Pauli-Luba nski vector Wa=1 2bcdaPbJcd=ihPa (The \i" appears in the last two equations only when we use the antihermitian form of the generators GABandJab.) Although any massless representation of the conformal group satises the above conditions (see excercise IIB2.3), the twistor representationsatises the unusual property that helicity is realized as a linear transformation on thecoordinates: For the twistors the implicit denition of helicity can be solved explicitlyto give h= 1 4fA;Ag=1 2AA+1=1 2(p!p.!.) 100 II. SPIN which is exactly the U(1) transformation of U(2,2)=SU(2,2) U(1). (This is similar to SU(2) in terms of \twistors": See excercise IC1.1.) On functions of pand p.,i t eectively just counts half the number of p's minus p.'s. Excercise IIB7.2 These results are pretty clear from symmetry, but we should do some algebra to check coecients: aUse the denition of the action of the Lorentz generators on a vector operator in vector and spinor notations, [Jab;Vc]=V[ab]c;[J;V . ]=V (. C) ; [J.. ;V . ]=V (.C. ).  to derive J .. =1 2(CJ.. +C.. J) bExpressJabandPain terms of the twistors p;p.;!;!.(with normalization ofJabxed by its action on the twistors themselves), and plug into PJ =ihP to derive the above expression of hin terms of twistors. The simple form of the helicity in the twistor formalism is another consequence of it being a covariantized lightcone formalism. In the lightcone frame, there is still a residual Lorentz invariance; in particular, a rotation about the spatial direction inwhich the momentum points leaves the momentum invariant. This is another deni- tion of the helicity, as the part of the angular momentum performing that rotation. (Only spin contributes, since by denition the momentum is not rotated.) Since theproduct of two Lorentz transformations is another one, this rotation can be inter- preted as a transformation acting on the Lorentz transformation to the lightcone frame, i.e., on the twistor, such that the momentum is invariant. This is simply a phase transformation: g 0 =ei0 0ei g )p0=eip We can generalize the Penrose transform in a simple way to wave functions car- rying indices to describe spin: 1:::m. 1:::. n(x)=Z d2pd2p.p1


pmp. 1


p. n [exp(ix. pp. )+(p;p.)+exp(ix. pp. )(p;p.)] For the integral to give a nonvanishing result, the integrand must be invariant under the U(1) transformation generated by the helicity operator h: In other words, must B. POINCAR E 101 have a transformation under h, i.e., a certain helicity, that is exactly the opposite that of the explicit pfactors that carry the external indices to give a contribution to the integral, since otherwise integrating over the phase of pwould average it to zero. (Explicitly, if we derive the helicity by acting on the Penrose transform, this minus sign comes from integration by parts.) This means that (x) automatically has a certain helicity, half the number of dotted minus undotted indices: h=1 2(nm)[w=1 2(m+n)] as given by the above twistor operator expression acting on . (Alternatively, com- paring the x-space form of the Pauli-Luba nsky vector, its action plus that of the twistor-space one must vanish on jip, so the helicity is again minus the twistor- space helicity operator acting on the prefactor.) Since, after restricting to the appropriate helicity, the integral over this phase is trivial, we can also eliminate it by replacing the \volume" integral over the twistor or its complex conjugate (but not both) with a \surface" (boundary) integral: Z d2p!I pdp (Alternatively, we can insert a -function in the helicity.) The result is equivalent to the usual integral over the three independent components of the momentum. This generalization of the Penrose transform implies that (x) satises some equations of motion besides p2=0 ,n a m e l y p. :::. :::. =p . :::. :::. =0 which are also implied by Sab@b+w@a= 0 (see excercise IIB2.3). Besides Poincar e invariance, these equations are invariant under the phase transformation 0 :::. :::. =ei2h :::. :::.  that generalizes duality and chiral transformations. We also see that (anti-)self- duality and chirality are related to helicity. Another way to understand the twistor result is to remember its interpretation as a Lorentz transformation from the light cone: In the light cone frame, where p+. +is the only nonvanishing component of p., the above equations of motion imply the only nonvanishing component of 1:::m. 1:::. n is +:::+. +:::. +, which can be identied with +(forp+. +>0) or(forp+. +<0). REFERENCES 1E. Wigner, Ann. Math. 40(1939) 149; V. Bargmann and E.P. Wigner, Proc. Nat. Acad. Sci. US 34(1946) 211: 102 II. SPIN little group; more general discussion of Poincar e representations and relativistic wave equations for D=4. 2A.J. Bracken, Lett. Nuo. Cim. 2(1971) 574; A.J. Bracken and B. Jessup, J. Math. Phys. 23(1982) 1925; W. Siegel, Nucl. Phys. B263 (1986) 93: conformal constraints. 3E.C.G. St uckelberg, Helv. Phys. Acta 11(1938) 299. 4P.A.M. Dirac, Rev. Mod. Phys. 21(1949) 392: lightcone gauge. 5G. Nordstr om, Phys. Z. 15(1914) 504: dimensional reduction. 6O. Klein, Z. Phys. 37(1926) 895; V. Fock, Z. Phys. 39(1927) 226: mass from dimensional reduction. 7R.J. Dun, Phys. Rev. 54(1938) 1114; N. Kemmer, P r o c .R o y .S o c . A173 (1939) 91. 8M.H.L. Pryce, P r o c .R o y .S o c . A195 (1948) 62; S. Tani, Soryushiron Kenkyu 1(1949) 15 (in Japanese): free version of Foldy-Wouthuysen. 9L.L. Foldy and S.A. Wouthuysen, Phys. Rev. 78(1950) 29. 10M. Cini and B. Touschek, Nuo. Cim. 7(1958) 422; S.K. Bose, A. Gamba, and E.C.G. Sudarshan, Phys. Rev. 113(1959) 1661: \ultrarelativistic" version of Foldy-Wouthuysen transformation. 11R. Penrose, J. Math. Phys. 8(1967) 345, Int. J. Theor. Phys. 1(1968) 61; M.A.H. MacCallum and R. Penrose, Phys. Rep. 6(1973) 241: twistors. 12W. Pauli, unpublished;J.K. Luba nski, Physica IX(1942) 310, 325. C. SUPERSYMMETRY 103 :::::::::::::::::::: :::::::::::::::::::: :::::::::::::::::::: C. SUPERSYMMETRY :::::::::::::::::::: We'll see later that quantum eld theory requires particles with integer spin to be bosons, and those with half-integer spin to be fermions. This means that any sym- metry that relates bosonic wave functions/elds to fermionic ones must be generatedby operators with half-integer spin. The simplest (but also the most general, at least of those that preserve the vacuum) is spin 1/2. Supersymmetry is a major ingredi- ent in the most promising generalizations of the Standard Model. In this section welook at representations, generalizing the results of the previous sections for Poincar e symmetry. 1. Algebra From quantum mechanics we know that for any operator A h jfA;Aygj i=X n(h jAjnihnjAyj i+h jAyjnihnjAj i) =X n(jhnjAyj ij2+jhnjAj ij2)0 from inserting a complete set of states. In particular, fA;Ayg=0)A=0 from examining the matrix element for all states j i. This means the anticommuta- tion relations of the supersymmetry generators must be nontrivial. We are then led to anticommutation relations of the form, in Dirac (Majorana) notation, fq;qg=p=o rfq;qyg=pa ap 2 0 (We use translations instead of internal symmetry or Lorentz generators because of dimensional analysis: Bosonic elds dier in dimension from fermionic ones by half integers.) Note that this implies the positivity of the energy: trfq;qyg=p 2patr( a 0)=p 2patr(1 2f a; 0g)=p01p 2tr I Similar arguments imply that the supersymmetry generators are constrained, just as the momentum is constrained by the mass-shell condition. For example, in the massless case, fp=q;qp=g=p=p=p==1 2p2p==0)p=q=0 104 II. SPIN In four dimensions the commutation relations can be written in terms of irre- ducible spinors as fq;q. g=p. ;fq;qg=fq;qg=0 This generalizes straightforwardly to more than one spinor, carrying a U(N) index: fqi;qj. g=j ip.  2. Supercoordinates Since the momentum is usually represented as coordinate derivatives, we natu- rally look for a similar representation for supersymmetry. We therefore introduce an anticommuting spinor coordinate . Because of the anticommutation relations q can't be simply @=@ , but the modication is obvious: q=i@ @+1 2. @ @x. ; q.=i@ @.+1 2@ @x. We can also express supersymmetry in terms of its action on the \supercoordinates": Using the hermitian innitesimal generator q+.q., =; .=.; x. =1 2i(. +. ) Note that ( q)y=q.,(q)y=q.. We can also dene \covariant derivatives": derivatives that (anti)commute with (are invariant under) supersymmetry. These are easily found to be d=@ @+1 2. p . ; d.=@ @.+1 2p. Besides overall normalization factors of i, leading to the opposite hermiticity condition (d)y=d., these dier from the q's by the relative sign of the two terms. These changes combine to preserve fd;d. g=p.  as a result of which pis also a covariant derivative as well as being a symmetry generator (as for the Poincar e group), but now ( d)y=+d.. In classical mechanics, the fact that @=@x commutes with translations is \dual" to the fact that the innitesimal change dx, or the nite change xx0, is also invariant under translations. Furthermore, the d'Alembertian =(@=@x )2being Poincar e invariant is dual to the line element ds2=(dx)2being invariant. This allows the C. SUPERSYMMETRY 105 construction of the action from.x2. In the supersymmetric case the innitesimal invariants under the q's (and therefore p)a r e d;d .;d x. +1 2i(d). +1 2i(d. ) and the corresponding nite ones (by integration) are 0; .0.;x. x0. +1 2i0. +1 2i. 0 Although these can be used to construct classical mechanics actions, their quantiza- tion is rather complicated. Just as for particles of one particular spin, direct treatmentof the quantum mechanics has proven much simpler than deriving it by quantization of a classical system. Excercise IIC2.1 Check explicitly the invariance of the above innitesimal and nite dierencesunder supersymmetry. Now that we have a (super)coordinate representation of the supersymmetry gen- erators, we can examine the wave functions/elds that carry this representation. Such \superelds" can be Taylor expanded in the 's with a nite number of terms, with ordinary elds as the coecients. For example, if we expand a real (hermitian) scalar supereld (x;;)=(x)+  (x)+. .(x)+::: and also expand its supersymmetry transformation = +. .+1 2i. @ . +1 2i.@.+::: we nd the component eld transformations = +. .; =1 2i. @ . +:::;   .=1 2i@.+:::; ::: which mix the dierent spins. An alternative, and more convenient, way to dene the expansion is by use of the covariant derivatives. Using \ j"t om e a n\j=0", we can dene =j; =(d)j;  .=(d.)j; ::: There is some ambiguity at higher orders in because the d's don't anticommute, and this can be resolved according to whatever is convenient for the particular problem, avoiding eld redenitions in terms of elds appearing at lower order in :S i n c e the eld equations must be covariant under supersymmetry (otherwise there is no 106 II. SPIN advantage to using superelds), they must be written with the covariant derivatives. Then one denes the component expansions by choosing the same ordering of d's as appear in the eld equations (where relevant), which gives the component expansion of the eld equations the simplest form. It also gives a convenient method for deriving supersymmetry transformations, since the d's anticommute with the q's: [(d:::d)j]=[d:::d()]j=[d:::d(iq)]j=[ (iq)d:::d]j=[ (d)d:::d]j w h e r ew eh a v eu s e dt h ef a c tt h a t q=id+-stu, where the -stu is killed by evaluating at = 0, once it has been pulled in front of all the -derivatives. Covariant derivatives can also be used for integration, sinceR d=@=@ =dup to anx- derivative, which can be dropped when also integratingR dx. 3. Supergroups We saw certain relations between the lower-dimensional classical groups that turned out to be useful for just the cases of physical interest of rotational (SO(D 1)), Lorentz (SO(D1,1)), and conformal (SO(D,2)) groups. In particular, the Poincar e group, though not a classical group, is a certain limit (\contraction") of the groupsSO(D,1) and SO(D 1,2), and a subgroup of the conformal group. Similar remarks apply to supersymmetry, but because of its relation to spinors, these classical \su- pergroups" (or \graded" classical groups) exist only for certain lower dimensions, thesame as those where covering groups for the orthogonal groups exist. In higher di- mensions the supergroups do not correspond to supersymmetry, at least not in any way that can be represented on physical states. We'll consider only the graded generalization of the classical groups that appear in the bosonic case. The basic idea is then to take the group metrics and combine them in ways that take into account the dierence in symmetry between bosons and fermions: Unitary: . AB OrthoSymplectic:MAB Real:. AB pseudoreal (*): . AB whereis symmetric and antisymmetric, as before, while Mis graded symmetric: ForA=(a;) with bosonic indices aand fermionic ones , M[AB)=0:MabMba=MaMa=M+M=0 C. SUPERSYMMETRY 107 A g a i nw eh a v ei n v e r s em e t r i c s ,e . g . , MKIMKJ=I J With respect to the usual index-contraction convention (no extra grading signs when superscript is contracted with subscript immediately following), we should take theordering of indices on as JI. T h e r ei sn oa n a l o go ft h e tensor, at least for nite-dimensional groups, since it would have an innite number of indices when totally symmetric. However, \special" supergroups can still be dened by generalizing the denition of trace and determinantto supermatrices. One convenient way to do this is by using Gaussian integrals, since this is a common way that such expressions will arise. As a generalization of the bosonic and fermionic identities we therefore dene the \superdeterminant" (sdetM ) 1=NZ dzydz ezyMz where \N" is a normalization factor dened so sdet I = 1. By explicitly evaluating the integral, separating out the commuting and anticommuting parts, we nd (see excercise IB3.3) sdetAB CD =det A det(DCA1B)=det(ABD1C) det D The \supertrace" (see also excercise IA2.3c) then can be dened by generalizing the bosonic identity det(eM)=etrM: sdet(eM)=estrM str(MAB)=(1)AMAA=MaaM=tr Atr D follows, as in the bosonic case, from lns d e tM =str(M1M), which is derived by varying the Gaussian denition. A useful identity for superdeterminants can be derived by starting with the fol- lowing identity for the inverse of a matrix for which the range of the indices has been divided into two pieces: ab cd1 =(abd1c)1(cdb1a)1 (bac1d)1(dca1b)1 We have assumed all the submatrices are square and invertible; equivalent expressions, which are more useful in other cases, can be derived easily by multiplying and dividing by the submatrices: For example, ab cd1 =(abd1c)1a1b(dca1b)1 d1c(abd1c)1(dca1b)1 108 II. SPIN From either of these we immediately see AB CD1 =~A~B ~C~D )sdetAB CD =detA det ~D=1 det D det ~A The graded generalizations of the classical groups are then GL(mjn,C) [SL(mjn,C),SSL(njn,C)] U: [S]U(m +,mjn) [SSU(n +,njn++n)] OSp: OSp(mj2n,C) R: GL(mjn) [SL(mjn),SSL(njn)] *: [S]U*(2mj2n) [SSU*(2nj2n)] U&O S p R: OSp(m +,mj2n) *: OSp*(2mj2n) where \(mjn)" refers to mbosonic and nfermionic indices, or vice versa. In the matrices of the dening representation, the elements with one bosonic index and onefermionic are anticommuting numbers, while those with both indices of the same kindare commuting. In particular, the commuting parts give the bosonic subgroups: GL(mjn,C)GL(m,C) GL(n,C) SL(mjn,C)GL(m,C) SL(n,C) SSL(njn,C)SL(n,C) SL(n,C) U(m +,mjn)U(m +,m) U(n) SU(m +,mjn)U(m +,m) SU(n) SSU(n +,njn++n)SU(n +,n) SU(n ++n) OSp(mj2n,C)SO(m,C) Sp(2n,C) GL(mjn)GL(m) GL(n) SL(mjn)GL(m) SL(n) SSL(njn)SL(n) SL(n) U*(2mj2n)U*(2m) U*(2n) SU*(2mj2n)U*(2m) SU*(2n) SSU*(2nj2n)SU*(2n) SU*(2n) OSp(m +,mj2n)SO(m +,m) Sp(2n) OSp*(2mj2n)SO*(2m) USp(2n) C. SUPERSYMMETRY 109 When the commuting and anticommuting dimensions are equal, we can impose trace- lessness conditions on both bosonic parts of the generators separately (\SS": tr A = tr D = 0). This is related to the fact str(I)=0i ns u c hc a s e s . 4. Superconformal Since the conformal group is a classical group, its supersymmetric generalization should be a classical supergroup. Because the fermionic generators must include the supersymmetry generators, which are spinors, the representation of the conformal group that appears in the dening representation of the supergroup must be the spinor representation. However, we have seen that only for n 6 (where covering groups exist) and n=8 (where the spinor of SO(8) is another of its dening representations) can the spinor representation of SO(n) be dened by classical group restrictions. This implies that the superconformal group exists only in D 4a n dD = 6 . The relevant supergroups can be identied easily by looking at the bosonic sub- groups: D= 3 : OSp(Nj4) 4: S U ( 2 , 2jN) (or SSU(2,2j4)) 6 : OSp*(8j2N) (We consider only D >2, since the conformal group is innite-dimensional in D 2.) These three cases of D=3,4,6 are special for a number of reasons: In particular,these three supergroups can be related to SU(N j4) over the division algebras: the real numbers, complex numbers, and quaternions, respectively. (Similar remarks apply to their important classical bosonic subgroups: the conformal, Lorentz, and rotation groups. Attempts have been made to extend these results to the octonions for D=10, but with less success, and there seems to be no superconformal group forthat case.) However, just as in the case of the Hilbert space of quantum mechanics, the complex numbers seems to be the best of these \division algebras", having the analytic properties the real numbers lack, while avoiding the noncommutativity ofthe quaternions. We'll see later that nontrivial interacting (local, classical) conformal eld theories exist only in D 4. For example, for D=4 we nd that the bosonic generators are the conformal group and the internal symmetry group U(N) (or SU(4) for N=4), while the fermionic gener-ators include supersymmetry (N spinors) and its fraternal twin, \S-supersymmetry". As supersymmetry is the \square root" of translations, so S-supersymmetry is the square-root of conformal boosts. 110 II. SPIN Excercise IIC4.1 For D=4, write the (graded) commutation relations of the superconformal generators. Decompose them into representations of the Lorentz group, andnd their commutation relations. 5. Supertwistors We saw that a simple way to nd representations of SO(4,2) was to use the coordinate representation for SU(2,2): The resulting twistors gave all massless rep-resentations ( p 2= 0 for all helicities). This method generalizes straightforwardly to the superconformal groups: The generators are GAB=BA (For the SU case we should also subtract out the trace, but that generator commutes with the rest anyway.) The coordinates and their conjugate momenta satisfy [A;Bg=B A Ais then in the dening representation of the supergroup, while the wave function, which is a function of , contains more general representations. For D=3, the reality condition sets =,s ot h e's are the graded generalization of Dirac matrices. In fact, the anticommuting 's are the matrices of the SO(N) subgroup of the OSp(N j4). On the other hand, the commuting 's carry the index of the dening representation of Sp(4), so they are a spinor of SO(3,2), the 3D conformal group: They are the bosonic twistor, and can be used in a similar way to the 4Dtwistors discussed earlier. For D=4, there is a U(1) symmetry acting on under which G ABis invariant, generated by (1)AGAA, as in the bosonic case: This is the \superhelicity". For D=6,is pseudoreal. In general, for pseudoreal representations of groups it is often convenient to introduce a new SU(2) under which the pseudoreal representation Aand its equivalent complex conjugate representation . B . BAtransform as a doublet (SU(2) spinor). This is also obvious from construction, since half of the components are related to the complex conjugate of the other half. We then can write Ai=(A;i. B . BA)=. B. k . B. kAi; . B. kAi= . BACki;MAi;Bk=MABCik GAB=BiAi C. SUPERSYMMETRY 111 (Thus, OSp*(2mj2n)OSp(4nj4m), and SO*(2m) Sp(4n), USp(2n)SO(4m).) This means there is now an SU(2) symmetry on , generated by Gij=A (iAj) under which GABis invariant. This is the 6D version of superhelicity. In the D=6 light cone, the manifest part of Lorentz invariance is SO(D 2)=SO(4)=SU(2) SU(2). This is one of those SU(2)'s. We now concentrate on D=4 (although our methods generalize straightforwardly to D=3 and 6). The simplest way to nd (massless) representations of 4D super- symmetry is to generalize the Penrose transform. Just as twistors automatically satisfy the massless eld equations in D=4, supertwistors automatically satisfy their supersymmetric generalization. The supertwistor is the dening representation of SU(2,2jN). The SU(2,2) part is the usual twistor, while the SU(N) part is the usual fermionic creation and annihilation operators for SU(N). Thus, to relate superspace to supertwistors, we write p . =i@ . !pp.  qi=i@i+1 2.  i@ . !aip; qi.=i@i.+1 2i@.!ayip. This determines the Penrose transform from superspace to supertwistors:  :::. :::(x;)=Z d2pd2p.dNaip


p. 


[ei'+(p;p.;ai)+ei'(p;p.;ai)] '=(x. i1 2i.  i)pp. +iaip where we have used \chiral superelds" (trivial dependence on ) without loss of generality. (Instead of treating aias coordinates to be integrated, we can also treat them as operators; we then make the 's functions of ay, and replace the integration with vacuum evaluation h0jj0i.) As for ordinary twistors, this result can be related to the lightcone: For given momentum, we can choose the lightcone frame p= ; thenq i=ai, whileq i= 0 is a result of the supertwistor formalism automatically incorporating p=q=0 . Excercise IIC5.1 Find the Penrose transform for D=3. (Warning: The anticommuting part of the twistor is now like Dirac matrices rather than creation/annihilation operators.) 112 II. SPIN Taylor expanding in ai(and thusi, producing terms antisymmetric in i:::jand symmetric in ::: ), the states then carry the index structure ;i;ij;:::;~i;~, totally antisymmetric, and terminating with another singlet, where ~=1 N!i1


iNi1


iN; ~i1=1 (N1)!i1


iNi2


iN; ::: From our discussion of helicity in subsection IIB7, we see that the states also decrease in helicity by 1/2 for each a(i.e., ignoring ,e a c hicomes with a p, simply because it adds an undotted index). Taking the direct product with any helicity (coming fromthe explicit p 's and p.'s carrying the external Lorentz indices), we see that the states have helicity h;h1=2;h1;:::;hN=2, with multiplictyN n
for helicity hn=2: state helicity (Poincar e) multiplicity [SU(N)]  h 1 i h1 2 N ij h1N(N1) 2......... i1


inhn 2N n
...... ... ~ihN 2+1 2N ~ hN 21 This multiplet structure is carried separately by +and by, which are related by charge (complex) conjugation, one describing the antiparticles of the other, as forordinary twistors. (The existence of both multiplets also follows from CPT invariance,which is required for local actions, to be discussed in subsection IVB1. Here wegeneralized from the Penrose transform, which contained both terms as a consequence of being the most general solution to S abpb+wpa= 0, which is CPT invariant.) Because of the values of the helicities, we can impose a reality condition, identifyingall states with helicity jas the complex conjugates of those with j,o n l yf o rh= hN=2!h=N=4, whenNis a multiple of 4. We can also get larger representations by taking the direct product of these smallest representations of supersymmetry with representations of U(N), in which case the elds will carry those additional SU(N) indices. Excercise IIC5.2 For D=4, show that \supergravity", the supersymmetric theory with helicitiesof magnitude 2 and lower, can exist only for N 8. Show that the relevant C. SUPERSYMMETRY 113 representation for N=8, if real, is the same as the one (complex) for N=7. Find the analogous statements for \super Yang-M ills", with helicities 1 and less. The explicit form of the reality condition is somewhat complicated in terms of the chiral superelds, because they are really eld strengths of real gauge elds. (Consider, for example, expressing reality of A . in terms of fi nt h ec a s eo fe l e c - tromagnetism.) However, in terms of the twistor variables, charge conjugation can be expressed as C:!*;ai!(ai)y where the transformation on aiis required because it carries the SU(N) \charge". Since this violates \chirality" in these variables (dependence on aand notay), it is accomplished by Fourier transformation: C:(ai)!CZ d~ayie~ayiai[(~ai)]*C1 for some \charge conjugation matrix" C(in case the eld carries an additional index). REFERENCES 1P. Ramond, Phys. Rev. D3(1971) 86; A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86, Phys. Rev. D4(1971) 1109: 2D supersymmetry in strings. 2Yu.A. Gol'fand and E.P. Likhtman, JETP Lett. 13(1971) 323; D.V. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109; J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39: 4D supersymmetry. 3A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477; S. Ferrara, B. Zumino, and J. Wess, Phys. Lett. 51B (1974) 239: superspace. 4Gates, Grisaru, Ro cek, and Siegel, loc. cit. ; J. Wess and J. Bagger, Supersymmetry and supergravity , 2nd ed. (Princeton University, 1992);P. West, Introduction to supersymmetry and supergravity , 2nd ed. (World-Scientic, 1990);I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and super- gravity, or a walk through superspace (Institute of Physics, 1995). 5Berezin, loc. cit. (IA): superdeterminant. 6P. Ramond, Physica 15D (1985) 25: supergroups and their relation to supersymmetry. 7J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39: superconformal group in D=4. 8R. Haag, J.T. Lopusza nski, and M. Sohnius, Nucl. Phys. B88 (1975) 257: superconformal symmetry as the largest symmetry of the S-matrix. 114 II. SPIN 9A. Ferber, Nucl. Phys. B132 (1978) 55: supertwistors. 10T. Kugo and P. Townsend, Nucl. Phys. B221 (1983) 357; A. Sudbery, J. Phys. A17 (1984) 939; K.-W. Chung and A. Sudbery, Phys. Lett. 198B (1987) 161: spacetime symmetries and division algebras. A. ACTIONS 115 III. LOCAL In the previous chapters we considered symmetries acting on coordinates or wave functions. For the most part, the transformations we considered had constant param-eters: They were \global" transformations. In this chapter we will consider mostlyeld theory. Since elds are functions of spacetime, it will be natural to considertransformations whose parameters are also functions of spacetime, especially thosethat are localized in some small region. Such \local" or \gauge" transformations arefundamental in dening the theories that describe the fundamental interactions. :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: A. ACTIONS :::::::::::::::::::::::::::: A fundamental concept in physics, of as great importance as symmetry, is the action principle. In quantum physics the dynamics is necessarily formulated in termsof an action (in the path-integral approach), or an equivalent Hamiltonian (in theHeisenberg and Schr odinger approaches). Action principles are also convenient and powerful for classical physics, allowing all eld equations to be derived from a single function, and making symmetries simpler to check. 1. General We begin with some general properties of actions. (For this subsection we'll re- strict ourselves to bosonic variables; however, in the following subsection we'll ndthat the only modication for fermions is a more careful treatment of signs.) Gen-erally, equations of motion are derived from actions by setting their variation with respect to their arguments to vanish: S[]S[+]S[]=0 Here the variables are themselves functions of time; thus, Sis a function of functions, a \functional". A general principle of mechanics is \locality", that events at one timedirectly aect only those events an innitesimal time away. (In eld theory theseevents can be also only an innitesimal distance away in space.) This means that theaction can be expressed in terms of a Lagrangian: S[]=Z dt L[(t)] whereLat timetis a function of only (t) and a nite number of its derivatives. For more subtle reasons, this number of time derivatives is restricted to be no more than 116 III. LOCAL two for any term in L; after integration by parts, each derivative acts on a dierent factor of. The general form of the action is then L()=1 2. m. ngmn()+. mAm()+U() where \."m e a n s@=@t, and the \metric" g, \vector potential" A, and \scalar po- tential"Uare not to be varied independently when deriving the equations of motion. (Specically, U=(m)(@U=@m), etc. Note that our denition of the Lagrangian diers in sign from the usual.) The equations of motion following from varying an action that can be written in terms of a Lagrangian are 0=SZ dt mS m)S m=0 where we have eliminated . mterms by integration by parts (assuming =0a tt h e boundaries in t), thereby dening the \functional derivative" S=m, and used the fact that(t) is arbitrary at each value of t. For example, S=Z dt1 2.q2) 0=S=Z dt.q.q=Z dt(q)..q)S q=..q=0 Excercise IIIA1.1 Find the equations of motion for mfrom the above general action in terms of the external elds g,A,a n dU(and their partial derivatives with respect to). Sometimes the functional derivative is dened in terms of that of the variable itself:m(t) n(t0)=m n(tt0) wherem nis the usual Kronecker delta function, while (tt0) is the \Dirac delta function". It's not really a function, since it takes only the values 0 or 1, but a \distribution", meaning it's dened only by integration: Z dt0f(t0)(tt0)=f(t) If we apply this denition of the Dirac to= , we obtain the previous denition of the functional derivative. (Consider, e.g., S=Rdt f .) Such actions can be reduced to ones that are only linear in time derivatives by introducing additional variables. First, separate out the subspace where gis invertible, with coordinates q(m=(qi; )); the Lagrangian is then written as L(q; )=1 2.qi.qjgij(q; )+.qiAi(q; )+. A(q; )+U(q; ) A. ACTIONS 117 This Lagrangian gives equivalent equations of motion to L0(q;p; )=[.qipi+. A]+[1 2gij(pi+Ai)(pj+Aj)+U] wheregijis the inverse of gij. (Many other forms are possible by redenitions of p.) Eliminating the new variables pby their equations of motion gives back L(q; ). Note that this works only because p's equations of motion are algebraic: For example, eliminating xfrom the Lagrangian .xp+1 2p2by the equation of motion.x=pis illegal (it would give the trivial action S=R dt1 2p2), since it would require solving for the time dependence of x. On the other hand, pis given explicitly in terms of the other variables by its equations of motion without inverting time derivatives, so eliminatingit does not lose any of the dynamics. (It is an \auxiliary variable".) The result is a Hamiltonian form of the Lagrangian: L H() =i. MAM() +H() in terms of the Hamiltonian H,w h e r e=( q;p; ). It has the \gauge invariance" AM=@M() (where@M=@=@M), since that adds only a total derivative term i. . ClearlyA will introduce a modication of the Poisson bracket if it is not linear in  (e.g., aswhen we make independent nonlinear redenitions of coordinates and momenta on the usual form of the Lagrangian). To determine this modication we compare the equation of motion as dened by a Poisson bracket, .  M=i[M;H]=i[M;N]@NH with that following from varying the action, i. NFNM+@MH=0;FMN=@[MAN] to nd [M;N]=(F1)NM where \F1" is the inverse on the maximal subspace where Fis invertible. The variables in the directions where Fvanishes are \auxiliary", since they appear without time derivatives: Their equations of motion are not described by the Poisson bracket.In particular, if they appear linearly in Hthey are \Lagrange multipliers", whose variation imposes algebraic constraints on the rest of . 118 III. LOCAL Finally, we can make redenitions of the part of  describing the invertible sub- space so thatAis linear: AM=1 2N NM)LH() =1 2i. MN NM+H() where is a constant, hermitian, antisymmetric (and thus imaginary) matrix. For some purposes it is more convenient to assume this Hamiltonian form of the actionas a starting point. We now have the canonical commutation relations as [ M;N]= MN where MNis the inverse of NMon the maximal subspace: MN PN=PM for the projection operator  for that subspace. Excercise IIIA1.2 For electromagnetism, dene ~ =~E+i~B. Show that Maxwell's equations (in empty space) can be written as two equations in terms of ~ . Interpret the equation involving the time derivative as a Schr odinger equation for the wave function ~ , and nd the Hamiltonian operator. Dene the obvious inner productR d3x~ *
~ : What physical conserved quantity does this represent? (Note that, unlike electrons, the number of photons is not conserved.) Note that the requirement of the existence of a Hamiltonian formulation deter- mines that the kinetic term for a particle in the Lagrangian formulation go as.x2 and notx..x. Although such terms give the same equations of motion, they are not equivalent quantum mechanically, where boundary terms (dropped when using inte-gration by parts for deriving the equations of motion) contribute. Furthermore, theHamiltonian form of the action S=Z dt Hdx ipi shows that the energy Hrelates to the time in the same way the momentum relates to the coordinates, except for an interesting minus sign that is explained only by special relativity. A. ACTIONS 119 2. Fermions In nonrelativistic quantum mechanics, spin is usually treated as a quantum eect, rather than being derived from classical mechanics. Although it is possible to derivespin from classical mechanics, in general it is rather cumbersome, and involves rst introducing a large number of spins and then constraining away all the undesired ones, whereas in the quantum mechanics one can just directly introduce some particularrepresentation of the spin angular momentum operators. The one nontrivial exception is spin 1/2. We know from quantum mechanics that the spin variables for spin 1/2 are de- scribed by the Pauli matrices. Since they satisfy anticommutation relations, and are represented by nite-dimensional matrices, they are interpreted as fermionic. We have already seen that classical fermions are described by anticommuting numbers, so we begin by considering general quantization of such objects. We can now consider actions that depend on both commuting and anticommuting classical variables,  M=(m; ), where now refers to the bosonic variables and to the fermionic ones. The Hamiltonian form of the Lagrangian can again be written as LH() =1 2i. MN NM+H() When is invertible, the graded bracket is dened by (see subsection IA2) [M;Ng=h MN; MN PN=M P To describe spin 1/2, we therefore look for particle actions of the form SH=Z dt[.xipi+1 2i. i i+H(x;p; )] This corresponds to using M=(m; )=(i; i)=(xi;pi; i) mn= i;j=ijC; = ij=ij; m= n=0 The fundamental commutation relations are then [xi;pj]=ihi j;f i; jg=hij([x;x]=[p;p]=[x; ]=[p; ]=0 ) We recognize ias the Pauli matrices (the Dirac matrices of subsection IC1 for the special case of SO(3)), i=p hi. The free Hamiltonian is just H=p2 2m 120 III. LOCAL as for spin 0: Spin does not aect the motion of free particles. A more interesting case is coupling to electromagnetism: Quantum mechanically, the Hamiltonian can be written in the simple form H=f i[pi+qAi(x)]g2 mhqA0(x) in terms of the vector and scalar potentials AiandA0. The classical expression is not as simple, because the commutation relations must be used to cancel the 1 =hbefore taking the classical limit. This is an example of \minimal coupling", H(pi)!H(pi+qAi)qA0 However, this prescription works only if Hfor spin 1/2 is written in the above form: Using the commutation relations before or after minimal coupling gives dierent re- sults. The form we have used is justied only by considering the nonrelativistic limit of the relativistic theory. Excercise IIIA2.1 Use the multipication rules of the matrices to show that the quantum me- chanical Hamiltonian for spin 1/2 in an electromagnetic eld can be writtenas a spin-independent piece, identical to the spin-0 Hamiltonian, plus a term coupling the spin to the magnetic eld. 3. Fields The eld equations for all eld theories (e.g., electromagnetism) are wave equa- tions. Wave equations also follow from mechanics upon quantization. Although classical eld theory and quantum mechanics are not equivalent in their physical in- terpretation, they are mathematically equivalent in that they have identical wave equations. This is true not only for the free theories, but also for particles in external elds, and without direct self-interactions. This is no accident: Classical eld theoryand classical mechanics are two dierent limits of quantum eld theory. They are both called classical limits, and written as  h!0, but since  his really 1, this limit depends on how one inserts  h's into the quantum eld theory action. The wave equation in quantum mechanics is the Schr odinger equation. The cor- responding eld theory action is then simply the one that gives this wave equation as the equation of motion, where the wave function is replaced with the eld: S ft=Z d4x *(i@t+H) A. ACTIONS 121 As usual (cf. electromagnetism), the eld is a function of space and time; thus, we integrated4x=dt d3xover the three space and one time dimensions. The Hamilto- nian is some function of coordinates and momenta, with the replacement pi!i@i, where@i=@=@xiare the space derivatives and @t=@=@t is the time derivative. The Hamiltonian can contain coupling to other elds. For a general Hamilto- nian quadratic in momenta, in a notation implied by the corresponding Lagrangian quadratic in time derivatives, H=1 2gij(i@i+Ai)(i@j+Aj)+U wheregij,Ai,a n dUare now interpreted as elds, and thus depend on both xiandt, as does .I n t h e c a s e gij=ij,w ec a ni d e n t i f y AiandUas the three-vector and scalar potentials of electromagnetism, and we can add the usual action for electromagnetismto the action for . The action then can be varied also with respect to AandUto obtain Maxwell's equations with a current in terms of and *. We can also treat g ijas a eld, in which case it and parts of AandUare the components of the gravitational eld. Field theory actions can be quantized in the same ways as mechanics ones. In this case, we recognize the *. term as a special case of the.  term in the generic Hamiltonian form of the action discussed earlier. Thus, (xi)a n d *(xi) have replaced xiandpias the variables; xiis now just an index (label) on and *, just asiwas an index on xiandpi. The eld-theory Hamiltonian is then identied as Hft[ ; *] =Z d3xH;H= *H In eld theory the Hamiltonian will always be a space integral of a \Hamiltonian density"H. We can now dene the two classical limits of quantum eld theory. If we put in hin the generic way for actions, Sft!h1Sft then we dene the classical limit  h!0 as classical eld theory, since in that limit the classical eld equations are preserved. On the other hand, if we put in  h's as @i!h@i;@t!h@t which gives the usual  hdependence associated with the Schr odinger equation, then the classical limit  h!0 gives classical mechanics. This denes classical mechanics as the macroscopic limit, the limit of large distances and times. 122 III. LOCAL A convenient way to implement this limit is to introduce the mechanics action S=R dt(.xipi+H) into the eld theory, and then take the limit  h!0a f t e rt h e replacement S!h1S on the mechanics action instead of on the derivatives. The mechanics action can be introduced when solving the eld equations: The solution to the wave equation canbe expressed in terms of the propagator, which in turn can be written in terms of the mechanics action or Hamiltonian. More generally, we can dene actions that are not restricted to be quadratic in any eld. The Hamiltonian density H(t;x i) or Lagrangian density L(t;xi), S[]=Z dt d3xL[(t;xi)] should be a function of elds at that point, with only a nite number (usually no more than two) spacetime derivatives. This is the denition of locality used for gen- eral quantum systems in subsection IIIA1, but extended from derivatives in time toalso those in space. Although this condition is not always used in nonrelativistic eld theory (for example, when long-range interactions, such as Coulomb or gravita- tional, are described without attributing them to elds), it is crucial in relativisticeld theory. For example, global symmetries lead by locality to local (current) conser- vation laws. Locality is also the reason that spacetime coordinates are so important: Translation invariance says that the position of the origin is an unphysical, redundant variable; however, locality is most easily used with this redundancy. Field equations are derived by the straightforward generalization of the variation of actions dened in subsection IIIA1: As follows from treating the spatial coordinates in the same way as discrete indices, SZ dt d 3xm(t;xi)S m(t;xi) For example, S=Z dt d3x1 2. 2)S =..  4. Relativity Generalization to relativistic theories is straightforward, except for the fact that the Klein-Gordon equation is second-order in time derivatives; however, we are fa-miliar with such actions from nonrelativistic quantum mechanics. As usual, we need to check the sign of the terms in the action: Checking the positivity of the Hamil- tonian (i.e., the energy), we see from the general relation between the Lagrangian A. ACTIONS 123 and Hamiltonian (subsection IIIA1) that the terms without time derivatives must be positive; the time-derivative terms are then determined by Lorentz covariance. At this point we introduce some normalizations and conventions that will prove convenient for Fourier transformation and other reasons to be explained later. When-ever D-dimensional integrations are involved (as should be clear from context), we use Z dxZd Dx (2)D=2;Z dpZdDp (2)D=2 (xx0)(2)D=2D(xx0); (pp0)(2)D=2D(pp0) In particular, this normalization will be used in Green functions and actions. For example, these implicit 2 's appear in functional variations: SZ dx S ) (x)(x0)=(xx0) The action for a real scalar is then S=Z dx L; L =1 4(@)2+V() whereV()0, and we now write Lfor the Lagrange density. In particular, V= 1 4m22for the free theory. The free eld equation is then p2+m2=+m2=0 , replacing the nonrelativistic i@t+H= 0. For a complex scalar, we replace1 2! in both terms. We know from previous considerations (subsection IIB2) that the eld equation for a free, massless, Dirac spinor is
@ = 0. The generalization to the massive case (subsection IIB4) is obvious from various considerations, e.g., dimensional analysis; the action is S=Z dx (i@=+mp 2) in arbitrary dimensions, again using the notation @==
@. In four dimensions, we can decompose the Dirac spinor into its two Weyl spinors (see subsection IIA6): L= (i@=+mp 2) = (  . Li@. L+ . Ri@. R)+mp 2(  L R+ . L R.) For the case of the Majorana spinor, the 4D action reduces to that for a single Weyl spinor, S=Z dx[i . @ .  +mp 21 2(  + . .)] Note that in our conventions 0 . =1p 2(and similarly for the opposite indices, since a . =. a), so that the time derivative term is always proportional to y(i@0) ,a s nonrelativistically (previous subsection). 124 III. LOCAL A scalar eld must be complex to be charged (i.e., a representation of U(1)): From the gauge transformation 0=ei we nd the minimal coupling (for q=1 ) S=Z dx[1 2j(@+iA)j2+1 2m2jj2] This action is also invariant under charge conjugation C:!*;A!A which changes the sign of the charge, since *0=ei*. Excercise IIIA4.1 Let's consider the semiclassical interpretation of a charged particle as de- scribed by a complex scalar eld , with Lagrangian L=1 2(jr j2+m2j j2) aUse the semiclassical expansion in  hdened by r! h@+iqA; !peiS=h Find the Lagrangian in terms of andS(and the background eld A), order- by-order in  h(in this case, just  h0and h2). bTake the semiclassical limit by dropping the  h2term inL, to nd L!1 2[(@S+qA)2+m2] Vary with respect to Sandto nd the equations of motion. Dening p@S show that these eld equations can be interpreted as the mass-shell condition and current conservation. Show that Acouples to this current by varying L with respect to A. The spinor eld also needs doubling for charge. (Actually, the doubling can be avoided in the massless case; however, problems show up at the quantum level, related to the fact that there is no charge conjugation transformation without doubling.) The gauge transformations are similar to the scalar case, and the action again follows from A. ACTIONS 125 minimal coupling, to an action that has the global invariance ( = constant in the absence ofA): 0 L=ei  L; 0 R=ei  R Se=Z dx[ .  L(i@ . +A . )  L+ .  R(i@ . A . )  R+mp 2(  L R+ . L R.)] The current is found from varying with respect to A: J. = .  L  L .  R  R Charge conjugation C:  L$  R;A!A (which commutes with Poincar e transformations) changes the sign of the charge and current. Excercise IIIA4.2 Show that this action can be rewritten in Dirac notation as Se=Z dx (i@=A=+mp 2) and nd the action of the gauge transformation and charge conjugation on the Dirac spinor. As a last example, we consider the action for electromagnetism itself. As before, we have the gauge invariance and eld strength A0 . =A . @ .  F . ;. =@. A . @ . A. =Cf. . +C. . f;f=1 2@(. A). We can write the action for pure electromagnetism as SA=Z dx1 2e2ff=Z dx1 2e2f.. f.. =Z dx1 8e2FabFab dropping boundary terms, with the overall sign again determined by positivity of the Hamiltonian, where eis the electromagnetic coupling constant, i.e., the charge of the proton. (Other normalizations can be used by rescaling A . .) Maxwell's equations follow from varying the action with a source term added: S=SA+Z dx A. J . )1 e2@. f=J. Excercise IIIA4.3 By plugging in the appropriate expressions in terms of Aa(and repeatedly 126 III. LOCAL integrating by parts), show that all of the above expressions for the electro- magnetism action can be written as SA=Z dx1 4e2[A
A+(@
A)2] Excercise IIIA4.4 Find all the eld equations for all the elds, found from adding to SAall the minimally coupled matter actions above. The energy-momentum tensor for electromagnetism is much simpler in this spinor notation, and follows (up to normalization) from gauge invariance, dimensional anal- ysis, Lorentz invariance, and the vanishing of its trace. It has a form similar to that of the current in electrodynamics: T . . =1 e2ff. .  Note that it is invariant under the duality transformations of subsection IIA7 (as is the electrodynamic current under chirality). We have used conventions where eappears multiplying only the action SA,a n d not in the \covariant derivative" r=@+iqA whereqis the charge in units of e: e.g.,q= 1 for the proton, q=1 for the electron. Alternatively, we can scale A, as a eld redenition, to produce the opposite situation: A!eA:SA!Z dx1 8F2;r!@+iqeA The former form, which we use unless noted otherwise, has the advantage that the coupling appears only in the one term SA, while the latter has the advantage that the kinetic (free) term for Ais normalized the same way as for scalars. The former form has the further advantage that eappears in the gauge transformations of none of the elds, making it clear that the group theory does not depend on the value of e.( T h i s will be more important when generalizing to nonabelian groups in section IIIC.) Note that the massless part of the kinetic (free) terms in these actions are scale invariant (in arbitrary dimensions, when the dimension-independent forms are used),when the elds are assigned the scale weights found from conformal arguments in subsection IIB2. Excercise IIIA4.5 Using vector notation, minimal coupling, and dimensional analysis, nd the A. ACTIONS 127 mass dimensions of the electric charge ein arbitrary spacetime dimensions, and show it is dimensionless only in D=4 . An interesting distinction between gravity and electromagnetism is that static bodies always attract gravitationally, whereas electrically they repel if they are likeand attract if they are opposite. This is a direct consequence of the fact that the graviton has spin 2 while the photon has spin 1: The Lagrangian for a eld of integer spinscoupled to a current, in an appropriate gauge and the weak-eld approximation, is 1 4s!a1:::asa1:::as+1 s!a1:::asJa1:::as where the sign of the rst term is xed by unitarity in quantum eld theory. (Clas- sically the sign can also be related to positivity of the energy.) From a scalar eld in the semiclassical approximation (see excercise IIIA4.1 above), starting with Ja1:::as *$ @a1


$ @as where \A$ @B"m e a n s\A@B(@A)B", we see that the current will be of the form Ja1:::aspa1


pas for a scalar particle, for some . (The same follows from comparing the expressions for currents and energy-momentum tensors for particles as in subsection IIIB4 below. The only way to get vector indices out of a scalar particle, to couple to the vectorindices for the spin of the force eld, is from momentum.) In the static approximation, only time components contribute: We then can write this Lagrangian as, taking into account 00=1, (1)s1 4s!0:::00:::0+1 s!0:::0(p0)s whereE=p0>0f o rap a r t i c l ea n d <0 for an antiparticle. Thus the spin-dependence of the potential/force between two particles goes as ( E1E2)s. It then follows that all particles attract by forces mediated by even-spin particles, and a particle and its antiparticle attract under all forces, while repulsion will occur for odd-spin forces between two identical particles. (We can substitute \particles of the same sign charge" for \identical particles", and \particles of opposite sign charge" for \particle and its antiparticle", where the charge is the coupling constant appropriate for that force.) Excercise IIIA4.6 Show that the above current is conserved, @a1Ja1


as=0 (and the same for the other indices, by symmetry) if satises the free Klein- Gordon equation (massless or massive). 128 III. LOCAL 5. Constrained systems Constraints not only frequently appear in nonrelativistic physics, but are a general feature of relativistic particles, so we now give a brief description of how they are incorporated into actions. Consider a general action, with constraints, in Hamiltonianform: S=Z dt(.q mpm+H);H =Hgi(q;p)+iGi(q;p) (For simplicity, we consider all physical variables to be bosonic for this subsection, but the method generalizes straightforwardly paying careful attention to signs.) This action is a functional of qm;pm;i, which are in turn functions of t,w h e r emandirun over any number of values. We can think of this as describing a nonrelativistic particle with coordinates qand momenta pin terms of time t, but the form is general enough to apply to relativistic theories. The.qpterm tells us pis canonically conjugate to q;t h e rest of the action gives the Hamiltonian, usually quadratic in momenta. The variables iare \Lagrange multipliers", whose variation in the action implies the constraints Gi= 0. We then can interpret Hgias the usual (\gauge invariant") Hamiltonian. We also require that the transformations generated by the constraints close, and that theHamiltonian be invariant: [G i;Gj]=ifijkGk;[Gi;Hgi]=0 (More generally, we can allow [ Gi;Hgi]=ifijGj.) This says that the constraints don't imply any new constraints that we might have missed, and that the \energy"represented by H giis invariant under these transformations. We then nd that the action is invariant under the canonical transformations (q;p)=i[iGi;(q;p)])qm=i@Gi @pm; pm=i@Gi @qm 0=d dt =@ @t+iH =i(i)Gii. iGi+[jGj;iGi] )i=. i+jkfkji (with(d=dt) dened as in subsection IA1), where @=@t acts on the \explicit" t dependence (that in everything except qandp): For general expressions, the total time derivative and total variation are given by commutators as d dtA=@ @tA+i[H;A]; A =0A+i[iGi;A] A. ACTIONS 129 where0acts on everything except qandp. The action then varies under these transformations as the integral of a total derivative, which vanishes under appropriate boundary conditions: SH=Z dtd dt[(qm)pm+iGi]=0 The simplest example is the case with one constraint, which is linear in the vari- ables: If the constraint is p, the gauge transformation is q=, so we gauge q=0 and use the constraint p= 0. In general, this means that for every degree of freedom we can gauge away, the conjugate variable can be xed by the constraint. Thus, for each constraint we eliminate 3 variables: the variable xed by the constraint, its conjugate, and the Lagrange multiplier that enforced the constraint, which has no conjugate. (In the Lagrangian form of the action the conjugate may not appear ex- plicitly, so only 2 variables are eliminated.) As another example, for a nonrelativisticparticle constrained to a sphere, G=(x i)21, we can change to spherical coordinates, apply the constraint to eliminate the radial coordinate, and use the gauge invariance to eliminate the radial component of the momentum, leaving an unconstrained the-ory in terms of angles and their conjugates. In most cases in eld theory a similar procedure can be applied: The result is called a \unitary gauge". The standard example of a relativistic constrained system is in eld theory | electromagnetism. Its action can be written in \rst-order (in derivatives) formalism"by introducing an auxiliary eld G ab: F2!F2G2!F2(GF)2=2GFG2 where in the rst step we added a trivial term for Gand in the second step made a trivial redenition of G, so elimination of Gby its algebraic equation of motion returns the original Lagrangian. The Hamiltonian form comes from eliminating only Gijby its eld equation, since only F0icontains time derivatives: 2GFG2!(Fij)24G0iF0i+2 (G0i)2 =4. AiG0i+[ 2 (G0i)2+(Fij)2]4A0@iG0i which we recognize as the three generic terms for the action in Hamiltonian form, withG0ias the canonical momenta for Ai,a n dA0as the Lagrange multiplier. The constraint is Gauss' law, and it generates the usual gauge transformations. Thusiare also gauge elds for the gauge (time-dependent) transformations i(t). They allow construction of the gauge-covariant time derivative r=@t+iiGi;d dt=r+iHgi)r=i[iGi;r] 130 III. LOCAL It is convenient to transform the gauge elds away using these gauge transformations, soH=Hgi. However, with the usual boundary conditionsR1 1dtiis gauge invariant under the linearized transformations, so the most we could expect is to gauge ito constants. More precisely, the group element T exp iZ1 1dt i(t)Gi is gauge invariant, where \ T" is time ordering, meaning we write the exponential of the integral as the product of exponentials of innitesimal integrals, and order them with respect to time, later time intervals going to the left of earlier ones. (We treat Giquantum mechanically or use Poisson brackets when combining the exponentials.) This is the quantum mechanical version of the time development resulting from thecorresponding term in the classical action. It is also the phase factor coming from the innite limit of the covariant time translation e kr(t)=T exp iZt tkdt0i(t0)Gi ek@t as seen from reordering the time derivatives when writing ekr(t)as the product of exponentials of innitesimal exponents. This allows us to write the explicit gauge transformation ei(t)=T exp iZt t0dt0i(t0)Gi =e(
t)r(t)e(
t)@t;
t=tt0 )r0(t)=ei(t)r(t)ei(t)=e(
t)@tr(t)e(
t)@t=r(t0)=@t+ii(t0)Gi (where we dene @tto varytwhile keeping tt0xed). Thus, we can gauge to its value at a xed time t0. Another way to see this is that varying iin the action at a xed time gives Gi= 0 at that time, but the remaining eld equations imply. Gi=0 ,s oGi=0a l w a y s ,a n d iis redundant at other times. This means that if we carelessly impose i= 0 at all times, we must also impose Gi= 0 at some xed time. Note that this special gauge transformation itself has a very simple gauge trans- formation: Transforming the in  by an arbitrary nite transformation i(t), ei0(t)=eii(t)Giei(t)eii(t0)Gi consistent with the transformation law of r0(t) above. Thus, applying the trans- formation  to any gauge-dependent quantity gives a gauge-independent quantity 0(;), which is invariant under the local transformations (t) and transforms only under the \global" transformations (t0). Thus, xing the gauge (t)=0i se q u i v a l e n t to working with gauge-invariant quantities. A. ACTIONS 131 Fixing an invariance of the action is not unique to gauge invariances: Global invariances also need to be xed, although the procedure is so trivial we seldom discuss it. For example, even in nonrelativistic systems Galilean invariance needs to be xed: When analyzing a specic problem, we often choose some object to be at rest (velocity transformations), choose another to be oriented or moving in a specicdirection (rotations), and choose a specic event to happen at the origin of space and time (translations). Alternatively, we can work with Galilean invariants, just as in gauge theories we can work with gauge invariants; however, in practice, for explicitcalculations (as opposed to discussing general properties), it is more convenient to x the invariance, as this allows simplication of the equations. REFERENCES 1 M.B. Halpern and W. Siegel, Phys. Rev. D16 (1977) 2486: Classical mechanics as a limit of quantum eld theory. 2P.A.M. Dirac, P r o c .R o y .S o c . A246 (1958) 326; L.D. Faddeev, Theo. Math. Phys. 1(1969) 1: Hamiltonian formalism for constrained systems. 132 III. LOCAL ::::::::::::::::::::::::::: ::::::::::::::::::::::::::: ::::::::::::::::::::::::::: B. PARTICLES ::::::::::::::::::::::::::: The simplest relativistic actions are those for the mechanics (as opposed to eld theory) of particles. These also give the simplest examples of gauge invariance in rela- tivistic theories. Later we will nd that various properties of the quantum mechanicsof these actions help to explain some features of quantum eld theory. 1. Free For nonrelativistic mechanics, the fact that the energy is expressed as a function of the three-momentum is conjugate to the fact that the spatial coordinates are expressedas functions of the time coordinate. In the relativistic generalization, all the spacetime coordinates are expressed as functions of a parameter : All the points that a particle occupies in spacetime form a curve, or \worldline", and we can parametrize this curvein an arbitrary way. Such parameters generally can be useful to describe curves: A circle is better described by x();y()t h a ny(x) (avoiding ambiguities in square roots), and a cycloid can be described explicitly only this way. The action for a free, spinless particle then can be written in relativistic Hamil- tonian form as S H=Z d[.xmpm+v1 2(p2+m2)] wherevis a Lagrange multiplier enforcing the constraint p2+m2=0 . T h i sa c - tion is very similar to nonrelativistic ones, but instead of xi(t);pi(t)w en o wh a v e xm();pm();v()( w h e r e\ ."n o wm e a n s d=d). The gauge invariance generated by p2+m2is x=p; p =0; v =.  A more recognizable form of this invariance can be obtained by noting that any actionS(A) has invariances of the form A=ABS B;AB=BA which have no physical signicance, since they vanish by the equations of motion. In this case we can add x=(.xvp); p =.p; v =0 and set=vto get x=.x; p =.p; v =. (v) B. PARTICLES 133 We then can recognize this as a (innitesimal) coordinate transformation for : x0(0)=x();p0(0)=p();d 0v0(0)=d v();0=() The transformation laws for xandpidentify them as \scalars" with respect to these \one-dimensional" (worldline) coordinate transformations (but they are vectors with respect to D-dimensional spacetime). On the other hand, vtransforms as a \density": The \volume element" d v of the world line transforms as a scalar. This gives us a way to measure length on the worldline in a way independent of the choice of  parametrization. Because of this geometric interpretation, we are led to constrain v>0 so that any segment of the worldline will have positive length. Because of this re- striction,vis not a Lagrange multiplier in the usual sense. This has signicant physical consequences: p2+m2is treated neither as a constraint nor as the Hamil- tonian. While in nonrelativistic theories the Schr odinger equation is ( EH) =0 andGi = 0 is imposed on the initial states, in relativistic theories ( p2+m2) =0 is the Schr odinger equation: This is more like H =0 ,s i n c ep2already contains the necessaryEdependence. The Lagrangian form of the free particle action follows from eliminating pby its equation of motion vp=.x: SL=Z d1 2(vm2v1.x2) Form6= 0, we can also eliminate vby its equation of motion v2.x2+m2=0 : S=mZ dp .x2=mZp dx2=mZ ds=ms The action then has the purely geometrical interpretation as the proper time; how- ever, this last form of the action is awkward to use because of the square root, anddoesn't apply to the massless case. Note that the vequation implies ds=m(d v), relating the \intrinsic" length of the worldline (as measured with the worldline vol- ume element) to its \extrinsic" length (as measured by the spacetime metric). As aconsequence, in the massive case we also have the usual relation between momentum and \velocity" p m=mdxm ds (Note that p0is the energy, not p0.) Excercise IIIB1.1 Take the nonrelativistic limit of the Poincar ea l g e b r a : 134 III. LOCAL aInsert the speed of light cin appropriate places for the structure constants of the Poincar e group (guided by dimensional analysis) and take the limit c!0 to nd the algebra of the Galilean group. bDo the same for the representation of the Poincar e group generators in terms of coordinates and momenta. In particular, take the limit of the Lorentz boosts to nd the Galilean boosts. cTake the nonrelativistic limit of the spinless particle action, in the form ms. (Note that, while the relativistic action is positive, the nonrelativistic one isnegative.) Excercise IIIB1.2 Consider the following action for a particle with additional fermionic variables and additional fermionic constraint
p: S H=Z d(.xmpm1 2i. m m+1 2vp2+i
p) whereis also anticommuting so that each term in the action is bosonic. Find the algebra of the constraints, and the transformations they generateon the variables appearing in the action. Show that the \Dirac equation"
pj i= 0 implies p 2j i= 0. Find the Lagrangian form of the action as usual by eliminating pby its equation of motion. (Note 2=0 . ) Excercise IIIB1.3 Consider a \supercoordinate" Xmthat is a function of both a fermionic vari- ableand the usual : Xm(;)=xm()+i m() where the Taylor expansion in terminates because 2=0 . I d e n t i f y xwith the usualx,a n d with its fermionic partner introduced in the previous problem. In analogy to the way
pwas the square root of the -translation generator1 2p2, we can dene a square root of @=@ by the \covariant fermionic derivative" D=@ @+i@ @)D2=i@ @ We also want to generalize vin the same way as x, to make the action inde- pendent of coordinate choice for both and. This suggests dening E=v1+i and the gauge invariant action SL=Z dd1 2E(D2Xm)DXm B. PARTICLES 135 Integrate this action over , and show this agrees with the action of the previous problem after suitable redenitions (including the normalization ofR d). The (D+2)-dimensional (conformal) representation of the massless particle (sub- section IA6) can be derived from the action S=Z d1 2(.y2+y2) whereis a Lagrange multiplier. This action is gauge invariant under y=.y1 2.y;  =. +2.+1 2... If we varyto eliminate it and yas in subsection IA6, the action becomes S=Z d1 2e2.x2 which agrees with the previous result, identifying v=e2, which also guarantees v>0. Excercise IIIB1.4 Find the Hamiltonian form of the action for y: The constraints are now y2, r2,a n dy
r, in terms of the conjugate rtoy(see excercise IA6.2). Find the gauge transformations in the standard way (see subsection IIIA5). Showhow the above Lagrangian form can be obtained from it, including the gauge transformations. Using instead the corresponding twistor (subsection IIB6) to satisfy y 2=0 ,t h e massless, spinless particle now has a single term for its mechanics action: S=Z d1 4ABCD.zA.zB zCzD  Unlike all other relativistic mechanics actions, all variables have been unied into just z, without the introduction of square roots. Excercise IIIB1.5 Expressing zin terms of andx.as in subsection IIB6, show this action reduces to the previous one. 136 III. LOCAL 2. Gauges Rather than use the equation of motion to eliminate vit's more convenient to use a gauge choice: The gauge v= 1 is called \ane parametrization" of the worldline. Note that the gauge transformation of v,v=. , has no dependence on the coordi- natesxand momenta p, so that choosing the gauge v= 1 avoids any extraneous x orpdependence that could arise from the gauge xing. (The appearance of such de- pendence will be discussed in later chapters.) Since T=R d v, the intrinsic length, is gauge invariant, that part of vstill remains when the length is nite, but it can be incorporated into the limits of integration: The gauge v= 1 is maintained by. =0 , and this constant can be used to gauge one limit of integration to zero, completely xing the gauge (i.e., the choice of ). We then integrateRT 0,w h e r eT0( s i n c e originallyv>0), andTis a variable to vary in the action. The gauge-xed action is then SH;GF =ZT 0d[.xmpm+1 2(p2+m2)] In the massive case, we can instead choose the gauge v=1=m; then the equations of motion imply that is the proper time. The Hamiltonian p2=2m+ constant then resembles the nonrelativistic one. Another useful gauge is the \lightcone gauge" =x+ p+ which, unlike the Poincar e covariant gauge v=1 , x e scompletely; since the gauge variation(x+=p+)=,w em u s ts e t = 0 to maintain the gauge. Also, the gauge transformation is again xandpindependent. In lightcone gauges we always assume p+6= 0, since we often divide by it. This is usually not too dangerous an assumption, since we can treat p+= 0 as a limiting case (in D >2). We saw from our study of constrained systems that, for every degree of freedom we can gauge away, the conjugate variable can be xed by the constraint that generates that gauge invariance: In the case where the constraint is p, the gauge transformation isq=, so we gauge q= 0 and use the constraint p= 0. In lightcone gauges the constraints are almost linear: The gauge condition is x+=p+and the constraint is p=:::, so the Lagrange multiplier vis varied to determine p. On the other hand, varyingpgives p)v=1 so this gauge is a special case of the gauge v= 1. An important point is that we used only \auxiliary" equations of motion: those not involving time derivatives. (A slight B. PARTICLES 137 trick involves the factor of p+: This is a constant by the equations of motion, so we can ignore.p+terms. However, technically we should not use that equation of motion; instead, we can redene x!x+:::, which will generate terms to cancel any.p+ terms.) The net result of gauge xing and the auxiliary equation on the action is SH;GF =Z1 1d[.xp+.xipi+1 2(pi2+m2)] wherexa=(x+;x;xi), etc. In particular, since we have xed one more gauge degree of freedom (corresponding to constant ), we have also eliminated one more constraint variable (T, the constant part of v). This is one of the main advantages of lightcone gauges: They are \unitary", eliminating all unphysical degrees of freedom. Excercise IIIB2.1 Another obvious gauge is =x0, which works as well as the lightcone gauge as far as eliminating worldline coordinate invariance is concerned. (The sameis true for=n
xfor any constant vector n.) Unfortunately, the same is not true for the auxiliary equations of motion: After using the gauge condition, p 0appears without time derivatives, so it and vcan be eliminated by their equations of motion. Show this gauge is consistent only for p0>0. The resulting square root is awkward except in the nonrelativistic limit: Take it, and compare with the usual nonrelativistic mechanics. 3. Coupling One way to introduce external elds into the mechanics action is by considering the most general Lagrangian quadratic in derivatives: SL=Z d[1 2v1gmn(x).xm.xn+Am(x).xm+v(x)] I nt h ef r e ec a s ew eh a v ec o n s t a n t e l d s gmn=mn,Am=0 ,a n d=1 2m2.T h ev dependence has been assigned consistent with worldline coordinate invariance. The curved-space metric tensor gmndescribes gravity, the D-vector potential Amdescribes electromagnetism, and is a scalar eld that can be used to introduce mass by interaction. Excercise IIIB3.1 Use the method of the problem IIIB1.3 to write the nonrelativistic action for a spinning particle in terms of a 3-vector (or (D 1)-vector)Xi(;)a n d the fermionic derivative D. Find the coupling to a magnetic eld, in terms of the 3-vector potential Ai(X). Integrate the Lagrangian over . Show 138 III. LOCAL that the quantum mechanical square of i[pi+Ai(x)] is proportional to the Hamiltonian. Excercise IIIB3.2 Derive the Lorentz force law by varying the Lagrangian form of the action forthe relativistic particle, in an external electromagnetic eld (but at metric),with respect to x. This action also has very simple transformation properties under D-dimensional gauge transformations on the external elds: g mn=p@pgmn+gp(m@n)p; Am=p@pAm+Ap@mp@m;  =p@p )SL[x]+SL[x]=SL[x+](xf)+(xi) where we have integrated the actionRf idand setx(i)=xi,x(f)=xf.T h e s e transformations have a very natural interpretation in the quantum theory, where Z Dx eiS=hxfjxii Then thetransformation of Ais canceled by the U(1) (phase) transformation 0(x)=ei(x) (x) in the inner product h fj ii=Z dxfdxih fjxfihxfjxiihxij ii=Z dxfdxi f*(xf)hxfjxii i(xi) while thetransformation associated with gmnis canceled by the D-dimensional coordinate transformation 0(x)= (x+) 4. Conservation There are two types of conservation laws generally found in physics: In mechanics we usually have global conservation laws, of the form. Q= 0, associated with a symmetry of the Hamiltonian Hgenerated by a conserved quantity Q: 0=H=i[Q;H ]=. Q On the other hand, in eld theory we have local conservation laws, since the action for a eld is written as an integralRdDxof a Lagrangian density that depends only B. PARTICLES 139 on elds at x, and a nite number of their derivatives. The local conservation law implies a global one, since @mJm=0) 0=ZdDx (2)D=2@mJmd dtZdD1x (2)D=2J0=. Q=0 where we have integrated over a volume whose boundaries in space are at inn- ity (whereJvanishes), and whose boundaries in time are innitesimally separated. Equivalently, the global symmetry is a special case of the local one. A simple way to derive the local conservation laws is by coupling gauge elds: We couple the electromagnetic eld Amto arbitrary charged matter elds and demand gauge invariance of the matter part of the action, the matter-free part of the action being separately invariant. We then have 0=SM=Z dx (Am)SM Am+()SM  using just the denition of the functional derivative =. Applying the matter eld equationsSM== 0, integration by parts, and the gauge transformation Am= @m, we nd 0=Z dx @mSM Am )Jm=SM Am;@mJm=0 Similar remarks apply to gravity, but only if we evaluate the \current", in this case the energy-momentum tensor, in at space gmn=mn, since gravity is self-interacting. We then nd Tmn=2SM gmn gmn=mn;@mTmn=0 where the normalization factor of 2 will be found later for consistency with the particle. In this case the corresponding \charge" is the D-momentum: Pm=ZdD1x (2)D=2T0m In particular we see that the condition for the energy in any region of space to be nonnegative is T000 To apply this to the action for the particle in external elds, we must rst dis- tinguish the particle coordinates X() from coordinates xfor all of spacetime: The particle exists only at x=X()f o rs o m e , but the elds exist at all x.I n t h i s notation we can write the mechanics action as SL=Z dx gmn(x)Z d (xX)1 2v1. Xm. Xn +Am(x)Z d (xX). Xm+(x)Z d (xX)v 140 III. LOCAL usingR dx (xX()) = 1. We then have Jm(x)=Z d (xX). Xm Tmn=Z d (xX)v1. Xm. Xn Note thatT000( s i n c ev>0). Integrating to nd the charge and momentum: Q=Z d (x0X0). X0=Z dX0(. X0)(x0X0)=(p0) Pm=Z d (x0X0)v1. X0. Xm=Z dX0(. X0)(x0X0)v1. Xm=(p0)pm w h e r ew eh a v eu s e d p=v1. X(for the free particle), where pis the momentum conjugate to X, not to be confused with P. The factor of (p0)((u)=u=jujis the sign ofu) comes from the Jacobian from changing integration variables from toX0. The result is that our naive expectations for the momentum and charge of the particle can dier from the correct result by a sign. In particular p0,w h i c hs e m i - classically is identied with the angular frequency of the corresponding wave, can be either positive or negative, while the true energy P0=jp0jis always positive, as physically required. (Otherwise all states could decay into lower-energy ones: There would be no lowest-energy state, the \vacuum".) When p0is negative, the charge Q anddX0=dare also negative. In the massive case, we also have dX0=dsnegative. This means that as the proper time sincreases,X0decreases. Since the proper time is the time as measured in the rest frame of the particle, this means that the particle is traveling backward in time: Its clock changes in the direction opposite to that of the coordinate system xm. Particles traveling backward in time are called \antiparti- cles", and have charges opposite to their corresponding particles. They have positive true energy, but the \energy" p0conjugate to the time is negative. Excercise IIIB4.1 Compare these expressions for the current and energy-momentum tensor to those from the semiclassical expansion in excercise IIIA4.1. (Include the in- verse metric to dene the square of @mS+qAmthere.) B. PARTICLES 141 5. Pair creation Free particles travel in straight lines. Nonrelativistically, external elds can alter the motion of a particle to the extent of changing the signs of spatial components of the momentum. Relativistically, we might then expect that interactions could also change the sign of the energy, or at least the canonical energy p0. As an extreme case, consider a worldline that is a closed loop: We can pick as an angular coordinate around the loop. As increases,X0will either increase or decrease. For example, a circle in the x0-x1plane will be viewed by the particle as repeating its history after some nite , moving forward with respect to time x0until reaching a latest time tf, and then backward until some earliest time ti. On the other hand, from the point of view of an observer at rest with respect to the xmcoordinate system, there are no particles until x0=ti, at which time both a particle and an antiparticle appear at the same position in space, move away from each other, and then come back togetherand disappear. This process is known as \pair creation and annihilation". t tf ix x0 1 t Whether such a process can actually occur is determined by solving the equations of motion. A simple example is a particle in the presence of only a static electriceld, produced by the time component A 0of the potential. We consider the case of a piecewise constant potential, vanishing outside a certain region and constant inside. Then the electric eld vanishes except at the boundaries, so the particle travels in straight lines except at the boundaries. For simplicity we reduce the problem to two dimensions: A0=Vf o r 0x1L; 0otherwise for some constant V. The action is, in Hamiltonian form, SH=Z df.xmpm+v1 2[(p+A)2+m2]g and the equations of motion are .pm=v(p+A)n@mAn)p0=E (p+A)2=m2)p1=p (E+A0)2m2 v1.x=p+A)v1.x1=p1;v1.x0=E+A0 142 III. LOCAL whereEis a constant (the canonical energy at x1=1)a n dt h ee q u a t i o n.p1=::: is redundant because of gauge invariance. We assume E> 0, so initially we have a particle and not an antiparticle. We look only at the cases where the worldline begins at x0=x1=1 (lower left) and continues toward the right till it reaches x0=x1=+1(upper right), so thatp1=v1.x1>0 everywhere (no re ection). However, the worldline might bend backward in time (.x0<0) inside the potential: To the outside viewer, this looks like pair creation at the right edge before the rst particle reaches the left edge; the antiparticle then annihilates the original particle when it reaches the left edge, whilethe new particle continues on to the right. From the particle's point of view, it has simply traveled backward in time so that it exits the right of the potential before it enters the left, but it is the same particle that travels out the right as came in theleft. The velocity of the particle outside and inside the potential is dx 1 dx0=8 >>>< >>>:p E2m2 Eoutside p (EV)2m2 EVinside From the sign of the velocity we then see that we have normal transmission (no antiparticles) for E>m +VandE>m , and pair creation/annihilation when Vm>E>m)V> 2m The true \kinetic" energy of the antiparticle (which appears only inside the potential) is then(EV)>m. Excercise IIIB5.1 This solution might seem to violate causality. However, in mechanics as well as eld theory, causality is related to boundary conditions at innite times. Describe another solution to the equations of motion that would be inter- preted by an outside observer as pair creation without any initial particles : What happens ultimately to the particle and antiparticle? What are the al-lowed values of their kinetic energies (maximum and minimum)? Since many such pairs can be created by the potential alone, it can be accidental (and not acausal) that an external particle meets up with such an antiparticle. Notethat the generator of the potential, to maintain its value, continuously loses energy (and charge) by emitting these particles. REFERENCES 1 A. Barducci, R. Casalbuoni, and L. Lusanna, Nuo. Cim. 35A (1976) 377; L. Brink, S. Deser, B. Zumino, P. DiVecchia, and P. Howe, Phys. Lett. 64B (1976) 435; B. PARTICLES 143 P.A. Collins and R.W. Tucker, Nucl. Phys. B121 (1977) 307: world-line metric; classical mechanics for relativistic spinors. 2R. Marnelius, Phys. Rev. D20 (1979) 2091; W. Siegel, Int. J. Mod. Phys. A 3(1988) 2713: conformal invariance in classical mechanics actions. 3W. Siegel, hep-th/9412011, Phys. Rev. D52 (1995) 1042: twistor formulation of classical mechanics. 4R.P. Feynman, Phys. Rev. 74(1948) 939: classical pair creation. 144 III. LOCAL ::::::::::::::::::::::::: ::::::::::::::::::::::::: ::::::::::::::::::::::::: C. YANG-MILLS ::::::::::::::::::::::::: The concept of a \covariant derivative" allows the straightforward generalization of electromagnetism to a self-interacting theory, once U(1) has been generalized to a nonabelian group. Yang-Mills theory is an essential part of the Standard Model. 1. Nonabelian The group U(1) of electromagnetism is Abelian: Group elements commute, which makes group multiplication equivalent to multiplication of real numbers, or addition if we write U=eiG. The linearity of this addition is directly related to the linearity of the eld equations for electromagnetism without matter. On the other hand, the nonlinearity of nonabelian groups causes the corresponding particles to interact with themselves: Photons are neutral, but \gluons" have charge and \gravitons" haveweight. In coupling electromagnetism to the particle, the relation of the canonical mo- mentum to the velocity is modied: Classically, the covariant momentum is dx=d = p+qAfor a particle of charge q(e.g.,q= 1 for the proton). Quantum mechanically, the net eect is that the wave equation is modied by the replacement @!r =@+iqA which accounts for all dependence on A(\minimal coupling"). This \covariant deriva- tive" has a fundamental role in the formulation of gauge theories, including gravity.Its main purpose is to preserve gauge invariance of the action that gives the wave equation, which would otherwise be spoiled by derivatives acting on the coordinate- dependent gauge parameters: In electromagnetism, 0=eiq ;A0=A@) (r )0=eiq(r ) or more simply r0=eiqreiq (More generally, qis some Hermitian matrix when is a reducible representation of U(1).) Yang-Mills theory then can be obtained as a straightforward generalization of elec- tromagnetism, the only dierence being that the gauge transformation, and thereforethe covariant derivative, now depends on the generators of some nonabelian group. We begin with the hermitian generators [G i;Gj]=ifijkGk;Giy=Gi C. YANG-MILLS 145 and exponentiate linear combinations of them to obtain the unitary group elements g=ei; =iGi;i*=i)gy=g1 We then can dene representations of the group (see subsection IB1) 0=ei ; y0= yei;(Gi )A=(Gi)AB B For compact groups charge is quantized: For example, for SU(2) the spin (or, for internal symmetry, \isospin") is integral or half-integral. On the other hand, with Abelian groups the charge can take continuous values: For example, in principle the proton might decay into a particle of charge and another of charge 1 .T h e experimental fact that charge is quantized suggests already semiclassically that all interactions should be descibed by (semi)simple groups. Ifis coordinate dependent (a local, or \gauge" transformation), the ordinary partial derivative spoils gauge covariance, so we introduce the covariant derivative ra=@a+iAa;Aa=AaiGi Thus, the covariant derivative acts on matter in a way similar to the innitesimal gauge transformation,  A=iiGiAB B;ra A=@a A+iAaiGiAB B Gauge covariance is preserved by demanding it have a covariant transformation law r0=eirei)A=[r;]=@i[A;] The gauge covariance of the eld strength follows from dening it in a manifestly covariant way: [ra;rb]=iFab)F0=eiFei;Fab=FabiGi=@[aAb]+i[Aa;Ab] )Fabi=@[aAb]i+AajAbkfjki The Jacobi identity for the covariant derivative is the Bianchi identity for the eld strength: 0=[r[a;[rb;rc]]] =i[r[a;Fbc]] (If we choose instead to use antihermitian generators, all the explicit i's go away; however, with hermitian generators the i's will cancel with those from the derivatives when we Fourier transform for purposes of quantization.) Since the adjoint represen- tation can be treated as either matrices or vectors (see subsection IB2), the covariant 146 III. LOCAL derivative on it can be written as either a commutator or multiplication: For example, we may write either or [ r;F]o rrF, depending on the context. Actions then can be constructed in a manifestly covariant way: For matter, we take a Lagrangian LM;0(@; ) that is invariant under global (constant) group trans- formations, and couple to Yang-Mills as LM;0(@; )!LM;A=LM;0(r; ) (This is the analog of minimal coupling in electrodynamics.) The representation we use forGiinra=@a+iAi aGiis determined by how represents the group. (For an Abelian group factor U(1), Gis just the charge q, in multiples of the gfor that factor.) For example, the Lagrangian for a massless scalar is simply L0=1 2(ra)y(ra) (normalized for a complex representation). For the part of the action describing Yang-Mills itself we take (in analogy to the U(1) case) LA(Ai a)=1 8g2 AFiabFj abij whereijis the Cartan metric (see subsection IB2). This way of writing the action is independent of our choice of normalization of the structure constants, and so gives one unambiguous denition for the normalization of the coupling constant g.( I t i s invariant under any simultaneous redenition of the elds and the generators that leaves the covariant derivative invariant.) Generally, for simple groups we can choose to (ortho)normalize the generators Giwith the condition ij=cAij for some constant cA; for groups that are products of simple groups (semisimple), we might choose dierent normalization factors (but, of course, also dierent g's) for each simple group. For Abelian groups (U(1) factors) ij= 0, but then the gauge eld has no self-interactions, so the normalization of the coupling constant is dened only by matter terms in the action, and we can replace ijwithijin the above. Usually it will prove more convenient to use matrix notation: Choosing some convenient representation RofGi(not necessarily the adjoint), we write LA(Ai a)=1 8g2 RtrRFabFab The normalization of the trace is determined by R, and thus so is the normalization convention for the coupling constant; a change in the representation used in the action C. YANG-MILLS 147 can also be absorbed by a redenition of the coupling. For example, comparing the dening and adjoint representations of SU(N) (see subsection IB2), LA=1 8g2 DtrDFabFab=1 8g2 AtrAFabFab)g2 A=2Ng2 D In general, we specify our normalization of the structure constants by xing cRfor someR, and our normalization of the coupling constant by specifying the choice of representation used in the trace (or use explicit adjoint indices). As a rule, we nd the most convenient choices of normalization are cD=1;g =gD (see subsection IB5). Excercise IIIC1.1 Write the action for SU(N) Yang-Mills coupled to a massless (2-component) spinor in the dening representation. Make all (internal and Lorentz) indicesexplicit (no \ tr", etc.), and use dening (N-component) indices on the Yang- Mills eld. We have chosen a normalization where the Yang-Mills coupling constant gappears only as an overall factor multiplying the F 2term (and similarly for the electromagnetic coupling, as discussed in previous chapters). An alternative is to rescale A!gAand F!gFeverywhere; then r=@+igAandF=@A+ig[A;A], and theF2term has no extra factor. This allows the Yang-Mills coupling to be treated similarly to othercouplings, which are usually not written multiplying kinetic terms (unless analogies to Yang-Mills are being drawn), since (almost) only for Yang-Mills is there a nonlinear symmetry relating kinetic and interaction terms. Current conservation works a bit dierently in the nonabelian case: Applying the same argument as in subsection IIIB4, but taking into account the modied (innitesimal) gauge transformation law, we nd J m=SM Am;rmJm=0 Since@mJm6= 0, there is no corresponding covariant conserved charge. Excercise IIIC1.2 Let's look at the eld equations: aUsing properties of the trace, show the entire covariant derivative can be integrated by parts as Z dx tr (A[r;B]) =Z dx tr ([r;A]B);Z dx yr=Z dx(r )y 148 III. LOCAL for matricesA;Band column vectors ;. bShow Fab=r[aAb] cUsing the denition of the current as for electromagnetism (subsection IIIB4), derive the eld equations with arbitrary matter, 1 g21 2rbFba=Ja dShow that gauge invariance of the action SAimplies ra(rbFba)=0 Also show this is true directly, using the Jacobi identity, but not the eld equations. (Hint: Write the covariant derivatives as commutators.) eExpand the left-hand side in the eld, as 1 g21 2rbFba=1 g21 2@b@[bAa]ja wherejcontains the quadratic and higher-order terms. Show the noncovari- antcurrent Ja=Ja+ja is conserved. The jterm can be considered the gluon contribution to the current: Unlike photons, gluons are charged. Although the current is gaugedependent, and thus physically meaningless, the corresponding charge can be gauge independent under situations where the boundary conditions are suitable. 2. Lightcone Since gauge parameters are always of the same form as the gauge eld, but with one less vector index, an obvious type of gauge choice (at least from the point of viewof counting components) is to require the gauge eld to vanish when one vector index is xed to a certain value. Explicitly, in terms of the covariant derivative we set n
r=n
@)n
A=0 for some constant vector n a. We then can distinguish three types of \axial gauges": (1) \Arnowitt-Fickler", or spacelike ( n2>0), (2) \lightcone", or lightlike ( n2=0 ) , and \temporal", or timelike ( n2<0). By appropriate choice of reference frame, and C. YANG-MILLS 149 with the usual notation, we can write these gauge conditions as r1=@1,r+=@+, andr0=@0. One way to apply this gauge in the action is to keep the same set of elds, but have explicit ndependence. A much simpler choice is to use a gauge choice such as A0= 0 simply to eliminate A0explicitly from the action. For example, for Yang-Mills we nd A0=0)F0i=. Ai)1 8(Fab)2=1 4(. Ai)2+1 8(Fij)2 where \. " here refers to the time derivative. Canonical quantization is simple in this gauge, because we have the canonical time-derivative term. However, the gaugecondition can't be imposed everywhere, as seen for the corresponding gauge for the one-dimensional metric in subsection IIIB2, and in our general discussion in subsec- tion IIIA5: Here we can generalize the time-ordered integral for the temporal gauge to an integral path-ordered with respect to a straight-line path in the ndirection: e kn
r(x)=ei(x;xkn)ekn
@;ei(x;xkn)=P exp iZx xkndx0
A(x0) Applying this gauge transformation to n
r, as in subsection IIIA5, xes n
Ato a constant with respect to n
@; the eect on all of ris: r0(x)=ei(x;xkn)r(x)ei(x;xkn),r0(x+kn)=ekn
r(x)r(x)ekn
r(x) For example, for the temporal gauge, if we choose \ x" to be on the initial hypersurface x0=t0,t h e nw ec a nc h o o s e k=tt0so thatr0is evaluated at arbitrary time t: r0 a(x)=[ (ekr0(x)ra(x)ekr0(x))jx0=t0]jk=x0 By Taylor expanding in k, this gives an explicit expression for Aaat all times in terms ofAa,a n dFaband its covariant time-derivatives, evaluated at some initial time, but with simply A0(t;xi)=A0(t0;xi). Thus, we still need to impose the A0eld equation [ri;. Ai] = 0 as a constraint at some initial time. Excercise IIIC2.1 Show explicitly that the eld equations for Aifollowing from the action for Yang-Mills in the gauge A0= 0 (see excercise IIIC1.2 for J=0 )i m p l yt h a t thetime derivative of the constraint [ ri;. Ai]=0v a n i s h e s . In the case of the lightcone gauge we can carry this analysis one step further. In subsection IIB3 we saw that lightcone formalisms are described by massless eldswith (D2)-dimensional (\transverse") indices. In the present analysis, gauge xing alone gives us, again for the example of pure Yang-M ills, A +=0)F+i=@+Ai;F+=@+A;Fi=@Ai[ri;A] 150 III. LOCAL )1 8(Fab)2=1 4(@+A)21 2(@+Ai)(@Ai[ri;A]) +1 8(Fij)2 In the lightcone formalism @(@+) is to be treated as a time derivative, while @+can be freely inverted (i.e., modes propagate to innity in the x+direction, but boundary conditions set them to vanish in the xdirection). Thus, we can treat A as an auxiliary eld. The solution to its eld equation is A=1 @+2[ri;@+Ai] which can be substituted directly into the action: 1 8(Fab)2=1 2Ai@+@Ai+1 8(Fij)21 4[ri;@+Ai]1 @+2[rj;@+Aj] =1 4AiAi+i1 2[Ai;Aj]@iAj+i1 2(@iAi)1 @+[Aj;@+Aj] 1 8[Ai;Aj]2+1 4[Ai;@+Ai]1 @+2[Aj;@+Aj] We can save a couple of steps in this derivation by noting that elimination of any auxiliary eld, appearing quadratically (as in going from Hamiltonian to Lagrangian formalisms), has the eect L=1 2ax2+bx+c!1 2ax2j@L=@x =0+Ljx=0 In this case, the quadratic term is ( F+)2, and we have 1 8(Fab)2=1 8(Fij)21 2F+iFi1 4(F+)2!1 8(Fij)21 2(@+Ai)(@Ai)+1 4(F+)2 where the last term is evaluated at 0=[ra;F+a]=@+F++[ri;F+i])F+=1 @+[ri;F+i] )L=1 8(Fij)2+1 2Ai@+@Ai1 4[ri;@+Ai]1 @+2[rj;@+Aj] as above. In this case, canonical quantization is even simpler, since interpreting @as the time derivative makes the action look like that for a nonrelativistic eld theory, witha kinetic term linear in time derivatives (as well as interactions without them). The free part of the eld equation is also simpler, since the kinetic operator is now just . (This is true in general in lightcone formalisms from the analysis of free theories in chapter XII.) In general, lightcone gauges are the simplest for analyzing physical degrees of freedom (within perturbation theory), since the maximum number of de- grees of freedom is eliminated, and thus kinetic operators look like those of scalars. C. YANG-MILLS 151 On the other hand, interaction terms are more complicated because of the nonlo- cal Coulomb-like terms involving 1 =@+: The inverse of a derivative is an integral. (However, in practice we often work in momentum space, where 1 =p+is local, but Fourier transformation itself introduces multiple integrals.) This makes lightconegauges useful for discussing unitarity (they are \unitary gauges"), but inconvenientfor explicit calculations. However, in subsection VIB6 we'll nd a slight modication of the lightcone that makes it the most convenient method for certain calculations. (In the literature, \lightcone gauge" is sometimes used to refer to an axial gaugewhereA +is set to vanish but Ais not eliminated, and D-vector notation is still used, so unitarity is not manifest. Here we always eliminate both components andexplicitly use ( D2)-vectors, which has distinct technical advantages.) Although spin 1/2 has no gauge invariance, the second step of the lightcone formalism, eliminating auxiliary elds, can also be applied there: For example, for amassless spinor in D=4, identifying @ . =@as the lightcone \time" derivative, we vary  . (or ) as the auxiliary eld: iL= . @.  + . @ .  . @ .   . @. ) =1 @. @ .   )L= . 1 2 i@.   This tells us that a 4D massless spinor, like a 4D massless vector (or a complex scalar) has only 1 complex (2 real) degree of freedom, describing a particle of helicity +1/2and its antiparticle of helicity 1/2 (1 for the vector, 0 for the scalar), in agreement with our general discussion of helicity in subsection IIB7. On the other hand, in the massive case we can always go to a rest frame, so the analysis is in terms of spin(SU(2) for D=4) rather than helicity. For a massive Weyl spinor we can perform thesame analysis as above, with the modications L!L+im p 2(  + .  . ))L= . 1 2(m2) i@.   where we have dropped some terms that vanish upon using integration by parts and the antisymmetry of the fermions. So now we have the two states of an SU(2) spinor,but these are identied with their antiparticles. This diers from the vector: While for the spinor we have 2 states of a given energy for both the massless and massive cases, for a vector we have 2 for the massless but 3 for the massive, since for SU(2)spin s has 2s+1 states. 152 III. LOCAL Excercise IIIC2.2 Show that integration by parts for 1 =@gives just a sign change, just as for @. In general dimensions, massless particles are representations of the \little group" SO(D2) (the helicity SO(2) in D=4), as described in subsection IIB3. Massive particles represent the little group SO(D 1), corresponding to dimensional reduction from an extra dimension, as described in subsection IIB4. 3. Plane waves The simplest nontrivial solutions to nonabelian eld equations are the general- izations of the plane wave solutions of the free theory. We begin with general, free,massless theories, as analyzed in subsection IIB3. In the lightcone frame only p +is nonvanishing. In position space this means the eld strength depends only on x. This describes a wave traveling at the speed of light in the positive x1direction, with no other spatial dependence (i.e., a plane wave). We allow arbitrary dependence on x, corresponding to a superposition of waves with parallel momenta (but dierent values ofp+). While its dependence on only xsolves the Klein-Gordon equation, Maxwell's equations are solved by giving the eld strength as many upper + indices as possible, and no upper 's. Generalizing to interactions, we notice that the Yang-Mills eld equations and Bianchi identities dier from Maxwell's equations only by the covariantization of the derivatives (at least for pure Yang-Mills). Because Maxwell's equations were satised by just restricting the index structure, we can do the same for the covariant derivativesby assuming that only r +is novanishing on the eld strengths. In other words, we can solve the eld equations and Bianchi identities by choosing the only nontrivial components of the gauge elds to be those in r+. The nal step is to solve the relation between covariant derivative and eld strength. This is simple because the index structure we found implies the only non- trvial commutators are [@i;r+]=iFi+; [@;r+]=0 In particular, this implies that the gauge elds have no x+dependence, and only a very simple dependence on xi. We nd directly A+=xiFi+(x) whereFi+(x) is unrestricted (other than the explicit index structure and coordinate dependence). Of course, this result can also be used in the free theory, although it diers from the usual lightcone gauge. C. YANG-MILLS 153 4. Self-duality The simplest and most important solutions to the eld equations are those that are invariant under the \duality" symmetry that relates electric and magnetic charge: [ra;rb]=1 2abcd[rc;rd] Applying the self-duality condition twice, we nd abefefcd=+c [ad b] which requires an even number of time dimensions. For example, since the action is usually Wick rotated anyway for perturbative purposes, we might assume thatwe should do the same for classical solutions that are not considered as \small" uctations about the usual vacuum. (Such a Euclidean denition of eld theory has been considered for a mathematically rigorous formalism, called \constructive quantum eld theory", since the Gaussian path integrals for scalars and vectors arethen well-dened and convergent. However, other spins, such as for fermions orgravity, are a problem in this approach.) Alternatively, we can replace  abcdwith iabcdand complexify our elds. The self-duality condition, when combined with the Bianchi identities, implies the eld equations: For Yang-Mills, r[aFbc]=0) 0=1 2abcdraFbc=1 4abcdrabcefFef=raFad Since the self-duality condition is only rst-order in derivatives, it's easier to solve than the usual eld equations. Plane wave solutions provide a simple example of self-duality, since the eld strengths can easily be written as the sum of self-dual and anti-self-dual parts: In Minkowski space we dene the self-dual part as helicity +1 ( f.. ), and anti-self-dual as1(f). For example, for a wave traveling in the \1" direction, the F+2iF+3 components give the two self-dualities for Yang-Mills, describing helicities 1 (the two circular polarizations). Before further analyzing solutions to the self-duality condition, we consider ac- tions that use self-dual elds directly. This will allow us to describe not only theorieswhose only solutions are self-dual, but also more standard theories as perturbationsabout self-duality, and even massive theories. The most unusual feature of this ap- proach is that complex elds are used without their complex conjugates, since this is implied in D=3+1 by self-duality. (Alternatively, we can Wick rotate to 2+2 di-mensions, where all Lorentz representations are real.) There are two stages to this 154 III. LOCAL approach: (1) Use a rst-order formalism where the auxiliary eld is self-dual. The usual rst-order actions for spin 1/2 (Weyl or Dirac) already can be interpreted inthis way, where \self-duality" means \chirality". (2) For the massive theory, elimi-nate the non-self-dual eld (as an auxiliary eld, as allowed by the mass term), sothat the dynamics is described by the self-dual eld, which was formerly considered as auxiliary. The massless theory then can be treated as a limiting case. The simplest (and perhaps most useful) example is massive spin 1/2 coupled in a real representation to Yang-Mills elds: L= Tir. .+1 2p 2m( T + T. .) where the transposition (\T") refers to the Yang-Mills group index (with respect to which the spinors are column vectors). Note that must be a real representation of this group ( AT=A) for the mass term to be gauge invariant (unless the mass term includes scalars: see the following chapter). Even though and are complex conju- gates, they can be treated independently as far as eld equations are concerned, sincethey are just dierent linear combinations of their real and imaginary parts. (Com-plex conjugation can be treated as just a symmetry, related to unitarity.) Noticingthat the quadratic term for  has no derivatives, we can treat it as an auxiliary eld, and integrate it out (i.e., eliminate it by its equation of motion, which gives an explicit local expression for it): L! p 2 m[1 4 T(m2) +1 2 Tif ] w h e r ew eh a v eu s e dt h ei d e n t i t y r. r. =1 2fr. ;r. g+1 2[r. ;r. ]=1 2 if whose simplicity followed from being a real representation of the Yang-Mills group. (Of course, we could have eliminated instead, but not both.) For convenience we also scale by a constant !21=4pm to nd the nal result L!1 4 T(m2) 1 2 Tif  Now the massless limit can be taken easily. This action resembles that of a scalar, plus a \magnetic-moment coupling", which couples the \(anti-)self-dual" (chiral) spinor to only the (anti-)self-dual part fof the Yang-Mills eld strength. C. YANG-MILLS 155 For the same reason, the kinetic operator can be written in terms of just the self-dual part Sof the spin operator: L=1 4 T(m2ifS) This operator is of the same form found by squaring the Dirac operator: 2r=2=2(
r)2=(f a; bg+[ a; b])rarb=iFabSba except for the self-duality. The simple form of this result again depends on the reality (parity invariance) of the Yang-Mills representation; although this squaring trick can be applied for complex representations (parity violating), the coupling does not simplify. This is related to the fact that real representations are required for our derivation of the self-dual form. In the special case where the real representation is the direct sum of a complex one +with its complex conjugate (as for quarks in the Standard Model, or electrons in electrodynamics), we can rewrite the Lagrangian as Lc=1 2 T +(m2)  T +if  The method can also be generalized to the case of scalar couplings, but the action becomes nonpolynomial. For spin 1, we start with the massless case. We can write the Lagrangian for Yang-Mills as L=tr(Gf1 2g2G2 ) whereGis a (anti-)self-dual auxiliary eld. Although this action is complex, elim- inatingGby its algebraic eld equation gives the usual Yang-Mills action up to a total derivative term ( abcdFabFcd), which can be dropped for purposes of perturbation theory. For g=0 ,t h i si sa na c t i o nw h e r e Gacts as a Lagrange multiplier, enforcing the self-duality of the Yang-Mills eld strength. If we simply add a mass term Lm=1 4(m g)2A2 thenAcan be eliminated by its eld equation, giving a nonpolynomial action of the form L+Lm!1 2(@G)[(m g)2+G]1(@G)1 2g2G2 Just as the spin-1/2 action contained only a 2-component spinor describing the 2 polarizations of spin 1/2, this action contains only the 3-component G, describing the 3 polarizations of (massive) spin 1. 156 III. LOCAL Excercise IIIC4.1 Find the Abelian part of this action. Show the free eld equation is (m2)G= 0 (without gauge xing). 5. Twistors In four dimensions with an even number of time dimensions, the \Lorentz" group factorizes (into SU(2)2for D=4+0 and SL(2)2for D=2+2). This makes self-duality especially simple in spinor notation: For Yang-Mills (cf. electromagnetism in subsec-tion IIA7), [r 0;r 0]=iC f00(f=0 ) where we have written primes instead of dots to emphasize that the two kinds of indices transform independently (instead of as complex conjugates, as in D=3+1). For purposes of analyzing self-duality within perturbation theory, we can use a lightconemethod that breaks only one of the two SL(2)'s (or SU(2)'s), by separating out its indices into theand components: [r 0;r0]=0)r0=@0 where we have chosen a lightcone gauge: The vanishing of all eld strengths for the covariant derivative r0says that it is pure gauge (as seen by ignoring all but the x 0coordinates). We now solve [r[0;r ]0]=0)r 0=@ 0+i@0 i.e.,r 0@ 0has vanishing curl, and is therefore a gradient. We therefore have A0=0;A 0=@0;f00=i@0@0 These can also be written in terms of an arbitrary constant twistor (= above) as A0=@ 0(i );f00=@ 0@0(i ) The nal self-duality condition [ r 0;r 0] = 0 then gives the equation of motion 1 2+(@0)(@ 0)=0 Excercise IIIC5.1 Show that the sign convention for Wick rotation of the Levi-Civita tensor consistent with the above equations is Fab=1 2abcdFcd;F00=Cf00) C. YANG-MILLS 157 00 00=CC C00C0 0CC C00C 00 Excercise IIIC5.2 Look at the action Gffor self-dual Yang-Mills in the lightcone gauge, using the results above. Show that this action is equivalent to the lightcone action for ordinary Yang-Mills (subsection IIIC2), with some terms in the interaction dropped. At least for 4D Yang-Mills, advantage can be taken of the conformal invariance of the classical interacting theory by using a formalism where this invariance is man- ifest. We saw in subsection IA6 that classical mechanics could be made manifestly conformal by use of extra coordinates. Covariant derivatives can be dened in termsof projective lightcone coordinates, but the twistor coordinates z A(see subsections IIB6 and IIIB1) are more useful. The self-dual covariant derivatives then satisfy [rA;rB]=iCfAB in direct analogy to 4D spinor notation. This equation also can be solved by the lightcone method used above, but now this method breaks only the internal SL(2)symmetry, leaving SL(4) conformal symmetry manifest. More general self-dual eld strengths in this twistor space are also of the form f A:::B, totally symmetric in the indices. We also need to impose the constraint on the eld strength zAfAB=0 (and similarly for the more general case) to restrict the range of indices to the usual 4D spinor indices (in which the eld strengths are totally symmetric). Self-dualityimplies the Bianchi identity r [AfB]C=0 which also generalizes to the other eld strengths, and is the equivalent of the usual rst-order dierential equations (Dirac, Maxwell, etc.) satised by 4D eld strengths. As usual, it in turn implies the interacting Klein-Gordon equation, which in theYang-Mills case is 1 2r[ArB]fCD=i[fC[A;fB]D] The Bianchi together with the zindex constraint imply the constraint on coordinate dependence (zArA+ )fBC=0 which eliminates dependence on all but the usual 4D coordinates. These four equa- tions are generically satisied by self-dual eld strengths. The self-duality itself of the 158 III. LOCAL eld strengths is a consequence of their total symmetry in their indices, and the fact that they are all lower (SL(4)) indices. (The zindex constraint then reduces them to SL(2) Weyl indices all of the same chirality.) Excercise IIIC5.3 Derive the last three equations from the previous two (self-duality and zf=0). Excercise IIIC5.4 Show that non-self-dual Yang-Mills is conformally invariant in D=4 by extend-ing the (4+2)-dimensional formalism of subsections IA6 and IIIB1 (especiallyexcercise IIIB1.4): Show the eld strength F ABC=i1 2y[A[rB;rC]] satises the gauge covariances AA=[rA;]yA^ and Bianchi identities y[AFBCD ]=r[AFBCD ]=0 The duality transformation FABC!1 6ABCDEFFDEF then suggests the eld equations yAFABC=rAFABC=0 in addition to the usual constraint y2FABC=0 By reducing to D=4 coordinates with the aid of the above yFconditions, show Freduces to the usual eld strength, and the remaining equations reduce to the usual gauge transformation, Bianchi identity, duality, and eld equation. C. YANG-MILLS 159 6. Instantons Another interesting class of self-dual solutions to Yang-Mills theory are \instan- tons", so called because the eld strength is maximum at points in spacetime, unlikethe plane waves, whose wavefronts propagate from and toward timelike innity. Aparticular subset of these can be expressed in a very simple form by the 't Hooftansatz in terms of a scalar eld: In twistor notation, choosing the Yang-Mills gaugegroup GL(2) (in 2+2 dimensions, or SU(2) GL(1) for 4+0), iA A= @Aln)iAA=@Aln so the GL(1) piece is pure gauge, and has been included just for convenience. Note that this ansatz ties the SL(2) twistor index with the SL(2) gauge group indices ( ;), but in this notation the index that carries the spacetime (conformal) symmetry is free.Imposing the self-duality condition on the eld strength, and separating out the terms symmetric and antisymmetric in AB, we nd if AB=1 2@(A@B)1 1@A@B=0 The \eld equation" for is just the twistor version of the (free) Klein-Gordon equation, and its solution is the projective lightcone version of 4D point sources (seesubsection IA6): Since for any two 6D lightlike vectors yandy 0 y=e(x;1;1 2x2))y
y0=1 2ee0(xx0)2 we have the solution =k+1X i=1(y
yi)1;y2 i=0 withyg i v e ni nt e r m so f zas before, and yiare constant null vectors. \ k"i st h en u m b e r of instantons. (The one term for k= 0 is pure gauge.) The usual singularities in the Klein-Gordon equation at y=yiare killed by the extra factor of 1in the eld equation. Excercise IIIC6.1 Let's check the Klein-Gordon equation for y6=yidirectly in twistor space. We will need the identity y2 i=0)yiA[ByiCD]=0 noX i! 160 III. LOCAL in the product y
yi=1 2yAByiAB Prove this identity in two ways: aShow it follows from the denition y2 i=1 4ABCDyiAByiCD bShow it follows from plugging in the solution to the lightlike condition, yiAB=ziAziB cNow use the identity to show the above solution satises its eld equation by evaluating the zderivatives. We can rewrite this in the usual 4D coordinates by transforming from zAto andx0aszA=( ;x0) (see subsection IIB6): dzAAA=dx0A0+[ (d )1 ]zAAA=dx0A0i1 d  where in the rst step we have used the expression for zin terms of andx,a n di n the second we used the result that (zA@A+ )=0 We now recognize that the gauge transformation that gets rid of all but the \ x components" of A(whose existence is guaranteed by the condition zAfAB=0 )u s e s itself as the gauge parameter: dzAAA=i1 d +1 (dzAA0 A)  The net result is that Acan be reduced to an ordinary 4-dimensional expression by just setting =in the original expression. Then iA0= @0ln;  =X i1 ei(xxi)2 withyin terms of xand a scale factor (worldline metric) eas in subsections IA6 and IIIB1 (and dropping an overall factor that doesn't contribute to A). Note that, unlike the expression in twistor space, where conformal invariance is manifest, here Lorentz invariance is tied to the Yang-Mills symmetry. Excercise IIIC6.2 Show in 4D coordinates that the gauge-invariant quantity tr(f2) is nite at C. YANG-MILLS 161 the pointsx=xi,w h e r eAis singular. (This means that the gauge choice is singular, not physical quantities.) Another important property of instantons is that they give nite contributions to the action. In vector notation, we have Fab=1 2abcdFcd)S=1 8g2trZd4x (2)2FabFab=1 16g2trZd4x (2)2abcdFabFcd The last expression can be reduced to a boundary term, since 1 8tr F [abFcd]=1 6@[aBbcd] in terms of the \Chern-Simons form" Babc=tr(1 2A[a@bAc]+i1 3A[aAbAc]) Excercise IIIC6.3 Although the Chern-Simons form is not manifestly invariant, its variation is, up to a total derivative: aShow that its general variation is Babc=1 2tr[(A[a)Fbc]@[aAbAc]] bShow the gauge transformation of Bis Babc=1 2@[abc];ab=1 2tr(@[aAb]) If we assume boundary conditions such that Fdrops o rapidly at innity, then Amust drop o to pure gauge at innity: iAm!g1@mg Since instantons always deal with an SU(2) subgroup of the gauge group, we'll assume now for simplicity that the whole group is itself SU(2). Then the action can be givena group theory interpretation directly, since the integral over the surface at innity is an integral over the 3-sphere, which covers the group space of SO(3), and thus half the group space of SU(2). Explicitly, S= 1 82g2I d3m1 6mnpqBnpq !1 82g2I d3m1 6mnpqtr(g1@ng)(g1@pg)(g1@qg) 162 III. LOCAL =1 82g2I d3x1 6ijktr(g1@ig)(g1@jg)(g1@kg) where in the last step we have switched to coordinates for the 3-sphere, using the factR d4xmnpqfmnpq is independent of coordinate choice. In fact, in the case where gis a one-to-one map between the 3-sphere and the group SO(3), this last expression isjust the denition of the invariant volume of the SO(3) group space. In that case, the integral gives just the volume of the 3-sphere (2  2). In general, the map gwill cover the SU(2) group space an integer number qof times, and thus cover the SO(3) group space 2qtimes, so the result will be S=jqj 2g2 where we have used the fact that self-dual solutions have q>0 while anti-self-dual haveq<0. (Anti-)self-dual solutions give relative minima of the action with respect to more general eld congurations: 0trZ 1 2(Fab1 2abcdFcd)2=trZ (F21 2abcdFabFcd) )Sjqj 2g2 qis an integer, and thus can't be changed by continuous variations: It is a topological property of nite-action congurations. Thus the self-dual solutions give absolute minima for a given topology. (All these solutions will be given implicitly by twistor construction in the following subsection. Note that our normalization for the structureconstants of SU(2) diers from the usual, since we use eectively tr(G iGj)=ij instead of the more common tr(GiGj)=1 2ij, which would normalize the structure constants as in SO(3): fijk=ijk. The net eect is that our g2contains a relative extra factor of 1/2, in addition to the eective extra factors coming from our dierent normalization of the action.) Excercise IIIC6.4 Explicitly evaluate the integral for the instanton number qfor the solutions of the 't Hooft ansatz. Show that the asymptotic form can be expressed in terms of ( g)=x0.(detg6= 1 because of the GL(1) piece.) Note that there are boundary contributions not only at x=1but also around the singular points x=xi, which are of the same form but opposite sign. (Since the singular parts of Aare pure gauge, they cancel in F.) At the quantum level, instantons are important mostly because they are an ex- ample of elds that don't fall o rapidly at innity, and thus contribute toRFF. C. YANG-MILLS 163 However, once the restriction on boundary conditions is relaxed, there can be many such eld congurations. The instantons are then distinguished by the fact that theyare the minimal action solutions for a given topology; this makes them important fordescribing low-energy behavior. 7. ADHM Much more general solutions of this form can be constructed using twistor meth- ods. (In fact, they can be shown to be the most general self-dual solutions that fallo fast enough at innity in all directions.) The rst step of the Atiyah-Drinfel'd-Hitchin-Manin (ADHM) construction is to introduce a scalar square matrix in a larger group space U II0=(uI;vIi)(I0=(;i)) The indexis the usual two-valued twistor index, for SU(2) in Euclidean space or SL(2) in 2+2 dimensions. The other indices are (H) I(G) i SO(N) SO(N+4k) GL(2k) SU(N) (SL(N)) SU(N+2k) (SL(N+2k)) GL(k,C) USp(2N) (Sp(2N)) USp(2N+2k) (Sp(2N+2k)) GL(k) The index is for the dening representation of the Yang-Mills group H, which is any of the compact classical groups for Euclidean space, but is its real Wick rotationfor 2+2 dimensions. The index Iis for the dening representation of the group G, a larger version of H, where k is the instanton number. Finally, the index iis for a general linear group. We also have the matrix U I I0=(uI ;vI i0) For the SO and (U)Sp cases both Umatrices are real, for the SU case they are complex conjugates of each other, and for the SL case they are real and independent. We next relate the two U's by uI uI= ;uI vIi=vI i0uI=0;vI i0vIi=Cgii0 so they are almost inverses of each other, except that the \metric" gis not constrained to be a Kronecker . We then write the gauge eld as a generalization of pure gauge: iAA=uI @AuI 164 III. LOCAL (This is similar to the method used for nonlinear models of coset spaces G/H as discussed in subsection IVA3 below, except for g.) Self-duality then follows from requiring a certain coordinate dependence of the U's: This is xed by giving the explicit dependence of the v's as vIi=bIiAzA ;vI i0=bI i0AzA  where theb's are constants. The orthonormality conditions on the U's then implies the constraint on the b's bI i0(AbIiB)=0 as well as determining the u's in terms of the b's (with much messier dependence than thev's), and thus A. Note that the zdependence of ucan be written in terms of just x, as follows from rewriting the uvorthogonality as (after multiplying by z) uI bIiAyAB=uIbI i0AyAB=0 and noting scale invariance. Then the xcomponents of Acan also be written in terms of just x. We then can check the self-duality condition by calculating f:T h e orthonormality condition on the U's can be written as J I=uIuJ +vIigii0vJ i0 wheregii0is the inverse of gii0. Then schematically we have iF=@iA+iAiA =(@u)(@u)(@u)uu(@u) =(@u)vgv(@u) =u(@v)g(@v)u =ubgbu or more explicitly ifAB=(uI bIi(A)gii0(uJbJ i0B)) where self-duality is FA;B=CfAB. We can also directly show zAfAB=0 . C. YANG-MILLS 165 8. Monopoles Instantons are essentially 0-dimensional objects, localized near a point in 4- dimensional spacetime (or many points for multi-instanton solutions). Another type of solution is 1-dimensional; this represents a particle (with a 1D worldline). Unlikethe plane-wave solutions, which represent the massless particles already described explicitly by elds in the action, we now look for time-independent solutions, which describe massive (since they have a rest frame), bound-state particles. Looking at time-independent solutions is similar to the dimensional reduction that we considered in subsection IIB4 to introduce masses into free theories, only (1) this mass vanishes, and (2) we reduce the time dimension, not a spatial one. In our case, the dimensional reduction of a 4-vector (the Yang-Mills potential) gives a3-vector and a scalar, both in the adjoint representation of the group. Let's consider the reduction in Euclidean space, so the scalar kinetic term comes out with the right sign. Then the 4D Yang-Mills action reduces as 1 8F2 ab!1 8F2 ij+1 4[ri;]2 where we have labeled the scalar A0=and by dimensional reduction @0!0. Note that this is the same action that would have been obtained by starting out with Yang- Mills coupled to an adjoint scalar in four dimensions, either Minkowski or Euclidean, and choosing the gauge A0= 0. Thus, time-independent solutions to Euclidean Yang-Mills theory are also time-independent solutions to Minkowskian Yang-Mills coupled to an adjoint scalar (although not the most general, since the gauge A0=0 is not generally possible globally, especially when we assume time independence of even gauge-dependent quantities). In particular, this means that time-independent solutions to self-dual Yang-Mills are also solutions of Minkowskian Yang-Mills cou-pled to an adjoint scalar. This allows us to use the rst-order dierential equations and topological properties of self-dual Yang-Mills theory to nd physical bo und-state particles in this vector-scalar theory. Dimensionally reducing the (Euclidean) self-duality condition, we have [r i;]=1 2ijkFjk As for instantons, the simplest solutions are for SU(2). As for the 't Hooft ansatz, we look for a solution that is covariant under the combined SU(2) of the gauge group and 3D rotations: In SO(3) vector notation for both kinds of indices (using the SO(3) normalization of the structure constants [ iGi;iGj]=ijkiGk), i=xi'(r); (Ai)j=ijkxkA(r) 166 III. LOCAL (We know to use an tensor inAbecause of covariance under parity.) The self- duality equation then reduces to two nonlinear rst-order dierential equations (the coecients of ijandxixj=r2): 'r2A'=2A+rA0;r'0+r2A'=rA0+r2A2 After some massaging, we nd the change of variables ~'=1 r+r'; ~A=1 r+rA leads to the simplication ~'0=~A2; ~A0=~A~' ~'then can be eliminated, giving an equation for ~A. Making a nal change of variables, =~A1) 00( 0)2=1 we can guess the solution (with regularity at r=0 ) =k1sinh(kr))A =1 r2kr sinh(kr)1 ;' =1 r2[krcoth (kr)1] Excercise IIIC8.1 Repeat this calculation in spinor notation: aIn Euclidean space we can choose 0 0C0. Show that we can then write the 4-vector potential for the monopole as i(A) = xA+(r)+ x A(r) which is symmetric in neither nor . (Compare the 't Hooft ansatz in subsection IIIC6.) However, xis now symmetric from dropping x0. bImpose self-duality, where @x =1 2 ( )=  1 2CC  from subtracting out the @0x0piece. Derive the resulting equations for A, and show they agree with the above for A=1 2(A') In general, the Lagrangian of a Euclidean theory is the Hamiltonian of the Minkowskian theory (with the sign conventions we introduced in subsection IIIA1), C. YANG-MILLS 167 since Wick rotation changes the sign of the kinetic energy and not the potential en- ergy. In our case, this means the Minkowskian energy of the Yang-Mills + adjoint scalar theory can be evaluated in terms of the same topological expression we used for instantons: From the previous subsection, using S=R dt E and@0=0 ,w ee v a l u a t e in Euclidean space E=1 162g2I d2ii0jkB0jk;i0jkB0jk!ijktr(Fjk)=2tr([ri;]) =@itr(2) where we have used an integration by parts to simplify B. (Compare excercise IIIC6.3a.) Since at spatial innity i!xijkj r1 r2 ;Aij!ijkxk1 r2 and eectivelyH d2i!4rxi, we nd E=jkj 2g2 Also by similar arguments to those used for instantons, we see that any solutions with boundary conditions A!0,jj!jkjasr!1 have energy at least as great as this. There is also a topological interpretation to this energy: Writing it as E=1 162g2I d2iijktr(hiFjk) we see that the energy is proportional to the magnetic ux, i.e., the \magnetic charge" of the monopole. (The asymptotic value hiofpicks out a direction in isospace, reducing SU(2) to U(1).) As in electromagnetism, magnetic charge is quantized interms of electric charge. However, for compact gauge groups, electric charge is also quantized. (For the usual U(1), charges are arbitrary, but for SU(2), any component of the isospin is quantized.) The energy is thus quantized in terms of k:I t i s a multiple of the energy we found for the single monopole above. Excercise IIIC8.2 Perform a singular gauge transformation that makes hipoint in a constant (rather than radial) direction in isospin (SU(2)) space. Show that the isospin component of the asymptotic form of Adescribes a U(1) magnetic monopole: magnetic ux radiating outward from the origin. REFERENCES 1. O. Klein, On the theory of charged elds, in New theories in physics , proc. Warsaw, May 30 - June 3, 1938 (International Institute of Intellectual Co-operation, Paris, 1939)p. 77; 168 III. LOCAL W. Pauli, unpublished ( 1953): early work on \Yang-Mills" theory. 2C.N. Yang and R.L. Mills, Phys. Rev. 96(1954) 191; R. Shaw, The problem of particle types and other contributions to the theory of elemen- tary particles , Cambridge University Ph.D. thesis (1955); R. Utiyama, Phys. Rev. 101(1956) 1597: complete formulation of the (classical) theory. 3L. O'Raifeartaigh, The dawning of gauge theory (Princeton University, 1997): early history of Yang-Mills theory, including reprints and previously unpublished ma- terial. 4R.L. Arnowitt and S.I. Fickler, Phys. Rev. 127(1962) 1821; W. Kummer, Acta Phys. Austriaca 14(1961) 149. 5S. Weinberg, Phys. Rev. 150(1966) 1313; J.B. Kogut and D.E. Soper, Phys. Rev. D1(1970) 2901: lightcone eld theory. 6S. Coleman, Phys. Lett. 70B (1977) 59: Yang-Mills plane waves. 7R. Feynman and M. Gell-Mann, Phys. Rev. 109(1958) 193; L.M. Brown, Phys. Rev. 111(1958) 957; M. Tonin, Nuo. Cim. 14(1959) 1108; G. Chalmers and W. Siegel, hep-ph/9708251, Phys. Rev. D59 (1999) 045012; M. Veltman, hep-th/9712216, Acta Phys. Polon. B 29(1998) 783: description of spin 1/2 with only undotted spinors. 8A. Ashtekar, Phys. Rev. Lett. 57(1986) 2244, Phys. Rev. D36 (1987) 1587; T. Jacobson and L. Smolin, Phys. Lett. 196B (1987) 39, Class. Quant. Grav. 5(1988) 583; J. Samuel, Pramana 28(1987) L429: rst-order actions with self-dual auxiliary elds. 9C.N. Yang, Phys. Rev. Lett. 38(1977) 1377: reduction of self-dual Yang-Mills eld to single component. 10A.N. Leznov, Theor. Math. Phys. 73(1988) 1233, A.N. Leznov and M.A. Mukhtarov, J. Math. Phys. 28(1987) 2574; A. Parkes, hep-th/9203074, Phys. Lett. 286B (1992) 265: lightcone gauge for self-dual Yang-Mills. 11A.A Belavin, A.M. Polyakov, A.S. Shvarts, and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85:instantons. 12G. 't Hooft, unpublished;R. Jackiw, C. Nohl, and C. Rebbi, Phys. Rev. D15 (1977) 1642: 't Hooft ansatz for multi-instanton solutions. 13S.-S. Chern and J. Simons, Ann. Math. 99(1974) 48. 14M.F. Atiyah and R.S. Ward, Comm. Math. Phys. 55(1977) 117; M.F. Atiyah, V.G. Drinfel'd, N.J. Hitchin, and Yu.I. Manin, Phys. Lett. 65A (1978) 185; E. Corrigan, D. Fairlie, P. Goddard, and S. Templeton, Nucl. Phys. B140 (1978) 31; N.H. Christ, E.J. Weinberg, and N.K. Stanton, Phys. Rev. D18 (1978) 2013; M.F. Atiyah, Geometry of Yang-Mills elds (Scuola Normale Superiore, Pisa, 1979); C. YANG-MILLS 169 V.E. Korepin and S.L. Shatashvili, Math. USSR Izvestiya 24(1985) 307: general multi-instanton construction, using twistors. 15G. 't Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, JETP Lett. 20(1974) 194: monopoles in nonabelian theories. 16E.B. Bogomol'nyi, Sov. J. Nucl. Phys. 24(1976) 449; M.K. Prasad and C.M. Sommerfeld, Phys. Rev. Lett. 35(1975) 760: exact solution for monopole. 17W. Nahm, Phys. Lett. 90B (1980) 413: general monopole construction based on ADHM instanton construction. 170 IV. MIXED IV. MIXED In this chapter we consider ways in which gauge symmetry combines with global symmetries for new eects. The interplay between global internal symmetries of scalar and spinor theories and local symmetries of Yang-Mills is important for understanding mass generation for all spins, and is fundamental for the Standard Model. :::::::::::::::::: :::::::::::::::::: :::::::::::::::::: A. HIDDEN SYMMETRY :::::::::::::::::: Symmetries, especially local ones, are clearly very important in the formulation of interactions. However, symmetries are not always apparent in nature: For example, while most symmetries prefer massless particles, of all the observed particles the only massless ones are the graviton, photon, and neutrinos. Furthermore, of the massive ones, none with dierent properties have the same mass, although some are close (e.g., the proton and neutron). There are three solutions to this problem: (1) The symmetry is not a property of nature, but only an approximate symmetry. Some terms in the action are invariant under the symmetry, but other terms violate it. We can treat such \explicit symmetry breaking" by rst studying the symmetry for the invariant terms,and then treating the breaking terms as a perturbation. (2) Although the laws of physics are symmetric, nature is an asymmetric solution to them. In particular, such a solution is the \vacuum", or state of lowest energy, with respect to which all other states are dened. This case is called \spontaneous symmetry breaking". (3) The particles in terms of which these laws are formulated are not those observed in nature. For example, the hydrogen atom is most conveniently described in terms of a proton and an electron, but in its low-energy physics only the atom itself is observed as a separate entity: The U(1) symmetry related to charge is not seen from the neutral atoms. The more extreme case where such particles always appear in bound states is known as \connement". Generally, such broken symmetries are at least partially restored at high energies. For example, if the symmetry breaking introduces masses, or mass dierences between related particles, then the symmetry may become apparent at energies large with respect to those masses. Similarly, a hydrogen atom excited to an energy much larger than its lower energy levels will ionize to reveal its constituent particles. It often is possible to change to a set of variables that are invariant under a local symmetry. For example, if we can dene everywhere a variable that transforms as(x)=(x), then it can be used to everywhere undo the invariance. We can choose the \gauge" =, transforming to 0 everywhere, leaving no residual A. HIDDEN SYMMETRY 171 invariance, or we can work with composite, invariant variables: E.g., 0= eiis replaced (invertibly) with ^ = ei,s o^ 0=^ . 1. Spontaneous breakdown We rst consider symmetry breaking by the vacuum, known as \spontaneous breakdown". The action is invariant under the symmetry, but the vacuum state is not:Thus, the symmetry acting on the vacuum produces other zero-energy solutions to the eld equations, but this symmetry is not apparent when considering perturbation about the vacuum. In this case, although the symmetry is broken, there are obviousresidual eects, particularly if the breaking can be considered as \small" with respect to some other eects. The \Goldstone theorem" is an important statement about the eect of symme- try breakdown: If a continuous global symmetry is spontaneously broken, then thereis a corresponding massless scalar. The proof is simple: Consider a (relative) mini- mum of the potential, as the vacuum. By denition, we have spontaneous symmetry breaking if this minimum is not invariant under the continuous symmetry: i.e., ap-plying inntesimal symmetry transformations gives a curve of nearby states, which have the same energy, because the transformations are a symmetry of the theory. But the mass of a scalar, by denition, is given by the quadratic term in its potential,i.e., the second derivative of the potential evaluated at the vacuum value. (The rst derivative vanishes because the vacuum is a minimum.) So, if we look at the scalar dened to parametrize this curve of constant energy in eld space, its mass vanishes.(This eld may be a function of the given elds, such as an angle in eld space.) We can also formulate this more mathematically, for purposes of calculation: Consider a theory with potential V( i). (The Lagrangian is Vplus derivative terms. For simplicity we consider just scalars.) The masses of the scalars are dened by the quadratic term in the potential, expanding about a minimum, the vacuum. Thestatement of symmetry of the potential means that symmetry  i=i()) 0=V=i@iVf o r a l l i where we allow nonlinear symmetries, and @i=@=@i. Dierentiating, and then evaluating at this minimum, h@iVi=0at minimum =hi ) 0=h@j(i@iV)i=h(@ji)(@iV)i+hi@i@jVi=hiih@i@jVi 172 IV. MIXED where here the vacuum value hiclassically means to just evaluate at =hi.( S o classicallyhABi=hAihBi.) Spontaneous symmetry breaking means the vacuum breaks the symmetry: If this symmetry is broken, then hii6= 0, so it is a nontrivial eigenvector ofh@i@jVi(the mass matrix) with vanishing eigenvalue. So, we can write i=hii+hii+::: whereis a massless eld. The simplest example is a single free, massless eld, V=0 . T h e n is simply a constant. The simplest choice of vacuum is just hi= 0, which breaks the symmetry: L=1 4(@)2;  =constant;hi=0 Thenis a \Goldstone boson". The simplest nontrivial example, and a useful one, is a complex scalar with the potential V()=1 42(jj21 2m2)2 This is invariant under phase transformations =i. There is a continuous set of minima atjj=m=p 2. We choosehi=m=p 2; then the Goldstone theorem tells us that the imaginary part of is the Goldstone eld. Explicitly, separating the eld into its real and imaginary parts, =1p 2(m+ +i))V=1 42m2 2+1 42m ( 2+2)+1 162( 2+2)2 whereh i=hi= 0. We could also use the nonlinear separation of the eld into magnitude and phase, =(m+)ei=p 2: Thendrops out of the potential, and its transformation ( is invariant) is the same as that of the free massless scalar. If had been real, then only the discete symmetry $would have been broken, and there would be no Goldstone boson. Excercise IVA1.1 Write the complete action in terms of and. Note that this model would naively seem to have a tachyon (state with negative (mass)2) if we had expanded about hi= 0. However, since the vacuum is dened always as a minimum in the potential (or the energy), the true states always have nonnegative (mass)2. This is the case for positive spins for similar reasons: We saw in subsection IIB4 that free massive theories follow from massless ones by dimensional reduction from one extra spatial dimension. If we had used an extra time dimension instead, as required for the \wrong sign" for the mass term in p2+m2, there would also be wrong signs for Lorentz indices, resulting in kinetic terms with arbitrarily negative energy. A. HIDDEN SYMMETRY 173 2. Sigma models The Goldstone mechanism thus produces massive particles as well as massless ones, at least for polynomial potentials, to which we are restricted by quantum con- siderations, to be discussed later. We now look for approximations to polynomial scalar actions that eliminate the massive elds, but still take them into accountthrough their equations of motion, in the limit where their masses tend to innity. For example, in the above simple model, we can take the limit !1 ,w h i c ht a k e s the mass (m) to innity. In this limit, the potential energy can remain nite only if it vanishes:jj 2=1 2m2. (In quantum language, the potential's contribution to the path integral is just (R V) in that limit. Alternatively, we can neglect the kinetic energy forjjin comparison to the mass or potential, and then eliminate jjthrough its equation of motion in this approximation.) We can also enforce this limit directly by using a Lagrangr multiplier eld : L=1 2j@j2+ (jj21 2m2) The solution to the constraint is =mp 2ei, and the action then describes just a free, real scalar. A less trivial example is a nonabelian generalization of this example: Consider as a vector of an internal SO(n) symmetry. (The previous example was the case SO(2).) The Lagrangian is then L=1 4(@)2+1 2(2m2) The usual way to solve quadratic constraints without introducing square roots is to use the identity j(1 +ix)2j2=(j1+ixj2)2) (2x)2+( 1x2)2=( 1+x2)2 This is often used for trigonometric substitutions or simplifying integrals. For exam- ple, when an integrand has ap 1x2, substituting x=sin  eliminates the square root at the price of requiring trigonometric identities, which in turn are usually solved by making a second variable change to y=tan(=2). On the other hand, the above identity suggests making instead the variable change x=2y=(1 +y2), which actually gives the same result, more directly, as the previous two-step method. (This identity can also be used for nding integer solutions to the Pythagorean theorem: A right triangle with two shorter sides of integer lengths 2 mnandm2n2has the hypotenuse m2+n2,w h e r em;n are integers.) 174 IV. MIXED We then can solve the constraint 2=m2with the coordinates for the sphere in terms of an SO(n1) vector, =m 1+1 42;11 42 1+1 42 Then the kinetic term (now the whole action) becomes 1 4(@)2=1 4m2(@)2 (1 +1 42)2 Another way to obtain this result is to use the solution of subsection IA6 to the constraint 0=y2=(ya)22y+y)y=e(xa;1;1 2x2) Then the desired constraint (ya)2+(y1)2=1 follows from further constraining 1=y0=e1p 2(1 +1 2x2))e=p 2 1+1 2x2 )dy2=e2dx2=2dx2 (1 +1 2x2)2 yielding the above result for x==p 2. We thus have a nonpolynomial action, each term having derivatives. The original SO(n) symmetry is nonlinearly realized on the \angle" variables , and the vacuum ( hi= 0) spontaneously breaks the symmetry to SO(n-1). The constant macts as a dimensionful coupling, as seen by scaling !=m to give the kinetic term the standard normalization. A complex generalization of this model is described by the Lagrangian L=1 2jrj2+ (jj2m2) whereis now a complex n-component vector, ris a U(1)-covariant derivative (r=(@+iA)), and  is a Lagrange multiplier enforcing that has magnitude m. This model thus has a U(n) symmetry. Since Ahas no kinetic term ( F2), we can eliminate it by its algebraic eld equation: L!1 2j@j2+1 8m2(y$ @)2+ (jj2m2) where we have applied the constraint jj2=m2(or shifted  to cancel terms propor- tional tojj2m2). Since the U(1) gauge was not xed yet, we still have local U(1) A. HIDDEN SYMMETRY 175 invariance even without an explicit gauge eld. We can use this invariance to x the phase of one component of , and use the constraint from  to x its magnitude. In terms of the remaining (n 1)-component complex vector , =m 1+1 4jj2;11 4jj2 1+1 4jj2 )L=1 2m2j@j2+1 4(y$ @)2 (1 +1 4jj2)2 (Alternatively, we can solve the constraint and x the gauge rst, then eliminate Aby its eld equation.) This model is known as the CP(n 1) model (\complex projective"). Another example that will prove more relevant to physics is to generalize to an n n matrix: We then consider the Lagrangian L=tr[1 2(@)y
(@)+1 22(ym2I)2] (whereIis the identity matrix). Since yis hermitian and positive denite, the minimum of the potential is at y=m2I, and we can choose hi=mI using the SU(n) SU(n) U(1) invariance 0=ULUR1 (We can include the U(1) in either ULorUR.) The vacuum then spontaneously breaks this invariance to SU(n): h0i=hi)UL=UR In the large-mass limit, we get the constraint y=m2I)=mU; UyU=I;hUi=I; L!1 2m2tr[(@U)y(@U)] so the eld Uitself is now unitary. 176 IV. MIXED 3. Coset space The appearance of the scalar elds (Goldstone bosons) as group elements can be generalized directly in terms of the eective theory, without reference to massiveelds. Such a theory should be considered as a low-energy approximation to some unknown theory. Although the unknown theory may be better behaved at high energies quantum mechanically (see later), the low-energy eective theory can bedetermined from just (broken) symmetry. We therefore assume a symmetry group G that is broken down to a subgroup H by the vacuum. (I.e., the vacuum is invariant under the subgroup H, but not the full group G.) We are interested in only theGoldstone bosons, associated with all the generators of the group G less those of H. These elds are thus coordinates for the \coset space" G/H: They correspond to elements of the group G, but elements related by the subgroup H are identied. Explicitly, we rst write the eld gas an element of the group G, either by choosing a matrix representation of the group (as in the U(N) example above), or explicitly expanding over the group generators G I: g=ei; =I(x)GI We then \factor" out the subgroup H by introducing a gauge invariance for that subgroup: g0=gh; h =eih(x)H in terms of the H generators H, which are a subset of GI: GI=(H;Ti) whereTiare the remaining generators, corresponding to G/H. In particular, we can choose gauge=0)=iTi However, G should still be a global invariance of the theory, though not of the vacuum. We therefore assume the global transformation g0=g0g whereg0is an element of the full group G, but is constant in x. The vacuumhgi=I is then invariant under the global subgroup g0=h1, where thus his constant and g02H (i.e., G is spontaneously broken to H). We then attempt to construct a eld strength invariant under the global group as an element of the Lie algebra of G: g1@ag=@a+iA aH+iFi aTi=ra+iFi aTi A. HIDDEN SYMMETRY 177 This can be evaluated in the parametrization as multiple commutators, as usual: A andFare both nonpolynomial functions of , but with only one derivative. We have absorbedAinto a covariant derivative rbecause of the remaining transformation law under the local group H: (r+iF)0=h1(r+iF)h)r0=h1rh; F0=h1Fh w h e r ew eh a v ea s s u m e d[ H;Ti]Tj. (In particular, this is true for compact groups, where the structure constants are totally antisymmetric: Then fi=0)fi=0 . ) Then the action invariant under global and local transformations can be chosen as L=1 4m2tr(F2) In particular, the real vector model we gave in the previous subsection describes the coset space SO(n)/SO(n 1), the complex vector describes SU(n)/U(n 1), and the matrix model describes U(n) U(n)/U(n). Excercise IVA3.1 Use the coset-space construction to derive the specic models explicitly given in the previous subsection, as just identied. We have described how nonpolynomial actions quadratic in derivatives can arise as a low-energy approximation to polynomial theories. Further nonpolynomial termsquartic in derivatives (but no more than quadratic in time derivatives) can be usefulfor certain applications, but these arise from polynomial actions quadratic in deriva-tives (which are preferred for quantum reasons) only by quantum eects. 4. Chiral symmetry Later we'll examine a description of the strongly interacting particles (\hadrons") in which they are all considered as composites (bound states) of fermionic \quarks".However, this theory is extremely dicult to solve, so we rst consider treating thehadrons as fundamental instead. Since there are probably an innite number of kindsof hadrons (or at least some integer power of 10 40, considering the (Planck mass)2), this would require a formulation in terms of a \string" that treated all \mesons"(bosonic hadrons) as a single entity. That possibility also will be considered later;for now, we look at the simpler possibility of studying just the low-energy physics ofhadrons by using elds for just the lightest particles. So far, the only observed scalar particles have been strongly interacting ones. Some of the scalar mesons, especially the \pions", are not only the lightest hadrons, 178 IV. MIXED but can be considered close to massless on the hadronic scale. We therefore look for a description of pions (and some close relatives) in the massless approximation; then mass-generating corrections can be considered. Normally, quantum corrections can aect masses. The only way to guarantee masslessness at the quantum level is through some symmetry; we then can study this symmetry already at the classical level. We have seen that (unbroken) gauge invari- ance can require masslessness for all elds except the scalar and spinor. Masslessness for a spinor can be enforced by \chiral symmetry": If there is a U(1) symmetry for all irreducible spinors , then no mass terms (bilinears  1 2) can be constructed. (Generally, each spinor can have dierent U(1) charges, as long as no two charges add to zero. Of course, this U(1) can be a subgroup of a larger chiral symmetry group.) The only way a scalar can be guaranteed masslessness is if it is a Goldstone boson. Wetherefore look for a description of pions as Goldstone bosons of some spontaneously broken symmetry. (Supersymmetry is another possibility to enforce massless scalars, but only if there are also massless fermions, which is not the case for hadrons.) For simplicity, we consider the coupling of scalar mesons to quarks. We could instead couple mesons to \baryons" (fermionic hadrons), thus treating only hadrons, but the principles would be the same, only the indices would be messier. Combining C invariance with chiral symmetry, and including a meson potential for spontaneoussymmetry breaking, we can write the action for just the quarks and scalar mesons as S=Z dx tr L L=[q y. Li@.qL+qT Ri@.q*R.]+[1 2(@)y
(@)+1 42(y1 2m2I)2] +[q LqT R+q*. Ryqy L.] whereis an m m matrix (m \ avors"), qLandqRare n m matrices (n \colors"), and  is the \Yukawa coupling". Besides color symmetry (local if we had bothered to write in the Yang-Mills elds for the \gluons", by @!r on the quarks), we have the (global) U(m) L U(m)Rchiral ( avor) symmetry q0 L=qLUL;q0 R=qRU*R;0=U1 LUR including the (global) U(1) \baryon number" symmetry UL=UR=ei)q0 L=eiqL;q0 R=eiqR;0= If we think of baryon number as an SO(2) symmetry, then charge conjugation is just the re ection that completes this to an O(2) symmetry (see excercise IIA1.2): C:qL$qR;!T A. HIDDEN SYMMETRY 179 From this, and the usual CP CP:qL!qL*;qR!qR*;!* we nd the parity symmetry P:qL$qR*;!y (where for CP and P we also transform the coordinates as usual). As before, the vacuum hi=mp 2Ibreaks the avor symmetry to the diagonal subgroupUL=UR, which commutes with parity (and is therefore no longer \chiral"). It also gives masses to the quarks (since chiral symmetry is broken); this is a general feature of spinors coupled to scalars under spontaneous breakdown. In the limit !1 (where the mass of all bosons but the Goldstones becomes innite, but the quark mass M=mis xed), the Goldstone bosons are described by the unitary matrixU, which transforms as U0=U1 LUUR. An interesting special case is m=1 (one avor). The Goldstone boson of axial U(1) can be identied with the 0. In the limit !1 , the Lagrangian becomes (with atrno longer needed) L=(qy. Li@.qL+qT Ri@.q*R.)+1 4m2(@)2+Mp 2(eiqT RqL+eiqy. Lq*R.) writingfor the neutral pion eld.  = M=m is still the coupling of the pion to the quarks, as can be seen by rescaling !=m to give the kinetic term the usual normalization. (The coupling mis known as the \pion decay constant", and is usually denotedf. If we include leptons with the quarks, then this coupling also describes the decay of the pion into two leptonic fermions.) In this case, the (broken) axial U(1) transformations are q0 L=eiqL;q0 R=eiqR;0=2 The corresponding axial current (determined, e.g., by coupling a gauge vector) is J.  A=(qy.  Lq LqT Rq*.  R)m2@.  This current is still conserved, since the eld equations aren't changed by the prop- erties of the vacuum. The linear term is characteristic of expanding the Goldstoneeld about the spontaneously broken vacuum; it corresponds to the fact that that eld has an inhomogeneous transformation under the broken symmetry. However, in reality the pion is not exactly massless, so we should add to the previous action a mass term for the pion, which explicitly violates the symmetry. (It 180 IV. MIXED is then a \pseudogoldstone boson".) In the general chiral symmetry model, where the Goldstone bosons are described by a unitary matrix, a simple term that gives them masses while preserving the polar (parity-preserving) diagonal symmetry UL=URof the vacuum is, for some constant , Lm=t r(+yp 2mI) Since this explicitly breaks the axial U(m) symmetries, the corresponding currents are no longer conserved. In the U(1) case, we can also add just a mass term Lm=1 42; =m2m2  (for some constant ), which is the leading contribution from the general term above. The change in the eld equation for now violates the conservation law as @
JA= This explicitly broken conservation law is known as \Partially Conserved Axial Cur- rent" (PCAC). 5. St uckelberg By denition, only gauge-invariant variables are observable. Although in general a change of variables to gauge-invariant ones can be complicated and impractical, there are certain theories where such a procedure can be implemented very simply as part of the normal gauge-xing. Not surprisingly, the only nonlinearity in theseredenitions involves scalars. The simplest cases of such redenitions are free theories, and are thus contained in our earlier discussion of general free, massive gauge theories. The simplest of these is the massive vector. As described in subsection IIB4, the Lagrangian and gaugeinvariance are L= 1 8F2+1 4(mA+@)2 A=@;  =m whereFabis the Abelian eld strength. Note that the scalar is pure gauge: It is called a \compensator" for this gauge invariance. Since it has a nonderivative gauge transformation, it can easily be gauged to zero at each point, by just choosing = =m. This means that without loss of generality we can consider the theory in terms of just the gauge-invariant eld A0=A+1 m@ A. HIDDEN SYMMETRY 181 This \composite" eld can also be considered as a eld redention or gauge transfor- mation onA. The lagrangian simplies to L=1 8F02+1 4m2A02 Later we'll see that it is often more useful to keep as an independent eld. Excercise IVA5.1 Choose the gauge A0=. Show that then can be eliminated by its equa- tion of motion, leaving only the transverse 3-vector Ai, with Lagrangian 1 4Ai(m2)Ai. Show the relation to the lightcone gauge of subsection IIIC2, using the dimensional reduction langauge of subsection IIB4. The original Lagrangian can also be considered an unusual coupling of a massless vector to a massless scalar: Remember that the massless scalar is the simplest example of a Goldstone boson, with the spontaneously broken global symmetry =T; T =1 where we have dened the symmetry generator Tto act inhomogeneously on .W e then couple the \photon" to this charge: After a trivial rescaling of the gauge eld, L=1 8m2F2+1 4(r)2;r=@+AT wheremis the \charge" with which Acouples to, which in this case happens to have dimensions of mass. The electromagnetic current in this case is simply J=1 2r, whose conservation is the scalar eld equation = 0 (with gauge-covariantized ). Because the spontaneously broken symmetry of the corresponding Goldstone model is now gauged, expanding about hi= 0 is no longer a physical statement about the vacuum, since is no longer gauge invariant. (As we saw, we can even choose= 0 as a gauge condition.) Therefore, from now on, when we make a statement such as \ hi= 0" in such a case, it will be understood to refer to choos- ing= 0 as the value about which to perform perturbation expansions (e.g., for separating actions into kinetic terms and interactions). Note that the St uckelberg action can be generated starting from the action with justA0, and performing a gauge transformation that is not an invariance: A0!A0+1 m@ Dropping the prime from A, this transformation is just the inverse of the one we used to eliminate the scalar. If we start from an action that has also a coupling of A0to matter, we see that conserved currents decouple from :Z A0
J!Z A
J1 mZ @
J 182 IV. MIXED More precisely, if the only term in the action for vector + matter that is not gauge invariant is the vector mass term (1 4m2A02), then the above gauge transformation aects only that term. 6. Higgs We have seen that spontaneous symmetry breakdown can generate masses for spinors. We also saw how a massless vector could become massive by \eating" a would-be Goldstone scalar, in the simplest case of a scalar without self-interactions. We'll now examine more interesting models: Yang-Mills theories, which describe self- interacting vectors, must couple to self-interacting scalars to become massive. We can expect, by considering the linearization of any Yang-Mills theory coupled to scalars, that we will need more scalars than massive vectors, since each vector needs to eat a scalar to become massive, and some scalars will become massive and thus uneaten. (Only would-be Goldstone bosons can be eaten, as seen by linearization to the St uckelberg model.) For the simple (and most useful) example of U(n) for the gauge group, an obvious choice for the scalar \Higgs" eld is an n n matrix. (SU(n) can be treated as a slight modication.) The simplest such model is the one studied in subsection IVA2: We now consider one of the SU(n) symmetries (together withthe U(1)) as the local \color" symmetry to which the Yang-Mills elds couple, and the other SU(n) as the global \ avor" symmetry (where we use the names \color" and \ avor" to distinguish local and global symmetries, not necessarily related tochromodynamics). The Lagrangian for this \Gervais-Neveu model" is then L=tr[ 1 8g2F2+1 2(r)y
(r)+1 42(y1 2m2I)2] wherer=@+iA,a n dAaandare n n matrices (but Aaare hermitian). Now y is gauge invariant (although not invariant under the avor group), so we still have hyi=1 2m2I as a gauge-invariant statement (but hi=mp 2I,o rhyi=1 2m2I, still makes sense only for purposes of gauge-dependent perturbation expansions). Since any complex matrix can be written as =UH=p 2, whereUis unitary and His hermitian, we can choose the \unitary gauge" U=I(i.e.,=y). As for the Stuckelberg case, this is equivalent to working in terms of the gauge-invariant elds (dened by using this Uas a gauge transformation) A0=U1(i@+A)U; 0=1p 2H=U1 A. HIDDEN SYMMETRY 183 whereUcan be dened by 1p 2H=p y; U =p 2H1 This is well-dened as long as His invertible, which is true for small perturbations about its vacuum value hHi=mI If the perturbation is so large that Hhas vanishing eigenvalues, then this is equivalent to looking at states so far away from the vacuum that some of the broken symmetry is restored. Expanding about the vacuum ( H!mI+H), the Lagrangian is now L=tr[1 8g2F02+1 4m2A02+1 4(@H)2+1 42m2H2 +A0
1 4(Hi$ @H)+1 2mA02H+1 42mH3+1 4A02H2+1 162H4] Thus all particles are now massive. As for the Goldstone case, we can take the limit !1 to get rid of all the massive scalars, which in this case leaves just the massive vectors, adding only the mass term to the original Yang-Mills action. This was clearfrom the nonlinear model that resulted from that limit, by coupling that eld ( U) to Yang-Mills directly. Excercise IVA6.1 Find the chiral action for this model of the type described in subsection IIIC4, where the massive vectors are described by self-dual tensors instead of vectors. Excercise IVA6.2 Consider again this model, for the case n=2. We modify this example bydropping the U(1) gauge eld, so we have just SU(2). Since SU(2) is pseu- doreal, we can further restrict the Higgs eld to satisfy the reality condition *=CC . Thus, both color and avor groups are SU(2), and is the usual matrix representation of the 4-vector of SO(4)=SU(2) SU(2) (see subsection IIA5). Repeat the analysis given above. Excercise IVA6.3 Consider again the gauge group SU(2), but now take the Higgs eld in the adjoint representation, with no avor group (i.e., a real 3-vector). Show that only 2 of the 3 vectors get mass, leaving a residual U(1) gauge invariance. Explain this in terms of the gauge transformations of the 3-vector. (Hint: think 3D rotations.) 184 IV. MIXED REFERENCES 1Y. Nambu, Phys. Rev. Lett. 4(1960) 380; Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122(1961) 345, 124(1961) 246: introduced into relativistic physics massless bosons associated with a broken symmetry. 2J. Goldstone, Nuo. Cim. 19(1961) 154; J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127(1962) 965: found the theorem relating the two. 3M. Gell-Mann and M. L evy, Nuo. Cim. 16(1960) 705: -models. 4S. Weinberg, Phys. Lett. 18(1967) 188: nonlinear-models. 5H. Eichenherr, Nucl. Phys. B146 (1978) 215; V.L. Golo and A.M. Perelomov, Lett. Math. Phys. 2(1978) 477, Phys. Lett. 79B (1978) 112;E. Cremmer and J. Scherk, Phys. Lett. 74B (1978) 341: classical CP(n). 6T.H.R. Skyrme, P r o c .R o y .S o c . A260 (1961) 227; J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95; E. Witten, Nucl. Phys. B223 (1983) 422, 433: higher-derivative terms. 7H. Yukawa, Proc. Phys.-Math. Soc. Japan 17(1935) 48. 8R. Marshak and E.C.G. Sudarshan, Proc. Padua-Venice conference on mesons and recently discovered particles , September, 1957; Phys. Rev. 109(1958) 1860; Feynman and Gell-Mann, loc. cit. (IIIC): chiral symmetry in weak interactions (for coupling to vector currents). 9Gell-Mann and Levy, loc. cit. ; Nambu, loc. cit. ; K.-C. Chou, Sov. Phys. JETP 12(1961) 492: PCAC. 10Stuckelberg, loc. cit. (IIB). 11P.W. Anderson, Phys. Rev. 112(1958) 1900, 130(1963) 439: nonrelativistic \Higgs" eect in condensed matter theory. 12F. Englert and R. Brout, Phys. Rev. Lett. 13(1964) 321; P.W. Higgs, Phys. Lett. 12(1964) 132; G.S. Guralnik, C.R. Hagen, and T.W.B. Kibble, Phys. Rev. Lett. 13(1964) 585; A.A. Migdal and A.M. Polyakov, Sov. Phys. JETP 24(1967) 91: \Higgs" eect. 13J.L. Gervais and A. Neveu, Nucl. Phys. B46 (1972) 381. B. STANDARD MODEL 185 ::::::::::::::::::: ::::::::::::::::::: ::::::::::::::::::: B. STANDARD MODEL ::::::::::::::::::: In this section we discuss the \Standard Model", the minimal theory that de- scribes all the observed particles and forces (except gravity). We also consider somefeatures of \Grand Unied Theories" (GUTs), an extension of the Standard Model that uses fewer multiplets. 1. Chromodynamics One way in which particles naively described by the action can be hidden from observation is if the force binding them is too strong to allow them to exist freely.Such a condition is often called \infrared slavery" since this alleged property of theforce is a long-range feature, preventing the constituent particles from escaping to innity. This \connement" is not a classical phenomenon, and its occurence even at the quantum level has not yet been proven. Therefore, in this section we'll simplyassume connement, and describe the resultant symmetry properties, leaving thedynamical properties for later chapters. The assumption of \color" connement is that the color forces are so strong that they bind any objects of color to other such objects; thus, only \colorless" states, thosethat are singlets under the color gauge group, can exist freely. Composite elds that are invariant under the local group transformations can be obtained by multiplying matter elds or Yang-Mills eld strengths, perhaps using also covariant derivatives,and contracting all color indices. The color gauge group is generally assumed to be SU(n): usually SU(3), but sometimes larger n for purposes of perturbation in 1/n. Larger n is also used for unication, but in that case the Higgs mechanism is used toreduce the group of the massless vectors to SU(3) (times Abelian factors). Another feature of these conned states, to be considered later, is their geomet- rical structure. The observed spectrum and scattering amplitudes of the \hadrons"(strongly interacting particles) indicates a stringlike identication of at least the ex-cited states. (The ground states may behave more like \bags".) This picture also ts in with connement, since the spatial separation of the quarks and antiquarks in excited states would force the gluons that convey their interactions (and self-interact)to conne themselves as much as possible by collapsing into \strings" connecting thequarks. Thus, we describe a meson with an \open string", with a quark at one end and an antiquark at the other. Similarly, an excited glueball would no longer be a ball, but rather a \closed string", forming a closed loop. 186 IV. MIXED We will need to reconsider also the discrete symmetries, C, P, and T, and their combinations. First of all, we note the \CPT theorem": All local, hermitian, Poincar e invariant actions are CPT invariant. This is easy to see from the fact that CPT only changes the overall sign of the coordinates, which is eectively the same as changing the sign of each derivative, as well as giving a 1 for each vector index on a eld. Since CPT also gives signs to dotted spinors and not undotted ones, we also get 1's for vector combinations of indices on spinors (  .; signs cancel when contracting spinor indices on pairs of dotted or undotted spinors). Thus, all these signs cancel becausePoincar e invariance requires an even number of vector indices (in even numbers of dimensions, from contracting with  abandabcd). Alternatively, and even more simply, in D=4 we can attribute it to having even numbers both of undotted spinor indices and of dotted spinor indices, since we can dene CPT by associating a sign with each dotted index (including those that appear as part of a vector index). Consequently,from now on we ignore T and consider only C, P, and CP. Although we have considered C (and thus CP) in the context of electromagnetism, nonabelian gauge elds require some (simple) generalization, since they carry charge themselves. We start with the general coupling of massless fermions to nonabeliangauge elds: L= y. (i@ . +A . )  where is a column vector with respect to the gauge group, and Aah e r m i t i a n matrix. The CP transformation of the fermions then determines that of the vectors, needed for invariance: CP: 0= *.; 0*.= ;@0 . =@.;A0 . =AT. (remember ( )* ., but ( )* = .because of the factor of C), where we have chosen to represent parity on the coordinates as acting on the explicit @rather than on the arguments of the elds. The transformation on the vector is thus parityon the vector index, combined with charge conjugation A 0 a=AT a=Aa*: The minus sign can be associated with change in sign of the coupling (as for the Abelian case), while the complex conjugation takes into account the charge of the vector elds themselves. (As discussed in subsection IB2, G!G* is an invariance of the algebra, where g!g*a n dg=eiG.) Although these terms, as well as the F2term for the vectors, are CP invariant, this invariance can be broken by coupling to scalars: The Yukawa coupling LY= T +h:c: B. STANDARD MODEL 187 for some matrix of scalars, would require the CP transformation 0=* (up to perhaps some unitary transformation), but unlike the vectors there is no guar- antee that under complex conjugation the matrix =iMifor real scalars iand constant matrix (Yukawa couplings) Miwill preserve this form, i.e., satisfy 0iMi=iMi* since the matrices Mican be complex. The basic assumption of \chromodynamics", or in the quantized version \quan- tum chromodynamics (QCD)", is that we have a nonabelian gauge theory withoutfundamental scalars that couple directly (but scalars will show up when we intro- duce electroweak interactions). Namely, we assume only Yang-Mills for the \color" gauge group SU(n), specically n=3, with the usual action, minimally coupled tospin-1/2 \quarks" in the dening representation of the color group, which may have masses. (These masses are actually generated by weakly interacting Higgs bosons, whose coupling we consider in the next subsection; for now we include just the re- sulting mass terms.) Such an action is automatically invariant under CP and T. We furthermore assume invariance under charge conjugation: Just as an irreduciblereal scalar describes particles that are their own antiparticles, and needs doubling (or complexication) to dene charge, an irreducible (massive) spinor cannot describe distinguishable particle and antiparticle. Therefore, for every quark eld q L(\L" for \left") we have an \antiquark" eld qR(\R" for \right"), and they transform into each other under charge conjugation, just as a scalar transforms into its complex conjugate. (A spinor can't transform into its complex conjugate under C, since Ccommutes with spacetime symmetries, like Lorentz transformations.) Besides this doubling, and the n colors of the quarks, we also assume a further multiplicity of m dierent \ avors" of quarks. Gauge invariance requires the quark masses be indepen-dent of color, and C invariance requires the mass terms couple q LtoqR, but these terms violate an otherwise global U(m) U(m) avor symmetry. The action is then of the form tr[1 8F2+(qy LirqL+qy RirqR)+(Mp 2qT RqL+h:c:)] where we have written qLandqRas matrices with respect to SU(n) color ( Uc)a n d U(m) U(m) avor ( UfLandUfR) such that they transform as q0 L=UcqLUfL;q0 R=Uc*qRUfR* 188 IV. MIXED and thus the covariant derivatives can be written as raqL=(@a+iAa)qL;raqR=(@aiAa*)qR whereAaare hermitian, traceless, n n matrices. (By denition, charge conjuga- tion takes a representation of an internal symmetry into the complex-conjugate one.)Charge conjugation is then C:q L$qR;Aa!Aa* while parity is P:q L;R!qL;R.;Aa!Aa While the color symmetry is a local symmetry, the avor symmetry is broken, inducing the transformation on the mass matrix M0=UfLMU1 fR This transformation allows the mass matrix Mto be chosen real and diagonal: Any matrix can be written as a hermitian one times a unitary one. A UfRtransformation, as a eld redenition, then can be made to cancel the unitary factor in M;t h e n a unitary transformation UfL=UfRcan be made to diagonalize M(while keeping it hermitian). These diagonal elements are then simply the masses of the m dif- ferent quark avors. The most symmetric case is M= 0, which leaves the entire U(m) U(m) chiral symmetry unbroken. (See subsection IVA4.) The least symmet- ric case is where all the masses are nonzero and unequal, leaving as unbroken only the subgroup U(1)m,w i t hUfL=UfR.( I n g e n e r a l , UfL=UfRif all masses are nonvanishing.) The minimal form of this action, besides making CP and T automatic, also au- tomatically extends the discrete symmetry C to an O(2) symmetry, whose \parity" transformation is C and whose continuous SO(2)=U(1) symmetry is the U(1) part of the U(m) avor symmetry, which is not broken by the mass term. It corresponds to a charge called \baryon number": Up to an overall normalization factor, it simply counts the number of quarks qL;qR.(which form a Dirac spinor) minus the number of \antiquarks" qR;qL.. However, such an O(2) symmetry can be dened separately for each avor, since (after Mhas been diagonalized) the action can be written as a sum of independent terms for each avor. In particular, each avor has its ownseparately conserved quark number. These avor conservation laws, at the classical level, are broken only by the weak interactions, which we have not included in the above action. (Gravity and electromagnetism do not violate them.) B. STANDARD MODEL 189 Since connement is a quantum eect, the details of hadronic scattering cannot be discussed within classical eld theory. However, we saw that low-energy properties ofmesons (and similarly for baryons) could be described by eective Lagrangians. Thefact that hadrons are made of quarks can be used to obtain a bit more information even at the classical level, especially if the relevant quarks are heavy. (Heavy with respect to what is unfortunately also a question that can be answered only at thequantum level.) For example, in a nonrelativistic approximation, low-energy proper- ties of hadrons can be found from just the quantum numbers, spin-spin interactions, and masses of the quarks, while their velocities are ignored, and the gluons are ne-glected altogether. In such an approximation, reasonably accurate predictions are made for the masses and magnetic moments of the ground-state hadrons. Actually, the claim that color nonsinglet states can never be observed needs a bit of stipulation: There may be a \quark-gluon plasma" phase of hadronic matter that can exist only at extremely high temperatures or pressures. Thus, a hypothet-ical observer during the rst moments of the universe might observe \free" quarks and gluons. Similarly, a small enough observer, living inside an individual hadron, might see individual quarks and gluons, since the size of his equipment would bemuch smaller than what we consider \asymptotic" distances. Conversely, we could consider the possibility of a new chromodynamic force, other than the one respon- sible for the hadrons of which we are composed, that has a connement scale thatis astronomical (extremely low energy), so that earthly laboratories would t insidethe new \hadrons". Thus, any statement about the observability of color must be a dynamical one, and does not follow as an automatic consequence of the appearance of a nonabelian group: Just as for the Higgs eect, connement can be repealed underappropriate circumstances, and the observability of color depends on the details of the dynamics, and in particular on the values of the various parameters (momenta and couplings). 2. Electroweak The weak and electromagnetic interactions are mediated by observed spin-1 par- ticles, some of which have charge and mass. Specically (see subsection IC4), the massive vectors form a triplet ( W+,W,Z), while there is only one massless vector (the photon). This suggests a gauge group of SU(2) U(1), with a Higgs eect that leaves only U(1) unbroken. From the table of known fundamental fermions, we can see that they fall into doublets and singlets of this SU(2), with the U(1) charge being that 190 IV. MIXED of electromagnetism. (This SU(2) U(1) unication of the weak and electromagnetic interactions is called the \Glashow-Salam-Weinberg" model.) We saw in subsection IVA4 a very simple model of spontaneously broken chiral U(m) U(m) symmetry where masses were generated for quarks. In subsection IVA6 we saw how the same scalars could generate masses for vectors, by coupling to one of the U(m)'s. We now combine those two models, specializing to the case m=2, but with two slight modications: (1) Since the dening representation of SU(2) is pseudoreal, we can impose a reality condition on the Higgs eld, which is in the (1 2,1 2) representation of SU(2) SU(2): *=CC This makes it a vector of SO(4)=SU(2) SU(2) (See excercise IVA6.2.) It's also the reality condition satised by an element of (the dening representation of) SU(2). (See subsection IIA2.) This is not surprising, since the group product U0=ULUURallows the interpretation of a group element itself as a representation of chiral symmetry. This is the situation described in subsection IVA2 ( !Uin the large-mass limit), but in this case yis automatically proportional to the identity (it gives the square of the 4-vector), so in general an SO(4) 4-vector can be written as the product of a scalar with an SU(2) element. This reality condition breaks the chiral U(1) U(1) to the diagonal U(1) that leaves the Higgs invariant. (2) The gauged SU(2) is still one of the two chiral SU(2)'s, but the gauged U(1) must now be a subgroup of the other SU(2), since the Higgs is now invariant under the usual U(1)'s. Thus, the ungauged SU(2) is explictly broken, and this accounts for the mass splittings in the doublets of known fundamental fermions. Rememberthat observables are singlets of gauged nonabelian groups (except perhaps for Abelian subgroups), so any observed internal SU(2) must be a global symmetry, even when it's broken. As described in subsection IVA6, these singlets can be constructed as composite elds resulting from the gauge transformation obtained from the SU(2) part of. Using the electromagnetic charges of the various particles, we thus determine their SU(2) U(1) representations: For spin 1, we have W=(1,0) and V=(0,0), where the rst entry is the \isospin" and the second is the U(1) charge. For spin 0, we have=( 1 2,1 2), choosing the U(1) generator as the diagonal one from UR. Finally, for spin 1/2, we have for the leptons lL=(1 2,1 2), which combines with to pro- duce (0,0)(0,1), andlR=(0,1). Similarly, for the quarks we have qL=(1 2;1 6), and qR=(0,1 61 2). The Lagrangian is then L=L1+L0+L1=2 B. STANDARD MODEL 191 L1=1 8g02F2(V)+1 8g2tr F2(W) L0=tr[1 4(r)y(r)+1 42(y1 2m2)2] L1=2=tr( yir )+tr+ 00  qT RqL+lT RlL1 0 +h:c: where the fermions =(qL;qR;lL;lR), and the SU(2) U(1) covariant derivative acts as r=@+iWi1 2V1 00 1
rqL=@qLiqLW+i1 6VqL rqR=@qR+i1 2VqR 1 3I+1 00 1
 rlL=@lLilLWi1 2VlL rlR=@lR+iVlR For simplicity we have ignored the indices for color (and its gauge elds, treated in the previous section), families (treated in the following subsection), and spin. We have also used matrix notation with respect to the local SU(2) (gauged by W)a n d the global SU(2) (explicitly broken in L1=2by the gauging of a U(1) subgroup, the Yukawa couplings, and the chirality of the massless neutrinos): Thus Wis a traceless hermitian 22 matrix,is also 22 but satisfying the \reality" condition given above (traceless antihermitian plus real trace), qL,qR,a n dlLare complex 22 matrices, andlRis a 2-component column. (By denition, the diagonal parts of Wandare electromagnetically neutral.) The quark Yukawa coupling is diagonal in the brokenSU(2) to preserve the local U(1) symmetry. (The trhere is trivial for the lepton Yukawa term, but we have left it for generalization to more than one family.) In the unitary gauge for the local SU(2), = 1p 2'I;h'i=m where'is a single real scalar, the simplications to the Lagrangian are L0!1 4(@')2+1 8'2trf[W1 2V1 00 1
]2g+1 82('2m2)2 L1=2!tr( yir )+1p 2'tr+ 00  qT RqL+lT RlL1 0 +h:c: We then can expand 'about its vacuum value m: The lowest order terms give masses for most of the vectors and fermions: The massless fermions are the neutrinos, while the massless vector gauging the unbroken U(1) (a combination of the original U(1)with a U(1) subgroup of the SU(2)) is the photon (of electromagnetic fame). The 192 IV. MIXED mass of the remaining vectors accounts for the weakness and short range of the \weak" interactions. Excercise IVB2.1 Diagonalize this Lagrangian with respect to the mass eigenstates. For conve-nience, normalize g= 1p 2g0cos W;g0=g0sinW whereWis the \weak mixing (Weinberg) angle". aFind explicitly the masses for all the particles in the Standard Model (rst family for fermions) in terms of the couplings m;;g 0;W;;. Show from the experimental values for the vector masses given in subsection IC4 that sin2W:223. bFind all the other couplings of the mass eigenstates. Show that, with the conventional electric charge assignments, 1 e2=1 2g2+1 g02 (Hint: Rather than rescaling the vectors, note that the generated mass term, and the given couplings of VandW, suggest dening W0=W1 2V1 00 1
)V= +k1Z;p 2W0= +k2Z in the conventions of subsection IIA1, for the new elds Zand photon ,f o r appropriate constants ki.) Note that, unlike the strong (chromodynamic) or purely electromagnetic (or even gravitational) interactions, the weak interactions break every discrete spacetime sym- metry possible. (The others break none. CP violation will be discussed in thefollowing subsection. Of course, CPT is always preserved.) Sometimes this is at- tributed to the presence of a chiral symmetry, used to reduce 4-component spinors to 2-component; however, we have already seen that in general chiral and parity symme-tries are unrelated. (You can have either without the other. This fact will be further discussed in subsections IVB4 and VIIIB3.) A better explanation is to attribute P and C to doubling, which converts spinors from 2-component to 4-component: 2-component spinors are the simplest description of helicity/spin 1 2;4 - c o m p o n e n t spinors are useful only to manifest a larger symmetry, when it exists. The weak inter-actions violate parity because the neutrino is not doubled, and because the fermions that are doubled no longer have a symmetry relating their two halves. B. STANDARD MODEL 193 3. Families In the Standard Model (and its simpler generalizations) there is no explanation for the existence of more than one family of fermions. However, the existence of 3 families does have interesting consequences. Most of these follow from the form of the Yukawa couplings, and thus the fermion masses. In subsection IVB1 we consideredredenitions of the fermion elds as unitary avor transformations. These allowed us to obtain the simplest form of the mass matrices, since they were not avor singlets, and thus transformed. We now perform similar transformations, but only on thefamily indices, since transformations that don't commute with the gauge symmetries would complicate the other terms in the action. Now ignoring spin, color, and local avor indices, and using matrix notation for the family indices, the fermions transform as q 0 L=qLUqL;q0 R=qRUqR*;l0 L=lLUlL;l0 R=lRUlR* whereqL,qR,lL,a n dlRhave m components for the m families. qRare the 2 components of the (explicitly broken) global avor doublet qR.W et h u sh a v e 5U ( m ) symmetries, all broken by the Yukawa couplings: These eld redenitions induce transformations on them, 0 =UqLU1 qR; 0=UlLU1 lR As in subsection IVB1, UqRandUlRcan be used to make  and  hermitian. ThenUlLcan be used to make  diagonal, also as in subsection IVB1, leaving a U(1)m symmetryUlL=UlR, corresponding to separate conservation laws for electron number (including its neutrino), muon number, and tauon number for the 3 known avors. However, the quark sector works a bit dierently: We can use UqLto diagonalize  + or , but not both. This leaves another U(1)msymmetryUqL=UqR+=UqR.I f  + has been diagonalized, then 1 of the m U(1)'s, corresponding to total quark number (baryon number) conservation, leaves  invariant, while the remaining m 1 U(1)'s can be used to eliminate some of the phases of the complex o-diagonal components of . The remaining global avor symmetries are thus m lepton U(1)'s and 1 quark U(1). The remaining Yukawa couplings are the real, diagonal , describing the m masses of the massive leptons (the neutrinos remain massless), the real, diagonal +, giving the m masses of half of the quarks, and the hermitian  , consisting of m diagonal components, describing the masses of the other quarks, m(m 1)/2 magnitudes of the o-diagonal components, and (m 1)(m2)/2 phases of the o- diagonal components. These phases violate CP invariance: CP, besides its aect on 194 IV. MIXED the coordinates, switches each spinor eld with its complex conjugate. Since the complex conjugate term in the action uses the complex conjugates of the 's, this symmetry is violated whenever any of the components have imaginary parts (after taking into account all possible symmetries that could compensate for this, as we have just done). Note that CP is violated only for 3 families or more. (C and Pare separately violated for any number of families by the SU(2) U(1) coupling: As discussed in subsection IVB1, C invariance of the strong interactions is the symmetry q L$qR.) Since we can choose to transform away the phases in the subsector of the 2 lighter quark families, the large masses of the heavier quarks s uppress this eect, accounting for the smallness of CP violation. Since observed particles are mass eigenstates, it's convenient to perform a further unitary transformation (the \Cabibbo-Kobayashi-Maskawa matrix") that diagonal-izes the mass matrix. Although this is clearly possible by the arguments of subsec- tion IVB1, it is not part of the unitary transformations considered in this subsection because it does not commute with the SU(2) gauge symmetry: After such a trans-formation, we nd that the components of each SU(2) quark multiplet are linear superpositions of dierent families. Excercise IVB3.1 Perform this diagonalization explicitly for the case m=2 (two fam ilies), using the two lightest families of quarks and leptons as listed in subsection IC4. Which particles mix? Parametrize this mixing by an angle  c(the \Cabibbo angle"). An important experimental result with which the Standard Model is consistent is the suppression of \ avor-changing neutral currents (FCNC)". The two electrically neutral gauge elds in this model, the Zand the photon , couple to currents that are neutral with respect to the U(1) symmetries associated with each of the quark ( avor)numbers. This is true by construction before the unitary CKM transformation, but this transformation also leaves these two currents invariant (the \Glashow-Iliopoulos- Maiani mechanism"). Thus, at the classical level we do not see eects such as the decayK 0!Z!+, which would violate this \conservation law". Furthermore, the quantum corrections are suppressed (though nonvanishing) for similar reasons:For example, the lowest-order nonvanishing quantum correction comes from replacing theZwith aW +Wpair. Without the CKM matrix, this contribution would vanish; treating CKM, and its resulting contribution to quark masses, as a perturbtation, theresulting contribution is suppressed by a factor of m 2 q=m2W. The absence of FCNC is an important constraint on generalizations of the Standard Model. B. STANDARD MODEL 195 4. Grand Unied Theories The Standard Model gives a description of the weak and electromagnetic inter- actions that describes the spin-1 particles in terms of gauge elds, and accounts forall masses by the Higgs eect. However, it does not give any unication, in the sensethat we still have 3 groups (SU(3), SU(2), and U(1)) for 3 interactions (strong, weak,and electromagnetic), and a large variety of spin-1/2 elds that are unrelated ex-cept by color and broken SU(2) avor. Grand Unied Theories unify this symmetryby forcing all 3 gauge groups to be subgroups of a simple group, which is broken toSU(3) SU(2) U(1) by Higgs (and then broken to SU(3) U(1) by more Higgs). This means introducing new spin-1 particles that are unobserved so far because of theirvery large masses. On the other hand, the known fermions are then grouped togetherin a small number of multiplets without introducing new fermions (except perhapspartners for the neutrinos to allow them to have small masses). Unfortunately, thisrequires a more complicated (and ambiguous) Higgs sector, with separate spin-0 mul-tiplets and couplings for rst breaking to SU(3) SU(2) U(1) and then breaking to SU(3) U(1); we won't discuss those Higgs elds here. The simplest such model uses the group SU(5). Recall the SU(3) SU(2) U(1) representations of each family of fermions: q L=( 3;1 2;1 6);qR+=(3;0;1 3);qR=(3;0;2 3);lL=( 1;1 2;1 2);lR=( 1;0;1) where the rst argument is the dimension of the SU(3) representation ( 3b e i n gt h e complex conjugate of the 3), the second is the SU(2) isospin, and the third is the U(1)charge. An SU(3) SU(2) U(1) subgroup of SU(5) can be found easily by taking the 5-component dening representation and picking 3 components as the deningrepresentation of SU(3) and the other 2 for that of SU(2): I.e., consider a tracelesshermitian 55 matrix as an element of the SU(5) Lie algebra, and take (SU(5) )!SU(3) 1 3IU(1) 0 0 SU(2) +1 2IU(1) or in other words 5!(3;0;1 3)(1;1 2;1 2) From this we recognize the fermions as falling into a 510, where the 10 is the antisymmetric product of two 5's, which consists of the antisymmetric product of thetwo 3's (a 3), the antisymmetric product of the two SU(2) doublets, and the product of one of each: 5!(3;0; 1 3)(1;1 2;1 2)=qR+lL 196 IV. MIXED 10!(3;0;2 3)(1;0;1)(3;1 2;1 6)=qRlRqL A more unifying model is based on SO(10). A U(5) subgroup can be found from the spinor representation by dividing up the set of 10 Dirac matrices into two halves, and taking complex combinations to get 5 sets of anticommuting creation andannihilation operators. (See excercise IC1.2.) The Dirac spinor is then (1; 5 2)(5;3 2)(10;1 2)(10;1 2)(5;3 2)(1;5 2) in terms of the SU(5) representation and the U(1) charge. This Dirac spinor is reducible into Weyl spinors 16 16; in fact, ip 2 1is just (1)Y+1=2in terms of the U(1) charge Y. (The SO(10) generators are even in oscillators, and thus do not mix even levels with odd.) We then have 16!(1;5 2)(10;1 2)(5;3 2) Ignoring the U(1) charge, these are the multiplets found for each family in the SU(5) GUT, plus an extra singlet. A simple way to understand this extra singlet is to look at a dierent path of breaking to SU(3) SU(2) U(1): Looking at the vector (dening) representation of SO(10), we can break it up as 6+4 (in the same way we broke up the 5 of SU(5)as 3+2) to get the subgroup SO(6) SO(4)=SU(4) SU(2) SU(2). We can also see that a Dirac spinor of SO(10) (16  16) will be a Dirac spinor of SO(6) (4 4)times (not plus) a Dirac spinor of SO(4), while the Dirac spinor of SO(4) is a deningrepresentation of one SU(2) (( 1 2,0))plus a dening representation of the other SU(2) ((0,1 2)). Thus, 16!(4;1 2;0)(4;0;1 2) w h e r ew eh a v eu s e dt h ef a c tt h a t 1(used for projection to Weyl spinors) of SO(10) is proportional to the product of all the -matrices, and thus the product of 1's for SO(6) and SO(4). Looking at this model (\Pati-Salam model") as an alternative to SU(5) (but with a semisimple, rather than simple, group, so it unies only spin 1/2, not spin 1), we now look at breaking SU(4) !U(3)= SU(3) U(1) (using 4=3+1, as we did 5=3+2 for SU(5)), and breaking one SU(2) !U(1). We then nd (4;1 2;0)!(3;1 3;1 2;0)(1;1;1 2;0) =qLlL (4;0;1 2)!(3;1 3;0;1 2)(3;1 3;0;1 2)(1;1;0;1 2)(1;1;0;1 2) =qR+qRlRlR B. STANDARD MODEL 197 where the arguments are the SU(3) representation, the U(1) charge from SU(4), the SU(2) isospin, and the U(1) charge from the broken SU(2). If we choose theU(1) charge of SU(3) SU(2) U(1) as1/2 times the former of these two U(1) charges plus 1 times the latter, this agrees with the result obtained by way of SU(5). However, we now see that all the left-handed fermions are contained within oneSU(4) SU(2) SU(2) multiplet, and the right-handed within another, but with a partner for the neutrino. Also, one of the SU(2)'s is that of SU(3) SU(2) U(1), while the other is the other SU(2) of the Standard Model, which was broken explic-itly there to U(1), whereas here it is broken spontaneously. Thus, there is a local chiral SU(2) SU(2) avor symmetry. SO(10) .& SU(5) SU(4) SU(2) L SU(2)R &.  1016GeV? SU(3) SU(2)L U(1)R # 100 GeV SU(3) U(1) Furthermore, the SU(4) SU(2) SU(2) model is invariant under C: In general, C is just a permutation symmetry. In this case, it simply switches the two multiplets of each family, C:( 4;1 2;0)$(4;0;1 2) Combining with the usual CP, this model is thus also invariant under P: P:( 4;1 2;0)$(4;0;1 2)*i:e:; (4;1 2;0)$ .(4;0;1 2) But both C and P are broken spontaneously on reduction to the Standard Model. However, SO(10) lacks C and P invariance (contrary to some statements in the liter- ature), since there is only a single complex representation for each family of fermions (and thus no nontrivial C; of course, there is still CP, at least for the vector-spinorcoupling, as always). In fact, the C of SU(4) SU(2) SU(2) is just an SO(10) trans- formation: Although SO(10) is not O(10) (which is why it lacks a C), it still includes re ections in an even number of \axes", since re ection in any pair of axes is a rota- tion (just as for SO(2)). Thus, breaking 10 !6 + 4 includes not only SO(6) SO(4), but also the re ection of an odd number of the \6" axes together with an odd number of the \4" axes | a combined \parity" of both SO(6) and SO(4). (They are all the 198 IV. MIXED same up to continuous SO(6) SO(4) transformations.) This parity of the internal space is the C given above. (We saw a similar situation for O(2) in subsection IVB1.) Since GUTs unify quarks and leptons, they allow decay of the proton. However, since this requires simultaneous decay of all 3 quarks into 3 leptons, it is an extremely unlikely (i.e, slow) decay, but barely within limits of experiment, depending on themodel. Proton decay is still unobserved: This eliminates the simplest version of theSU(5) GUT. REFERENCES 1 The rise of the standard model: Particle physics in the 1960s and 1970s ,e d s .L .H o d - deson, L. Brown, M. Riordan, and M. Dresden (Cambridge University, 1997): interesting accounts of the development of the Standard Model from many of the people responsible. 2J.J.J. Kokkedee, The quark model (Benjamin, 1969); J.L. Rosner, Plenary report on recursive spectroscopy (theory), in Proc. of the XVII Int. Conf. on High Energy Theory , London, July, 1974, ed. J.R. Smith (Science Research Council, 1974) p. II-171; A. De R ujula, H. Georgi, and S. Glashow, Phys. Rev. D12 (1975) 147: nonrelativistic quark model. 3H. Fritzsch and M. Gell-Mann, Proc. XVI International Conference on High Energy Physics , eds. J.D. Jackson and A. Roberts, Chicago and Batavia, Sep. 6-13, 1972 (Na- tional Accelerator Laboratory, 1972) v. 2, p. 135; H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys. Lett. 47B (1973) 365; S. Weinberg, Phys. Rev. Lett. 31(1973) 494, Phys. Rev. D8(1973) 4482: QCD, with connement. 4W. Pauli, Niels Bohr and the development of physics (McGraw-Hill, 1955) p. 30; G. Luders, Ann. Phys. 2(1957) 1: CPT theorem. 5E. Fermi, Ric. Scientica 4(1933) 491; Nuo. Cim. II(1934) 1; Z. Phys. 88(1934) 161: weak interactions, without vector bosons. 6T.D. Lee and C.N. Yang, Phys. Rev. 104(1956) 254: parity violation in weak interactions. 7S.L. Glashow, Nuc. Phys. 22(1961) 579; A. Salam and J.C. Ward, Phys. Lett. 13(1964) 168; S. Weinberg, Phys. Rev. Lett. 19(1967) 1264: unication of weak and electromagnetic interactions. 8N. Cabibbo, Phys. Rev. Lett. 10(1963) 531; M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49(1972) 282. 9S.L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D2(1970) 1285. 10J. Pati and A. Salam, Phys. Rev. Lett. 31(1973) 275: earliest GUT, but semisimple. 11H. Georgi and S. Glashow, Phys. Rev. Lett. 32(1974) 438: SU(5) GUT; earliest with simple group. 12H. Georgi, in Particles and Fields | 1974 , proc. AIP conference, Division of Particles and Fields, Sep. 5-7, 1974, Williamsburg, ed. E. Carlson (American Institute of Physics, B. STANDARD MODEL 199 1975) p. 575; H. Fritzsch and P. Minkowski, Ann. Phys. 93(1975) 193: SO(10) GUT. 13G.G. Ross, Grand unied theories (Benjamin/Cummings, 1984). 200 IV. MIXED :::::::::::::::::::: :::::::::::::::::::: :::::::::::::::::::: C. SUPERSYMMETRY :::::::::::::::::::: In section IIC we studied some general properties of supersymmetry in arbitrary dimensions, and its representations in D=4. We now consider 4D interactions, by introducing gauge elds dened on superspace, and their actions. A complete dis-cussion of supersymmetry would require (at least) a semester; but here we give more than just an overview, and include the basic tools with examples, which is enough for many applications. Quantum aspects of supersymmetry will be discussed in chapters VI and VIII, supergravity in chapter X, and some aspects of superstrings in chapter XI. 1. Chiral We rst consider some eld equations that appear in all free, massless, super- symmetric theories. Of course, since the theory is massless it satises the massless Klein-Gordon equation by denition:  = 0. From our earlier discussion of general properties of supersymmetry, we also know that p. q. =p. q = 0. These don't look covariant, but noticing that pqdiers from pdonly byterms (because of the index contraction), which already vanishes, we have the eld equations p. d. =p. d=0 These equations imply the Klein-Gordon equation, as seen by hitting them with anotherdand using the anticommutation relations fd;d. g=p . .T h e y i m p l y stronger equations: By evaluating at = 0, we nd p.  . =p.  =0 the usual for massless spin 1/2. Another equation that can be imposed is the \chirality" condition d.=0 This requires that be complex, otherwise we would also have d=0a n dt h u s p= 0 by the anticommutation relations. The component expansion is given completely by just the d's and not the d's: j=A; (d)j= ; (d2)j=B whereAandBare complex scalars, and we use the normalization d2=1 2dd C. SUPERSYMMETRY 201 All other components are x-derivatives of these, since the d's can be pushed past the d's (producing p's) until they annihilate . Another way to state this is to use the fact d=eU=2@eU=2; d.=eU=2@.eU=2;U=. p .  to solve the chirality constraint as (x;;)=eU=2^(x;) where ^is independent of : It is dened on \chiral superspace". (In this equation U generates a complex coordinate transformation.) Another way to solve the chirality constraint is to use the covariant derivatives: Since ddd = 0 by anticommutativity (and similarly for d's), d.=0)=d2 where is a \general" (unconstrained) complex supereld. It is the \prepotential" for the eld . From the anticommutation relations we nd [d.;d2]=p.d Since this must vanish on , we nd dd2=d.d2=0)p . d2=0)d2=constant (We can safely ignore this constant, at least when considering the free theory: It corresponds to a term in the action linear in the elds.) This eld equation, togetherwith the chirality constraint, is sucient to determine the theory: Ais the usual free (complex) scalar, is the usual free spinor, and Bis a constant. To describe interactions of this (\scalar") multiplet, we keep the chirality condi- tion, since that greatly simplies the eld content of the supereld. In fact, this isclearly the simplest o-shell supereld we can dene, since it already has the smallestnumber of fermions (as do the coordinates of chiral superspace). (\O shell" meansall components less gauge degrees of freedom.) This means that the equation d 2=0 will be generalized, since it implies the Klein-Gordon equation. The simplest way to do this is by constructing an explicit action, our next topic. 202 IV. MIXED 2. Actions The construction of actions in superspace is dierent from ordinary theories be- cause the geometrically simple objects, the potentials, are constrained, while the un-constrained objects, the prepotentials, can be awkward to work with directly. (This problem is magnied with extended supersymmetry, whose actions we don't consider here.) We start with the simplest supermultiplet, the chiral supereld. Since chiral superelds are dened on chiral superspace, a natural generalization of a potential (nonderivative) term in the action to superspace is S 1=Z dx d2f()+h:c: in terms of some function (not functional) fof chiral superelds (the \superpoten- tial"). We can ignore any dependence because it contributes only total derivatives: =eU=2^(x;))f()=eU=2f(^) Integration over is dened as in subsection IA2; however, now we can replace partial derivatives with covariant ones, since the modication is again only by total deriva- tives: Z dx d2=Z dx d2 with an appropriate normalization. This turns out to be the most convenient one, since it allows covariant manipulations of the action, and the integration can be per- formed covariantly: Since we know that the result of integration gives a Lagrangian that depends only on x, up to total derivative terms, we can evaluate it as Z dx d2f()=Z dx[d2f()]j =Z dx[f0(j)(d2)j+f00(j)1 2(d)j(d)j] (suppressing indices on multiple 's). This gives the result directly in terms of com- ponent elds, using the covariant method of dening the component expansion: In the conventions of the previous subsection, this part of the action becomes Z dx[f0(A)B+f00(A)1 2  ] We now consider integration over the full superspace. As a generalization of the above, we can write Z dx d4K(;)=Z dx(d2d2K)j C. SUPERSYMMETRY 203 Supersymmetric versions of nonlinear models can be written in this way; here we consider just the case where Kis quadratic, which is the one interesting for quantum theory. Since a function of just (or just ) will give zero in the d4integral, we choose K=)S0=Z dx d4 Explicitly, L0=d2d2()=d2(d2)=(1 2)+(i@. d. )d(d2)(d2) !A1 2A+ i@.  . BB where we have used the commutation relations of the covariant derivatives to push all d's past d's to hit . Clearly, this term by itself reproduces the results derived in the previous subsection based on kinematics, so it is the desired massless kinetic term. This could also be seen by deriving the supereld equations of motion by varying the action. Since is constrained, it can't be varied arbitrarily; varying instead the prepotential (=d2 ), we ndd2= 0 (and the complex conjugate). We can now see the in uence of adding the superpotential term to the action: The result of combining the two terms, and then eliminating the auxiliary eld Bby its equation of motion, is S0+S1!L=A1 2A+ i@.  . +jf0(A)j2+[f00(A)1 2  +h:c:] For example, a quadratic fgives mass to the physical scalar and spinor. This action is invariant under modied supersymmetry transformations, where the aux iliary elds are replaced by their equations of motion there also; those transformations then become nonlinear in the presence of interactions. Note that the scalar potential ispositive denite; this is a consequence of supersymmetry, since it implies that the energy is always positive. Excercise IVC2.1 Find the explicit form of the component-eld action for arbitrary K( i;i) andf(i) for an arbitrary number of chiral superelds i, including all in- dices. Eliminate the auxiliary elds from the action, and nd the modied supersymmetry transformations. Show by direct evaluation that the action isstill invariant. As a notational convenience, we can drop the \ j" after expanding a superspace action in components: For example, we can write simply =d; B =d2 204 IV. MIXED After performing the -integration as above by using derivatives dand d,a n dt h e n \evaluating" these derivatives on by writing andB, the component action is expressed completely in terms of such superelds and only spacetime derivatives @ . . This component action is independent of (the Lagrangian is independent up to total spacetime derivatives): This is the statement of supersymmetry invariance. Thus, wecan choose to evaluate at =0 ,o r=, or whatever; it is irrelevant. It is then understood that the relation to the usual component actions is simply to treat the supereld as a component eld, since the -derivatives (in dandR d) have been eliminated. From now on we will generally drop the j's. Since chiral superelds are essentially independent of , not only integration is modied, but also (functional) variation. Since a chiral supereld is (up to a trans- formation) an arbitrary function on chiral superspace, we dene S[]=Z dx d 2()S  for an arbitrary variation of a chiral supereld , and similarly for varying .I n evaluating such variations, we make use of the identities Z dx d4L=Z dx d2d2L d2d2=1 2 (d2d2d2=1 2d2) Thus, to vary a general action, it is convenient to rst integrate over , and then vary in the naive way: For example, S=Z dx d4+Z dx d2f()+h:c: ) 0=S =d2+f0() Excercise IVC2.2 Check for this action that the component expansion of the supereld equationsof motion agree with the variation of the corresponding component action. 3. Covariant derivatives The supersymmetric generalization of nonabelian gauge theories can be derived by similar methods. We rst write the supersymmetry covariant derivatives collectivelyas d A=(d;d.;@.)=EAM@M C. SUPERSYMMETRY 205 @M=(@;@.;@m)=@=@zM;zM=(;.;xm) Unlike the nonsupersymmetric case, the \vielbein" EAMhasdependence even in \ at" superspace, and thus the \torsion" Tis nonvanishing: [dA;dBg=TABCdC T .  . =T.  . =i . . ;r e s t =0 We now gauge-covariantize all the supersymmetry-covariant derivatives: rA=dA+iAA The covariant eld strengths are then dened as [rA;rBg=TABCrC+iFAB From our analysis of general representations of supersymmetry in D=4 in subsection IIC5, we know that the simplest supersymmetrization of Yang-Mills is to include a spinor with the vector, in terms of physical degrees of freedom. (The spinor and vector each have two physical degrees of freedom, one for each sign of the helicity.)O shell, Fermi and Bose components must still balance, so there must also be an auxiliary scalar. From dimensional analysis, the eld strengths must therefore satisfy F =F.. =F . =0 ;F ;. =iCW. ;F.;. =iC.. W whereWjis the physical spinor. The constant piece of the torsion implies stronger relations among the eld strengths than in nonsupersymmetric theories. For super Yang-Mills we nd fromthe Jacobi identity for the covariant derivatives the Bianchi identity for the eld strengths r [AFBC)=T[ABjDFDjC) Specically, the dimension-1 constraints above on the eld strengths imply the dimen- sion-3/2 algebraic constraint that denes W,a sw e l la s F .;. =C1 2r(.W. )+C.. 1 2r(W) They also imply that Wis covariantly chiral and satises a \reality" condition, r.W=0;rW+r.W.=0 The most straightforward way to derive these results is to just evaluate the Jacobi identities directly. We begin with a weaker set of conditions, both of dimension 1, that 206 IV. MIXED will be found (in the following subsection) to be necessary and sucient for solving explicitly. One directly determines the vector derivative in terms of the spinor ones: F . =0)ir.=fr;r.g Since one could always dene the vector covariant derivative this way, imposing this condition simply eliminates redundant degrees of freedom. The remaining constraint (including its complex conjugate) allows coupling of super Yang-Mills to the chiral supereld: r.=0) 0=fr.;r. g=iF..  It also implies the maintenance of certain free identities, such as rr=1 2[r;r]+1 2fr;rg=Cr2 (Such constraints appear also for rst quantization, e.g., in superstring theory, when- ever a supersymmetric system is put in a background of a supersymmetric gauge eld of higher superspin. This should not be confused with background eld equations imposed by any gauge system put in a background of the same type: see subsection VIB8.) Thus, our minimal set of constraints can be written directly in terms of the eld strengths as F=F.. =F . =0 but for our purposes it will prove more convenient to write them directly as (anti)com- mutators: fr;rg=fr.;r. g=0;fr;r. g=ir .  The solution to the dimension-3/2 Jacobis are then [r(;fr;r )g]=0)trivial [r(;fr);r. g]+[r. ;fr;rg]=0) [r;r. ]=CW. for some eld W, simply applying the constraints to drop fr;rgand replace fr;r. gwithr . . Similarly, we nd from the dimension-2 Jacobis fr(;[r);r . ]g+[r . ;fr;rg]=0)rW. =0 [r.;fr;r. g]+fr;[r. ;r.]g+fr. ;[r;r.]g=0 C. SUPERSYMMETRY 207 ) [r.;r . ]=i(Cf.. +C.. f);f=1 2r(W);rW+r.W.=0 where we separated the last equation into its (Lorentz) irreducible pieces. (The dimension-5/2 and 3 identities are redundant.) Excercise IVC3.1 Explicitly evaluate all the remaining Jacobi identities, and show that theyimply no further conditions on W . 4. Prepotential We saw in the previous subsection that coupling super Yang-Mills to matter gave directly one of the minimal constraints on the super Yang-Mills elds themselves. Hence, as for ordinary Yang-Mills, the denition of the gauge theory follows fromconsidering the transformation of matter, and generalizing it to a local symmetry. As for self-dual Yang-Mills (see subsection IIIC5), the vanishing of some eld strengths implies that part of the covariant derivative is pure gauge: fr ;rg=0)r=e de However, sincefr;r. g6= 0, this gauge transformation (\prepotential") is com- plex. We therefore have the covariantly chiral supereld r.=0;r.=e d.e )=e ^; d .^=0 Alternatively, we could combine this exponential with that already contained in the free spinor derivative: r=eU=2 @eU=2+ ; =eU=2+ ^; @ .^=0 U+ 2 is the analog of the covariant derivative for the Yang-Mills prepotential. This is a hint at supergravity: Uis just the at piece of the supergravity prepotential. We thus see that supersymmetry automatically gives gravity the interpretation of the gauge theory of translations. Component expansions are now dened with Yang-Mills-covariant derivatives: r= ;r2=B rW=f+iCD;r2W=ir. W.  w h e r ew eh a v eu s e dt h eB i a n c h ii d e n t i t i e sf o r W,a n df(not to be confused with F) is the usual Yang-Mills eld strength (in spinor notation). The \vector multiplet" 208 IV. MIXED thus consists of the component elds Aa(the gauge eld whose strength is f),W, andD(auxiliary). (As explained earlier, we drop all j's.) Note that the gauge parameter is real, while the matter multiplet is (covariantly) chiral. The resolution of this apparent inconsistency is that solving the constraintsintroduces a new gauge invariance: r 0 A=eiKrAeiK;r=e de )e 0=eie eiK;d=0 0=eiK;  =e ^) ^0=ei^ This suggests the denition of a new (\chiral") representation, where we use the obvious eld ^and the chiral gauge parameter  replaces the real one K:M a k i n ga nonunitary similarity transformation, brA=e rAe )br.=d.;br=eVdeV;eV=e e ^=e ;^=e )d.^=0;^=(^)yeV br0 A=eibrAei;eV0=eieVei Alternatively, we can also include Uin the transformation as above; then UandV appear only in the combination U+V. Excercise IVC4.1 Show that the explicit expression for the eld strength Win terms of the prepotential Vin the chiral representation is W=id2(eVdeV) Show this expression is chiral. Excercise IVC4.2 In the Abelian case, give an explicit component expansion of the prepotential V, such that the vector potential Aa, the physical spinor W, and the auxiliary eldDappear as independent components. Note that the other components do not appear explicitly in component expansions when gauge-covariant ex- pansion (r:::j) is used. The component (nonsupersymmetric) gauge where these components are set to vanish is the \Wess-Zumino gauge", and is the  part of the radial gauge of subsection VIB1 below. Excercise IVC4.3 For some purposes (like quantization) we need the explicit form of an in-nitesimal gauge transformation of V. Show this can be written as V=iL V=2[( + ) +coth(LV=2)()] C. SUPERSYMMETRY 209 (Hint: Consider eVeV,a n dt h i n ko f as an operator, as for the expansion ofr=eVdeV.LAwas dened in subsection IA3.) 5. Gauge actions Generalization of actions to super Yang-Mills theory is straightforward. Matter coupling is achieved simply by replacing the chiral superelds of the matter multiplets with Yang-Mills-covariantly chiral superelds. The coupling can be seen explicitly in the chiral representation: In the kinetic term, =(^)yeV^ while in theR d2term allV-dependence drops out because of gauge invariance. (The superpotential is a gauge invariant function of the 's, and the transformation to the chiral representation is a complex gauge transformation. The fact that the gauge transformation is complex is irrelevant, since the superpotential depends only on  and not .) Component expansion can be performed covariantly by replacing d's with r's in the denition of integration: Since the Lagrangian is a gauge singlet, this is the same acting on it, although individual terms in the expansion dier because the elds are not singlets. Similarly, d2can be replaced with r2also when performing  integration for purposes of varying an action with respect to a chiral supereld. This is equivalent to gauge covariantizing the functional derivative (e.g., by transforming from a chiral representation) as (x;) (x0;0)=r2(xx0)4(0) Usually we will drop the \ b"'s onand, when the representation is clear from the context by the use of explicit V's. The action for super Yang-Mills itself follows from dimensional analysis: Since eachintegral is really a derivative, d2integration has mass dimension +1, the same as a spacetime derivative. Since the Lagrangian for a physical spinor, in this caseW, has a single such derivative, dimensional analysis says the action must be SsYM=1 g2trZ dx d21 2WW where the (covariant) chirality of Wallows integration over chiral superspace. (Sim- ilar analysis applies to the matter multiplet, whereRd4takes the place of a for the scalar.) ReplacingRd2!r2, we evaluate the component expansion as SsYM=1 g2trZ dx(1 2ff+Wir. W. D2) 210 IV. MIXED Another term we can write, for superelectromagnetism (supersymmetrization of an Abelian gauge theory) is the \Fayet-Iliopoulos term" SFI=Z dx d4V=Z dx D which involves only the auxiliary eld D. (The analog for the chiral scalar supereld isR dx d2.) Excercise IVC5.1 Derive the supersymmetric analog of the St uckelberg model of subsection IVA5, by coupling an Abelian vector multiplet to a massless chiral scalar multiplet using the symmetry generator Tdened there. ( G!iTin trans- formation laws, covariant derivatives, etc., on ,w h e r eT=1)T2=0 . ) aTo couple the gauge eld it is necessary to start, as usual, with a (quadratic) matter action that is globally invariant under this symmetry: S0=Z dx d41 2()2 (At this point this is the usual, since only the cross-term survives, but this will not be the case for the covariantly chiral superelds.) Find the super- symmetric gauge coupling, and express the resulting action in terms of Vand ^. bUse this result to nd the mass term for Vin the gauge ^=0 . Another interesting form of the action uses a generalization of the Chern-Simons form dened in the discussion of instantons in subsection IIIC6. In superspace, the calculation of the eld strength with curved indices is modied to rM=EMArA=@M+iAM;i[rM;rNg=FMN=EMAENBFAB where we have left sign factors from index reordering in the last equation implicit. Although the curved-index expressions are not as useful (for example, for seeing whichcomponents vanish by constraints), we can see easily that some arguments used in nonsupersymmetric theories carry over to superspace. Thus, we can dene the super Chern-Simons form by 1 8tr F [MNFPQ)=1 6@[MBNPQ ) BMNP =tr(1 2A[M@NAP)+i1 3A[MANAP)) Converting to at superspace (again with some implicit sign factors), BABC=EAMEBNECPBMNP =tr(1 2A[AdBAC)1 4A[ATBC)DAD+i1 3A[AABAC)) C. SUPERSYMMETRY 211 In terms of this expression, the super Yang-Mills action can be written simply in terms of the spinor-spinor-vector part B . cofBABCas SsYM; 1=1 2i1 g2trZ dx d4B;.. Note that the fact that the curl of Bis gauge invariant implies that Btransforms under a gauge transformation as the curl of something, and thus the integral of anypart ofBis gauge invariant (up to possible torsion terms: see the excercise below). Furthermore, we can drop the F . = 0 constraint on the Ain this action; it follows from variation with respect to A . . One simple way to check this action is to use the chiral representation A.= 0: Then only the A. $ d. Aand (A . )2terms contribute, andA . =id. A, whileW=d2A,s oR d2integration givesR dx d2W2. Excercise IVC5.2 Derive the expression for BABCdirectly using only at indices: aStart with F[ABFCD)expressed in terms of TandA, and write it as a total derivative plus torsion terms. bDo the same for the gauge transformation of B. Show that the torsion terms do not contribute to B.; ;.. The multiplets and couplings we have considered are sucient to write a super- symmetric generalization of the Standard Model. Unfortunately, supersymmetry pro- vides no unication. To get the right symmetry breaking, it turns out to be necessary to provide a supersymmetry multiplet for each particle of the Standard Model: Thespin-1 gauge bosons are accompanied by spin-1/2 \gauginos" (\gluinos", \photino",\Wino", \Zino"), the spin-1/2 leptons by spin-0 \sleptons", the quarks by \squarks",and the spin-0 Higgs' by spin-1/2 \Higgsinos". Furthermore, since a reality condition can't be imposed on chiral scalar multiplets, the Higgs scalars are themselves doubled. Ultimately, the success of supersymmetry depends on the experimental detection ofthese particles. 6. Breaking The methods of the section IVA can be generalized straightforwardly to supersym- metric theories: Goldstone bosons and Higgs elds become supermultiplets, etc. How- ever, to obtain realistic models supersymmetry itself must be broken, since fermions and bosons with similar mass and other properties are not observed in nature. Morespecically, since gravity is observed, any supersymmetric theory of the world must 212 IV. MIXED include supergravity, and thus the breaking must be spontaneous. (Explicit breaking would violate gauge invariance.) Then the gravitino, which gauges supersymmetry,will become massive by a superhiggs mechanism, by eating a Goldstone fermion. (Seesubsections XB6-7. If the graviton and gravitino are treated as composites, then this fermion could also be a composite.) We saw in subsection IIC1 that energy is always nonnegative in supersymmetric theories. In particular, from the same arguments used there we see that a state can be invariant under supersymmetry ( qj i=q yj i= 0) if and only if it has zero energy. Any such state can be identied as the vacuum, since no state has lower energy.This means that the only way to guarantee spontaneous supersymmetry breaking isto choose a theory which has no zero-energy state. (Note that energy is uniquely dened by the supersymmetry algebra; there is no possibility of adding a constant as in nonsupersymmetric theories.) In theories with extended supersymmetry, therelation between supersymmetry and energy applies for each supersymmetry; thussupersymmetry is either completely broken spontaneously or completely unbroken. (An exception is central charges, which modify the supersymmetry algebra: See the following subsection.) Furthermore, physical scalars appear at = 0 in matter multiplets, while auxiliary elds appear at higher order. Since supersymmetry breaking requires dependence in a vacuum value of a supereld, this means an auxiliary eld must get a vacuumvalue. A simple example of spontaneous supersymmetry breaking is the O'Raifeartaigh model; it has the Lagrangian L O'R=Z d43X i=1ii+Z d2(1+m23+ 12 2)+h:c: To study symmetry breaking we ignore derivative terms, since vacuum values are constants. Then the scalar eld equations are:  Bi! Bi+@if=0:B1++A2 2=B2+mA 3+2A1A2=B3+mA 2=0  Ai!Bj@i@jf=0: 2A2B2=mB 3+2A2B1+2A1B2=mB 2=0 (where@i=@=@Aion the superpotential f(A)). Since there is no solution for Bi=0 , supersymmetry breaking is required. In general, for superpotential f(), the eld equations for B=0a r ef0(A) = 0, so a linear term is always needed for supersym- metry breaking. C. SUPERSYMMETRY 213 With Abelian vector multiplets, a Fayet-Iliopoulos termR d4Vcan also generate such breaking, since it also is a linear term of an auxiliary eld. Excercise IVC6.1 Evaluate the Lagrangian R d4for covariantly chiral by using covariant -integration,R d4=r2r2. For the case of U(1) gauge theory, add the action for the gauge supereld with a Fayet-Iliopoulos term, and nd the potential for the physical scalars by eliminating the auxiliary eld Dby its eld equation. For simplicity (as in this chapter), we may want to ignore supergravity; however, we still need to take account of its contribution to breaking global supersymmetry viathe superhiggs eect. The net low-energy contribution from the supergravity elds (assuming no cosmological constant is generated) is to introduce eective explicit supersymmetry breaking: Although the original theory is locally supersymmetric, weneglect the supergravity elds but not their vacuum values (in particular, those ofthe auxiliary elds). In particular, if the supergravity elds are bound states, then this procedure is essentially the classical introduction of nonperturbative quantum eects. Thus we consider adding terms to the classical action that break supersymmetry explicitly. The easiest way to do this is to introduce constant superelds (\spurions");this allows us to continue to take advantage of the superspace formalism (at both the classical and quantum levels). Since we are neglecting (super)gravity, and in particular its nonrenormalizability (see chapter VII), we consider only terms that willpreserve the quantum properties of the unbroken theories. This will clearly be thecase if we consider only the usual terms, with some elds replaced by spurions: This is equivalent to using background (xed) elds, in addition to (but in the same way as) the usual eld variables, performing all (classical/quantum) calculations as usual,and then setting the background elds (specically, the auxiliary elds, which areresponsible for breaking supersymmetry) to constants. Thus, introducing constant (in x) chiral and real spurion elds '= 2c;V=22r in terms of complex and real parameters candr, in addition to the true elds and V, we have terms of the form Z d2[';'2;'3;'W2;(d2dV)W];Z d2d2VeV 214 IV. MIXED (and complex conjugates). These terms can preserve the usual gauge invariances, and can be shown to also preserve the desirable quantum properties of supersymmetry:The condition is that replacing the spurion eld by 1 (instead of its above value)gives either 0 or a conventional term (one with coupling constant of nonnegative mass dimension). Another way to introduce these spurions (except perhaps for the Vcrossterm, which is less useful) is as coupling constants , rather than as elds: Instead of introducing new terms to the action, we generalize the old ones, so the constant part of each coupling is the usual coupling, while its -dependent terms produce the breaking. Excercise IVC6.2 Find the component expansions of the above explicit breaking terms. Whatare the mass dimensions of the constants candrin the various cases? Excercise IVC6.3 Expand the Lagrangian L=Z d 4+Z d2(1 63+')+h:c: in components. Find the masses. 7. Extended The supersymmetry we discussed earlier in this chapter, with a single spinor coor- dinate, is called \simple (N=1) supersymmetry"; the generalization to many spinors is called \extended (N >1) supersymmetry" (for N spinor coordinates). N=1 supersym- metric theories, at least for spins 1, are most conveniently described by superspace methods. (There are also some denite advantages for N=1 supergravity at the quan- tum level.) On the other hand, the technical diculties of extended superspace often outweigh the advantages. (The main advantage of extended superspace is proving cer-tain properties of the quantum theories. Of course, extended supersymmetric theoriesare complicated in any case.) Alternative formulations of extended supersymmetry are either (1) on shell, (2) in terms of components (ordinary spacetime, not super- space), or (3) in simple superspace (manifesting only one of the supersymmetries). By going half way, using N=1 superelds to describe extended supersymmetry, some of the advantages of the superspace approach can be retained. In this subsectionwe will list some of the extended supersymmetric actions for lower spins in N=1 superspace form. These actions can be obtained by: (1) using extended superspace to derive the component eld equations (usually using dimensional reduction: see C. SUPERSYMMETRY 215 subsections XC5-6), and combining components into N=1 superelds, or (2) writing the extra supersymmetries in N=1 superspace form, and using them to determine the action. The simplest example is N=2 supersymmetry. As for any extended supersymme- try, its algebra can be modied by including Abelian generators Z(with dimensions of mass), called \central charges": fqi;qj. g=j ip . ;fqi;qjg=CCijZ;fqi.;qj. g=C.. CijZ;[Z;q]=[Z;q]=0 (wherei=1;2). In terms of dimensional reduction (for N=2, from D=5 or 6; see subsections XC5-6), the origin of these generators can be understood as the higher- dimensional components of the momentum. N=2 supersymmetry is sometimes called\hypersymmetry", and N=2 supermultiplets, \hypermultiplets". Our rst example is the free, massive N=2 scalar multiplet: Since we already know the eld content (see subsection IIC5), it's easy to write the free Lagrangian L sm;N =2=Z d4i0i0+1 2Z d2mi0j0i0j0+h:c: where the index \ i0" is for an extra SU(2) (not the one acting on the supersymmetry generators), broken by the mass term, and the mass matrix mi0j0is symmetric while mi0j0=Ck0i0mk0j0is hermitian. In other words, it represents a 3-vector of this SU(2), and thus a generator of the preserved U(1) subgroup, which we have used to denethe central charge: Z i0=mi0j0j0 The other N=2 multiplet of low spin is the vector multiplet. It also has a simple Lagrangian, LsYM;N =2=1 g2trZ d2W2+Z d4 whereis covariantly chiral and in the adjoint representation of the Yang-Mills gauge group. In the Abelian case, we can also add an N=2 Fayet-Iliopoulos term, LFI;N =2=Z d4c0V+Z d2c++h:c: where (c0;c+;c)(c=c+*) is a constant 3-vector of the SU(2) of the N=2 super- symmetry: The 3 scalar auxiliary elds of this N=2 multiplet form a 3-vector of the SU(2). Unlike the previous example, this multiplet has all the auxiliary elds neededfor an o-shell N=2 superspace formulation: Not only do the physical components balance between bosons and fermions (4 of each), but also the auxiliary ones (also 4 of each). 216 IV. MIXED These 2 N=2 multiplets can be coupled: The scalar multiplet action is modied to Lsm;N =2=Z d4i0i0+1 2Z d2i0j0i0(+M)j0+h:c: where now i0is also a representation of the Yang-Mills group (not necessarily ad- joint), with respect to which it is covariantly chiral. However, the same SU(2) matrix that appears in the mass matrix mi0j0=Mi0j0now also appears with the N=2 super Yang-Mills elds, rAi0=dAi0+iAn AGni0j0j0; =nGn whereGnare the usual Yang-Mills group generators. (Without loss of generality, we can choose i0j0=1 00 1
;t h e n+0is some arbitrary representation of the Yang-Mills group, while 0is the complex conjugate.) Note that the mass term appears in exactly the same way as an Abelian N=2 vector multiplet that has been replaced by a vacuum value for its physical scalars. This can also be seen from the commutationrelations for the N=2 super Yang-Mills covariant derivatives (see below), since the scalars appear in exactly the same way as the central charge. By our earlier helicity arguments, the only N=3 supersymmetric theory with spins 1 is N=3 super Yang-Mills. The anal ogous statement also holds for N=4, while no such theories exist for N >4. Since theories with N supersymmetries are a subset of those with only N 1 supersymmetries, N=3 and N=4 super Yang-Mills must be the same: Counting states of supersymmetry representations, we see that this theoryis the same as N=2 super Yang-Mills coupled to one N=2 scalar multiplet in the adjoint representation (in direct analogy to N=2 super Yang-Mills in terms of N=1 multiplets). In terms of N=1 multiplets, this is super Yang-Mills plus 3 adjoint scalarmultiplets. The action then follows from the above results (without central charges and Fayet-Iliopoulos terms): L sYM;N =4=1 g2tr Z d2W2Z d4II+Z d21 6IJKI[J;K]+h:c: where \I" is a U(3) index. (The U(1) part of the U(3) symmetry involves also a phase transformation of the 's.) As for o-shell N=1 supersymmetry, much information on extended supersym- metric gauge theories can be gained by examining the properties of the covariant derivatives and their eld strengths. In fact, this is more true in the extended case, where the \obvious" constraints often imply eld equations (which is more than one would want for an o-shell formulation). The empty-space covariant derivatives are C. SUPERSYMMETRY 217 the direct generalization of N=1: Introducing N 's asi(and complex conjugate i.), where \i" is an N-valued index with as much as a U(N) symmetry, dA=(di;di.;@.);di=@ii1 2i.@.;di.=@i.i1 2i@. Ti;j.  . =Tj. ;i . =ij i . . ;r e s t =0 Excercise IVC7.1 Find the superspace representation of the extended supersymmetry generators(which anticommute with these covariant derivatives). For N=2, include the central charge. By denition, extended super Yang-Mills has only spins 1 and less. Dimensional analysis then gives the unique result, including physical elds only, fr i;rj. g=j iir .  fri;rjg=Ciij [ri.;ir . ]=C. .iWi  [r.;r . ]=Cif.. +C.. if (and complex conjugates of some of these equations). This corresponds directly to our discussion in subsection IIC5, where we saw that a general representation looked like antisymmetric tensors ;i;ij;:::of U(N), corresponding to helicities h, h 1/2, h1,... . In this case, h=1, and these helicities come from the surviving on-shell components of f;Wi;ij;... . For N=4 we have self-duality with respect to charge conjugation (see also subsection IIC5), ij=1 2ijklkl Excercise IVC7.2 Analyze the Bianchi identities of these covariant derivatives. Show that for N>2 they imply the eld equations. Find a component action that yields these eld equations for N=4. An interesting simplication of extended superspace occurs for self-duality: Con- straining f=Wi=ij=0 and dropping the self-duality condition for N=4 (so ij6= 0), we nd all commutators involvingri.are trivial: fri;rj. g=j iir . ;fri.;rj. g=[ri.;r . ]=0 218 IV. MIXED while all the remaining commutators have a similar form: fri;rjg=Ciij; [ri;ir . ]=CiW i. ; [r.;r . ]=Cif..  The latter result suggests we combine the internal and dotted spinor indices as A=(.;i) so that we can combine the nontrivial equations as [rA;rBg=iCfAB The former equations then allow us to interpret the remaining covariant derivatives ri.as a subset of the SL(2 jN) generators that rotate the Aindex, which form a subgroup of the superconformal group (S)SL(4 jN). We therefore restrict ourselves to the chiral superspace described by the coordinates zA=(x.;i) The net result is that we have a superspace with no torsion, with coordinates that represent half of the supersymmetries as translations and the other half as rotations. By comparison with our treatment of the self-dual bosonic theory in subsections IIIC5-7, we see that we can extend trivially all our results for the bosonic case to the(extended) supersymmetric case by simply extending the range of the indices. In par-ticular, we also have a chiral twistor superspace: Extending the range on the twistorcoordinates z Aused there soAis now an SL(4jN) index, the superconformal group is now manifest, and all the methods and results there (e.g., the ADHM construction)apply automatically to the supersymmetric case. REFERENCES 1 Wess and Zumino, loc. cit. ( I I C ,r e f .2 ) : free scalar and vector multiplets, Wess-Zumino gauge. 2J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52: interacting chiral scalar multiplets. 3J. Wess and B. Zumino, Nucl. Phys. B78 (1974) 1: coupling Abelian vector multiplet to scalar multiplets. 4S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413; A. Salam and J. Strathdee, Phys. Lett. 51B (1974) 102: nonabelian vector multiplets. 5J. Wess and B. Zumino, Phys. Lett. 66B (1977) 361; J. Wess, Supersymmetry-supergravity, in Topics in quantum eld theory and gauge theories , VIII GIFT Int. Seminar on Theoretical Physics, Salamanca, Spain, Jun 13-19, C. SUPERSYMMETRY 219 1977 (Springer-Verlag, 1978) p. 81: covariant derivatives for super Yang-Mills. 6P. Fayet and J. Iliopoulos, Phys. Lett. 51B (1974) 461. 7S.J. Gates, Jr. and H. Nishino, Int. J. Mod. Phys. A 8(1993) 3371: Chern-Simons superform. 8L. O'Raifeartaigh, Nucl. Phys. B96 (1975) 331. 9L. Girardello and M.T. Grisaru, Nucl. Phys. B194 (1982) 65: general soft explicit supersymmetry breaking, with spurions. 10L.A. Avdeev, D.I. Kazakov, and I.N. Kondrashuk, hep-ph/9709397, Nucl. Phys. B510 (1998) 289;N. Arkani-Hamed, G.F. Giudice, M.A. Luty, and R. Rattazzi, hep-ph/9803290, Phys. Rev.D58 (1998) 115005: spurions in couplings. 11H.P. Nilles, Phys. Rep. 110(1984) 1: review of supersymmetry phenomenology. 12P. Fayet, Nucl. Phys. B113 (1976) 135: extended supersymmetry. 13P. Fayet, Nucl. Phys. B149 (1979) 137: central charges. 14F. Del Aguila, M. Dugan, B. Grinstein, L. Hall, G. Ross, and P. West, Nucl. Phys. B250 (1985) 225; L. Girardello, M. Porrati, and A. Zaaroni, hep-th/9704163, Nucl. Phys. B505 (1997) 272: semi-realistic N=2 models. 15F. Gliozzi, J. Scherk, and D.I. Olive, Nucl. Phys. B122 (1977) 253; L. Brink, J.H. Schwarz, and J. Scherk, Nucl. Phys. B121 (1977) 77: N=4 super Yang-Mills. 16R. Grimm, M. Sohnius, and J. Wess, Nucl. Phys. B133 (1978) 275; M.F. Sohnius, Nucl. Phys. B136 (1978) 461; E. Witten, Phys. Lett. 77B (1978) 394: covariant super Yang-Mills derivatives for extended supersymmetry. 17W. Siegel, hep-th/9205075, Phys. Rev. D46 (1992) R3235; hep-th/9207043, Phys. Rev. D47 (1993) 2504; loc. cit. (IIIB, ref. 3): conformal chiral superspace. 18Gates, Grisaru, Ro cek, and Siegel, loc. cit. 220 V. QUANTIZATION PART TWO: QUANTA Many important new features show up in eld theory at the quantum level. Prob- ably the most important is \renormalizability", which states that all the parameters (masses and couplings) that appear as coecients of terms in the action must havenonnegative mass dimension (when the massless part of the kinetic term has no di- mensionful coecient). Since the action is dimensionless,R d 4xhas dimension4, and the elds have positive dimension, this allows only a small number of terms for any given set of elds. This one condition gives relativistic quantum eld theory more predictive power than any known alternative. There are many perturbation expansions that can be applied to quantum eld theory. One is the mechanical JWKB expansion, which is an expansion in derivatives.Of the inherently eld theoretical expansions, the simplest is to expand directly in elds, or equivalently, in the coupling constants. This expansion is the basis of per- turbative quantum eld theory. However, this expansion does not preserve gauge invariance term by term. On the other hand, the terms in this expansion can be collected into small subsets that do preserve gauge invariance. There are three suchregroupings, discussed in the four following chapters, and they are based on pertur- bation expansions: (1) the eld theoretic JWKB (\loop") expansion, (2) expansions in spin or helicity, and (3) expansions in internal symmetry (color or avor). V. QUANTIZATION For the most part, integrals are hard to evaluate, in particular the path integrals of exponentials that appear in quantum theory. The only exponentials that are generally easy to integrate are Gaussians, and the products of them times polynomials, which can in turn be evaluated as derivatives of Gaussians. Such integrals are the basis of perturbation theory: We keep the quadratic part of the action, but Taylor expand theexponential of higher-order terms. Eectively, this means that we not only expand in orders of  hto perturb about the classical theory, but also expand in orders of the coupling constants to perturb about the free theory. This makes particularlyuseful our analysis of relativistic quantum mechanics (as free eld theory). The JWKB expansion for the wave function (or S-matrix) expands the exponent in powers of h, dividing it onto three qualitatively dierent parts: (1) negative powers of  h (generally 1 =honly), which describe the classical theory (they dominate the classical limit h!0), whose physical implications have been considered in previous chapters; A. GENERAL 221 (2) h-independent, where almost all of the important (perturbative) quantum features appear (including topological ones, and quantum breaking of classical symmetries);and (3) positive powers, which give more quantum corrections, but little new physics, except when summed to all orders. These are generally known as \trees", \one-loop", and \multiloop", because of their graphical interpretation. :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: A. GENERAL :::::::::::::::::::::::::::: In the Schr odinger approach to quantum mechanics one solves a dierential equa- tion. The Feynman approach is complementary: There one performs an integral.Integrals are solutions to dierential equations (e.g., f 0=g)f=R g), but usually dierential equations are easier to solve than integral equations. However, there is animportant exception: Gaussian integrals are easy, and so are their boundary condi-tions. In eld theory the most important approximation is one where the integrandis approximated as a Gaussian, and the exact integral is evaluated as a perturbationabout that Gaussian. Of course, solving the corresponding dierential equation isalso easy, but in that case the integral is easier because it corresponds to workingwith the action, while the dierential equation corresponds to working with the eldequations. A major advantage of Feynman's approach is that it allows space and time to be treated on an equal footing. For example, as in classical electrodynamics, wecan solve the wave equation inside a spacetime volume in terms of conditions on theboundary of that volume: It is not necessary to choose the spatial boundary at innityso that it can be ignored, and divide the temporal boundary into its \future" and\past" halves so that all conditions are \initial" ones imposed at the past boundary.It is not even necessary to distinguish between preparation (\if") and measurement(\then") when describing probabilities: We can instead ask the probability of a givenwave function describing the whole boundary. This is a particular advantage forrelativistic quantum eld theory, where space and time are more closely related thanin nonrelativistic theories. We now \review" Feynman's approach for general quantumsystems, and quantum mechanics in particular, so that it can be applied withoutfurther explanation when we come to quantum eld theory. 1. Path integrals In Feynman's path integral approach to quantum mechanics (based on an analogy of Dirac), the action is the starting point for quantization. The basic idea is to begin 222 V. QUANTIZATION with the basic quantity in quantum mechanics, the transition amplitude, and write it as an integral of the action hfjii=Z D eiS[] whereR Dis a \functional integral": Integrate over (t)f o re a c h t(with some appropriate normalization). The boundary conditions in tare dened by the choice of initial and nal states. In this subsection we will dene this integral in a more explicit way by breaking up the time interval into discrete points and taking the continuum limit; in the next subsection we will study ways to evaluate it using itsgeneral properties. The path integral can be derived from the usual Hamiltonian operator formal- ism. Considering for simplicity a single coordinate q, the wave function is given in coordinate space by (q)=hqj i;j i=Zdq p 2 (q)jqi w h e r ew eu s et h ec o n v e n i e n tn o r m a l i z a t i o n s Zdqp 2jqihqj=1=Zdpp 2jpihpjh hqjq0i=p 2(qq0);hpjp0i=p 2(pp0)i for coordinate and momentum space. To describe time development, we work in the Heisenberg picture, where time dependence is in the operators (and thus their eigenstates): (q;t)=hq;tj i Time development is then given completely by the \propagator" or \Green function" G(qf;tf;qi;ti)hqf;tfjqi;tii) (qf;tf)=Zdqip 2G(qf;tf;qi;ti) (qi;ti) Excercise VA1.1 Let's review the relationship between time development in the Heisenberg and Schrodinger pictures. Using the usual relation h jQ(t)jih (t)jQj(t)i between the time-independent states j iand time-dependent operators Q(t) of the Heisenberg picture and the time-dependent states j (t)iand time- independent operators Qof the Schr odinger picture, dene time-dependent eigenstates in two ways: Qjqi=qjqi)hq(t)j (t)ihqj i (q;t)hqj (t)ihq;tj i A. GENERAL 223 Given the time development of a state j (t)i=U(t)j i nd the development of O(t),jq(t)i,a n djq;ti, and show in particular that jq(t)i6=jq;ti. Which is the eigenstate of Q(t)? In general, even for time-dependent Hamiltonians, we can nd the innitesimal time development explicitly from the denition of the time derivative and the time- dependent Schr odinger equation: [i@tH(i@q;q;t)]hq;tj=0 )hq;t+j=hq;tjf1iH[P(t);Q(t);t]ghq;tjeiH[P(t);Q(t);t] and similarly for hp;t+j(wherePandQare the Hilbert-space operators). To derive the path-integral formalism, we then iterate this result to obtain nite timedevelopment by inserting unity innitely many times, alternating between coordinate and momentum, hq f;tfjqi;tii=Zdp0p 2dq1p 2dp1p 2:::hqf;tfj::: :::jp1;ti+3ihp1;ti+3jq1;ti+2ihq1;ti+2jp0;ti+ihp0;ti+jqi;tii to obtain successive innitesimal exponentials, Zdp0p 2dq1p 2dp1p 2:::hqfjeiH:::eiHjp1ihp1jeiHjq1ihq1jeiHjp0ihp0jeiHjqii where the time dependence follows from the previous equation. However, note that all the implicit time dependence of the Heisenberg picture drops out, because weextracted the e iH's, putting all the factors of each matrix element at the same time: Although each matrix element is evaluated at time earlier than the one to its immediate left, each is of the form ha;tjeiH[P(t);Q(t);t]jb;ti=hajeiH[P;Q;t ]jbi (wherejbijb;tii, etc.), leaving only any explicit time dependence that may appear in the Hamiltonian, eectively translating the other t's!ti. Then we only need to know hqjpi=eipq;hpjqi=eipq to evaluate the matrix elements in the path integral as Zdp0p 2dq1p 2dp1p 2:::expfi[qip0+H(p0;qi)q1p0+H(p0;q1)+q1p1+H(p1;q1)+:::]g 224 V. QUANTIZATION More explicitly, this result is hqf;tfjqi;tii=Z Dp Dq eiS;D p D q =N1Y n=0dpnp 2N1Y n=1dqnp 2 S=N1X n=0f(qn+1qn)pn+[H(pn;qn)+H(pn;qn+1)]g q0=qi;qN=qf;tfti=2N The classical picture is a segmented path, with the particle traveling along a straight line segment from point qnto pointqn+1with momentum pn:E a c hqis associated with a point, while each pis associated with the line segment connecting two consecutive points. In the \continuum" limit !0,N!1 ,tftixed, S=Ztf tidt[.qp+H(p;q;t )] (We have dropped some terms in hqjHjpiandhpjHjqifrom reordering the operators QandPinH(P;Q) to applyPjpi=pjpiandQjqi=qjqi. These commutator terms alternate in sign, combining to give terms of order 2, and can be dropped in the continuum limit.) More generally, we can evaluate an arbitrary transition amplitude as A=hfjii=Zdqfp 2dqip 2 f*(qf)hqf;tfjqi;tii i(qi)=Z Dp Dq f*(qf)eiS i(qi) where now Dp Dq =N1Y n=0dpnp 2NY n=0dqnp 2 Note that we can combine the initial and nal wave function, as (qi;qf) f*(qf) i(qi))A =Z Dp Dq (qi;qf)eiS The complex conjugation of fvs. iis due to the complex conjugation involved in time reversal (as seen, e.g., when comparing an eigenstate of pat the inital time to the same eigenstate at the nal time). In eld theory, where the \ p's andq's" are functions of space as well as time, if we choose the boundary in space also to be nite, so that the space and time boundaries form a single connected and closed boundary, then is simply a function of the q's over all that boundary. Finite time development can also be written in operator form: From the above derivation of the path integral, by integrating back out the insertions of unity im- mediately after extracting the innitesimal exponentials and translating the time of each matrix element to zero, we nd hq;tfjq;tii=hqfjU(tf;ti)jqii A. GENERAL 225 U(tf;ti)=eiH(tf)


eiH(ti+)eiH(ti)T exp iZtf tidt H (t) which denes the \time-ordered product" T. This is eectively a Schr odinger-picture expression (all the P's andQ's are at the initial time), and can also be derived in that picture by solving for the time dependence of any state j (t)i.I fHis time independent and we have a (orthonormal) basis of eigenstates of H,w ec a nw r i t e HjIi=EIjIi)U(t;t0)=X IjIihIjei(tt0)EI 2. Semiclassical expansion The path integral formulation is especially suited for semiclassical approxima- tions: The Bohr-Sommerfeld quantization rule follows from the fact that the func-tional integral is invariant under S!S+2n,s i n c eSappears only as e iS;i nt h a t sense the action is more like an angle than a single-valued function. The JWKB ex- pansion follows from S!S=hand expanding in  h. This expansion can be interpreted as an expansion in (space and time) derivatives, since it leads in the usual way to the identication p=ih@=@x andE=ih@=@t . One way to apply it is: (1) Find a classical solution to the equations of motion. This gives the leading contributionin h. (2) Expand about the classical solution as = cl+p h
. Expanding in
, the rst term gives the classical contribution, while the linear term vanishes by the equations of motion. The quadratic term gives an  h-independent contribution to the exponential, which is easy to integrate since it is a Gaussian. The boundary conditions are
=0a ttiandtf. The normalization of the integral can be deter- mined by comparing the free case, or considering the limit where the initial and nal times converge. (3) Higher orders in  hcan be evaluated by perturbation about the Gaussian integral: Polynomials times Gaussians are also easy to integrate. As an example, consider the free particle. The separability of the action translates into factorization of the functional integral, so the result can be found from the one- dimensional case. As usual, L=1 2m.x2)xcl(t)=xi+xfxi tfti(tti) where we have written the classical solution in terms of the variables appropriate to the initial and nal states, namely xifor an initial state localized there at time ti, andxf;tffor the nal state. Since the classical action is itself quadratic, so is its expansion: S=Scl+
S; S cl=1 2m(xfxi)2 tfti;
S=Z dt1 2m(.
x)2 226 V. QUANTIZATION Since
Sis independent of xf;xi, the functional integral of ei
Scan be determined easily by checking the limit tf!ti. The nal one-dimensional result is then hxf;tfjxi;tii=s im tftieim(xfxi)2=2(tfti) w h e r ew eh a v eu s e dp 2(x) = lim !01pex2=2 (one way of dening a Dirac function) to normalize hxf;tjxi;ti=p 2(xfxi) Note that we have been sloppy about the denition of the \integration measure": In going from the Hamiltonian form of the action to the Lagrangian form, we ignoredsomemdependence. Specically, if we start with the Hamiltonian form, as derived in the previous subsection, and derive the Lagrangian form by integrating out p,w e nd the 1=minH=p 2=2mleads to N1Y n=0dpnp 2N1Y n=1dxnp 2!mN=2N1Y n=1dxnp 2 Them(N1)=2then cancels similar factors from the N1x-integrals, while the re- mainingpmis that found in the nal result above. If we had considered a more general Hamiltonian, as in subsection IIIA1, where p2appeared as1 2gij(x)pipj, then we would have obtained a measure of the form (for i=1;:::;D ) [detg (x0)det g (xN)]1=4N1Y n=1dDxn (2)D=2p det g (xn) ( W eh a v ea v e r a g e d gasg(x)p2!p g(xn)g(xn+1)p2 n,s i n c exnis associated with the pointnwhilepnis associated with the link from nton+ 1.) Such measure factors are easy to recognize, since they are always local: If we included it in the action, itwould be a term proportional to lnY ndetg (xn)=1 X nlndetg (xn)(0)Z dt ln det g (x(t)) In practice we just drop all such factors throughout the calculation, and x the normalization at the end of the calculation. Since the Lagrangian form follows from the Hamiltonian form, which was properly normalized, we know such factors will cancel anyway. Auxiliary elds can require similar factors for proper normalization; A. GENERAL 227 then such factors are simply the Jacobians from the eld redenitions from a form where they appeared with trivial quadratic terms. The Gaussian integral for the free particle can also be performed explicitly, by using the discretized Hamiltonian path integral of the previous subsection. Besidesthe Gaussian integrals considered in subsection IB3, we will also need to evaluate Gaussians with linear terms: Zd Dx (2)D=2exTSx=2+jTx=(det S )1=2ejTS1j=2 ZdDz*dDz (2i)DezyHz+zyj+jyz=(det H )1ejyH1j from shifting the integration variables ( x!x+S1j, etc.) to eliminate the linear terms, then using the previous results. These identities are sucient for explicit eval- uation of path integrals for quadratic actions. They are also useful for perturbationabout Gaussians: For functions multiplying the Gaussians, xcan be replaced with @=@j (and similarly for z) and then pulled outside the integral. (If a linear term is not included, it can be introduced, and the result can be evaluated at j=0 . ) Excercise VA2.1 Generalize the above results for integration of Gaussians with linear terms to the cases with fermionic and mixed (subsection IIC3) integration variables. Excercise VA2.2 The path integral for the free, nonrelativstic particle can be evaluated muchmore easily using the Hamiltonian form of the action. First consider the Gaussian integral Z 1 1dx eipxx2=2 as a special case of the Gaussians already evaluated, and use it to derive the identity Z1 1dx eipx=2(p) (Thethus acts as a regulator to make the integral well dened.) Then use the discretized expression of subsection VA1, and evaluate the xintegrals rst. All but one of the pintegrals then can be trivially evaluated, the last giving a Fourier transform. Excercise VA2.3 Evaluate the path integral for the harmonic oscillator, to nd the result hxf;ti+tjxi;tii=r im! sin!texp( im![1 2(x2 f+x2 i)cos !txfxi] sin!t) 228 V. QUANTIZATION The two lowest orders in  hare then given by the above Gaussian integrals, the classical contribution coming from the classical action S(x0) evaluated at the classical solutionS0(x0) = 0, and the rst quantum correction coming from the determinant ofS00(x0). (The fact that the \ S" in this exponent is imaginary will be treated in subsection VA5.) Higher orders in  hcome from expanding the exponential in the S000(x0) and higher-derivative terms. Excercise VA2.4 Consider the nonrelativistic JWKB expansion for the propagator (for an ar- bitrary Hamiltonian H) to the rst two orders in  h, writing it as GpeiS=h aShow the corresponding orders in the time-dependent Schr odinger equation att>0 can be written as the classical equation of motion for the action S and the (probability) current conservation law for the (probability) density  (\Hamilton-Jacobi equations"), H=. S;@ @qi @H @pi +.=0 when the argument pofHis evaluated at pi=@S @qi (Assume a symmetric ordering of p's andq's in the quantum H.) Compare therelativistic case examined in excercise IIIA4.1. bThe propagator is expressed in terms of qandq0,w h e r eG(q;q0;t)(qq0) att= 0, so the rst order in  his found by using the solution to the Hamilton- Jacobi equations to write the classical action in terms of the \nal" position qand initial position q0. (In principle; in general even the classical equations may be too dicult to solve analytically.) However, the Hamiltonian is givenas a function of pandq. Show that the change in variables from q;ptoq;q 0 gives @H @pi=(M1)ij@2S @qj 0@t; (M)ij=@2S @qi 0@qj Show that =det(i1 hM) (the \van Vleck determinant") solves the current conservation law, using the explicit expression for ( det M )M1g i v e ni ns u b s e c t i o nI B 3 . C h e c kt h en o r - malization, using the initial condition for propagators (or comparing to the free case). A. GENERAL 229 cConsider the special case of the harmonic oscillator, and show the result agrees with that of the previous problem. (Hint: First solve the classical equationsof motion for x(t), then rewrite it in terms of x i=q0andxf=q;p l u gi n t o Scl=Sand apply the above.) 3. Propagators This free amplitude we just evaluated is the free propagator or Green function for the Schr odinger equation. Explicitly, we dene G(q;t;q0;t0)(tt0)hq;tjq0;t0i where we have included the factor (tt0)( 1f o rt>t0, 0 otherwise) to enforce that the nal time is later than the initial time (retarded propagator). This satisies thefree case of the general dening equation of the propagator [@ t+iH(i@q;q)]G(q;t;q0;t0)=[@t0+iH(i@q0;q0)]G(q;t;q0;t0) =p 2(qq0)(tt0) w h e r ew eh a v eu s e d @t(tt0)=(tt0) and the facts that Gwithout the factor is a homogeneous solution of the Schr odinger equation (no 's) and becomes a inxfor small times. The propagator then gives a general solution of the Schr odinger equation as hq;tj=Zdq0 p 2hq;tjq0;t0ihq0;t0j) (q;t)=Zdq0 p 2G(q;t;q0;t0) (q0;t0) In particular, for (q;t0)=p 2(qq0)a ts o m et i m e t0for some point q0, (q;t)= G(q;t;q0;t0) at all later times. For the example of the free particle in one dimension, G(x;t;x0;t0)=(tt0)r im tt0eim(xx0)2=2(tt0) This solution for the propagator is not unique; as usual, a rst-order dierential equation needs one boundary condition. Another way to say it is that the inhomoge-neous dierential equation is arbitrary up to a solution of the homogeneous equation. We have eliminated the ambiguity by requiring that the propagator be retarded, as incorporated in the factor (tt 0); using instead(t0t) would give the advanced propagator. This has an interesting translation in terms of the Fourier transform, 230 V. QUANTIZATION which is another way to solve dierential equations, and which replaces the so-called \time-dependent" Schr odinger equation with the \time-independent" one. Dening ~ (p;E)=Zdqp 2dtp 2ei(qxEt) (q;t) and similarly for ~G,w eh a v e ,f o ra n y H(q;p) without explicit time dependence, i(EH)~G(p;E;p0;E0)=p 2(pp0)(EE0) ) ~G=i EHp 2(pp0)(EE0) t < 0 t > 0H+ie H-ieE Now inverse Fourier transforming, we have an ambiguity in integrating Epast t h ep o l ea t E=H. We therefore shift the pole slightly o the real axis, so we can integrate exactly on the real axis. Closing the contour by adding to the real axis a semicircle of innite radius in either the complex upper- or lower-half-plane, wherever convergent, we nd ZdE 2eiEti EHi=(t)eiHt which gives either the retarded or advanced propagator depending on the choice of sign for the inntesimal constant (retarded for EH+i). To perform the inverse Fourier transform, we note that the exponent needs an innitesimal negative part to make the integral convergent: Z dt eiEt()(t)eiHtt=i EHi Excercise VA3.1 Show thati x+ii xi=2(x) by three methods: aUse the above result for the Fourier transform. A. GENERAL 231 bShow that this is the contour integral denition of the function, which is actually a distribution, by integration, multiplying by an arbitrary (nonsin-gular) function and integrating along the real axis. (Hint: Push the polesonto the real axis, shifting the contours along with them, to nd the integralof a single function along the dierence of two contours.) cProve the identity (checking the normalization) lim !02 x2+2=2(x) After doing the Eintegrals, we have the half-transformed propagator for the particle ^G(p;t;p0;t0)=(tt0)p 2(pp0)ei(tt0)p2=2m in the retarded case, which we also could have found easily by solving the dierential equation directly. The remaining pintegrals are then simple Gaussians. 4. S-matrices In the interacting case, the amplitude we get from the path integral is the inter- acting propagator. However, to be able to take the limit describing time developmentbetween innite initial and nal times, we need to choose boundary conditions such that the initial and nal basis states have the time dependence of free particles, de- scribed byH 0, assuming that the particle behaves freely at such asymptotically large times. This is called the \interaction picture", to distinguish from the Heisenbergpicture, where the states have no time dependence, and the Schr odinger picture, where the states have the complete interacting time dependence. We thus evaluatethe limiting amplitude A= lim ti!1 tf!+1h f(tf)j i(ti)i= lim ti!1 tf!+1Zdqfp 2dqip 2 f*(qf;tf)hqf;tfjqi;tii i(qi;ti) for the interaction-picture states j (t)i, relating the interaction-picture coordinate basis 0hq;tjto the Heisenberg-picture basis hq;tj(with initial conditions 0hq;0j= hq;0jhqj): 0hq;tj=hqjeitH0)hqf;tfjqi;tii=0hqf;tfjeitfH0T eiRtf tidt H eitiH0jqi;tii0 (q;t)= 0hq;tj i)A =h fjSj ii;S= lim ti!1 tf!+1eitfH0T eiRtf tidt H eitiH0 232 V. QUANTIZATION The \S(cattering)-matrix" Sthen describes the time development between innite initial and nal times, appropriate for describing scattering from a potential of nite spatial extent. The fact that time development conserves probability ( H=Hy) is re ected in the corresponding unitarity condition for the S-matrix: SyS=1 A more complicated condition is causality : The basic idea is that interactions take place in chronological order. (A stronger statement of causality will be found in the relativistic case: that any interaction should take place at a spacetime point, rather than just at a single time. It follows from this weaker one in relativistic theories,since event B is later than event A in every Lorentz frame only when B is in A'slightcone.) Causality is the condition that the Hamiltonian at any time involves only variables evaluated at that time. ( H(t) is a function of only (t), all at the same timet,w h e r e=p;qare the quantum variables appearing in the Hamiltonian.) A nice way to describe the interactions is by introducing a classical background as we did for the semiclassical expansion of path integrals, such as by (t)!(t)+(t), whereis just some function. The important point is that we have shifted (t)b y (t)a tt h es a m e t, so as not to disturb causality. We then consider the eect on the S-matrix of modifying the background by a function localized (nonvanishing) at some particular time t, and a function  0localized at t0, such that t>t0.P i c k i n g out thepieces in the time-ordered product, we can therefore write S[++0]=U(f;t)V(t)U(t;t0)V(t0)U(t0;i) S[+]=U(f;t)V(t)U(t;t0)U(t0;i) S[+0]=U(f;t)U(t;t0)V(t0)U(t0;i) S[]=U(f;t)U(t;t0)U(t0;i) whereU(t0;i) is the time-development operator from time tito timet0(including the factor with H0),V(t0) is the extra factor in the time development at time t0resulting from the function 0localized there, etc. Then we easily nd S[++0]=S[+]S1[]S[+0] ) (S1[+]S[++0]I)(S1[]S[+0]I)=0 ) (t) S[]y (t0)S[] =0fort>t0 A. GENERAL 233 using the innitesimal functions and0to dene derivatives. In general, it is not possible to solve the Schr odinger equation for the propagator or the S-matrix exactly. One approximation scheme is the perturbation expansion in orders of the interaction: H=H0+V)T (eiR dt H)=ei(tfti)H0+Ztf tidt ei(tft)H0[iV(t)]ei(tti)H0 +Ztf tidtZt tidt0ei(tft)H0[iV(t)]ei(tt0)H0[iV(t0)]ei(t0ti)H0+::: )SfihfjSjii=hfjii+Z1 1dthf;tj[iV(t)]ji;ti +Z1 1dtZt 1dt0hf;tj[iV(t)]ei(tt0)H0[iV(t0)]ji;t0i+::: The rst term in Sis just the identity (i.e., the free piece). All the other terms consist of a string of interactions ( iV) connected by free propagators ( eitH0,w h e r e tis the time between the interactions), with each interaction integrated over all time (subject to time-ordering of the interactions), and the initial/nal state (wave function) evaluated at the initial/nal interaction time. Excercise VA4.1 Assume the initial and nal states are eigenstates of the free Hamiltonian: H0jii=Eijii;H 0jfi=Efjfi AssumingVhas no explicit time dependence, explicitly evaluate the time integrals in the S-matrix, eectively Fourier transforming from time to energy, to nd Sfi=hfjii2i(EfEi)hfj(EH0)1 EH+i(EH0)jiijE=Ei (Hints: Redene the integration variables to be the times between interac- tions. Taylor expand 1 =(EH+i)i nVfor comparison.) In eld theory we want to express any state in terms of a basis of products of 1-particle states, so we can calculate the behavior of these specied particles. We try to do this by using eld variables (the \ q's" of eld theory): Each eld operator should produce a single particle. Unfortunately, this is not the case: An asymptotic state of given 3-momentum created by such a eld operator is not necessarily an eigenstate of the energy, because such a state can be either 1-particle or n-particle,due to interactions. The propagator for the eld is then of the form ^G(p;t;p 0;t0)(pp0)X I *I(p) I(p)ei(tt0)EI(p) 234 V. QUANTIZATION EI(p)=nIX i=1EI;i(pi);nIX i=1pi=p where \EI;i(pi)" is the energy of a 1-particle state (theP Iwill include an integral in general). However, as long as all particles have masses, such an asymptotic 1-particle state is distinguishable as that of lowest energy E0: The higher-energy states are n-particle states to which this particle can couple. (If some of the n-particle stateswere lower energy, the 1-particle state could decay into them, and thus the 1-particlestate would be unstable, and not asymptotic. With massless particles things are morecomplicated: Then 1-particle states are more dicult to dene and to measure.) Inprinciple, we could dene the 1-particle states by constructing the corresponding operator, consisting of the eld plus terms higher order in the elds; in practice, this is rather complicated. A simpler way to make the asymptotic states unambiguous is by modifying the denition of the S-matrix: S= lim ti!1 (1+i) tf!+1(1+i)eitfH0T eiRtf tidt H eitiH0 introducing factors of 1 + ifor some positive , which may be chosen small for convenience. (Actually, we can generally replace 1 + iwith justiif it is not too confusing: The result is the same.) The eect is seen by considering a matrix elementof particular elds that may be a superposition of dierent energies E, but evaluated between states of energy E 0.S i n c eEE0, the time dependence of any such matrix element is proportional to Sfi lim ti!1 (1+i) tf!+1(1+i)ei(tfti)(EE0)=1forE =E0 0forE>E 0 (For simplicity we have assumed energy conservation, so initial and nal energies are the same.) Alternatively, we can simply impose E=E0directly in the denition: S= lim ti!1 tf!+1eitfH0H(tf);H0T eiRtf tidt H (t) H(ti);H0eitiH0 where the free Schr odinger equation H0=E0denesE0for the initial and nal states, andH;H 0is evaluated by examining the asymptotic time-dependence of the time- development operator with respect to tiandtf: Normally eld theory is calculated in energy-momentum space, working with the spacetime Fourier transform of the above,where this amounts to simply comparing energies E f=Ei=E0. A. GENERAL 235 If we know some details of the interaction, this modication may be irrelevant: In particular, in local quantum eld theory interactions happen at a point in space and time. For example, consider the inner product between a 1-particle state in its rest frame and a related n-particle state, which appears in the same propagator. Because of locality, the wave function for the n-particle state, when evaluated in position space(which is where the theory is local) is simply the product of n 1-particle wave functions evaluated at the same point. But we know that for small relative momenta (where a nonrelativistic approximation holds) that the individual wave functions propagate as j jjtt 0j(D1)=2 from the form of the free 1-particle propagator. (Or, we can use dimensional analysis, and consider the spread of a particle of restricted range of momenta from a conned region: Thenj j21=Vand the volume Vjtt0jD1.) This implies that the n-particle wave function will fall o as the nth power of that, so in the limit of large times the 1-particle state will dominate. In a relativistic theory the length scaleassociated with this fall-o will be associated with the masses involved, and thus at a subatomic scale. 5. Wick rotation In the previous subsection we ensured convergence in the denition of the S-matrix by eectively making the \coordinate change" t!(1i)t=eit in the denition of the limit (1i)t!1 ) t!(1 +i)1 This aected the time-development operator as eiHt!eiHtt forH> 0 to pick out the ground state H= 0. The same eective substitution was made in subsection VA3 in dening the contour integral for the propagator: ZdE 2eiE(1i)ti EH=ZdE 2eiEti (1 +i)EH =ZdE 2eiEti E(1i)H=ZdE 2eiEti EH+i 236 V. QUANTIZATION which is the same as the substitution E!(1 +i)E=eiE since essentially E=i@=@t . In general, having to do contour integrals and keep track of i's in propagators is inconvenient. Fortunately, there is a simple way in practical calculations to get ridof not only the i's but (almost) all the other i's as well. The method is known as \Wick rotation". The basic idea is to extend the above complex rotation from angle to angle=2: t!it=e i=2t; E!iE pushing the contour even farther away from the singularities. Thus, the Schr odinger equation is changed to a \diusion equation" (to describe, e.g., Brownian motion): (i@tH) =0) (@t+H) =0 For example, for the free particle the resulting equation has no i's. The time- independent Schr odinger equation then becomes (EH) =0) (iEH) =0 The result for the propagator is then Z1 1dE 2eiEt 1 HiE=(t)eHt Now noiprescription is needed, since the pole was moved away from the real axis. Similar remarks apply to the inverse Fourier transform Z1 1dt eiEt(t)eHt=1 HiE Excercise VA5.1 Find the Wick-rotated retarded propagator G(x0;t0;x;t) for the free (1D) particle, satisfying (@t+H)G=(@t0+H0)G=p 2(xx0)(tt0) Furthermore, if we dene the S-matrix directly in this Wick-rotated space S= lim ti!1 tf!+1etfH0T eRtf tidt H etiH0 A. GENERAL 237 then the limiting procedure is unambiguous even in eld theory, since lim ti!1 tf!+1e(tfti)(EE0)=1forE =E0 0forE>E 0 Another important eect is on actions. For example, in the mechanics path integral for a particle with kinetic term T=1 2m.x2in a potential U(x), we integrated eiS:S=Z dt(UT) Upon Wick rotation, this becomes eS:S=Z dt(U+T) The major change on the exponent Sis that it is now not only real, but negative denite. (For physical purposes, we assume the potential has a lower bound, which can be dened to be nonnegative without loss of generality.) Thus, the semiclassicalapproximation we made earlier, called the \stationary phase" approximation, has now become the \steepest descent" approximation, namely tting e S=hto a Gaussian, which is approximating the integral by the places where the integrand is largest. Wethus write S(x)=S(x 0)+1 2(xx0)2S00(x0)+:::; S0(x0)=0;S00(x0)>0 for one variable, with the obvious generalization to many variables. Explictly, we then haveZdxp 2heS(x)=h1p S00(x)eS(x)=h S0(x)=0 plus higher orders in  h, expressed in terms of higher derivatives of S.I n t h e c a s e of many variables, S00is replaced with a determinant, as for the Gaussian integrals of subsection IB3, and for functional integrals, with a functional determinant. (But sometimes the functional determinant can be replaced with an ordinary determinant: See excercise VA2.4.) So now we can rst calculate everything in Wick-rotated spacetime, where everyt- ing is real (more precisely, classical reality properties are preserved quantum mechan- ically), and then Wick rotate back to nd the correct result in physical spacetime. In particular, the appropriate 's, still needed to correctly position the singularities in physical spacetime, can be restored by rotating back through an angle1 2: inverseWick :t!(i+)t=ei(=2)t; E!(i+)E=ei(=2)E 238 V. QUANTIZATION REFERENCES 1N. Wiener, J .M a t h .a n dP h y s .S c i . 2(1923) 132: Euclidean path integrals for Brownian motion. 2P.A.M. Dirac, P h y s .Z .d e rS o w j . 3(1933) 64: proposed path integrals for quantum mechanics. 3R.P. Feynman, Rev. Mod. Phys. 20(1948) 367: formulated path integral approach to quantum mechanics. 4R.P. Feynman, Phys. Rev. 84(1951) 108: path integrals in phase space. 5R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, 1965): review of path integrals; interesting side stu, like D=2. 6R. Shankar, Principles of quantum mechanics , 2nd ed. (Plenum, 1994): good quantum mechanics text that includes path integrals. 7J.H. van Vleck, Proc. Natl. Acad. Sci. USA 14(1928) 178. 8P.A.M. Dirac, Proc. Roy. Soc. A136 (1932) 453, P.A.M. Dirac, V.A. Fock, and B. Podolosky, P h y s .Z .S o w j . 2(1932) 468: interaction picture. 9J.A. Wheeler, Phys. Rev. 52(1937) 1107; W. Heisenberg, Z. Phys. 120(1943) 513, 673: S-matrix. 10E.C.G. St uckelberg, Helv. Phys. Acta 19(1946) 242; E.C.G. St uckelberg and D. Rivier, Helv. Phys. Acta 24(1949) 215; E.C.G. St uckelberg and T. Green, Helv. Phys. Acta 24(1951) 153: causality in quantum eld theory. 11N.N. Bogoliubov, Doklady Akad. Nauk USSR 82(1952) 217, 99(1954) 225: explicit condition of causality on S-matrix, using functionals. 12L.S. Brown, Quantum eld theory (Cambridge University, 1992) p. 293: decoupling of asymptotic multiparticle states in eld-theory propagators. 13M. Kac, On some connections between probability theory and dierential and integralequations, in Proc. 2nd Berkeley Symp. Math. Stat. Probability , ed. J. Neyman, 1950 (University of California, 1951) p. 189;E. Nelson, J. Math. Phys. 5(1964) 332: Wick rotation of path integral in quantum mechanics. 14G.C. Wick, Phys. Rev. 96(1954) 1124: Wick rotation in quantum eld theory. B. PROPAGATORS 239 ::::::::::::::::::::::: ::::::::::::::::::::::: ::::::::::::::::::::::: B. PROPAGATORS ::::::::::::::::::::::: Classically we distinguish between particles and elds (waves). This can be con- sistent with a classical limit of a quantum theory if there is a conserved charge associ- ated with the classical particles, with respect to which the classical elds are neutral. Then we can have continuous worldlines for the particles: The statement that theworldlines do not end or split is associated with charge conservation. The interaction between the particles and elds is described by modifying the particle (mechanics) action (and not the action for the elds). If we look at just the mechanics action, the modication is the same as considering external elds (like external potentials in nonrelativistic mechanics), since we are ignoring the eld action, which is needed forthe eld equations. The action for the elds then can be added separately. Coupling to such external elds is a simple way to study properties of particles without applying eld theory. For example, in nonrelativistic mechanics it helps to explain charge and spin, whichdon't appear explicitly in the free Schr odinger equation. 1. Particles All the information in quantum mechanics is contained in the propagator, which gives the general solution to the Schr odinger equation, and can be obtained by the Feynman path integral. Here we discuss the free propagator for the spinless particle (whose classical description was given in section IIIB), which is the starting point forrelativistic perturbation theory. We consider quantization rst in the Lorentz covariant gauge v= 1. Except for theTintegration, the same methods can be applied as in the nonrelativistic case. The simplest expression (and ultimately the most useful one) is obtained by Fouriertransforming with respect to x: In comparison to the multidimensional nonrelativistic result ^G(p i;t;p0i;t0)=(pp0)(tt0)ei(tt0)p2=2m (where here (pp0)=( 2)(D1)=2D1(pip0i)f o rD1 spatial dimensions), the relativistic result is ~G(p;p0)=Z dT (pp0)(T)eiT(p2+m2)=2 (where now (pp0)=( 2)D=2D(pap0a)f o rDspacetime dimensions). There are several simple yet important dierences from the nonrelativistic case: (1) The dependence on the mass mis dierent. In particular, we can set m= 0 only in 240 V. QUANTIZATION the relativistic case. (2) There is an additional integrationR dT, because the variable T, which is the remaining part of v, survives the gauge v= 1. This is analogous to the time integral in the nonrelativistic case for ~G(pi;E;p0i;E0), if we set the energy to zero. This is as expected, since the relativistic classical mechanics diers from thenonrelativistic one mainly by constraining the \Hamiltonian" 1 2(p2+m2)t ov a n i s h . This interpretation also leads to the \zero-energy" version of the inhomogeneous(proper-)time-independent Schr odinger equation for this case, i 1 2(m2)G(x;x0)=(xx0) (3) The propagator is automatically \retarded" in the \proper time" T, as a conse- quence of the positivity condition v> 0, which was motivated by the geometrical interpretation of vas the worldline metric. When used in this manner to write the propagator in terms of a Gaussian, Tis known as a \Schwinger parameter". Generally, it is convenient to remove the momentum -function (which resulted from translational invariance) as G(x;x0)=
(xx0)) ~G(p;p0)=(pp0)
(p)
(p)=Z dT (T)eiT(p2+m2)=2 where we have simply written
( p) for the Fourier transform of
( x) (dropping the tilde). Performing the Tintegral, using the same methods as for the tintegral in the nonrelativistic case, we have the nal result
(p)=i 1 2(p2+m2i) Actually, this result is almost obvious from solving the relativistic wave equation. The only part that is not obvious is the \ iprescription": how to perform the contour integration upon Fourier transformation. In the nonrelativistic case, we saw twoobvious choices, corresponding to retarded or advanced propagators; the classicalaction did not distinguish between the two, although the retarded propagator has the obvious convenience of determining later events from earlier ones. On the other hand, in the relativistic case the choice of propagator was xed from classical considerations.Tis restricted to be positive, and the iis needed to make the Tintegral converge. Excercise VB1.1 Take the nonrelativistic limit of the relativistic propagator, and compare withthe propagator of nonrelativistic quantum mechanics. Explain the dierencein terms of the nonrelativistic limit of the classical mechanics action. B. PROPAGATORS 241 Excercise VB1.2 Perform the analysis of excercise VA2.2 for the relativistic particle. First replace the integration over Tby a sum: Instead of dividing up the time into 2Nintervals of length and taking the limit N!1;!0, with 2Nxed, sum 2P1 N=0, and then take the limit !0. (2Nis nowTinstead oftfti, and we integrate over it instead of keeping it xed.) Perform all xintegrals and then all but the last pintegral before summing over N. Again, the entire calculation is much easier than using the Lagrangian (second-order) form ofthe path integral. To understand this point better, we examine the Fourier transformation with respect to time. In contrast to the nonrelativistic case, there are now two poles, at p 0=!; ! =p (pi)2+m2:
=i !1 p0(!i)1 p0+(!i) where nowpa=(p0;pi). These are also the two classical values of the canonical energy (as opposed to the true energy, which is the absolute value), which we saw previously corresponded to particles and antiparticles. With our prescription for integrating around the poles, using the same methods as in the nonrelativistic case, we then nd ^G(pi;t;p0i;t0)=( 2)D=2D1(pip0i)1 !ei!jtt0j =( 2)D=2D1(pip0i)1 ![(tt0)ei!(tt0)+(t0t)ei!(tt0)] We now see that the particles ( p0=!) have a retarded propagator, while the antipar- ticles (p0=!) have an advanced propagator. This is the quantum version of the classical result we saw earlier, that particles travel forward in time, while antiparticlestravel backward. We next compare quantization in the lightcone gauge. Whereas in the covariant gauge the analog to the nonrelativistic time twas the \proper time" T, the analog is now the lightcone \time" .S i n c e=x +=p+,w eh a v eE=pp+(E=i@=@ vs.p=i@=@x+), and thus
(p)=i E1 2(pi2+m2)+i=i 1 2(p2+m2i) as before. Note that this derivation was almost identical to the nonrelativistic one: Unlike the covariant gauge, we did not have to add in Tas a separate variable of integration (but not path integration). However, this Schwinger parameter is useful for evaluating momentum integrals and analyzing momentum dependence. This is atypical characteristic of unitary gauges: They are more useful for keeping track of degrees of freedom. 242 V. QUANTIZATION 2. Properties As in electrodynamics, the free scalar satisifes a dierential equation second-order in time, so the propagator is used dierently from nonrelativistic quantum mechanics to give a general solution to the wave equation. We use the identity I dD1mA$ @mB=Z dDx@
(A$ @B)=Z dDx[A(m2)BB(m2)A] where \H dD1m" is the integral over the closed surface bounding the volume inte- grated over inR dDx. In practice we take the volume to encompass all spacetime in the limit, neglect the part of the boundary at spacelike innity, and choose the parts of the boundary at timelike innity to be surfaces at constant time, so the boundary integrals are over just space: I dD1mA$ @mB=Z dD1xA$ @tBj1 1=Z dD1xA$ @tBj1Z dD1xA$ @tBj1 We then have the solution for the wave function inside the volume in terms of that on the boundary: (m2) =0;i1 2(m2)G(x;x0)=i1 2(0m2)G(x;x0)=(xx0) )IdD10m (2)D=2G(x;x0)1 2i$ @0 m (x0)= (x) where the wave equation for is the Klein-Gordon equation. Similarly, this denes a conserved current from any two wave functions @
( 1*$ @ 2)= 1*(m2) 2 2(m2) 1*=0 or, evaluating the integral over a volume innite in space but innitesimal in time, the conserved charge d dtZ dD1x 1*$ @t 2=0 This leads to the covariant inner product h1j2i=(p0)ZdD1x (2)D=2 1*1 2i$ @t 2 where the(p0) appears because the contour integral gives a + at later times (positive energy) and aat earlier times (negative energy). Explicitly, we nd for the inner product of plane waves p(x)=hxjpi=eip
x )hpjp0i=( 2)D=21D1(pip0i)(p0)1 2(p0+p00) B. PROPAGATORS 243 We have used p2+m2=p02+m2= 0, which also implies that jp0j=jp00j:T h u s , the inner product vanishes if the waves have opposite-sign energy, while for the same sign(p0)1 2(p0+p00)=jp0j. The result then can be written manifestly covariantly as hpjp0i=(pp0) 2[1 2(p2+m2)]=(p0p00)!(2)D=21D1(pip0i) Similarly, the solution for the wave function in terms of the Green function gives only positive-energy contributions from the part of the surface at earlier times, and only negative-energy contributions from the part of the surface at later times. Moregeneral on-shell wave functions, since they depend on only D1 spatial momenta and the sign of the energy, can be written as a restricted Fourier transform (x)=Z dp2[ 1 2(p2+m2)]eip
x~ (p) )h 1j2i=Z dp2[1 2(p2+m2)]~ 1(p)*~ 2(p) (Here ~ (p)* means to complex conjugate after Fourier transforming to p-space; oth- erwise, we need to change the sign of the argument.) In particular, for a plane wave we have ~ p0(p)=(pp0) 2[1 2(p2+m2)] It will prove useful later to have a collection of solutions to the homogeneous and inhomogeneous Klein-Gordon equations, and compare them in 4-momentum space and time-3-momentum space. Using the previous nonrelativistic and relativistic re- sults, we nd
:i=(p2+m2i))(t)ei!t+(t)ei!t=ei!jtj
*:i=(p2+m2+i))(t)ei!t+(t)ei!t=ei!jtj
R:i=(p2+m2ip0))(t)ei!t(t)ei!t=2i(t)sin(!t)
A:i=(p2+m2+ip0))(t)ei!t(t)ei!t=2i(t)sin(!t)
+:(p0)2(p2+m2))(t)ei!t+(t)ei!t=ei!t
:(p0)2(p2+m2))(t)ei!t+(t)ei!t=ei!t where we have omitted certain common factors (see above).
satisfy the homo- geneous equation, while the rest satisfy the inhomogeneous one. (These are easily checked in the mixed space, where the Klein-Gordon operator is (@2 t+!2).) This table makes explicit which sign of the energy propagates in which time direction, aswell as the linear relations between the momentum-space expressions. In particular, we see that
+propagates just the positive-energy states, while
propagates just the negative-energy ones. 244 V. QUANTIZATION Excercise VB2.1 The relativistic propagator uses a particular choice for integrating around thetwo poles in the complex energy plane, as encoded in the iprescription. If we ignored the classical determination of that prescription, there would be four simple choices, integrating either above or below the two poles. Show these four choices can be enforced by replacing iinp 2+m2iwith i;i; ip0;ip0 and derive the results of the table above. Give explicit expressions for the four propagators in position space in four dimensions for the massless case. We can check the propagator's behavior by explicit evaluation, using plane waves: (p0)ZdD1x0 (2)D=2
(xx0)1 2i$ @t0 p(x0)=(p0)1 2(p0+i@t)1 !ei!jtt0jei~p
~xip0t0 =[p0(tt0)] p(x) where we have used the previous result for ^G(~p;t;~p0;t0)( a n dt h u s
( ~p;t)). Again we see that the propagator propagates positive-energy solutions forward in time and negative-energy backward. This propagator also applies to relativistic eld theory. (See subsection IIIA3 for nonrelativistic eld theory.) In comparison to the nonrelativistic case, the propagator is nowi=1 2(p2+m2) instead ofi=(1 2m~p2E), and this determines the kinetic term in the eld theory action: S0=Z dx1 21 2(m2)=Z dx1 4[(@)2+m22] To make the functional integral of eiS0converge, we replace m2!m2i,w h i c h is the same iprescription found in rst-quantization. Note that we have used a real eld*=. (A complex eld can be used by doubling =(1+i2)=p 2.) This is possible only in the relativistic case because we have both positive-energy solutions eiEtas well as negative ones e+iEt. (In other words, the relativistic Schr odinger equation is a second-order dierential equation, so we get two i's to make the kinetic operator real.) Reality simply means identifying particles with antiparticles. (E.g., there is no \antiphoton" distinct from the photon.) B. PROPAGATORS 245 3. Generalizations More generally, we will nd propagators of the form (in momentum space)
=i K;K =Ky corresponding to free actions S0=Z dx1 2K whereK=1 2(m2) in the case just considered. Then the inner product is dened as above in terms of the Green function by using the relation (p0)I dD1mAyMmB=iZ dDx[(KAy)BAyKB] to dene h1j2i=ZdD1m (2)D=2 1yMm 2 and thus (x)=(p0)IdD10m (2)D=2G(x;x0)M0 m (x0) whereMm=(p0)1 2i$ @mabove. (For the usual equal-time hypersurfaces, we use M0=M0. There may be additional implicit matrix factors in the Lorentz-invariant inner product AyB.) This inner product gives a nonnegative norm on physical bosonic states, but on physical fermionic states it is negative for negative energy, because ordering the initial state to the left of the nal state (the wrong ordering for quantum mechanics) produces a minus sign from the anticommutativity of the fermions. (Fromthe explicit integral, this appears because Kis generally second-order in derivatives for bosons, but rst-order for fermions, so M mhas one factor of p0for bosons and none for fermions.) For physical elds, the (free) eld equation will always imply the Klein-Gordon equation (after gauge xing for gauge elds). Thus, the propagator can always be written as
=i K=iN(p) 1 2(p2+m2i) in terms of some matrix kinematic factor N(p). Using this expression for the propa- gator in our above position-space inner product, this implies p(x)=^ (p)eip
x)N(p)Mm(p)^ (p)=(p0)pm^ (p) If we choose a basis that is orthonormalized with respect to all quantum numbers other than momenta (spin/helicity and internal symmetry), we then have hp;ijp0;ji=ij(pp0) 2[1 2(p2+m2)] 246 V. QUANTIZATION If we ignore coordinate/momentum dependence and focus on just the other quantum numbers, then it is clear that the extra factor of Nis just the sumP ijiihij(so Njii=jii, etc.). The positive-energy propagator is then given by a sum over all positive-energy states:
+=N(p)(p0)2[1 2(p2+m2)])N(p)=[(p0)]2sX i^ y i(p)^ i(p) where we have included an extra sign factor for negative energy and half-integer spin from the reordering of states, as explained above. The fact that Kis not simply the Klein-Gordon operator is a consequence of unphysical (gauge/aux iliary) degrees of freedom appearing in the action: Then Nis a projection operator that projects out the auxiliary degrees of freedom on shell, and the gauge degrees of freedom on ando (in unitary gauges), as represented above by a sum over physical states. However,more general N's are sometimes used that include unphysical degrees of freedom; these must be canceled by \ghosts", similar unphysical degrees of freedom of theopposite statistics. Excercise VB3.1 Demonstrate all these properties for spin (helicity) 1 2(see chapter II): aFor the massless case, use twistors for the solution to the eld equation to nd (N) . (p)=pp. =(p0)p . ; (M . ) . = . .  bDo the same for the massive Dirac spinor, to nd N=(p0)(p=+mp 2);Mm= m (Hint: Consider the rest frames for p0>0a n d<0.) Excercise VB3.2 Use the construction of excercise VB1.2 to dene the path integral for the spinning particle of excercise IIIB1.2. Show that in the covariant gauge v=1 , = constant, the propagator can be written as
(p)Z d dT  (T)eT1 2p2i
pi
p 1 2p2=i
p up to some arbitrary normalization factor, where =Rd  is the only gauge invariant part of (asT=Rd v is forv). We will nd that quantum corrections modify the form of the propagator. In particular, it may modify the position m2of the pole inp2and its residue, as well B. PROPAGATORS 247 as adding terms that are analytic near that pole. For example, consider a scalar propagator of the form
(p)=iN 1 2(p2+m2i)+R whereNis a constant and Ris analytic in p. By the procedure of \renormalization", Ncan be set to 1, and m2can be set to its original value (see chapter VII). Alter- natively, we can cancel Nin the normalization of external states, and redene the masses of these states to coincide with what appears in the propagator. Excercise VB3.3 Use this propagator to dene the inner product between two plane waves,and evaluate it explicitly. Show that Rgives no contribution, and the plane waves need factors ofp Nto maintain their normalization. (Hint: What is the wave equation corresponding to
, and how is it related to Mm?Y o u c a n also consider the relation of Nto
+.) Away from the pole, at higher values of p2thanm2, there will also be cuts cor- responding to mutiparticle states. Although these higher-energy intermediate statesin the propagator will contribute to the time development even for on-shell states (those satisfying p 2+m2= 0 asymptotically), in S-matrix elements we can ignore such contributions on external lines, using our modied denition for evaluating theasymptotic limit for the S-matrix (see subsection VA4). Excercise VB3.4 Consider the general scalar propagator
(p)=iZ 1 0d() 1 2(p2+2i); ()=(m)+(2m)() which contains a pole at mass mand contributions from multiparticle states at mass 2mand higher. Fourier transform from energy to time. Use this propagator to dene the time development of a momentum eigenstate satis- fying the free wave equation asymptotically, using the 1 + iprescription of subsection VA4 to dene the asymptotic limit: (t) lim ti!1 (1+i)Z G(t;ti)1 2i$ @ti 0(ti) and show that () does not contribute: (m2) 0=0) (t)= 0(t) 248 V. QUANTIZATION 4. Wick rotation As in the nonrelativistic case, the iprescription can also be xed by the innites- imal Wick rotation (see subsection VA5) t!(1i)t; E!(1 +i)E)1 p2+m2!1 p2+m2i However, in the relativistic case, a nite Wick rotation gets rid of not only i's but also the annoying minus signs associated with the Minkowski metric. We now replaceall timelike coordinates, including proper time, with spacelike coordinates: t!it; !i In addition, for every vector eld V awe replace V0!iV0(Vi!Vi) and similarly for tensor elds. (Here we have dened Wick rotation in the rst- quantized sense: on all explicit coordinates and momenta, as well as on explicit Lorentz indices. For example, in the eld theory action we rotate the explicit deriva- tives and the integration measure, rather than the arguments of the elds.) Euc. Min.-w+ie +w-iep0 Note thatEas dened in the nonrelativistic case was p0,w h i c hi st h es a m ea s p0only in Minkowski space: p0!ip0;p 0!+ip0 Furthermore, there is some apparent ambiguity in how to change the integration measure, corresponding to how the integration contours are rotated (i.e., changes in the limits of integration). In particular, we see from subsection VA5 that thecontour rotation for Eis actually in the opposite direction of that for t, consistent with Fourier transformation. (Eectively, we keep the extra iforRdpfrom rotating p 0, while dropping the 1f r o mp0$p0because of the usual absolute value in the B. PROPAGATORS 249 Jacobian in real changes of variables.) The net result is the naive change forR dx, while that forR dppreserves the inverse Fourier transform: Z dx!iZ dx;Z dp!iZ dp;(x)!i(x); (p)!i(p) When manipulating explicit expressions, the factors of ion coordinates/momenta and elds can be transferred to the constant tensors contracting their indices: Thenet eect is that the Wick rotation is equivalent to changing just , the integration measures, the at-space metric, and the Levi-Civita tensor: !i;@ @!i@ @ Z d!iZ d;Z dx!iZ dx;Z dp!iZ dp mn!mn;abcd!iabcd So now the inner product is positive denite: We have gone from Minkowski space to Euclidean space. For example, for the relativistic particle in the gauge v= 1, the propagator is now
(p)=Z dT (T)eT(p2+m2)=2=1 1 2(p2+m2) The integral is automatically convergent because p2+m2is now positive denite. If we examine our transformation of timelike components in terms of the complex p0plane, we see that we have just rotated the contour from the real axis to the imaginary axis,through an angle =2, which avoids the poles at p 0=(!i). (There is actually a slight cheat in the massless case, since the two poles converge near vanishing 3- momentum, where != 0, but this problem can be avoided by an appropriate limiting procedure.) Note that in the relativistic case the Euclidean propagator is completely real also in momentum space; even the overall ihas been killed. (Compare the nonrelativistic case in subsection VA5, where i=(HE)!1=(HiE).) This follows from rst Wick rotating in position space, then performing the Fourier transform as usual (avoiding an extra ifrom rotating theRdtin the Fourier transform). Excercise VB4.1 Find the propagator in time-3-momentum space, ^G(pi;t;p0i;t0), after Wick rotation. (I.e., Wick rotate
( p) rst, then Fourier transform.) Excercise VB4.2 Find the massless propagator in 4D Minkowski coordinate space, including thei,b y 250 V. QUANTIZATION aFourier transforming the Schwinger-parametrized momentum-space propaga- tor in Minkowski space (including the \Minkowski" ), bdoing the same entirely in Euclidean space, and then Wick rotating the time back to Minkowski space, and cFourier transforming both of the above cases without using the Schwinger parameter, rst doing the energy integrals as in the previous section. (Hint:Use rotational invariance to point ~xin a particular direction to simplify the angular integration.) Excercise VB4.3 Although the propagator in momentum space is most useful for scatter-ing of plane waves, its position-space dependence is more useful for bound states/scattering of localized sources: aEvaluate the Wick-rotated propagator in arbitrary dimensions D for large x=p x2by (1) Fourier transforming the Schwinger-prametrized form, (2) per- forming the (Gaussian) pintegration before the T, and (3) using the steepest descent approximation on the exponential to approximate the Tintegral (but don't bother sticking the power of Tmultiplying the exponential in it as a log). Why does this approximation correspond to large x?( I tm a yb eu s e f u l to make the redenition T!(x=m).) You should nd the result
(x)p 2m(D3)=2x(D1)=2emx After Wick rotating back, the exponential becomes a phase inside the light- cone, while outside it gives exponential damping, as in quantum mechanical barrier penetration. (In the massless case, there is damping away from the lightcone both inside and outside, but only by powers, determined by dimen-sional analysis.) bShow the above result is exact in D=1 and 3 by (1) Fourier transforming without the Schwinger parameter, (2) performing a contour integral over the magnitude of pby closing it appropriately and picking up the contributions at the poles at p=im, and (3) for D=3, xing x min a particular direction and doing the angular part of the pintegration. (This method can also be used to obtain the exact corrections to the above in terms of elementary functions for higher odd D.) For the physical case of the potential produced by a staticpoint source in 3 spatial dimensions, we nd
( x)=p 2emx=x,s oa ts h o r t range the Coulomb potential is unmodied, while it is exponentially damped at range 1=m. B. PROPAGATORS 251 cCheck the results for bby taking the massless limit, and comparing to the analogous result using the method of a, but doing the Tintegral exactly for those cases. (Warning: The result is innite in D=1, and some type of \regularization" must be used to subtract an innite constant, leaving a nite x-dependent remainder.) The corresponding eect on the action, where we path-integrated eiS:S=Z d1 2(vm2v1.xm.xnmn) is to integrate the Wick rotated expression eS:S=Z d1 2(vm2+v1.xm.xnmn) This method also applies to relativistic eld theory. Wick rotation t!itof the kinetic term eiS0:S0=Z dx1 4[mn(@m)(@n)+m22] now gives eS0:S0=Z dx1 4[mn(@m)(@n)+m22] This has an interesting consequence in the complete action (including interac- tions). Positivity of the energy implied the non-time-derivative terms in the action had to be positive, but the time-derivative terms are now the same sign in the Wick-rotated action, eectively the same as adding an extra spatial dimension: The energy T+Uis the same as the Wick-rotated Lagrangian ( T+U!T+U). So we can replace the condition of positivity of the energy with positivity of the Wick-rotatedaction, which we need anyway for path-integral quantization. (Note that for the particle this again requires v0.) Unfortunately, the simple results for Wick rotation obtained here for spin 0, although they generalize to spin 1, do not work so simply for other spins. (Consider,e.g., trying it in spinor notation.) However, the method can still be applied to the coordinates, and simplies momentum integrals, since one can avoid contours, i's, and Minkowski minus signs. REFERENCES 1 V.A. Fock, P h y s .Z .S o w j . 12(1937) 404; E.C.G. St uckelberg, Helv. Phys. Acta 14(1941) 322, loc. cit. (IA); Y. Nambu, Prog. Theo. Phys. 5(1950) 82, R.P. Feynman, Phys. Rev. 80(1950) 440: quantization of relativistic mechanics in terms of proper time. 252 V. QUANTIZATION 2J. Schwinger, Phys. Rev. 75(1949) 651, 82(1951) 664: his parameters. 3J. Polchinski, Commun. Math. Phys. 104(1986) 37: gives a more rigorous discussion of path-integral quantization of the relativistic particle. 4Stuckelberg, loc. cit. (IA); R.P. Feynman, Phys. Rev. 76(1949) 749, 769: relativistic propagator. C. S-MATRIX 253 :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: C. S-MATRIX :::::::::::::::::::::::::::: Arelativisitic quantum eld theory is dened by three properties: (1) Poincar e invariance (\relativistic") is the basic result of special relativity. Its consequences have been accurately observed both macroscopically and (sub)microscopically, andno violations are known. (2) Unitarity (\quantum") is the main mathematical result of quantum mechanics: Any quantum theory can be considered as the corrections to the classical theory implied by unitarity. This is one way to dene perturbationtheory, and is equivalent to the usual (JWKB) expansion in  h.( T h e o t h e r m a j o r axioms of quantum mechanics are concerned with the physical interpretation of the quantities calculated, such as the preparation and measurement of states.) Quantum mechanics also has been accurately veried, with no observed violations. (3) Causal- ity(\eld theory") appears in many areas of physics, formulated in many ways. The strongest way to state causality, in a way independent of special relativity and quan- tum mechanics, is as locality: All interactions happen at a point; there is no action at a distance. This means that any force applied by an object at one point in spacetimeon another elsewhere(/when) must be mediated by yet another object that carries the eect of that force between the two. The most accurate verications of this principle have been through the predictions of relativistic quantum eld theory. We now dene the perturbation expansion of the S-matrix, and give its general properties. 1. Path integrals Since general transition amplitudes involve also wave functions, we also Taylor expand them: In eld theory, this is an expansion in the number of particles. We therefore write []=1X N=01 N!Zdm 1:::dn N (2)ND= 2 N(x1;:::;xN)M1m:::MNn(x1):::(xN) where we have used the covariant inner product of subsection VB3. (The surfaces of integration are at t=1.) We have also used the freeN-particle wave function N, K1 N=:::=KN N=0 which is sucient to describe particles at t=1. Amplitudes for such asymptotic states are elements of the S-matrix, as dened in the interaction picture (see subsec- tion VA4). In practice we choose a particular value of N, and use a basis element for 254 V. QUANTIZATION N, namely the product of N 1-particle wave functions: N(x1;:::;xN)=NY i=1 Ni(xi)+permutations) []=NY i=1h Niji In principle, if there are bound states in the theory, we can consider similar wave functions, but besides we expand in the composite eld describing the bound state. It should be possible to discover such states by looking at the properties of the amplitudes of the states. (For example, a two-particle bound state would show up in the amplitude describing the scattering of two particles.) Since in a covariant approach we treat particles and antiparticles on an equal footing, [ ] should include both the initial and nal wave function. We therefore want to evaluate the path integral A=Z D eiS[] [] Separating out the free and interacting pieces of the (gauge-xed) action, S=S0+SI=Z dx1 2K +SI and using the integration identity Zdup 2euMu= 2f(u+v)=Zdup 2euMu= 2eu@vf(v)e@vM1@v=2f(v) atv= 0, we can evaluate the path integral as A=exp iZ dx1 2 1 K  eiSI[] [] =0 iZ dx1 2 1 K =Z dx dx01 2 (x)
(xx0) (x0) We have absorbed the proportionality constant into the denition of D,a sw ed i d for semiclassical expansion in subsection VA2. In this case, this normalization is xedby the \free" part of the S-matrix. It will prove convenient to distinguish the ends of propagators that attach to S I from those that attach to , so using the identity f(@x)g(x;x+y)=f(@0 x+@0 y)g(x0;y0)(x0=x; y0=y+x) evaluated at y= 0, we rewrite this expression as A=exp iZ 1 2 1 K + 1 K '+1 2 '1 K ' eiSI[] ['] ='=0 C. S-MATRIX 255 We then evaluate the derivatives that act on only one or only the other: A=exp iZ 1 K ' Z[]~ ['] ='=0 ~ []= exp iZ 1 2 1 K  [] Z[]= exp iZ 1 2 1 K  eiSI[]A YYZ e-iSI Then we move the dierential operators into the wave functional: A=^   Z[] =0 ^   =exp iZ 1 K ' ~ ['] '=0 Z[] (the \generating functional" for the S-matrix) contains all propagators with SI[]'s attached at both ends, and forms the basis of the perturbation expansion. From the integration identity above, we can also write it as Z[']=Z D ei(S0[]+SI[+']) Eectively, we have just taken the functional integralR DeiS[]and separated the eld into a \quantum eld" (the integration variable) and a \background eld" ', where'includes the asymptotic states, which propagate to innity, while vanishes at innity (or at least goes to a constant) fast enough to allow the usual integration by parts in performing the functional integral. Thus 'gives the boundary value of the eld. This is essentially the same as the general prescription for path integralsg i v e ni ns u b s e c t i o nV A 2 ,e x c e p tt h a tw et a k e 'to be arbitrary for convenience of functional dierentiation, and we drop free 'terms, which were incorporated into ~ (i.e., we expand just S I). ~ [] is the result of contracting some pairs of the one-particle wave functions with propagators. This gives the usual inner product of those one-particle states: Since for any one-particle wave function satisfying the free eld equations we have the propagator identity Zdm (2)D=2 (x)Mm
(xx0)= (x0) 256 V. QUANTIZATION such contractions give Zdmd0n (2)D (x)Mm
(xx0)M0 n 0(x0)=Zdm (2)D=2 (x)Mm 0(x)=h j 0i where the inner product vanishes unless the states have opposite energies (i.e., one is incoming and one is outgoing). This is the boring part of the S-matrix element: It represents the corresponding particles not interacting at all. (For example, in the free caseSI=0w eh a v e Z=1 ,a n dA=~ j=0consists of only such inner products.) For most purposes we just factor out such free inner products, and consider only processeswhere all particles interact. Finally, the conversion from ~ t o ^ replaces all the on-shell inner products with integrals over all spacetime: Using the propagator identity from above, ^   =1X N=01 N!ZdNDx (2)ND= 2~ N(x1;:::;xN) (x1)::: (xN) where ~ Nappears in ~ e x a c t l ya s Nin : ~ []=1X N=01 N!Zdm 1:::dn N (2)ND= 2~ N(x1;:::;xN)M1m:::MNn(x1):::(xN) Usually we represent Nas the product of single-particle wave functions, ^ N  =NY i=1Z Ni  )AN=NY i=1Z Ni  Z[] =0 in which case ^ simply replaces each eld inZwith one of these wave functions.
!8 >>< >>:
:S I()SI 1: ()SI hji: () A Z e-iSI SIyÆ | æ 1/K 1 C. S-MATRIX 257 Thus, of the three types of propagators, only the ones that connected two factors ofSIremain, all inside Z; the ones that connect the wave functions to Zhave been replaced with just spacetime integrals, while those connecting the wave functions toeach other have become the usual spatial integrals for the Hilbert-space inner product. Excercise VC1.1 Show that the amplitude can also be written as A=   Z[] 0[] =0; 0=ehji=2    =1X N=01 N!ZdNDx (2)ND= 2 N(x1;:::;xN) (x1)::: (xN) making all inner products of non-scattering particles explicit. We can inter- pret 0as the free \vacuum wave functional" and Z 0as the interacting vacuum wave functional. 2. Graphs Before giving applications of these rules, we consider a few general properties. A convenient way to describe the terms in the expansion of the two exponentials in Zis pictorially, by \Feynman diagrams/graphs". Each factor of SIis a \vertex" of the graph, represented by a dot in the diagram; each factor of the propagator 1 =K is a \link" in the graph, represented by a line connecting the two dots representingthe two factors of S Ion which each =' acts. (Both derivatives can also act on the same factor of SI, giving a loop.) So any term in the expansion of Zis repre- sented by a diagram consisting of a bunch of dots (interaction vertices) connected by lines (propagators). When we want to \draw" the amplitude Aitself, we also draw additional lines, each with one end attached to a vertex and one end unattached.These \external lines" represent the one-particle wave functions coming from N, and not propagators (\internal lines"). While in the diagram for Zeach vertex can have'dependence, in the diagram for Athere is none, and the number of lines (internal and external) coming from any vertex explicitly indicates the order in 'of the corresponding term in SI. The physical interpretation of these diagrams is simple: The lines represent the paths of the particles, where they act free, while the vertices represent their collisions, where they interact. These diagrams are generally evaluated in momentum space: Wethen can associate a particular momentum with each line (propagator), and momen-tum is conserved at each vertex. An arrow is drawn on each line to indicate the 258 V. QUANTIZATION direction of \ ow" of the momentum. (Otherwise there is a sign ambiguity, since complex conjugation in position space changes the sign of the momentum.) Then the sum of all momenta owing into (or all out of) a vertex vanishes. The momentum associated with a line is then interpreted as the momentum of that particle, with the arrow indicating the direction of ow of the proper time .(pchanges sign with .) When evaluating an S-matrix element, the fact that the external-line wave func- tions satisfy the free wave equation means the external momenta are on-(mass-)shell (p2+m2= 0); on the other hand, this is not true of the momenta on the internal lines, even though those particles are treated as free. Some graphs in the S-matrix are disconnected: They can be divided into separate parts, each with a subset of the particles that interact with each other but not with the other subsets. For convenience, we consider only the connected graphs: If wewrite Z[']=e iW['];Ac=~  i ' (i)W[']j'=0 thenWis the generating functional for the connected S-matrix Ac. To prove this relation between ZandW,w e r s tn o t et h a ti ti sj u s tt h ec o m b i n a t o r i c so ft h e graphs, and has nothing to do with spacetime. Therefore, it is sucient to consider the simple (unphysical) case where the action has no derivatives. Since the propagatoris then local, connectedness is equivalent in this case to locality. We then observe that the lack of derivatives allows the functional integral to be factorized explicitly into ordinary integrals at each point in spacetime: Z[']=Z D e iR dx L ((x);'(x))=Y xZ d(x)eiL((x);'(x)) =Y xZ('(x)) =eiR dxW('(x))=eiW['] Thus, this Wis local, and therefore connected; this implies Wis connected in the general case. The simplest kind of connected graph is a \tree" graph, which is a graph that has no closed paths; the rest are called \loop" graphs. \One-particle-irreducible" (1PI) graphs are dened to be those connected graphs that can't be disconnected by severing a single propagator. It then follows that anyconnected graph can be represented as a generalized tree graph, whose \vertices" (including two-point vertices) are actually 1PI graphs. We then dene the \eective action" [] to be the classical action plus all 1PI loop graphs. Note that the vertices of the original action are the 1PI tree graphs; thus is also the classical kinetic term plus all (tree and loop) 1PI graphs. Actually, since the 1PI tree graphs are iS I,w e dene the classical part of to be S, but the quantum part to be the quantum 1PI C. S-MATRIX 259 part ofW. Of course, the eective action is nonlocal. However, the tree graphs that follow from this action are exactly all the connected graphs of the original action: This is clear for all but the 2-point vertices from the denition. For the propagator and its relation to the 2-point 1PI loop graphs, we simply compute the expression following from : Denoting the 2-point 1PI loop operator as A, the kinetic operator of isK+A. The propagator following from is then 1 K+A=1 K1 KA1 K+1 KA1 KA1 K::: But this is exactly the result of the complete propagtor (including loop graphs) fol- lowing from the original action. Z W W WG G G G This quantum modication of the propagator leads us to reanalyze our prescrip- tion for evaluating the S-matrix: For example, even in the simplest case, where thisAis just a constant, the full propagator diers from the free propagator by a change in the mass. This means the mass of asymptotic states should also be changed, which invalidates part of our evaluation of the S-matrix in the previous subsection. Similar problems occur when Ais proportional to K, which changes the normalization of asymptotic states. There are two ways to x these problems: (1) We compensate bymodifying the kinetic term in the classical action, replacing KwithKminus such local contributions from A. Treating these new terms as part of the interaction in our derivation of the S-matrix, so our normalization and mass in the free propagatorare unchanged, these \interaction" terms cancel the unwanted terms in the quantum propagator, so it then has the same residue and pole as the free one. This procedure is known as \renormalization", and will be discussed further in chapter VII, primarily 260 V. QUANTIZATION for the purpose of eliminating innities. (2) Alternatively, we modify our derivation of the S-matrix to take the full propagator into account. The easiest way to see thischange is to remember that by denition the S-matrix follows from treating the eec-tive action as a classical action (except for its nonlocality and nonhermiticity), butkeeping only the \tree" graphs. Then clearly (a) the quadratic part 0of is used to dene the asymptotic states, and (b) instead of eliminating all free propagatorsexcept those connecting factors of S I, we eliminate all full propagators (found from 0) except those connecting factors of I, the nonquadratic part of . In other words, we modify our earlier denition of the S-matrix by dropping all graphs that have any quantum correction to external lines. Thus, this procedure can also be applied inthe case of renormalization; in fact, it should be applied in general, simply becauseit allows us to immediately ignore many graphs. It also allows us to avoid confusionresulting from attaching wave functions of the wrong mass to propagator corrections:E.g., in momentum space, we would have to interpret ambiguous factors such as (K)A(1=K):::, where the factor of (K) comes from a plane-wave wave function. GGG GG GG GGG This analysis of the quadratic part of also leads us to examine the terms of lower order: constant and linear. The constant term is just a normalization, andshould be dropped. (This is not true in the case of gravity, where a constant term inthe Lagrangian is not gauge invariant by itself.) The linear term describes the decayof a particle into the vacuum: It implies we have the wrong vacuum. A linear term necessarily has no derivatives (otherwise it is a boundary term, which vanishes by our boundary conditions); it is part of the \eective potential" (a generalization of thepotential energy, whose contribution to a classical mechanics Lagrangian contains notime derivatives; see subsection VIIB2). The existence of a linear term means thatthe minimum of the eective potential, i.e., the true vacuum, is not described byvanishing elds. To correct this situation we therefore apply the same procedure asfor the classical action (chapter IV): (1) Shift the appropriate elds by constants, to put us at the minimum of the potential, and (2) use the new quadratic terms in the potential to determine the true masses of states dened by perturbation aboutthis new vacuum. Again eliminating any constant terms, the resulting has onlyquadratic and higher-order terms. C. S-MATRIX 261 To summarize, the general procedure for calculating Feynman graphs is: (1) Cal- culate the eective action, i.e., the 1PI graphs. (2) Shift the scalars to put them at the minimum of the eective potential, dropping the resultant constant, to reveal the true masses of all particles. (3) Calculate the S-matrix from diagrams without external- line corrections, with external wave functions whose normalization and masses aredetermined by the zeroes of the kinetic operators in the shifted . Another use of the eective action, besides organizing the calculation of the S- matrix, is for studying low-energy behavior: This means we apply an expansion in derivatives, as in rst-quantized JWKB (see subsection VA2). Of most interest is thelowest order in the approximation, where all elds are eectively constant: This gives the eective potential. (In practice, \all elds" means just the scalars, since constant spinor elds are not generally useful, while higher spins are described by gauge elds,whose constant pieces can be set to vanish in an appropriate formulation: E.g., the constant piece of the metric tensor can be attributed to a scalar | see subsection IXA7.) However, the denition of \1PI" graphs is ambiguous, depending on how wedene \particle": For example, if we include auxiliary elds in the eective action (as in supersymmetry, but also for bound-state problems: see subsections VIIB3 and 6), the result at xed order in any expansion parameter ( h, coupling, etc.) is dierent, since the auxiliaries get contributions at each order, so elminating them by their eective-action eld equations mixes orders. (E.g., B 2+hBf(A)!h2f2.) This is crucial when the composite elds dened by these auxiliaries, and thus the auxiliaries themselves, obtain vacuum values. Therefore, the eective action is most useful for these purposes when, for appropriate choice of elds and denition of  h, a useful rst-quantized semiclassical expansion can be found. Another important use of the eective action is that it is gauge invariant (even in the nonabelian case, when using the background-eld gauge; see subsection VIB8): Sometimes simplications due togauge invariance are thus easier to see in the eective action than in the S-matrix. We now consider an interesting topological property of graphs: For any graph, if we draw an extra propagator from a vertex to itself or another, that gives an extra \loop" (closed circuit) and no extra vertices. Adding a 2-point vertex to the middleof a propagator gives an extra propagator and no extra loops. Adding an external line to a vertex changes nothing else. Since any nontrivial (not a lone propagator) connected graph can be built up this way from a lone vertex, we nd PV=L1 forPpropagators, Vvertices, and Lloops (and Eexternal lines). The same result follows from counting momentum integrals: In momentum space there is an inter- 262 V. QUANTIZATION nal momentum, and corresponding integral, for each propagator, and a momentum conservation condition, and corresponding function, for each vertex. The only in- dependent momenta are the external ones (associated with each 'inZ[']) and one momentum vector for each loop. Thus, after integrating out all the delta functions,except for an overall momentum conservation function for each connected graph, we are left with integrations over only the loop momenta. So, we are again led to theabove result. 3. Semiclassical expansion We can dene perturbations by inserting  h's in various ways, as discussed in sub- section IIIA3. The  hthat denes classical mechanics yields an expansion in derivatives on \matter" elds (those that describe classical particles in the limit  h!0, as op- posed to the \wave" elds). This expansion is covariant as long as the  hmultiplies covariant derivatives. However, it can't be applied to Yang-Mills elds, and it doesn'tcorrespond to a diagrammatic expansion. On the other hand, the  hthat denes clas- sical eld theory is an expansion in the number of \loops", and allows us to groupgraphs in gauge-invariant sets, since gauge transformations are not  h-dependent: As in quantum mechanics, we can perform a JWKB expansion by appropriately insertingh: Z[']=Z D exp i h(S0[]+SI[+']) =exp ihZ dx1 2 '1 K ' eiSI[']=h The order in  hhas a simple graphical interpretation. We see that there is a factor of hfor each propagator and a factor of 1 =hfor each vertex. Thus, by the above topological identity, for each connected graph the power in  his one less than the number of loops. We therefore write Z[']=eiW[']=h;W =1X L=0hLWL whereW0generates the connected \tree" graphs, which have no loops. We know that the leading term in the JWKB expansion is associated with the classical theory. We can make this more explicit in the eld theory case by nding thegeneral classical (perturbative) solution to the eld equations from the tree graphs.Graphically the solution is very simple: We replace one 'on each tree graph with a propagator, and associate the end of the propagator with the position of the classical C. S-MATRIX 263 eld (x). If we then act on this , which is a sum over all tree graphs, with K, it cancels the propagator, leaving a bunch of 's (also sums over all tree graphs) connected at x, with the appropriate vertex factor. In other words, we nd K= SI[]=, the classical eld equations. K Ff f fF Ff f f f= To prove this, it's convenient to again use functionals, to automatically keep track of all combinatorics. The quantum eld equations can be derived from the generalidentity Z D f[]=0 since we only integrate functionals fthat are assumed to fall o fast enough as !1 to kill all boundary terms. (This follows from the perturbative denition of the functional integral.) In particular, for any action ~S, 0=Z D i h ei~S=h=Z D~S ei~S=h For our present purposes, we choose ~S=S0[]+SI[+'] ) 0=Z DS0[] +SI[+']  ei~S=h=Z D K+ih ' ei~S=h )Khi'+W['] '=0 where \hi'" is the expectation value of the eld in a background: hi'=RD ei~S=h RD ei~S=h (and, of course,RD ei~S=h=eiW=h). We now examine the classical limit  h!0 of this result by noting that if we impose the free eld equation on the background K'=0)~S =S[+']  264 V. QUANTIZATION whereS[]=S0[]+SI[] is the usual action: In other words, the eld equations following from ~Sare just the usual eld equations for the complete eld ~=+'; ' = lim x!1~ since we chose our boundary conditions so !0a sx!1 (includingjtj!1 , whereas'!0 only at spatial innity). We then apply the stationary-phase approx- imation (or, after Wick rotation, the steepest-descent approximation) lim h!0Z D f []ei~S=h= f[]ei~S=h ~S= =0 forf=and 1 to nd K+W0['] '=0) ='1 KW0['] ' lim h!0h~i'=~jS[~]=~=0 Thus,  is the solution to the classical eld equations with boundary condition  !', and can be found directly from the classical (0-loop) part of Wby replacing one eld with a propagator. A similar result holds for W0itself, by taking the classical limit as above for f[]=1 : W0[']=S0[]+SI[+'] when evaluated at the result of varying the above with respect to either argument:  )S[+'] =K'=0  ' )W0['] '=K The former follows directly from the limiting procedure; the latter we have just proven equivalent by evaluating the limit for h~i'. Since by denition the eective action is related to Win exactly the same way that Sis related to W0(the trees from give the full W), we also have W[']= 0[]+I[+'] [+'] =~K'=0,W['] '=~K where ~Kis the kinetic operator appearing in 0, the quadratic part of . (Some care must be taken for the fact that the poles and residues of ~Kinp2may dier from those ofK, as discussed in the previous subsection.) C. S-MATRIX 265 In practice, if one wants to make use of the classical eld equations perturbatively, one looks at tree graphs with a specic number of external lines: For example, in a scalar theory with 3interaction (assuming hi=0 ) , K+1 22=0;=1X n=1n)Kn+1 2n1X m=1mnm=0 gives a recursion relation for the term  nthat isnth order in '. Excercise VC3.1 Consider the relativistic Schr odinger (Klein-Gordon) equation for a scalar wave function in an external scalar potential : (K+) =0 (If you nd it less confusing, you can consider the nonrelativistic case K= ~p2=2mE,w h e r eH=~p2=2m+.) Find the perturbative solution for the quantum mechanical (one-particle) S-matrix for (see subsection VA4). Show that this agrees with the contribution to the eld-theoretic S-matrix for the Lagrangian L( ;)= *(K+) +L() coming from tree graphs with an external line, an external * line, and an arbitrary number of external lines. Excercise VC3.2 Consider, instead of the background eld ', a \current" source Jthat attaches propagators to external lines. The current is eectively just a one-point in- teraction (it caps loose ends of propagators), so it can be introduced into the generating functional by the modication SI[]!SI[]+Z dx J We now have eiW[J]=h=Z D exp i h S0[]+SI[]+Z dx J Z[J] is thus the Fourier transform of eiS[]=hwith respect to the conjugate variablesandJ. aDerive the \Schwinger-Dyson equations" S[] (x) =ih=J+J(x)! eiW[J]=h=0 266 V. QUANTIZATION bFind the classical limit W0[J]=S[]+Z J at  =W0[J] J,J=S[]  (i.e.,W0[J] is the \Legendre transform" of S[]). Find the corresponding relation for [ ]a n dW[J]. cShow that the free part of W[J]i sg i v e nb y Wfree=1 2Z dx J1 KJ)'free=1 KJ Note thatJcan be replaced with 'inW[J]e v e r y w h e r e except the free term, since in all other terms Jappears only in the combination (1 =K)J. Show how this can be done in such a way as to reproduce the results above for the solution of the classical eld equations in terms of W[']. Warning: Before this substitution we use K'+J= 0, but afterwards we apply K'=0 ;a l s o beware of integration by parts, since 'does not vanish at 1, so the naive substitution !+'is not very helpful. (Historically, Zwas introduced as a functional of J. However, the only two applications of Feynman diagrams, the S-matrixAand the eective action , both required that the external- line propagators resulting from the Feynman rules for Z[J] be \amputated". Therefore, we use background elds exclusively. The resulting derivations, generalities, and applications are at least as simple as and often a little simpler than the corresponding ones with current sources.) 4. Feynman rules It is usually most convenient to calculate Feynman diagrams in Wick-rotated (to eliminate i's) momentum space (where massive propagators are simpler). The \Feynman rules" are then read o of the action as S=Z dx1 2K +SI[])Z[]=eW[]=expZ 1 2 1 K  eSI[] where inZ[] we simply replace each eld with a single-particle wave function in all possible permutations, since for the case of an N-particle amplitude we usually write the wave function as the product of N single-particle wave functions (although more generally it is a linear combination of these): AN=^ N  Z[] AN;c=^ N  W[] ;^ N  =NY i=1Z Ni  C. S-MATRIX 267 We Fourier transform as (x)=Z dp eip
x(p); (p)=Z dx eip
x(x) (Of course, (x)a n d(p) are dierent functions, but the distinction should be clear from context.) In practice we choose the single-particle wave functions to be eigen-states of the momentum, so i(x)=eipi
x^ i(pi); i(p)=(ppi)^ i(pi)(p2 i+m2=0 ) where ^ is some simple factor (e.g., 1 for a scalar). Then the external line factor terms become Z dx i(x) (x)=^ i(pi) (pi) while for propagator terms Z dx1 2 (x)1 K(i@) (x)=Z dp1 2 (p)1 K(p) (p) and for vertex terms Z dx  1(x):::n(x)=Z dp1:::dpn1(p1):::n(pn)(p1+:::+pn) where each of the 's in the vertex may represent a eld with derivatives; then we replacei@on(x)w i t hpon(p). Thus, e.g., we have AN=Y^ Ni(pi) (pi) Z[] Excercise VC4.1 Use the denition  ~(p)~(p0)=(pp0) to show that g  (p)= ~(p) where we now use tildes to indicate Fourier transformation. Note that there is some ambiguity in the normalization of external line factors, associated with the numerator factor in the propagator
=1 K=N(p) 1 2(p2+m2) 268 V. QUANTIZATION For nonzero spin (or internal symmetry), we have already discussed the normalization analogous to that for scalars, namely X^ y(p)^ (p)=N(p) (withfor negative energy and half-integer spin). However, there is already some freedom with respect to coupling constants: Even for scalars, if a coupling appears in the kinetic term as a factor of 1 =g2, then eectively the kinetic operator is K=K0=g2, whereK0is the usual (coupling-independent) one. Thus N=g2N0,s o ^ =g^ 0, meaning coupling dependence in external lines. Alternatively, this external-line fac- tor of the coupling can be included in the denition of probabilities in terms of amplitudes, which already includes nontrivial factors because of the use of (non-normalizable) plane waves. Furthermore, quantum eects modify the form of thepropagator: Such eects can be absorbed near the pole p 2=m2by a eld redef- inition, but often it is more convenient to leave them. Then Nwill again have a constant, (but more complicated) coupling-dependent factor, which must be canceledin either the external-line factors or probabilities. (However, note that these questionsdo not arise in calculations of the functionals W[]o r [].) In tree graphs all momentum integrals are trivial, with the momentum conserva- tionfunctions at each vertex, and the functions of the external lines, determining internal momenta in terms of external momenta. In loop graphs there is a momentum integral left for each loop, over the momentum of that loop. The amplitude will al-ways have an overall function for momentum conservation for each connected piece of the graph. Since we are always interested in just the connected graphs, we pull this conservation factor o to dene the \T-matrix": Including the factor of \ i"f r o m Wick rotating back to Minkowski space, S connected =iX p T In general there will be combinatoric factors associated with a graph. These follow automatically from the functional expressions, but can also be seen from the symmetries of the graph. Here \symmetries" means ways in which the graph can be twisted, with external lines xed, such that the graph looks the same, includingthe types of particles propagating along the lines. For example, a graph with 2vertices that are connected by n identical propagators would get a factor of 1/n! for that symmetry. There are also sign factors from fermions: Permutation of external fermion lines gives minus signs, because it involves permutation of anticommutingelds in the functionals. Each fermion loop gets a minus sign for the same reason. C. S-MATRIX 269 (This is related to the fact that fermionic integration gives determinants instead of inverse determinants.) Explicitly, it comes from evaluating expressions of the form (ignoring momentum dependence and external elds)     


     ( )


( ) =           


      where the propagator derivatives ( = )(= ) give no signs connecting up successive vertex factors  , but the last one does in closing the loop. The general rules for contributions to the (unrenormalized) eective action [ ] are then: (A1) 1PI graphs only, plus S0(R1 2K ). (A2) Momenta: label consistently with conservation, withR dpfor each loop. (A3) Propagators: 1 =1 2(p2+m2), or 1=K, for each internal line. (A4) Vertices: read o of SI. (A5) External lines: attach the appropriate (o-shell) elds andR dp,w i t h(Pp). (A6) Statistics: 1/n! for n-fold symmetry of internal/external lines; 1 for fermionic loop; overall 1. (If we want to calculate W[] instead, then simply replace step 1 with \Connected graphs only".) The next step is to analyze the vacuum: (B1) Find the minimum of the eective potential (for scalars). (B2) Shift (scalar) elds to perturb about minimum; drop constant in potential.(B3) Find resulting masses; nd wave function normalizations. Renormalization is performed either before or after this step, depending on thescheme. Finally, the trees from are identied with the complete amplitudes from S. Thus, T-matrix elements are given by: (C1) Connected \trees" of (shifted, renormalized) : (A2-4) for L=0 with S!. (C2) \Amputate" external 0-propagators. (C3) External lines: 1, or appropriate to 0wave equation ~K =0(P^ y^ =N). (C4) External-line statistics: No symmetry factors; 1 for fermion permutation. Note that is usually simpler than Twith respect to treatment of external lines: InTwe often have contributions from graphs which are identical except for permutation of external lines from identical elds. In only one such graph need be considered, since the statistics of the attached external elds automatically takes care of this symmetry. (We then also drop the 1/n!, or at least reduce it.) 270 V. QUANTIZATION The simplest nontrivial tree graphs are 4-point amplitudes. These are conve- niently expressed in terms of the Mandelstam variables (see subsection IA4). For symmetry, it is now more convenient to label all external momenta as owing out of the diagram (or all owing in); then s=(p1+p2)2;t =(p1+p3)2;u =(p1+p4)2 Now positive energies travel to x0=+1, while negative energies go to 1.W e also use the convention that sis dened in terms of the momenta of the two initial particles (and we also use this same denition when there are more than two nal particles);tanduare then more or less interchangable, but if initial and nal particles are pairwise related we choose tin terms of the momenta of such a pair. For example, the simplest nontrivial theory is 3theory, with K=1 2(+m2)!1 2(p2+m2) SI=Z dx1 6g3=Z dp1dp2dp31 6g(p1)(p2)(p3)X p The four-point S-matrix amplitude at tree level (order g2) then comes from using the following contributions to the factors in A(calculated in Euclidean space): ^ 4= (p1) (p2) (p3) (p4) eR (=)(1=K)(=)=2=:::+Z dk1 2 (k)1 1 2(k2+m2) (k)+::: eSI=:::+1 2S2 I+::: Using (=(p))(k)=(pk), keeping only the connected part, and integrating out thefunctions (except (Ppexternal )), we are left with Sc=(p1+p2+p3+p4)g21 1 2(m2s)+1 1 2(m2t)+1 1 2(m2u) (There would be an extra ifor thein Minkowski space.) Note the symmetry factor 1=3! for the3coupling, which is canceled upon taking 3 functional derivatives for the vertex factor. The T-matrix then comes from just factoring out the : T=g21 1 2(m2s)+1 1 2(m2t)+1 1 2(m2u) But this contribution to Wis given by a single term: W=g2Z dp1dp2dp31 2[1 2(p1)(p2)]1 1 2(m2s)[1 2(p3)(p1p2p3)] C. S-MATRIX 271 or, in position space, W=g2Z dx1 2(1 22)1 1 2(m2)(1 22) (The1 2's correspond to the various symmetries: switching a pair connecting to the same vertex, or switching the two pairs.) 1 23 4s 1+2t1 23 41+3 1+4u1 23 4++ Excercise VC4.2 Find the 5-point tree amplitude for 3theory. What order in gis the n-point tree? The momentum integrals are real in Euclidean space: There are no singularities in the integrand, since p2+m2is always positive (although there are some subtleties in the massless case). Thus, all these integrals are most conveniently performed in Euclidean space. However, eventually the result must be analytically continued back to Minkowski space: x0!ix0,w h i c hm e a n s p0!ip0(butp0!ip0,b e i n ga l s o careful to distinguish n m!n mandmn!mnfor indices on elds) via a 90rotation. This returns some of the idependence. There are also i's associated with integration measures: Since all the momentum integrals for the S-matrix elements have already been performed, all that remains is a factor of ito go with(Pp) for each connected graph. There can also be idependence in external line factors. Remember that negative p0indicates a particle traveling backward in time; the true motion of such a particle is opposite to that of the arrow indicating momentum ow. Thus, external lines with arrows pointing into the diagram and positive p0,o r out of the diagram and negative p0, both indicate initial states, arriving from t=1. Conversely, external lines with arrows pointing into the diagram and negative p0,o r out of the diagram and positive p0, both indicate nal states, departing to t=+1. A related issue is particles vs. antiparticles. If a particle is described by a real eld, it is identied as its own antiparticle; but if it is described by a complex eld, then it is identied as a particle if it has a certain charge, and as an antiparticle if it has the opposite charge. (For example, a proton is positively charged while an 272 V. QUANTIZATION antiproton is negatively.) Of course, this is convention, since a complex eld can always be replaced by two real elds, and we can always relabel which is the eldand which the complex conjugate; generally there should be a useful conservation law(symmetry) associated with these complex combinations (usually electric charge),and the one called \particle" is the one more common to the observer. For example,suppose we have a complex scalar external eld/wave function (p). Forp 0>0t h i s describes a particle propagating to x0=+1. Similarly, *(p)f o rp0>0 describes anantiparticle propagating to x0=+1. On the other hand, *f o rp0<0 describes a particle propagating from x0=1, whileforp0<0 describes an antiparticle propagating from x0=1. 5. Semiclassical unitarity As in nonrelativistic quantum mechanics, the only conditions for unitarity are that: (1) the metric (inner product) on the Hilbert space is positive denite (so allprobabilities are nonnegative), and (2) the Hamiltonian is hermitian (so probabilitiesare conserved). Both of these conditions are statements about the classical action.The second is simply that the action is hermitian, which is easy to check. The rst isthat the kinetic (quadratic) terms in the action, which dene the (free) propagators,have the right sign. This can be more subtle, since there are gauge and auxiliary degrees of freedom. Therefore, the simplest way to check is by using the lightcone formalism. We see from the analysis of subsection IIB3 for eld equations, or for actions in subsection IIIC2 (for spins 1, and more generally below in chapter XII) that after lightcone gauge xing and elimination of auxiliary degrees of freedom the kineticterms for physical theories always reduce to 1 4for bosons and1 4 (=i@+) for fermions, where there is a sum over all bosons and fermions, and each term in the sum has a eld with a single hermitian component. Complex elds can multiply theircomplex conjugates, but these can always be separated into real and imaginary parts.There are never crossterms like A B, since after eld redenition, i.e., diagonalization of the kinetic operator, this gives A0A0B0B0, so one term has the wrong sign. Similar remarks apply to massive elds, but with replaced bym2(as seen, e.g., by dimensional reduction), or we can treat the mass term as part of the interactions. Now we only need to check that the single-component propagators of these two cases dene positive-denite inner products. Since multiparticle inner products areproducts of uniparticle inner products, it's sucient to look at one-particle states.We therefore examine the S-matrix dened in subsection VC1 for the special case of C. S-MATRIX 273 1 particle at t=1 going to 1 particle at t=+1, using the free, massless lightcone Lagrangians given above. For the boson we found that this matrix element was simplythe inner product between the two states, appearing in the formR dd 0
0.T h i s worked only because the propagator had the right sign. (It is essentially + ei!jtj.) Thus the sign we use is required for unitarity. A simple way to treat the fermion is to use supersymmetry: Since the boson and fermion kinetic terms are spin independent in the lightcone formalism, we can look at any supersymmetric theory, and check the boson and fermion kinetic terms there. Ifthe boson term agrees with the one we just checked, then the fermion term is OK bysupersymmetry (which preserves unitarity by the lightcone-like supertwistor analysisof subsection IIC5). Alternatively, we can use the same method applied to the boson:The propagator now has an extra factor of 1 =(i@ +), or 1=p+in momentum space. Sincep+is always positive for positive energy, these states also appear with the correct-sign norm. To analyze negative energy (antiparticles), we note that in thederivation of the path integral nal states always appear to the left and initial states to the right. In the fermionic case this is important because it will introduce an extra sign: Since 1@+ 2=+ 2@+ 1 (with two signs canceling from reordering fermions and integration by parts), the right sign will always be produced with correct ordering of initial vs. nal states (i.e., positive vs. negative energy), independent of the helicity (or whether electronvs. positron, etc.) For similar reasons, it is clear that integral spin is always described by commut- ing (bosonic) elds, while half-integral spin is always described by anticommuting(fermionic) elds: The number of undotted minus dotted indices on a eld is always odd for half-integer spin, even for integer, and a derivative carries one dotted and one undotted index, so contraction of all indices means an even number of derivatives forinteger spin and odd for half-integer. Without loss of generality, we then can separateeach eld into its real and imaginary parts. Then for each real eld integration byparts gives (i@) n=(1)n[(i@)n]=(1)n+(i@)n) (1)n+=1 where (1)is the statistical factor for (1 for bosons,1 for fermions), and we have inclueed the appropriate i's for hermiticity of the action. Thus, integer spin is associated with bosons (( 1)n=1=(1)), and half-integer with fermions (( 1)n= 1=(1)). This is the \spin-statistics theorem". By using real elds with a 274 V. QUANTIZATION diagonal kinetic term, we have implicitly assumed kinetic terms appear only with the correct sign: For example, for a complex bosonic eld =A+iB)y@=Ai$ @B=Bi$ @A and thus has indenite sign. Thus, spin and statistics follows from Poincar ei n v a r i - ance, locality, and unitarity. If we drop unitarity, we get \ghosts": We'll see examples of such wrong-statistics elds when quantizing gauge theory. Note that demanding unitarity (the right sign of the kinetic term) is the same as demanding positivity of the true energy, as least as far as the kinetic term is concerned: The energy is given by the Hamiltonian of the eld theory; if the kinetic term changes sign, the corresponding contribution to the Hamiltonian does also. (Compare thediscussion of Wick rotation of the action in subsection VB4.) Using anticommuting elds to describe fermions is more than a formality. In gen- eral, the signicance of describing states by quantizing classical elds that commute or anticommute has two purposes: (1) to avoid multiple counting for indistinguish-able particles, and (2) to insure that two identical fermions do not occupy the same state. Thus, when describing two particles in dierent states, the phase associated with (anti)commutation is irrelevant: A \Klein transformation" can be made thatmakes anticommuting quantities commute for dierent states, and anticommute (i.e., square to zero) only for the same state. However, such transformations are nonlocal, and locality is crucial in relativistic eld theory. (See excercise IA2.3e.) 6. Cutting rules For some purposes it is useful to translate the three dening properties of relativis- tic quantum eld theory into graphical language. Poincar ei n v a r i a n c ei st r i v i a l ,s i n c e the propagators and vertices are manifestly Poincar e covariant in covariant gauges. Unitarity and causality can also be written in a simple way in functional notation. We rst note that the inner product for free multiparticle wave functions can be written very simply in momentum space as h ji= y[]eD+[]
 =0;D +=1 2Z dp  (p)
+(p) (p)
(p)=iN(p) 1 2(p2+m2i))
+(p)=(p0)2[1 2(p2+m2)]N(p) where andare products of positive-energy single-particle states, and
+projects onto the positive-energy mass shell. (The exponential takes care of the usual combi- natoric factors.) We have written a generic propagator, with numerator factor N(=1 C. S-MATRIX 275 for scalars). (Without loss of generality, we have assumed a real basis for the elds, soNcan be taken as real.) The S-matrix amplitude then can be written in operator language as []= y[][])A =Z D []eiS[]=   Z[]=h jSji We have used the fact that positive-energy states propagate forward in time and negative backwards to write the usual Hilbert-space inner product in terms of initialand nal states of positive energy. The S-matrix operator Sappears because and satisfy the free equations of motion, and Sperforms time translation from t=1 fortot=+1for to include interactions. The unitarity condition is then (see subsection VA4) S yS=1)Z[]yeD+Z[]=1 while causality is  (x) S[]y (y)S[] =0) (x) Z[]yeD+ (y)Z[] =0forx0>y0 This causality relation, which already holds in nonrelativistic eld theory, can be strengthened by using Lorentz invariance: If ( xy)2>0 (spacelike separation), then x0<y0can be Lorentz transformed to x0>y0. Thus, the above expression vanishes everywhere except on or inside the backward lightcone with respect to xy.T h e s e functional forms of unitarity and causality (and Poincar e invariance) can also be used as a basis for the derivation of the functional integral form of Z[]i nt e r m so ft h e action, rather than relying on its relation to the Hamiltonian formalism. The fact that these conditions are satised by Feynman diagrams follows easily from inspection. We examine them using the explicit expression for Zfollowing from the functional integral: Z[]=eDeiSI;Z []y=eD*eiSI D=1 2Z dp (p)
(p) (p);
(p)=iN 1 2(p2+m2i) D*=1 2Z dp (p)
(p)* (p);
(p)* =iN 1 2(p2+m2+i) These expressions can be translated straightforwardly into position space as Z dp (p)
(p) (p)=Z dx dx0 (x)
(xx0) (x0) 276 V. QUANTIZATION etc. From the results at the end of subsection VB2, we see that the propagators satisfy the relations
+(x)=
(x)
A(x)=
* (x)+
R(x)
+(x)=
(x);
(x)=
(x);
R(x)=
A(x) and, of course,
R(x)=0f o rx0<0. We will now see that the cancelations in the unitarity and causality relations occur graph by graph: There are contributionsconsisting of a sum of terms represented by exactly the same diagram, with each term diering only by whether each vertex comes from ZorZ y.F i r s t ,t h i sa  e c t st h es i g n of the diagram, since each vertex from Zygets an extra sign from the eiSIinZy,a s compared to the eiSIinZ. Second, this aects which propagators appear: (x;y)in:(Z;Z)!
(xy) (Zy;Z)!
+(xy) (Z;Zy)!
(xy) (Zy;Zy)!
*(xy) Now if we sum over two diagrams diering only by whether xappears in ZorZy, the result is proportional to Y
(xyi)Y
(xzj)Y
+(xyi)Y
*(xzj) whereyiare fromZandzjare fromZy.H o w e v e r ,
(xy)
+(xy)=
(xy)
*(xy)=
R(yx)=0for x0>y0 and therefore the two diagrams cancel if x0is the latest of all the vertices. In the unitarity relation we sum over whether a vertex occurs in ZorZyfor each vertex, including the latest one, so that condition is easily satised. (The only diagram that survives is the one with no particles, which gives 1.) In the causality relation we perform this sum for each vertex except y(since=(y)a c t so n l yo n Z), but since y0<x0,y0is not the latest vertex, so again the latest one is summed over. Excercise VC6.1 Consider an arbitrary 1-loop graph. Why would replacing all the propagators
with advanced propagators
A(or all with retarded propagators
R)a l lt h e way around the loop in the same direction give zero? Use this result, and the relation between the various propagators, to show that any one-loop diagram (with normal propagators) can be expressed as a sum of products of tree C. S-MATRIX 277 graphs, with some summations of external states (\Feynman tree theorem"). How does this dier from the cutting rule for unitarity? (Hint: Look at thesigns of the energies of external states.) There is one ne point in this construction: We may use Feynman rules from a complex action, such as those used for massive theories in subsection IIIC4, or whenusing complex gauges (see section VIB). In that case, since the S-matrix Sis gauge- independent, and the original action Swas real (before eliminating complex elds or choosing complex gauges), it is legal to use the Hermitian conjugate actionS yto dene the Feynman rules for Sy(andZy): When multiplying SyS,w eu s et h eu s u a l rules to nd the second factor S, and the conjugate rules to nd the Sused in the rst factorSy: SyS=[S(Sy)]yS(S) The result of conjugating the S-matrix then will be to complex conjugate twice, and return rules identical to those used for S, except for the dierences noted above for real actions. That means that the above proof of the cutting rules goes through unmodied, where we use the same complex rules in the entire diagram, regardless of whether they are associated with ZorZy. In particular, this means that vertices from the two parts of the graph will dier only by sign (conjugating just the iineiSI,n o tt h e SI), and propagators will dier only by their (momentum-space) denominators, not their numerators. This is particularly important for the complex elds of subsectionIIIC4, since otherwise even the types of indices carried by the elds would dier. 7. Cross sections In quantum physics, the only measurables are probabilities, the squares of abso- lute values of amplitudes. Since we calculate amplitudes in momentum space, proba- bilities are expressed in terms of scattering of plane waves. They are more naturally normalized as probabilities per unit 4-volume (or D-volume in arbitrary dimension),since plane waves are uniformly distributed throughout space. This can be seen ex-plcitly from the amplitudes: Because of the total momentum conservation -function that appears with each connected S-matrix element S fi, we have for the probability P Sfi=iX p Tfi )P=jSfij2=jTfij2X p (0) =jTfij2X pVD (2)D=2 278 V. QUANTIZATION where we have found the coordinate D-volume by the Fourier-transform denition of the-function: (0) =Z dx1=ZdDx (2)D=21=VD (2)D=2 A \cross section" is dened as a probability for the scattering of two incoming particles into some number of outgoing particles. The scattering is \elastic" if the two nal particles are the same as the two initial particles (they exchange only 4-momentum), \inelastic" otherwise. Generally one particle is in a beam directed at a target (at rest in the lab frame) containing the other \incoming" particle, but in some experiments two beams are directed at each other. In either case the cross section isdened by the rate at which one particle interacts divided by the ux of the other particle, where flux =rateof arrival area=(density )(relativevelocity ) and thus the \dierential cross section" (yet to be integrated/summed over nal states) is d=P VD1 12v12 The spatial density is the integrand of the spatial integral that denes the inner product: From subsection VB2, for bosonic plane waves we have h j i=(p0)ZdD1x (2)D=2 1 2i$ @t =ZdD1x (2)D=2! )=! (2)D=2 where!=jp0j. (The same result can be obtained for fermions, when their external line factors are appropriately normalized.) The expression for dis actually independent of the frame, as long as the 3- momenta of the two particles are parallel. This is the case for the most frequentlyused reference frames, the center-of-mass frame and the \lab frame" for either particle (where that particle is at rest, as is the lab if that particle is part of a target). Then (! 1!2v12)2=!2 1!2 2~p 1 !1~p2 !22 =j!2~p1!1~p2j2=1 2(p1[ap2b])2=2 12 2121 4[s(m1+m2)2][s(m1m2)2] using again the Mandelstam variables and ijintroduced in subsection IA4. Finally, we include the \phase space" for the nal states to obtain d=jTfij2(2)DD(Pp) 12Y fdD1p (2)D=21!Y f:ni d e n t1 n! C. S-MATRIX 279 where the rst product is over all nal one-particle states, and the second is over each set ofnidentical nal particles. The normalization again follows from the inner- product for plane waves: By Fourier transformation, dD1x=!dD1p=(2)D1. It also appears in the \cut propagator"
+used in unitarity, as in the previous subsection: ZdDp (2)D=2(p0)2[1 2(p2+m2)] =ZdD1p (2)D=21! The simplest and most important case is where two particles scatter to two parti- cles. (This includes elastic scattering.) The \dierential cross section" d=d , where d is the angular integration element for p3, is found by integrating d(in D=4) over d3p4anddj~p3j. The former integration is trivial, using the function for 3-momentum conservation. The latter integration is almost as trivial, integrating the remaining  function for energy conservation: d3p3X p0 =d dj~p3j(~p3)2@Pp 0 @j~p3j1 (j~p3jj~p3j0) X p0=!1!2+q (~p3)2+m2 3+q (~p1+~p2~p3)2+m2 4 )@Pp0 @j~p3j=1 2(sm2 3m2 4)!3m2 3!4 j~p3j!3!4 wherej~p3j0isj~p3jevaluated as a function of the remaining variables atPp0=0 . W e then nd d d =( 2)2jTfij2j~p3j3 12[1 2(sm2 3m2 4)!3m2 3!4] The center-of-mass frame (see subsection IA4) is the simplest for computations. In that frame the dierential cross section simplies to d d =( 2)2jTfij234 12s Another convenient form for the dierential cross section is d=dt , tradingfortand integrating out the trivial dependenceRd=2. In the center-of-mass frame we have d =2d(cos  )=s 1234dt Sincetis Lorentz invariant, we therefore have in allframes (that conform to our earlier requirement for the 12factor ind) d dt=1 2(2)3jTfij21 2 12 280 V. QUANTIZATION For example, for the 4-point scalar example considered in subsection VC4, we have d dt=(4)3g4 s(s4m2)1 sm2+1 tm2+1 um22 S S† D D_D_D+D+ D* For some purposes (such as considerations of unitarity and causality, as in the previous subsection) it is useful to draw the Feynman diagrams for the cross sectionitself (or actually jTj 2). In such a diagram we draw one of the diagrams from S and one fromSy, separating the two by a line (dashed, zig-zag, or shaded on one side, according to your preference), and connecting allthe external lines (initial and nal) on one side to the corresponding ones on the other. The result is a bubblediagram with a \cut": The \cut propagators" are
+=2(p0)N[1 2(p2+m2)] (or
, depending on how we label the momenta), corresponding to the propagator
=N=1 2(p2+m2), if we sum over all polarizations; otherwise Nis replaced by a term in the sum N=P^ y^ . The momenta of the cut propagators may not be integrated over, depending on whether they represent nal states whose momentaare summed over (i.e., not measured; in practice the momenta of initial particles are always measured). The only other dierence in the Feynman rules from the S- matrix is that in Wick rotating back Sgets the usual m 2!m2iwhileSygets m2!m2+i, and each connected graph in Sgets ani(Pp) while each inSy gets ai(Pp). The algebra for the cross section is thus identical to that of a vacuum bubble (although the momentum integration is not, and the cut propagators lack the usual denominators). In particular, instead of summing over just physicalpolarizations in a cut vector propagator, which corresponds to using a unitary gauge, we can include ghosts in the external states, and use any gauge: This follows from the cutting rules derived for unitarity. 8. Singularities We know from free theories that any propagator has a pole at the classical value of the square of the mass. This statement can be extended to the interacting theory: The C. S-MATRIX 281 (\Landau") singularities in any Feynman diagram are exactly at classically allowed (on shell) values of the momenta. The simplest way to see this is to write the propagators in a way reminiscent of the classical theory, where the appearance of the worldline metric in the action resultsin the (Wick-unrotated) Schwinger parametrization of the propagator, i 1 2(p2+m2)=Z1 0d ei(p2+m2)=2 For simplicity we consider a scalar eld theory with nonderivative self-interactions. The corresponding form of a Feynman diagram, written in momentum space by Fourier transformation, is then Z dx0 idpijdijeiP hiji[ij(p2 ij+m2)=2(xixj)
pij] wherei;jlabel vertices (and external endpoints), hijilabels links (propagators), and dx0indicates integration over just vertices and not free ends of external lines. Integration over these x0's produces functions for momentum conservation: X jpij=0 i.e., the sum of all momenta owing into any vertex vanishes. (Note pij=pji.) This constraint can be solved by replacing momenta associated with each propagator withmomenta associated with each loop (and keeping momenta associated with external lines). For example, for planar diagrams, we can write p ij=pIJ=kIkJ whereI;Jlabel loops: pIJlabels the propagator by the two loops on either side, rather than the vertices at the ends (and similarly for ij=IJ). Similar remarks apply to nonplanar diagrams, but the parametrization in terms of loop momenta kIis more complicated because of the way external momenta appear. (In the planar case,the above parametrization can also be used for external momenta, and automatically enforces overall momentum conservation.) The Feynman integral then becomes Z dk 0 IdIJeiP hIJiIJ[pIJ(k)2+m2]=2 wherekare loop (k0) and external momenta after solving the conservation conditions. We can now treat the exponent of the Feynman diagram in the same way as a classical mechanics action, and nd the corresponding classical equations of motion by 282 V. QUANTIZATION varying it. By the stationary phase approximation (or steepest descent, after Wick rotation of ), the classical solutions give the most important contribution to the integrals (at least for weak coupling). This approximation is related to long-distance (i.e., infrared) behavior (see excercise VB4.3a). Taking acount of the fact that the 's are constrained to be positive (by treating = 0 separately or making a temporary change of variables =2oreto an unconstrained variable), we nd IJ(p2 IJ+m2)=0;X JIJpIJ=0 or, in terms of the original variables, ij(p2 ij+m2)=0;pij=xixj ij;X jpij=0 These are known as the \Landau equations". Their correspondence with classical congurations of particles follows from treating as the proper time, as seen from p=
x=
. (Actually, it is the generalization discussed in section IIIB of proper time to include the massless case; in the massive case s=m.) The equation (p2+m2)=0 says that either the particle for the line is on-shell or there is no such line (it has vanishing proper length), while the equationP Jp= 0 says that the sum of p around a loop vanishes, another statement that p
=
x. Note that these physical singularities are all in physical Minkowski space: In Euclidean space, p2+m2is positive denite, so it never vanishes (except for constant, massless elds p=m= 0). Furthermore, in Euclidean space, one can always rotate any momentum to any direction, whereas in Minkowski space one can never Lorentztransform to or through either the forward or backward lightcone. Thus, calculating in Euclidean space makes it clear that S-matrix elements for positive-energy states are given by the same expressions as those for negative-energy states: To comparetwo amplitudes that are the same except for some nal particles being replaced with initial antiparticles, or vice versa, we just change the sign of the energy. This is called \crossing symmetry". In the case where all particles are reversed, it is CPT invariance (\CPT theorem"). 9. Group theory Although the manipulation of spin indices in Feynman diagrams is closely tied to momentum dependence, the group theoretic structure is completely independent, and can be handled separately. Therefore, it is sucient to consider the simple example of scalars with a global symmetry. (The more physical case of chromodynamics will C. S-MATRIX 283 be the same with respect to group theory, but will dier in dependence on momentum and spin.) The simplest case is the U(N) family of groups. We choose an action of the form L=1 g2tr[1 4(@)2+V()] whereis a hermitian NN matrix. This action is invariant under the global U(N) symmetry 0=UU1 A simplication we have chosen is that the interaction has only a single trace; this is the case analogous to pure Yang-M ills theory. (We also included the coupling constant as in Yang-Mills.) gluon quarkcolor flavor When we draw a Feynman diagram for this eld theory, instead of a single line for each propagator, we draw a double (parallel) line, each line corresponding to one of the two indices on the matrix eld. Because of the trace in the vertex, the propagatorlines connect up there in such a way that eectively we have continuous lines thattravel on through the vertices, although the two lines paired in a propagator go theirseparate ways at the vertex. These lines never split or join, and begin or end only onexternal elds. We can also draw arrows on the lines, pointing in the same directioneverywhere along a single line, but pointing in opposite directions on the two linesin any propagator pair: This keeps track of the fact that appears in the trace always multipled as and never as  T. A physical picture we can associate with this is to think of the scalar as a bound-state of a quark-antiquark pair, with one line associated with the quark and another with the antiquark; the arrows are thenoriented in the direction of time of the quark (which is the opposite of the directionof time for the antiquark). The quark is thus in the dening representation of U(N)(and the antiquark in the complex conjugate representation). Group theory factors 284 V. QUANTIZATION are trivial to follow in these diagrams: The same color quark continues along the extent of a \quark line"; thus, there is a Kronecker for the two indices appearing at the ends of the quark line (at external elds); each quark is conserved. Excercise VC9.1 Using the quark-line notation, where the lines now represent avor, draw all4-point tree graphs, with 3-point vertices, representing scattering of K +K! +(see subsection IC4) via exchange of other mesons in that U(6) multi- plet. What are the intermediate states (names of mesons) in each channel? Note that the \ avor ow" of both diagrams can be represented by a singlediagram, by separating all pairs of intermediate lines to leave a square gap inthe middle: This \duality diagram" represents the mesons as strings; the gapbetween quarks represents the \worldsheet". Excercise VC9.2 Consider quark-line notation for doing group theory in general: (Calculate using only graphs, with numerical factors | no explicit indices or Kronecker's, except for translating denitions.) aWrite the structure constants for U(N) as the dierence of two diagrams, by considering a vertex of the form tr(A[B;C]). bFind pictorially the resulting expression for the Cartan metric (see subsection IB2): Show that it is the identity times 0 for the U(1) subgroup and 2N forthe SU(N) subgroup (as found previously in subsection IIIC1). cAlso use these diagrams to prove the Jacobi identity (i.e., nd the resulting 6 diagrams and show they cancel pairwise). dDerive diagrammatically the value of d ijkof excercise IB5.3b for the dening representation. Excercise VC9.3 Consider the group theory factors for the above scalar theory, with only acubic interaction. Draw all the 1PI 1-loop diagrams with 4 external double-lines, and rewrite the corresponding factors in terms of traces and productsof 4 elds, including factors of N. (Be careful to include all permutations ofconnecting propagators to vertices.) In general, we nd that connected Feynman diagrams may include diagrams that are disconnected with respect to the above group-theory diagrams, where we considerthe two group-theory lines on an external line to be connected. Such diagrams corre-spond to multiple traces: There is a factor of the form tr(G iGj:::Gk) corresponding C. S-MATRIX 285 to each connected group-theory graph, where i;j;:::;k are the group-theory indices of the external lines (actually double indices in the previous notation). However, all connected trees are group-theory connected. Furthermore, they are all \planar": Any connected U(N) tree can be drawn with none of the quark lines crossing, and all the external lines on the outside of the diagram, if the external lines are \color-ordered"appropriately. Of course, one must sum over permutations of external lines in the T-matrix because of Bose symmetry. However, in calculating W[] one need consider only one such planar graph, with the external lines color-ordered; the Bose symmetryof the elds in Wautomatically incorporates the permutations. (Similar remarks apply to loop and nonplanar graphs.) As an example, consider the U(N) generalization of the 4-point tree example of subsection VC4. The Lagrangian is now (scaling the coupling back into just thevertex) L=trf 1 4[(@)2+m22]+1 3g3g where the interaction term now has a combinatoric factor of1 3instead of1 6because it is symmetric only under cyclic permutations. The result for Wis now modied to W=g2trZ dx1 22 1 1 2(m2)2 This analysis for U(N) can be generalized to SU(N) by including extra diagrams with lines inside propagators short-circuited, representing subtraction of traces. It can also be generalized to SO(N) and USp(2N): In those cases, antisymmetry or symmetry of the matrices means the lines no longer have arrows, and we include diagramswhere the lines inside the propagators have been \twisted", with signs appropriate to symmetrization or antisymmetrization. Generalization for all the above groups to include dening representations is also straightforward: Such elds, like the true quarks in QCD, carry only a single group-theory line. For example, sticking with our simpler scalar model, we can generalize to a scalar theory with the same grouptheory as chromodynamics, with \free" quark elds, appearing as scalar elds in the dening representation of the group: L!L+tr[ 1 2(@ )y
(@ )+ yf() ]; 0=U U1 f where is an NM matrix with M avors, and Ufis the avor symmetry. (This color+ avor symmetry was treated in subsections IC4 and IVA4.) This eld has apropagator with a single color line (with an arrow); however, we can also use another double-line notation, where propagators carry one line for color and another for avor. This method can be generalized to arbitrary representations obtained by 286 V. QUANTIZATION direct products of dening representations, (anti)symmetrizations, and subtractions of traces, by giving the propagators the corresponding number of lines (though usually two lines are sucient for the interesting cases). REFERENCES 1P.A.M. Dirac, P r o c .R o y .S o c . A114 (1927) 243; W. Heisenberg and W. Pauli, Z. Phys. 56(1929) 1, 59(1930) 168; E. Fermi, Rev. Mod. Phys. 4(1932) 87: birth of (interacting) quantum eld theory (QED). 2S. Tomonaga, Prog. Theor. Phys. 1(1946) 27; J. Schwinger, Phys. Rev. 74(1948) 1439, loc. cit. (VB, ref. 2, rst ref.), 76(1949) 790: gave relativistic procedure for treating quantum eld theory. 3Feynman, loc. cit. (VB, ref. 4): same as above, with diagrams. 4F.J. Dyson, Phys. Rev. 75(1949) 486, 1736: related Tomonaga-Schwinger and Feynman approaches; gave procedure to all loops. 5J. Schwinger, Proc. Natl. Acad. Sci. USA 37(1951) 452, 455: generating functionals for eld theory. 6Feynman, loc. cit. (VA): path-integrals for eld theory. 7J. Schwinger, Quantum electrodynamics (Dover, 1958): reprints of most of the important quantum eld theory papers of the 40's and 50's. 8S.S. Schweber, QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga (Princeton University, 1994). 9J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127(1962) 965; G. Jona-Lasinio, Nuo. Cim. 34(1964) 1790: eective action. 10Y. Nambu, Phys. Lett. 26B (1968) 626: tree graphs are classical eld theory. 11D.G. Boulware and L.S. Brown, Phys. Rev. 172(1968) 1628: expression for classical eld in terms of tree graphs. 12S. Coleman, Secret symmetry: An introduction to spontaneous symmetry breakdown and gauge elds ,i nLaws of hadronic matter , proc. 1973 International School of Subnuclear Physics, Erice, July 8-26, 1973, ed. A. Zichichi (Academic, 1975) part A, p. 139:relation of Legendre transform to 1PI graphs via path integrals. 13Dyson, loc. cit. (ref. 2); Schwinger, loc. cit. (ref. 4). 14W. Pauli, Phys. Rev. 58(1940) 716: spin-statistics theorem. 15R.E. Cutkosky, J. Math. Phys. 1(1960) 429: cutting rules for unitarity. 16T. Veltman, Physica 29(1963) 186: cutting rules for causality. 17R.P. Feynman, Acta Phys. Polon. 24(1963) 697: tree theorem. 18L.D. Landau, Nucl. Phys. 13(1959) 181. C. S-MATRIX 287 19S. Coleman and R.E. Norton, Nuo. Cim. 38(1965) 438: classical interpretation of Landau singularities. 20W. Siegel, hep-th/9601002, Int. J. Mod. Phys. A 13(1998) 381: Landau rules from classical mechanics action. 21R.J. Eden, P.V. Landsho, D.I. Olive, and J.C. Polkinghorne, The analytic S-matrix (Cambridge University, 1966):singularities of Feynman diagrams. 22M. Gell-Mann and M.L. Goldberger, proc. Fourth International Conference on High Energy Physics (University of Rochester, 1954); J.M. Jauch and F. Rohrlich, The theory of photons and electrons , 1st ed. (Springer- Verlag, 1955) p. 161:crossing symmetry. 23H. Harari, Phys. Rev. Lett. 22(1969) 562; J. Rosner, Phys. Rev. Lett. 22(1969) 689; T. Matsuoka, K. Ninomiya, and S. Sawada, Prog. Theor. Phys. 42(1969) 56: quark lines for group theory factors. 24J.E. Paton and Chan H.-M., Nucl. Phys. B10 (1969) 516: traces for group theory in planar graphs. 25G. 't Hooft, Nucl. Phys. B72 (1974) 461: double-line notation in eld theory. 288 VI. QUANTUM GAUGE THEORY VI. QUANTUM GAUGE THEORY We now consider special features of spin and gauge invariance, and introduce some special methods for dealing with them. In quantum theory, gauge xing is necessary for functional integration: Gauge invariance says that the action is independent of some variable; integration over that variable would thus give innity when evaluatingamplitudes for gauge-invariant states. Eliminating that variable from the action (a \unitary gauge") solves the problem, but not always in the most convenient way. (If it were, we wouldn't have introduced such a redundant description in the rst place.)Note that such innities already appear for global symmetries: For example, the functional integral with wave function 1 (vacuum-to-vacuum amplitude) is innite by translation invariance. This innity is easier to understand for a nontrivial amplitudein momentum space, as a factor of a momentum-conservation -function (which is either1or 0, but necessarily 1for the vacuum amplitude, which has vanishing momentum because the vacuum is translationally invariant). ::::::: ::::::: ::::::: A. BECCHI-ROUET-STORA-TYUTIN ::::::: We have seen the relationship of gauge invariances to constraints in subsection IIIA5. In this section we consider the quantization of constrained systems, and its application to gauge theories. The Becchi-Rouet-Stora-Tyutin (BRST) method isnot only the most powerful, but also the easiest way to gauge x: It replaces the gauge symmetry with an unphysical, fermionic, global symmetry that acts only on unphysical degrees of freedom. 1. Hamiltonian Physical observables commute with the constraints. Thus, time development is described by the gauge-invariant Hamiltonian Hgi, or we can set the gauge elds i equal to some arbitrary functions fias a \gauge choice": i=fi)H=Hgi+fiGi In quantum mechanics, physical states should be annihilated by the constraints. How- ever, more generally we can require only that these states satisfy the constraintsthrough expectation values: h jG iji=0 This condition is satised by dividing up the constraints into: (1) a subalgebra G0 that annihilates all physical states, (2) complex \lowering operators" Gthat also A. BECCHI-ROUET-STORA-TYUTIN 289 annihilate these states, and are a representation of the subgroup generated by G0, and (3) their hermitian conjugate \raising operators" G+=Gy. (We treat +,, and 0 here as multivalued indices.) Thus, G0j i=Gj i=h jG0=h jG+=0 In the Abelian case, it then follows that, although G+do not annihilate these states, they generate gauge invariances: j i=G+j+i;G 0j+i=Gj+i=0 preserves the inner product of such states as well as the constraints on j i. Unfortunately, things get more complicated in the nonabelian case. For example, the gauge invariance and constraints above are no longer compatible: 0=Gj i=GG+j+i=[G;G+]j+i=if++G+j+i6=0 A convenient way to deal with this problem is to replace the nonabelian algebra Gi with a single operator, which is therefore Abelian. We dene a BRST operator Q that imposes all constraints Giby adding a classical anticommuting \ghost" variable ci, and its canonical conjugate bi, fbi;cjg=j i for each constraint: Q=ciGii1 2cicjfjikbk The second term has been added to insure the Poisson bracket or commutator fQ;Qg=0 so that its crossterm cancels the square of the rst term, while its own square vanishes by the Jacobi identity f[ijlfk]lm= 0. Quantum mechanically, the BRST operator is nilpotent: quantummechanically fQ;Qg2Q2=0 We can also describe the ghost dependence by the \ghost number" J=cibi) [J;Q]=Q (Quantum mechanically, we need to normal order these expressions for QandJ.) The BRST operator provides a convenient method to treat more general gauges than=f, such as ones where the gauge elds become dynamical, which will prove 290 VI. QUANTUM GAUGE THEORY useful particularly in relativistic theories. Now the original physical observables A will satisfy [Gi;A]=[bi;A]=[ci;A]=0) [Q;A]=[J;A]=0 and similarly the physical quantum mechanical states j iwill satisfy (G0;G;b0;b;c+)j i=0)Qj i=Jj i=0 where we have used the fact the only nonvanishing structure constants are f000,f0++, f0,f+++,f,a n df+k. (In the quantum case there are also some subtleties due to normal ordering.) We also have the gauge invariances A=fQ;g;j i=Qji for arbitrary operators  and (unrelated) states ji,s i n c et h e Q-terms won't con- tribute when evaluating matrix elements with states annihilated by Q: (h 1j+h1jQ)(A+fQ;g)(j 2i+Qj2i)=h 1jAj 2i The gauge invariances are consistent with the constraints because of the nilpotence of the BRST operator. (So Q(j i+Qji) still vanishes, etc.) States satisfying Qj i=0;j i=Qji (i.e., we identify states that dier by Qon something) are said to be in the \coho- mology" of Q(\BRST cohomology"), and operators satisfying [Q;A]=0; A =fQ;g are said to be in its \operator cohomology". (The latter cohomology also has a classical analog.) Excercise VIA1.1 Assume that each physical state can be represented as a physical observable(Hermitian operator) acting on a ground state, which is itself physical: j i=Aj0i;A =A y;Qj0i=0 Show how this relates the gauge parameters and cohomologies of j iandQ. The BRST operator incorporates the ghosts that are necessary to generalize treat- ment of the constraints to the nonabelian case: For example, to reproduce the gauge transformations of the Abelian case, we choose ji=b+j+i;Qj+i=0)j i=Qji=^G+j+i A. BECCHI-ROUET-STORA-TYUTIN 291 where ^Gi=fQ;big=Giicjfjikbk are the \gauge-xed" constraints, which include an extra term to transform the ghosts as the adjoint representation. They reduce to just Giin the Abelian case, but add ghost terms to the gauge transformation law otherwise. In particular, the Hamiltonian is a physical operator describing the energy and the time development, so we can write H=Hgi+fQ;g;[Q;Hgi]=0)T eiR dt H =T eiR dt Hgi +fQ;g for some. This includes gauge xing for the gauge i=fidiscussed above, using: =fibi)H=Hgi+fi^Gi;L =(.qmpmi.cibi)+H The ghost terms in ^Gionly aect the time development of unphysical states in this gauge. For example, when calculating S-matrix elements, the result is independent of the gauge choice , as long as both the gauge-invariant Hamiltonian Hgiand the states are BRST invariant. ( Hgicommutes with Q, the initial and nal states are annihiliated by it.) It is also independent of the gauge choice jiforj i!j i+Qji. (Such a \residual gauge invariance" persists even though the asymptotic states satisfy the free eld equations.) In the cases of interest in relativistic physics, the constraints always consist of a linear term depending only on the canonical momenta p(conjugate to the fundamental variablesq), at least after some redenitions, plus higher-order terms, which can be treated perturbatively. Therefore, as the simplest nontrivial example, we consider a model with a single variable q,w i t h Hgi=0;G =p)Q=cp If we assume boundary conditions on the wave functions such that they can be Taylor expanded in q(they can always be expanded in c), we can write =1X n=0(n+cn)1 n!qn and similarly for . Examining  =Q, we see that we can easily gauge n=0f o r alln.L o o k i n g a t Q = 0, we then nd that n=0f o ra l l nexceptn=0 ,s oo n l y the constant piece of survives. In other words, the cohomology is given by Q =0; =Q)p =b =0 292 VI. QUANTUM GAUGE THEORY So, solving for the cohomology of Q=cpis the same as solving the constraint p =0 without ghosts. Excercise VIA1.2 Find the cohomology of the BRST operator Q=cay+cya for creation and annihilator operators satisfying [a;dy]=[d;ay]=fc;byg=fb;cyg=1 (the other commutators vanishing), by expanding in creation operators ay,by, cy,dyabout a vacuum state destroyed by the annihilation operators a,b,c,d. (This is the common alternative to the boundary conditions used for Q=cp above.) 2. Lagrangian To obtain more interesting gauges we need some extra bosonic variables, such as the gauge elds ithat we lost along the way, and their canonical conjugates (\Nakanishi-Lautrup elds") Bi, [Bi;j]=ij i as well as their corresponding ghosts (\antighosts") ~ ci. We can do this in a trivial way by including constraints that set Bto zero: Q=ciGii1 2cicjfjikbk+~biBi;J =cibi~ci~bi where ~ciis conjugate to ~bi, f~ci;~bjg=j i As a simple example, consider =ibi)fQ;g=i^Gii~bibi The action now includes the gauge elds and all the ghosts as dynamical variables: L=(.qmpm+. iBii.cibii. ~bi~ci)+Hgi+fQ;g For this gauge we can eliminate band~bby their equations of motion; assuming Gi is only linear in p, we then can eliminate pto return completely to a Lagrangian formalism: L=Lgi(q;). iBii(rtci). ~ci A. BECCHI-ROUET-STORA-TYUTIN 293 whereLgirepresents the original gauge-invariant action (which depended on both q and, including time derivatives), and rtis the covariant (time) derivative: rtci=.ci+cjkfkji The gauge condition (from varying B)i sn o w. = 0, generalizing the non-derivative gauges found without the antighosts and Nakanishi-Lautrup elds. Correspondingly, the ghost term is now second order in derivatives. Excercise VIA2.1 Consider the general gauge choice =ibi+[Fi(q;p)+Ei(B)]~ci )fQ;g=i^Gii~bibi+(Fi+Ei)Bi+ci[Gi;Fj]~cj whereFiare some arbitrary functions of the original variables, and Eiare functions that eectively average over the types of gauges produced by Fi. Find the gauge-xed Hamiltonian and Lagrangian. In the case where Eis linear inB, eliminate B,b,a n d ~bfrom the Lagrangian by their algebraic equations of motion. Now that we understand the principles, all these manipulations can be performed directly in the Lagrangian formalism. This will have the advantage that in eldtheory the Lagrangian is manifestly Lorentz covariant, while the Hamiltonian (or the Lagrangian in the Hamiltonian form .qp+H) is not, because of the way it singles out time derivatives (and not spatial ones). (Consider, e.g., electromagnetism.) Similarly, the gaugeG i= 0 is usually not Lorentz covariant. We can work with just the original variablesq;plus the new variables B;c; ~c, and dene Qby the transformation it induces (as derived from the Hamiltonian formalism): Qqm=ciiqm;Q i=i(.ci+cjkfkji);Q ci=i1 2cjckfkji;Q ~ci=Bi;Q Bi=0 whereiis the gauge transformation induced by Gi([Gi;] in the Hamiltonian for- malism). Note that the BRST transformations of the original variables are exactly the same as the gauge transformations, with the gauge parameters replaced with the corresponding ghosts. We can also consider the Nakanishi-Lautrup elds Bas original variables, with the fact that they don't occur explicitly implying they have constraints B= 0. Alternatively, we can treat the antighosts ~ cas pure gauge degrees of freedom, with their own nonderivative gauge transformation ~c=~that allows them to be completely gauged away. 294 VI. QUANTUM GAUGE THEORY The Lagrangian can be gauge-xed directly as L=Lgi+QL where in the case just considered L=. i~ci gives the same Las above for the. = 0 gauge. In the simpler case described earlier (the gauge = constant) L=(ifi)~ci This gives the result, for the simplest choice f=0 , L=Lgi(q;)+iBii(.ci+cjkfkji)~ci!Lgi(q;0)i.ci~ci after eliminating the Lagrange multipliers Bandby their algebraic equations of motion. Note that  L=i~cicorresponds to the Hamiltonian formalism's  = 0. Thus, in the Hamiltonian formalism we never quantize with H=Hgi+G, but can use just Hgiand  = 0, which is equivalent to using Hgi+fQ;gfor any , while in the Lagrangian formalism we can never quantize with just Lgi(q;), or even Lgi(q;0), and  Lis never zero, but must be chosen so as to break the gauge invariance. However, the extra term for Lgi(q;0) is just the.cbterm found from converting Hgito the Lagrangian formalism. Excercise VIA2.2 Repeat excercise VIA2.1 directly in the Lagrangian formalism. (Find  L, etc.) All our results for quantization apply equally well in the path-integral formalism, which can be applied to either the Hamiltonian or Lagrangian. (Of course, for eldtheory we will be interested in applying BRST to path integrals for Lagrangians.)We then evaluate matrix elements as A=Z D []e iS[];S=Sgi+Q; = gi+Q ;QSgi=Q gi=0 Sgiand gidepend on just the physical elds (no ghosts), so they are gauge invari- ant as well as BRST invariant. For S-matrices, since is an asymptotic state, theBRST operator used for its constraint and gauge invariance can be reduced to itsfree part:Qthen acts on only the gauge elds. The statement of gauge invariance of giis then equivalent to the requirement that gauge elds appear in it only as their A. BECCHI-ROUET-STORA-TYUTIN 295 Abelian eld strengths. For example, the usual gauge vector Adescribing electro- magnetism appears in single-particle factors in the wave functional ( [ ]=Q 1[] as in subsection VC1) only as: 1[A]=h ajAai; A a=@a; a==0 ) 0= 1=h ajAai=h@a aji using@0h ji= 0 (where the relativistic inner product hjiwas dened in subsection VB2). The transversality of ais equivalent to coupling to the Abelian eld strength, since @a a=0) a=@b ab)h ajAai=1 2h abjFabi in terms of an antisymmetric-tensor external-line factor ab. 3. Particles We have seen that the relativistic particle (with or without spin) is a simple example of a contrained system. For the simplest case, spin 0, the BRST operator follows simply from the single constraint: Q=c1 2(m2) Unlike the nonrelativistic case, the relativistic \Hamiltonian" is identied with this constraint. Since we know constraints are treated by the BRST operator, we can consider writing the eld theory action in terms of it: S0=Z dx dc1 2Q Using the explicit cdependence of the eld  = ic , we nd the usual scalar kinetic term.is thus the usual eld, while is an \antield", which has opposite statistics to(fermion instead of boson). We'll see in chapter XII that Qcan be constructed straightforwardly for arbitrary spin, and has a simple expression in term of generalizedspin operators. (As in nonrelativistic theories, spin is easier to treat directly in quantum mechanics rather than by rst-quantization of a classical system.) The kinetic term then generally can be written as a slight modication of the above. Thenthe antields will be found to play a nontrivial function, rather than just automatically dropping out as in this case. From the constraints and their algebra for spin 1/2 (see also excercise IIIB1.2) we nd the BRST and ghost-number operators: Q=c 1 2(m2)(
@imp 2)2b+~; J =cb++~~ 296 VI. QUANTUM GAUGE THEORY whereand its conjugate are bosonic ghosts, and we have added a nonminimal term with boson ~(conjugate ~)a n df e r m i o n (conjugate ) to allow gauges general enough for rst-quantization: [;]=[~;~]=f;g=1 For convenience, we also have chosen (and)t o anticommute with , f; ag=f; ag=0 to avoid having to replace imp 2with 1im; this has the natural interpretation of treatingandas bosonic (ghost) components of the matrices (see subsections XIIA4-5,B5). Note that [ Q;] = 0, butfQ;Ag6=for anyA,s ois in the operator coho- mology ofQ. Normally, this would imply innite copies of the physical states in the cohomology, since applying a \translation" with the ghost variable gives a new state in the cohomology from any given one. The nonminimal variables allow us toavoid this problem by combining with ~to produce harmonic oscillator creation and annihilation operators: = 1p 2(a+ay);~=1p 2i(aya); =1p 2(~a~ay);~=1p 2i(~ay+~a) [a;~ay]=[ ~a;ay]=1;r e s t =0 This allows us to dene a ground state aj0i=~aj0i=0 which breaks the translation symmetry of . In chapter XII we'll show in a more general framework how the  Q type of action then reproduces the Dirac action. 4. Fields As described in subsection VIA2, we can perform gauge xing through BRST, including the introduction of ghosts, directly on the Lagrangian at the classical level. Also, the BRST transformations on the physical elds are just the gauge transforma- tions with the gauge parameters replaced by ghosts, and the BRST transformation on the ghosts is quadratic in ghosts times the structure constants, while on the antighostsit gives the Nakanishi-Lautrup elds, and it annihilates the NL elds. In the case of Yang-Mills we then have QAa=[ra;C];Q C =iC2;Q ~C=iB; QB =0 A. BECCHI-ROUET-STORA-TYUTIN 297 while for matter transforming as =iwe have Q=iC where we have used matrix notation for the group algebra, as usual. There are two minor dierences from the transformation rules we used in our general discussion previously: (1) We have included an extra \ i" in our denition of the relativistic Q, for a convenience that will become apparent only when we relate relativistic rst- andsecond-quantization (see chapter XII). (2) There is a relative sign dierence for QC because now Qis second-quantized while G iis still rst-quantized (i.e., matrices). More explicitly, we have, e.g., C2=1 2fC;Cg)QCi=1 2CjCkfkji;Q=iCiGi The gauge-xed action is then the gauge-invariant action plus the BRST trans- formation of some function : Sgf=SgiiQ For example, consider Yang-Mills in the most common type of gauge, where some function of Ais xed: =trZ 1 2~C[f(A)+1 2B])Lgf=Lgi1 2B[f(A)+1 2B]1 2i~C@f @A
[r;C] for some constant .F o r=0 ,Bis a Lagrange multiplier, enforcing the gauge f(A) = 0, while for 6= 0, we can eliminate Bby its auxiliary eld equation: 1 2B[f(A)+1 2B]!1 4f2 Examples will be given in the following section. In eld theory gauge-xing functions always have linear terms, as do gauge trans- formations. Furthermore, there always exist \unitary gauges", where no ghosts are required. The ghost terms in general gauges serve simply to provide the appropriateJacobian factor for the eld redenition that transforms from the general gauge to the unitary gauge, which appears at the quantum level from functionally integrating out the ghosts. The simplest example is the trivial gauge invariance that occurs inthe St uckelberg model: QA=@C; Q =C; Q ~C=iB; QC =QB=0 which we can x with iQ(~CO)=BO+i~COC 298 VI. QUANTUM GAUGE THEORY for some eld-independent operator O. Functionally integrating out Bstill sets=0 , but produces an inverse functional determinant of O(from redenition of ,o rf r o m (O)), canceled by that from integrating out the ghosts. The advantage of BRST is that all this can be treated at the classical level, in terms of the classical action,without regard to functional integration, while directly giving a solution that can be expressed immediately in terms of Feynman rules. Excercise VIA4.1 Show that the gauge xing iQ(~CO+~CAB+CB+CCB) whereO,A,B,a n dCare eld-independent operators, gives a result equivalent to the previous, by considering functional determinants or eld redenitions. Excercise VIA4.2 Show that the Lagrangian AAB+CBD!AABD by the eld redenition D!D+B 1B for bosonsA,B,C,Dand operatorsA,B. This is the classical equivalent of det(AB)=det(A)det(B). These methods apply straightforwardly to supersymmetric theories in superspace: From the gauge transformations of subsection IVC4, QeV=iCeVieVC;QC=iC2;Q C=iC2 Q~C=iB; Q ~C=iB;QB=QB=0 whereC,~C,a n dBare chiral superelds, and C,~C,a n d Btheir hermitian conjugates. In practice, this BRST approach is sucient for gauge xing. In particular, this is true for the fundamental elds used in the standard model (including gravity),which have spin2. Therefore, we'll use mostly this approach in the rest of this text. However, some observed hadrons have much higher spin. The rst-quantized approach of chapter XII gives a natural and direct way of understanding ghosts and BRST for the elds describing such particles, and translates directly into the treatment of Zinn-Justin, Batalin, and Vilkovisky (ZJBV) for eld theory. A. BECCHI-ROUET-STORA-TYUTIN 299 REFERENCES 1Feynman, loc. cit. (VC, ref. 17); B. DeWitt, Phys. Rev. 162(1967) 1195, 1239; L.D. Faddeev and V.N. Popov, Phys. Lett. 25B (1967) 29; S. Mandelstam, Phys. Rev. 175(1968) 1580: ghosts. 2N. Nakanishi, Prog. Theor. Phys. 35(1966) 1111; B. Lautrup, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34(1967) No. 11, 1. 3C .B e c c h i ,A .R o u e t ,a n dR .S t o r a , Phys. Lett. 52B (1974) 344, Ann. Phys. 98(1976) 287;I.V. Tyutin, Gauge invariance in eld theory and in statistical physics in the operator formulation, Lebedev preprint FIAN No. 39 (1975), in Russian, unpublished; M.Z. Iofa and I.V. Tyutin, Theor. Math. Phys. 27(1976) 316; T. Kugo and I. Ojima, Phys. Lett. 73B (1978) 459; L. Baulieu, Phys. Rep. 129(1985) 1: BRST. 4E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. 55B (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Lett. 69B (1977) 309; E.S. Fradkin and T.E. Fradkina, Phys. Lett. 72B (1978) 343; M. Henneaux, Phys. Rep. 126(1985) 1: Hamiltonian BRST. 300 VI. QUANTUM GAUGE THEORY ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: B. GAUGES ::::::::::::::::::::::::::::: There are two important properties of gauges we have examined: (1) Gauges which eliminate some degrees of freedom, such as lightcone or unitary gauges, are simpler classically, which makes them easier to understand physically. (2) Gauges that manifest as many global invariances as possible, such as the Fermi-Feynman gauge, will be found later to simplify quantum calcuations, because the explicit momentum dependence of the propagator or vertices is simpler, and keeping a symmetry manifestmakes it unnecessary to check. In this section we'll examine these gauges in greater detail, especially as they relate to intereacting theories. We'll study also some special gauges, with nontrivial interaction terms, that have both of these properties to some extent. In particular, they are manifestly Lorentzcovariant, but avoid many of the complications associated with ghosts. 1. Radial We know from nonrelativistic classical and quantum mechanics that the equations of motion can be solved exactly only for certain simple external eld congurations. One particular case we have already emphasized is that of an action quadratic in the dynamical variables, i.e., the harmonic oscillator and its generalizations. Higher-orderterms are then treated as perturbations about the exact solution. Such an expansion in the coordinates xis the particle version of the JWKB expansion in  h: Calling the \classical" part of x\y", we substitute x!y+p hxand Taylor expand in x.( F r o m now on we'll drop the  h's, and just remember to perturb about the quadratic terms.) For the scalar eld we write !+x
@+1 2xmxn@m@n+::: where@:::@ is implicitly evaluated at y. For the gauge elds we would like to be a bit more clever: For example, for the electromagnetic potential Amwe know we can always add a constant, so Am(y)i s irrelevant, while for @AonlyFmn=@[mAn]is gauge invariant. This means we want to choose a gauge best suited to this calculation: a gauge that both eliminates as many as possible of the lower-order terms, and expresses A(y+x) in terms of only F(y) and its derivatives. Similarly, we should have a Taylor expansion for charged eldsin terms of covariant derivatives. The appropriate gauge, which easily can be found explicitly, is the \radial gauge" x
A(y+x)=0 B. GAUGES 301 (Note that, unlike F,Adepends on xindependently of y, not just as y+x,s i n c et h e gauge condition itself is x-dependent. We write A(y+x) only to indicate that Ais evaluated at position y+x.) One way to solve this condition is to use the identity xnFnm=(x
@+1 )Am[rm;x
A](@=@=@x ) which follows from the denition of F. Using the gauge condition, we then can write Am=1 x
@+1xnFnm Alternatively, we can replace xeverywhere (including the argument y+x)b yx,a n d then identify x
@=@=@ to nd xnFnm(y+x)=@Am(y+x) Integrating both sides over from 0 to 1, we nd Am(y+x)=Z1 0d xnFnm(y+x) Note in particular that A(y)=0 . Another way to dene this gauge is to consider gauge covariant translation from ytoy+xto produce a gauge transformation from an arbitrary gauge to the radial gauge. Writing the covariant derivative at yas D=D+iA(y);D =@=@y we know from subsection IIIC2 that 0(y+x)ex
D (y)=eiex
D (y)=ei (y+x) so that covariant translation produces a 0(y+x) that is the same as (y+x)u pt o a gauge transformation. Thus the gauge-transformed can be written as a covariant Taylor expansion (for purposes of perturbation) about y: 0(x+y)=1X n=01 n!xa1


xan(Da1


Dan 0)(y) In particular, 0(y)= (y). However, we want to dene a covariant derivative with respect to x(noty), so that r=@+iA0(x+y);r 0(y+x)=(D )0(y+x)=ex
D(D )(y) 302 VI. QUANTUM GAUGE THEORY Using @ (y)=0 we nd the solution r=ex
DDex
Dmodex
D@ex
D where the latter term vanishes on 0(y+x), so the right amount of it can be added to the former expression to cancel any Dterms. The result is r=ex
D(@+D)ex
D This implies x
A0(y+x) = 0 directly: Contracting both sides with x,t h eT a y l o r expansion of the right-hand side terminates after the rst couple of terms. Taylor expanding the uncontracted expression, we have A0 a(y+x)=1X n=01 n!1 n+2(x
D)nxbFba(y) We can also write @+iA0(x+y)=r=ex
Dex
D[@+iA(x+y)]ex
Dex
D=ei[@+iA(x+y)]ei and r=ex
Dey
@Dey
@ex
D Excercise VIB1.1 Show this Taylor expansion is equivalent to that obtained from the rst method used in this section to solve the gauge condition. (Hint: Look out for hiddenxandydependence | How does x
@on 0orF0relate tox
D? Also beware of notation: In the rst construction we did not use a gauge transformation, so no primes were used.) Thus, to just quadratic order in x, the mechanics action for a relativistic particle in external elds (subsection IIIB3) becomes SLZ df1 2v1mn.xm.xn+1 2.xmxnFnm(y) +v[(y)+xm(@m)(y)+1 2xmxn(@m@n)(y)]g To this approximation the classical equations of motion are linear and can be solved exactly. It can also be used to nd exact solutions for constant electromagnetic elds. B. GAUGES 303 2. Lorentz For purposes of explicit calculations in perturbation theory, it's more convenient to use gauges where Lorentz covariance is manifest. \Lorentz gauges" are a class ofgauges using f=@
A (and similarly for other gauge elds) as the gauge-xing function. From the discussion of subsection VIA4, we have from the usual BRST as applied to Yang-Mills L gf=1 8F2 abiQ1 2[~C(@
A+1 2B)] =1 4A
A1 4(@
A)21 2[Aa;Ab](i@aAb+1 2AaAb)1 2B@
A1 4B21 2i~C@
[r;C] After eliminating Bby its eld equation, the kinetic terms are 1 4A
A1 4(@
A)2+1 4(@
A)21 2i~CC In particular, for = 1 we have the \Fermi-Feynman" gauge, which gives the nicest propagators. (It is also the gauge that follows automatically from a rst-quantizedBRST construction, which will be described in chapter XII.) More generally, we ndthe propagator from inverting the kinetic operator: For the ghosts this is always 2 =p 2, but forAa, 2[abp2+(1 1)papb]1=2ab p2+(1)papb (p2)2 For= 0 this is the \Landau gauge", which has the advantage that the propagator is proportional to the transverse projection operator. (It kills terms proportional top a.) However, = 1 is clearly the simplest, and the 1 =p4term can cause problems in perturbation theory. Excercise VIB2.1 In the Abelian case, consider making a gauge transformation on the gauge- xed action (including matter), with 1B. Show that the only eect is to change the value of the coecient of theB2term. Find a similar transformation for the form of the action where Bhas been eliminated. This shows explicitly the decoupling of the longitudinal mode of the photon. Excercise VIB2.2 Show that for general AandB (abA+papbB)1=1 A abpapbB A+p2B 304 VI. QUANTUM GAUGE THEORY Note that in the Abelian case the lightcone gauge is a special case of the Landau gauge. (An analogous situation occurs in the classical mechanics of the particle for the gauges of the worldline metric, as discussed in subsection IIIB2.) Here we have 0=na(@bFab)=n
@(@
A)(n
A) In the lightcone formalism, this is the eld equation that comes from varying the auxiliary eld. In the lightcone gauge n
A= 0, it implies @
A= 0 (and thus also Aa=0 ) ,s i n c e n
@is invertible. This is particularly useful in D=4, where we can generalize from the lightcone to a Lorentz-covariant form by using twistors: From subsection IIB6, p2=0)p.=(p0)pp.;n2=0)n.=(n0)nn. Massless spinors are described on shell in momentum space by =p;  .=p. where external-line factors for Feynman diagrams are given by setting =1 . F o r massless vectors, we have p
A=n
A= 0 (butn
p6= 0), so depending on whether the helicity is +1 (self-dual eld strength) or 1 (anti-self-dual eld strength), we nd, respectively, f.. p.p. ;f=0)A. =np.  n p  fpp; f.. =0)A. =pn.  n. p.  The normalization of Ahas been chosen compatible with jj=1a n dAaA*a=1f o r evaluating cross sections. In a general Landau gauge the arbitrary gauge-dependentpolarization spinors n ,n.can be chosen independently for each external line, since gauge invariance means independent gauge parameters for dierent momenta. (This method is known as \spinor helicity".) However, in a lightcone gauge the polarizationspinors are constant. The lightcone gauge condition is thus again a stronger gauge condition than Lorentz gauges, as expected from the fact that it has fewer derivatives. This dier- ence shows itself in various ways: (1) In perturbation theory on shell, in the lightconeframe the Landau gauge condition 0 = p
A=p +AkillsAbut says nothing aboutA+, which can be eliminated by the residual gauge invariance A+=p+to obtain the lightcone gauge. (2) In perturbation theory o shell, more derivatives in B. GAUGES 305 the gauge transformation imply more derivatives in the ghost kinetic operator. Thus, more ghost degrees of freedom are introduced to cancel the extra unphysical degrees of freedom in the gauge eld. (3) Lorentz gauges also have a nonperturbative ambi- guity (the \Gribov ambiguity") that axial gauges avoid: Nonperturbative solutions to the gauge condition can be found that dier from the perturbative one, in thenonabelian case. Specically, it is possible to nd a nontrivial gauge transformation g(r 0=g1rg) such that 0=@
A0=i@
g1(rg)for @
A=0 even when gis required to satisfy boundary conditions that it approach the identity at innity (except in the Abelian case, where g=ei)=0)= 0). This is not the case for axial gauges, where 0=n
A0=n
g1(rg)for n
A=0)g1(n
@g)=0)g=I by simply integrating from innity. 3. Massive In subsection IIB4 we described the introduction of mass for the vector by dimen- sional reduction, giving the St uckelberg formalism for a massive (Abelian) gauge eld. The gauge-invariant action (subsection IVA5) and BRST transformation laws (sub- section VIA4) followed from adding an extra dimension and setting the corresponding component of the momentum equal to the mass: Lgi=1 8F2 ab+1 4(mAa+@a)2 QAa=@aC; Q =mC; Q ~C=iB; QB =0 where the scalar is the extra component of the vector. There are two obvious covariant gauges for such a vector: (1) The \unitary gauge" f= simply gauges away the scalar. Since the scalar has a nonderivative gauge transfor- mation, the ghosts do not propagate: The gauge-xing term iQ(~C)=B+im~CC simply eliminates the scalar and ghosts as auxiliary elds. The net result is that we could have simply chosen gauge  =0 306 VI. QUANTUM GAUGE THEORY and ignored ghosts because of 's nonderivative transformation law. Thus the gauge- xed Lagrangian is just the result of adding a mass term to the massless Lagrangian: Lgf=1 8F2 ab+1 4m2A2 But the propagator is 2[ab(p2+m2)papb]1=2ab p2+m2+papb m2(p2+m2) Notice that the second term is higher in derivatives than the rst; this can cause some technical problems in perturbation theory. (2) The Fermi-Feynman gauge works similarly to the massless case. We then modify the gauge-xing function to f=@
A+m so iQ[1 2~C(@
A+m+1 2B)] =1 2B(@
A+m+1 2B)1 2i~C(m2)C )Lgf=1 8F2+1 4(mA+@)2+1 4(@
A+m)21 2i~C(m2)C =1 4A
(m2)A1 4(m2)1 2i~C(m2)C The propagators are again simpler. The vector has D propagating components instead of just the D1 physical ones; the 2 ghosts cancel and the extra component in A. Excercise VIB3.1 Generalize the Fermi-Feynman gauge for the St uckelberg formalism to the \renormalizable gauges" with gauge-xing function f=m @
A+ Show that the ghosts and have mass, while the vector propagator has the form 2 ab+papb m21 p2+m2papb m21 p2+2 This shows explicitly the second unphysical bosonic mode of mass to cancel the 2 ghosts, as well as the 3 transverse physical modes of mass m.L o o ka t the cases =8 < :0 (Landau gauge) m(Fermi-Feynman gauge) 1(unitary gauge) B. GAUGES 307 These two choices of gauge also exist for Yang-Mills theories exhibiting the Higgs mechanism, since those models give the St uckelberg model when linearized about the vacuum values of the elds. The advantages are the same: The unitary gaugeeliminates as many unphysical degrees of freedom as possible (see subsection IVA6 for an example), while the Fermi-Feynman gauge gives the simplest propagators. Excercise VIB3.2 Work out the Fermi-Feynman gauge for an arbitrary Higgs model, generalizing the analysis for the St uckelberg case. 4. Gervais-Neveu We next consider pure Yang-Mills theory for the gauge group U(N), but use a complex gauge-xing function f0=@
A+iA2 whereAais a vector of hermitian N N matrices, and A2AaAa.( T h e h e r m i t i a n conjugate,i!i, gives similar results.) The gauge-xed Lagrangian (in the action S=g2trR L)i st h e n LA=1 8F2+1 4f2 0=1 4A
AiAaAb@bAa1 4AaAbAaAb (where is the free D'Alembertian) while the ghost action can be written as LC=1 2i~Cr2C1 2~CCf 0 whereracts onCas if it were in the dening representation (i.e., rC=@C+iAC, not [A;C]). This \Gervais-Neveu gauge" already has the simplication that some of the terms in the Yang-Mills self-interaction have been canceled. Excercise VIB4.1 Consider the \anti-Gervais-Neveu gauge", where the same gauge-xing term is added with opposite overall sign. aShow the resulting Lagrangian can be written as LAtr[(@=A=+iA=2)2] where the trace is with respect to both (N N) internal and (4 4) Dirac matrices. Thus, spin can be treated in a manner closely analogous to internal symmetry. 308 VI. QUANTUM GAUGE THEORY bShow the propagator can be written in the form of the product of 2 (massless) Dirac-spinor propagators. cStarting with the complex rst-order formulation of Yang-Mills of subsection IIIC4, show that the action can be written in a way that replaces the above44 matrices with 22 matrices, as L Atr[^G2+^G(@A*+iAA*)] in rst-order form, where now ^Gis neither traceless nor symmetric in spinor indices (its trace is the Nakanishi-Lautrup eld), or in second-order form as LAtr[(@A*+iAA*)2] (Note that this diers from the above Dirac form, as expanded in 2 2 matri- ces, because it includes the Chern-Simons term.) Next, consider a model where the Yang-Mills elds couple to scalars that are also represented by NN matrices, but that are in the dening (N-component) represen- tation of the gauge (\color") U(N), while also being in the dening representation of a second, global (\ avor") U(N). (See subsection IVA6.) This complex eld thus has 2N2real components compared to the N2gauge vectors, and the 2N2ghosts. We also choose a Higgs potential such that the masses of the scalar and vector come out the same (but we can also specialize to the massless case). The scalar Lagrangian is then (again with g2trin the action) L=1 2yr2+1 4R2;R =y1 2m2 Finally, we modify the gauge-xing function to f=f0+iR With this choice, the ghost terms are unmodied ( Ris gauge invariant), but the scalar self-interaction is completely canceled (including the mass term). The total Lagrangian is then L=(LA+1 4m2A2)+(1 2yr2+i1 2yf0)+i(1 2~Cr2C+i1 2~CCf 0) Since the scalar Lagrangian is identical in form to that of the ghosts, and neither has self-interactions, functional integration over them will produce canceling func- tional determinants, because they have opposite statistics. This is a re ection of the fact that both sets of elds now describe unphysical polarizations, since both describe B. GAUGES 309 massless states in a theory where all physical states are massive (as seen, e.g., in a unitary gauge). This has the great advantage that, for this particular model, both the scalar elds and the ghosts can be dropped altogether, while the Lagrangian L!LA+1 4m2A2 completely describes the physical massive vector and scalar states. This was possible only because of the use of a complex gauge condition: The longitudinal component of the vector is now imaginary, which xes the wrong sign associated with the Minkowskimetric. A related characteristic of this gauge is that we nowhere needed to change the vacuum value of any eld, unlike other gauges for actions where there is a Higgs eect. We now note that this result for the massive case (and its massless limit) actually can be obtained more easily than the result for the pure Yang-Mills case: Since the nal result has no ghosts, it is in a unitary gauge, where the vector not only \eats" the usual compensating scalar, but \overeats" by absorbing the physical scalar. Theappropriate gauge condition is still complex and involves the scalars, but is now linear : gauge  =hi= 1p 2mI wherey, treated as independent, is unxed. (As for the usual unitary gauge Im = 0, i.e.,=y, there are no propagating ghosts, since the gauge transformation of  has no derivatives.) In this gauge the action becomes quadratic in y: L!1 2yi(@
A+iA2)1p 2m+1 4(y1p 2m1 2m2)2 In fact,yappears as an auxiliary eld (taking the place of the Nakanishi-Lautrup eld), so we can eliminate it by its equation of motion:  y)y=mp 2+p 2 mif0)L=LA+1 4m2A2 This procedure is analogous to that used for the lightcone gauge, where one compo- nent of the gauge eld is xed and one is eliminated as an auxiliary eld: A closer analogy will be found in subsection VIB6. Of course, such gauges generalize to other Higgs models, but results will not be as simple when the vector and scalar masses dier: Excercise VIB4.2 Make the coecient of the R2term inLarbitrary, so the masses of the vector and scalar are unequal, but choose the same gauge =hi.F i n d t h e propagator, and compare with that of excercise VIB3.1. 310 VI. QUANTUM GAUGE THEORY 5. Super Gervais-Neveu Nonhermitian gauges are also useful in supersymmetric theories: Here we consider the supersymmetric analog of the massive model of the previous section. Although we work in N=1 superspace, the model turns out to automatically have an N=2supersymmetry. Just as the bosonic model ended with only a vector eld describing only physical polarizations, we now want a real scalar supereld to have only physical polarizations. Since such a supereld has 8 bosonic components and 8 fermionic, whilemassless N=1 multiplets have 2+2 physical polarizations, we need 1 vector multiplet plus 3 scalar multiplets. Since the bosonic model had a complex scalar representation, 2 of these scalar multiplets must form the analogous dening dening representation of local global groups, so the last must be a real (adjoint) representation of the local group. The model is then given by (where again S=g 2trR dx L ) Lgi=Z d2W2Z d4(eV0eV0++eV++eV) Z d2(+1 4m2)0+h:c: where we have included the only possible scale-invariant potential term, and intro- duced a Higgs mechanism by an N=2 Fayet-Iliopoulos term, which we chose to writein terms of the chiral scalar. (See subsection IVC7.) The BRST transformations (which also imply the gauge transformations) for this action are (see subsection VIA4) Qe V=iCeVieVC; QeV=eV(QeV)eV=iCeVieVC QC=iC2;Q C=iC2;Q~C=iB; Q~C=iB;QB=QB=0 Q+=iC +;Q  0=i[C; 0];Q =iC Q+=i+C; Q 0=i[C;0];Q =iC Our nonhermitian choice for the BRST gauge-xing function is =Z d2~C(d2eV+0)Z d2~C(d2eV+0) Note thateVis an element of the algebra as well as a \nonunitary element" of the group, only because we chose the group U(N) (as was the case for A2in the bosonic version). The gauge-xing and ghost terms are then iQ=Z d2B(d2eV+0)+Z d2B(d2eV+0) Z d4(~CeVC+~CeVC)+Z d2C~C0+Z d2C~C0 B. GAUGES 311 where we have used the eld equation enforced by the Lagrange multipliers Band B(or, equivalently, made eld redenitions of the Lagrange multipliers to generate terms proportional to their constraints). Excercise VIB5.1 Make a component analysis of this theory: aExpand the gauge-invariant action in components. bDo the same for the gauge-xing terms. cCompare the bosonic part of both the gauge-invariant and gauge-xed actions to those of the previous subsection, after elimination of auxiliary elds. We now see that the ghost terms are identical in form to those for , under the identication (+;;+;)$(C;~C;~C;C) (but beware signs from ordering of ghosts). So again the ghosts cancel the (N=2) matter elds, leaving only the N=2 vector multiplet. But we can also eliminatethe N=1 matter half of this N=2 multiplet using the Nakanishi-Lautrup Lagrange multiplers: The nal simple result for the gauge-xed action is thus L=Z d 2W2Z d4[eV(d2eV)eVd2eV+1 4m2(eV+eV)] A further simplication results from the redenition (again possible only for U(N)) eV!1+V This also simplies the BRST (and gauge) transformation for V: QV=i(CC)+i(CVVC) whose linear form resembles the bosonic case. Using the expression (see excercise IVC4.1) W=id2eVdeV!id21 1+VdV for the eld strength, the Lagrangian becomes L=Z d4 1 21 1+V(dV)d21 1+VdV+1 1+V(d2V)(1 +V)d21 1+V +1 4m2 V+1 1+V Although the nonabelian vector multiplet has nonpolynomial self-interactions in any gauge, this gauge simplies the lower-point interactions, which are the ones more 312 VI. QUANTUM GAUGE THEORY frequently used for a xed number of external lines. Expanding this action to cubic order, we use the identity dd2d=d.d2d.=1 2+fd2;d2g for the kinetic term, and d.d2=d2d.+i@.d for the gauge-xing part of the cubic term, with integration by parts. (For the gauge- invariant term, some work can be saved by using the equivalent W2form.) The result is L=Z d4f1 4V(m2)V+[1 4m2V3+(d.V)Vi@.dV]+O(V4)g Not only are there fewer terms than with linear gauge conditions, but these terms have fewer spinor derivatives, which yields viewer nonvanishing contributions in loops (see subsection VIC5). As for the bosonic model of the previous subsection, this analysis also applies for the unbroken case m=0 . Excercise VIB5.2 Find the corresponding form of the kinetic and cubic terms without the re- denitioneV!1+V. Excercise VIB5.3 Gauge x by using the unitary gauge +==mp 2 to obtain the same result. Excercise VIB5.4 Look at the super anti-Gervais-Neveu gauge, or super anti-Fermi-Feynman gauge, changing the sign of the gauge-xing term for the vector multiplet (see excercise VIB4.1). aShow that in the massless case the kinetic operator becomes, instead of , Kd41 4! ddd d where we now use 4-component spinor indices. bShow that the resulting propagator is of the form, in supercoordinate space,
(x;;x0;0)d4 24(0)4(xx0)ln[(xx0i1 2 0)2] where \xx0i1 2 0" (see subsection IIC2) is the supersymmetry invariant. (Hint: Consider a \supersymmetry-covariant" Fourier transform in both x and. Warning: If derived by Fourier transformation, the integral is infrared divergent, and requires dropping an innite constant.) B. GAUGES 313 6. Spacecone We have just seen that gauge independence allows complex gauge conditions, which make the action complex. (In subsection IIIC4, we also used complex auxiliaryelds, with a similar eect.) In this subsection we introduce a complex analog of thelightcone, the \spacecone", which will greatly simplify Feynman diagram calculationswith massless elds. The spacecone gauge condition is A 2iA3=0 or more generally n
A=0;n2=0;n
n*>0 (but onlyna,n o tn*a, appears in the action). While this gauge is spacelike (in the sense that only spatial components of the gauge eld are xed), it is also null, byvirtue of being complex. Thus, although algebraically like the lightcone, it allowscanonical quantization with the usual time coordinate. In fact, it is just a Wickrotation of the lightcone. We then eliminate A 2+iA3as an auxiliary eld. The spacecone is a new gauge to add to our list of axial gauges n
A=0f r o m subsection IIIC2, and the related gauges for scalars from subsections IVA5-6, VIB3-4: axial gauges non-null null (+auxiliaryfieldeq: ) (partly )temporal timelike :A0=0 lightcone :A+=0; =  A spacelike Arnowitt -Fickler :A1=0spacecone :At=0; =  At scalar unitary :=yGervais -Neveu :=hi; =  y In fact, at least for the free theories, the gauges for the scalars can be considered as dimensional reductions (from 1 or 2 extra dimensions) of those for the vector, asused for deriving the St uckelberg formalism in subsection IIB4, where the spacelike components of the vector associated with gauge xing become scalars: Arnowitt- Fickler!unitary, spacecone !Gervais-Neveu. The main advantages of the spacecone over the lightcone are special to D=4, so we now review the lightcone in a way specialized to physical spacetime. Starting withthe gauge condition (see subsections IA4 and IIA3 for notation) A t=0 and eliminating Atby its eld equation, the Lagrangian for pure Yang-Mills becomes L=A+@t@tA1 4(F+)2+1 4(Ftt)2 314 VI. QUANTUM GAUGE THEORY F+=@+A@A++i[A+;A] Ftt=@A++@+A+i1 @t([A+;@tA]+[A;@tA+]) We simplify the Lagrangian by using the self-dual and anti-self-dual combinations: Dropping also the tsuperscripts on @for simplicity, F=1 2(FttF+)=@A+i1 @[A;@A] L=A+@@A+F+F =A+1 2Ai@ @A+ [A+;@A]i@+ @A [A;@A+] +[A+;@A]1 @2[A;@A+] Excercise VIB6.1 Label all the elds and derivatives in the above forms of the Lagrangian, F in particular, in spinor notation. Excercise VIB6.2 Show that the eld redenitions A!(@)1, when applied to just the rst two terms of the above Lagrangian, produce a local action describing the self-dual eld equations of the lightcone formalism of subsection IIIC5 (taking intoaccount the dierence between the lightcone and spacecone). Compare the results of excercise IIIC5.2. Thus, by treating the latter two terms separately from the former two, Yang-Mills can be treated as a perturbation about self-dual Yang-Mills. Another simplication for massless D=4, and closely related to the use of helicity, is twistors. For our Feynman diagram calculations for spins 1, almost all spinor algebra involves objects carrying at most two spinor indices (spinors, vectors, self-dualtensors), so we use the twistor matrix notation of subsection IIB6. In particular, in ageneral class of gauges the external line factors for Yang-Mills elds in this notation (see subsection VIB2) read A= +=ji[pj hpi)f*jp][pj;f =0 for + helicity or A==jpi[j [p])fjpihpj;f*=0 for,w h e r e()aare the polarization 4-vectors for helicity 1i nt e r m so fat w i s t o r ;., which can vary from line to line, and whose choice denes the gauge, as a special B. GAUGES 315 case of the Landau gauge. (Positive helicity is the same as self-duality, negative is anti-self-dual. The Landau gauge condition is generally applied in arbitrary Lorentz gauges to external lines, to eliminate the redundant longitudinal degrees of freedom.) One special case is the lightcone gauge n
A=0;n =ji[j in terms of an arbitrary constant lightlike vector n. A more convenient gauge is the spacecone gauge, which can be written in terms of two twistors: n=j+i[j These two twistors are sucient to dene a complete reference frame: We can convert all spinor indices into this basis, as =  etc. This corresponds to using two lightlike vectors to dene the spacecone gauge, n=ji[j. For simplicity, we write ji=ji; then a vector in this basis can be written as p=p+j+i[+j+pji[j+ptji[+j+ptj+i[j if we use the normalization h+i=[+] = 1 E.g., for massless momentum p=jpi[pj, p+=hpi[p];p=h+pi[p+];pt=h+pi[p];pt=hpi[p+] We will also drop the superscript tin contexts where there is no ambiguity. This basis is related to our previous spinor basis up to phase factors, jiji,j]j. ], and we assume them to be commuting (rather than anticommuting); these changes are more convenient for dealing with twistors (commuting spinors). The advantage of the spacecone is that we can Lorentz covariantize the Feyn- man rules by identifying these two lightlike vectors with physical on-shell masslessmomenta. We need two such \reference" vectors because we are not allowed to have n=pon any line. Since only j+iappears in the external line factors for helicity +1, and onlyj]i nt h o s ef o r1, the simplest choice is to pick the momentum of one external line with helicity +1 to dene j] for all lines with helicity 1, and pick the momentum of one line with helicity 1 to denej+ifor lines with helicity +1. (In the presence of massless external spinors, we can also choose a helicity +1/2 line 316 VI. QUANTUM GAUGE THEORY to denej], etc.) Our above normalization means that we have chosen the phase h+i=[+] = 1 as allowed by the usual ambiguity of twistor phases, while our choice of the magnitude h+i[+] =h+j[+j
jij ] = 1 is a choice of (mass) units. In explicit calculations, we restore generality (in particular, to allow momentum integra-tion) by inserting appropriate powers of h+iand [+] at the end of the calculations, as determined by simple dimensional and helicity analysis. (This avoids a clutter of normalization factorsp h+i[+] at intermediate stages.) For example, looking at the form of the usual spinor helicity external line factors, and counting momenta inthe usual Feynman rules, we see that any tree amplitude (or individual graph) in pureYang-Mills must go as hi 2E+[]2E whereEis the number of external lines with helicity . We now return to external line factors. The naive factors for the above Lagrangian are 1, since the kinetic term resembles that of a scalar. However, this would lead tounusual normalization factors in probabilities, which are not obvious in this complex gauge. Therefore, we determine external line factors from the earlier spinor helicity expressions for external 4-vectors. In Lorentz gauges (  )awould be the polarization for helicity1 for the complete 4-vector, but in the spacecone formalism only A appear. Furthermore, in the spacecone we nd (+)=j+i[+j
j+i[pj h+pi=0 ()+=ji [j
jpi[j [p]=0 since by antisymmetry h++i=[] = 0, so that A+carries only helicity +1 and Aonly1. (This statement has literal meaning only on shell, but we can make this convenient identication more general by using it as a denition of helicity o shell.) The appropriate external line factors for these elds are thus (+)+=ji [j
j+i[pj h+pi=[p] h+pi ()=j+i[+j
jpi[j [p]=h+pi [p] Note that these factors are inverses of each other, consistent with leaving invariant (the inner product dened by) the kinetic term. An exception is the external line factors for the reference momenta themselves, wherejpi=jifor helicitygives vanishing results. However, examination of the B. GAUGES 317 Lagrangian shows this zero can be canceled by a 1 =@in a vertex, since p=p=0f o r the reference momenta by denition. (Such cancelations occur automatically from eld redenitions in the lightcone formulation of the self-dual theory.) The actualexpressions we want to evaluate, before choosing the reference lines, are then p p(+)+=h+pi[p+] h+pi[p][p] h+pi=[p+] h+pi p+ p()=hpi[p] h+pi[p]h+pi [p]=hpi [p] Evaluating the former at jpi=jiand the latter atjpi=j+i,w eg e t1i nb o t hc a s e s . In summary, for reference lines: (1) use only the 3-point vertex of the corresponding self-duality ( for helicity), and use only the term associating the singular factor with the reference line (the other term and the other vertices give vanishingcontributions); (2) including the momentum factors on that line from the vertex, the external line factor is 1. 7. Superspacecone To generalize these results to high-energy (massless) QCD, we consider supersym- metric QCD, i.e., Yang-Mills coupled to massless fermions in the adjoint representa- tion. For tree graphs, this is equivalent to ordinary massless QCD except for grouptheory, which can be evaluated separately. We rst apply the spacecone approach to the component action for supersymmetric QCD: The modication of this action for ordinary massless QCD is trivial (replacing the adjoint quark current with dening).From this we derive the \superspacecone" formalism, rewriting the action more sim-ply in terms of spacecone superelds. (This form can also be derived from the usual superspace, but we will not consider that here.) We now combine the spacecone approach to pure Yang-Mills of subsection VIB6 with the spacecone version of the lightcone treatment of the massless spinor in sub-section IIIC2. We modify the lightcone to the spacecone for the quarks by instead eliminating and . as auxiliary. For later convenience, we also write the remaining fermionic elds as  . ! +; ! Then we directly nd the terms in the Lagrangian L=A+@@A+F+F+i +(@r1 @r+) F=@A1 @([A;i@A]+f +; g) 318 VI. QUANTUM GAUGE THEORY where the quark term in Fcomes from the quark coupling to Atwhen using its equation of motion to solve for Ftt. Collecting terms, we have L=L2+L3+L4 L2=A+1 2A+ +1 2 i@ L3=@ @A ([A;i@A]+f +; g)+@ @  [A; ] L4=([A+;i@A]+f +; g)1 @2([A;i@A+]+f +; g)[A+; ]1 i@[A; +] where terms with have only a single sum over it. Although this Lagrangian is much messier than the original covariant one, one again saves work by expandingterms once in the action rather than repeatedly for each Feynman diagram. External-line factors for the spinors follow from the covariant ones of subsection VIB2 as they did for the spacecone vectors of subsection VIB6. We thus have +=[p]; =h+pi Compared with those for A,w es e e + has an extra factor of p=h+pi[p]a s compared with A+A, as expected from the extra factor of 1 =(i@) in the kinetic operator. Similarly, if we choose to use external quark lines as reference lines, we use p p +=[p+];p+ p =hpi which also reduce to 1 for the appropriate reference momenta. Noting that the bosonic and fermionic terms are the same except for factors of i@, we can combine them into chiral superelds that depend on only two anticom- muting coordinates, +and(reallyand. ): S=1 g2trZ dx d+dL;Z d+d=d+dordd+;fd+;dg=i@ d=0 ;j=A;dj=  (no sum on). These spinor derivatives (and their corresponding supersymmetry generators) describe only spatial supersymmetry, since they contain no time deriva-tives. Then, using the identity d d[;]j=[A;i@A]+f +; g we easily combine the terms in the Lagrangian Linto the superspacecone Lagrangian L: L=+1 2 i@+@ @ [;]+[+;d]1 @2[;d++] B. GAUGES 319 The last term can also be written as [+;]d+d @2[+;] Excercise VIB7.1 Introduce another pair of chiral superelds as auxiliary. Show the above Lthen can be rewritten in local form, with no spinor derivatives, where the kinetic term resembles the covariant one for a massless spinor, while the inter- action term contains no derivatives and is only cubic. (Hint: dd=(i@)a r e projection operators for chiral superelds.) Thus this Lagrangian resemblesthe Chern-Simons one that appears on the 3D boundary for the topological term in Yang-Mills (see subsection IIIC6). Expand the action in components, and separate out the pure Yang-Mills part. Excercise VIB7.2 Repeat excercise VIB6.2 to obtain the superspacecone action for selfdual super Yang-Mills, quadratic in  +and linear in . Use the eld redenition !d+';d'=0 and integrate the action over just (by acting with d) to obtain a \chiral" action, with no spinor derivatives, and superelds that are functions of just +integrated over just +. After further redenitions as in VIB6.2, obtain an action identical to the nonsupersymmetric one obtained there, except for theR d+. Expand in components, and relate to the nonsupersymmetric case. 8. Background-eld A more general type of gauge choice is the background eld gauge. As we saw in subsection VC1, the generating functional can be written in a form where the quan- tum eld is expanded about a background eld in the interaction part of the classical action. The basic steps of the background eld gauge method are: (1) choose gauge xing that is gauge invariant in the background gauge eld, (2) show that the quan- tum/background splitting of the entire gauge-invariant action is also gauge invariantin the background gauge eld, and (3) show that the eect of splitting the kinetic term in the gauge-invariant action can be neglected. (Only the interaction terms should have been split.) The result is then that the eective action , which dependsonly on the background elds, is gauge invariant in them. This gauge invariance is a strong condition which not only simplies the eective action but allows a \back- ground gauge" to be chosen for it that is independent of the \quantum gauge" applied 320 VI. QUANTUM GAUGE THEORY to the path integral: The background elds and quantum elds can be in dierent gauges. For example, for a relativistic treatment of the quantum corrections to bound states whose constitutents are nonrelativistic (such as the hydrogen atom), it is con- venient to use a Fermi-Feynman gauge (convenient for relativistic matter coupling to electromagnetism or chromodynamics) for the quantum elds and a Coulomb gauge(convenient for static or nonrelativistic matter) for the background elds. A simple way to formulate the background expansion is in terms of the covariant derivative: A!~A=A+A)r ! D +iA;D=@+iA whereAis the quantum eld (the variable of path integration) and Dis the \back- ground covariant derivative" in terms of the background eld A. We then nd for the eld strength F ab!i[Da+iAa;Db+iAb]=Fab+D[aAb]+i[Aa;Ab] and similarly for the action. Matter elds are split as usual, !~='+ We now have two gauge invariances, corresponding to the two gauge elds. Both transformations are dened to have the same, usual form on r=D+iA(and on ~='+), and thus both leave the action inert, but (1) the \background gauge invariance" is dened to transform the background elds covariantly background :D0=eiDei('0=ei');r0=eirei(~0=ei~) )A0=eiAei(0=ei) and thus the quantum eld transforms as a matter (non-gauge) eld, while (2) the \quantum gauge invariance" is dened to leave the background elds inert quantum :D0=D('0=');r0=eirei(~0=ei~) )A0=ei[(iD+A)ei][0=ei('+)'] The latter then determines the new BRST transformations QAa=[Da+iAa;C];Q C =iC2;Q ~C=iB; QB =0 [Q=iC('+)] The key to the background eld gauge is to break the quantum invariance, so a propagator can be dened, but preserve the background invariance, so the path B. GAUGES 321 integral is gauge invariant. Since Q, the BRST operator for the quantum gauge invariance, is now background gauge invariant, we need only choose a gauge-xing function  that is also background gauge invariant. Many gauges are possible: The basic rule is to modify any normal gauge condition simply by replacing any partial derivatives @with background covariant derivatives D. For example, for a Lorentz covariant gauge @
A!D
A We then gauge x in the usual way, and now the gauge-xing terms and the ghost terms are background gauge invariant, as long as we dene the ghosts to transform covariantly: background :C0=eiCei; ~C0=ei~Cei;B0=eiBei For example, for Lorentz gauges the ghost term is modied, by the modication of the gauge condition and the quantum BRST transformation, as ~C@
rC!~CD2C+~CD
i[A;C] Furthermore, even axial gauges are modied: For example, even though the gauge conditionA0= 0 allows elimination of a component of the quantum eld, it doesn't aect the background eld, which now appears in the ghost Lagrangian ~C@0C!~CD0C Since the S-matrix is gauge-independent (when BRST is used to perform gauge xing, as we have), we can use the background eld gauge version of the generatingfunctional (now using to represent all quantum elds and 'all background), Z[']=Z D e i~S; ~S=S0[]+SI[+']iQ[;']=^S(S0[+']S0[]) ^S=S[+']iQ;S []=S0[]+SI[] whereS[] is the original gauge-invariant action, Q is the gauge-xing as de- scribed above, and ^Sis the sum of this gauge xing and the background-expanded gauge-invariant action. We have thus separated the total action ~Sappearing in the background-gauge-xed generating functional into the background-gauge-invariant part ^Sminus the noninvariant terms S0[+']S0[]. As usual, the classical part of the eective action [ '] is given by adding the kinetic term S0['] of the (gauge-invariant) classical action to the 1PI tree graphs, which are just the vertices for the background elds. Thus, class[']=~Sj=0+S0[']=^Sj=0=S['] 322 VI. QUANTUM GAUGE THEORY We now note that, as far as calculating just the eective action is concerned, we can drop all terms in the gauge-xed action independent of or linear in :A n y independent term contributes only classically; any linear term will generate one- particle reducible graphs (\tadpoles"). This means we can drop the noninvariant termsS0[+']S0[]f r o m ~S. Thus, the Feynman rules for calculating are: (1) Use the classical gauge-invariant action S['] for the classical contribution to ; and (2) for the quantum contribution, use all the 1PI loop graphs coming from ^S. The result is background gauge invariant, since ^Sis manifestly so. Another important feature of the quantum-gauge-xed background eld action is that it is background-gauge-invariant order-by-order in the quantum elds. In fact, every term in the corresponding ordinary gauge action has been replaced by one (or more, if there are Fterms) background-gauge-invariant term. Excercise VIB8.1 Consider the Fermi-Feynman background-eld gauge for the quantum eld of pure Yang-Mills theory. Write all terms (both gauge-invariant and gauge- xed) quadratic in the quantum eld. Show that these combine as 1 4A
Ai1 2Aa[Fab;Ab] where =(D)2(and \DA"m e a n s\ [D;A]", etc.). Since all external lines are associated with background elds, if we draw graphs in such a way as to exhibit only the quantum elds, they will all look like vacuum graphs: graphs with no external lines. However, any particular such vacuum graph will represent many of the original graphs, since the background lines can be attachedin many ways. Furthermore, in background eld gauges any such vacuum graph, con- sidered as a contribution to the eective action, will be gauge invariant with respect to the background gauge transformations, since it results from the non-backgroundgauge true vacuum graph by the replacement of the ordinary derivative with the background covariant derivative @!D (plus perhaps some noniminimal Fterms), including in the propagator. In particular, the complete one-loop contribution to is given by the vacuum graph with no quantum interactions: It can be obtained from just the part of the ^Sthat is quadratic in the quantum elds. B. GAUGES 323 Excercise VIB8.2 Consider an arbitrary gauge-invariant Yang-Mills action S[~A]w i t h ~A=A+A in terms of the background eld Aand quantum eld A.T a y l o r e x p a n dt h e action inAas S[~A]=S[A]+AS[A] A+::: The innitesimal quantum gauge transformation mixes dierent-order terms in the expansion. Show that the term quadratic in Ais invariant under an Abelian quantum gauge transformation only if the background satises the eld equations, S[A]=A= 0. Similar remarks apply to BRST transforma- tions and the gauge-xed action. (Since quadratic actions, even in backgroundelds, yield only a propagator, they can be described by rst-quantization:Thus gauge invariance implying background eld equations occurs whenevera gauge eld appears as both a quantum mechanical state and a backgroundeld, for example in string theory. See subsection XIIB7 for a simpler exam-ple.) The S-matrix is then given in the usual way from [ '], after adding another gauge-xing term for the background gauge invariance. Since the total S-matrix is given by just the trees following from treating as a classical action, we need only agauge-xing term for the physical elds, and we can ignore background ghosts. (Ofcourse, quantum ghosts were already used to calculate .) This background gaugexing is independent of the quantum gauge xing. In particular, we can choosedierent quantum and background gauges: For example, when treating spontaneouslybroken gauge theories, it's often more convenient to choose a Fermi-Feynman quantumgauge and a unitary background gauge; i.e., we expand the background elds aboutthe physical vacuum to make the physical states obvious, but leave the quantum elds unexpanded to avoid complicating the Feynman rules. This also avoids the complication of having to expand about the vacuum twice, since vacuum values getquantum corrections to those appearing in the classical action. The gauge invariance of the eective action in the background-eld formalism is a big advantage over other quantum gauges, where the eective action is only BRST invariant, since gauge invariance is a much stronger constraint than BRST invariance: Gauge symmetry is local, while BRST is only global. Thus, the background-eldgauge produces a much simpler eective action. In other words, the background-eld gauge produces an eective action without ghosts: Although we can drop ghostterms from the eective action in general, because there are no physical externalghost states (since we calculate only the \tree" graphs of the eective action), the 324 VI. QUANTUM GAUGE THEORY result is not normally BRST invariant; but in the background-eld gauge it is still BRST invariant, since it is gauge invariant. This means that the background-eldgauge yields not only simpler results, but fewer calculations: Many terms can be determined by \gauge covariantization". Excercise VIB8.3 Consider an eective action for Yang-Mills plus matter in a backgro und-eld gauge. Its gauge invariance can be used to derive \Ward-Takahashi identi- ties". (These were originally expressed as properties of the S-matrix, but are much simpler to understand in terms of the eective action.) aShow that the part of the eective action quadratic in the Yang-Mills elds, and independent of the matter elds, is invariant under the Abelian gauge transformations. (Hint: Taylor expand.) Thus, in such gauges the quantum correction to the gluon propagator is transverse. bBy the same method, nd a relation between any quantum 3-point vertex coupling matter to Yang-Mills and the corresponding matter prop agator cor- rection. (Note a simpler case: Since the renormalization counterterms are local, gauge invariance just says that the coecients of the two corresponding counterterms are the same, i.e., they occur in the combination  r= .) However, this does not mean we can completely ignore BRST and ghosts by using background-eld gauges: Although the eective action is gauge invariant and ghost free, ghosts and BRST still appear in the (quantum-gauge-xed) classical action. In practice this means, as far as calculating the Feynman graphs that contribute to the eective action, that in the background-eld gauge calculations are about one loop simpler than in other gauges. For example, for one-loop graphs we eectively cal- culate free one-loop vacuum bubbles (including ghosts) covariantly coupled to back- ground elds: There are fewer of the complications of nonabelian theories, since the quantum elds appear only as non-gauge elds with covariant couplings and no self- interactions. However, already at two loops we have self-interactions of the quantum elds, which include the same kinds of terms that would have appeared had we not used a background-eld formalism. Another complication is that BRST invariance is not as restrictive as gauge invari- ance: It can be shown that in general gauges at the quantum level BRST invariance is preserved only up to \wave-function renormalizations" (rescalings) of the quan- tum elds. However, in the background-eld gauge wave-function renormalizations of the quantum elds can be ignored, since the quantum eld is a dummy vari- able: There are no external quantum elds, so all such factors cancel. (Actually, we B. GAUGES 325 can also ignore wave function renormalization counterterms in non-background-eld gauges, since when calculating S-matrix elements such divergences will be canceled by corresponding divergences in the external-line factors. In general, external-line normalization factors may be nontrivial even when wave-function renormalization is performed, depending on the renormalization scheme.) An exception is Abelian gauge theories, such as QED: Because the gauge-invariant action for just the gauge elds is free, background eld gauges are identical to ordinary gauges. Also, the ghosts decouple (for linear gauge conditions). 9. Nielsen-Kallosh So far we have considered only gauges where the gauge-xing term is the square of the gauge-xing function. More generally, we'll need gauge-xing terms of the formfOffor some operator O. Straightforwardly, we can write iQ 1 2~C[f(A)1 2O1B]=1 2B(f1 2O1B)+1 2i~C@f @A
[r;C] However,Bis no longer auxiliary, so we can't eliminate it by its eld equation. But we can diagonalize the Lagrangian by the corresponding redenition, B!B+1 Of (The Jacobian of such redenitions is unity, the determinant of a triangular matrix of the form1 0x 1
.) The gauge-xing terms are then 1 4BO1B+1 4fOf The inverse operator is inconvenient for Feynman rules. We know that integrating outBgives a functional determinant, so O1can be replaced by an Oif we change the statistics of the Nakanishi-Lautrup eld. However, this is a bit formal, since tech- nicallyOmust be symmetric between the two B's, while it should be antisymmetric between two fermions. A useful example is gauge xing for super Yang-Mills in superspace. Gauge xing for massless Yang-Mills is actually more dicult than for the massive (Higgs) case, considered in subsection VIB5. We'll look at the Abelian theory, to determine what kind of gauge xing we need to dene the propagator. (With slight generalization,this is also sucient for the background-eld gauge: See the following subsection.) In that case the BRST transformations are QV=i(CC);Q ~C=iB; Q~C=iB; QC =QB=QC=QB=0 326 VI. QUANTUM GAUGE THEORY whereC,~C,a n dBare chiral. Using the result (the Abelian case of excercise IVC4.1) W=id2dV the gauge-invariant kinetic term is (rearranging derivatives and using integration by parts; see subsection VIB5) L0=Z d21 2WW=Z d41 2Vdd2dV=Z d4V(1 4d2d2)V To gauge-x to the Fermi-Feynman gauge we choose L1=iQZ d4[(~C+~C)V+~C(1 2)1B] =Z d4[(~CC~CC)(B+B)VB(1 2)1B] (droppingd4integrals of totally chiral or totally antichiral terms, which vanish). If we were to simply redene Bby B!Bd2d2V; B!Bd2d2V the gauge-xing terms would diagonalize as (using d2d2d2=1 2d2) (B+B)VB(1 2)1B!Vd2d2VB(1 2)1B giving the desired result for V: At this stage the total result is L=L0+L1!Z d4[1 4VV+~CC~CCB(1 2)1B] BecauseBis complex, the replacement of Bwith a fermionic supereld can be performed classically, just like the rest of the gauge-xing procedure. We thusintroduce ghosts for a trivial gauge invariance as described in subsection VIA4: QD=E; Q E=iD; QE =QD=0 We have treated the ghosts and their hermitian conjugates independently; alterna- tively, we can consider Dand Eas not being the conjugates of DandE. The gauge xing is simply L 2=iQ(ED)=DDiEE We next make the redenition D!D+(1 2)1d2B; D!D+(1 2)1d2B B. GAUGES 327 which has the eect Z d4[DDB(1 2)1B]!Z d4DD+Z d2D B +Z d2DB which vanishes, after using the now-algebraic eld equations from varying Band B. Alternatively, we can make this eld redenition instead of the previous eld redenition: We then have the terms Z d4[DD(B+B)V]+Z d2D B +Z d2DB!Z d4Vd2d2V after using the still-algebraic Bequations. The net result L=L0+L1+L2!Z d4(1 4VV+~CC~CCiEE) is that the original nonlocal Bterm has been replaced classically with the local EEterm, which yields the same determinant upon quantization, but gives simple Feynman rules more directly. (The determinant is nontrivial in the background-eldgauge. A similar procedure can be applied to gauge xing for spin 3/2.) Excercise VIB9.1 Perform the analogous quantization for the nonabelian case of pure super Yang-Mills (no matter), using the super Gervais-Neveu gauge. Compare withthe limitm!0 of the model considered in subsection VIB5, and show the Vpart of the action agrees. Excercise VIB9.2 Use this method to produce a gauge-xing term (n
A) (n
A) for a gauge vectorAin terms of a parameter and constant vector n. Find all propaga- tors. Look for simplifying special cases of andn. 10. Super background-eld Although in principle the background-eld formalism is the same for supersym- metric theories as nonsupersymmetric, there are some technical dierences because ofthe nonlinearity in the prepotentials. (Similar remarks apply to nonlinear models.) The basic idea is that we want to expand the full covariant derivative in quantum elds about background-covariant derivatives: As for the nonsupersymmetric case,r!D +iA, but nowAis further expressed in terms of Dand the prepotential because of the constraints. The generalization in this case (and for nonlinear mod- els) is easy because the solution to the constraints makes the prepotentials appear 328 VI. QUANTUM GAUGE THEORY as (complex) group elements: Because of the closure of group multiplication, we can write g!gBgQ in terms of quantum ( gQ) and background ( gB) group elements (elds). More explic- itly, for our case we write (see subsection IVC4) e !e Be Q and thus for the covariant derivatives r!e QDe Q absorbing the background prepotential completely into the background covariant derivative D=e Bde B In other words, as the name suggests, the full covariant derivative rhas been ex- panded about an arbitrary background, described by B. (This is even clearer in the supergravity case, where we simply replace the at-space dwith the curved-space D,s i n c edis more than a partial derivative, and already contains the at-space part of the metric tensor.) For purposes of quantization, it is most convenient to goto a chiral representation for the quantum eld. For the background eld we need notbe so specic, since it is hidden in the background covariant derivatives. The resultis then r !eVDeV;r.!D.;r.!ifD.;eVDeVg whereVis the quantum eld. Excercise VIB10.1 Solve the rest of the commutator algebra to nd expressions for all the eldstrengths in terms of VandD A. The rest of the quantization procedure then follows as for the nonsupersymmetric case, except for the Nielsen-Kallosh ghost described in the previous subsection. Inparticular, for the terms in the gauge-xed classical action quadratic in the quantumeldV, W !i1 2[D.;fD.;eVDeVg]=WiD2DV+i1 2D2[V;DV]+O(V3) )S2V=Z dx d4V(1 2DD2D+i1 2WD+D2D2)V B. GAUGES 329 Pushing theDin the rst term to the right, we nd 1 2DD2D=i1 4(D.D.D+D.D.D)D2D2 Using integration by parts on all the derivatives in the second term so they act to the left, then switching the V's so they again act to the right, D.D.D+D.D.D!D.D.D+DD.D. =D.D.D+D.DD.+[D;D.]D. =i+2W.D. where =DaDa. The nal result is similar to the bosonic case (excercise VIB8.1): S2V=Z dx d41 4V(+2iWD+2iW.D.)V (This result is invariant under integration by parts because of the Bianchi identity DW+D.W.=0 . ) Ghosts and matter are quantized straightforwardly: For matter we have r.=r=0) !'+; !eV('+); D.'=D'=0 T h ea c t i o nt h u sl o o k st h es a m ea su s u a l( ( '+)eV('+), etc.), except that all chiral superelds are now background-chiral. For the standard ghosts we have for the ghostactionS C=R dx d4LC(remembering there are no background ghosts, and using the full nonlinear transformation law from excercise IVC4.3) LC=(~C+~C)LV=2[coth(LV=2)(CC)+(C+C)] = ( ~C+~C)(CC)+O(V) !(~CC~CC)+O(V) the same as in non-background gauges, except again the ghosts are background-chiral. Now the Nielsen-Kallosh ghost of the previous subsection is nontrivial: We again have LNK=iEE but these ghosts also are background-chiral. This means they contribute to the eec- tive action only at one loop, through \vacuum bubbles". REFERENCES 1Fock, loc. cit. (VB); Schwinger, loc. cit. (VB, ref. 2, second ref.): radial gauge. 330 VI. QUANTUM GAUGE THEORY 2Fermi, loc. cit. (VC); Feynman, loc. cit. (VB, ref. 4). 3A.A. Abrikosov, I.M. Khalatnikov, and L.D. Landau, Doklady Akad. Nauk USSR 95 (1954) 773:Landau gauge. 4P. De Causmaecker, R. Gastmans, W. Troost, and T.T. Wu, Nucl. Phys. B206 (1982) 53;F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans, W. Troost, and T.T. Wu,Nucl. Phys. B206 (1982) 61; Z. Xu, D.-H. Zhang, and L. Chang, Nucl. Phys. B291 (1987) 392; J.F. Gunion and Z. Kunszt, Phys. Lett. 161(1985) 333; R. Kleiss and W.J. Sterling, Nucl. Phys. B262 (1985) 235: spinor helicity. 5V.N. Gribov, Nucl. Phys. B139 (1978) 1. 6G. 't Hooft, Nucl. Phys. B35 (1971) 167; K .F u j i k a w a ,B . W .L e e ,a n dA . I .S a n d a , Phys. Rev. D6(1972) 2923; Y.-P. Yao, Phys. Rev. D7(1973) 1647: renormalizable gauges. 7Gervais and Neveu, loc. cit. (IVA). 8G.F. Chew and M. Levinson, Z. Phys. C 20(1983) 19: proposed Feynman rules similar to anti-Gervais-Neveu gauge in a context related tostring theory. 9W. Siegel, hep-th/9502163, Phys. Rev. D52 (1995) 1035: super Gervais-Neveu. 10M. Mangano and S.J. Parke, Phys. Rep. 200(1991) 301: review of modern methods for tree graphs. 11E.T. Newman and R. Penrose, J. Math. Phys. 3(1962) 566: use of \null tetrad" in gravity, complex basis vectors in terms of constant spinors. 12G. Chalmers and W. Siegel, hep-ph/9801220, Phys. Rev. D59 (1999) 045013: spacecone. 13W. Siegel and S.J. Gates, Jr., Nucl. Phys. B189 (1981) 295; S. Mandelstam, Nucl. Phys. B213 (1983) 149; L. Brink, O. Lindgren, and B.E.W. Nilsson, Nucl. Phys. B212 (1983) 401: lightcone superelds. 14Feynman, loc. cit. (VC, ref. 17); DeWitt, loc. cit. (VIA); J. Honerkamp, Nucl. Phys. B48 (1972) 269; G. 't Hooft, The background eld method in gauge eld theories, in Functional and probabilistic methods in quantum eld theory , proc. 12th Winter School of Theoretical Physics, Karpacz, Feb. 17-Mar. 2, 1975, v. 1, Acta Univ. Wratislav. 368(1976) 345; L.F. Abbott, Nucl. Phys. B185 (1981) 189: background-eld gauge. 15J.C. Ward, Phys. Rev. Lett. 78(1950) 182; Y. Takahashi, Nuo. Cim. 6(1957) 370. 16N.K. Nielsen, Nucl. Phys. B140 (1978) 499; R.E. Kallosh, Nucl. Phys. B141 (1978) 141. 17M.T. Grisaru, W. Siegel, and M. Ro cek, Nucl. Phys. B159 (1979) 429: super background-eld gauge. C. SCATTERING 331 ::::::::::::::::::::::::: ::::::::::::::::::::::::: ::::::::::::::::::::::::: C. SCATTERING ::::::::::::::::::::::::: We have seen how covariant expansions of the S-matrix can be based on various denitions of  h. Covariant expansions can also be based on spacetime quantum numbers: For example, we can perturb in mass; this is equivalent to adding low-energy corrections to the high-energy approximation. Also, the rst-quantized versionof the hexpansion, which expands in powers of momenta, is eectively an expansion in inverse powers of mass (low-energy approximation). The only other spacetime property of a particle is spin, or helicity for massless particles in D=4. It is possible to dene expansions in terms of it by describing theleading order by a complex action. This violates semiclassical unitarity at that order;however, the loop expansion violates unitarity at tree order also, so the expansionis still useful as long as unitarity returns once the expansion has been summed.Furthermore, we have already seen that gauges where unitarity is not manifest havesome advantages over unitary gauges. In particular, the Gervais-Neveu gauge uses acomplex gauge condition. 1. Yang-Mills We rst consider calculations for massless theories; these are simpler than massive ones in D=4 because the little group of the Lorentz group is SO(D 2) instead of SO(D1), and is thus Abelian: We can label the spin of a state by an integer or half- integer, the helicity, by use of the spacecone formalism. To simplify notation, we dropthe transverse index ( p t!p), and distinguish 4-momentum Pfrom its transverse component pby using upper- and lower-case. We also use color ordering; i.e., we examine only planar diagrams for each permutation of external lines. We begin by summarizing the spacecone rules for pure Yang-Mills found in sub- section VIB6: The Lagrangian appearing in the action S=g2trRdx L , writing derivatives as momentum operators for later convenience, is L=A+(1 2P2)A+(p pA+)[A+;pA]+(p+ pA)[A;pA+]+[A+;pA]1 p2[A;pA+] Twistor notation (see subsection IIB6) is used: p+=hpi[p];p=h+pi[p+];p =h+pi[p];p=hpi[p+] h+i=[+] = 1 332 VI. QUANTUM GAUGE THEORY The propagator and vertices are read from Lin the usual way, but in addition we have further simplication from the choice of external line factors +=[p] h+pi;=h+pi [p];p p=p+ p =1 whereand are the reference lines, with + and helicity, respectively (not to be confused with the earlier notation for spinor indices =(; )). However, the reference momenta for helicitiesare taken from lines with helicities : P=ji[j;P =j+i[+j The reference external line factors occur only in the above combinations, because only 1 term of 1 of the 3-point vertices contributes to each. The simplest examples are classes of diagrams that vanish by virtue of their \maximal helicity violation": By simple counting of +'s and 's, we see that the tree graphs with the fewest external 's, those with only self-dual vertices (++ ), have a single external. Thus the all + amplitude vanishes automatically. Furthermore, the diagrams with a single external must have that line chosen as one of the reference lines. However, by the above rules that line can carry only the anti-self-dual vertex (+), so those amplitudes also vanish. The simplest nonvanishing amplitude is ++ . We consider the case where the helicities are cyclically ordered as ++ ; we label them 1234, and choose 1 and 4 as the reference lines; this amplitude can be denoted as + .(P4=j+i[+], P1=ji[j: The positive-helicity reference line gives the reference momentum for negative helicity, and vice versa.) We label all external momenta as owing outward.There are only three diagrams; however, the + reference line uses only the ++ vertex, while the reference line uses only the + vertex, so the 4-point-vertex diagram vanishes, as does the diagram with both reference lines at the same vertex. Thus, we are left with only 1 graph. 12+ 3 4+ Furthermore, we know that the 3-point vertices contribute only 1 term to the reference line, so this graph has only 1 term. This means we can immediately write down the answer (dropping the factors of gat each vertex): 2+3p2p31 1 2(P3+P4)2=[2] h+2ih+3i [3]h+2i[2]h+3i[3]1 h34i[34]1 h+i[+] C. SCATTERING 333 =[12]2h34i [34][41]h14i where we have restored helicity and dimensions, and used p1=p4=0 . W eh a v e omitted the usual group theory factor (see subsection VC9). (Note that the propaga- tor is1=1 2P2, because of the signature for the spacecone components. This extra sign cancels that coming from the fact that one vertex has cyclic ordering and oneanticyclic with respect to group theory, i.e., the commutators in the action.) Using the identities, following from overall momentum conservation, (P 1+P4)2=(P2+P3)2) [41]h14i= [23]h32i X jpi[pj=0)h 34i[14] =h32i[12] this can be put in the standard form [12]4 [12][23][34][41] Excercise VIC1.1 Using similar manipulations, cast it into the form h34i4 h12ih23ih34ih41i Excercise VIC1.2 Repeat the calculation for the + +(color-ordered) amplitude. The corresponding dierential cross section is very simple: Using hpqi*=[pq])j hpqij2=j[pq]j2=P
Q and momentum conservation, we nd (after including the coupling g) jTj2=g4s2 t2or g4t2 s2 depending on the orientation of the diagram with respect to time, for this color- ordered contribution. (Depending on the color quantum numbers of the external states, this can be the only contribution.) Then (see subsection VC7) d dt=2 ( 2)3g41 t2ort2 s4 A more complicated example is the +++ amplitude. Again taking color- ordered (planar) amplitudes, we choose the amplitude cyclically ordered as +++ with lines labeled 12345, picking 1 and 5 as the reference lines, which we denote as 334 VI. QUANTUM GAUGE THEORY ++ . Again dropping all graphs with a reference line at a 4-point vertex or 2 references lines at a 3-point, all 5 graphs with a 4-point vertex are killed, and only 3 of the remaining 5 survive. (We also need to consider various combinations of + andindices, but only 1 survives for each graph because of the chirality of 3-vertices with reference lines.) 2+ 3+ 4 5+ +12+3+ ++4 512+3+ ++4 5 1+ + Since 3-point vertices with (without) a reference line have 1 (2) terms, we are left with only 6 terms. The initial result for the amplitude is then 2+3+42 4p3 4 p 2 p2p3 p3 (P2
P3)(P4
P5)p2p2 4 p 2+1 p2p3 p3 (P1
P2)(P4
P5)+p2 2p4 p 2+1 p2p3 p3 (P1
P2)(P3
P4)3 5 where we have used the fact that the reference lines have trivial momenta: 1 for the component withindex opposite to its helicity, 0 for the remaining components. The two terms for each diagram simplify to one, using p p=[p+] [p])p 2 p2p 3 p3=[2+][3][3+][2] [2][3]=[23] [2][3] (applying the cyclic identity) with our normalization. Using this result, we nd the similar result p 2+1 p2p 3 p3=h+i[3] +h+2i[23] h+2i[2][3]=h+4i[34] h+2i[2][3] applying momentum conservation. We next translate the momentum denominators into twistor notation, and also substitute the spacecone expressions for the polariza-tions and numerators. canceling identical factors in numerator and denominator (but no further use of identities), the amplitude becomes (+ = 5, =1 ) h+4i 3 h+2ih+3i[4]2 h23i[4+]+[4][34] h2i[4+]+h+2i[2] h2ih34i =h+4i3 h+2ih+3i h+2i[4] h2ih23i+h+2i[2] h2ih34i =h+4i3 h2ih23ih34i=h45i4 h12ih23ih34ih45ih51i C. SCATTERING 335 applying momentum conservation twice, restoring normalization, and replacing the numerals for. Excercise VIC1.3 Using the spacecone gauge, evaluate all diagrams contributing to the six-point gluon (Yang-Mills) scattering tree amplitude (T-matrix) with color-ordered helicities ++++, that correspond to the symmetric diagram with a central 3-point vertex each of whose legs is connected to another 3-point vertex, eachof which carries 2 of the external lines. These results can be generalized to arbitrary (color-ordered) n-point tree ampli- tudes with twohelicities, labeled iandj, and the rest + (\Parke-Taylor ampli- tudes"): The result is (in an obvious notation), including now the coupling ( g) n2, h+1


+i1i+i+1


+j1j+j+1


+ni=(g)n2hiji4 h12ih23i


hn1;nihn1i 2. Recursion A simple way to derive higher-point amplitudes is using the classical eld equa- tions. (See subsection VC3. In the literature, the eld has often been mistaken forthe current, since =J ,J=. As usual, these are distinguished by the fact the eld always has an external propagator, while the current has it amputated, sinceK+:::=J.) The steps are: (1) Calculate the rst few terms in the series (enumerated by the number of external lines). (2) Guess the general result. (3) Provethat it is correct by induction, using the classical eld equations. Of course, the sec-ond part is the hardest in general (at least when one simplies the third step by usingspacecone methods), and has been possible for just a couple of cases, only because the results for those cases are so simple. Since these results are for o-shell elds, and not S-matrix elements, they are gauge dependent: For example, if they are insertedinto larger diagrams, the same choice of reference lines must be used. The solution to the classical eld equations is given by tree graphs with all external lines but one (the eld itself) amputated and put on shell. (The usual external-linewave functions describe the asymptotic eld, which is free.) The two cases withknown solutions are those where all the on-shell lines have the same helicity, or onedierent. Note that the eld A has aassociated with the opposite end of its external propagator. We then see in the former case, with all +'s on on-shell lines, thatAvanishes because there are no fully-amputated diagrams, even o-shell, with only +'s externally (again counting +'s and 's on vertices). Similarly, for the latter case, with only one on an on-shell line, we see that Ahas only ++vertices; 336 VI. QUANTUM GAUGE THEORY but setting that one on-shell to be a reference line (which by denition must be on-shell), it is not allowed such a vertex, so Avanishes also in this case. By similar reasoning, we see that A+in the former case consists entirely of ++ vertices; and in the latter case consists of all ++ except for one+( n o+ +), which must have thereference line directly attached. The appearance of only the self-dual eld ( A+) and almost only the self-dual vertex (++) means that in both cases one is essentially solving equations in the self-dual theory: If we take just the kinetic term and ++ vertex from the action, and make the eld redenitions (see excercise VIB6.2) A+=p; A=p1^ we obtain (after integration by parts and rearrangement inside the trace) L+;++=^(1 2P2+[p;p ]) These redenitions make the ++ vertex local. ^appears only as a Lagrange mul- tiplier, and its variation gives the self-dual eld equation 1 2+i(@ .)(@ .)=0 (which diers from the result of subsection IIIC5 by an ifrom the use of pinstead of @in the eld redenition, and ! from the use of the spacecone instead of the lightcone). We now consider in more detail the simpler (former) example (the one which does not directly give a nontrivial scattering amplitude). As a slight simplication, we look at the recursion relation for the eld as dened in the self-dual theory. The recursion relation is now (see subsection VC3), scaling the coupling out of the kinetic term, (1;n)=g 1 2P2(1;n)n1X i=1(1;i)(i+1;n)[p(1;i)p(i+1;n)p(1;i)p(i+1;n)] P(j;k)kX m=jPm where we again use color ordering, number the external lines cyclically, and (j;k) denotes the eld with on-shell lines with momenta PjthroughPk. (Thus, on the left-hand side of the equation the eld has non-shell lines, while on the right-hand C. SCATTERING 337 side the two elds have iandni.) Plugging in the twistor expressions for the vertex momenta, we nd p(1;i)p(i+1;n)p(1;i)p(i+1;n)=iX j=1nX k=i+1h+ji[jk]h+ki If we are clever we can guess the general result from explicit evaluation of the lower-order graphs; instead we nd in the literature, after the above redenition, (i;j)=(g)N1 1 h+iihi;i+1i


hj1;jih+ji whereNis the number of background momenta ( Pi;:::;Pj)f o r(i;j). For the initial- condition case N= 1 this is simply the statement that the external line factor for  is now =+ p=1 h+pi2 The induction hypothesis is also easy to check: The product of the two 's from the induction hypothesis gives the desired result by itself up to a simple factor: (1;i)(i+1;n)=1 g(1;n)hi;i+1i h+iih+;i+1i (The algebra of the color indices works as usual.) We then perform the sum over i before that over jandk(the complete sum is over all i;j;k with 1ji<kn), making use of the identity habi h+aih+bi+hbci h+bih+ci=haci h+aih+ci)k1X i=jhi;i+1i h+iih+;i+1i=hjki h+jih+ki Multiplying this by the vertex momentum factor gives a sum over j<k ofhjki[jk]= Pj
Pk, canceling the external propagator, yielding the desired result. 3. Fermions We have seen in subsection VIB7 how these methods can be applied to massless spinors. Rather than applying the rules directly, in this subsection we examine therelation of the results in QCD to those in pure Yang-Mills theory. We also saw insubsection VIB7 how supersymmetry could be used to relate dierent QCD ampli- tudes. However, in practice supersymmetry relations give only a few useful relations, and only ones that can already be seen directly from the spacecone rules, which givemore results than can be seen by supersymmetry alone. 338 VI. QUANTUM GAUGE THEORY The simplest relations that follow from supersymmetry are the vanishing of tree graphs with fewer than two negative helicities, which we saw in subsection VIC1 follows automatically from the spacecone rules. The remaining useful supersymme- try relation for tree graphs is the relation between Parke-Taylor amplitudes for pure Yang-Mills and those with one external line each of positive and negative helicityreplaced with spinors or scalars. The easiest way to see this result is to make use of the conventions of the selfdual theory, as in the preceeding subsection. In Parke- Taylor amplitudes only one vertex is a non-selfdual vertex, which accounts for thesimplicity of these amplitudes. (Tree amplitudes with only selfdual vertices van- ish.) Furthermore, after transforming to the selfdual conventions, all (nonvanishing) selfdual vertices are identical | independent of spin. Furthermore, the nonselfdual3-point vertex with one negative-helicity gluon chosen as a reference line (the only non-selfdual vertex we'll need for this relation) is independent of the spins of the remaining two lines. Consequently, the only dierence between the two amplitudeswe are relating comes from the dierence in normalization of external line factors for gluons and quarks (and scalars). We will not review the superspace formulation of selfdual supersymmetric theories here. The main features will be evident from the example of supersymmetric QCDthat we now examine in more detail. The main result follows from treating the selfdual eld of the nonsupersymmetric theory as a spacecone (or lightcone) supereld. Dimensional analysis then tells us that the eld of helicity hhas dimension 1h. The appropriate redenitions of the spacecone elds are then A +!pA+; +!p +;!; ! ;A!1 pA for the Yang-Mills elds A, spinors , and scalars . The resulting external line factors are then simply h+pi2h After these redenitions, the kinetic terms, selfdual (++ ) vertices, and antiselfdual vertices forgluon reference line (referencing positive helicity) are L2=A+1 2P2A+ +1 2P2 L3;sd=(pA+)([pA+;A]+fp +; g)+(p +)[pA+; ] L3;sd;=p+ p2A ([pA+;A]+fp +; g) for supersymmetric QCD. (In the A3term in the last line we have used integration by parts, and dropped a ( p+=p)Aterm that vanishes for the reference line: There (p+=p2)=1n o w ,s o( p+=p)= 0 vanishes for that line since p!0.) C. SCATTERING 339 We now see easily that the terms L2andL3;sdthat dene the selfdual theory are independent of whether boson or fermion is chosen for the positive helicity elds and the negative helicity one (only the helicities of the elds must add up to 0 for L2and 1 forL3;sdfor Lorentz invariance). Thus, supersymmetry is a much stronger restriction in a selfdual theory than a nonselfdual one. Finally, the current that couples to thereference line ( p +=p2)Ais also the same for bosons and fermions. We therefore have, for example, the relation (;1 2;+1 2;+


+) =h13i h12i(+


+) for the color-ordered tree amplitudes (where we have labeled helicities 1b y). This follows from choosing line 1 as reference line (for positive helicity, from a line with negative helicity). For example, from our result for the 4-gluon tree, we havethe 2-quark, 2-gluon tree (; 1 2;+1 2;+) =h12i2h13i h23ih34ih41i Excercise VIC3.1 Repeat these calculations using scalars in place of the spinors. 4. Masses The spacecone formalism yields the simplest method for deriving S-matrix ele- ments in massless theories (at least for trees; for loops it may be preferable to use background eld gauges, with a Lorentz gauge, like Gervais-Neveu, for the quantum gauge and spacecone for the background gauge). The anal ogous method for the mas- sive case is to use actions based on self-dual elds, as described in subsection IIIC4. The advantage of these two methods is that they use elds that are representations ofthe little group, so in the massive case elds have 2s+1 components and only undotted spinor indices (SO(3)=SU(2)), while in the massless case they have only 2 components and no indices (SO(2)=U(1)). Although the actions used are more complicated, thisis just a re ection of the fact that algebra that is usually done repetitively in graphs has been performed once and for all in the action. However, in the massive case the simplication is not as drastic as in the massless one: S-matrix elements are just simpler in massless theories, with many vanishing;the spacecone method takes advantage of this simplication in the nal results by simplifying the intermediate steps. The massive examples we will consider in this subsection, taken from QED, are somewhat simple in any case, so we will stick to 340 VI. QUANTUM GAUGE THEORY the older methods (although the uses of methods based on self-duality are still being explored). The major dierence in simplicity between the massless and massive cases (in any approach) is in the external line factors. The ambiguity in the explicit expressionsfor the external line factors is just the little group: In the massless case the solutionsto the eld equations (one solution and its complex conjugate) are unique up to aphase factor, which is why the twistor formalism is so useful. In the massive case thesolutions (2s+1) are ambiguous up to an SU(2) transformation, which means they aremessy for any choice. Just as in the massless case the twistor is part of the Lorentztransformation from an arbitrary frame to the lightcone frame, in the massive casethe solutions are part of the transformation to the rest frame. In other words, theexternal line factors simply convert Lorentz indices to little-group indices; this makesindices trivial for the massless case (in D=4), and not as nice for the massive. The result is that in practice whenever any of the external particles are massive their external line factors are left as implicit in S-matrix elements, and only theirsquares are explicitly evaluated, in cross sections. This was common in older experi-ments (especially QED), since recent experiments are mostly at energies so high thatmasses of external, stable particles are usually neglected. This adds to the algebra,since it means that Lorentz algebra is performed in each of n 2terms in the cross section rather than nterms in the S-matrix. Furthermore, the algebra is usually simplied by considering experiments where polarization is determined in neither the preparation of the initial states nor themeasurement of the nal states. This was also common in older experiments, whendevices for polarization were not well developed. The result is that one averages overinitial states and sums over nal states, producing the same algebraic factors thatappear in the propagator, as described in subsection VB3:
is replaced with
+. One then applies the rules for Feynman diagrams for cross sections, as described insubsection VC7. The standard S-matrices in QED are the 4-point tree graphs, with 2 3-point vertices and 1 internal propagator. There are just 2 graphs to consider, with variouslabelings of momenta: (1) The graph with 4 external fermions (electrons/positrons)connected by 1 internal photon describes both Mller (electron-electron) and Bhabha(electron-positron) scattering, 2 labelings each. (2) The graph with 2 external photonsand 2 external fermions, as a continuous line that includes the 1 internal fermion,describes Compton (electron-photon) scattering as well as electron-positron creation/-annihilation, also 2 labelings each. In each case, the 1 S-matrix diagram results in 2 C. SCATTERING 341 cross section diagrams, each with 2 momentum labelings (for a total of 2 2=4): 1 diagram from multiplying similar terms and 1 from cross-terms. In Dirac spinor notation the Lagrangian for QED is (see subsection IIIA4) 1 8F2+ (i@=eA=+mp 2) w h e r ew eh a v es c a l e dt h e\ e" into the vertex. The Feynman rules are now (Fermi- Feynman gauge): Photon propagator: ab=1 2p2 Fermion propagator: ( p=+mp 2)=1 2(p2+m2) Vertex: e a (We use the Fermi-Feynman-gauge propagator also for dening the cut propagator; ghosts decouple in QED.) The cross section diagrams contain closed fermion loops, resulting in traces of products of matrices (with a 1 for each loop by Fermi-Dirac statistics). The algebra is manageable for the present case, using the 4D -matrix identities from subsection IIA6: a a=2; aa= a=a=; aa=b= a=a
b; aa=b=c= a=c=b=a= tr(I)=4;t r (a=b=)=2a
b; tr (a=b=c=d=)=a
bc
d+a
db
ca
cb
d The traces encountered in the above processes are of the form N1=tr( aA bB)tr( aC bD) N2=tr( aA aB bC bD);N 3=tr( aA bB aC bD) whereA=a=+mp 2, etc. Using the above identities, these are evaluated as N1=4m4+2m2(a
b+c
d)+2 (a
cb
d+a
db
c) N2=4m4+m2[2(a+c)
(b+d)a
c4b
d]+(a
bc
d+a
db
ca
cb
d) N3=2m4+m2(a
b+a
c+a
d+b
c+b
d+c
d)+2a
cb
d Excercise VIC4.1 Generalize the above identities and expressions for the N's to arbitrary di- mension D. 342 VI. QUANTUM GAUGE THEORY s 134 2t 134 2 Our rst example is e+e!e+e(\Bhabha scattering"). We have aligned all momenta to be that of the electrons (i.e., minus that of the positrons), so that all numerator factors are p=+mp 2without signs. Specically, we have chosen p1for the (positive-energy) momentum for the incoming electron, p2for the incoming positron,p3for the outgoing electron, and p4for the outgoing positron. With these conventions, s=(p1p2)2=(p3p4)2;t=(p1p3)2=(p2p4)2 u=(p1+p4)2=(p2+p3)2(p2 i=m2;s +t+u=4m2) )p1
p2=p3
p4=1 2sm2;p 1
p3=p2
p4=1 2tm2;p 1
p4=p2
p3=1 2u+m2 N1(ijkl) N3(ijkl)ijkl a b iljk a b For the squared amplitude we have for the average over initial polarizations and sum over nal 1 4X poljTj2=N1(1342) t2+N1(1243) s2+N3(1243) +N3(1342) st =1 2f(s)+f(u) t2+1 2f(t)+f(u) s2+f(u) st not including the overall factor of e4,w h e r e f(x)(x2m2)(x6m2) Every other Nterm is the result of switching s$t(p2$p3,o rp1$p4)i nt h e previous, since that is the relation of the 2 Feynman graphs contributing to the S- matrix. The N1terms are the squared diagrams, while the N3's are the cross terms. The \"i nN3comes from Fermi-Dirac statistics, switching two fermion lines. C. SCATTERING 343 Finally, adding the appropriate factors to get the dierential cross section (see subsection VC7) d dt Bhabha=(2)3e4 s(s4m2)f(s)+f(u) t2+f(t)+f(u) s2+2f(u) st The probabilities jTj2foree!ee(\Mller scattering"), or e+e+!e+e+,a r e related by crossing symmetry s$u(p1$p3orp2$p4): d dt Mller=(2)3e4 s(s4m2)f(s)+f(u) t2+f(s)+f(t) u2+2f(s) tu A convenient frame for any of these cross sections is the center-of-mass frame (subsection IA4). In these cases all the external masses are equal, so the Mandelstam variables have simple expressions in terms of the energy (which is the same for all 4particles) and the scattering angle: s=4E 2;t =4(E2m2)sin2 2;u =4(E2m2)cos2 2 su 1342 1342 Another famous example is e !e (\Compton scattering"). Now we label p1for the incoming electron, p3for the incoming photon, p2for the outgoing electron, andp4for the outgoing photon, so the Mandelstam variables are s=(p1+p3)2=(p2p4)2;t=(p1p2)2=(p3+p4)2 u=(p1+p4)2=(p2p3)2(p2 1=p2 2=m2;p2 3=p2 4=0 ;s+t+u=2m2) )p1
p2=1 2tm2;p 1
p3=1 2(sm2);p 1
p4=1 2(um2) p2
p3=1 2(um2);p 2
p4=1 2(sm2);p 3
p4=1 2t N2(ijkj) – N3(ijkl)ijk ab ijk lab 344 VI. QUANTUM GAUGE THEORY The probability is 1 4X poljTj2=N2(1;1+3;2;1+3 ) (sm2)2+N2(1;1+4;2;1+4 ) (um2)2 N3(1;1+4;2;1+3 )+N3(1;1+3;2;1+4 ) (sm2)(um2) =1 2m4+m2(3s+u)su (sm2)2+1 2m4+m2(3u+s)su (um2)2m2(t4m2) (sm2)(um2) where now every other term comes from switching s$u(p3$p4), and the cross section is, after some rearrangement, d dt Compton=(2)3e4 (sm2)2" 4m41 sm2+1 um22 +4m21 sm2+1 um2 um2 sm2+sm2 um2 A useful frame is the lab frame (i.e., the rest frame of the electron), where in terms of the initial and nal (positive) energies ( EandE0) and scattering angle of the photon we have s=m2+2mE; u =m22mE0;t =2m(E0E);1 E01 E=2sin2 2 m By crossing symmetry, s$t,w eg e te+e!2 (\pair annihilation") and 2 !e+e(\pair creation"): d dt annihil.=(2)3e4 s(s4m2)" 4m41 tm2+1 um22 +4m21 tm2+1 um2 um2 tm2+sm2 tm2 d dt creation=(2)3e4 s2" 4m41 tm2+1 um22 +4m21 tm2+1 um2 um2 tm2+sm2 tm2 Excercise VIC4.2 Calculate all the corresponding massless cross sections using the spaceconegauge. Show they agree with the m=0c a s eo ft h ea b o v e . C. SCATTERING 345 5. Supergraphs In supersymmetric theories the easiest way to calculate Feynman diagrams is in superspace. Supersymmetric cancelations are then automatic, and new special prop- erties of supersymmetric theories are revealed. The derivation of the \supergraph" rules is similar to that of subsection VC1, except for some ne points in the treat-ment of chiral superelds. The path integral required the explicit evaluation of onlya Gaussian for perturbation. Since we dropped proportionality constants, this was equivalent to substituting the solution to the classical, free eld equations back into a quadratic action. For real scalar superelds (used for super Yang-Mills) this is trivial,but chiral scalar superelds (used for scalar multplets) satisfy the chirality constraint,and have superpotential terms: integrals over chiral superspace (R dx d 2), not the full superspace (R dx d4). We want to make use of the identity for evaluating the path integral (see subsec- tion VC1) Zdup 2euMu= 2f(u+v)=Zdup 2euMu= 2eu@vf(v)e@vM1@v=2f(v) Then the \action" we need to integrate is ~S=Z dx d4Z dx d2(mp 2)1 22+h:c: Z dx d2 '+h:c: consisting of the (derivative part of the) kinetic term, mass term, and (minus the) term that acts on eSI[']. Solving the eld equations (see subsection IVC2) d2+mp 2+ '=d2+mp 2+ '=0 we nd =1 1 2(+m2) d2 'mp 2 ' ; =1 1 2(+m2) d2 'mp 2 ' The propagator exponentR1 2(=' )(1=K)(=' ) thus becomes (putting back '!) Z dx d4  1 1 2(+m2) +Z dx d41 2  mp 21 1 2(+m2) +h:c: Before writing the Feynman rules, we rst note that functional dierentiation with respect to a chiral supereld, as follows from the above variation, gives  (x;)(x0;0)=d24(0)(xx0) 346 VI. QUANTUM GAUGE THEORY This means that there will be an extra d2at theend of any chiral propagator and an extrad2at the end. We could associate these directly with the propagator, but we will use one factor of d2to convert aR d2intoR d4at any superpotential vertex, and similarly for the complex conjugate. Therefore, we include such factors explicitly as a separate Feynman rule for the ends of chiral propagators. However, this meansthepropagator (and similarly for ) gets an extra factor of d 2=1 2to compensate for the fact that we include two d2factors, whereas it really had only one because its integral was only d2. Furthermore, we Fourier transform xas usual, but not , basically because there is no translation invariance in , but also for a better reason to be explained soon. The Feynman rules of subsection VC4 are then modied as: (A21 2) Theta's: one for each vertex, with anR d4. (A30) Propagators: VV:1 1 2(p2+m2)4(0) :1 1 2(p2+m2)4(0) :mp 2d2 1 2p21 1 2(p2+m2)4(0) :mp 2d2 1 2p21 1 2(p2+m2)4(0) (A41 2) Chiral vertex factors: d2on theend(s) of every chiral propagator, d2on the end(s), but drop any one such factor at superpotential vertex. We next explain how integrations are performed on any connected graph. Con- sider any two vertices directly connected by a propagator. All the spinor derivatives acting on its 4(0) can be removed from that propagator by integration by parts. We then can use that function to trivially integrate over 0,r e m o v i n gt h eRd40and that4(0), and replacing 0everywhere with . Eectively, those two vertices have been contracted to the same point in space, eliminating that propagator as far asdependence is concerned. We can repeat this procedure until allvertices are contracted to a single point. However, we are then left with a \tadpole" for each loop: Contracting propagators this way sequentially around a loop identies all the vertices of that loop, and leaves the loop as a single propagator with both ends atthat point. To evaluate this tadpole, we note that [d 2d24(0)]j0==1 (0derivatives can be converted into minus derivatives when acting directly on the; this is basically integration by parts.) Fewer derivatives give 0; more can be C. SCATTERING 347 reduced to terms of 4 or less. This completes all the evaluation in space, leaving an expression in terms of elds (some with d's acting on them) with dierent momenta, times the usual momentum factors, with the usual momentum integrals, but all atthe same, with a singleR d 4. This means that the generating functionals Wand are completely local in . There is a further consequence of this evaluation. We have obtained terms withR d4, but none withR d2. However, to do it we had to introduce the factors d2=(1 2p2)i n t ot h epropagators. On the other hand, such a factor can easily be killed by a d2from a vertex: We sandwich the d2between a d2from each vertex, using the identity d2d2d2=1 2p2d2, and return the d2to one vertex. The only time we can't do that everywhere is if every vertex is a superpotential (so every propagator in the graph is and every external eld is ), since otherwise we can inductively borrow d2's from some non-R d2vertex. Any such 1PI graph always vanishes, be- cause there are exactly enough d's left to make the tadpoles nonvanishing, leaving anR d4of a product of 's with nod's, which vanishes. On the other hand, for a tree graph there is exactly one d2=p2left, which converts theR d4to anR d2. The net result is that not only are Wand local in , but only their classical parts can haveR d2terms, and the spurious d2=p2factors (which should not appear in massive theories) are always canceled. In particular, this implies that all UVdivergences areR d 4terms: All terms in the superpotential are unrenormalized (no loop corrections) to all orders in perturbation theory. Excercise VIC5.1 Calculate all the contributions to W[;] from 4-point trees in massive super- 3theory, and write the result in both p-a n dx-space (in analogy to the nonsupersymmetric example at the end of subsection VC4). Improvements again result from using background-eld gauges. We have already seen in subsection VIB10 the modication to the quantization for supersymmetricbackground-eld gauges. The background-eld expansion can be dened by solvingthe constraints on the full covariant derivatives in terms of quantum prepotentialsbut background potentials (A A,n o tV), essentially by covariantizing dAto the back- groundrA.T h e nrcan be manipulated (integration by parts, etc.) in the graphs i nt h es a m ew a ya s dwas, leaving only Aa(notA) and its derivatives ( W,e t c . )a s background elds. This leads to improved power counting, and can be used to prove\superrenormalizability" (niteness beyond one loop) for N=2 extended supersym-metric theories, and niteness for N=4. 348 VI. QUANTUM GAUGE THEORY As for other gauges (background-)chiral superelds need special treatment, now to get the most out of background gauge invariance. Variation can be dened in the obvious way, but now we also need the covariantized identity D2D2=1 2(+i[W;D]) from pushing the D's to the right and using the commutation relations. The func- tional integral over the quantum background-chiral superelds can also be performed in the same way as for other gauges, the only modications being background co-variantization (including the above \nonminimal" term for ), and the fact that we can no longer neglect the \vacuum" contribution (one-loop diagrams with only background elds externally). Specically, if we look at the general derivation of the Feynman rules in subsection VC1, we see it gave rules for all graphs except theone-loop vacuum bubble, since this graph has no (quantum) vertices. These rules, as adapted to superspace earlier in this subsection, are now modied only by the covari- antization just discussed, which only adds background potentials (not prepotentials)and eld strengths to propagators and vertices. The background-covariantized prop-agators then can be further expanded about the free . The net result is that in alldiagrams except (perhaps) these chiral one-loop vacuum bubbles the background elds appear only in the form of potentials and eld strengths. These vacuum bubblesthen can be evaluated by the usual methods, since the formerly neglected Gaussianpath integral for these \quantum-free" elds is just the usual one-loop path integral for a chiral supereld with external Yang-Mills superelds, only the external elds are now identied as background instead of quantum. In some cases, this last calculationcan be further simplied to again yield an expression directly in terms of potentialswithout explicit prepotentials (see subsection VIIIA6 below). REFERENCES 1 M.T. Grisaru, H.N. Pendleton, and P. van Nieuwenhuizen, Phys. Rev. Lett. 15(1977) 996;M.T. Grisaru and H.N. Pendleton, Nucl. Phys. B124 (1977) 333: supersymmetry identities for trees. 2M.T. Grisaru and W. Siegel, Phys. Lett. 110B (1982) 49: generalization to one loop. 3S.J. Parke and T. Taylor, Nucl. Phys. B269 (1986) 410, Phys. Rev. Lett. 56(1986) 2459;F.A. Berends and W.T. Giele, Nucl. Phys. B306 (1988) 759: Parke-Taylor amplitudes. 4W.A. Bardeen, Prog. Theor. Phys. Suppl. 123(1996) 1; D. Cangemi, hep-th/9605208, Nucl. Phys. B484 (1997) 521; C. SCATTERING 349 G. Chalmers and W. Siegel, hep-th/9606061, Phys. Rev. D54 (1996) 7628: relation of Parke-Taylor amplitudes to self-dual Yang-Mills. 5R. Brooks (May, 1993), unpublished; N. Berkovits and W. Siegel, hep-th/9703154, Nucl. Phys. B505 (1997) 139: use of rst-order actions with self-dual auxiliary elds for perturbation about self-duality. 6Feynman, loc. cit. (VB, ref. 4): Feynman diagrams for QED, from the horse's mouth. The original derivation, not count-ing his later-published rst-quantized path-integral approach. Basically a mechanicspoint of view. 7H.J. Bhabha, P r o c .R o y .S o c . A154 (1936) 195. 8C. Mller, Ann. Phys. 14(1932) 531. 9O. Klein and Y. Nishina, Z. Phys. 52(1929) 853: Compton scattering. 10A. Salam and J. Strathdee, Phys. Rev. D11 (1975) 1521; K. Fujikawa and W. Lang, Nucl. Phys. B88 (1975) 61; J. Honerkamp, M. Schlindwein, F. Krause, and M. Scheunert, Nucl. Phys. B95 (1975) 397;S. Ferrara and O. Piguet, Nucl. Phys. B93 (1975) 261; D.M. Capper, Nuo. Cim. 25A (1975) 259; R. Delbourgo, Nuo. Cim. 25A (1975) 646, J. Phys. G 1(1975) 800; R. Delbourgo and M. Ram on Medrano, Nucl. Phys. B110 (1976) 473; W. Siegel, Phys. Lett. 84B (1979) 193, 197: early supergraphs. 11Wess and Zumino, loc. cit. (IVC, ref. 2); J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310; Capper, loc. cit. ; Delbourgo, loc.cit. (rst ref. above); B. Zumino, Nucl. Phys. B89 (1975) 535; P.C. West, Nucl. Phys. B106 (1976) 219; D.M. Capper and M. Ram on Medrano, J. Phys. G 2(1976) 269; S. Weinberg, Phys. Lett. 62B (1976) 111: nonrenormalization theorems for chiral superelds, from components or old-fashionedsupergraphs. 12Grisaru, Siegel, and Ro cek, loc. cit. (VB, ref. 14); M.T. Grisaru and W. Siegel, Nucl. Phys. B201 (1982) 292; Gates, Grisaru, Ro cek, and Siegel, loc. cit. ; M.T. Grisaru and D. Zanon, Phys. Lett. 142B (1984) 359, Nucl. Phys. B252 (1985) 578, 591:supergraphs as done today; more general nonrenormalization theorems. 350 VII. LOOPS VII. LOOPS Although our analysis so far is sucient to evaluate the lowest-order term in the hexpansion (\trees"), certain new features arise at higher orders. :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: A. GENERAL :::::::::::::::::::::::::::: When innities were rst found in perturbative quantum eld theory, they were thought to be a serious problem. A prescription can be given to remove these in- nities, called \regularization". It was later shown that this regularization can bedened in such a way as to preserve all the desirable physical properties of the theory, called \renormalization". Unfortunately, it seems that the original innities, exiled by renormalization from any nite order of perturbation theory, return to plague eld theory when all orders of perturbation theory are summed. Therefore, renormaliza- tion should not be considered a cure to the disease of innities, but only a treatmentthat allows divergent theories to be more useful. 1. Dimensional renormalization \Perturbative renormalization" is dened to preserve the three properties that dene relativistic quantum eld theory (Poincar e invariance, unitarity, causality). The most general prescription is to start with a classical theory that is causal, Poincar e invariant, and satises the semiclassical part of unitarity (as described in subsection VC5). This gives the tree graphs of the theory. One then applies unitarity to dene a perturbation expansion, determining the higher orders (loop diagrams) from thelowest (trees). Although the usual loop diagrams are divergent, there is enough ambiguity in the unitarity condition to allow removal of the divergences. The only practical way to implement this procedure is to slightly modify the di- vergent graphs one obtains from the naive use of the Feynman graph rules (obtained, e.g., from path integral quantization of a classical action). The steps are: (1) \Reg- ularize" each divergent graph by modifying the momentum integrals, introducing aparameter(s), the \regulator(s)", giving a nite result that reproduces the original divergent integral in a certain limit. (2) \Renormalize" each regularized graph by subtracting out the \divergent part" of the regularized graph, keeping only the \- nite part". Once the graph has been rendered nite, the regulator can be dropped. One then has to check that the method of removing divergences, order-by-order in the perturbation expansion, preserves the three properties of relativistic quantum A. GENERAL 351 eld theory. The easiest way to do this is to use both a regularization scheme and a subtraction scheme that preserve these properties manifestly. The standard way forthe subtraction scheme to do this is to change the coecients of terms in the classical action by (real) constants that depend on both the regulator and  h. Since the classical action already satises Poincar e invariance, causality, and the semiclassical part of unitarity, this will automatically preserve all the desired properties. In this manner of renormalization, there thus remains only two steps to prove that a theory can be renormalized: (1) the existence of a regularization that manifestly preserves the three properties, and (2) that the modication of the action by making the coecients regulator- and  h-dependent is sucient to cancel all divergences that might reappear in the limit as the regulator is removed. The latter step, discussed in subsection VIIA5 below, can be further divided into substeps, proving: (a) The ultraviolet divergence in any graph corresponding to scal- ing all internal (integration) momenta to innity (the \supercial" divergence) comes from a term in that graph polynomial in external momenta, and can therefore be can- celed by a local term from the action; and (b) recursively in the number Lof loops, if the renormalization procedure has already been successfully applied through L1 loops, the resulting modication of the action is sucient to cancel all divergences appearing at Lloops except the supercial ones. The former substep is the one that detemines whether the theory can be renormalized. The former step is satised by dimensional regularization, the standard method of regularization in relativistic quantum eld theory (and for practical purposes beyond one loop, the only one). It is dened by writing the theory under consideration in arbitrary dimensions D, and treating integrals over loop momenta as analytic func-tions of D. These integrals are then analytically continued from lower D, where they are (ultraviolet) convergent. The resulting expressions have pole singularities in D at integer D, so these poles can be subtracted out as the divergent parts. There are two main reasons why dimensional regularization is so useful: (1) Most classical actions can be written as easily in arbitrary dimensions as in D=4. (The important exception is those that in some way involve the Levi-Civita tensor  ab:::c. The diculty with such theories is not a drawback of dimensional regularization, but a general property of quantum eld theory, and is related to the quantum breakdown of classical symmetries, to be discussed later.) In particular, this means it manifestly preserves gauge invariance (which is a part of unitarity), the property of relativistic quantum eld theory most dicult to preserve. It is also the only workable scheme of regularization to do so. 352 VII. LOOPS (2) It requires only one regulator, the number of dimensions D itself. (Most other regularization schemes require at least one regulator for each loop.) This is the main reason why this scheme is the only practical method of regularization at higher loops.(An enormous number of regularization schemes have been proposed, almost all ofwhich work well at one loop, but all of them are more dicult than dimensionalregularization already at two loops, and even worse at three loops and higher.) The renormalization scheme based on this regularization is also very simple: DeningD=D 02(whereD0is the physical dimension, usually 4), we can Laurent expand any L-loop amplitude in , starting at 1 =L.T h e s e 1=nterms arise from n or more divergent D-momentum integrals. If we cancel all the negative powers of , we can take the limit D!D0(!0) by just dropping all the positive powers of , i.e., evaluating the remainder at = 0. The procedure is to modify the coecients of the terms in the \classical" action (couplings, masses, and eld normalizations) by making them -a n dh-dependent, giving them  hL1=n(\counter")terms. Such terms can cancel any local divergence at Lloops. One then has to show that they also can- cel all nonlocal divergences at higher loops resulting from this divergence appearingin anL-loop subgraph. Thus, the procedure is recursive: (1) apply the counterterms obtained from calculations at less than Lloops to cancel subdivergences; (2) cancel the remaining, local, supercial divergences by introducing new L-loop counterterms. The form of the supercial divergence can be determined by evaluating the divergencecoming from the region of momentum space where all loop momenta go to innityat the same rate. The supercial divergence is determined by 1 =terms of this loop and of subloop divergences; however, if the 1 =piece of a prospective counterterm vanishes at a certain loop order, so do all higher powers at that loop order. Thus, sim-ple power counting (as well as global and local symmetries) is sucient to determinewhat counterterms can appear at any particular loop order. These rules are sucient for evaluating momentum integrals to the point where renormalization can be applied. However, further simplications are possible wherespin is involved: Techniques specic to D=4 are useful to simplify algebra in general,and required to preserve manifest supersymmetry in particular. These methods treatspin indices as 4D, in contrast to the vector indices on momenta (and coordinates),which are analytically continued away from D=4 by the denition of dimensionalregularization. This is natural in 4D N=1 supersymmetric theories formulated insuperspace (or 4D N >1 in N=1 superspace), since there spins 1 are described by scalar prepotentials: There the simple prescription is to continue in the dimensionof the commuting coordinates ( x), while xing the dimension of the anticommuting A. GENERAL 353 coordinates ( ). These rules, for either supersymmetric or nonsupersymmetric the- ories, have a simple physical interpretation for integer D <4: dimensional reduction. The reduced theories dier from those produced by simple dimensional regularization: Vectors get (4D) extra scalars; spinors may become multiple spinors. For super- symmetric theories this is again natural, since vector and scalar multiplets remainirreducible after dimensional reduction. Unfortunately, vectors in nonsupersymmet- ric theories reduce to vectors plus scalars that are not related by any symmetry, so their renormalization is independent. However, the complications produced by theextra renormalizations are usually smaller than the algebraic simplications resulting from the restriction to 4D spin algebra, especially for lower loops. Another compli- cation is that Levi-Civita ( ) tensors can't be treated consistently in the dimensional reduction scheme. Although serious in principle, in practice this is not a problem as long as axial anomalies cancel, which is required anyway for unitarity. (See subsection VIIIB3. Also, axial anomalies are easier to calculate using Pauli-Villars regularizationthan with any form of dimensional regularization.) 2. Momentum integration The rst step in performing momentum integration is to make all integrals Gaus- sian by exponentiating propagators using Schwinger parameters 1 1 2(p2+m2)=Z1 0d e(p2+m2)=2 The general momentum integral in an arbitrary Feynman diagram is then A=N(p;i@x)ZdLDk (2)LD=2Z dPekT
A()k=2kT
B(;p)+ikT
xC(;p;m ) x=0 whereA;B;C are rst-order in ;Bis rst-order in pwhileCis second-order in p;m; andLis the number of loops and Pthe number of propagators. Also, we have used matrix notation with repsect to the P-dimensional -space, with respect to which Ais a matrix,Bis a vector, and Cis a scalar. Finally, Nrepresents all \numerator" factors (propagator numerators and vertices; everything but propagator denominators) and has been brought outside the integral by Fourier transformation (i.e., introducing x to produce a generating functional for all numerators). The momentum integrals are now Gaussian, and can be evaluated by the methods of subsection IB3 (and VA2): A=N(p;i@x)Z dP(det A )D=2e(Bix)TA1(Bix)=2C x=0 354 VII. LOOPS Some of the integrations can be performed by various scalings of subsets of the 's. For example, to see the supercial divergence of the graph we scale all of the 's and insert the identity as 1=Z1 0d  X ii! ;i=i whereare \Feynman parameters". The amplitude then becomes A=N(p;i@x)Z d P1LD=2dP 1X  [det A ()]D=2 e[B()TA()1B()=2C()]iB()T
A()1xxT
A()1x=2jx=0 (This method of introducing Feynman parameters is equivalent to directly changing variables from 's toand one less , and nding the Jacobian.) Thexderivatives in Nmust now be taken. For a contribution from these deriva- tives of order n,w eh a v eintegrals of the form Z d P1LD=2ne[BTA1B=2C]= (P1 2LDn)[C1 2BTA1B]P+LD=2+n where we have used the denition of the function (z)=Z1 0d z1e This integral converges only for Re z > 0, but we can extend it to (almost) all zby analytic continuation: Using integration by parts, z(z)=Z1 0d ed dz=(ez)j1 0+Z1 0d ze= (z+1 ) in the convergent region. Analytic continuation then says to evaluate the integral for (z) in the region 0Re z >1a s (z+1 )=zin terms of the integral for ( z+1 ) , and so on:Z1 0d z1e= nY k=01 z+k!Z1 0d z+ne Thus, (z) has simple poles in zat all the nonpositive integers. Excercise VIIA2.1 Using the identity ( z+1 ) =z(z), derive the following special cases for nonnegative integer n: (n+1 )=n!; (n+1 2)=(n1 2)(n3 2):::1 2p=(2n)! n!22np A. GENERAL 355 Feynman parameter integrations give more complicated functions. A simple but common example is the Beta function B(x;y)=Z1 0d x1(1)y1=(x)(y) (x+y) The latter expression for the Beta function can itself be derived by similar methods: Excercise VIIA2.2 Derive the following Bfunction identities: aUse the integral representation of the function to write an expression for (a)(b) as an integral over two Schwinger parameters, and introduce a scal- ing parameter (as with general two-propagator Feynman graphs, except herethere is no momentum). Show the result is ( a+b)B(a;b), whereBis given by the integral denition, thus proving the Beta function can be expressed in terms of Gamma functions. bUse the integral denition of Bto prove B(x;x)=2 12xB(1 2;x))(x) (2x)=212xp (x+1 2) Use this result to nd the expression in excercise VIIA2.1 for ( n+1 2). cDerive the identity Z1 0d a1(1 +)ab=B(a;b) from the substitution =z=(1z). Use this to show (z)(1z)=c s c (z) by changing variables =eu, and closing the contour in the complex plane to pick up the contributions from the poles. We thus have (z+n)(1zn)=(1)n(z)(1z) (which is also seen inductively from ( z+1 )= (z)). In general UV divergences will come from powers of 1 =in a resulting from integration over some scaling parameter. We thus need an expression for the Laurent expansion of ( z). This can be obtained directly from the integral expression: Using the denition of eas a limit, (z) = lim n!1Zn 0d z1(1 n)n= lim n!1nzB(z;n+1 )=1 zlim n!1nznY m=11 1+z m 356 VII. LOOPS from the change of variables =n, and the above identities. Dening the \Euler- Mascheroni constant" by = lim n!1 lnn+nX m=11 m! =0:5772156649 ::: )e z= lim n!1nznY m=1ez=m we then can write (z)=1 ze z1Y n=1ez=n 1+z n which is an alternate denition of . We then have ln(1z)= z+1X n=1[ln(1z n)z n]) ln(1z)= z+1X n=2(n) nzn; (y)=1X m=11 my by Taylor expansion of the ln,w h e r eis the \Riemann zeta function". Excercise VIIA2.3 Derive the following identities from the previous: aFind the rst two terms in the Laurent expansion of ( z): lim z!0(z) = lim z!01 z(z+1 )=1 z +O(z); =Z1 0d(ln)e bDo the same for expansions about other integers: (n+1+)=n!" 1+ +nX m=11 n! +O(2)# (n+)=(1)n1 n!1 " 1+ +nX m=11 n! +O(2)# cUsing the cscrelation in excercise VIIA2.2c and the above expansion of ln(1z), show that cot z =1 z 121X n=1(2n) 2nz2n! and thus(2n) can be written as a rational number times 2n. A. GENERAL 357 3. Modied subtractions The convenient normalization for the quadratic part of the gauge-invariant action for an arbitrary eld theory we use is S0=(1 22)(D4)=2 g2ZdDx (2)D=21 2K for a real eld and some coupling constant g, where for bosons K=1 2(+m2)+::: The explicit factor of 1/2 cancels the factor of 2 obtained when varying the action with respect to. (Similar permutation factors are used for interaction terms.) Equiva- lently, it gives the natural normalization for Gaussian functional integration over  in the eld theoretic path integral. For complex elds we instead have K without the 1/2, since then andcan be varied (or integrated over) independently. The \renormalization mass scale" has been introduced to preserve the mass dimension of gin arbitrary spacetime dimension; it appears naturally with the nor- malization1 22because the kinetic operator contains1 2p2and1 2m2. Generally there will be more than one coupling, but only one . For some purposes it may be con- venient to scale the elds so that the coupling and dependence appear only in the interaction terms. We will usually suppress the dependence, since it is determined by dimensional analysis, and is relevant only for quantum corrections, where D6=4 becomes important. Note that our normalization diers from that normally chosen in the literature. It has been chosen to give the normalization appropriate for Gaussian integrals, which appear in both the rst- and second-quantized theories. The rst dierence is thefactor of (2) D=2for coordinate and momentum integration (rather than the usual 1 for coordinate and (2 )Dfor momentum); the second is the factor of 1/2 multiplying the kinetic operator p2+m2(contained in K), rather than 1. Our normalization is more natural not only for Gaussian integration and Fourier transformation, but also slightly simplies perturbative eld theory calculations, allowing one to ignore spuri- ous factors of 2 and especially 2 . The net eect is only to change the normalization of coupling constants, since such factors can be absorbed into the 1 =g2sitting in front. For example, the most accurately experimentally veried prediction of quantum theory is the anomalous magnetic moment of the electron, to be discussed later. The result for the total magnetic moment, in various normalizations, to second order inperturbation theory, is  mag=1+e2=1+e2 m 2=1+e2 ft 82 358 VII. LOOPS where \e" is for our normalization (obviously the simplest), \ em" is the normalization you rst learned in classical mechanics (the one that gives e2 m=r2as the electrostratic force between two electrons), and \ eft" is the one you would see in other quantum electrodynamics courses. To complicate matters, you may have also seen the denition =e2 m=e2 ft=4, of which the only merit is supposed to be that 1 =is very close to the integer 137, which is s illy since 1=e2is even closer to the integer 861. (Actually, is the natural expansion parameter for nonrelativistic quantum mechanics, which is basically 3D, but our e2is more natural for the loop expansion, which is inherently 4D.) We have also used units  h= 1; restoring it introduces the further complication =e2 m=hm=e2 fthft=4because of the dierence in the semiclassical expansions for quantum mechanics and quantum eld theory. Furthermore, for nonabelian groups we have an extra factor of1 2compared to the standard normalization because we normalize trD(GiGj)=ijinstead oftrD(GiGj)= 1 2ij: The latter originated from the case of SU(2), where it cancels thep 2i nt h e diagonal generator (and in the others, if one uses hermitian ones rather than raisingand lowering). Unfortunately, for SU(N) with N3 this historical normalization introduces ap 2, while not canceling factors likep N(and making trD(GiGj)N - dependent would wreak havoc when considering subgroups, as well as for raising andlowering operators). Thus, in the above notation, nonabelian :g 2=g2 ft 162 Although we have chosen a normalization for the coupling constants that is nat- ural for Gaussian momentum integration, and for symmetry with respect to Fourier transformation, in divergent integrals it has the disadvantage of having 's (Euler- Mascheroni constant) in the nite parts. Another natural normalization that getsrid of this irrational number from all graphs is to divide out the angular part of the integrals: The volume of the unit D 1-dimensional sphere (the surface of unit radius in D-dimensional Euclidean space), is easily evaluated with Gaussians: 1=Zd Dk (2)D=2ek2=2=( 2)D=2Z dD1 Z1 0dk kD1ek2=2 =1 2(D 2)D=2Z dD1 We might therefore choose our normalization to cancel this factor in momentum integrals, along with the (2 )Dfrom Fourier transformation. Then the action, e.g., for a massless scalar, might be normalized as (with conventional kinetic term) S0=D4 g2Z dDx(RdD1 ) (2)D1 2(@)2=D4 g2ZdDx (4)D=2(D 2)(@)2 A. GENERAL 359 This diers from our previous normalization by a factor of 1 =(D 2), which is 1 in exactly D=4, but diers innitesimally far away. The result is that the two schemeswill dier eectively by nite renormalizations: For example, in divergent one-loops graphs the 1 =divergences will have the same coecient in the two schemes, but the nite remainders will dier by constants, since eectively the coupling has beenredened by a factor of 1+ O(). Thus, the same result can be achieved by modiying the counterterm to be proportional to 1 =+constant . Hence, the earlier version of dimensional regularization is called \minimal subtraction (MS)", while the modi- cation inspired by the volume of the sphere is called \modied minimal subtraction ( MS)". We now examine explicitly the dierence between the two schemes. As we can see from our previous example, momentum integrals from scaling various subsets of Schwinger parameters in a multiloop diagram will produce function factors of the form, with this new normalization, Y i[(D 2)]Li(niLiD 2) for some integers ni,w h e r eP iLi=L.I n e v e n d i m e n s i o n s D0(especially 4), we can use (z+1 ) =z(z) to write each Li-loop factor as a rational function of =(D0D)=2t i m e s [(1)]Li(1 +Li)(e )LieLi =1 where we have written only the dependence. Thus all 's cancel. This modica- tion allows some further simplication by eliminating extra terms arising at 2 loops involving(2) (but this is only 2=6; see excercise VIIA2.3). Similar results can be obtained by using (1 ) instead of (D 2). Another subtraction scheme, the \G scheme", is dened by normalizing momen- tum integration so that the coecient of the 1-loop massless propagator correction in3theory in 4 dimensions (see subsection VIIB4 below) is exactly 1=(without extra nite terms) times a power of1 2p2, or that up to a sign in higher even dimen- sions. (The normalization factor must be positive, and also nite and nonvanishing as!0.) As for MS, we can also pull out rational factors to get just the (1 + n)'s. The net eect of these two schemes, as compared to MS, is to modify  h, which appears asRdx=hin the classical action or  hRdpfor loop integrals, as MS : h!(D 2)ho r (1)h G: h!(1)D0=2 (2D 2)B(D 21;D 21)ho r(12) (1 +)[(1)]2h 360 VII. LOOPS (If we want to be picky, we can also normalize the former forms appropriately for D=D0, by including an extra factor of 1 =(D0 2)f o r MS and (D0 21)=(D02) for G.) This particular x for eliminating irrational numbers works only for those arising at one or two loops: In general, because subdivergences produce expressions of the form (1 + )=LatLloops, we encounter nite terms involving (L)a tLloops, which is irrational (worse than just 's,e's,p 2's, etc.) for odd L.S o , f o r e x a m p l e , (3) appears at 3 loops, and it can be shown that (3) (and higher (n)) can't always be canceled. The \momentum subtraction scheme (MOM)", rather than simplifying numer- ically, is designed to give a more physical interpretation of the coupling constants appearing in the action: It is dened so that they take their on-shell values. Thus, it is particularly suited to low-energy calculations, which involve an expansion aboutthe mass shell. For example, consider the quantum kinetic operator K+
K,a p - pearing in the quadratic part of the eective action. It depends on the momentum through one variable: p 2orp=, etc. We then can consider Taylor expanding it in this variable about its classical on-shell value. (For reasons to be explained later, this can be dangerous for massless elds, when it requires infrared regularization.) This is equivalent to expanding in powers of the classical kinetic operator Kitself. The MOM prescription is then to use subtraction terms Kto cancel the terms in the quantum correction
Kto the kinetic operator that are linear in K:
K=a+bK+O(K2))K=abK )KrenK+
K+K=K+O(K2))1 Kren=1 K+O(K0) The two renormalizations are related directly to the \wave function" and mass renor- malizations, one being proportional to the entire kinetic term, the other to the con-stant (mass) term. The result is that the renormalized propagator has the same pole and residue as the classical one. Note that the MOM scheme, unlike the others, does not introduce an independent mass scale : Only physical masses set the scale. This is a consequence of the fact that the MOM scheme is designed for studying low-energy (near-mass-shell) behavior, while the others are more suited for studying high-energy behavior. This will be important for our explicit calculations later, when we see that MOM is more useful forQED, which is better dened (and thus more useful), in terms of perturbation theory, at low energies, while QCD is better dened at high energies. More precisely, the on- shell values of QED masses and couplings are observed experimentally, whereas those A. GENERAL 361 of QCD are almost meaningless, since the corresponding particles are not observed as asymptotic states. On the other hand, in QCD the introduction of the arbitrary scaleallows the denition of a more physical mass scale, and its arbitrariness can actually improve the accuracy of perturbative calculations. 4. Optical theorem Writing the S-matrix as S=1+iT, unitarity can be written as 1=SyS=( 1iTy)(1 +iT)=1+i(TTy)+TyT)TyT=i(TyT) (Actually, the more useful statement is in terms of S=eiT, since thenTrepresents the connected graphs, but the result of the argument is the same.) Summation of a probability over nal states then yields the \optical theorem": X fjTfij2=X fhijTyjfihfjTjii=hijTyTjii=2ImTii Applying unitarity in terms of the cutting rules (subsection VC6), we see that this condition can be applied to Tiidiagram by diagram, using any combination of parts ofTfiandTyfithat t together to form the graph considered for Tii.( F o re x a m p l e , the probability coming from a tree graph with two nal states is the imaginary part of a one-loop graph with two intermediate states.) Separating out the momentum-conservation -function for a connected S-matrix element, we have nally T=X p T)X fX p jTfij2=2ImTii The simplest example of an experimental measurement of an interaction is a decay rate. (The only particle properties contained in the free Lagrangian are mass and spin.) At the tree (classical) level, this is given directly by a cubic (for decay into two particles) or higher-order term in the Lagrangian. More generally, by theoptical theorem the decay probability is given by the imaginary part of the prop agator correction, evaluated on shell. We then nd the total decay probability per unit time by dividing the probabil- i t yb yt h es p a t i a ld e n s i t y times the spatial volume times the time duration, and summing over nal states: dP dt=X fP VD=2ImTii ! 362 VII. LOOPS using the expressions for ,a n dPin terms ofjTfij2, given in subsection VC7. The optical theorem can be applied similarly for the total cross section: =2 (ImTii)(2)D=2 12 The decay rate for a particle is frame dependent, but we usually pick the rest frame for massive particles, where !=m. Alternatively, we can dene the total decay probability per unit proper time: !=mdt ds)dP ds=2ImTii m (wheresandtshould not be confused with the Mandelstam variables). For the massless case, we can use instead the parameter , as it appears in classical mechanics in the gauge v= 1, or as the classical value of the Schwinger parameter from the Landau equations: pa=dxa d)dP d=2ImTii Now the decay rate of a particle can also be associated with the imaginary part of the mass, since M=mir)j j2jeiMtj2=e2rt so the wave function for the particle at rest automatically includes a decay factor. The probability of decay, normalized by dividing by j j2,i st h u s dP dt=d(1j j2) j j2dt=2r The analogous statement in momentum space is found by Fourier transforming the propagator/wave function from time to energy: eiMt!1 EM=1 Em+1 Em(ir)1 Em+::: This expansion is in terms of the free propagator 1 =(Em) and the connected graphs ir. We now check that this result agrees with that obtained from the eective action. The quantum propagator has a pole at p2=M2for some complex constantM: lim p2!M2
=1 1 2(p2+M2) We have normalized the propagator at the pole by rescaling the eld; we also keep the real part of the mass the same as the classical value through renormalization of A. GENERAL 363 the mass term. (Only the real part can be renormalized consistently with unitarity.) The on-shell condition is then at physical (real) momentum p2=m2, so the kinetic operator on-shell is 1 2(p2+M2)=1 2m2+1 2(mir)2=1 2r2imr Remembering that \interaction" terms from contribute with a minus sign to am- plitudes, we then have dP dt=2ImTii m=2mr m=2r 5. Power counting We now consider why subtracting out divergences, as poles in D, can be imple- mented by giving singular D-dependence to the coupling constants. This is based on dimensional analysis, which tells us how divergent a graph is at large momenta: the\ultraviolet" (UV) divergence. (There are also infrared divergences, which occur for physical reasons, and do not require renormalization. They occur only for massless particles, and will be considered later.) Consider rst any 1-loop 1PI graph. In mo-mentum space, it has an integral over the loop momentum,R d Dp. It will diverge if the integrand (before introducing Schwinger parameters) goes as pDor slower to innite momentum (UV limit). In this limit we can ignore masses. If we dierentiatethis graph with respect to any of the external momenta, it will become more conver- gent, since the power of momenta in the integrand decreases. (The numerator of the integrand is a polynomial, while each factor in the denominator depends on the loopmomentum.) With enough derivatives, it becomes convergent. This means that the divergent part of the graph is a polynomial in the external momenta. Similar remarks apply to any 1PI graph, if we consider the divergence coming from letting all loop mo- menta go to innity, known as the \supercial divergence". Of course, the supercial divergence is also polynomial in the coupling constants, as is the graph as a whole;but the supercial divergence is also polynomial in the masses, since dierentiation with respect to them has the same eect as with respect to external momenta. We can determine several more properties of this local, but divergent, contribution to the eective action. First of all, it is Poincar e invariant and invariant under all global symmetries of the classical action, since the eective action is invariant for all values of D, so poles in D are also. (Consider, e.g., contour integration in D to pick out the pole.) If we use the background eld gauge (or consider only Abelian gaugeelds), then it is also gauge invariant. (However, possible exceptions are conformal invariance and invariances involving 1, since those are not invariances of the classical action for all D.) The other property we need is that the coecient of the divergence 364 VII. LOOPS is real. This follows from the fact that the S-matrix satises unitarity, as preserved manifestly by dimensional regularization. As discussed in the previous subsection, we have unitarity SyS=I;S=I+iT)i(TyT)=TyT As we saw in subsection VC6, this identity actually can be applied to a single graph, where the element of Ton the left-hand side of the equation is that graph, while on the right-hand side the summation over intermediate states gives a sum where eachterm divides the graph into two parts, one for Tand one forT y.W e t h e n s e e t h a t at any loop the imaginary part of a 1PI graph in Tis given by \sewing" together diagrams from lower loops. This means that any new divergence at any number of loops must be real, since sewing doesn't introduce new (UV) divergences: Sewed lines are on shell, and phase space for on-shell states is always nite. (The 3-momentumof each state is bounded by the energy, and each positive energy of an outgoing state is bounded by the total energy of the system.) Since Poincar e and gauge invariances, locality, and semiclassical unitarity were used as properties to determine the classical action, this suggests that the divergentterms in the eective action might all be of the form of terms already in the classical action. Such a property is called (perturbative) \renormalizability". When it holds, all innities can be absorbed by a redenition of the coupling constants (and masses)appearing in the classical action. This is physically important because all innities are dened only up to nite pieces: For example, in dimensional regularization, we saw in subsection VIIA3 that the D-dependent normalization of the classical actionis ambiguous, resulting in an ambiguity in the nite pieces left over after subtrac- tion of 1=terms. Since we now know that supercial divergences are local, we see that renormalization can produce arbitrary nite, local terms in the eective action,corresponding to the divergent terms. But if the divergent terms are all of the same form as in the classical action, all such nite terms can be absorbed by a redention of the coupling constants. On the other hand, if such nite terms did not already appear in the classical action, we would be forced to introduce them, to make the renormalization procedure unambiguous. (Of course, we could give an unambiguousprescription by denition, but from the point of view of another prescription this would be the same as including the extra terms in the classical action, and using the rst prescription to arbitrarily x the nonzero values of the couplings.) Thus, thecondition of renormalizability is necessary to prevent the appearance of an i nnite number of coupling constants, which would result in the loss of predictability. (If divergences require only a nite number of such couplings to be added, we simply A. GENERAL 365 include those, to obtain a renormalizable theory with a number of couplings that is nite, although larger than that with which we started.) Since dimensional analysis determines the form of the divergent terms of any mo- mentum integral, it also detemines which theories are renormalizable. By appropriaterescaling of elds by constants, write the classical action in a form where the deriva-tive parts of the kinetic terms have no dependence on any couplings. Then dene the couplings to be the coecients of the interaction terms. It is easy to see that the renor- malizable theories are the ones that include all terms which satisfy all the propertiesrequired of the classical action (including preservation of all appropriate symmetriesthat are manifestly preserved by dimensional regularization), where all couplings haveengineering dimensions that are nonnegative powers of mass: Consider rst the casewhere all couplings are dimensionless (and there are no masses, or at least we ignorethem at high energies for purposes of considering UV divergences). Then the theoryis renormalizable simply because there are no dimensionful parameters around, soany local term must be of the form of those originally in the classical action. If wenow introduce couplings with positive mass dimension, then perturbatively they canoccur only to nonnegative power in any diagram, so any divergence thus produced again has a coecient with nonnegative mass dimension. Since the elds themselves have positive mass dimension, there are only a nite number of such terms possible.On the other hand, if we were to allow couplings with negative dimension, then termswith arbitrarily high powers of such couplings would also allow arbitrarily high pow-ers of elds, and thus lead to nonrenormalizability. (By similar arguments, theorieswith only couplings of positive mass dimension, called \superrenormalizable", canhave divergences only to a certain nite number of loops.) In particular, in D=4 the derivative part of the kinetic term for bosons is of the formR d 4x@@ ,a n df o rf e r m i o n sRd4x @ , so bosonic elds have dimension 1 and fermions 3/2. That means that bosons can appear only quartically and fermions onlyquadratically. More specically, renormalizable theories can only have terms of theform ; 2;3;4;@;2@;@@ ; 2; @ ; 2 (where each c a nb ea n yb o s o nw i t ha n ys p i n ,a n de a c h any fermion). There can also be constant terms (eld-independent), which we always drop, since they don'tcontribute to perturbative amplitudes (after appropriate normalization). The terms,and their relations, are restricted by Lorentz, gauge, and internal symmetries. Thepotential for scalar elds must also be bounded from below, to allow the existence ofa vacuum (state with lowest energy); otherwise nothing would be stable, continually 366 VII. LOOPS decaying into states of lower energy (i.e, the energy of the scalars converting to other particles): Thus, 3terms for scalars requires also 4terms. Spin 1 can't couple minimally to spins >1. (One way to show this is to covari- antize the general eld equation of IIB1 to Sabrb+kra, and show the commutator algebra of this constraint, and +:::, doesn't close unless the spin 1o rt h ee x - ternal eld strength vanishes.) Furthermore, gauge invariance for spins >1p r e v e n t s them from having renormalizable gauge couplings in D=4: For example, we saw insubsection IIIA4 that spin-2 (gravity) couplings include terms of the form @@ . Renormalizability therefore restricts us to spins 0, 1/2, and 1. Using Poincar ea n d gauge invariance, the most general action is then of the form L=trn 1 8g2F2+ ir. .+1 4(r)2+V()+[1 2 (+mp 2) +h:c:]o where all group matrices are implicit: They may appear in all elds, in m, and even ing, which has independent values for the dierent factors of the Yang-Mills gauge group. Also, the matrices may dier for the same eld in dierent terms: A(inr) has dierent Yang-Mills representations on dierent scalar and spinor elds, and  appears with some matrix in its Yukawa coupling  . (Of course, all matrices must be chosen consistently with gauge and global invariances.) The potential V()i sn o higher than quartic. Note that the Higgs mechanism is required to give nonabelian gauge elds mass: A2is not gauge invariant, and ( r)2is the only way to dress it up with scalars in a renormalizable way. (For the Abelian case we can use St uckelberg elds, withr=@+mAT as in subsection IVA5. In the nonabelian case, introducing scalars by a gauge transformation, as for St uckelberg, results in a nonrenormalizable (eirei)2term.) If we ignore gauge invariance, the A2term produces unitary-gauge propagators with bad high-energy behavior (see subsection VIB3), which leads to thesame nonrenormalizable behavior in the absence of a Higgs mechanism. Excercise VIIA5.1 Show by power counting that interacting renormalizable theories with poten-tials that are bounded from below exist only in D 4. Show that for D 2 there are an innite number of possible renormalizable terms in the action. What are the kinds of renormalizable terms possible in D=3? Excercise VIIA5.2 Superrenormalizable theories aren't realistic, but they give oversimplied ex-amples of many quantum features of eld theory. aWhat theories are superrenormalizable in D=3? Show that the only super- renormalizable interaction in D=4 is (scalar)  3. A. GENERAL 367 bLet's do power counting (dimensional analysis) for 4D 3theory. Write the action in the form S=1 g2Z dx[1 4(m2)+3] sog2counts loops. ( !ggives the form where gcounts vertices.) What are the dimensionless terms
S=(g2)L1Z dx n for alln(including the vacuum bubbles n=0 ;o fc o u r s e , L0)? Since super- cial divergences are polynomial in everything (elds, momenta, couplings,masses), this gives the maximum number of loops L(n) for a supercial di- vergence to appear in an n-point 1PI amplitude. Make a similar analysis for 3D 4theory. cFind all the divergent 1PI diagrams in 4D 3theory. (Hint: There are 5, or 6 if we include the trivial 1-loop graph with no vertices.) Such global and local symmetry requirements can also be applied to the eective action. In background-eld gauges is gauge invariant (see subsection VIB8), whichrestricts the form of the eective potential, and even nonlocal terms. In QED chargeconjugation, in addition to switching 2-spinors of opposite charge, changes the signof the electromagnetic potential: Consequently, any pure- Aterm in must be even inA's (\Furry's theorem"). Such classical symmetries can be applied at the quantum level only in the absence of \anomalies", quantum violations (discussed in chapterVIII below). However, even the anomalies themselves are restricted by symmetries: Anomalies occur only in symmetries that can't be manifestly preserved by regular- ization, which means only conformal or axial symmetries. Thus, when the couplingsof gauge vectors are parity invariant, the axial anomaly (which violates parity bydenition) is irrelevant. 6. Infrared divergences Although ultraviolet (UV) divergences represent a serious problem, in the sense that they strongly restrict which theories can be useful, and require renormalization,infrared (IR) divergences are merely a consequence of poor semantics: The denitionof the S-matrix assumes the existence of well-dened one-particle asymptotic states.Unfortunately, these do not exist when massless particles are present, even in classicalmechanics: (1) Any particle can be accompanied by an arbitrary number of massless 368 VII. LOOPS particles with vanishing 4-momentum, and such a collection of particles can be indis- tinguishable from the lone particle if the (measured) quantum numbers are the same.(These are physical states, since p a=0!p2= 0.) (2) Any massless particle can be indistinguishable from an arbitrary number of massless particles, with the sametotal 4-momentum, and each with the same sign energy, if their 4-momenta are allproportional, since then they are all traveling in the same direction at the same speed. (This situation is not important for QED, since the photon can't decay directly into two photons.) Experimentally, because detectors have nite accuracy, in the rst case there can be such \soft" particles with total energy below some small upper limit, and in the sec- ond case there can be such \colinear" particles within some small angle of resolution. In principle this means we should change our denition of asymptotic states accord-ingly; in practice this is too complicated, but to any particular order in perturbationtheory only a nite number of such additional massless particles will couple. Theprocedure is then to (1) infrared regularize the S-matrix amplitudes (by dimensionalregularization, or introducing masses for all particles, or keeping massless particles o-shell); (2) calculate probabilities/cross sections, including contributions from soft and colinear particles, as a function of some upper limit on their energy/angle (rep-resenting the experimental accuracy); and (3) remove the regularization. No infraredrenormalization is necessary. (Examples will be given in later sections. Of course, fortotal cross sections, all energies and angles are integrated over anyway.) In general,such a procedure must be applied to both initial and nal states (the\Kinoshita-Lee- Nauenberg theorem"), but in QED (as opposed to QCD) it is sucient to treat only the nal ones in cases of physical interest. The reason why innities appear in cross sections if we ignore this careful pre- scription, and in S-matrix elements in any case, is the long range of forces mediated by massless particles. A cross section, although it represents a probability, is nor-malized in such a way that it is an area, representing the eective cross-sectionalarea of a particle being targeted by another particle. The range of the interactionsets the scale of this area; this is related to the mass of the particle mediating theinteraction. Since massless particles produce innite-range forces, the result is in- nite cross sections. One might expect that these innities would appear only in total cross sections, where the momenta of particles in the nal state are integrated over.However, by the optical theorem, this total cross section is given by the imaginarypart of an S-matrix amplitude, which thus must also have this innity. A. GENERAL 369 The fact that these infrared divergences are physical also follows from the fact that these kinematic situations occur in classical mechanics: In subsection VC8 we saw that physical singularities occur in S-matrices for momenta that are allowed classically. REFERENCES 1P.A.M. Dirac, Proc. Cambridge Phil. Soc. 30(1934) 150; W. Heisenberg, Z. Phys. 92(1934) 692; V. Weisskopf, Kon. Dan. Vid. Sel. Mat.-fys. Medd. XIV (1936) 1: charge renormalization. 2W. Pauli and M. Fierz, Nuo. Cim. 15(1938) 167; H. Kramers, Nuo. Cim. 15(1938) 134: mass renormalization. 3Dyson, loc. cit. (VC): renormalization to all loops. 4A. Salam, Phys. Rev. 82(1951) 217, 84(1951) 426: completed Dyson's proof. 5Bogoliubov and Shirkov, loc. cit. : general approach to renormalization. 6G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; C.G. Bollini and J.J. Giambiagi, Nuo. Cim. 12B (1972) 20; J.F. Ashmore, Lett. Nuo. Cimento 4(1972) 289; G.M. Cicuta and E. Montaldi, Lett. Nuo. Cim. 4(1972) 329: dimensional regularization. 7G. 't Hooft and M. Veltman, Diagrammar, in Particle interactions at very high energies , proc. 2nd Summer Institute on Particle Interactions at Very High Energies, Louvain,Belgium, Aug 12-25, 1973, eds. D. Speiser, F. Halzen, J. Weyers (Plenum, 1974) part B, p. 177; and in Under the spell of the gauge principle , ed. G. 't Hooft (World Scientic, 1994) p. 28:dimensional renormalization. 8Siegel, loc. cit. (VIC, ref. 10, next to last), W. Siegel, Phys. Lett. 94B (1980) 37; Gates, Grisaru, Ro cek, and Siegel, loc. cit. : regularization by dimensional reduction. 9J. Polchinski, Nucl. Phys. B231 (1984) 269: simplest proof of renormalization; uses renormalization group, doesn't use geometry of graphs (\forests" or \skeletons", etc.) or \Weinberg's theorem"; not yet simply gener-alized to gauge theories. 10Feynman, loc. cit. (VB, ref. 4): Feynman parameters. 11E.T. Whittaker and G.N. Watson, A course of modern analysis ,4 t he d .( C a m b r i d g e University, 1927) p. 235: detailed discussion of functions. 12A.A. Vladimirov, Theor. Math. Phys. 36(1978) 732; W.A. Bardeen, A.J. Buras, D.W. Duke, and T. Muta, Phys. Rev. D18 (1978) 3998: MSscheme. 370 VII. LOOPS 13K.G. Chetyrkin, A.L. Kataev, and F.V. Tkachov, Nucl. Phys. B174 (1980) 345: G scheme. 14E. Feenberg, Phys. Rev. 40(1932) 40; N. Bohr, R.E. Peierls, and G. Placzek, 1947 manuscript, in Niels Bohr collected works , v.9, ed. S.R. Peierls (North-Holland, 1986) p. 487:optical theorem. 15W.H. Furry, Phys. Rev. 51(1937) 125. 16F. Bloch and A. Nordsieck, Phys. Rev. 52(1937) 54; D.R. Yennie, S.C. Frautschi, and H. Suura, Ann. Phys. 13(1961) 379: infrared divergences. 17T. Kinoshita, J. Math. Phys. 3(1962) 650; T.D. Lee and M. Nauenberg, Phys. Rev. 133B (1964) 1549. 18R. Gastmans and R. Meuldermans, Nucl. Phys. B63 (1973) 277; W.J. Marciano and A. Sirlin, Nucl. Phys. B88 (1975) 86: earliest applications of dimensional regularization to infrared divergences. B. EXAMPLES 371 ::::::::::::::::::::::::::: ::::::::::::::::::::::::::: ::::::::::::::::::::::::::: B. EXAMPLES ::::::::::::::::::::::::::: We now give some explicit examples of the evaluation of S-matrices and contri- butions to the eective action | momentum integration, regularization, and renor-malization | and some examples of their application. 1. Tadpoles . . . The simplest examples of dimensional regularization are one-loop \tadpoles", graphs with only one external line. By the Schwinger parameter method describedin subsection VIIA2, we nd A 1(x;m2)=Z dk eik
x 1 1 2(k2+m2)=Z1 0d D=2e(m2+x2=)=2 Further evaluation requires Taylor expansion in x(which we'll need anyway to eval- uate a specic integral of k:::k= (k2+m2)): A1=1X n=01 n!(1 2x2)n(1D 2n) (1 2m2)1D=2n The mass dependence, as well as the argument of the function, are as expected by dimensional analysis:R dDkk2n=k2is ultraviolet divergent (large k)f o rD2(1n), and infrared divergent (small k) in the limit m!0f o rD2(1n). The ultraviolet divergence is re ected in ( z), which has poles at the nonpositive integers. To analyze the massless case, we evaluate the integral for D< 2(1n)a n d m> 0, where it is nite and well-dened, analytically continue to the region ReD> 2(1n) (but not exactly at the points where Dis an even integer), take the limit m! 0 there, and nally analytically continue this vanishing result to all D. Therefore, all massless tadpoles can be taken to vanish in dimensional regularization: Z dkka:::kb 1 2k2=0 or more generally Z dkka:::kb (1 2k2)a=0 372 VII. LOOPS This includes negative a, particularly integrals of polynomials of momenta. Such an integral can result from \measure factors", as discussed in subsections VA2 and VC1: For example, if an auxiliary eld appears in the action with its quadratic term multiplied by a function of other elds, then functionally integrating it out of the action results in a functional determinant (in addition to replacing it in the classicalaction by the solution to its eld equation). This is represented in terms of Feynman graphs as one-loop diagrams whose propagators are all those of the aux iliary eld, namely 1. The result is then regularized as Z dx1(0)!Z dk1=0 consistent with the fact that such factors would cancel corresponding factors we should include in the functional integration measure. (In other words, since we can always arrange to have all (0) factors cancel, we ignore them.) On the other hand, massive tadpoles contribute both divergent and nite pieces under minimal subtraction: For example, for D=42, A 1(0;m2)=Z dk1 1 2(k2+m2)= ( 1D 2)(1 2m2)D=21 =1 2m2f1 +[ +1ln(1 2m2)]g (see excercise VIIA2.3b), using A=el n(A).T h e can be killed by using an MS or G scheme (see subsection VIIA3): At 1-loop order any version of those schemes has the eect of just canceling the (but dierences appear at 2 loops: see subsection VIIB7 below). To include the dependence of the coupling, we just replace everywhere (see also subsection VIIA3) ln(1 2m2)!lnm2 2 (and similarly for any momentum factors such as ln(1 2p2)t h a tm i g h ta p p e a rm o r e generally); eectively we are using units1 22= 1. Note that we are not allowed to Taylor expand in m: Doing so before integration would give an incorrect result; after integration it's impossible. Similar remarks apply to the exponential eik
xinA1if we interpret it as the denition by Fourier transformation of the propagator in position space. Excercise VIIB1.1 Find the 2D massless propagator in position space by Fourier transformation. (But don't Taylor expand in x.) Note that this Fourier transform is innite, and requires \renormalization" (of a constant of integration). Compare this B. EXAMPLES 373 with the result obtained by solving the integral form of the Klein-Gordon (Laplace) equation (i.e., Gauss' law in D=2). Two-loop tadpole integrals are not much more dicult if one line is massive, or two are massive with the same mass. (Again, tadpoles with only massless lines can betaken to vanish in dimensional regularization.) If two propagators are massless, then they can be treated rst as a one-loop propagator graph: By dimensional analysis, the result of that one-loop subintegral must be a power of the momentum squared. (The explicit result will be calculated in the following subsection.) We therefore consider more general one-loop tadpole integrals with more complicated propagators that mayresult from subintegrations in a higher-loop graph. For example, we consider ^A 1(a;x;m2)=Z dk eik
x (a) [1 2(k2+m2)]a Using the denition of the function, we can write (a) [1 2(k2+m2)]a=Z1 0d a1e(k2+m2)=2 Performing the resultant Gaussian momentum integration and Taylor expanding in x, we easily nd ^A1(a;x;m2)=1X n=01 n!(1 2x2)n(aD 2n) (1 2m2)aD=2n A more complicated example is A1(a;b;m2)=Z dk(a) (1 2k2)a(b) [1 2(k2+m2)]b =Z1 0d1d2a1 1b1 2Z dk e[1k2+2(k2+m2)]=2 =Z1 0d1d2a1 1b1 2(1+2)D=2e2m2=2 We then introduce a scaling parameter (also described in VIIA2), scaling i=i in the insertion 1=Z1 0d (12)=Z1 0d 1(112) 374 VII. LOOPS and integrating the over2to get2=11. This gives (with 1=) A1(a;b;m2)=Z1 0d a1(1)b1Z1 0d a+bD=21e(1)m2=2 =(a+bD 2) (1 2m2)a+bD=2B(a;D 2a) When two of the propagators in the two-loop tadpole graph have the same non- vanishing mass, we consider directly the two-loop integral A1;2(a;b;c;m2)=Z dk1dk2(a) [1 2(k1+k2)2]a(b) [1 2(k2 1+m2)]b(c) [1 2(k2 2+m2)]c (This integral also represents the physically less interesting 2-loop \vacuum bubble": no external lines, and thus eld independent.) Introducing the Schwinger parameters and performing the momentum integration, we nd Z1 0d3a1 1b1 2c1 3[23+1(2+3)]D=2e(2+3)m2=2 Since1does not appear in the exponential we integrate over it rst directly, using the second integral form for the Beta function, from excercise VIIA2.2c. Then 2and 3can be handled by introducing a scaling parameter for them only, leading to the previous types of integrals. The result is then A1;2(a;b;c;m2)=(a+b+cD) (1 2m2)a+b+cDB(a+bD 2;a+cD 2)B(a;D 2a) 2. Eective potential A propagator in an external eld represents a certain class of Feynman tree dia- grams. Thus, some tree graphs can be described by quantum mechanics. (In principle this means we can start from classical mechanics and rst-quantize, by either operator or path-integral methods. However, as we'll see in chapter XII, in practice we save some eort if we start directly with the quantum mechanics.) If we take the ends of such a propagator and sew them together, we can describe arbitrary 1PI 1-loop graphs by the background eld method. While tree graphs describe classical eld theory, one-loop graphs contain many of the important quantum properties, partly because they are the lowest-order quantum correction, and partly because they are associated with the functional determinant part of the (second-quantized) path inte- gral. (In terms of the exponent, classical is the only negative power in  h,1 - l o o pi s h-independent, and higher loops are positive powers.) B. EXAMPLES 375 In quantum mechanics, the expansion in  his an expansion in derivatives (since it appears only as pa=ih@a). In terms of the contribution of one-loop graphs to the eective action, this means an expansion in the number of derivatives acting on the elds. This denition can be applied in general in quantum eld theory, without reference to quantum mechanics. However, the simplest one-loop calculations of thisexpansion are most easily expressed in quantum mechanical terms. In practice, this means expanding the external elds in xabout some xed point, expanding the exponentiated (by a Schwinger parameter) propagator about the part Gaussian in p andx, and using any of the usual methods to exactly evaluate the matrix element of a polynomial times a Gaussian. Since we generally want arbitrary orders in a eld and some of its lower derivatives for this method to have any advantage over the usual diagrammatic methods, in thisapproach one generally cuts o the expansion at the approximation that gives just the Gaussian. This means we can keep up to two derivatives of an external scalar, but only a constant eld strength for an external gauge vector. (See subsection VIB1.)The simplest, and most useful, example is a constant scalar eld. The part of the eective action that consists of only scalars without derivatives is called the \eective potential", since it generalizes the potential term of the classical action. This potentialdetermines the quantum corrections to spontaneous symmetry breaking and the Higgs eect, and this is important for describing mass generation for all spins. Consider a complex scalar running around a loop, under the in uence of an ex- ternal real scalar. The Lagrangian is L= *[ 1 2(+m2)+] +L where the form of Lwon't be important for calculating the loop. A constant external scalar eld is eectively the same as a mass term, modifying m2!m2+2. Thus the eective potential in this case can be evaluated by summing tadpoles: V=1X n=11 n(1)nnZ dp[1 2(p2+m2)]n for our complex scalar; for a real scalar running around the loop there would be an extra factor of 1/2. We can integrate before summing: V=1X n=1(1)nn(nD 2) n!(1 2m2)n+D=2 Using the identities (from Taylor expansion in a=b,a n d (z+1 )=z(z)) (a+b)x=1X n=0x n anbxn;x n =(x+1 ) n!(x+1n)=(1)n(nx) n!(x) 376 VII. LOOPS we have V=(D 2) (1 2m2+)D=2(1 2m2)D=2 We can also integrate after summing: Using the identities ln(a+b)lnb=Za 0du u+b=Za 0du1 b1X n=0(1)nx bn =1X n=11 n(1)nanbn Za 0du u+b=Za 0duZ1 0d e(x+b)=Z1 0d  e(a+b)eb
we have V=Z1 0d Z dp e[+(p2+m2)=2]e(p2+m2)=2 which gives the same result. For D=4, we nd (after subtracting divergent counterterms, and some correspond- ing nite pieces, corresponding to a MOM type of subtraction) V=1 2(1 2m2+)2ln 1+2 m2 Since this modies the classical potential, it demonstrates that quantum eects can generate spontaneous symmetry breaking where there was none classically, or vice versa (the \Coleman-Weinberg mechanism"). For more complicated cases we need a more general procedure: The basic idea is that any Gaussian integral gives a (inverse) determinant, of which we must take(minus) the logarithm for the eective action, and we use lndet =tr ln . (The trace includes integration over xorp.) After subtracting out the eld-independent part (vacuum bubble), this gives an expression as above: For a general kinetic operatorH=H 0+:::(generallyH0=1 2(p2+m2)), we want = tr ln (H1)tr ln (H1 0) =Z1 0d Z dxhxjeHeH0jxi H(andH0) is now treated as an operator, in terms of the coordinate operator X and momentum operator P,a n dXjxi=xjxi. External elds depend on X, but are Taylor expanded about x: e.g., (X)=(x)+(Xx)
@(x)+::: We then can use translation invariance to write hxjeH[P;Xx;(x)]jxi=h0jeH[P;X; (x)]j0i WhenHis quadratic in PandX, we can use (see excercise VA2.4) hxjeHjyi=s det@2(S) @x@yeS whereSis the classical \action" corresponding to the \Hamiltonian" H. (Further examples will be given in subsection VIIIB1.) B. EXAMPLES 377 3. Dimensional transmutation The 2D version of the CP(n) model described in subsection IVA2 is an interesting model in that it demonstrates generation of bound states at the one-loop level. ItsLagrangian is: L= 1 2jrj2+ (jj21 g2) wheregis now dimensionless. For the eective potential for the Lagrange multiplier f r o maloop, we nd (modifying the calculation of the previous subsection for D=2) V1= ln 1 22 1 after including the renormalization mass scale to make the argument of the loga- rithm dimensionless, and the coupling dimensionless in all dimensions. Now the coupling can be absorbed into the denition of this scale: Adding to the classical term V0==g2, the total eective potential for  up to one loop is V=1 g2+ln 1 22 1 = ln 1 2M2 1 whereMis the \renormalization group invariant mass scale": M2=2e1=g2 Since this was the only place the coupling gappeared in the action, the mass scale M has now replaced it completely. This replacement of a dimensionless coupling ( g)w i t h ad i m e n s i o n f u lo n e( M) is called \dimensional transmutation". It is also a common feature of quantum high-energy behavior (see below); its importance at low energiesdepends on whether the classical theory already has dimensionful parameters (like masses). Varying the eective potential to nd the minimum, which we identify as the (quantum) vacuum value of the eld , lnhi 1 2M2 =0)h i=1 2M2 Because  has a vacuum value, now has a mass (as seen by expanding  about its vacuum value). Furthermore, since  now has more than just linear terms in the eective action, it is no longer a Lagrange multiplier. In fact, by calculating amassiveloop with two external 's, we see that  is now a massive physical scalar also. Without a Lagrange multiplier, is now unconstrained, so it has an additional physical degree of freedom. This leads to a restoration of the spontaneously broken 378 VII. LOOPS U(N) symmetry. This is related to gaining mass, since we no longer have Goldstone bosons associated with the symmetry breaking. Finally, if we calculate a massive  loop with two external gauge vectors, we see that at low energies there is an F2term, soAis now a physical, massive vector instead of an auxiliary eld. 4. Massless propagators For the massless one-loop propagator corrections, we also introduce a scaling parameter to convert to Feynman parameters (see the examples of subsection VIIB1, or the general method in subsection VIIA2), with the result A2(x;p2)=Z dk eik
x 1 1 2(k+1 2p)21 2(k1 2p)2 =Z1 0d 1D=2Z1 0d1d2(112) expf1 8p2[1(12)2]i1 2(12)p
x11 2x2g Making the change of variables 1=1 2(1 +); 2=1 2(1) the amplitude takes the form A2=Z1 0d1 2(eip
x=2+eip
x=2)Z1 0d 1D=2exp[1 8(12)p211 2x2] The integrals can be simplied if we make use of gauge invariance: For example, the electromagnetic current for a complex scalar is of the form *$ @, so the gauge eld couples to the dierence of the momenta of the two scalar lines, which is 2 kfor the above as applied to the scalar-loop correction to the photon propagator. On theother hand gauge invariance, or equivalently current conservation, says that such a vertex factor should give a vanishing contribution when contracted with the external momentum, which is pin that case. Checking this explicitly, we do in fact nd Z dkk
p 1 2(k+1 2p)21 2(k1 2p)2=Z dk1 1 2(k1 2p)21 1 2(k+1 2p)2 =0 (even with an arbitrary additional polynomial factor in the numerator), using the facts that the integral of the sum is the sum of the integrals when regularized, and B. EXAMPLES 379 that massless tadpoles vanish. (This also tells us that replacing the numerator k
p withk2+1 4p2gives 0. Furthermore, without an extra numerator factor the integral vanishes by antisymmetry under k!k.) Thus, ifxis proportional to pinA2,t h e only contribution is from the x= 0 term in the Taylor expansion. This implies that the dependence on xis only through the combination u=(p
x)2p2x2 so we can evaluate the integral by either of the substitutions x2!0;p
x!puo rp
x!0;x2!u=p2 We'll consider now the latter choice. (The former gives the same result: See the excercise below.) Again, since we need to Taylor expand in xanyway to nd the result for a particular numerator, we expand and perform the integration: A2=1X n=01 n!u 2p2n (1 8p2)n+D=22(2D 2n)Z1 0d(12)n+D=22 Performing the change of variables 2= to convert the remaining integral to a Beta function, and using the identities (1 2)=p; (z)(1z)=c s c (z) (see the excercises in subsection VIIA2), the nal result is A2=1 23=2csc(D 2)(1 8p2)D=221X n=01 n!(n+D 21 2)f1 16[p2x2(p
x)2]gn From thecscfactor we see the integral is divergent for all even D: These are ultraviolet divergences for D4 and infrared ones for D4; dimensional regularization does not carefully distinguish between the two, although the dierence can usually be toldby examining momentum dependence (here from the exponent D=22). Also notice that the two can be mixed up by the conversion to Feynman parameters. Excercise VIIB4.1 Evaluate the general massless one-loop propagator correction using x 2!0, p
x!pu. aShow it gives the same result as p
x!0,x2!u=p2by using the and B identities in subsection VIIA2. bShow it can also be written as (for convenience of expansion about D=4) A2=(1 2p2)D=22(D 21)(2D 2)1X n=0(n+D 21) n!(2n+D2)f1 4[p2x2(p
x)2]gn 380 VII. LOOPS As discussed in subsection VIIB1, sometimes certain subdiagrams of higher-loop diagrams can be evaluated explicitly, particularly propagator corrections that them- selves involve only massless propagators. Furthermore, such a formula might be used recursively in appropriate diagrams. For example, a higher-loop diagram that is itself a propagator correction might reduce, as a nal integration, to something of the form Z dk(a) (1 2k2)a(b) [1 2(k+p)2]b=(a+bD 2) (1 2p2)a+bD=2B(D 2a;D 2b) again using the above methods, nding similar integrals to the previous. Excercise VIIB4.2 Let's examine this integral more carefully. aEvaluate it in two dierent ways: rst, by the method used above; second, by Fourier transforming each factor using Z dk eik
x(a) (1 2k2)a=(D 2a) (1 2x2)D=2a (derive this also) and its inverse, simply multiplying the resulting factors in xspace, and inverse transforming. bShow that the MS scheme cancels 's and(2)'s in iterated massless propa- gator corrections to all orders in by examining (D 2)Z dk1 (1 2k2)n1+L1[1 2(k+p)2]n2+L2 whereLiare the numbers of loops in the propagator subgraphs (show this by dimensional analysis) and niare other integers. Show the G scheme does the same. Excercise VIIB4.3 Calculate the \phase space" for nmassless particles VP=Z"nYdD1pi (2)D=21!i# (2)D=2D pnX pi! wherepis the total momentum of the nparticles, by using the optical theorem: aConsider the scalar graph with nmassless propagators connecting 2 vertices. Show, both by induction in the number ( n1) of loops, and by Fourier trans- formation (as in the previous problem), that this graph (for distinguishable particles) gives [(D 21)]n [n(D 21)][n(n1)D 2] (1 2p2)n(n1)D=2 B. EXAMPLES 381 bWick rotate back to Minkowski space ( p2<0) and take the imaginary part to obtain the result for continuous real D> 2 VP=2[(D 21)]n [n(D 21)][(n1)(D 21)](1 2p2)n+(n1)D=2 which simplies in D=4 to VP=21 (n1)!(n2)!(1 2p2)n2 (Hint: (1 2p2i)r=(1 2p2)reir.) 5. Massive propagators Another way to distinguish infrared divergences is by introducing masses (being careful not to break any invariances, or restoring them in the massless limit). Forexample, we again evaluate the one-loop propagator correction, without numeratorfactors, but with dierent masses on the internal propagators. By the same steps asbefore, the Feynman parameter integral is ^A 2(p2;m2 1;m22)=Z dk1 1 2[(k+1 2p)2+m2 1]1 2[(k1 2p)2+m2 2] = ( 2D 2)1 2Z1 1dBD=22;B=1 8p2(12)+1 4(m2 1+m2 2)+1 4(m2 1m2 2) Now theintegral is harder for all D, but the masses eliminate the IR divergences (and the UV divergences are already explicit in the ), so we immediately expandaboutD=42: ^A 2()1 2Z1 1d(1lnB) We then use integration by parts Z1 1d lnB=(l nB)j1 1Z1 1d d dlnB B=a2+b+c=a(+)();=bp b24ac 2a=m2 1m2 2212 p2 )d dlnB= ++ =2++ ++  in terms of 12(s) of subsection IA4 for s=p2. Note that in Euclidean space 212=q (p2+m2 1m2 2)2+4m2 2p2=q (p2+m2 2m2 1)2+4m2 1p2p2+jm2 1m2 2j 382 VII. LOOPS )1 where the strict inequality holds for both masses nonvanishing. The integrals then take the simple form Z1 1d+ ++  =+ln+1 ++1 +ln1 +1 Putting it all together, ^A2= ( 1+)1 ln(1 2m1m2)+2+1 2+ln+1 ++1 +1 2ln1 +1 (We can cancel the (1 + ) by nonminimal subtraction.) By analytic continuation from Euclidean space, taking p2from positive to negative along the real axis, we see there is no ambiguity at p2=0o r(m1m2)2,a n d ^A2remains real until we reach p2=(m1+m2)2, where it gets an imaginary part (whose sign is determined by (m1+m2)2!(m1+m2)2i), corresponding to the possibility of real 2-particle intermediate states. Excercise VIIB5.1 Let's consider some special cases: aShow for equal masses m1=m2=mthat this result simplies to ^A2(p2;m2;m2)= ( 1+)1 ln(1 2m2)+2+l n1 +1 =s p2+4m2 p2 bConsider the case with one internal particle massless, m1=m,m2=0 ,a n d nd ^A2(p2;m2;0) = (1 + )1 ln(1 2m2)+2p2+m2 p2lnp2+m2 m2 cShow both these results agree with the previously obtained massless result in the limitm!0. However, note that both these cases, unlike the massless case, are IR convergent at p2=(m1+m2)2. Excercise VIIB5.2 Find the phase space for 2 massive particles, again using the optical theorem(as in excercise VIIB4.3). The calculation is easier if one takes the imaginary part before performing the Feynman parameter integration: Show the result is then V P=1 (D 21)Z +d(B)D=22 B. EXAMPLES 383 which simplies in D=4 to VP=212 1 2p2 In particular, show from the explicit parameter integral expression for the propagator that the only cut is at p2(m1+m2)2, as expected from the optical theorem. Evaluate in general D for the massless case and show it agrees with excercise VIIB4.3. In subsection VIIA3 we considered the application of the MOM subtraction scheme to propagator corrections. We assumed the propagator corrections were Tay- lor expandable in the classical kinetic operator. From the above explicit expression for the 1-loop correction in scalar theories, we see this is possible except near the branch point at p2=(m1+m2)2, i.e., when the external particle (whose propaga- tor we're correcting) has a mass equal to the sum of the internal ones. To analyze this more carefully, let's recalculate the propagator correction, performing the Taylor expansion before evaluating the integrals. We consider the case with one vanishing mass,m1=m,m2= 0, to generate an IR divergence. Assuming the external mass is alsom, we expand around the branch point in p2+m2. The Feynman parameter integral is then, to linear order in p2+m2,i nt e r m so f =1 2(1 +), ^A2(p2;m2;0) = ()Z1 0d 1 2m22 1+1 p2+m2 m2 (1 +)(1 2m2)Z1 0d1 2(1)12p2+m2 m2 = ( 1+)(1 2m2)1 1 121 21 12p2+m2 m2 (1 +)(1 2m2)1 UV+2 +1 21 IR+2p2+m2 m2 where we have distinguished the UV divergence (in the integral for 0) from the IR one (in the integral for 0). After including the (1 22)in the coupling, the (1 2m2)converts each 1 =into a 1=ln(m2=2). (Of course, we can choose =m for convenience.) Note that this infrared divergence was a consequence of trying to Taylor expand about a branch point due to a massless particle. Excercise VIIB5.3 Do MOM subtraction for external mass M=m1+m2,w i t h neither internal mass vanishing, and show there is no divergence other than the UV divergence of the minimal scheme. 384 VII. LOOPS Later, we will encounter propagator corrections in gauge theories with massive internal lines, and with various numerators. Here, we examine these purely from the point of view of the integrals. First, consider Aa=Z dkka 1 2[(k1 2p)2+m2 1]1 2[(k+1 2p)2+m2 2] Sincepis the only external momentum for a propagator, by Lorentz invariance we have Aa=pa1 p2p
A so it is sucient to evaluate the integral of p
A. In analogy with the earlier massless expression, we look at 1 1 2[(k1 2p)2+m2 1]1 1 2[(k+1 2p)2+m2 2]=k
p+1 2(m2 2m2 1) 1 2[(k1 2p)2+m2 1]1 2[(k+1 2p)2+m2 2] from which we nd Aa=pam2 1m2 2 2p2[^A2(p2;m2 1;m22)^A2(0;m2 1;m22)] in terms of our result ^A2above for the integral without numerator. As a more complicated (but important) example, we examine Aab=Z dkkakb 1 2[(k+1 2p)2+m2]1 2[(k1 2p)2+m2] Following our procedure of the previous example, we note Z dk(p
k)k 1 2[(k+1 2p)2+m2]1 2[(k1 2p)2+m2] =Z dkk 1 2[(k1 2p)2+m2]k 1 2[(k+1 2p)2+m2] =Z dkk+1 2p 1 2(k2+m2)k1 2p 1 2(k2+m2)=pZ dk1 1 2(k2+m2) Thus transversality again determines the amplitude in terms of a scalar: ^Aab=Z dkkakb 1 2[(k+1 2p)2+m2]1 2[(k1 2p)2+m2]ab 1 2(k2+m2) =(abp2papb)A(p2;m2) (This amplitude actually will be more useful than Aab.) We also have the identity Z dk1 2(k2+m2)+1 8p2 1 2[(k+1 2p)2+m2]1 2[(k1 2p)2+m2] B. EXAMPLES 385 =1 2Z dk1 1 2[(k1 2p)2+m2]+1 1 2[(k+1 2p)2+m2]=Z dk1 1 2(k2+m2) Taking the trace of the previous expression, (D1)p2A(p2)=Z dkk2 1 2[(k+1 2p)2+m2]1 2[(k1 2p)2+m2]D 1 2(k2+m2) =(1 4p2+m2)Z dk1 1 2[(k+1 2p)2+m2]1 2[(k1 2p)2+m2](D2)Z dk1 1 2(k2+m2) =(1 4p2+m2)^A2(p2;m2;m2)(D2)A1(0;m2) in terms of the 3propagator and tadpole graphs evaluated earlier. This result can be reorganized if we make use of the p=0c a s e : 0=m2^A2(0;m2;m2)(D2)A1(0;m2) (which also follows easily from the earlier explicit expression for ^A1(a;0;m2)). We then nd A=1 4(D1)^A2(p2;m2;m2)1 D1m2^A2(p2;m2;m2)^A2(0;m2;m2) p2 Excercise VIIB5.4 Check that these results are consistent in the massless limit with the expres-sions obtained in the previous subsection, by relating the rst two terms in A 2(x;p2) for arbitrary D. Excercise VIIB5.5 Calculate the one-loop propagator corrections for  and Ain the 2D CP(n) model. 6. Renormalization group An interesting, useful, and simple application of the propagator correction is to study the high-energy behavior of coupling constants. For example, we have seen that, by a change in normalization of gauge elds A!A=g, gauge couplings can be moved from the covariant derivative to the kinetic term: r=@+igA!@+iA,L0=1 8(@A+ igAA )2!1 8g2(@A+iAA)2. Thus, quantum corrections to gauge couplings can be found from just the propagator (kinetic-operator) correction. A simpler example is ascalar eld; a  4self-interaction has a dimensionless coupling in D=4, like Yang-Mills. However, unlike Yang-Mills, this model has no cubic coupling, and thus no 1-loop propagator correction. Furthermore, in Yang-M ills the one-loop prop agator correction 386 VII. LOOPS contribution to the eective action gives a multiloop contribution to the propagator itself, from the expansion of 1 =(K+A). This corresponds to the graph consisting of a long string of these corrections connected by free propagators. There is a 1-loop4-point correction in  4theory, and this graph resembles a propagator correction, but with two external lines at each vertex instead of one. Such corrections can also be strung together, resembling the Yang-Mills string, but with no free prop agators inserted. Since all the intermediate states in this graph are 2-particle, it is 1PI, so theeect of this string is not contained in just the 1-loop eective action, even though it is an iteration of a 1-loop eect. This diculty can be avoided by introducing the  4interaction through an aux- iliary eld, just as it appears in supersymmetric theories (see subsection IVC2): L=1 2(1 2+)1 2g2 where we have neglected the mass term since we will be concentrating on the high- energy behavior. Here the coupling is introduced through the auxiliary-eld \kinetic" term. The diagrams just discussed now appear through the 1-loop correction to theauxiliary-eld prop agator: Since its free propagator is just a constant, it can be contracted to a point in these multiloop diagrams. The denition of 1PI graphs has now changed, since we can now cut auxiliary-eld prop agators, which would not exist in the usual  4form of the action. This modication of the eective action simplies the analysis of quantum corrections to the coupling, as well as making it more analogous to gauge theories. Note in particular the change in interpretationalready at the tree level: We have used the conventional normalization of 1/n! forfactors of nin the potential, since canceling factors of n! arise upon functional dierentiation. However, the result of eliminating from the classical action produces 1 84instead of1 244. The reason is that in the diagrams with there are 3 graphs contributing to the 4- -point tree, corresponding to propagators in the s,t,a n du \channels". (See subsection VC4.) Although this is a trivial distinction for the trees, this is not the case for the loops, where the propagator string consists of pairs of  particles running in one of these three channels. The contribution to the 1-loop eective action for is then given by the above calculations, after including the factors of 1/2 for symmetries of the internal andexternal lines, and the usual 1 for the eective action: L 0=1 4()B(1;1)(1 2) B. EXAMPLES 387 where as usual =2D=2. Expressing the Beta function in terms of the Gamma function, and expanding as in previous subsections, ()B(1;1)(1 2)1 +[ +2ln(1 2)] Renormalizing away the constant pieces, we nd for the classical action plus this part of the 1-loop eective action L+L0=1 2(1 2+)1 2h 1 g1 2ln 2i =1 2(1 2+)+1 4ln M2
 where the renormalization group invariant mass scale Mis given by M2=2e2=g Thus, the constant coupling 1 =ghas been replaced by an eective \running coupling" 1 2ln(p2=M2), with energy dependence set by the scale M. (This is sometimes called the \renormalization group", the group being related to scale invariance, which is broken by the introduction of the mass scale M.) We saw the same dimensional transmutation occuring in the eective potential in massless theories in subsection VIIB3. The form is similar because both are re- lated to the appearance of the renormalization mass scale from the breaking of scale invariance by quantum corrections, at either low or high energy: In both casesdimensional transmutation comes from a nite ln  2term arising from the innite renormalization. The dierence is that in the eective potential case we ignore higherderivatives, so the  2must appear in a ratio to scalar elds, while in the high energy case we look at just the propagator correction, so it appears in the combination 2=p2. (More complicated combinations will appear in more general amplitudes.) Excercise VIIB6.1 Generalize this model to include internal symmetry: Write an analog to thescalar analog to QCD discussed in subsection VC9, where the \quark" now carries color and avor indices, while the \gluon" (classically auxiliary) carries just color. Find M, especially its dependence on the numbers nof colors and mof avors. Write the same model with the gluons replaced by \mesons" carrying just avor indices (so that classical elimination of theauxiliary elds yields the same action), and repeat the calculation. What arethe dierent approximation schemes relevant to the two approaches? 388 VII. LOOPS 7. Overlapping divergences We now perform some 2-loop renormalizations. Our rst example is part of the propagator correction in 4theory. By restricting ourselves to the mass renormal- ization (coecient of the mass term), we need evaluate the graph only at vanishingexternal momentum. (It is then equivalent to a vacuum bubble in  3theory, or a tadpole graph in the mixed theory.) Furthermore, we consider the case where some of the elds are massless. In such a theory, we encounter (a special case of) the 2-loop graph of subsection VIIB1, where 1 propagator is massless and 2 are massive. Ex- panding in , and keeping only the divergent terms (1 =2and 1=), we nd (including a symmetry factor of 1/2 for the 2 massive scalar lines for real scalars) T2=1 2(3D) (1 2m2)3DB(2D 2;2D 2)B(1;D 21) =(1 2m2)12 2(1)(12)[()]2 1 4m2[()]2[13+2ln(1 2m2)] To this we need to add the counterterm graph, coming from inserting into the 1-loop massive tadpole T1(with 2 external lines) the counterterm
4(for renormaliz- ing the4term) from the 1-loop divergence in the 4-point graph with 1 massive and 1 massless propagator. (Since the massless tadpole vanishes in dimensional regular- ization, we need not consider the counterterm from the 4-point graph with 2 massive propagators.) From section VIIB5, we use the corresponding integral for a 1-loop propagator correction A,w h i c hi s A= ()+finite)
4=() We use a \modied minimal subtraction", using the ( ) as the subtraction instead of just the 1 =part of ()1= . B. EXAMPLES 389 The 1-loop massive tadpole without coupling is T1=(1D 2) (1 2m2)1D 2=(1 2m2)1 1()1 2m2()[1 +ln(1 2m2)] Combining these results, the divergent part of the 2-loop propagator correction, with 1-loop coupling counterterm contributions included, is T2+
4T1=1 4m2[()]2[13+2ln(1 2m2)+2+22ln(1 2m2)] =[ ()]2(1)1 4m2 Thus, thelnm2divergences cancel, as expected. (Divergences must be polynomial in masses as well as couplings.) If we had kept momentum dependence, we would have seenln p2's associated with the ln m2's: So, we have also canceled nonlocal diver- gences. The surviving local divergence is the supercial divergence, to be canceledby the 2-loop mass counterterm. Excercise VIIB7.1 Calculate the p 2part of the 2-loop kinetic counterterm by writing the above 2-loop propagator graph with nonvanishing external momentum, introducing the Schwinger parameters, doing the loop-momentum integration, taking thederivative with respect to p 2, and then evaluating at p=0 . W h yi st h e r en o subdivergence (1 =2)? Excercise VIIB7.2 Calculate the complete (all graphs, innite and nite parts of the) 2-loop propagator correction for massless4. (See excercise VIIB4.3a.) For our next example we consider massless 3theory, and work in 6 dimensions, where the theory is renormalizable (instead of superrenormalizable, as in 4 dimen-sions). For the 2-loop propagator correction, there are only two graphs (plus 1-loop 390 VII. LOOPS graphs with 1-loop counterterm insertions), one of which is simply a 1-loop propagator graph inserted into another. The other graph is P=Z dk dq1 1 2(k+q)21 2(k+1 2p)21 2(k1 2p)21 2(q+1 2p)21 2(q1 2p)2 (with a symmetry factor of1 2for real scalars). This graph can be rewritten as iterated propagator corrections by use of integration by parts in momentum space. This is legalized by dimensional regularization, since boundary terms vanish in low enoughdimensions. All invariants can be expressed as linear combinations of the propagator denominators (there are 5 of each, not counting the square of the external momentum p 2), so any product of momentum times derivative acting on the integrand will give terms killing one denominator and squaring another, except for p2terms, which can be canceled by appropriate choice of the momentum multiplying the derivative: Z dk dq@ @k
k+q 1 2(k+q)21 2(k+1 2p)21 2(k1 2p)21 2(q+1 2p)21 2(q1 2p)2=0 This operation eectively gives the factor @ @k
(k+q)!(D4) +(q1 2p)2 (k+1 2p)2+(q+1 2p)2 (k1 2p)2(k+q)2 (k+1 2p)2(k+q)2 (k1 2p)2 We thus have (D 22)P=P1P 2 P1=Z dk1 [1 2(k+1 2p)2]21 2(k1 2p)2Z dq1 1 2(q+1 2p)21 2(q1 2p)2 P2=Z dk1 [1 2(k+1 2p)2]21 2(k1 2p)2Z dq1 1 2(k+q)21 2(q+1 2p)2 The former term is the product of two 1-loop propagator graphs, the latter is the insertion of one 1-loop propagator graph into another. Both graphs can be evaluated by repeated application of the generalized massless one-loop propagator correction (with arbitrary powers of free propagators) given at the end of subsection VIIB4. The result can be expressed as P1=(D3)(P0)2 1 2p2 P2=cP1;c =[(D3)]2(5D) [(3D 2)]2(3D 25)(D 21) in terms of the 1-loop propagator correction P0. We therefore modify our minimal subtraction so that P0has the simplest form (G scheme): P0=1 61 (1 2p2)1 B. EXAMPLES 391 whereD=62, and we calculated the coecient of the 1 =term and threw in a normalization factor that canceled the rest: h!N h N=1 3(D6)(2D 2)B(D 21;D 21)=( 12 3)(12)(12) (1 +)[(1)]2 Further evaluating c, we nd c=1 312 (13 2)(13)[(12)]2(1 + 2) [(1 +)]2(13)(1) Using the expansion of ln(1z)i nt e r m so f and(n), it is easily checked that this combination of 's is 1+ O(3), so we can just drop them. Collecting our results, we have P=1c D 22P1=1 361 232 1 1+1 312 (13 2)(13) (1 2p2)12 Excercise VIIB7.3 Calculate the same graph in four dimensions. It's nite there, so no countert- erms are necessary. However, in this case integration by parts gives a factor of 1=, and each of the two resulting graphs has an additional factor of 1 =2. The result then has a factor of 1 minus the previously obtained combinationof 's, which we already saw was of order  3. The nal result is thus obvious except for a factor of a rational number: 6(3)1 1 2p2 (The on-shell infrared divergence is as expected from power counting.) We next calculate the counterterm graphs. These are the ones that cancel the subdivergences coming from the 1-loop 3-point subgraphs. We therefore need thedivergent part of this subgraph. This is easy to evaluate by our previous methods:The result of Schwinger parametrization, scaling, etc., doing all integration exactlyexcept over the Feynman parameters is Z dq1 1 2q21 2(q+k+1 2p)21 2(q+k1 2p)2 =Z1 0d3(1X )(3D 2)[1 2+(1+)(k+1 2p)2+1 2(1)(k1 2p)2]D=23 =1 21 +finite 392 VII. LOOPS by simply replacing the factor in brackets by 1 (since it is raised to the power), w h e r ew eh a v eu s e d Z1 0d3(1X )=Z1 0d+Z1+ 0d=1 2 Since we know the divergence is momentum-independent, we can obtain the same result from a (infrared regularized) tadpole graph with its propagator raised to thethird power: In the notation of subsection VIIB1, 1 (3)^A1(3;0;m2)=(3D 2) (3)(1 2m2)D=23=1 21 +finite The contribution of the 2 counterterm graphs (or one for the eective action if we drop the symmetry factor) is thus 2
3P0=2 (1 21 )P0 Collecting terms, we have P+2
3P0=1 121 212 3 1 1+1 312 (13 2)(13) (1 2p2)12+1 61 2(1 2p2)1 After a little algebra, dropping terms that vanish as !0, we nd P+2
3P0=(1 2p2)[1 121 21 181 1 12(ln1 2p2)2+1 9ln1 2p223 216] Note that modifying minimal subtraction is equivalent to redening1 22,w h i c h we have set to 1, but which appears only in the ln's, asln(1 2p2)!ln(p2=2). Thus, modifyingN, which appears only in the combination N(1 2p2), is the same as shifting ln1 2p2: N!Nea)ln1 2p2!ln1 2p2a For example, choosing a=2 3, P+2
3P0!(1 2p2)[1 121 21 181 1 12(ln1 2p2)25 72] Only theO() part of the normalization factor aects the nal result: More generally, (1 2p2)!N (1 2p2))ln1 2p2!ln1 2p21 lnN Since after adding counterterms, which cancel nonlocal divergences arising from subdi- vergences,ln's appear only in nite terms, only the O()p a r to flnNwill contribute. Thus, we can approximate any normalization factor as Nea;a = +rational B. EXAMPLES 393 as far as the renormalized results are concerned. (The identies this normalization as modied minimal subtraction, as the MS or G schemes.) This is just the statement of the renormalization group, that the nal result in minimal subtraction schemes depends only on the choice of scale: The complete normalization factor is really Ntotal=N(1 22))ln1 22!ln1 22+1 lnN However, the higher-order terms can be convenient for intermediate stages of the calculation. In this particular case, the nonlocal divergences appearing before cance-lation are of the form (1 =)lnp 2,s ot h eO(2)p a r to fNcontributes at intermediate stages. For example, replacing the original Nwith N0=( 12 3)(12)(1) would have given the same result even before cancelation, since the change is by another combination of 's that give 1 + O(3). Excercise VIIB7.4 Complete the 6D calculation of the exact 2-loop propagator correction in 3theory, including the missing graph and counterterms, to nd the total renormalized 2-loop propagator and its 2-loop counterterms. REFERENCES 1Goldstone, Salam, and Weinberg; Jona-Lasinio; loc. cit. (VC): eective potential. 2S. Coleman and E. Weinberg, Phys. Rev. D7(1973) 1888. 3A. D'Adda, M. L uscher, and P. Di Vecchia, Nucl. Phys. B146 (1978) 63; E. Witten, Nucl. Phys. B149 (1979) 285: quantum CP(n). 4L.M. Brown and R.P. Feynman, Phys. Rev. 85(1952) 231; G. Passarino and M. Veltman, Nucl. Phys. B160 (1979) 151; W.L. van Neerven and J.A.M. Vermaseren, Phys. Lett. 137B (1984) 241: reduction of 1-loop integrals with numerators to scalar 1-loop integrals. 5A.C.T. Wu, Mat. Fys. Medd. Dan. Vid. Selsk. 33(1961) 72: scalar box (1-loop 4-point) graph. 6R.P. Feynman, unpublished;L.M. Brown, Nuo. Cim. 22(1961) 178; F.R. Halpern, Phys. Rev. Lett. 10(1963) 310; B. Petersson, J. Math. Phys. 6(1965) 1955; D.B. Melrose, Nuo. Cim. A 40(1965) 181: reduction of scalar 1-loop graphs to D-point graphs. 7G. 't Hooft and M. Veltman, Nucl. Phys. B153 (1979) 365; G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. C 46(1990) 425; 394 VII. LOOPS A. Denner, U. Nierste, and R. Scharf, Nucl. Phys. B367 (1991) 637: further evaluation and simplication of 1-loop graphs. 8K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159: integration by parts in momentum space for exact evaluation of higher-loop masslesspropagators. 9D.I. Kazakov, Phys. Lett. 133B (1983) 406, Theor. Math. Phys. 58(1984) 223, 62 (1985) 84; D.I. Kazakov and A.V. Kotikov, Theor. Math. Phys. 73(1988) 1264: more techniques for exact evaluation of such graphs. 10Dyson, loc. cit. (VC): treatment of lowest-order overlapping divergences. 11Salam, loc. cit. (VIIA): overlapping divergences to all loops. C. RESUMMATION 395 :::::::::::::::::::::: :::::::::::::::::::::: :::::::::::::::::::::: C. RESUMMATION :::::::::::::::::::::: So far we have only assumed that connement arises nonperturbatively in 4D QCD. However, to connect with known, successful results of perturbation theory, weneed to understand how the same methods used to give these perturbative results can be generalized to include the nonperturbative ones. The simplest method would be to take the perturbation expansion as is, and nd a good method for evaluating(or perhaps redening) its sum, with the hope that summation to all orders by itself would reveal features invisible at nite orders. Besides the technical diculties associated with such an approach, the main prob- lem is that the summation of the perturbation expansion does not converge. Parts ofthis problem can be solved by appropriate redenitions, but other parts indicate a se- rious problem with perturbation theory, caused by the very renormalization that was supposed to solve the main problem of nite-order perturbation theory (innities). 1. Improved perturbation We saw in the previous section that dimensional transmutation replaced the di- mensionless coupling constant with a mass scale. In principle, we would like to explicitly make this replacement as the basis of our perturbation expansion, notonly to make the perturbative parameter physical, but also to take into account the running of the original coupling. Unfortunately, this is not possible in practice; however, we can choose the arbitrary (unphysical) renormalization scale to be in the range of energies in the problem at hand, so that the ln(p 2=2) corrections are small. A change in scale from one value of to another is related to a resummation of graphs: Although the one-loop term in the eective action containing ln(p2=2) comes from a single 1PI amplitude, it contributes an innite number of terms at dif- ferent loop orders to the propagator when inserted into any higher-loop 1PI graph,as 1=(K+A)=1=K(1=K)A(1=K)+:::. Although K+Adepends only on M,K depends only on gandAdepends only on . Thus, any redenition of that leaves the physical quantity Munchanged requires a corresponding redenition of g: M 2=2e1=g2)g2(2)=1 ln 2 M2 and thus changing redistributes the contributions to 1 =(K+A) (and therefore to the summation of graphs in any amplitude) over the dierent loop orders. For exam- ple, if the amplitude is most sensitive to the momentum in a particular propagator (independent of loop momenta), and we choose 2p2, then although we can't use 396 VII. LOOPS the resummed perturbation expansion directly, we can at least push most of it into the lower orders. Things get more complicated at higher loops: It becomes dicult to associate the running of the coupling with the resummation of a particular subset of all the graphs. However, we already know that this eect can be derived from the breaking of scale invariance by renormalization. For example, let's consider Yang-Mills theory, sincegauge invariance restricts it to have only a single coupling parameter. (This makes it the simplest case conceptually, although not computationally. Here we use only the fact that it has a single coupling; its explicit renormalization constants won't be considered until the next chapter. As an alternative, we can consider the scalar QCD analog of subsections VC9 and VIIB6, a  4theory with an auxiliary eld, if we ignore mass renormalization, or arbitrarily renormalize the mass to zero.) For convenience of dimensional analysis, we use only coupling constants that are dimensionless in all dimensions, by scaling with an appropriate power of . (In general, we can do this even for masses.) The classical Yang-Mills action, before and after the addition of counterterms, is then Sclass=1 g2(1 22)Z dx tr1 8F2;Sclass+
S=1 ^g2Z dx tr1 8F2 where 1 ^g2=1 (1 22)" 1 g2+1X n=11 ncn(g2)# ;cn(g2)=1X L=n(g2)L1cnL for some numerical constants cnL. (We can also include  h's asg2!g2h.) We use 1 22to produce the combination (1 2p2=1 22)in graphs. (In practice, one uses units 1 22= 1 until the end of the calculation, and restores units.) The dependence is then given by varying for xed ^g: 2@ @2g2g2(g2);2@ @2^g20 where theg2term is the classical contribution, will be found to be independent of (except indirectly through g2), and ^g's independence from is the statement that the physics is independent of the choice of (i.e., ^gdepends on only Mand). We then nd 0=22@ @21 ^g2)=g4@ @g2(g2c1);@ @g2(g2cn+1)=@ @g2cn Thus, the coecients of the 1 =terms determine those of both the higher order terms and. C. RESUMMATION 397 T h i sg i v e su sa ne x p r e s s i o nf o r , =1X L=1(g2)L+1L;L=Lc1L Sinceg2is itself unphysical, the information we can get from analyzing the running of this coupling is arbitrary up to redenitions. For example, assume that all Lare nonvanishing, and write the denition of as (inD=4!=0 ) 2@ @21 g2=f=1 g4=1X L=0(g2)LL+1 Then under a redenition g2!g2(g02)w eh a v e 2@ @21 g02=@(1=g2) @(1=g02)1 f(g2(g02))f0(g02) Now we consider a \perturbative" type of redenition, as results from changing renor- malization prescriptions, so g2gets only \O(h)" corrections: Taylor expanding g2=g02+k1g04++k2g06+O(g08) )1 g2=1 g02+constant +O(g02) we nd @(1=g2) @(1=g02)=1+O(g04);f (g2(g02)) =f(g02)+O(g04) )f0(g02)=f(g02)+O(g04) Thus, the rst two coecients of (1and2) are unaected, while terms found at 3 loops and beyond can be modied arbitrarily, and even be set to vanish. In the moregeneral case of more than 1 coupling, it is sometimes possible to eliminate also some of the 2-loop contributions. Therefore, to consider the general behavior of the coupling as a function of energy ( 2), it is sucient to solve the equation 2@ @2g2=1g42g6 (using, e.g., the change of variables t=ln2andu=1=1g2)a s 2 M2=e1=1g21 g2+2 12=2 1 398 VII. LOOPS withM2as the constant of integration. Using an allowed type of redenition for g2, and also redening the arbitrary constant of integration M2, we can simplify this to 1 g2!1 g22 1;M2!M2e2=2 1 )=1g4 12 1g2;2 M2=e1=1g2(g2)2=2 1 (This redenition changes the range of what g2is called negative and what positive. However,g2is just a parameter, not a physical coupling: As far as the unitarity of the kinetic term is concerned, only the residues near the poles of the propagator are relevant. Also, our allowed class of redenitions do not aect behavior for small g2, and thus perturbation theory.) Excercise VIIC1.1 Let's analyze this solution in more detail: aGraph the function y(x)=eaxxb(or graphln y to make it simpler) for a andbpositive, negative, and vanishing, to study the behavior of the function 2(g2). The analysis can be simpliifed (and the behavior for dierent values of aandbrelated) by considering g2positive and negative, and the symmetries a!a; x!x; y!(1)by b!b; x!x; y!(1)b1 y Note thatg2can be nonpositive for some values of 2: For example, even for 2=0 ,w eh a v e g2=1=1ln(2=M2), which is negative for <M or for >M . What happens for 26=0 ? bAfter applying the above redenition, apply the second redenition 1 g!1 g+2 1g Find the new and2(g2). Compare to the behavior of 2(g2)b e f o r et h i s redenition, for the cases 2=1<0, noting the \duality" symmetry g$ (1=2)=g. Excercise VIIC1.2 Consider some theory with a single dimensionless coupling g2, but now also a single mass m. By the above methods we nd 2@ @2m2=m2[1m(g2)] C. RESUMMATION 399 (Themdependence follows from dimensional analysis.) Solve for m2as a function of g2,a sa ni n t e g r a lo v e r g2in terms of andm. Show that after an appropriate redenition 2@ @2m2=m2(1m1g2) for some constant m1.S o l v e f o r m2explicitly in terms of g2,w h e nw eh a v e also redened to1g4+2g6. Then make the nal redenition 1 =g2! 1=g22=1used to simplify M2. For purposes of perturbation theory, it is useful to invert this: For small g2,w e have approximately 1 g21ln2 M2+2 1ln 1ln2 M2 This implies that the terms in the eective action that carry the Mdependence are given by MtrZ dx1 8F 1ln M2+2 1ln 1ln M2 F (We can also replace ! in this limit, ignoring i's in comparison to ln's.) The general class of coupling redenitions we considered are allowed by pertur- bation theory: If we knew the exact solution to a eld theory, we would be more restrictive, requiring invertibility. However, in perturbation theory, given two renor-malization prescriptions related by some such coupling redenition, we might know this redenition only perturbatively, and perhaps only to a few orders. Even if we knew it exactly, and knew it to be noninvertible, it still might not be clear which ofthe two prescriptions was the correct one, if either. Therefore, the renormalization group alone is sucient to draw conclusions about the behavior of a theory only at \small" (1) coupling. Similar remarks apply to propagators, S-matrix elements, etc. Consider any func- tionG nappearing as the coecient of nelds in a term in the eective action. Comparing the unrenormalized ^Gnto the renormalized Gn, Gn(g2;2)=Zn(^g2(1 22);)^Gn(^g2;);2@ @2^G=0 )2d d2G=2@ @2+@ @g2+n  G=0; =2@ @2lnZ whereZis a wave-function renormalization factor (not required for pure Yang-Mills in the background gauge; or we can examine ratios of such quantities where the Z's cancel, which are more physical, such as S-matrix elements). 400 VII. LOOPS 2. Renormalons The perturbation expansion in general can't be resummed in the naive way be- cause the number of diagrams increases as n!(constant )natnloops. The simplest example of this is a self-interacting scalar in D=0: Z=Z1 1dp 2e1 221 4g24=1X n=0g2nZn Since there is no momentum integration, each diagram is just 1 (times some permu- tation factors), so Znjust counts the number of diagrams at nloops. We use g2so the coupling is similar to that in Yang-Mills: As usual, we can rescale !=gto recognizeg2as h: 0=g)1 22+1 4g24=1 g2(1 202+1 404) (Of course, we can be more explicit by writing  hg2in place of just g2or h, but the eect is identical, since they both appear only in that combination.) This integral can be evaluated exactly at any order of perturbation theory: Zn=Z1 1dp 21 n!(1 44)ne1 22=1 n!(1)n1p(2n+1 2)1p 2(n1)!(4)n where we have used the Stirling approximation for ( z) at largez. Excercise VIIC2.1 Find the following properties of the function for large argument: aDerive the Stirling approximation lim z!1(z)r 2 zz ez by applying the method of steepest descent to the integral denition of (z+ 1). (See subsections VA2 and VA5.) bUse this approximation, and limz!1(1 +1 z)z=e,t os h o w lim z!1(az+b)p 2(az)az+b1=2eaz Thus we might as well apply the steepest descent approximation directly to the original integral: Using also an integral for  h(=g2in this case), Zn=Idh 2ihn+1Z Dp h eS=h C. RESUMMATION 401 we rst apply steepest descent to the integral, yielding the usual rst two terms in the JWKB expansion. Then the  hintegral can be approximated as ( n) by keeping only the part of the contour on the positive real axis: Idh 2ihn+1eS=h S= =01 2i(n)1 Sn S= =0 )ZnX S= =0 S6=01 2i det2S 21=2 (n1)!1 Sn (S= 0 solutions contribute only to Z0. A similar result can be obtained by simulta- neously using steepest descent for the  hintegral, yielding a \classical value" of  hin terms ofS.) In the present case, the nontrivial classical solutions are S=1 22+1 44)=i which gives the same Znas previously (being careful to sum the two terms for the two solutions). Thus, we see that in general we have to sumP1 n=0n!(h=S)n,w h i c h does not converge. Furthermore, this divergence is associated with nite-action (\in- stanton") solutions to the classical equations of motion. The simplest example of a resummation problem is the one-loop propagator cor- rection. We have seen that the classical and one-loop kinetic terms can be combined to give a kinetic operator of the form 1K(p2)ln(p2=M2) in massless theories, or at high energy in massive theories, where Kis the classical kinetic operator. The free (or asymptotic) theory has solutions where this kinetic operator has a zero (the propagator blows up). Besides the classical solution at K(p2) = 0, there is another atp2=M2: 1 1K(p2)ln p2 M2=1 1K(p2)ln 1p2+M2 M2M2 1K(M2)1 p2+M2 This might be expected to be a bound state, called a \renormalon" because of its rela- tion to the renormalization group. However, the residue of this pole in the propagatorcan have the wrong sign, indicating the appearance of a ghost (\Landau ghost"), and thus a violation of unitarity. Excercise VIIC2.2 The Landau ghost itself is not necessarily a problem in quantum eld theory,although it indicates the possibility of such problems. Examine the behavior of this ghost after taking into account the 2-loop correction (  2), before and after the simplifying redenition of the previous subsection, for all the various 402 VII. LOOPS signs of1and2. Since the expression for 2(g2) can't be inverted, use the fact that the propagator follows from the coupling g2(2)a s
g2(p2) p2 (Field redenitions can't remove the momentum dependence of couplings.) Then new poles (or other singularities) in the propagator correspond to the limitg2!1 , so ndp2(g2)t h e r e . . . . . . . This causes problems similar to those from instantons when the quantum prop- agator is inserted into another graph. We set external momenta to vanish, as anapproximation for high energy for the loop momenta, or to evaluate low-energy quan-tities such as anomalous magnetic moments. In any one-loop 1PI graph with n1-loop propagator insertions and lexternal lines, we get an integral at high energy of the form (e.g., in QCD or the scalar analog with auxiliaries of subsections VC9 and VIIB6) Z d 4k(k2)l 1lnk2 2n (1)nZ1 0du e(l2)uunn! 1 l2n where we changed variables to u=ln(k2=2) (remembering ( n+1 ) =n!). We have used eectively an infrared cuto by approximating the u-integral from 0 to 1 instead of1 to1. If we look instead at the low-energy (of the loop momentum) behavior, now taking l1 massive classical propagators with 1 massless propagator (to insure IR convergence) with ninsertions, we nd Z d4k(k2+m2)(l1)(k2)1 1lnk2 2n n 1Z0 1du eu(u)n=n!n 1 Since the former comes from UV behavior it's called a \UV renormalon", while the latter coming from IR behavior is called an \IR renormalon". The essential dierenceis the relative factor of ( 1) n. In fact, the former expression is also the high-energy limit of the latter (neglecting masses then), so the complete integral ( ufrom1 to 1,s ok2from 0 to1) can be approximated as the sum of the UV renormalon and IR renormalon contributions. C. RESUMMATION 403 3. Borel Since renormalons and instantons cause the perturbation expansion to diverge by af a c t o ro fn!, we look for a method to formally sum such series, by relating them to series that do converge. In general, we consider the series A(h)=1X n=0hnan and dene the \Borel transform" as: ~A(z)=Zr+i1 ri1d(1=h) 2iez=hA(h) (for some real number rto the right of all singularities of A) in anticipation of instanton-like contributions. The inverse is A(h)=Z1 0dz ez=h~A(z) The inverse Borel transform is related to the Laplace transform (with the variable changex=1=h) and the Mellin transform ( x=1=handy=ez). Evaluating explicitly for the above series, ~A(z)=(z)a0+1X n=0zn1 n!an+1 So the Borel-transformed sum comverges faster by a factor of n!, which is just what we need for perturbation theory. The idea for resumming the perturbation expansion is to rst do the Borel sum, then inverse Borel transform the resulting function. Of course, this procedure does not necessarily x the original problem, which mightmerely be translated into problems of convergence or ambiguity for integration of theinverse transform. In particular, we need ~A(z) to be well dened along the positive real axis. We saw that generically the sums involved were approximately of the form A(h) 1X n=1hn(n1)!(k)n In that case ~A(z)1X n=0(1)nznkn+1=1 z+1 k Whenk< 0, this leads to a singularity in the integral dening the inverse Borel transform. It can be \regularized" by choosing a contour that goes around the pole, 404 VII. LOOPS but the choice of contour is ambiguous, and choosing an arbitrary linear combination of the two contours introduces a free parameter. Explicitly, we have A(h)=A0(h)+e1=jkjh whereA0is the result of a particular prescription (e.g., principal value), and is the new parameter. The term is clearly nonperturbative, since each term in its Taylor expansion in  hvanishes. This new parameter can be interpreted as a new (nonper- turbative) coupling constant in the theory, just like ambiguities in renormalization ofnew counterterms in perturbatively nonrenormalizable theories. Now we more carefully analyze the explicit sums we found in the previous sub- section. The rst example isp hZfor D=0: ~A(z)=Zr+i1 ri1d(1=h) 2iez=hZ1 1dp 2eS=h=Z1 1dp 2(zS)=1p 2X S=z(S0)1 (The contribution from S= 0 is articial, coming from our using A=p hZinstead ofZ.) So, this integral can be explicitly evaluated. (For example, for the action we used above, we can explicitly solve for atS=z.) However, there is then a problem in inverting the Borel transform: Near z=z0S(0) for classical solutions 0,w e have S()S(0)+1 2S00(0)(0)2)S0()S00(0)(0);zz01 2S00(0)(0)2 ) (S0)1[2S00(0)]1=2(zz0)1=2 Therefore, there are cuts with branchpoints at classical values of the action, leading to ambiguities in the result for A(h). We thus see that new coupling constants are introduced for each solution to the classical eld equation with positive action. (For our D=0 example S< 0, and there is no problem, but more realistic examples, like Yang-Mills instantons, have S>0.) In the case of renormalons, we see from the previous subsection that the large- n behavior gives singularities at z=N= 1for positive integer N.T h i s i n t e g r a l a l s o i s easier to evaluate after Borel transforming: We consider a one-loop graph, but replace one internal line with the \full" quantum propagator coming from the 1-loop eective action (the same as summing a string of 1-loop propagator insertions), while using massive propagators for the remaining lines. We thus examine rst the transform ofthe quantum propagator Zd(1=h) 2iez=h1 k21 1 h+1ln(k2=2)=1 k2k2 21z C. RESUMMATION 405 by closing the contour on the left. Then inserting this transformed propagator into the complete diagram, Z d4k1 (k2+m2)l11 k2k2 21z m2 21z (11z)(l2+1z) using the integrals of the subsection VIIB1. This expression is the sum over nof the UV/IR renormalon example at the end of the previous subsection, except that we havedone the summation over nas the rst step (and used the Borel transform to assist in the evaluation). The rst has poles at z=N= 1for positive N,r e p r e s e n t i n g the IR renormalon, which are relevant for 1>0, but the second has poles at z=(N+l3)=1for positive N(andl3 for the original diagram to be UV convergent), representing the UV renormalon, which are relevant for 1<0. To the one-loop approximation for the -function we have used, the singularities are just poles, but if the two-loop propagator insertions are used, these singularities become the branchpoints for cuts. The new coupling constants that appear nonperturbatively can be given a physical interpretation in terms of vacuum values of polynomials of the elds. The basic ideais analogous to perturbative tadpoles: In that case corrections to S-matrices due to vacuum expectation values of scalar elds can be expressed by propagators that end at a \one-point vertex", whose coecient is the vacuum value of the eld: h(x)i=Z D e iS(x)=c in position space for some constant c, or in momentum space as Z D eiS(p)=c(p) Similarly, we could expect graphs to have two propagators that end at a two-point vertex representing the vacuum value of the product of the two elds associated with the ends of the two propagators, and so on for higher-point vertices. For example, for a2vertex in a scalar theory, it would correspond to a contribution of the form Z D eiS(x)(y)=(c2+c0)+::: in position space, or in momentum space Z D eiS(p)(q)=(c2+c0)(p)(q)+::: wherec2is the contribution from hi2,s oc0represents (
)2=h2ihi2. Such vacuum values do not appear in perturbation theory for higher than one-point; we get 406 VII. LOOPS only one(p) for each connected part of any graph. However, such contributions would be expected to give similar contributions to those we have found for renormalons: By dimensional transmutation, a contribution to an amplitude of the form en=1hmust appear in the combination en=1h!en=1h2 p2n =M2 p2n This is the type of contribution expected from a propagator with tadpole insertions, or in the same way from any other type of vacuum value. In particular, in QCDthere are no fundamental scalar elds, but only scalar elds can get vacuum values, by Lorentz invariance. Thus, the vacuum values come from composite scalars, like tr(F 2), qq,e t c . Note that renormalons are a feature of renormalizable theories: They do not appear in superrenormalizable or nite theories. In particular, the path-integral methods of \constructive quantum eld theory" have been used to show that cer- tain interacting eld theories in lower dimensions can be proven rigorously to exist| superrenormalizable theories with unique vacuua. 4. 1/N expansion Perturbation theory is insucient to evaluate all quantities in quantum physics, since (1) such expansions don't always converge; (2) if they do converge, they might not converge to the complete result; and (3) even if they do give the complete answer,their summation might not be practical. There are many perturbative expansions in quantum eld theory. When we say \perturbation theory" in this context, we generally mean an expansion in the number of elds (or, in diagrammatic terms, number of vertices), since in the path integralwe kept the exact quadratic part of the action but expanded in powers of the inter- action terms (cubic and higher). (This is usually also an expansion in the coupling constants, depending on how we dene the elds, which can be redened by factors of the couplings.) One disadvantage of this expansion is that it violates manifest gauge invariance: Nonabelian gauge transformations are nonlinear in the elds, andthus mix diagrams with dierent numbers of elds. (These are the internal elds; external elds are asymptotic, and approximated as free.) Graphs that are related by gauge transformations must be added together to obtain gauge-invariant, and thusphysically meaningful, expressions. Also, in practice individual graphs contain \gauge artifacts" that complicate them in certain gauges, but cancel in gauge-invariant ob- jects, like S-matrix elements. C. RESUMMATION 407 There can be a large number of graphs contributing to a particular physical process (given set of external states) at any particular loop order. There is another gauge-invariant expansion that can be applied to Yang-Mills theory to subdivide thesesets of graphs, based on the freedom of choice of the Yang-Mills group itself: Wehave seen that the classical groups are dened in terms of N N matrices, where N is arbitrary. Clearly, S-matrix elements must depend on N, even if the external states arerestricted to be group singlets or representations of an N-independent subgroup, sincethe number of internal states increases as some polynomial in N. We now examinehow this can be used to dene a perturbation expansion in terms of N. We have already seen in subsection VC9 that the group theory of any graph can be detached from the momentum and spin (so we considered there a simple modelof scalars). We also saw there that the group theory of such matrices is most con- veniently graphed by a double-line notation, where each line acts group-theoretically as a bound (anti)quark, reducing the group theory to trivial Kronecker 's. We now notice that in some loop graphs, depending on how the lines are connected, some ofthe quark lines form closed loops. Again the group theory is trivial: There is a factorof N for each such loop, from the sum over the N colors. We can also give a physicalpicture to these numerical factors: Since we draw the scalar propagator as quark andantiquark lines with nite separation, think of the scalar as a (very short) string,with a quark at one end and antiquark at the other. This gives a two-dimensionalstructure to the diagram, by associating a surface with the area between the quarksand antiquarks (including the area at the vertices). We can extend this picture byassociating a surface also with the area inside (i.e., on the other side of) each closedquark loop. In particular, for any \planar" diagram, i.e., any diagram that can bedrawn on a sheet of paper without crossing any lines, and with all external lines onthe outside of the diagram, the entire diagram forms an open sheet without holes,and with the topology of a disc (simply connected). It is also clear that, for a xednumber of loops and a xed number of external lines, a planar diagram has the great-est number of factors of N, since crossing lines combines quark loops and reduces thepower of N. We can be more quantitative about this N dependence, and relate it to the topol- ogy of the graph. In subsection VC2 we saw the number of propagators, vertices, andloops were related by PV=L1. This relation treats a Feynman diagram as just a graph, points connected by lines. We now consider a diagram as a polyhedron,with propagators as the edges, and closed quark loops as the faces, as dened by ouruse of matrices for elds. We then have as an additional relation for closed surfaces 408 VII. LOOPS \Euler's theorem", F=PV2(H1) (in terms of the \Euler number" 2(H1) =VP+F), whereFis the number of faces and His the number of \handles": 0 for the sphere, 1 for the torus (doughnut), etc. This follows from the previous relation: First combining them as L=F+2H1 we note that \cutting" any handle along a loop (without separating the pieces) pro- duces 2 faces; in other words, introducing two faces (as a \lens") into a loop thatcircles a handle changes the surface without changing the diagram, replacing 1 han- dle with 2 faces. The last relation then follows from the case with no handles, where each face gives a loop, except that the no-loop case corresponds to 1 face (or startwith a less trivial case, like a cube, if that's easier to picture and count momenta for). Using the fact that the g 2appears in Yang-Mills the same way as  h,a n dt h a t each face gets a factor of N, we nd the gand N dependence of any graph is (g2)L1N(L1)2(H1)=(Ng2)L1N2(H1) We thus see that eectively Ng2is the coupling squared suited to planar graphs, counting the number of loops, while 1 =N2is a new coupling squared, counting the number of handles. Therefore, we can sum over both Ng2and 1=N2: Each Feynman graph is a particular order in each of these two couplings. The sum of all graphs at xed orders in both couplings gives a gauge-invariant subset of the graphs contribut-ing to a particular S-matrix element. (This is sometimes called \color decomposition". Note thatg 2is the coupling normalized for matrices of the dening representation, which was required here to dene the 1/N expansion, while Ng2=1 2g2 Ais the coupling normalized for the adjoint: If we had used matrices for the adjoint representation, a factor 1=g2 Awould appear in front of the action, because of the dierence in normal- ization of the trace of the matrices.) Excercise VIIC4.1 Consider4theory in D=4, where is now an NN hermitian matrix. Gener- alize the auxiliary-eld prop agator correction calculation of subsection VIIB6 to leading order in 1/N, showing the N-dependence at all steps. Show thatnow, to this leading order, both the N- and g-dependence of the eective action can be absorbed into M. We can also consider more complicated models, such as chromodynamics, with elds appearing in the dening representation of the group, such as quarks. When C. RESUMMATION 409 a quark eld makes a closed loop, it looks like a planar loop of a gluon, except that the closed quark line is missing, along with a corresponding factor of N. Thus, there is eectively a \hole" in the surface. Since only one factor of N is missing, a hole counts as half a handle. We can also draw a avor-quark line for the quark propagator alongside the color-quark line. Since this line closes in quark-eld loops, we also geta factor of M (for M avors) for each quark loop. The fact that the 1/N expansion is topological (the power of 1/N is the number of holes plus twice the number of handles) closely ties in with the experimental ob- servation that hadrons (in this case, mesons) act like strings. Thus, we can expand in 1/N as well as in loops. While the leading order in the loop ( Ng 2) expansion is classical (particle) eld theory, the leading order in the 1/N expansion is classical open-string theory (planar graphs). However, seeing the dynamical string propertiesrequires summing to all orders in Ng 2for leading order in 1/N. Thus, 1/N acts as the string coupling constant. (N appears nowhere else in the action describing string states, since they are all color singlets.) The experimental fact that the hadronic spectrum and scattering amplitudes follow so closely that of a string (more on this later) indicates that the perturbative expansion in 1/N is accurate, i.e., that quantum corrections are \small" in that sense. One application of the smallness of 1/N (largeness of N) is the \Okubo-Zweig-Iizuka rule": A planar graph describesclassical scattering of open strings (mesons). It corresponds topologically to a disc, which is a sphere with one hole, and is therefore order 1/N. Compare this to two planar graphs connected by a handle. It describes classical scattering of open stringswith one intermediate closed string (glueball), where the handle is a closed-string propagator connecting two otherwise-disconnected classical open-string graphs. It corresponds to a cylinder, which is a sphere with two holes, and is therefore order1/N 2.I n t e r m s o f avor lines, the latter graph diers from the former in that it has an intermediate state (the glueball) with no avor lines. The OZI rule is that amplitudes containing an intermediate glueball are always smaller than those with anintermediate meson. This rule also has been veried experimentally, giving a further justication of the 1/N expansion (though not necessarily of string behavior). Generalizing to groups SO(N) and USp(2N) gives more varied topologies: Since the left and right sides of propagators are no longer distinguishable, the string surfaceis no longer orientable (the surface no longer has two distinguishable sides), so we can also have unorientable surfaces such as M obius strips and Klein bottles. One can also perform a separate expansion in the number M of avors. 410 VII. LOOPS The fact that the leading (planar) contributions are of order ( Ng2)L1requires a modication of the Borel transform of the previous subsection: We now identify h=Ng2 instead of just  h=g2, so we can use the 1/N expansion in conjunction with the Borel transform. In particular, this means removing the factor of N from 1and absorbing it into h. The result is that the position of the renormalon singularities in the zplane is independent of N. However, the same is not true for the instantons: A one-instanton solution corresponds to choosing a single component of nonvanishing in our scalar model, so that the classical solution 0for the action S[0] has no N-dependence. (Choosingproportional to the identity matrix yields an N-instanton solution.) The analog in the Yang-Mills case is using just a (S)U(2) subgroup of the full U(N) to dene the instanton. (Note that the structure constants for U(N) are N-independent for the dening representation: See excercise IB5.2.) Then S=g2=NS=h.T h er e s u l t for the positions of the singularities in zis then at integer multiples (positive or negative, depending on considerations given in the previous subsection) of z0,w h e r e z0=1=1 for renormalons NS[0] for instantons where1and the one-instanton action Sare N-independent. The net result is that instantons are unimportant for large N. Thus, if we take the 1/N approach of using a resummation to dene a string theory, the instantons do not take a role in dening the string. (They might return in another form when considering classical solutions to the string theory, or their contribution might bejust a small part of the total nonperturbative contribution.) On the other hand,approaches that analyze just the low-energy behavior of a theory can make use of the instantons: If the physical value of N is small, or the U(N) theory is spontaneously broken to give a small eective N at low energies (as in GUTs), then instantons maybe treated as the dominant nonperturbative contribution to low-energy eects such as chiral symmetry breaking. This can be sucient for studying low-energy bound states, but is insucient for studying connement, whose physical denition is theexistence of bound states of very high energy. REFERENCES 1 E.C.G. St uckelberg and A. Petermann, Helv. Phys. Acta 26(1953) 499; M. Gell-Mann and F.E. Low, Phys. Rev. 95(1954) 1300; C.G. Callan, Phys. Rev. D2(1970) 1541; C. RESUMMATION 411 K. Symanzik, Commun. Math. Phys. 18(1970) 227: renormalization group. 2G. 't Hooft, Nucl. Phys. B61 (1973) 455: renormalization group via dimensional regularization. 3G. 't Hooft, Can we make sense out of \quantum chromodynamics", in The whys of subnuclear physics , proc. 1977 Int. School of Subnuclear Physics, Erice, ed. A. Zichichi (Plenum, 1979) p. 943:arbitrariness of all but rst two nonvanishing coecients in . 4L.D. Landau and I. Pomeranchuk, Doklady Akad. Nauk USSR 102(1955) 489: Landau ghost. 5F.J. Dyson, Phys. Rev. 85(1952) 631: divergence of perturbation expansion in quantum eld theory. 6L.N. Lipatov, Soviet Physics JETP 45(1977) 216: D=0 renormalon. 7G. 't Hooft, loc. cit. (ref. 3), Phys. Lett. 109B (1982) 474; B. Lautrup, Phys. Lett. 69B (1977) 109; G. Parisi, Phys. Lett. 76B (1978) 65, Nucl. Phys. B150 (1979) 163; Y. Frishman and A. White, Nucl. Phys. B158 (1979) 221; C. DeCalan and V. Rivasseau, Comm. Math. Phys. 82(1981) 69; F. David, Nucl. Phys. B234 (1984) 237; A.H. Mueller, Nucl. Phys. B250 (1985) 327, Phys. Lett. 308B (1993) 355; L.S. Brown, L.G. Yae, and C.-X. Zhai, hep-ph/9205213, Phys. Rev. D46 (1992) 4712: renormalons. 8A.S. Wightman, Should we believe in quantum eld theory, in The whys of subnuclear physics ,ibid. p. 983: review of the relation of renormalons to constructive quantum eld theory. 9't Hooft, loc. cit. (VC): 1/N. 10S. Okubo, Phys. Lett. 5(1963) 165; Zweig, loc. cit. (IC); J. Iizuka, Prog. Theor. Phys. Suppl. 37-38 (1966) 21; J. Iizuka, K. Okada, and O. Shito, Prog. Theor. Phys. 35(1966) 1061. 11G. Veneziano, Phys. Lett. 52B (1974) 220: expansion in (1/) the number of avors. 412 VIII. GAUGE LOOPS VIII. GAUGE LOOPS Gauge invariance plays an important role in quantum corrections. It not only simplies their form, but leads to new eects. In particular, it not only improveshigh-energy behavior, but can eliminate divergences altogether, in the presence ofsupersymmetry. In general, the rst thing to calculate in quantum eld theory is the eective action. Once this has been calculated, other properties can be determined: thevacuum, S-matrix, etc. In particular, in spontaneously broken theories, the eectiveaction should be calculated with the symmetric (unbroken) vacuum, which has simplerFeynman rules; once the eective action has been calculated, vacuum values of theelds can be determined, and the S-matrix can be calculated as a perturbation aboutthis quantum vacuum. (The alternative of dening Feynman rules for the classicalbroken vacuum and then calculating quantum corrections doubles the work in ndingvacuum values.) ::::::::::::::::::::::: ::::::::::::::::::::::: ::::::::::::::::::::::: A. PROPAGATORS ::::::::::::::::::::::: We rst consider propagator corrections in some specic theories with spin. In the following calculations we assume the gauge coupling appears only as an overallfactor in the classical action: It thus also counts loops, so our 1-loop graphs are coupling-independent. All the integrals have been performed in subsections VIIB4-5; all that remains is the numerator algebra, which follows the examples of subsectionVIC4. As we have seen, such corrections are important in analyzing high-energybehavior; as we'll see in the following section, they are also important for low energy.(Of course, for massless particles the two are related by conformal invariance, evenwhen quantum corrections break it.) 1. Fermion Our rst calculation is the one-loop correction to the electron kinetic operator in QED: The S-matrix element is A2e=Z dk a(k=+1 2p=+mp 2) a 1 2(k1 2p)21 2[(k+1 2p)2+m2] A. PROPAGATORS 413 At this loop level the only dierence between using D-dimensional -matrix algebra (dimensional regularization) and 4-dimensional (dimensional reduction) is an unphys- ical nite renormalization, so for simplicity we'll use the latter method. Then the numerator is k=+1 2p=p 2m The result of the integral is then A2e=p=m2 2p2[^A2(p2;0;m2)^A2(0;0;m2)] + (1 2p=p 2m)^A2(p2;0;m2) in the notation of subsection VIIB5. The UV divergent part follows from ^A2(p2;0;m2)=1 +finite The contribution to is minus the S-matrix element, but the counterterm has a second minus sign to cancel the divergence:
S=h Z dx (1 2i@=p 2m) The calculation for the quark self-energy in QCD is the same except for group-theory factors (see subsection VIIIA5). Excercise VIIIA1.1 Repeat the calculation with D-dimensional -matrix algebra. What is the dierence in the nite part, and why doesn't it matter? In subsection VIIB5 we considered MOM subtraction (see subsection VIIA3) for scalar propagators. The analysis in this case is similar, but now we expand in p= instead ofp2:
K=a+b(mp 2p=)+O[(mp 2p=)2] However, since
Kis normally expressed as functions of p2times 1 and p=, we need to translate: Using (mp 2+p=)(mp 2p=)=1 2(p2+m2),
K=a+b0(mp 2p=)+c1 2(p2+m2)+O[(mp 2p=)(p2+m2);(p2+m2)2] =a+(b0+2mp 2c)(mp 2p=)+O[(mp 2p=)2] We next reevaluate the fermion propagator correction, to linear order inmp 2p=. Starting with ~A(x;p2;m2 1;m22)=Z dk eix
k 1 1 2[(k+1 2p)2+m2 1]1 2[(k1 2p)2+m2 2]=Z d2D=2eE E=1 21 x2+ix
1 2p+1 8(12)p2+1 4[(m2 1+m2 2)+(m2 1m2 2)] 414 VIII. GAUGE LOOPS we keep only linear order in xandp2+m2,a n ds e tm1=m,m2= 0 (switching back to=1 2(1 +)): Eix
(1 2)p+1 2(1)(p2+m2)+1 2m22 To clearly separate UV divergences (from 0) and IR divergences (from 0), we scale ! 2)E(1 2)ix
p+1 2(1 1)(p2+m2)+1 2m2 ~AZ1 0d 1em2=2Z1 0d 2[1(1 2)ix
p][11 2(1 1)(p2+m2)] The integrals are easily performed in either order: ~A (1 +)(1 2m2)1 UV1 12+1 21 (1)(12)ix
p + 1 21 IR+1 12p2+m2 m2+ 1 41 IR+3 21 121 21 1 ix
pp2+m2 m2 and in the limit !0, ~A (1 +)(1 2m2)1 UV+2+1 2ix
p+ 1 21 IR+1p2+m2 m2 + 1 41 IR+1 ix
pp2+m2 m2 The electron propagator correction to linear order inmp 2p=is then A2e(1 +)m2 2 1 2+ 1 41 IR+1p2+m2 m2 p= +1 UV+2+ 1 21 IR+1p2+m2 m2 (1 2p=2mp 2) (1 +)m2 2 (1 2) mp 2 31 UV+5 +1 UV+21 IR+5 (mp 2p=) The 1=UVterms are the same as the 1 =terms obtained above for minimal subtrac- tion. In the MOM scheme, this entire contribution ( O(K0)a n dO(K1)) is canceled by counterterms. A. PROPAGATORS 415 2. Photon We next calculate the spin-1/2 contribution to the photon (or gluon) self energy: The S-matrix element is Z dktr[ a(k=1 2p=+mp 2) b(k=+1 2p=+mp 2)] 1 2[(k1 2p)2+m2]1 2[(k+1 2p)2+m2] The result of the trace (again using 4-dimensional algebra) is 2[kakbab1 2(k2+1 4p2+m2)]1 2(abp2papb) The rst part is the expression appearing in ^Aabin subsection VIIB5, once we recog- nize itsabterms as the average of the denominator factors, yielding tadpoles. The integral thus gives (abp2papb)(2A1 2^A2)(1 lnp2)(1 3)(abp2papb) for the divergent and high-energy terms. Using Aa(p)(abp2papb)Ab(p)=1 2Fab(p)Fab(p) in terms of the linearized eld strength F, the corresponding contributions to the unrenormalized one-loop eective action are (including a factor of1 2for identical external lines) 1h2 3Z dx1 8Fab(1 ln )Fab (neglecting the \1" part ofln()) and the counterterm is thus
S=h (2 3)Z dx1 8FabFab in the case of QED. For QCD, we must include the group-theory factor tr(GiGj) multiplying FiabFj ab. (Examples will be given in the following subsections.) This propagator correction is easier to analyze in the MOM scheme than the electron propagator, since there are no internal massless particles, and thus no IRdivergence to distinguish from the UV one. We therefore just take the explicit ex- pressions for the integrals from subsection VIIB5 and Taylor expand in p 2about 0 (or actually in 1 =of VIIB5.1a, substituting for p2only at the end). The low-energy 416 VIII. GAUGE LOOPS part of the renormalized eective action for the photon, exhibiting the momentum dependence of the coupling, is then 0+1;2 ;rZ dx1 8Fab1 e2+2 15m2 Fab where we have applied MOM subtraction by canceling constant (innite and nite) constributions to the coupling. Excercise VIIIA2.1 Evaluate this contribution to the unrenormalized eective action to this or- der. Show that the constant contributions to the coupling (to be canceled byrenormalization) are 1 e2!1 e2+2 3[1  ln(1 2m2)] 3. Gluon The most interesting case is the propagator of the Yang-M ills eld, in a theory of Yang-Mills coupled to lower spins. There is an important simplication in this calculation in the background eld gauge: Writing the classical Yang-Mills Lagrangian astrF2=g2, the covariant derivative appears as r=@+iAwithout coupling constant, so the gauge transformation of Ais coupling independent, as in general for the matter elds. (In terms of a group element g,0=gandr0=grg1.) The eective action is gauge invariant, which means the only divergent terms involving the Yang-Mills eld are the gauge-covariantized kinetic (less mass) terms of the various elds. Thedivergences for the non-gauge elds are not so interesting, since they can be absorbed by rescaling those elds (\wave-function renormalization"), but the divergence of thetr F 2=g2term can be absorbed only by rescaling the coupling gitself. (On the other hand, if we use r=@+igA, then renormalization of grequires the opposite renormalization of Ato preserve gauge invariance.) Thus this divergence is related to the UV behavior of this coupling (as discussed in subsection VIIB6, and furtherlater). The important point is that there is no wave-function renormalization for the Yang-Mills eld (since there is no corresponding gauge-invariant counterterm), so the coupling-constant renormalization (like mass renormalizations) can be found from just the propagator correction, while in other gauges one would need also a much messier vertex (3-point) correction: BRST invariance is not enough to give the resultfrom a single graph. A. PROPAGATORS 417 We now consider the contributions of spins 0 (including ghosts) and 1 (including gluon self-interactions), and redo the spin-1/2 contribution in a way that resemblesthe bosons. It is based on the observation that there is a universal form for the gauge-covariantized Klein-Gordon equation for spins 0,1/2,1, which can also be shown bysupersymmetry. The kinetic operator in a background Yang-Mills eld is K= 1 2(iFabSba) where now =(r)2is gauge covariantized. This form is true in arbitrary dimensions. For spin 0 it is obvious. For spin 1/2, we use the fact that the one-loop contributionto the functional integral is the trace of the logarithm of the propagator, as follows from Gaussian integration, Z D D  e  K =det K =etr ln K where the trace is over all indices, including the coordinates. Then the contribution to the eective action from kinetic operator Kis 1/2 the contribution from K2.( S e e also excercise VIA4.2.) We then use (see subsection IIIC4) 2r=2=2(
r)2=(f a; bg+[ a; b])rarb= +iSabFab w h e r ew eh a v eu s e d S(1=2) ab=1 2 [a b];f a; bg=ab In the case D=4, this is equivalent to the result obtained in subsection IIIC4 in terms of just the undotted spinor, but there the 1/2 is automatically included because there are half as many elds, so the range of the trace is half as big. For spin 1, we usethe result of the background-eld version of the Fermi-Feynman gauge: At quadraticorder in the quantum elds, from excercise VIB8.1 we have 1 8F2+1 4(@
A)2!1 8(D[aAb])2+i1 4Fab[Aa;Ab] +1 4(D
A)2 =1 4A
Ai1 2Aa[Fab;Ab]=1 4A
(iFabSba)A where =(D)2contains only the background gauge eld, and in the last step we have written the quantum eld Aas a column vector in the group space and the background elds (like F) as matrices for the adjoint representation (which replaces commutators with multiplication), and used the explicit expression S(1) ab=j[aihb]j;hajbi=ab 418 VIII. GAUGE LOOPS To this order in the quantum elds, the kinetic operator for the two ghosts looks just like that for two physical scalars, but gives a contribution to the eective action of opposite sign because of statistics. This method can be used for arbitrary one-loop graphs with external gluons, and easily generalizes to massive elds; we now specialize to propagator corrections.There are two kinds of vertices, the spin-0 kind and the vertex with the spin operator.Sincetr S ab= 0, we get only graphs with either 2 spin vertices or none. There is only one spin graph, with 2 internal free propagators; the 2 spinless graphs includesuch a graph but also a tadpole, which vanishes by dimensional regularization in themassless case. Since the spinless graphs give the complete result for internal spin-0,their sum is separately gauge invariant; the spin graph is obviously so, since it isexpressed directly in terms of the eld strength. (We refer here to the Abelian partof the gauge invariance, which is all you can see from just 2-point graphs.) As far as Lorentz index algebra is concerned, we need to evaluate only tr(S abScd). For the vector, we have tr(S(1) abS(1) cd)=2b[cd]a For spin 1/2, the traces are the same as in D=4 except for overall normalization; using earlier identities, or using the same methods for this case directly, tr(S(1=2) abS(1=2) cd)=1 4tr(I)b[cd]a wheretr(I) is the size of the spinor. Excercise VIIIA3.1 Let's look at other ways to interpret the last two identities: aUse the double-line notation (subsection VC9) for the dening representation of the orthogonal group to derive the above expression for the trace of two S(1)'s. bUse the fermion action of IIIC4 in terms of just undotted spinors for D=4. Evaluate tr(S(1=2) S(1=2) ) using both bra-ket notation and double-line notation for SL(2,C). Show the result is the same as from vector notation (by relating Fabandf). All diagrams will also have a group-theory factor of tr(GiGj)ij. We'll be interested mostly in SU(N) for Yang-Mills theory (as appropriate to describe color inthe Standard Model for N=3, or arbitrary N for applying the 1/N expansion). Thenthe most interesting representations are the adjoint (for the gluons and their ghosts) A. PROPAGATORS 419 and the dening (for the quarks). As explained in subsection IB2, or as follows from the double-line notation of subsection VC9, we use the normalization trD(GiGj)=ij)trA(GiGj)=2Nij Finally, there are the momentum-space integrals, which have already been eval- uated in subsection VIIB4 for the massless case (which is sucient for determiningthe high-energy behavior, and thus the UV divergences) and VIIB5 for the massivecase. The integral for the spin graph is the same as that for  3theory (using the Sab vertex from1 21 2iFabSab). As labeled there, the external line has momentum p and the internal lines k1 2p. Then the vertex factors in the spinless graph with two propagators are both simply k(from1 2=1 2@2+1 2A
(i@)+1 2(i@)
A+1 2A2), givingAab, while the addition of the tadpole, with vertex factor , converts it to ^Aab. (By comparison, the tadpole graph that was apparently avoided in the Dirac- spinor calculation of the previous subsection appeared anyway after evaluating thetrace algebra.) This contribution also gets an overall tr(I) factor, simply counting the number of degrees of freedom. Note that the scalar factor Athat appears in ^A ab is the sum of a divergent term proportional to the 3graph and a convergent term that vanishes in the massless case. We now combine all factors to obtain the contributions to the two-gluon part of the unrenormalized 1-loop eective action (including the 1 for getting the eective action from the S-matrix, a 1 for internal fermions, either spin1 2or ghost, the1 2 for identical external gluon lines, the1 2for the spinor to compensate for squaring the propagator, and yet another1 2for identical internal lines if the group representation was real.) The result is the sum of contributions of the form 1;2g=htrZ dx1 8Fab(1 2cR)(1)2s[1 D1B1()4s2B2()]Fab wherecRis the group theory factor from the trace, which for the interesting cases is cR=2f o rNN(dening) 2Nfor adjoint (real) This result applies to spins s=0;1 2;1, with the understanding that it is the result for two polarizations, so there is an implicit extra factor of1 2for a single scalar, while for massive spin 1 (spontaneously broken gauge theories) the third polarizationin the (background-eld) Fermi-Feynman gauge is carried by a scalar eld. (Theresult above for s=1 is the sum of the contributions from the vector eld and the two 420 VIII. GAUGE LOOPS fermionic ghosts.) The functions B1andB2are the spinless and spin contributions, related to the massive 3propagator correction ^A2(p2;m2;m2)a s B2(p2)=^A2(p2;m2;m2);B1(p2)=B2(p2)+4m2B2(p2)B 2(0) p2 Note that B1B 21 lnp2 as far as divergent (at D=4) or high-energy (i.e., massless) terms are concerned. Also note that all contributions exactly cancel if all spins are in the adjoint and have the same mass, and appear in the ratio 1:4:6 for spins 1 (including ghosts),1 2,0 :F o rt h e massless case, this is N=4 super Yang-Mills, which is also the the massless sector of the dimensional reduction of the open superstring from D=10. The massless sector ofthe reduction of the open bosonic string from D=26 yields Yang-Mills plus 22 adjoint scalars, which cancels near D=4 up to nite terms. In examining the contribution of this term to the running of the coupling constant with energy, we see that the vectors contribute with opposite sign to lower spins. Inparticular, in terms of the coecient  1(of subsection VIIC1), only nonabelian vec- tors make positive contributions (since Abelian vectors are neutral). This means that nonabelian vectors are responsible for any weakening in a coupling at high energies,known as \asymptotic freedom", an important experimental feature of the strong interactions (see section VIIIC). Note that while the sign of  1for4theory, using the method of subsection VC9, is independent of the coupling (since all 1-loop correc-tions are coupling-independent when the coupling appears as an overall factor in the classical action, like  h), changing the sign of the coupling changes its sign relative to  1: The result is that this theory can be made asymptotically free only if its potential has the wrong sign (negative for large ). Thus, although nonabelian vectors are required for asymptotic freedom in physical theories, \wrong-sign 4"c a nb eu s e d as a toy model for studying features associated with asymptotic freedom (especially resummation of the perturbation expansion: see section VIIC). Note that for (massless) fermions that couple chirally to vectors (as in electroweak interactions), cRconsists of the contribution from a complex representation but not its complex conjugate: Only one of the two Weyl spinors of the Dirac spinor contributes. The result is that the contribution to the vector propagator is half that of the parity- invariant case. This fact follows from comparing the calculations of the chiral andnonchiral cases without the squared-propagator trick: In Dirac (4-component) spinor notation, the 1's drop out of the calculation; in Weyl (2-component) spinor notation, the left- and right-handed-spinor diagrams are identical except for (internal) group A. PROPAGATORS 421 theory. (Things are more complicated for higher-point functions, because the group theory gives more than just trR(GiGj): See subsection VIIIB3.) Excercise VIIIA3.2 Find the conditions for exact cancelation if spins1 2and 0 include both adjoint and dening representations. Find the weaker conditions if only the divergent(and therefore also high-energy) terms cancel. Excercise VIIIA3.3 Use the optical theorem to nd the decay rate for a massive vector (e.g., Zboson) into massive particle-antiparticle pairs of various spins. Excercise VIIIA3.4 Find the propagator correction for internal particles of dierent masses on each of the two lines (e.g., for a W boson propagator). In the case of QCD, with color gauge group SU(N c)a n dNf avors of quarks in the dening representation of color, the divergent and high-energy contributions tothis term in the unrenormalized 1-loop eective action are 1;2g;QCDht rZ dx1 8Fab1 3(2Nf11Nc)(1 ln )Fab At higher loops the eective action will still be gauge invariant in background-eld gauges (for the quantum elds), so the renormalization of the Yang-Mills couplingcan still be determined from just the gluon propagator correction. On the other hand,in other gauges a three-point vertex must also be calculated: It can be shown thatthe gauge-xed classical action, including counterterms, is BRST invariant only up to wave-function renormalizations; i.e, the most general counterterms needed (with a BRST preserving regularization) are BRST-invariant terms with additional mul-tiplicative renormalizations of the quantum elds. Thus, BRST invariance, unlikegauge invariance, is not strong enough to relate the gluon coupling and wave-functionrenormalizations. Not only does this mean evaluating many more graphs, but graphswhich make the propagator correction look easy by comparison. (This is not so dif-cult for just the one-loop divergences we have considered, but the diculty growsexponentially with the number of loops.) However, in the background-eld gauge the L-loop propagator correction has (L1)-loop vertex subdivergences, similar to those in other gauges. The net re- sult is: (1) We still have a BRST-invariant \classical" action, containing the same(counter)terms that appear in other gauges (including quantum ghosts), but covari-antized with respect to background gauge elds (and including coupling to other 422 VIII. GAUGE LOOPS background elds). However, the coecients need be calculated only to order L1 for theL-loop eective action, one loop less than in other gauges. (2) In addition, we have background-eld-only terms in the classical action whose L-loop coecients do need to be calculated, but with a relatively small amount of additional eort,due to gauge invariance. Thus renormalization consists of two steps: (1) addingBRST-invariant counterterms for the quantum elds (background covariantized) tocancel subdivergences, and (2) adding gauge-invariant counterterms for the back-ground elds (which can be interpreted as vacuum renormalization for the quantumelds) to cancel supercial divergences. Consequently, background-eld gauges save about one loop of diculty as far as renormalization is concerned. Furthermore, similar simplications occur for calculations of nite parts (e.g., eective potentials),because of simplications from gauge invariance. 4. Grand Unied Theories The best result of GUTs is their prediction that the gauge couplings of the Stan- dard Model coincide at some high energy, as a consequence of the running of thecouplings with energy. (Mixed results have been obtained for masses, arguably be-cause renormalization group arguments are accurate only for high energies, and thusleptons with large masses. A \failed" prediction is proton decay, which has alreadyeliminated the nonsupersymmetric SU(5) model with minimal Higgs.) The numericaldetails of this prediction are model dependent (and thus easy to fudge, given enoughfreedom in choice of nonminimal elds), but the fact that all three couplings comeclose together at high energies is already strong evidence in favor of unication. Thus we make only the crudest form of this calculation, using only the one-loop results of the previous subsection. The main assumption is that there is a \desert"between the Standard Model unication scale (around the masses of the intermediate vector bosons W and Z) and the Grand Unication scale M GUT, with no fundamental particles with masses in that range (although, of course, a huge number of hadronsappear there). This allows us to crudely approximate all fundamental particles belowthat region (i.e., those of the Standard Model) as massless, and all above as innitelymassive. In particular, in the framework of the minimal SU(5) GUT, this means all the fermions are treated as massless. Therefore the calculation is to use the one-loop results to calculate the running of the couplings in the Standard Model, and use the relation of the gauge couplingsin the SU(5) GUT to identify those of the Standard Model in terms of that of this A. PROPAGATORS 423 GUT. From the previous subsection, the running of the couplings is given by 1 g2(p2)1 g2 01lnM2 GUT p2 ; 1=X R;s1 2cR(1)2s(4s21 3) for two helicities of spin s(with an extra factor of1 2for only 1 helicity of spin 0), whereg0g(M2 GUT). If we useg1;g2;g3to label the couplings of U(1), SU(2), and SU(3) that are identied with the single SU(5) gauge coupling at the unication scale, then their relation to those of the Standard Model (as normalized in excercise IVB2.1) is 1 g2 1=6 51 g02=6 5cos2W e2;1 g2 2=1 g2=2sin2W e2;1 g2 3=1 g2 s wheregsandgthe usual SU(3) and SU(2) couplings, and the factor of6 5is because the U(1) generator (see subsection IVB4) satises trD(G2)=5 6in terms of SU(5) matrices. (We generally normalize to trD(G2) = 1 for each generator. Physical couplings are preserved if changes in normalization of generators are accompanied by changes in coupling normalization so as to preserve giGi.) Then the values of the 1's for the Standard Model are 1;1=041 10=41 10; 1;2=22 341 6=19 6; 1;3=1 14+0=7 where we have listed the contributions from spins 1,1 2(for 3 families), 0, respec- tively. (Note that the spinors contribute the same to each because they are all eectively massless: They don't notice the SU(5) breaking. Also, we can ignoreSU(2) U(1) breaking when calculating these 's, since we have neglected the corre- sponding masses.) Excercise VIIIA4.1 Calculate the contribution of the spinors to the  1's, in terms of both SU(5) and SU(3) SU(2) U(1) multiplets. (Note the chiral couplings for spinors, so forcRa complex representation and its complex conjugate might not both contribute.) The experimental values of the couplings (in the MS prescription) at =MZ 91 GeV are1 e2804;s i n2W:231;1 g2 s106 Unfortunately, taking any two of the equations for 1 =g2 igives widely varying answers: e.g., MGUT10152GeV 424 VIII. GAUGE LOOPS Alternatively, since we have used only two parameters to t three experimental num- bers, we can try to predict the value of any one of e,W,o rgsfrom the rest: e.g., fromeandgswe can nd sin2W:207 which shows the same disagreement (but looks better than the exponentiated error forMGUT). The result is not very accurate, since we have made many approximations, which can be improved with some eort: Two-loop corrections add lnln terms to the one- looplnterms; including the mass dependence of the eective couplings also adds signicant corrections. But the most important approximation assumption we made was the desert: Undiscovered particles, such as new fermions, nonminimal Higgs, or supersymmetric partners, change even the one-loop expressions 1. Specically, since by denition the unication scale is where the masses of all unobserved vectors reside, these new particles will all have spins 0 or1 2, and thus make the 's more negative. In particular, supersymmetrization yields a result consistent with experiment, with MGUT2:21016GeV (This has been interpreted as the only experimental verication of supersymmetry.) Excercise VIIIA4.2 Let's examine the eects of supersymmetry: aSupersymmetrize the Standard Model contributions to 1by adding the su- persymmetric partners to each spin: 1 !11 2,1 2!1 200, 0!1 200 (where the Higgs scalars have doubled because chiral scalar superelds can't satisfy reality conditions) to nd the result 1;1=063 5=33 5; 1;2=661=1; 1;3=96+0=3 bSolve for 1=g2 0,ln(M2 GUT=M2 Z)( a n dt h u s MGUT), andsin2Win terms of 1 =e2 and 1=g2 s. Then plug in to nd the numerical values. cShow the consistency condition relating the 3 couplings is 1;21;3 g2 1+1;31;1 g2 2+1;11;2 g2 3=0 and that the closest integer values for the couplings from the above data, 1 g2 1= 742;1 g2 2= 371;1 g2 3= 106 A. PROPAGATORS 425 satises it exactly. (OK, so this is just a numerical coincidence, considering experimental inaccuracies and theoretical approximations, but isn't it stillnice?) Also, note that 1 g2 1=21 g2 2)sin2W=3 13 dDrop the contributions of the Higgs (and its superpartners) to the 's in both the supersymmetric and nonsupersymmetric cases, and reevaluate sin2W, showing both give the same (poor) value. (Thus, Higgs can make a dierence.) 5. Supermatter Although the problem with infrared renormalons may be only technical, the ap- pearance of this same problem in several dierent approaches (including a nonpertur- bative one; see later) strongly suggests that the \correct" approach to quantum eld theory, in the sense of a practical method for unambiguously (i.e., with predictivepower) calculating perturbative and nonperturbative eects, might be to consider only theories that are perturbatively nite. In this subsection we will analyze gen- eral properties of supersymmetric eld theory using superspace, and in particularimproved UV behavior, concentrating on nite theories. Finite supersymmetric theories must be in particular one-loop nite. This turns out to be enough to guarantee niteness to all loops: Two-loop niteness is automatic, while an appropriate renormalization prescription is required to guarantee niteness is preserved order by order in perturbation theory. (No constraints on the couplingconstants are needed beyond those found at one loop, but without the renormalization prescription innities cancel between dierent loop orders.) Of course, wave-function renormalizations are gauge dependent: N=1 supersymmetric gauges eliminate someof these unphysical divergences (and gauges with higher supersymmetry more), as do background-eld gauges even in nonsupersymmetric theories. So, \nite theory" in general gauges refers only to the \physical" divergences | those that aect the high-energy behavior of the theory, namely those that appear in couplings and masses. Because of the nonrenormalization of chiral terms in the action (see subsection VIC5), it might seem that the corresponding couplings and masses are always un- renormalized. However, the kinetic terms of chiral superelds can receive quantumcorrections, and the true couplings are dened by eld redenitions that eliminate these rescalings. This means that all such renormalizations are related, and given by the wave function renormalizations. The only other couplings are the Yang-Mills 426 VIII. GAUGE LOOPS ones, whose renormalization is also given by kinetic terms in background-eld gauges. Thus, all \physical" renormalizations in supersymmetric theories can be found from just propagator corrections. In particular, this means that if the eective action is calculated with background-eld supergraphs, then it is completely nite in a nite theory. A possible exception to our statement of all physical renormalizations coming from propagator corrections would seem to be the Fayet-Iliopoulos tadpole termR d4V. However, massless tadpoles vanish in dimensional regularization, and massive ones require real representations, which cannot generate explicit-prepotential terms. (Inparticular, at more than one loop such terms never appear in the background-eld gauge for any representation.) The simplest one-loop propagator correction is to .( T h ecorrection van- ishes, sinceR d42= 0: See subsection VIC5.) There are two graphs to consider, one with two internal propagators, and one with internal andVVpropagators. Thedalgebra for the two graphs is identical: Both get a d2and a d2inside the loop, exactly enough to give a nonvanishing graph (using [ d2d24(0)]j0==1 ) . T h e r e is also a symmetry factor of1 2for the two propagators, and a 1f o rt h em i x e d graph because the two dierent types of internal propagator have opposite sign (and, as usual, an overall 1 to get from the T-matrix). Thus, the supersymmetry (spin) part of the algebra is almost trivial in this case. On the other hand, the internal group theory is slightly messy, so we treat the general case immediately: We take vector multiplets Vifor an arbitrary group (though we will need a semisimple group for niteness, since Abelian groups are not even asymptotically free). SumsP Gare over each simple subgroup (or each Abelian factor), since they can have independent coupling constants gRGfor representation R (especially gAGfor the adjoint, which we use for the pure super Yang-Mills term for deniteness; except for the Abelian factors, where a nontrivial representation should be substituted). Similarly, sumsP Rare over irreducible representations of the group; RIJis the corresponding projection operator. For the simple (or single-component Abelian) factors of the group Gij=cAGGij is used (but again, with a dierent normalization for the Abelian factors). We also use the group theory identities (normalizations) from subsection IB2, now generalized A. PROPAGATORS 427 to these nonsimple groups and reducible representations: GiIKGjKJRJI=X GcRGGij=X GcRG cAGGij GiIKGjKJij G=X RkRGRIJ=X RcRGdAG cAGdRRIJ Then from the Lagrangian L=Z d4I(eV)IJJ+Z d21 6IJKIJK+h:c: X G1 g2 AGZ d21 2WiWj Gij (ignoring mass terms) the result is simply 1;=hZ d4IMIJ^A2J;MIJ=X R;Gg2 AGcRGdAG cAGdRRIJ1 2IKLJKL where again ^A2is the operator representing the one-loop propagator correction (T- matrix) for self-interacting scalars. (Of course, this operator may vary depending on the internal masses; here we are concerned mostly with the divergences and leading high-energy behavior, which is mass-independent. As usual, we can rescale the gauge elds by their couplings in the Lagrangian; this moves these couplings from the prop-agators into the vertices, giving the same result for this term in , since it has no V's.) Of course, this is the identical group theory that appears in the nonsupersymmetric case; we have been more general here because we want to consider exact cancelation, while in the nonsupersymmetric case simplicity is usually more important. 6. Supergluon The supergluon self-energy calculation is similar to the nonsupersymmetric cases considered in subsections VIIIA2-3. Examining the Feynman rules, we see that those for the vector multiplets are similar to the nonsupersymmetric ones for vectors (asexpected), while those for the scalar multiplets are similar to those for spinors: d 2 and d2are analogous to (in 2 2 matrix notation) @and@*, etc. There are now only two kinds of loops to consider, vector and scalar multiplets: As for the nonsupersymmetric case, ghosts in background-eld gauges couple the same as matter, since at one loop the only coupling is to background elds and thus covariant, even for ghosts. For the real scalar supereld describing the quantum 428 VIII. GAUGE LOOPS vector multiplet, looking at the terms in the action quadratic in V(from subsection VIB10) S2V=Z dx d41 4V(+2iWD+2iW.D.)V we see that vertices have only 1 spinor derivative at most. However, we need at least 4 spinor derivatives (2 d's and 2 d's) per loop (see subsection VIC5), since the result of reducing any loop to a point in space always leaves the tadpole -integral [d:::d4(0)]j0=, which vanishes for fewer than 4 derivatives. Thus, a Vloop in a super Yang-Mills background vanishes for fewer than 4 external lines. This means the entire contribution of quantum super Yang-Mills to the supergluon prop agator correction (or 3-point correction from real representations) in the background-eldgauge comes from the 3 ghosts (including the Nielsen-Kallosh ghost), which couple t h es a m ea s3 scalar multiplets in the adjoint representation. Thus, for example, we see without evaluating a single graph that this correction vanishes for N=4 superYang-Mills, which has also 3 physical adjoint scalar multiplets. (See subsection IVC7.) For the scalar multiplets, we can nd the analog of the squared-propagator trick: The easiest way is by the method of subsection IIIC4, which automatically takes care of factors of 1 2, and can be applied classically, without worrying about functional determinants. This method requires we consider the massive theory at intermediate stages of the calculation, although the mass can be dropped at the end. The only resulting limitation is that we must restrict to real representations of the gauge group.(In other words, the couplings must preserve parity: For these terms, CP invariance is automatic, and reality means C invariance, so P invariance is implied.) However, this is a restriction of the usefulness of the squared-propagator trick anyway: Otherwise weget expressions like ( @=+iA=)(@=iA=*) which do not yield useful simplications. (They require as much work as without the trick.) In such cases we are stuck with doing the calculations the hard way. This is not just a technical diculty, it is a consequenceof the nal result being messier in such cases: For example, for real representations there is no possibility of anomalies. However, we can separate the generators into the real (scalar) and imaginary (pseudoscalar) ones: Then this trick simplies thereal (polar vector) couplings but not the imaginary (axial vector) couplings. (As for Pauli-Villars in subsection VIIIB2 below, but also for the physical elds before taking the mass to vanish after the trick has been applied, the mass term can be chosen topreserve the polar symmetries and thus violate the axial ones.) However, by comparison of the propagator correction for complex and real rep- resentations without (the supersymmetric version of) the squared-propagator trick, we see that the only dierence between the two is in the (Yang-Mills) group theory. A. PROPAGATORS 429 Thus, we can calculate for real representations rst, using the trick, and then for complex representations by simply replacing the group-theory factor in the result forthe real ones. Repeating the procedure of subsection IIIC4 with spinors replaced with chiral superelds, we begin with the Lagrangian ( S=R dx L ) L=Z d 4+mp 2Z d21 22+Z d21 22 where the chiral superelds are covariantly chiral (or background-covariantly chiral) r.=r=0 Treating as auxiliary (the 2term has no Yang-Mills coupling, as can be seen, e.g., in an \antichiral" representation), we eliminate it by its algebraic eld equation =p 2 mr2 After a trivial rescaling !21=4pm (and usingR d4=R d2r2) we obtain the action L=Z d21 2(r2r21 2m2)=Z d21 4(m2+i[W;r]) (=rara) using an identity from subsection VIC5. In the chiral vacuum-bubble loop, we no longer have an explicit chiral supereld to convertr2r2to +:::. However, using the chiral representation r2=d2,w ec a n write the kinetic operator as d2r2=d2d2+d2(r2d2) to separate the truly free part from the background interactions. Then quantization can be preformed as usual (see subsection VIC5): Essentially, we can now use thefree 0+m2=p2+m2as kinetic operator, since at each vertex there is a d2to project back to chiral superelds. Of course, in general we need only one projector in any trace over a subspace: In this case that result is obtained by integrating the d2's by parts in the loop back and forth across the free propagators, since sandwichinganyr 2d2between them produces d2(r2d2)d2=1 2( 0+i[W;r])d2 430 VIII. GAUGE LOOPS Repeating the procedure till only one d2is left, the Feynman rules for this loop become propagator :1 1 2(p2+m2)4(0) one vertex : d2(r2d2) other vertices :1 2( 0+i[W;r]) We thus see that one vertex has at most 3 derivatives ( d2d) while the other has at most 1 ( d): d2(r2d2)=d2[iAd+1 2i(dA)1 2AA] 1 2( 0+i[W;r]) =iWd+1 2i(dW)1 2[W;A]+iAa@a+1 2i(@aAa)1 2AaAa exactly the minimum needed. (Thus, there are insucient derivatives for a tadpole contribution to the propagator.) The result for this diagram is then the same as thecorresponding diagram in bosonic ' 3theory, with a group theory factor tr(GiGj), and replacing '(p)'(p)w i t h Z d4d40[iWi(p;0)d0 4(0)][iAj(p;)d2d4(0)] =Z d41 2Wi(p;)Aj (p;)=Z d21 2Wi(p;)Wj (p;) usingd=d0, integration by parts, [ d2d24(0)]j0==1 ,a n dW=d2A(chiral representation). Written in the notation of subsection VIIIA3, the 2-supergluon part of the unrenormalized 1-loop eective action is then 1;2sg=ht rZ dx d21 2W(1 2cR)^A2W for a scalar multiplet, and exactly 3 times that for a vector multiplet, including the massive case. Thus, cancelations again survive the introduction of masses. Also, if the masses of the various scalar multiplets are equal the entire propagator correction is canceled in such theories, while for unequal masses only the divergence, and thecorresponding leading (logarithmic) high-energy term, is canceled. Excercise VIIIA6.1 Show this result agrees with the restriction to N=1 supersymmetric theories of the component result of subsection VIIIA3. Excercise VIIIA6.2 Take the result of subsection VIIIA3 literally for allspinss(arbitrarily large). Using the fact that multiplets with N+1 supersymmetries can be written as 2 multiplets with N supersymmetries, diering in maximum helicity by 1/2, A. PROPAGATORS 431 recursively nd the result for general s(now labeling maximum helicity) for all values of N1, and show it vanishes for N 3. Excercise VIIIA6.3 Calculate the chiral scalar contribution to the one-loop supergluon propagator correction without the squared-propagator trick. (Hint: There are 8 spinorderivatives in the loop. Integrating them by parts o one propagator produces3 terms, since the number of d's and d's inside must be equal, because what's left is always spacetime derivatives on [ d 2d24(0)]j0=.) Generalizing the group theory as in the previous subsection, we have the total result 1;VV=hZ dx d2X G1 2Wi(1 2MG)Gij^A2Wj ;MG=X RcRG cAG3 (For Abelian factors, irrelevant for niteness, we should take the cAGfactor out of Gijand put it into MG;t h e ncAG= 0 for Abelian groups, so MG!P RcRG>0.) Therefore, combining with the results of the previous subsection, the conditions forniteness are X RcRG cAG=3;X R;Gg2 AGcRGdAG cAGdRRIJ=1 2IKLJKL In particular, for the case of N=4 super Yang-Mills written in terms of N=1 superelds (see subsection IVC7), we have 3 adjoint chiral scalars IwithI=iI0,w h e r eiis the adjoint label and I0=1;2;3( w h i c ha p p e a r e da st h el a b e l Iin subsection IVC7, where the adjoint label was implicit in matrix notation). Then IJK=gAfijkI0J0K0 and the above two niteness conditions reduce to I0 I0=3;J0 I0=1 2I0K0L0J0K0L0(j i=fiklfjkl) As explained in the previous subsection, in the general case the niteness conditions may receive quantum corrections at 3 loops and beyond, depending on the model and renormalization prescription, but no new conditions are added. Presently there is no deep understanding for the niteness of these models (at least, not deep enough to always avoid the quantum corrections to the niteness con- ditions). Note that they are nite for arbitrary values of the couplings, up to thetwo above restrictions: For example, we can scale all the couplings by a common 432 VIII. GAUGE LOOPS factor. Thus, they are nite order-by-order in perturbation theory (loops). Non- supersymmetric theories can also be nite, but only for specic numerical values of the coupling, i.e., not for arbitrarily small values of the coupling, and thus not order-by-order in the loop expansion; they therefore suer from the renormalon problem.(The renormalon-like behavior of instantons is not a problem in the framework of the 1/N cexpansion.) The niteness of theories with extended supersymmetry has been explained by various arguments (in particular, for N=2 there are no divergences be-yond 1 loop even for theories that are just renormalizable), but none of these appliesto the general case of simple supersymmetry. To obtain more realistic models, we may want to consider adding \soft" super- symmetry breaking terms (those which have little eect on high-energy behavior), asintroduced in subsection IVC6, to these nite theories. Finiteness can be maintained, but the conditions become considerably more complicated in the general case. Note that spontaneous breaking of supersymmetry is not allowed, because the rst condi-tion prohibits U(1) factors (with c AG=0 ;t h u sn oR d4Vterms), while the second prohibits gauge-singlet matter (with cRG=0 ;t h u sn oR d2terms). 7. Bosonization A common method in eld theory is to consider simpler models where calculations are easier, and see if they are analogous enough to give some insight. In particular, two-dimensional models sometimes have perturbative features that are expected only nonperturbatively in four dimensions: For example, we saw in subsection VIIB3the generation of bound states at one loop in the 2D CP(n) model. Of course,some of the features may be misleadingly simple, and may have no analog in D=4. Two-dimensional theories, especially free, massless ones, are also useful to describe the quantum mechanics of the worldsheet in string theory (see chapter XI). In thissubsection we consider free, massless 2D theories: Essentially, this means just thescalar and the spinor, since there are no transverse dimensions to give gauge eldsnontrivial components. Spinor notation is very simple in D=2, since the Lorentz group is SO(1,1)=GL(1). For that purpose it's convenient to use lightcone notation. 2D matrices can be chosen as +=0 i0 0
; =0 0i 0
; 1=1p 2i 00 i
; =0 ii 0
=p 2 0 =   ; =(i i ) A. PROPAGATORS 433 In general, even-D matrices can be constructed as direct products of D/2 sets of 2D matrices, so tr(I)=2D=2. (For details, see subsection XC1.) The Lagrangian for a massless, complex spinor can be written this way as L= i@= =  (i@ ) + (i@) (This also follows from truncation of 4D spinor notation.) Note that and transform independently under proper Lorentz transformations, as do their real and imaginary parts. Thus, we can not only impose a reality condition, but also a chirality condition, dropping or : A single real component is enough to not only dene a spinor Lorentz representation, but also construct an action. In position space, the propagator for a massless scalar is (see excercise VIIB1.1) 1 1 22(i)2(xx0)=ln[(xx0)2] up to a real, dimensionful constant: We use units =1 . T h e\i"f o rt h ecomes from Wick rotation back to Minkowski space. The propagator for a massless spinor isthen ( =2@+@, shortening the double 's and 's from spinor to lightcone vector notation) 1 i@2(i)2(xx0)=i@fln[(xx0)2]g=i (xx0) which has the unusual consequence (its wave equation) @1 (xx0)=2i2(xx0) with an appropriate -prescription implicit: Excercise VIIIA7.1 Show (e.g., by an innitesimal Wick rotation) that the correct iprescription for the spinor propagator is i (xx0)i(tt0)=(tt0)i (xx0)i+(t0t)i (xx0)+i and that it satises the wave equation. ( tt0can be replaced with ( xx0) in the above.) In D=2, the theory of a massless spinor eld is equivalent to that of a massless scalar eld (\bosonization"). The explicit correspondence between the spinor andscalar is shown by separating the scalar into its left-propagating and right-propagating 434 VIII. GAUGE LOOPS parts: This can be accomplished by dierentiating with respect to xa n dt h e ni n t e - grating back. We thus write =(+)+() where()is a function of only xand notxon-shell, and has propagator lnx. The product of two fermion propagators, for  (x) (x0)a n d  (x0) (x), as would follow from multiplying  (x) (x)w i t h  (x0) (x0) in the functional integral (including the1 for the fermion \loop", from reordering the 's) is i (xx0)i (x0x) =@@0 fln[(xx0)2]g This shows that @()has the same propagator as   ,w i t h=(; )n o t summed, so we can equate @()=   which gives an explicit expression for the scalar in terms of the spinor after integration overx. The inverse relation is (quantum mechanically, not classically) =ei() which can be checked by a sum of multiloop diagrams: The equivalent of connecting (x)t o (x0) by a single propagator is to expand ei()(x)andei()(x0)each tonth order and connect xandx0withn()propagators, and sum over n(with appropriate combinatoric factors). The ()propagators are ln[i(xx0)] with the \i" determined by Wick rotation. The propagator of the composite fermions is then:1X n=01 n!fln[i(xx0)]gn=i (xx0) Although this gives the appearance of a scalar being the bound state of spinors, and vice versa, even in the free theory, there is a simpler interpretation, even clas- sically: Massless particles in D=2 travel at the speed of light in one of two possible A. PROPAGATORS 435 directions. Thus, a collection of free \left-(or right-)handed" massless particles travels along together, not separating, and thus acting like a bound state. (As shown in sub- section VC8, singularities in perturbative quantum eld theory directly correspond to congurations in classical mechanics.) Bosonization extends to massive fermions: The \massive Thirring model" L= (i@ ) + (i@) +mp 2(  + )+g   is equivalent to the \sine-Gordon model" L=1 2[1 4(@)2+1 22(1cos  )] with the above relation between the spinor and scalar elds, and 1 2=1+2g;2 2m (Note in particular the free massive fermion for = 1.) In this case the bound states are dynamical. Note that the relation is between strong coupling in one theory and weak in the other (\duality"). 8. Schwinger model The simplest interacting model in D=2 is the \Schwinger model", massless QED. This theory is even simpler than scalar theories because its interactions occur onlythrough a massless gauge vector, which has no physical polarizations in two dimen- sions (D2=0). The most interesting feature of the Schwinger model is that all amplitudes with external vectors can be calculated exactly. In fact, the only nonvanishing 1PI vectoramplitude is the one-loop propagator correction, which gives just a mass term. In that sense the theory is trivial, and describes just a massive vector. However, the methods of calculation are instructive. We rst consider some simple methods ofcalculation of just the propagator correction, and then show that it is the only 1PI vector graph. One method we have already considered is dimensional regularization; from subsection VIIIA2-3 we have the contribution to the eective action (correctingfor the 2D normalization tr(I)=2 ) 1=Z dx F1F where we write Fab=abFin D=2. Although this calculation needs no renormal- ization, regularization is still necessary to allow naive manipulation of the integrand: 436 VIII. GAUGE LOOPS Using dimensional regularization, we see from the result of subsection VIIIA3 that we get a factor of1 D14s2inD=2+2, canceling the 1 =pole from the scalar integral. It can also be calculated in position space, using the methods of the previous subsection. The Lagrangian in lightcone notation is L=1 4e2F2+[ (i@ +A ) + (i@+A) ] F=@A @ A We can calculate separately the contributions of and to fermion loops. The \photon" propagator correction consists of the product of two fermion propagators,as given in the previous subsection. We then nd for the eective action (includinganother1f o rT!a n da 1 2for identical external lines), after including a nite counterterm to restore gauge invariance, 1 2A(@+)21 1 2A+1 2A+(@)21 1 2A+A+A=F1F (after integration by parts). This same calculation also gives the \axial anomaly": Consider an axial vector gauge eld Bthat couples to the current   (not summed), in addition to A's coupling to   .( I n D = 2 , Wa=abVb)W=V.) The contribution to the 1-loop eective action with one of each vector externally is, after including acounterterm to preserve Agauge invariance (and therefore break Bgauge invariance), B (@+)21 1 2AB+(@)21 1 2A+BA++B+A=(@
B)1 1 2F The anomaly is the breaking of Bgauge invariance, B=@)=Z @
 B=2Z F An anomaly is by denition a quantum eect: As we have seen from the 2D axial anomaly, it is related to a divergence that violates naive classical arguments,since the regulator itself violates the symmetry. In the axial case there is no actualdivergent term in the eective action, but a nite term results from a =type of cancelation. Dimensional analysis immediately reveals that the propagator correctionis the only graph in D=2 that can contribute such a term from the fermion loop.(Fermion propagators go as 1 =p, while the vertex is a constant: The electric charge has dimension in D 6=4.) A. PROPAGATORS 437 The complete one-loop eective action for the vectors then follows directly from the complete anomaly for the axial current, and the vanishing of the anomaly for the polar current: By separating out the anomalous term in the eective action, =Z F1F+
; J= A;
J=(
) A @
J=0;@J=2F)@
(
J)=@(
J)=0)
J=0)
= 0 (up to an irrelevant constant), where @J=ab@aJbis the curl of the polar current, but also the divergence of the axial current. (There are some questions of boundary conditions in solving the divergence- and curl-free conditions as
J= 0, but these are resolved by working in Euclidean momentum space.) Similar remarks apply to external gravity: From a similar calculation, replacing the vector current with the energy-momentum tensor, we nd @mTmn=0;@mm nTnppm@mR)Tm mR;R1R whereRis the 2D curvature (which is just a scalar, as the vector eld strength is a pseudoscalar). While in the vector case the nite local counterterm was chosen topreserve polar gauge invariance and thus violate axial, for the tensor case a term is chosen to preserve local conservation of energy-momentum and thus violate conformal invarianceT mm= 0. (The above expressions are linearized, but the results can be generalized to fully nonlinear gravity.) Excercise VIIIA8.1 Calculate the gravitational anomalies from a massless spinor loop in D=2, using the classical expressions (as follow from dimensional and Lorentz anal-ysis) T 1 2 i$ @ ;T +=0 (If you work in terms of you can dene the perturbative eld habsuch that =hab=Tab.) The simple form of the eective actions in the Schwinger model is a consequence of bosonization: Thus, including coupling to electromagnetism and gravity, the action for the massless spinor is equivalent to L=1 4+(F+R) Integrating out the scalar generates the above eective actions classically . Excercise VIIIA8.2 The above action is dual to the mass term of the St uckelberg action: 438 VIII. GAUGE LOOPS aConsider the rst-order Lagrangian L=G2+Ga(mAa+@a) Eliminating the auxiliary eld Gaby its eld equation yields the usual mass term for the St uckelberg model. Show that if we vary instead and solve the resulting constraint on G, we obtain (the nongravitational part of) the previous action. bGeneralize this construction to D=4, where the eld dual to the St uckelberg scalar is now an antisymmetric tensor gauge eld. (See excercise IIB2.1.) REFERENCES 1A.A. Slavnov, Theor. Math. Phys. 10(1972) 99; J.C. Taylor, Nucl. Phys. B33 (1971) 436: graphical form of unitarity for nonabelian gauge theories (later simplied as BRST). 2G. 't Hooft, Nucl. Phys. B33 (1971) 173, G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189, 50(1972) 318; B.W. Lee and J. Zinn-Justin, Phys. Rev. D5(1972) 3121, 3137, 3135, 7(1973) 1049: renormalization proof for nonabelian gauge theories. 3V.S. Vanyashin and M.V. Terent'ev, Sov. Phys. JETP 21(1965) 375: rst 1-loop gluon coupling renormalization, but neglecting ghosts. 4I.B. Khriplovich, Sov. J. Nucl. Phys. 10(1970) 235: rst complete 1-loop gluon coupling renormalization. 5G. 't Hooft, unpublished comment after K. Symanzik's talk at Colloquium on renor- malization of Yang-Mills elds and applications to particle physics , Marseille, June 19 -23, 1972; H.D. Politzer, Phys. Rev. Lett. 30(1973) 1346; D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30(1973) 1343: asymptotic freedom. 6R.J. Hughes, Phys. Lett. 97B (1980) 246: analyzed 1-loop gluon propagator corrections in terms of spinless + spin contributions. 7T.L. Curtright, Phys. Lett. 102B (1981) 17: extended this analysis to higher spins, supersymmetry, and massless sectors of reducedstrings. 8Z. Bern, G. Chalmers, L. Dixon, and D.A. Kosower, hep-ph/9312333, Phys. Rev. Lett. 72(1994) 2134; G.D. Mahlon, hep-ph/9312276, Phys. Rev. D49 (1994) 4438: generalization of Parke-Taylor amplitudes to one loop. 9Z. Bern, L. Dixon, D.A. Kosower, hep-ph/9602280, Ann. Rev. Nucl. Part. Sci. 46(1996) 109: review of modern methods for one-loop graphs. 10H. Georgi, H. Quinn, and S. Weinberg, Phys. Rev. Lett. 33(1974) 451: test of running couplings in SU(5) GUT. 11S. Dimopoulos, S. Raby, and F. Wilczek, Phys. Rev. D24 (1981) 1681: test of running couplings in supersymmetric SU(5) GUT. A. PROPAGATORS 439 12P. Langacker and N. Polonsky, Phys. Rev. D47 (1993) 4028: a review of accurate tests of supersymmetric and nonsupersymmetric GUTs, empha- sizing the improvement from supersymmetry. 13A. Parkes and P. West, Phys. Lett. 138B (1984) 99; D.R.T. Jones and L. Mezincescu, Phys. Lett. 138B (1984) 293: one-loop niteness implies two-loop niteness. 14D.I. Kazakov, Phys. Lett. 179B (1986) 352; A.V. Ermushev, D.I. Kazakov, and O.V. Tarasov, Nucl. Phys. B281 (1987) 72; O. Piguet and K. Sibold, Phys. Lett. 177B (1986) 373: one-loop niteness implies all-loop niteness. 15D.R.T. Jones, L. Mezincescu, and Y.-P. Yao, Phys. Lett. 148B (1984) 317: one-loop nite theories with soft breaking. 16I. Jack and D.R.T Jones, hep-ph/9405233, Phys. Lett. 333B (1994) 372: one-loop niteness of softly-broken theories implies two-loop niteness. 17Avdeev, Kazakov, and Kondrashuk; Arkani-Hamed, Giudice, Luty, and Rattazzi; loc. cit.(IVC): one-loop niteness of softly-broken theories implies all-loop niteness. 18S. Hamidi, J. Patera, and J.H. Schwarz, Phys. Lett. 141B (1984) 349, S. Hamidi and J.H. Schwarz, Phys. Lett. 147B (1984) 301; D.R.T. Jones and S. Raby, Phys. Lett. 143B (1984) 137, J.E. Bjorkman, D.R.T. Jones, and S. Raby, Nucl. Phys. B259 (1985) 503; J. Le on, J. Perez-Mercader, M. Quiros, and J. Ramirez-Mittelbrunn, Phys. Lett. 156B (1985) 66; X.-D. Jiang and X.-J. Zhou, Phys. Lett. 197B (1987) 156; D. Kapetanakis, M. Mondrag on, and G. Zoupanos, hep-ph/9210218, Z. Phys. C 60 (1993) 181; D.I.Kazakov, M.Yu.Kalmykov, I.N.Kondrashuk, and A.V.Gladyshev, hep-ph/9511419, Nucl. Phys. B471 (1996) 389: realistic nite models. 19Mandelstam; Brink, Lindgren, and Nilsson; loc. cit. (VIB): niteness proof for N=4 using lightcone superelds. 20Grisaru and Siegel, loc. cit. ( V I C ,r e f .1 2 ) : nonrenormalization theorems for extended supersymmetry. 21P. Howe, K.S. Stelle and P. West, Phys. Lett. 124B (1983) 55: niteness proof for N=2 (+ matter) and N=4 using nonrenormalization theorems. 22V.A. Novikov, M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Nucl. Phys. B229 (1983) 381: niteness proof for N 2 based on instantons. 23P. Jordan, Z. Phys. 93(1935) 464; M. Born and N.S. Nagendra Nath, Proc. Ind. Acad. Sci. 3(1936) 318; A. Sokolow, P h y s .Z .S o w j . 12(1937) 148; S. Tomonaga, Prog. Theo. Phys. 5(1950) 544: early attempts at bosonization. 24W. Thirring, Ann. Phys. 3(1958) 91. 25T.H.R. Skyrme, P r o c .R o y .S o c . A262 (1961) 237; D. Mattis and E. Lieb, J. Math. Phys. 6(1965) 304; B. Klaiber, The Thirring model, in Lectures in theoretical physics , eds. A.O. Barut and W.E. Brittin (Gordon and Breach, 1968) v. X-A, p. 141; 440 VIII. GAUGE LOOPS R.F. Streater and I.F. Wilde, Nucl. Phys. B24 (1970) 561; J. Lowenstein and J. Swieca, Ann. Phys. 68(1971) 172; K. Bardak ci and M.B. Halpern, Phys. Rev. D3(1971) 2493; G. Dell'Antonio, Y. Frishman, and D. Zwanziger, Phys. Rev. D6(1972) 988; A. Casher, J. Kogut, and L. Susskind, Phys. Rev. Lett. 31(1973) 792, Phys. Rev. D10 (1974) 732;A. Luther and I. Peschel, Phys. Rev. B9(1974) 2911; A. Luther and V. Emery, Phys. Rev. Lett. 33(1974) 598; S. Coleman, Phys. Rev. D11 (1975) 2088; B. Schroer, Phys. Rep. 23(1976) 314; S. Mandelstam, Phys. Rev. D11 (1975) 3026; J. Kogut and L. Susskind, Phys. Rev. D11 (1975) 3594: bosonization (mostly for sine-Gordon $Thirring). 26J. Schwinger, Phys. Rev. 128(1962) 2425. B. LOW ENERGY 441 :::::::::::::::::::::::: :::::::::::::::::::::::: :::::::::::::::::::::::: B. LOW ENERGY :::::::::::::::::::::::: In general, the only loop corrections that can be evaluated exactly in terms of el- ementary functions are the one-loop propagator corrections. However, limiting forms of vertex corrections, for various low- or high-energy limits, explicitly yield the most important pieces for certain applications. 1. JWKB Some low-energy contributions to the eective action can be obtained by vari- ous quantum mechanical JWKB approximations. This involves an expansion of the external eld about its vacuum value in spacetime derivatives (momenta). Such an expansion makes sense if this eld is massless, since then small spatial momentummeans also small energy, in the relativistic sense. (Otherwise one needs to expandnonrelativistically, about ~p= 0 butE=m. Such treatments were considered in sub- section IIB5, and will be applied to loops in subsection VIIIB6.) It also can be useful when the mass of the external eld is small compared to the mass scale relevant tothe interactions, such as for chiral symmetry breaking in the low-energy description of light mesons (subsection IVA4). On the other hand, the elds we are integrating out must be massive, with a mass greater than the energy we want to investigate: Otherwise, the internal particleswould show up as poles (and cuts) in the amplitudes, where Taylor expansion in momenta would be a poor approximation. The basic principle for analyzing the behavior of such a theory in a certain energy range is thus to rst nd contributionsto the eective action where: (1) only particles with masses of lower energy appear on external (background) lines, and (2) only particles with masses of higher energy appear on internal (quantum) lines. These contributions are approximated by Taylorexpansion to nite order in external momenta, yielding a local eective action. We could then consider nishing the functional integration by integrating out the lighter particles on internal lines: However, in this approximation it would be inaccurate toconsider such particles in loops, since there they would include energies above the approximation scale. Thus, the eective action obtained by integrating out just the heavier elds is useful only when the lighter elds are treated classically. We applythe same approximation scheme to the classical action: Eliminate the heavier elds by their classical equations of motion, and Taylor expand their propagators in momenta to the desired order to get a local result. 442 VIII. GAUGE LOOPS In subsection VIIB2 we saw the simplest example, the eective potential: In that case the constant background scalar eld acted as just a correction to the mass. We now consider more complicated cases, where spin and gauge invariance play roles for the internal or external elds. In particular, adding coordinate dependence to the background elds means we need to consider more general propagators for quadratickinetic operators, such as harmonic oscillators. We saw in subsection VIB1 the most general relativistic particle action for a scalar in external elds that was quadratic in xand.x. We now consider such actions in more detail: They are the most general ones for which we can derive one-loop results to allorders in the external elds (i.e., without performing the JWKB expansion beyond the rst quantum correction, which requires Taylor expanding the exponential in terms that are beyond quadratic, thus expanding in the number of external elds). Without loss of generality, we can consider Lagrangians that are homogeneous of second order in xand.x: Terms linear in.xare boundary (in )t e r m s ,a n dw e r e already eliminated by a gauge transformation (radial gauge). Terms linear in xcan be removed by a translation, in the presence of an x 2term (which is needed to bound anxterm in the potential). (Both these kinds of terms can be restored trivially at the end.) A constant term is also trivial, giving a contribution to the classical action that is just that times T(after integrationRT 0d), and can be treated separately. (It doesn't contribute to the equations of motion.) The remaining contribution to the mechanics action is then of the form (as usual, in the gauge v=1 ) S=ZT 0d1 2[.x2+xA.x+xBx])..x+A.x+Bx=0 )S=ZT 0d1 2(.x2x..x)=1 2(x.x)jT 0 whereAis an antisymmetric matrix and Bsymmetric. The steps to this contribution to the one-loop eective eld action are then: (1) Solve the equations of motion, whichare homogeneous second-order dierential equations. (2) Change variables from the two parameters used for each xtox(0) andx(T). (Second-order dierential equations require two initial conditions, or one initial and one nal.) (3) Find S(x(0);x(T)), including separately the contribution from the constant term in the Lagrangian. (4) Find the propagator for \time" T, including the e iSand the van Vleck determinant. (See excercise VA2.4.) (5) Integrate the propagator over Tto nd . (See subsection VIIB2.) For example, consider in QED the contribution to from a fermion loop. If we are interested in only the properties of photons, then this gives the entire contribu- tion to the functional integral from integrating out the fermions: This contribution, B. LOW ENERGY 443 plus the classical (free) Maxwell action, gives a nonlocal \classical" action of self- interacting photons, which can itself be quantized to give the exact QED result for external photons. Although this one-loop eective action is too dicult to calculateexactly, the rst-quantized JWKB approximation can give an accurate descriptionat energies small compared to the electron mass. Note that we are simultaneouslyapproximating to the rst quantum correction in JWKB expansions of both the eld(second-quantized) type (one-loop) and the mechanics (rst-quantized) type. The mechanics action for a massive particle in a constant external electromagnetic eld strength (the lowest nontrivial order, but also the highest that keeps the action quadratic), in the radial gauge for the background eld and ane parametrization ofthe worldline, is (see subsection VIB1) S=Z d 1 2(.x2+xaFab.xb+M2) To include spin, we identify (see subsection VIIIA3) M2=m2iSabFab Since the only appearance of spin operators in the calculation of the propagator (denominator) is this constant matrix, it commutes with everything, so we can treatit as a number till the last step. The equation of motion ..x+F.x=0 is easily solved in matrix notation. (Hint: Solve for.xrst.) Finding x i=x(0) and xf=x(T) in terms of our integration parameters and inverting, then expressing.x(0) and.x(T)i nt e r m so f xiandxf(andTandF), and making use of the antisymmetry ofF, the result is S=1 4(xfxi)Fc o t h (FT 2)(xfxi)1 2xfFxi+1 2M2T The propagator is then given by (see subsections VA2 and VIIB2) hxfjeiTHjxii=s det@2(iS) @xf@xieiS Plugging in, and then Wick rotating T!iT, we nd for the propagator with ends tied together hxjeTHjxi=r detiF 1eiFTeM2T=2 444 VIII. GAUGE LOOPS Finally, the contribution to the eective action is (see subsection VIIB2) =cZ dxZ1 0dT Ttr r detiF 1eiFTeM2T=2r detI Tem2T=2! =cZ dxZ1 0dT TD=21em2T=2"r detiFT 1eiFTtr(eiS
FT=2)tr(I)# wherec=1 2for fermions, for statistics and squaring the propagator. (The \ det"i s for the vector indices on Fab,t h e\tr" is for the spin indices from powers of Sabin \S
F".) Excercise VIIIB1.1 Explicitly evaluate the determinant and trace for D=2. Excercise VIIIB1.2 Expand in Fand show the resulting F2terms agree with those obtained in subsection VIIIA2-3. Excercise VIIIB1.3 Consider the quadratic action S=Z d1 2[.x2+x(aaT).xxaaTx] where the matrix acommutes with its transpose ([ a;aT] = 0). Solve the eld equations for S(xi;xf;T). FindhxjeTHjxi. 2. Axial anomaly The axial anomaly comes from a nite graph, as we have already seen in subsection VIIIA8 for the case D=2. However, the naive manipulations that would show the graph to preserve gauge invariance involve evaluating the nite dierence between divergent graphs, each of which needs regularization. Although in some cases the graph can be evaluated explicitly, and then shown to be anomalous, it is generally easier, and more instructive, to analyze the anomaly by itself. The axial anomaly is associated with the use of tensors. In renormalizable theories in D=4, these occur only through 1's for spinors. (In nonrenormalizable theories, or in D=2, tensors can occur in scalar theories. There is also the term abcdFabFcd, which is a total divergence, and has no eect in perturbation theory.) In general even dimensions, the massless kinetic term for a spinor is invariant under transformations generated by 1, but the mass term is not. Chiral symmetry is thus B. LOW ENERGY 445 related to masslessness; this is also true for conformal invariance, so it's not surpris- ing that quantum corrections can break both. (In fact, in supersymmetric theoriesconformal symmetry is related to a particular chiral symmetry by supersymmetry, so breaking of one requires breaking of the other if supersymmetry is to be preserved.) Dimensional regularization manifestly preserves neither conformal nor chiral in- variance; no regularization does. The existence of these anomalies proves the impos- sibility of such a regularization. Furthermore, dimensional reduction has dicultydealing with 1; it even has inconsistencies in the presence of axial anomalies. On the other hand, Pauli-Villars regularization is especially convenient for dealing with axial anomalies because it regularizes by introducing masses. Thus, it breaks chiralsymmetry explicitly but softly, conveniently parametrizing the breaking by mass pa- rameters. We therefore will use Pauli-Villars regularization for the single purpose of evaluating the axial anomaly. The basic idea of Pauli-Villars regularization is to include massive \ghost" elds which would cancel graphs from physical elds if they had the same mass. But themasses of the ghosts are used as regulators; after subtracting local divergences, the regulator mass is taken to innity. In our case, as we'll see by explicit evaluation, the anomaly itself is nite, so no subtraction is necessary. The graph whose anomaly we want to evaluate is a one-loop 1PI graph with external vectors and a massless internal spinor. Of the vectors, all but one is a \polar" vector, coupling to  a , while the last is an \axial" vector, coupling to  1 a . These are the currents associated with the symmetries 0=ei and 0=e 1 ( 2 1=1 2). We add to this graph a similar one, but with a massive spinor, and give the second graph an overall relative minus sign. Since the massbreaks chiral invariance, we have explicitly broken the gauge invariance of the axial vector, while preserving those of the polar vectors. Note that this is a feature of the regularization: If a regularization existed that preserved chiral symmetry, then wecould freely move the 1around the graph from one vertex to the next using the usual naive anticommutation relations, thus moving also the anomaly from one vertex to the next (i.e., violating gauge invariance in any vector we choose). 446 VIII. GAUGE LOOPS Gauge invariance is represented by vanishing divergence of the corresponding current: At each vertex we have the couplingR A
J, with gauge invariance A=@, implying@
J= 0 by integration by parts, where Jmay be polar or axial depending on the vertex. These currents are conserved classically. We know they are also conserved quantum mechanically in the absence of 1's, since dimensional regularization and renormalization preserve the gauge invariance of the eective action. In graphical terms, taking the divergence at a vertex kills a prop agator (since @
Jis proportional to the eld equations of the internal eld), and this can be shown to lead to vanishingof the graph. However, with the Pauli-Villars regulator, the classical conservation of the axial current is explicitly broken. The result is that the complete axial anomaly can be found by looking at just the contribution coming from this explicit classical violationof current conservation (inserted into the one-loop graph). (The classically vanish- ing contributions are actually nonvanishing because of the anomaly, but they cancel between the physical and regulator elds, precisely because the regularization allowsthe naive manipulations that justify dropping them.) We therefore want to evaluate the anomaly @ aJa(x)@a Aa(x) where we start with a term in the classical actionR A
J, so classically J=S=A , and then evaluate its quantum correction by looking at J=A in terms of the one-loop part of the eective action . Classically, we nd a contribution from only the regulator, @
J@
(p 2i 1 )=2m 1 So, all we need to evaluate is a one-loop diagram with the axial vector coupling to the regulator replaced with a pseudoscalar couplingR  1 , and look at the graphs with one external pseudoscalar and the rest polar vectors. Clearly this is the same as coupling the pseudoscalar to the propagator of a bosonic spinor regulator in an external vector eld: @
J=2mtr 11 ir=+mp 2! =2mt r 1(ir=+mp 2)1 r=2+1 2m2 =p 2m2tr 11 r=2+1 2m2 where the trace is in the -matrix space. In the limit m!1 , graphs with more external lines vanish more rapidly. On the other hand, we need at least D/2 fac- tors ofSab(D -matrices) to give a nonvanishing -matrix trace. Thus, the leading B. LOW ENERGY 447 contribution will be, using 2r=2= +iSabFabfrom subsection IIIC4, @
J=p 2m2tr 11 1 2(m2 0)1 2iSabFab1 1 2(m2 0)


1 2iSabFab1 1 2(m2 0) with D/2+1 propagators, where 0=(@a)2. Thus, the only Feynman diagram we actually need to evaluate is the one-loop 1PI diagram with external and internal scalars. The limit internal m!1 is the same as the limit external p!0. (The result does not depend on the internal momentum, which is integrated over, nor the external mass, which would appear only in external propagators.) Thus, this is just an eective potential calculation. We therefore have the integral (see subsection VIIB1) Z dk1 [1 2(k2+m2)]D=2+1=1 (D 2+1 )1 2m2 )@
J=2p 2 (D 2)!tr[ 1(1 2iSabFab)D=2] Excercise VIIIB2.1 Check this result by using the expression from subsection VIIIB1 for thepropagator in a constant external electromagnetic eld (strength). To evaluate in arbitrary even D, we note that the normalization of 1is such that we can choose ( 1)2=1 2) 1=(i)D=22(D1)=2 0 1


D1 tr(I)=2D=2;01


D=1)tr[ 1(1 2iSabFab)D=2]=1p 2(1 2)D=2ab


cdFab:::Fcd )@
J=21 2D=2(D 2)!ab


cdFab:::Fcd Thus, for example, for the Schwinger model (D=2) we have @
J=2F in agreement with subsection VIIIA8, while for D=4 @
J=1 4abcdFabFcd 3. Anomaly cancelation When the anomaly occurs in a current that couples to a gauge eld, unitarity is destroyed, since gauge invariance implies current conservation. This is a potential 448 VIII. GAUGE LOOPS problem, since axial vector couplings occur in the Standard Model. (Actually, they are \VA": (vector)(axial vector).) The only way to avoid this problem is to have an anomaly cancelation between the dierent spinors: The coecient of the anomalyis given purely by group theory, as tr(A;fB;Cg), whereA;B;C are the matrices representing the couplings of the three vectors to all spinors, and the anticommuta-tor comes from Bose symmetrization (from the crossed and uncrossed graph in theS-matrix, or the single contribution multiplying commuting elds in the eective ac-tion). We therefore require this trace (which represents the sum over all spinors)to vanish. (See excercises IB5.3 and VC9.2d for an example of the calculation ofthis trace.) The representations in the Standard Model have been chosen so this cancelation occurs in each family. We already know in terms of Dirac notation that axial anomalies appear only in the presence of 1's. An absence of 1's is equivalent in terms of Weyl no- tation to the use of a (pseudo)real representation for undotted Weyl spinors. Forexample, consider a real representation that is reducible to a smaller (by half) rep-resentation \R" and its complex conjugate \ R": Then we can complex conjugate the complex-conjugate representation to produce a dotted Weyl spinor that is the same representation as the undotted spinor. The undotted and dotted spinor canthen be combined, as usual, to form a Dirac spinor, which transforms as the complexrepresentation, without 1's, and thus the same goes for the coupling of the gauge vector: R R! R R.! R So, in Dirac notation we can see that such representations do not contribute to anoma- lies because of the absence of 1's. Similar remarks apply to general real or pseudo- real representations: We can take an arbitrary (pseudo)real representation and makeaMajorana spinor, as R! R R.! R where now  R.is simply the complex conjugate of RsinceR=R. This cancelation also can be seen directly in terms of Weyl spinors: The (pseudo)- reality of the representation is charge conjugation invariance (which is equivalent toparity invariance for spin-1 couplings to spinors, since such couplings are always CPinvariant). Anomaly cancelation is then a generalization of Furry's theorem (seesubsection VIIA5). Real and pseudoreal representations of the generators (includingcomplex + complex conjugate) are antisymmetric, up to a unitary transformation, since they are hermitian: G T=G*=UGU1 B. LOW ENERGY 449 (so =iG preserves reality or pseudoreality). Thus tr(A;fB;Cg)=tr(AT;fBT;CTg)=tr(A;fB;Cg))tr(A;fB;Cg)=0 In particular, any mass term (without Higgs) T requires a real representation (so its variation yields G+GT= 0); a pseudoreal representation won't work because it uses an antisymmetric metric which, when combined with C,m a k e s T vanish by symmetry (since is anticommuting). A related way to see in Weyl (or Dirac) notation that real representations are nonanomalous is to use the same squared-propagator trick we used for the propagator correction in subsection VIIIA3 (or re-lated complex action from subsection IIIC4), which resulted in simplied Feynmanrules only for real representations: With those rules, there are no potentially diver-gent 3-point graphs other than those that already occur for scalars (as part of thecovariantization of the propagator divergence). The absence of 1's is a special case of parity invariance. However, even par- ity invariance is not enough to enforce cancelation of anomalies, since some parityinvariant theories have axial gauge vectors, which couple to axial currents  1 a , and the appearance of these 1's can be sucient to introduce anomalies. In these anomalous cases, even if there is a C, the charge conjugation argument above doesnot apply because the C following from the usual CP and the obvious P does notsimply replace A!A T, but is some other permutation of similar representations. Thus, in general P (and C) invariance is unrelated to anomaly cancelation: We canhave one without the other. Having real representations (i.e., no 1's) is a special case of both. Excercise VIIIB3.1 Consider chiral symmetry (as in subsection IVA4 or IVB1) for a single avor|U ( 1 ) L U(1)R. Now gauge that symmetry: aIn Weyl spinor notation, write the action for massless Weyl spinors L, R each coupled to their own gauge vector. Clearly there is one anomaly for 3 externalALa's, due to L, and another for ARa, due to R, and no mixing. Now assume the left and right coupling constants are equal (so the anomaliesare equal). Write the resulting symmetry transformations on all elds underC P ,C ,a n dP . bRewrite this theory in Dirac notation. Using P, nd the combinations of A L andARthat are (polar) vector and axial vector. Relate the anomaly cal- culations in the two notations. Show that dropping the axial vector gives 450 VIII. GAUGE LOOPS (massless) QED. Find the theory that results from dropping the vector in- stead: Give the gauge symmetry, and show it is anomalous, and explain the anomaly (vs. the cancelation of the anomaly in QED) in both Weyl and Dirac language. cGeneralize all the above results to U(n) L U(n)R. (Note that C will now include complex conjugation on the hermitian matrices for the vectors, sothat P won't.) The simplest way to prove anomalies cancel in the Standard Model is to use our previous results for GUTs (subsection IVB4): (1) One way is to consider the GUT gauge group SU(4) SU(2) SU(2). First, we note that tr(G i) = 0 for all of the representations used, so there are no mixed anomalies. Then we see that the SU(4) couplings are the usual \color"-type couplings, without 1's (i.e., 44), so it has no anomalies. On the other hand, SU(2) has only (pseudo)real representations, so neither SU(2) has anomalies. Thus, anomalies cancel in the SU(4) SU(2) SU(2) GUT. Finally, breaking to SU(3) SU(2) U(1) (which also spontaneously breaks parity) leaves an extra singlet per family, which decouples, showing the cancelation for the Standard Model. (2) Another way is to start with SO(10), which is anomaly free for any represen- tation of fermions: tr(Gab;fGcd;Gefg)=0 simply because there is no combination of Kronecker 's with the appropriate sym- metry (and similarly for SO(N), except for N=2 or 6, where such a term can be produced with the tensor). Breaking to the Standard Model again drops just a singlet (as does breaking to SU(5), showing its anomaly cancelation; breaking to SU(4) SU(2) SU(2) drops nothing, again showing its cancelation). In general, prov- ing anomaly cancelation requires (a) using such arguments about real representations,or (b) the absence of anomalies for certain groups (namely, only SU(N) for N >2, or U(1), can have anomalies), or (c) explicitly calculating the relevant traces. 4.0!2 When an anomalous axial symmetry appears only as a global symmetry classi- cally, unitarity is preserved, since no gauge eld couples to that current. This can be a useful way to explain approximate global symmetries. The fact that the anomaly is always a total derivative (because of the tensor and the Bianchi identity for F) means that the global symmetry is not broken perturbatively. (However, when the B. LOW ENERGY 451 external vectors are nonabelian, there can be contributions from eld congurations like instantons: See subsection IIIC6.) In subsection IVA4, we saw that the neutral pion (0), the lightest hadron, could be considered as the pseudogoldstone boson of an axial U(1) symmetry. We also want to consider the pion as a bound state of a quark and antiquark: If we knew the wave function, we could write the coupling,and calculate directly the decay of the neutral pion into two photons (  0!2 )v i a quark-antiquark annihilation, or at least nd the leading low-quark-energy contribu- tion from the -function part of the wave function (in the relative coordinates of the quark and antiquark), corresponding to the coupling to  1 . (An expansion of the wave function in derivatives of the function would give coupling to currents containing derivatives.) Lacking such detailed information, the best we can do is extend the nonlinear  model approach, which is to look for the terms in the phenomenological Lagrangian (expressed in terms of composite meson elds, not fundamental quark elds) with fewest derivatives (i.e., those most important at low energy), applying the conditionof (approximate) chiral symmetry. Specically, the global axial symmetry  0=2, whereA0=Afor the photon eld, along with the electromagnetic gauge invariance forA, under which the neutral pion eld is invariant, would suggest couplings of pion to photon involving only @andF. However, the anomaly allows the existence of another term: Since by denition (from considering coupling to an unphysical axial gauge eld) the anomaly is given from a local axial transformation, while the pion eld transforms in a trivial way under this transformation, we can attribute the anomaly to the pion coupling as =2;  =Z @
J;= 0+
;(
) = 0 ) 0=Z 1 2@
J=Z 1 2D=2(D 2)!ab


cdFab:::Fcd Thus, in four dimensions we nd the contribution 0=Z 1 8abcdFabFcd Using the abelian form of the Chern-Simons form (subsection IIIC6), we also can write this as 0=Z 1 6abcd(@a)Bbcd (In the nonabelian case, we can neglect the surface term only if the vacuum value of has already been subtracted.) Adding this term to those found previously (the  andAkinetic terms, as well as the quark terms that dene the normalization of the 452 VIII. GAUGE LOOPS eld through its coupling), the decay rate for 0!2 can be calculated (including the 2 relevant avors of quarks, and 3 colors, using the values of their electromagnetic charges), and is found to agree closely with the experimental value. The global anomaly in the nonperturbative case can be applied to the strong interactions (QCD), although not as straightforwardly: Considering the external vec-tors to be gluons (so there is an implict trace above over the group indices), 0gives a coupling of a neutral meson to a pseudoscalar glueball, as discussed in subsection IC4. If the vacuum gives a nontrivial value to tr(abcdFabFcd)( a sf o ri n s t a n t o n s ) ,t h i s also leads to anomalous CP violation in the strong interactions. 5. Vertex One-loop triangle graphs can't be evaluated in terms of elementary functions. However, in QED the most important eects are at low energy. We therefore will evaluate the eective action in the quantum mechanical version of the JWKB ex-pansion, as an expansion in derivatives. The resulting approximation to the eectiveaction thus will be local, but include terms of higher dimension than the classical action, whose coecients are therefore nite and unrenormalized: By dimensional analysis, this means their coecients will have powers of the inverse electron mass,which can be considered as the expansion parameter. (See also subsection VIIB7, where a scalar 1-loop vertex divergence was evaluated.) The propagator corrections have been found already in subsection VIIIA1; now we calculate the vertex correction. The integral is Aa;3;QED =Z dkNa D Na= b(k=+p=0+mp 2) a(k=+p=+mp 2) b;D=1 2k21 2[(k+p0)2+m2]1 2[(k+p)2+m2] Without loss of generality, we can drop terms that vanish by the free fermion eld equations; this corresponds to canceling them by fermion eld redenitions. We then evaluate the numerator by applying the identities p2=p02=m2;q =p0p) (p+p0)2=4m2q2 B. LOW ENERGY 453 v= v==1 2v2 vv= as well as the identities of subsection VIC4 for b::: b, and the eld equations p==mp 2 on the far right and p=0=mp 2on the far left, to obtain N=(k=+p=) (k=+p=0)+m2 2 +mp 2(2k+p+p0) (k=+p=) (k=+p=0)=(k=+p=+p=0) (k=+p=+p=0)p= p=p=0 p=0p=0 p=k= p=p=0 k= =[1 2k2+k
(p+p0)2m21 2q2] (k+p+p0)(k=+2mp 2)+m2 +mp 2(p+p0)m2 2 +mp 2k )N =(1 2k2 kk=)+[k
(p+p0) (p+p0)k=+mp 2k](m2+1 2q2) For the momentum integral we evaluate A3(x;m2;q2)=Z dkeik
x D=Z d3D=2eE E=1 21 x2+ix
1 2[(1+2)(p+p0)+(12)q]+1 2[(1+2)2m2+12q2] again on the fermion mass shell. We also have Z d3=Z1 0d 2Z1 0d3 1X  =Z1 0d 2Z1 0d1Z11 0d2 using, e.g., the denitions Zb adx (x)f(x)=(a)(b)f(0);Z1 1dx (xa)(bx)f(x)=Zb adx f(x) As for the fermion propagator, we clearly separate UV and IR divergent integrals by the changes of variables =1+2; =12)Z d3=Z1 0d 2Z1 0d1 2Z d followed by ! 2;!)Z d3!Z1 0d 2Z1 0d 51 2Z1 1d which modies the integral to A3=Z1 0d Z1 0d 121 2Z1 1d eE E=1 2f1 2x2+ix
(p+p0+q)+[m2+1 4(12)q2]g 454 VIII. GAUGE LOOPS We now expand to O(x2)a n dO(q2). Theintegral is then trivial (the integrand becomes quadratic in ), theintegral gives the usual, and the integral is similar to the case of the fermion propagator. The result is A31 2(1 +)(1 2m2) 1 6q2 m2+ 11 6q2 m21 IR+ix
(p+p0) +1 8(x
(p+p0))2+1 4m2x2 +1 24(x
q)2+1 UV1 4m2x2 This leads to the expression for the vertex correction A3;QED(1 +)(1 2m2) 1+1 3q2 m21 IR+1 21 UV+7 2+1 12q2 m2 +1 4 11 6q2 m2p+p0 m=p 2 w h e r ew eh a v eu s e d a a=D in evaluating the contribution from the k2term. (Remember that all algebra from indices on the elds is done in 4 dimensions, while all algebra from indices on momenta is done in Ddimensions. Since the two parts of the calculation are usually done separately, this should cause no confusion; however, the dierence in evaluating a ais the main thing to watch.) Using the on-shell identity 4mp 2 =fp=+p=0; g+[q=; ]=(p+p0)+[q=; ] we can rewrite this as (again keeping only O(q2)) A3;QED(1 +)(1 2m2)1 IR+1 21 UV+5 2 + 1 31 IR+1 4q2 m2 +1 4[q=; ] m=p 2 The next step is to cancel the UV divergence by adding the counterterm for electron wave-function renormalization from subsection VIIIA1: A3;QED;r =A3;QED +A3;QED 1 31 IRlnm2 2 +1 4q2 m2 +1 4[q=; ] m=p 2 Equivalently, we can take the q= 0 piece ofA3;QED, and note that it combines with A2eof subsection VIIIA1 to gauge-covariantize the term proportional to p=!p=+A=. (The unrenormalized eective action is thus automatically gauge invariant, as is the counterterm.) At this point we can see the anomalous magnetic moment: Combining the tree and 1-loop result (including coupling), and writing as spinless + magneticmoment contributions, we have +e 2A3;QED;r 1+e2 1 31 IRlnm2 2 +1 4q2 m2 (1 4)p+p0 m=p 2+(1+e2)1 4[q=; ] m=p 2 B. LOW ENERGY 455 We can translate these 1-loop corrections into a contribution to the eective action as (with the usual 1 for the eective action) 1;3;QED;r = (p0)A3;QED;r (p)
A(q) We then note, again using the spinor (free) eld equations to imply ( p+p0)
q=0 , toO(q2), 4mp 2qa bq[aAb]qa(p+p0)bq[aAb]=q2(p+p0)
A The low-energy part of the renormalized eective action exhibiting up to order q2=m2 corrections to the coupling is then, in gauge invariant form, 0+1;2e;rZ dx  i@=A=+mp 2e2 2p 2miSabFab e2 m2 1 31 IRlnm2 2 +1 4 a(@bFab) 6. Nonrelativistic JWKB As for other processes, the application of quantum eld theory to bound states has two steps: (1) Calculate the (gauge-invariant) eective action; (2) nd solutions tothe eld equations following from the eective action (\on-shell" states). For bound states such solutions are nonperturbative; however, their determination is easier for nonrelativistic systems, since we can ignore production and annihilation of additional nonrelativistic (massive) particles in the second step because their eect already has been included as small corrections to the eective action. The Lamb shift is the (eld theoretic) quantum contribution to the energy levels of the hydrogen atom, which is described accurately even at one loop. The relativistic solution is found by perturbing the relativistic eective action in derivatives about the nonrelativistic one, whose solutions are the usual exact ones of the nonrelativisticSchrodinger equation. For atoms the electron speed p=m is of the order of (= 2e 2), so the loop and derivative expansions are in the same small parameter. The eective action is more conveniently calculated with manifestly relativistic methods, since the internal (\virtual") particles can be relativistic (especially thosethat contribute to the UV divergences). On the other hand, the solutions to the eld equations are more conveniently calculated in a representation that takes better ad- vantage of the nonrelativistic expansion, since the external particles are nonrelativis-tic. Therefore, the second step begins by performing a eld redenition that converts the manifestly Lorentz invariant eective action to a form recognizable as nonrel- ativistic eld theory with low-energy relativistic and loop corrections. (Originally 456 VIII. GAUGE LOOPS the Lamb shift was calculated without this transformation. Higher-order calculations then required use of the relativistic Bethe-Salpeter equation, which made collection of terms of a given order more dicult.) In subsection IIB5 we considered the gen- eralized Foldy-Wouthuysen transformation and its application to minimal coupling; we now apply it to the nonminimal coupling introduced by loop corrections. (In theliterature this step has been performed on the Feynman diagrams themselves; how- ever, as usual we can save some eort by working directly with the eective action.) Here the nonminimal correction to the transformation is easy, since the nonminimalterms are already near the order to which we work. We rst perform some dimensional analysis, using the fact that the leading be- havior is given by the usual nonrelativistic Schr odinger equation. Then the only parameters in units  h= 1 (but there is no cin the nonrelativistic theory with just Coulomb interaction) are the mass mand speede 2, so (in the notation of subsection IIB5) ime2;0me4 (neglecting the rest mass contribution). It is then convenient to reorganize the ex- pansion in 1 =mto relate to the expansion in e2: For example, we can identify the two by choice of units 1 me2)i1;01 m along with c= 1 (since we will include relativistic corrections). The relativistic form of the Schr odinger equation is obtained by multiplying 2 0 in front of the kinetic operator of the electron in a background electromagnetic eld, as obtained from the eective action. Approximating the proton as innitely massive(for which we can partially correct by using the reduced mass for the electron), we take the electric eld as described by the usual static \scalar" potential, and drop the magnetic eld along with the \vector" potential. We therefore modify the expansion of subsection IIB5 by (1) reorganizing the 1=mexpansion according to our dimensional analysis, (2) using only a static electric background, and (3) working directly in terms of matrices: We can either plug in the Dirac case of the spin operators into the expressions of subsection IIB5, includingthe reality-restoring transformation of subsection IIB4, S ab!1 2 [a b];S1a!1p 2 a or just expand the Dirac operator directly, 2 0(=+mp 2)=02 0 ii+p 2m 0 B. LOW ENERGY 457 (and similarly for the loop correction terms). From the reuslts of the previous sub- section, we thus choose to order 1 =m4 E1=p 2 0;O0=2 ipi 0;E1=m0 O3=m2e21p 2i iF0i;E4=m2e2 1 31 IRlnm2 2 +1 4 @iF0i (others vanishing), where we have included explicit mdependence so that the coe- cientsEnandOnare of order m0according to our above dimensional analysis (so our expansion in mmakes sense). Using tanhxx1 3x3 the relevant commutators from IIB5 are then, for the nonvanishing generators to this order mG=1 2f[G;
E]+LGcoth(LG)OgE1 )G1=1 2O0E1 G3=1 2([G1;E1]+1 3[G1;[G1;O0]])E1 G4=1 2O3E1 and for the transformed kinetic operator F0=E+tanh(1 2LG)O )F0 1=E1+1 2[G1;O0] F0 3=1 2[G3;O0]1 24[G1;[G1;[G1;O0]]] F0 4=E4+1 2[G1;O3]+1 2[G4;O0] Remembering that E1commutes with even and anticommutes with odd, we have identities like (LG1)nO0=(1)n(n1)=2(O0)n+1(E1)n; (E1)2=1 Substituting for GintoF0: F0 1=E1+1 2(O0)2E1 F0 3=1 8[O0;[O0;E1]]1 8(O0)4E1 F0 4=E4+1 2fO0;O3gE1 458 VIII. GAUGE LOOPS The nal result is, using (O0)2=(pi)2;m e2[O0;E1]=2O3E1 and settingE1=1 on the right for positive energy, F0 1=m01 2(pi)2 F0 3=1 4m[1 2(@iF0i)iSijfF0i;pjg]+1 8(pi)4 F0 4=m2e2 1 31 IRlnm2 2 @iF0i+i1 2SijfF0i;pjg As expected from dimensional analysis, F0 1is the nonrelativistic result, F0 3is the lowest-order relativistic correction, and F0 4is the lowest-order part of the one-loop correction. Putting it all together, to this order we have F00(pi)2 2m(pi)4 8m3 1 8m2 18 3e21 IRlnm2 2 @iF0i+1+2e2 4m2iSijfF0i;pjg Excercise VIIIB6.1 Find the additional terms in F0to this order when the electromagnetic eld is arbitrary (magnetic eld, time derivatives of background), assuming thesame dimensional analyis for the background. REFERENCES 1 W. Pauli and F. Villars, Rev. Mod. Phys. 21(1949) 434. 2J. Steinberger, Phys. Rev. 76(1949) 1180; Schwinger, loc. cit. (VB, ref. 2, second ref.): triangle graph, for 0!2 . 3J.S. Bell and R. Jackiw, Nuo. Cim. 60A (1969) 47: identied axial anomaly. 4S. Adler, Phys. Rev. 177(1969) 2426: explained axial anomaly. 5S.L. Adler and W.A. Bardeen, Phys. Rev. 182(1969) 1517: showed axial anomaly is only at 1 loop. 6H. Georgi and S.L. Glashow, Phys. Rev. D6(1973) 429: anomaly cancelation for general GUTs. 7H. Bethe, Phys. Rev. 72(1947) 339: calculation of Lamb shift. 8H. Fukuda, Y. Miyamoto, and S. Tomonaga, Prog. Theo. Phys. 4(1949) 47, 121; N.M. Kroll and W.E. Lamb, Jr., Phys. Rev. 75(1949) 388; Y. Nambu, Prog. Theo. Phys. 4(1949) 82; J.B. French and V.F. Weisskopf, Phys. Rev. 75(1949) 1240: relativistic calculation of Lamb shift. 9W. E. Caswell and G. P. Lepage, Phys. Lett. 167B (1986) 437: nonrelativistic eective actions. B. LOW ENERGY 459 10L.S. Brown, loc. cit. (VA), p. 511; A. Pineda and J. Soto, hep-ph/9711292, Phys. Lett. 420B (1998) 391: Lamb shift with only dimensional regularization. 11Quantum electrodynamics , ed. T. Kinoshita (World Scientic, 1990): higher-loop QED. 460 VIII. GAUGE LOOPS ::::::::::::::::::::::: ::::::::::::::::::::::: ::::::::::::::::::::::: C. HIGH ENERGY ::::::::::::::::::::::: The sign of the one-loop correction to the gauge coupling is opposite in QCD to that of QED: The photon coupling is weak at \low" energies (actually, any observable energy, since the coupling runs so slowly), while the gluon coupling is weak at highenergies (with respect to the hadronic mass scale). Thus, typically perturbation inloops is used to study high-energy behavior of QCD, while the low-energy behaviorawaits the discovery of a general nonperturbative approach. 1. Conformal anomaly Symmetries of the classical action that are violated at the quantum level are called \anomalous". There are two major sources for such \anomalies" in renormalizablequantum eld theory: (1) There are anomalies associated with the totally antisym-metric matrix  a1:::aD, called \axial" (see subsections VIIIA8 and VIIIB2-4). When they occur, they are found in graphs with at least (D+2)/2 external lines. They areassociated with graphs that have no divergences, yet require regularization. (2) The existence of divergences requires the introduction of a mass scale even in theories that are classically conformal. If anywhere, these show up at least in the most diver-gent graphs, the propagator corrections. Normally, both kinds of anomalies will rstappear at one loop. When anomalies are associated with global symmetries, they provide a natural way to explain approximate symmetries, in the sense of the perturbative approx-imation. However, when they occur in local symmetries, they destroy the gaugeinvariance needed to prove unitarity. The latter type of theory therefore must be avoided by applying the condition of anomaly cancelation in local symmetries. We have already seen the appearance of the conformal anomaly in our renormal- ization of divergent loop graphs: The introduction of a renormalization mass scale breaks the scale invariance of a classically scale-invariant theory. The simplest ex- ample, and generally the most important, is the one-loop propagator correction. Ifwe examine only high-energy behavior, then we can neglect masses from the classicalaction. Using dimensional regularization, the generic eect on the eective action of the complete one-loop propagator correction is to modify the kinetic term of an arbitrarymassless theory to 1 2K1 g2+1ln2  C. HIGH ENERGY 461 whereis an arbitrary-spin eld that we have normalized !=gfor some appro- priate coupling g(like the Yang-Mills coupling if is the Yang-Mills vector), Kis the classical kinetic operator, is the renormalization mass scale, and 1is a constant determined by the one-loop calculation. As long as 1is nonvanishing (i.e., the theory is not nite) we can rewrite this as 1 2K 1lnM2 where M2=2e1=1g2 is a renormalization-independent mass scale: Any physical measurement will observe gandin only this combination. A choice of dierent renormalization mass scale is equivalent to a nite renormalization of g2, such that Mis unchanged. In the case where gis dimensionless (the relevant one, since we are studying the conformal anomaly), the coupling constant has undergone dimensional transmutation, being replaced with a dimensionful constant. If there is more than one coupling constant, things are more complicated, but the same phenomenon occurs: One dimensionless coupling is replaced with a mass.A particularly interesting case is pure Yang-Mills theory: Then we can write an important contribution to the eective action as F 1lnM2F whereFis now the complete nonabelian eld strength, and i st h es q u a r eo ft h e covariant derivative. Since this contribution by itself gives the complete 1-loop con- formal anomaly, the rest of the 1-loop eective action is conformally invariant. (All itsMdependence cancels.) Note that the (one-loop) anomaly itself is local: If we perform an innitesimal conformal transformation on the one-loop part of the eective action, this variation gives a local quantity. This is clear from the way this anomaly arose in dimensional regularization: If there were no innities, there would be no anomaly, since the naive conformal invariance of the classical theory would be preserved at each step. However,to regularize the divergence we needed to continue the theory to arbitrary dimensions, and the theory is not conformal away from 4 dimensions. The scale variation of a 4D conformal action in 4-2 dimensions is proportional to times that action, as follows from dimensional analysis; this scaling can be associated with the nonvanishing (engineering) dimension of the coupling away from D=4. (Usually, we write the coupling as g ,w h e r egis dimensionless. The elds have engineering dimension 462 VIII. GAUGE LOOPS independent of D, dened by the value in D=4: E.g., in r=@+A,Ahas the same dimension as @.) However, the one-loop eective action is coupling independent; thus, when dimensionally regularized but unrenormalized, it's scale invariant. Forexample, in the propagator correction discussed above, we get a regularized term 1 1K(1 2), which is scale invariant but divergent. On the other hand, the counterterm added to make it nite is from the 4D conformal action, and thus is notscale invariant in D 6=4; so the breaking of scale invariance can be associated entirely with the counterterm. (I.e., the anomaly coming from the renormalized, nonlocaleective action is equal to that coming from the innite, local counterterm.) Sincethe counterterm is local, the anomaly is local. It's also nite, since it's proportional to(from the variation) times 1/ (from the divergent coecient of the counterterm). In our propagator example, we have 1 1K(1 2)+1 (1 22)1K1K ln2 A similar situation occurs for the axial anomaly with Pauli-Villars regularization: After regularization, the anomaly comes entirely from the regulator graph, which is not only nite by power counting, but local in the innite-mass limit because that is the zero-momentum (eective potential or JWKB) limit. There is a physical signicance to the sign of the constant 1.( W e s a w s o m e evidence of this already in our analysis of renormalons in section VIIC.) Instead of thinking in terms of the renormalization-independent mass scale M, we can treat g as an eective energy-dependent (\running") coupling, 1 g2(p2)=1 g2+1lnp2 2 In the case 1>0 the coupling gets weaker at high energy (\asymptotic freedom"), while for1<0 the coupling gets stronger at high energy (until it reaches the Lan- dau ghost). (For low energy the situation is generally more subtle, since we usuallyhave complications from physical masses.) For QCD, this weakening of the cou-pling at high energy allows the separation of an amplitude into a nonperturbativelow-energy piece (describing the observed particles, the bound-state hadrons), whichis determined experimentally, and a perturbative high-energy piece (describing the non-asymptotic, fundamental \partons", gluons and quarks), which can be calculated. (This sometimes goes under the somewhat misleading name of \perturbative QCD".)This strongly contrasts with QED, where the weakening of the coupling at low en-ergy means both fundamental particles (photons, electrons, etc.) and bound states(positronium, atoms, etc.) can be treated perturbatively, and the only experimentally C. HIGH ENERGY 463 determined quantities are the values of masses and the electron charge (coupling at low-energy). Thus, in QED one in principle can calculate anything, while in QCD one is restricted to parts of certain amplitudes. (Various nonperturbative methodsalso have been developed for QCD, but so far they have successfully calculated only afew low-energy constants, as used in models, i.e., masses and low-energy couplings.) Although experimental verication of these results is sucient to conrm the QCDdescription of hadrons, a practical description of hadronic cross sections at all ener-gies would seem to require a string model that can incorporate behavior attributedto both strings and partons. 2. e e!hadrons If quarks and gluons are conned, how can QCD be useful? QED is useful be- cause the coupling is small: e21=861 is the perturbation parameter in relativistic (quantum eld theory, or 4D) calculations, =2e21=137 in nonrelativistic (quantum mechanics, or 3D). Energy levels of the hydrogen atom can be calculated quite accurately, without the question of freely existing electrons and protons comingup. The speed of the bound electron is also , another way to understand why pair creation/annihilation and other relativistic or multiparticle eects are small, and canbe treated perturbatively. Therefore, the real usefulness of a eld theory depends not on how \physical" the choice of elds is, but how accurate the perturbation expansion is. \Nonperturba-tive" results may give some nice qualitative features, but they are ultimately uselessunless they can be used as the basis of a new perturbation expansion. (Attemptsat nonperturbative approaches to 4D quantum eld theory continue, but so far theresults are meager compared to perturbative results, or to nonperturbative results inquantum mechanics or 2D quantum eld theory.) The simplest application of QCD is to the production of hadrons by a photon created by the annihilation of an electron and a positron. The total cross section forsuch an event is given (according to the optical theorem) by the imaginary part ofquark contributions to the photon propagator: Since hadrons are made up of partons (quarks and gluons), we assume a sum over hadrons can be written as a sum over partons. This assumption, that hadrons can be described by a resummation of theperturbation expansion, should be good at least at high energies, where the partons'asymptotic freedom takes eect (and perhaps at lower energies by an appropriateextrapolation). To lowest order for the process under consideration this is a 1-loop 464 VIII. GAUGE LOOPS graph, with a quark in the loop. If we compare this to the production of, e.g., muon- antimuon pairs (but not back to electron-positron pairs, because that includes the crossed diagram) by the same procedure, and we neglect masses (at high enoughenergies), then the only dierence should be in the group theory: Hadron production should be greater by a factor of the number of colors times the sum over avors of the square of the quark's electric charge: RP(e +e!h) P(e+e!+)NcX fq2 f Experimentally this relation is conrmed for Nc= 3, if the only avors included in the sum are those with masses below the photon energy ((2 mf)2<s). This result can be extended to the case where the momenta of hadrons are ob- served (not summed over): Although individual partons are not observed as asymp- totic states, the dominant contribution to the cross section at high energies is givenby the conversion of the quarks into hadrons by the creation from the vacuum of parton pairs with energies, and angular deviation from the partons created by the photon, smaller than experimental accuracy. We treat all partons as approximatelymassless, with respect to the energy scale of the photon. Thus, each parton created by the photon starts out initially as free, is then accompanied by parallel partons of small energy to form hadrons, and then these hadrons may further decay, but with asmall angular spread with respect to the directions of each of the initial partons. Such collections of nal-state hadrons are called \jets". For high-energy electron-positron annihilation, the dominant hadronic decay mode of this o-shell photon is thus intotwo jets. This experimental result is further verication of QCD, and in particular a jet is the most direct observation of a parton. Of course, even for asymptotic states the directness of experimental observations varies widely: For example, compare aphoton or electron to a neutrino. A closer analogy is unstable particles: For example, the neutron is observed as a constituent of the nucleus (as quarks are constituents of hadrons), but eventually decays outside (as quarks \decay" into jets of hadrons). C. HIGH ENERGY 465 A similar analysis can be applied to the creation of any electroweak boson by annihilation of a lepton with an antilepton. 3. Parton model We have already seen that in quantum eld theory coupling constants are usually energy-dependent. However, the dependence is only logarithmic, and thus can betreated as perturbative unless the relevant energy scale is within a few orders of magnitude of the mass scale that appears by dimensional transmutation. In QED, the value quoted for the electron charge is at the scale of the electron mass m.U s i n g the result of subsection VIIIA2 (or VIIIA3) for the 1-loop propagator correction, we nd (neglecting higher-loop corrections) M QED m=e3=4e2=2:837890(82)10280)MQED=1:450159(42)10277GeV (where for fun we have included the 1-standard-deviation uncertainties for this 1-loop result as the gures in parentheses; the ein the exponent is the electron charge). Since the mass of the observable universe is of the order of 1080GeV, and the Planck mass (beyond which a particle will gravitationally collapse from its Compton radius falling within its Schwarzschild radius) is of the order of \only" 1020GeV, there is little worry of observing the QED Landau ghost, even if QED were correct to that scale. On the other hand, the mass scale for QCD is (in the MS scheme) MQCD:2GeV (This result depends on renormalization scheme, and is also an eective mass in the sense that the usual experimental energy scale is among the quark masses, so the high-energy approximation of the renormalization group is inaccurate, and the fullpropagator correction with quark mass dependence should be used.) This indicates that perturbative QCD is inadequate to describe properties for which the energy of the quarks is low, such as hadron masses (although nonrelativistic quark models havehad partial successes). However, in certain processes a single \parton" (quark or gluon) in a hadron is given a high energy with respect to the other partons, usually a quark by electroweak interaction. In those cases, the \strong" (chromodynamic) interaction of that par-ton with the others in its original hadron is negligible: It has been liberated. The approach is then to factor the amplitude into a piece with the electroweak and high- energy (\hard") chromodynamic interactions of this parton, which can be calculated 466 VIII. GAUGE LOOPS perturbatively, and the low-energy (\soft") chromodynamic part of the remaining partons, which is left as an unknown, to be experimentally determined. (Thus, thehard part is the easy part, while the soft part is the dicult part.) The predictivepower is thus limited to the dependence of the amplitude on the energy of this parton,and on the particulars of the electroweak particles involved. Another possible complication would be the eect of exciting many partons within a hadron, indirectly through the rst parton's interactions with the rest: Then onewould have several terms to sum in an amplitude, each with a dierent unknownsoft factor, making the approach useless. Originally, it was thought that the highenergy alone was enough to explain the parton acting as free once liberated from the hadron (based on \intuitive" arguments), but soon it was realized that this possibility depended totally on the high-energy behavior of the theory: It requires the decrease ofthe coupling with increasing energy, asymptotic freedom (or superrenormalizability,or niteness with eective asymptotic freedom through the Higgs eect). Based onthis property, one can show from the usual perturbation expansion that one soft factor(per each hadron with an excited parton) is sucient as a leading approximation, a property known as \factorization". This feature is a consequence of the fact that the dominant contributions to Feynman graphs in this high-energy limit are those wherethe values of the momenta of some of the partons are those corresponding to theirclassical mechanics, as described in subsection VC8 and VIIA6. This new approximation scheme is eectively a perturbation expansion in the inverse of the energy being channeled into this parton. One neglects terms that aresmaller by such powers (including those from masses and renormalons), but incorpo-rates logarithms through the renormalization group and other loop corrections to thehard factor. Since available energy scales are much nearer to M QCDthan toMQEDin QED, such an approximation scheme tends to break down around two loops, where the corrections compete with the neglected terms, ambiguities in renormalizationschemes, and the relative size (convergence) of successive terms in the expansion. Al-though the accuracy of the predictions of this approach cannot compare numericallywith those of QED, it is the only method to describe such processes that can layclaim to being a theory, and provides direct experimental evidence of the validity of QCD, both as a qualitative description of nature and as a valid perturbation scheme. (As in the previous subsection, we also have processes where all the partons appearonly in intermediate states, or eectively so for nal states in total cross sections viathe optical theorem, so factorization is unnecessary.) C. HIGH ENERGY 467 The most eective application of factorization is to \Deep(ly) Inelastic Scattering (DIS)". (An equivalent method for this process is the \operator product expansion", but unlike factorization there is no useful generalization of it to general processes.) In this process a high-energy photon (or intermediate vector boson) is exchanged between a lepton (usually an electron) and a quark. (This is the leading-electroweak-order interaction of a lepton with a hadron.) The quark and rest of the hadron do not interact again: Color singlets are obtained by the creation of soft partons from the vacuum, which split from their own singlets and eventually combine withthe separated quark and hadron. For this process one calculates only the total cross section, at least as far as all the strongly interacting particles are concerned (\inclusive scattering") but again this can be generalized to the observation of jets (\exclusivescattering"). Applying the optical theorem, and ignoring the leptons, the leading contribution to this process is given by the tree graph for scattering of a vector boson o a quark, where the intermediate quark has a cut propagator. This is theperturbatively calculated hard part, which is later attached to the soft factor. Thus the hard part is the lepton-parton cross section, while the soft part is the \parton distribution", giving the probability of nding a parton in the hadron with a particular fraction(0,1) of its momentum p. To leading order this fraction is determined by kinematics: Since the hadron and scattered parton are treated as on-shell andmassless, p 2=(q+p)2=0)=xq2 2q
p so the \(Bjorken) scaling variable" xis a useful dimensionless parameter even when ( a th i g h e ro r d e r s ) 6=x. The energy scale is set by the square of the momentum q of the vector boson. hard softk pxpk xp»q There are several approximations used in this analysis, all of which can be treated as the beginnings of distinct perturbation expansions: (1) The hard part is expanded 468 VIII. GAUGE LOOPS in the usual (loop/coupling) perturbation expansion of eld theory. The leading con- tribution is that of the naive (pre-QCD) parton model (\leading order"), where thequark that scatters o the photon is treated as free with respect to the strong inter- actions. One-loop corrections (\next-to-leading-order") introduce the running of the coupling associated with asymptotic freedom, which justies the validity of the par-ton picture. (Two-loop corrections lead to various ambiguities, and have not proven as useful yet.) This is usually the only perturbation expansion considered, because such corrections are logarithmic in the energy of the exchanged parton (rather thanpowers), and thus more important and easier to isolate from the data. Furthermore, by the usual renormalization group methods such logarithmic corrections can be re- duced by careful choice of renormalization scale (  2close toq2inln(q2=2)). (2) In calculating the hard part \light" quarks are approximated as massless. One can rec- tify this by also perturbing in the masses, as a Taylor expansion in the square of each mass divided by the square of the vector boson's energy ( m2=q2). (3) In the explicit calculation the momentum pof the excited parton is assumed to be pro- portional to the momentum pof the initial hadron. In the rest frame of the initial hadron (which is massive in real life), this corresponds to the nonrelativistic approx- imation of motionless quarks; one quark is then set into relativistic motion by the photon, liberating it from the hadron. This approximation can be corrected by aJWKB expansion (expressed in operator language, the operator product expansion), also known as an expansion in \twist" (eectively, the power of momentum trans- verse top). However, this means a separate soft part for each term in the expansion: Since these are determined experimentally, such an expansion would lead to a loss of predictability. Thus generally (with few exceptions), parton model predictions are restricted to high enough energies ( q 2) that such corrections can be neglected. (4) Ex- pansions in renormalons (see subsections VIIC2-3) introduce new coupling constants, eectively nonperturbative corrections to the otherwise perturbative hard part. Likeall but the rst of these expansions, this leads to correction terms that are down by powers of 1=q 2. (It may be possible to absorb this type of correction into the previous one, since in principle the hard parts should contain all nonperturbative correctionsby denition.) The other common application of the parton model is to \Drell-Yan scattering": In this case two hadrons scatter producing, in addition to hadrons, a photon (or other electroweak boson) that decays into a lepton-antilepton pair. To lowest order, therelevant diagram is the same as for DIS (crossing some of the lines). Because both of the initial particles are hadrons, 2 soft parts are required; however, each of these is the same as that used in DIS (\universality") so they do not need to be redetermined. In C. HIGH ENERGY 469 fact, there is a direct progression from e+eto DIS to Drell-Yan: The above diagrams are similar except for the number (0 !1!2) of soft parts (corresponding to the number of initial hadrons); the leading contribution comes from the same diagram, rotated to various positions (crossing). softhardsoft pxpp' x'p' xpx'p' »q AabB fBb fAadsab More generally, we can consider not only hard parts involving identied quarks in the initial state of the hard part, but also in the nal state, by examining jets.Thus, for soft parts we have not only the \parton distribution functions", which are probabilities found from amplitudes for an initial hadron !parton + anything (summing over anything), we have \fragmentation functions", which are probabilitiesfrom amplitudes for parton !nal hadron + anything. In principle these are related by crossing symmetry: The diagrams are similar to the previous, with the partons connecting to the hard part, but the external hadron lines may be either initial or nal(and the opposite for the corresponding parton with respect to the hard subgraph). As for the parton distributions, the fragmentation function for any particular parton and hadron is measured in one particular experiment, then used universally. (The simplest is deep inelastic scattering for the parton distribution, and e +eannihilation with one of the two jets !hadron + anything for fragmentation.) Then the cross section is generally of the form dA:::B=X a:::bZ da


dbfAa(1)


fBb(n)da:::b(i) wheredA:::B is the observed (dierential) cross section, a:::b label the dierent par- tons andA:::B their hadrons (we leave o the labels for non-strongly interacting 470 VIII. GAUGE LOOPS particles), the sum is over dierent kinds (and avors) of partons (and perhaps over dierent hard parts, if corrections down by powers are desired), ais the momentum fraction for parton aof hadronA's momentum, fAais either the parton distribu- tion function for A!a+X(X= \anything") or the fragmentation function for a!A+X,a n dda:::bis the hard cross section (calculated perturbatively), which is just the original with all the hadrons replaced by partons. For the parton distri-butions we integrateR 1 0d, while for fragmentation we integrateR1 1d, or change variables to the hadron's fraction of the parton's momentum =1=and integrateR1 0d. Note that, while physical cross sections are independent of the renormalization mass scale, the same is not true of the hard cross sections calculated perturbatively in the above factorized expressions, since they are expressed in terms of unphysicalquark \states". However, these hard parts satisfy renormalization group equations, ascalculated in the usual perturbative way. (Of course, nontrivial contributions requirecalculating beyond leading order.) This implies corresponding renormalization groupequations (see subsection VIIC1), the \evolution" or \Gribov-Lipatov-Dokshitzer-Altarelli-Parisi (GLDAP) equations", to be satised by the parton distributions, sothatdependence cancels in the complete cross sections. This determines the energy dependence of the parton distributions. The equations take the form  2d d2fAa(;2)=X bZ1 d fAb(;2)Pba ;g2(2) wherefAadescribesA!a+X,fAbdescribesA!b+X0, the \splitting functions" Pba describeb!a+X00(X=X0+X00), and the sum is over the intermediate parton b.F o r hadronAwith momentum p, the intermediate parton bhas momentum p, and parton ahas momentum p,s o=isa's fraction of b's momentum. The kinematics are such that 0x1 (momentum is lost to X's asA!b!a). The splitting functions are calculated perturbatively from the corresponding renormalization groupequation for the hard part, since the combined dependence must cancel in the physical cross section. For similar reasons, the hard cross sections are infrared divergent; the soft parts of the complete cross sections deal with low energies. This leads to complicationsbeyond next-to-leading order, due to the fact that the renormalization group scale , which relates to ultraviolet divergences (high-energy behavior), and the \factorizationscale", which relates to infrared divergences (it determines the division between hardand soft energies), are in principle independent scales. This allows an ambiguity infactorization prescriptions, in addition to the usual ambiguity in UV renormalization C. HIGH ENERGY 471 prescriptions. (In more general processes there can be other energy scales than just q2, each with its own factorization scale, further complicating matters.) REFERENCES 1R.P. Feynman, Phys. Rev. Lett. 23(1969) 1415; J.D. Bjorken and E.A. Paschos, Phys. Rev. 185(1969) 1975: parton model. 2J.C. Collins, D.E. Soper, and G. Sterman, Factorization of hard processes in QCD, in Perturbative quantum chromodynamics , ed. A.H. Mueller (World-Scientic, 1989) p. 1; G. Sterman, Introduction to perturbative QCD, in Perspectives in the standard model , proc. TASI '91, Boulder, Colorado, June 2-28, eds. R.K. Ellis, C.T. Hill, and J.D.Lykken (World-Scientic, 1992) p. 475;Sterman, loc. cit. ; G. Sterman, hep-ph/9606312, Partons, factorization and resummation, in QCD & be- yond , proc. TASI '95, Boulder, Colorado, June 4-30, ed. D.E. Soper (World-Scientic, 1996) p. 327:reviews of factorization. 3G. Sterman and S. Weinberg, Phys. Rev. Lett. 39(1977) 1436: justication for jets from perturbative QCD. 4V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15(1972) 438, 675; Yu.L. Dokshitzer, Sov. Phys. JETP 46(1977) 641; G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. 472 IX. GENERAL RELATIVITY PART THREE: HIGHER SPIN Higher-spin (unstable) particles have been observed experimentally. Whether they are considered elementary depends on how their theory is formulated. In partic-ular, a description of hadrons in terms of strings would have many advantages, such as unication of all hadrons, manifestation of duality symmetry, and calculability through an accurate perturbation scheme. Gravity and supergravity also include higher-spin particles. String theory might also yield some solutions to some of their problems, especially renormalizability and unication of all particles. Such gravitational strings would dier from hadronicstrings in their mass scale and in the appearance of massless particles, including the graviton itself (in contrast to the massive \pomeron", the analog of the graviton in hadronic strings). Gravitational strings might require supergravity. For these and other reasons supergravity and strings are two of the major areas of research in theoretical high energy physics today (although not the only ones). Most of the discussion of this part is introductory, and can be covered earlier, but it is notessential to the course; however, its inclusion in a eld theory text is essential at least for reference. IX. GENERAL RELATIVITY Before discussing supergravity we need to study ordinary gravity. Both can be treated as generalizations of Yang-Mills theory. We use this approach rather than the traditional one, based on the metric, which is insucient for describing spinors or supersymmetry: There is no useful denition of distance in anticommuting directions in curved (super)space. Gravity is the only observed long-range (massless) force mediated by a higher- spin (2) eld. It is relevant for astrophysics, cosmology, and unication, all of which have applications to the particles of lower spin. :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: A. ACTIONS :::::::::::::::::::::::::::: We begin with the general principles that dene pure gravity as a nonabelian gauge theory, and use them to derive actions and couple to matter. A. ACTIONS 473 1. Gauge invariance General relativity can be described by a simple extension of the methods used to describe Yang-Mills theory. The rst thing to understand is the gauge group. We start with coordinate transformations, which are the local generalization of translations,since gravity is dened to be the force that couples to energy-momentum in the same way that electromagnetism couples to charge. However, these are not enough to dene spinors. This is easy to see already from the linear part of coordinate transformations:Whereas SO(3,1) is the same Lie group as SL(2,C), GL(4) (a Wick rotation of U(4))does not have a corresponding covering group; there is no way to take the square root of a vector under coordinate transformations. So we include Lorentz transformations as an additional local group. We therefore have a coordinate transformation group,which includes translations and the orbital part of Lorentz transformations, and a local Lorentz group, which includes the spin part of Lorentz transformations. Clearly the coordinates x mthemselves, and therefore their partial derivatives @m, are not aected by the (spin) Lorentz generators. We indicate this by use of \curved" vector indices m;n;::: . On the other hand, all spinors should be acted on by the Lorentz generators, so we give them \ at" indices ;;:: , and we also have at vector indicesa;b;::: for vectors that appear by squaring spinors. Flat indices can be treated the same way as in at space, with metrics Candabto raise, lower, and contract them. Some gravity texts, particularly the more mathematical ones, emphasize the use of \index-free notation". An example of such notation is matrix notation: Matrix notation is useful only for objects with two indices or fewer, as we saw in our treatment of spinor indices in chapter II. Such mathematical texts consider the use of indicesas tantamount to specifying a choice of basis; on the contrary, as we have seen in previous chapters, indices in covariant equations usually act only (1) as place holders, indicating where contractions are made and how to associate tensors on either side ofequations, and (2) as mnemonics, reminding us of representations and transformation properties. Thus, the full content of the equation can be seen at a glance. In contrast, many mathematical-style equations (when indeed equal signs are actually used) saylittle more than \ A=B", with the real content of the equation buried in the text of preceding paragraphs. We therefore dene the elements of the group as g=e ; =m@m+1 2abMba 474 IX. GENERAL RELATIVITY where@macts on all coordinates, including the arguments of the real gauge parameters mandaband any elds. Mab=Mbaare the second-quantized Lorentz generators: They act on all at indices, including those on aband any elds that carry at indices. As a shorthand notation, sometimes we will also write 1 2abMba=IMI (and similarly for other appearances of the antisymmetric index pair ab). In general, the relation between rst- and second-quantized group generators is the same as the relation between active and passive transformations, and the relation between a matrix representation and the corresponding coordinate representation, as discussed in subsection IC1, where in this case the elds are the coordinates. In particular, the second-quantized Lorentz operators Mabh a v et h es a m ea c t i o na st h e rst-quantized Lorentz operators Sabintroduced in subsection IIB1: For any eld , Mh j=h jS, etc. The action of the Lorentz generators on vector indices is thus given by [Mab;Vc]=V[ab]c)I[MI;Va]=1 2bc[Mcb;Va]=abVb This implies the commutation relations [Mab;Mcd]=[c [aMb]d] In explicit calculations, only two indices in the commutator will match, and they reduce to simple expressions such as [M12;V2]=22V1; [M12;M23]=22M13 As for derivatives, when acting on functions instead of operators we can write the action of the Lorentz generator as simply MabVcwithout the commutator. When spinors are involved in four dimensions, it's simpler to convert all at indices to spinor indices. In that case, we can write =m@m+IMI;IMI=1 2abMba=1 2M+1 2.. M. . [M; ]= (C) )I[MI; ]=1 2 [M ; ]=  [M;M ]=( (M)) in terms of the SL(2,C) generators M=M.N o t et h a t( M)y=+M.. because (Mab)y=Mab. We have used conventions consistent with OSp generators 1 2BC[MCB; Ag=AB B;AB=(ab;C;C.. ) A. ACTIONS 475 Relating vector to spinor indices as usual as Va=V., etc., then xes the Lorentz subgroup of the OSp group as (see excercise IIB7.2a) M .. V . =V.C C. . V . C C.. =1 2(C.. V(. C) +CV (.C. ). )=1 2(C.. M+CM.. )V . )M .. =1 2(C.. M+CM.. ) ) .. =C.. +C..  For most of the remaining discussion of gravity, we'll limit ourselves to bosonic elds in vector notation, which is easy to generalize to arbitrary dimensions. For spinors, we must either choose a dimension and use its corresponding spinor notation (forD6), or work in mixed spinor-vector notation (which is much messier). Matter representations of the group work similarly to Yang-Mills. We dene such elds to have only at indices. Then their transformation law is 0=e where the transformation of a general Lorentz representation follows from that for a vector (or spinor, if we include them), as dened above. Alternatively, the transforma- tion of a vector could be dened with curved indices, being the adjoint representation of the coordinate group: V=Vm@m)V0=V0m@m=em@mVem@m However, as in Yang-Mills theory, it is more convenient to identify only the gauge eld as an operator in the group. In any case, only the adjoint representation (anddirect products of it) has such a nice operator interpretation. As an example of this algebra, we now work out the commutator of two transfor- mations in gory detail: We rst recall that the coordinate transformation commutator was already worked out in subsection IC2, using the usual quantum mechanical rela- tions (see also subsection IA1) [f;f]=[@;@]=0; [@;f]=(@f) for any function f. For the Lorentz algebra we will use the additional identities [M ab;@m]=[Mab;m]=[Mab;cdMdc]=0 476 IX. GENERAL RELATIVITY all expressing the fact the Lorentz generators commute with anything lacking free at indices (i.e., Lorentz scalars). The commutator algebra is then [m 1@m+1 2ab1Mba;n2@n+1 2cd2Mdc] =m [1[@m;n 2]]@n+m [1[@m;1 2ab 2]]Mba+1 2ab 1[Mba;1 2cd 2]Mdc =(n [1@nm2])@m+1 2(m [1@mab2]+ac [12]cb)Mba One ne point to worry about: We may consider spaces with nontrivial topologies, where it is not possible to choose a single coordinate system for the entire space. For example, on a sphere spherical coordinates have singularities at the two poles, wherevarying the longitude gives the same point and not a line. (However, the sphere can be described by coordinates with only one singular point.) We then either treat such points by a limiting procedure, or choose dierent sets of nonsingular coordinates ondierent regions (\patches") and join them to cover the space. 2. Covariant derivatives We can also dene covariant derivatives in a manner similar to Yang-Mills theory; however, since @mis now one of the generators, the \ @" term can be absorbed into the \A"t e r mo fr=@+A: ra=eam@m+1 2!abcMcb in terms of the\vierbein (tetrad)" eamand \Lorentz connection" !abc.N o w t h e a c t i o n of the covariant derivative on matter elds looks even more similar to the gaugetransformations: e.g., = m@m;ra=eam@m Va=m@mVa+abVb;raVb=eam@mVb+!abcVc  =m@m + ;ra =eam@m +!a I.e., the covariant derivative rais essentially Delements (labeled by \ a") of the gauge algebra. Excercise IXA2.1 Write the transformation law and covariant derivative of an antisymmetrictensor in spinor notation ( f ), and compare to vector notation as above. Note that the free index on the covariant derivative is at so that it transforms nontrivially under r0 a=erae A. ACTIONS 477 Explicitly, for an innitesimal transformation r=[;r]w eh a v e eam=(n@neamean@nm)+abebm !abc=m@m!abc+(eam@mbc+!ad[bdc]+ad!dbc) This commutator is the same as for [ 1;2] in the previous subsection, except for the two additional terms coming from the Lorentz generators acting on the free index on ra. In particular, the vierbein eamtransforms on its at index as the vector (dening) representation of the local Lorentz group, and on its curved index (and argument)as the vector (adjoint) representation of the coordinate group. Also, it should be invertible, since originally we had r=@+A: We want to be able to separate out the at space part as e am=m a+hamfor perturbation theory or weak gravitational elds. That means we can use it to convert between curved and at indices: Vm=Vaeam,Va=Vmema whereemais the inverse of eam. Furthermore, if we want to dene the covariant derivative of an object with curved indices, we can simply atten its indices, take the covariant derivative with r, and then un atten its indices. Flat indices are the natural way to describe tensors: (1) They are the only way to describe half-integral spin. (2) Even for integer spin, they correspond to the way components are actually measured. For example, consider nonrelativistic momentum in two at spatial dimensions, but in polar coordinates: Now using xmto represent just the nonrelativistic spatial coordinates, xm=(r;);pm=mdxm dt=(m.r;m. )=paeam eam=10 0r1 ;pa=(m.r;mr. ) Then the two components of pa(with the simplest choice of eam) are the usual com- ponents of momentum in the radial and angular directions. On the other hand, one component of pmis still the radial component of the momentum, while the other component of pmis the angular momentum | a useful quantity, but not normally considered as a component along with the radial momentum, which doesn't even have the same engineering dimensions. Excercise IXA2.2 Show that the above choice of eamactually describes at space: Use the fact thatpatransforms as a scalar under the coordinate transformations that ex- pressrandin terms of Cartesian coordinates xandy, and as a vector under 478 IX. GENERAL RELATIVITY local \Lorentz" transformations, which are in this case just 2D rotations, to transform it to the usual Cartesian p0a=(m.x;m.y). This direct conversion between curved and at indices also leads directly to the covariant generalization of length: In terms of momentum (as would appear in theaction for the classical mechanics of the particle), p m=mdxm ds;m2=p2=papbab)ds2=dxmdxnemaenbabdxmdxngmn Equivalently, the metric tensor gmnis just the conversion of the at-space metric ab to curved indices. Also, in terms of dierential forms, a=dxmema)ds2= a bab The eld strengths are also dened as in Yang-Mills: [ra;rb]=Tabcrc+1 2RabcdMdc where we have expanded the eld strengths over randMrather than @andMso that the \torsion" Tand \curvature" Rare manifestly covariant: M0=eMe=M)T0=eTe;R0=eRe The commutator can be evaluated as before, with the same change as for going from [1;2]t o[;ra] (i.e., now there are two free indices on which the Lorentz generators can act), except that now we rearrange terms to convert @m!eam@m!ra.M a k i n g the further denitions ea=eam@m;[ea;eb]=cabcec)cabc=(e[aeb]m)emc=eamebn@[men]c for the \structure functions" cabc, we nd the explicit expressions Tabc=cabc+![ab]c=eamebn(@[men]c+e[md!n]dc) Rabcd=e[a!b]cdcabe!ecd+![ace!b]ed=eamebn(@[m!n]cd+![mce!n]ed) If we ignore the action of ron curved indices (it doesn't act on them, but alternatively we could atten them, act, then curve them back), we can also write Tmna=r [men]a; [rm;rn]=1 2RmnabMba where rm=emara=@m+1 2!mabMba A. ACTIONS 479 is essentially a covariant derivative for the Lorentz group only. From this expression for the torsion we nd the following expressions for the curl and divergence of a vector in terms of curved indices: Dening edet eam )cbab=(e[bm@mea]n)enb=(ebm@mean)enbeam[(@mebn)enb]=@meameam@mlne we have eamebn@[mVn]=e[a(eb]mVm)cabcecmVm=r[aVb]TabcVc e@me1Vm=eam@mVa+cbabVa=raVa+TbabVa Excercise IXA2.3 Relate the two above identities by comparing (in D=4) r[aVbcd](generalizing r[aVb])a n draVaforVabc=abcdVd. In practice, a useful way to evaluate the commutator is to rst evaluate the com- mutators of the Lorentz generators with the whole covariant derivative, and then subtract out the double-counted [ M;M ] term. This is particularly convenient when considering some explicit solution to the eld equations with a reduced set of com-ponents (e.g., spherically symmetric), so that explicit indices may be lost except onthe Lorentz generators. Schematically, we then calculate [r1;r2]=[e1+!1;e2+!2] =f[e1;e2]+(e1!2)M2(e2!1)M1g+f!1[M1;r2]!2[M2;r1]!1!2[M1;M2]g This method turns out to be the simplest way to calculate explicit solutions (as opposed to discussing general properties). (For examples, see subsection IXC5 below.) The covariant derivative satises the Bianchi (Jacobi) identities 0=[r[a;[rb;rc]]] = [r[a;Tbc]drd+1 2Rbc]deMed] =(r[aTbc]d)rd+1 2(r[aRbc]de)MedT[abje(Tejc]frf+1 2Rejc]fgMgf)R[abc]drd )R[abc]d=r[aTbc]dT[abjeTejc]d;r[aRbc]deT[abjfRfjc]de=0 To make the transformation laws manifestly covariant we can dene instead =ara+1 2abMba 480 IX. GENERAL RELATIVITY which is just a redenition of the gauge parameters. The innitesimal transformation law of the covariant derivative is then ra=[ (eam)emb]rb+1 2(eam!mbc)Mcb=[brb+1 2bcMcb;ra] =(rab+cTcab+ab)rb+1 2(rabc+dRdabc)Mcb ) (eam)emb=rab+cTcab+ab;eam!mbc=rabc+dRdabc A \Killing vector" is a transformation that leaves the covariant derivative invari- ant. (The term is usually used to refer to just the general coordinate part mof the transformation, but we'll use it in a generalized sense to refer to the complete .) It represents a symmetry; the existence of Killing vectors depends on the particular space described by the covariant derivative. It then follows from the Jacobi identity for [1;[2;r]] that the Killing vectors form a group, the symmetry group of that space. Invariance of the covariant derivative requires: rab+cTcab+ab=0)r (ab)=cTc(ab);ab=1 2r[ab]1 2cTc[ab] rabc+dRdabc=0)rabc=dRdabc These equations are referred to as the \Killing equations". (Again, usually it is just the rst equation, on a, that is called by this name, but we'll use it to refer also to the equations for ab, which are needed to describe the symmetry when acting on spinors, etc.) Excercise IXA2.4 Express the Hamiltonian of the classical relativistic particle in terms of the vierbein: ea=eampm;H =1 2(abeaeb+m2) Doing the same for general coordinate transformations =aea,e x a m i n e the condition for invariance [ ;H] = 0 using the Poisson bracket. Using the commutation relations for the ea's, show that this implies the Killing equation r(ab)=cTc(ab). Excercise IXA2.5 Solve the Killing equations explicitly in the case of at space ra=@a. Show this gives the Poincar e group, including both orbital and spin pieces. A. ACTIONS 481 3. Conditions There are two kinds of conditions we can impose to eliminate some degrees of freedom: gauge choices and constraints . Gauge choices explicitly determine degrees of freedom that drop out of the action anyway. If the gauge is not completely xed, the form of the residual gauge transformations may change, since using particular gauge parameters to x the gauge, rather than eliminating those parameters, may just determine them in terms of the remaining parameters: We require that theresidual transformations do not violate the gauge conditions that have already been applied. Similar remarks apply to global symmetries: If they do not commute with the gauge transformations for the gauge that was xed, then they may aquire extragauge-transformation terms to preserve the gauge choiuce. On the other hand, con- straints are chosen to be covariant under the transformation laws, and thus do not alter them, while eliminating degrees of freedom that might otherwise appear in theaction (although not in all possible terms). Furthermore, the simplest explicit solu- tion to constraints can itself introduce new gauge invariances. (An example of this situation is supersymmetric Yang-Mills: see subsections IVC3-4.) In this subsectionthis analysis will be applied to Lorentz invariance: We already saw that global Lorentz transformations are included in coordinate transformations, and that local Lorentz invariance is unnecessary when only integer spin (and in particular, pure gravity) is treated. We now examine the consequences of eliminating this useful but redundant invariance and the gauge eld associated with it. Of course, we can eliminate local Lorentz transformations by hiding at indices: For the vierbein itself, we have the local Lorentz invariant g mn=abeamebn which is the inverse \metric tensor". However, we have seen that tensors with at indices have simpler coordinate transformations, and there is no way to get rid of at indices when spinors are involved. Furthermore, the metric has the constraint that it have Minkowski signature: This constraint is solved by expressing the metric in termsof the at-space Minkowsi metric and the vierbein. Thus, solving the constraint introduces local Lorentz invariance. (However, in this case the constraint does not eliminate degrees of freedom, but only limits their range.) The Lorentz transformations in  abare redundant to those in m. The extra gauge parameters also can be xed by an appropriate gauge choice: For example, consider the gauge eam=abebm)Lorentz gauge m[aeb]m=0 482 IX. GENERAL RELATIVITY A coordinate transformation takes us to a dierent Lorentz gauge, since the Lorentz gauge condition is not a scalar. This means that any coordinate transformation m must be accompanied by a Lorentz transformation abto preserve this gauge, where thisabis completely determined in terms of m. This is easy to see perturbing eam aboutm a: To lowest order we have simply 0=(m[aeb]m)ab+@[ab])ab@[ab] Excercise IXA3.1 Let's further analyze this gauge condition: aBy looking at the transformation of a vector, identify the specic terms in the Taylor expansions of mandabwhose coecients can be identied with global Lorentz transformations, in the approximation used above. bUsing the same methods as excercise IVC4.3, and writing in matrix notation eam=(eh)amfor some matrix h, solve explicitly for abin terms of mand eamto all orders. Similarly, the Lorentz connection !abcthat gauges the abtransformations is redundant to the vierbein that gauges m:!can be completely determined in terms ofeby constraining the torsion to vanish. To see this, we rst notice that in the general case the expression for the torsion in terms of the structure functions and connection can be inverted to give the connection in terms of the other two. One way to do this is to use the denition and permute the indices a!b!c(odd permutations are redundant because of the antisymmetry of the equation in the rsttwo indices): T abc=cabc+!abc!bac;Tbca=cbca+!bca!cba;Tcab=ccab+!cab!acb Using the antisymmetry of the connection in its last two indices, we add the rst and last equation and subtract the second to obtain !abc=1 2(~cbca~ca[bc]); ~cabc=cabcTabc Since the torsion is a covariant tensor, we can freely set it to vanish without aecting the transformation laws of the remaining objects (it's a covariant constraint, not agauge condition): T abc=0)!abc=1 2(cbcaca[bc]) From now on we assume this constraint is satised. This simplies the form of curls and divergences, which implies that rcan be integrated by parts in covariant actions A. ACTIONS 483 (see below). However, we have already seen that the torsion is nonvanishing in su- perspace (subsection IVC3): In that case the symmetry on at indices is constrained, so the connection has fewer components than the torsion, and can be determined by setting only part of the torsion to vanish. (See subsection XA1 below.) Excercise IXA3.2 Show explicitly that when the torsion vanishes the Killing equations from eam= 0 imply those from !mab=0 : r(ab)=0;ab=1 2r[ab])rabc=dRdabc Excercise IXA3.3 Consider using the group GL(D) on the at indices instead of SO(D 1,1). (This construction is not useful for fermions.) Compensate for the extra gauge invariance by replacing the Minkowski metric abwith a \ at"-index metric gab(and its inverse gab)t h a ti s coordinate dependent , but covariantly constant: =m@m+abGba;ab[Gba;Vc]=caVa;ab[Gba;Vc]=Vaac ra=ea+!abcGcb; [ra;rb]=Tabcrc+RabcdGdc gmn=gabeamebn;ragbc=0 where now there is no (anti)symmetry associated with the indices on Gab(or ab, etc.). As a result, gabtransforms nontrivially under both coordinate and GL(D) transformations. Use it (in place of ) to raise and lower at indices. aFind the explicit expressions for the torsion and curvature in terms of the vierbein and connection. Solve these, and rg= 0, for the connection in terms of the torsion and vierbein as !abc=1 2(~cbca~ca[bc])+1 2(ecgabe(agb)c); ~cabc=cabcTabc Show that there exists a GL(D) gauge gab=ab (assuminggabhas the right signature), that gauge has as a residual at-index invariance SO(D1,1), and the resulting covariant derivative is identical to that used earlier in this subsection. bShow that one can instead choose a GL(D) gauge eam=m a)gmn=gab 484 IX. GENERAL RELATIVITY and that this completely xes the GL(D) invariance. Since the vierbein has a curved index, the covariant derivatives are no longer covariant: Unlike the previous gauge, to maintain this gauge any coordinate transformation must be accompanied by a GL(D) transformation whose parameter is determined by the coordinate transformation parameter. Find the solution for abin terms ofmin the innitesimal case. Compare with the transformation law for curved indices (see subsection IC2). In this gauge the connection is known as the \Christoel symbols". The vanishing of the torsion simplies the Bianchi identities on the curvature: Tabc=0)R[abc]d=r[aRbc]de=0)Rabcd=Rcdab In terms of SU(N)-like Young tableaux, this means the curvature is of the form . For SO(N) Young tableaux, we subtract out the trace pieces: Rabcd!
where the rst term is the \Weyl tensor" Wabcd(traceless), the last two terms combine to give the \Ricci tensor" RabRacbc, and the last (singlet) term is the \Ricci scalar" RRaa=Rabab. They're simpler in spinor notation in D=4 : S i n c e[ ab]!() and (.. ), Rabcd!R ()(.. )R()( )=W(  )+C(( C))R in terms of Weyl W(  ), the traceless part of Ricci R ()(.. ), and the Ricci scalar R. Later we'll see that the Ricci tensor is xed exactly by the equations of motion. That leaves the Weyl tensor as the on-shell eld strength. As explained in subsection IIB7, it describes helicity 2. Excercise IXA3.4 Prove that Rabcd=Rcdabfollows from the Bianchi identity R[abc]d=0a n dt h e antisymmetry of Rabcdin bothabandcd. 4. Integration The antihermitian form of the group generators was a convenient choice because partial derivatives are antihermitian, and the generators of the Lorentz group (whichis real and orthogonal) are antisymmetric in the vector representation. Thus, the generators are real. However, the group elements are not unitary, since hermitian conjugation reorders  mwith respect to @m. The x comes from noticing that e=det eam)lne=emaeam=m@mlne@mm A. ACTIONS 485 )(e1)=e1 =e1m @m) (e1)0=e1e  where the derivatives @act on everything to the left, now includes just coordinate transformations, and we have exponentiated by the same method as for Lie groups in subsection IA3. (Note that if we expand the exponential in a Taylor series suchderivatives in all but the rst factor will hit 's, just as for those in e acting to the right.) Any function that transforms in this way is known as a \density" (see subsec- tion IIIB1 for the 1D case). We can easily see from the innitesimal transformationthat a density times any scalar is also a density. This allows invariant actions to be constructed as S=Z dxe 1L for any scalar L. For cases without spinors we can also use gdet gmn=e2)e1=pg wheregmnis the inverse of gmn. This can also be understood in terms of dierential forms, since a=dxmema) 4=dxmdxndxpdxqem0en1ep2eq3=d4xe1 0a(x0)= a(x))Z 4L0 =Z 4L under coordinate transformations. Excercise IXA4.1 Let's look at some properties of transformations acting backwards: aShow that for any function f f=[;f]=[f; ])ef=efe=e fe  and use it to show that the product of e1with any scalar transforms the same way as e1(i.e., is a density) under a nite coordinate transformation. bDerive fe = 1
e  (ef) (where the derivatives in each factor of act on everything to the left, but vanish on \1"). Excercise IXA4.2 We now examine nite transformations in terms of transformed coordinates (see subsection IC2): 486 IX. GENERAL RELATIVITY bShow that det@x0 @x =1
e  by evaluating Z dxe1(x)=Z dx0e01(x0);d x0=dx det@x0 @x cShow that det@~x @x =1
e  by similarly evaluating Z dxe1(x)=Z d~xe1(~x)=Z dxe01(x) From the results of subsection IXA2, we then have that covariant derivatives can be integrated by parts in such actions, since Tbab=0)Z dxe1raVa=Z dx @me1Vm Excercise IXA4.3 Derive the expression for the covariant divergence in terms of eand the partial divergence by assuming integration by parts: Z e1raVa=Z e1Vara Use this to nd a simple form for the covariant d'Alembertian on a scalar: r2=1pg@mpggmn@n Actions for matter are constructed in a similar way to Yang-Mills: Starting with the at-space action, replace ordinary derivatives with covariant derivatives. The new ingredient is the extra factor of e1. This prescription, as for Yang-Mills, is unam- biguous up to only eld-strength (curvature) terms, which can usually be eliminatedby symmetry requirements and dimensional analysis. (At least for low energies, we want terms of the lowest mass dimension.) This uniqueness (at low energies or long distances) is known as the \equivalence principle": Inertial \mass" (really energy,but also momentum), as determined by the kinetic term, is the same as gravitational mass, as determined by the coupling of the gravitational eld. A simple example of matter is a real scalar eld: S=Z e 11 4[(r)2+m22+aR2] A. ACTIONS 487 The constant acan sometimes be xed by symmetry: In the massless case, to preserve the global symmetry =,w em u s th a v e a= 0. (For self-interacting scalars, this generalizes to a global nonabelian symmetry.) To preserve conformal symmetry (see subsection IXA7), also for the massless case, we need a=1 4D2 D1. This form of actions in terms of scalar Lagrangians also suggests we modify the denition of functional variation for convenience and covariance: S=Z dxe1()S  or, equivalently, we use the covariant form of the function, (x) (x0)=e(x)(xx0) As in at space, the action for electromagnetism follows from gauge invariance: S=1 8e2Z e1F2 ab=1 8e2ZpggmngpqFmpFnq whereFmn=@[mAn]. Integration by parts then gives a simple form for Maxwell's equations. Such simple covariant equations of motion that don't require explicit expressions for the Lorentz connection appear only for antisymmetric tensors (which in practice means just spin 0 and 1 in 4D). Excercise IXA4.4 Methods related to dierential forms can be applied to these special cases: aRewrite the above action for electromagnetism in terms of Aaand covariant derivatives. Find the eld equations following from both forms of the action, and use this to nd a simple expression for the covariant divergence of an antisymmetric tensor with curved indices using just the metric. Compare the results of the previous excercise. bBy converting at indices on the covariant tensor abcdto curved, show that pgmnpqand1pgmnpq are also covariant tensors. Use these, and the covariance of the curl (see subsection IC2), to arrive at the same expression for the covariant divergence of an antisymmetric tensor. Another example is a Dirac spinor: S=Z e1 ( aira+mp 2) 488 IX. GENERAL RELATIVITY where aare the usual constant Dirac matrices, in terms of which the spin operator appearing inris the usual Mab!Sab=1 2 [a b]. In 4D, we can rewrite this in spinor notation by simply replacing @ . !r . in the at-space expressions given in subsection IIIA4, and replacing Mab!Ma sd e s c r i b e di ns u b s e c t i o nI X A 1 ,a sw e l l asR d4x!R d4xe1. 5. Gravity The Einstein-Hilbert action for gravity follows from choosing the only available scalar second-order in derivatives, the Ricci scalar: LG=1 4R=1 4Rabab This action normally has a coecient of 1 =2(compare Yang-Mills), but we'll gener- ally use (natural/Planck) units =1 ;t h e nis used only to parametrize expansion about the vacuum and dene the weak-eld limit. (Actually, Planck units normally useG= 1, whereas in our conventions =1!G=.) In any case, the 's can always be absorbed (unlike Yang-Mills) by a eld redenition of eam, and then appear only in the denition of the \vacuum" (perturbative ground state, or solution that denes the boundary conditions at innity): heami=2=(D2)am This makes eam@m, and thusdxmemaandds2, dimensionless. In this sense, gravity is a theory with \spontaneous breakdown" of conformal invariance: Coordinate trans- formations include conformal transformations, but this invariance is broken by thevacuum, which introduces a length scale ( ). Excercise IXA5.1 Consider the covariant derivative for nonvanishing torsion. By solving for the Lorentz connection in terms of the structure functions and torsion, express the covariant derivative in terms of the torsion-free covariant derivative rand the torsion. Thus, any action in terms of rcan be rewritten in terms of randT, so any theory with a nonvanishing torsion is equivalent to a similar one with vanishing torsion (assuming the action is only second-order in derivatives of the vierbein, and thus algebraic in the torsion). Take the commutator of two r's to nd the curvature in terms of the torsion-free curvature Rabcd. Write the Einstein-Hilbert action with nonvanishing torsion in terms of R, r,a n d T, to nd: R= R(Tabb)21 2TabcTbca+1 4TabcTabc2 raTabb A. ACTIONS 489 Since the last term vanishes upon integration, Tappears as an auxiliary eld, soRis equivalent to just R. Excercise IXA5.2 For some general applications, where the form of the vierbein is not specied,it is useful to have a more explicit expression for the action in terms of the vierbein. We found in subsection IXA2 that for vanishing torsion r aVa=e@m(e1eamVa)=eaVacabbVa Use this to show R=(cabb)2+1 2cabccbca1 4cabccabc2e@m[eam@n(eane1)] We can drop the last term in the action integral under appropriate boundary conditions. (Hint: Use the result of the previous excercise for !=0 . ) Excercise IXA5.3 In two dimensions there is a single Lorentz generator, Mab=abM)ra=ea+!aM; [ra;rb]=1 2abRM aShow that the connection and the only surviving part of the curvature then take the simple forms !a=abe@me1eam;R =2e@m[eam@n(e1ean)] =2e1 ea eae bDerive, for the sphere in spherical coordinates, eam=10 01 sin  (Hint: First use ds2=dxmdxngmnin 3D at space.) bUse these results to calculateRdxe1Rfor the sphere in two ways: (1) by showingRis a constant and pulling it out of the integral, and (2) by converting it into a boundary term, where the \boundary" consists of innitesimal circles around the coordinate singularities at the poles. (In general, even for spaces without true boundaries, one has to treat the boundaries of patches as such.) It's also possible to add a \cosmological term" to the gravitational action: Scos=Z dxe1 with the \cosmological constant" . This term has no derivatives, and is thus anal- ogous to a mass term. However, it only contributes to the nonpropagating spin-0 490 IX. GENERAL RELATIVITY mode of the vierbein (see later), so it doesn't give a physical mass, but does modify the vacuum. Excercise IXA5.4 Show that the action for gravity can be made polynomial ineamby a eld redenition (rescaling) of the form eam!ekeam whenktakes the values k=n+1 D2;n =2;3;4;::: and that the resulting action is order Dn+ 2 in the eld. In what cases (of n andD) is the cosmological term also polynomial? The variation of the curvature can be obtained directly by varying its denition in terms of [r;r]. We start with the denition eamabebm,ab(eam)emb and work in terms of the attened object ab. Then we drop its Lorentz piece, choosing ab=ba. We nd: ra=abrb+1 2abcMcb )1 2(Rabcd)Mdc=[r[a;rb]]=(r[ab]c)rc[ac1 2Rb]cdeMed+1 2r[ab]cdMdc+[ab]crc )r [ab]c+[ab]c=0; R abcd=r[ab]cd[aeRb]ecd )abc=r[bc]a )Rabcd=1 2fr[a;r[cgb]d]1 2([aeRb]ecd+ab$cd) e1=e1lne=e1emaeam=e1a a )(e1R)=2e1[(abrarb)+(Rab1 2abR)]ab whererara. Thus for pure gravity we have the eld equations SG=0)Rab1 2abR=0)Rab=R=0 while with a cosmological constant we haveS G+Scos=0)Rab1 2ab(R4) = 0)Rab1 DabR=0;R =4D D2 Note that calculating a variation is the same as performing a perturbation to lowest order: We will use this result in subsection IXB1. A. ACTIONS 491 Excercise IXA5.5 For gravity, a rst-order formalism follows from not imposing the torsion con- straint (see excercise IXA5.1), so either the torsion or the Lorentz connection can be treated as the auxiliary variable. aFind a rst-order action for gravity (in all D) by treating emaand!mabas the independent variables. In D=4, using mnpq, write this action as polynomial in these variables, eliminating the explicit e,t oo b t a i n SG=Z d4x1 16mnpqabcdemaenbRpqcd withRpqcdin terms of just !. bVary this action with respect to both eand!(independently) to nd the eld equations, expressed in terms of torsion and curvature, using [rm;rn] to nd the variation of Rpqcd(see subsection IXA2). Excercise IXA5.6 As discussed in subsection IIIC4 for Yang-Mills, in four dimensions we can write a complex rst-order action for gravity that yields the usual gravity action up to a surface term. For Yang-Mills, the complex action was obtained by starting with a normal rst-order formalism and replacing the auxiliary eld with its self-dual part. aStarting with the rst-order action of the previous problem, nd the analog for gravity by keeping just the part of !mabself-dual in ab, using spinor notation. bAssociate the coupling with the term quadratic in !(analogously to the Yang-Mills case). As for Yang-Mills, associate the self-dual theory with the limit!0. Find the equation for emathat follows from varying !in this case, and show that it is equivalent to setting the self-dual part of!mabto zero, where!is the usual torsion-free connection. Show this is equivalent to setting the self-dual part of the curvature Rmnabto vanish, in an appropriate gauge. (Technically, this means we must either complexify the elds, or Wick rotate to 4+0 or 2+2 space+time dimensions, where the Lorentz group factorizes.) 6. Energy-momentum In subsection IIIB4 we saw that in the same way as a current in electrodynam- ics or Yang-Mills is dened as the matter contribution to the gauge eld's equation of motion, SM=Aa=Ja(in that case SMexcludes only the pure Yang-Mills ac- tion), the \energy-momentum tensor" is dened as the matter contribution to the 492 IX. GENERAL RELATIVITY gravitational eld equation (in this case SMexcludes only the pure gravity action): SM=Z e1abTab=1 2Zpg(gmn)Tmn=1 2Zpg(gmn)Tmn T h ec a s ew h e r e abrepresents the invariances of the action implies restrictions on this tensor: Using the separate gauge invariance of the matter action gaugeSM=0a n d the matter eld equations SM=(matter ) = 0 (as for the Yang-Mills case), gauge variation of the gravity elds in SMimplies ab=( ab=ba)T[ab]=0: L o r e n t z 1 2r(ab))raTab=0: c o o r d i n a t e so coordinate invariance of the action implies local conservation of energy-momentum. For example, for a real scalar eld: S=Z e11 4[(r)2+m22+aR2] )2Tab=(ra)(rb)1 2ab[(r)2+m22]+a[(abrarb)+(Rab1 2abR)]2 Notice that for a6= 0, the energy-momentum tensor gets extra total-derivative terms which are separately conserved in at space (since they come from the R2term, which is separately covariant). Excercise IXA6.1 Using the action given in subsection IXA4 and the variation of the covariant derivative from subsection IXA5, nd the energy-momentum tensor for the Dirac spinor, and use its eld equations to show this tensor is conserved. Note that this is not the same as ordinary conservation @mTmn=0 :RpgT0n does not dene a conserved total energy-momentum. This is in contrast with the conserved current in electrodynamics, since we then can derive the usual global con- servation law 0=Z dDxe1raJa=Z dDx@me1Jmd dtZ dD1xe1J0 On the other hand, it's closely related to Yang-Mills, where Aa=raleads to raJa= 0 in terms of the derivative rcovariantized with respect to the Yang-Mills eld (as well as gravity, if in curved space), so @me1Jm=e1[iAm;Jm]6=0( s e e subsection IIIC1). However, if there is a Killing vector Ka, then the component of momentum in that direction is conserved: JaKbTba)raJa=(raKb)Tba+Kb(raTba)=0 A. ACTIONS 493 (Rememberr(aKb)= 0.) Some simple examples of this in at space are ( Ka)b=b a (translational invariance), for which the corresponding \charge" is the total momen- tum, and (Ka)bc=[b axc](Lorentz invariance), for which the charge is the total angular momentum. Including the variation of the gravitational action, we get the gravitational eld equations Rab1 2abR=2Tab Coordinate invariance of SGimpliesra(Rab1 2abR) = 0, which also follows from the Bianchi identities: In that sense gauge invariance is said to be \dual" to Bianchiidentities, one implying the other through variation of the action: In general, for anygauge eld with gauge parameter  =O; 0=S=Z dx(O)S ,OTS =0 where the \transpose" OTis dened by integration by parts. Positivity of the energy (contained in any innitesimal volume) is the condition T000. The addition of the cosmological term modies the left-hand side of the above equation of motion byadding a term 2  ab. Although there is no covariant denition of total energy-momentum, in the case where spacetime is asymptotically at (the metric falls o to the at metric sucientlyfast at innity), one can dene a noncovariant energy-momentum tensor t abfor gravity itself which is covariant with respect to coordinate transformations that themselvesfall o at innity. (See excercise IIIC1.2 for the analogous Yang-M ills case.) This tensor satisifes @ m(Tmn+tmn)=0( w h e r e Tmnis the usual tensor for matter), so the usual conservation laws can be derived for the total energy-momentum coming from integrating T+t. Many equivalent expressions exist for t.O n ew a yt od e r i v e i ti st o expand the eld equations order-by-order in has 1 2(Rab1 2abR)Labtab whereLabis the linearized part of the eld equations (see subsection IXB1) and tab is the quadratic and higher-order parts. By the linearized Bianchi identities, we know 0=@aLab@a(1 2Rab1 4abR+tab)=@a(Tab+tab) where we used the eld equations in the last step. Note that there is a great deal of ambiguity here: We could have linearized by expanding the metric around its atspace value instead of the vierbein, or by expanding R mnorRmninstead ofRab,e t c . 494 IX. GENERAL RELATIVITY Because of the expression in terms of Lab@@h, the integral of T+t, which gives the total energy-momentum vector, can be expressed as a surface term, just as Gauss' law in electrodynamics. Since space was assumed to be asymptotically at, only the quadratic part of tcontributes in the surface integral, which is why there is so much freedom in the denition of t.S i n c etis not covariant, the energy-momentum of the gravitational eld is not localized (coordinate transformations shift it around). However, since the total energy-momentum is invariant, one can ask questions about how much energy is radiated to innity, etc. 7. Weyl scale The simplest way to describe conformal transformations in eld theory is as a local scale transformation. If the theory is not coupled to gravity, we couple it to gravity as in Yang-Mills theory by replacing a Poincar e invariant Lagrangian L(@; ) withL(r; ) (where all elds have at indices), but also including the e1factor in the action. We then transform the elds as eam!eam; !w+(D2)=2 where  is the gauge parameter and w+D2 2is the engineering dimension (scale weight) of the eld . (See subsection IIB1.) Eectively, eamhas dimension 1, since it's the only eld with curved indices, and thus any derivative must appear in the combination eam@m, while the measure appears as dxe1. Of course, the action won't be locally scale invariant unless it is globally scale invariant, i.e., has only dimensionless coupling constants (and thus no masses). If the gravity-coupled theory is invariant under this local scale transformation, then the theory will be conformally invariant after decoupling gravity. This followsfrom the fact that the most general combined coordinate and local scale and Lorentz transformation that preserves the at-space vierbein e am=m ais exactly a conformal transformation. This is equivalent to our previous denition in terms of the scal-ing of the at-space ds 2under conformal transformations, since dx0mdx0ng0 mn(x0)= dxmdxngmn(x) under coordinate transformations. Excercise IXA7.1 Derive the usual conformal transformations by nding the most general localscale + Lorentz + coordinate transformation that preserves the at-space vierbein. A simple example is Yang-Mills theory. We look at the Yang-Mills eld with curved index, since its gauge transformation does not depend on the vierbein. ( A m= A. ACTIONS 495 @m+:::vs.Aa=eam@m+:::.) To avoid interference with the Yang-Mills gauge transformation, the Yang-Mills eld with curved index must be scale invariant. Then the action S=1 8e2Z e1eamebneapebqFmnFpq transforms with a factor 4D, and so is invariant in D=4o n l y . Excercise IXA7.2 Consider a more general gauge eld Aand eld strength Fdened by Am1


mN=1 (N1)!@[m1m2


mN];Fm1


mN+1=1 N!@[m1Am2


mN+1] whereAis totally antisymmetric in its Nindices. (Such theories were en- countered in excercise IIB2.1b.) aDene an action in terms of F2.I n w h a t d i m e n s i o n D(N) is it conformally invariant? bShow that this theory is related by a \duality transformation" (switching Bianchi identities and eld equations) to the theory with N0indices on a new A,w h e r eN0=D2N,a n dD(N0)=D(N). cExamine the cases N=D;D1;D2. Note that the scalar obtained by duality does not have an R2term in its action, and thus is conformal only in D=2. Gravity is not scale invariant, but it will prove useful to examine its scale breaking explicitly. To preserve gauge covariance and dimensional analysis, the scale transfor- mation law of the covariant derivative must take the form r0 a=ra+k(rb)Mab where the  scaling of eamwas dened above, and the linearity of !in  follows from the homogeneity of rine. (Alternatively, we could put in something more arbitrary, but it would be eliminated by the rest of the procedure anyway.) From the variation of commutation relations we then nd 1 2R0 abcdMdc=[r0 a;r0 b] =2[ra;rb]+( 1k)(r[a)rb]+k(r[arc)Mb]c+k2(r)2Mab )k=1;R0 abcd=2Rabcd+[c [arb]rd]c [ad b](r)2 If we make the redenition (at least for  positive) =2=(D2) 496 IX. GENERAL RELATIVITY then we nd the very simple scaling law for the integrand of the Einstein-Hilbert action: (e1R)0=e1(2R4D1 D2) Consider a eld theory without gravity that has a conformally invariant action. Spontaneous breakdown of scale invariance produces a Goldstone boson for that sym- metry, the \dilaton". Any theory can be made globally conformally invariant triviallyby performing a local scale transformation and making the parameter the dilaton eld. The dilaton can also act as a Higgs eld: If we couple the dilaton to conformal gravity (gravity with local Weyl scale invariance), the Higgs eect reduces conformal gravity to ordinary (Einstein) gravity. For example, if we introduce the dilaton intopure gravity by the local scale transformation above (in analogy to the St uckelberg model), S G=4D1 D2Z dxe11 4(1 4D2 D1R) Up to an (important) overall negative factor, this is the action for a conformal scalar. The dilaton eld is a compensator for local scale transformations, and acts as a Higgs eld for this gauge symmetry: By gauging it to its vacuum value hi=1 , we regain the usual form of the gravity action. (Alternatively, we can set hi= 1, and introduce through the proportionality constant in heamiam.) In this formalism, where we require the action to be locally scale invariant, the terms which were conformally invariant before coupling to gravity are easy to recognize: They're just the ones which have no-dependence. (This may require some eld redenition: typically rescaling the matter elds according to their weight as above.) The cosmological term becomes Scos=R e12D=(D2), which is a conformal self-interaction term for a scalar. Because what was the vierbein now appears only in the combination eam! 2=(D2)eam, there is now the local scale invariance eam!eam;!(D2)=2 since this transformation leaves the combination invariant. Gauge invariance of the matter action is then (using the innitesimal parameter  = 1 + ): 0=SMeamSM eam+D2 2SM  )Ta a=D2 2SM  Thus, conformal matter has vanishing Taa, since it decouples from . (Actually, we also need to scale the matter as above to achieve this decoupling, and there is a A. ACTIONS 497 corresponding SM= term in the above derivation, so the trace may vanish only after applying the matter eld equations, as in the derivation of raTab=0f r o m coordinate invariance in the previous subsection.) In particular, this is easy to check for the massless point particle, where Taa. Xm. Xngmn=0 . An interesting eect is obtained by eliminating the compensator by its eld equa- tion. (We'll consider just the classical theory here: In the quantum case, integrating out this eld produces an additional 1-loop contribution to the eective action.) Be- cause this manipulation involves integration by parts, we rst expand the compensator about its vacuum (asymptotic) value: =1+1 2)L=1 4[(D1 D21 4R)RR] Then eliminating by its eld equation, L!1 4 R1 R4D1 D2RR! This action still describes Einstein gravity, but is locally scale invariant (though not globally, because of the extraction of the vacuum value, and the way boundary terms were neglected). Of course, it is nonlocal, and the nonlocality becomes morecomplicated if nonconformal matter is included. Such terms also appear quantum mechanically: In two dimensions, dimensionally regularizing D=2+2 ,i naW e y ls c a l e invariant theory we can get a divergent, yet still Weyl scale invariant, contribution tothe eective action proportional to 1  R1 R4D1 D2RR! 1 R1 2R1R After renormalizing the divergent term, which is topological and thus locally scale invariant in exactly D=2, but not in D=2+2 , the remaining nite term contributes a conformal anomaly (see subsections VIIIA8 and C1). Excercise IXA7.3 The statement that the Rterm is topological in D=2 neglects boundaries. In general the topological invariant (the \Euler number") is (the \Gauss-Bonettheorem") =Zd 2x 21 2e1R+I1 2abtaDtb abtatb wheretais a tangent vector to the boundary Xm(), as for the worldline of the particle, and Dis the covariant dierential (as for the particle equation of motion and the radial gauge; see subsections IXB2 and 4 below): tm=v1. Xm;D ta=dXmembrbta=d vt
rta=d(. tavtbtc!bca) 498 IX. GENERAL RELATIVITY (We have used the usual counterclockwise contour, and our convention 01= 1, orxy= 1 in Euclidean space.) The additional term in is the angle subtended by the boundary with respect to the surface ( =2), as obtained from the cross product of tandt+Dt. We have written it in a form that is manifestly invariant under the reparametrization of ,a n dt h ev's cancel. (Of course, it is also manifestly coordinate invariant.) aProve that it is also scale invariant by showing that the connection part of theDexactly cancels the contribution of Rto the boundaries, leaving =Zd2x 21 2e1R patch boundaries+I1 2abtadtb abtatb where we have turned the Rterm into a boundary term, and its remaining contribution is from the fake boundaries at the borders of patches (or sur- rounding singularities; R=@!because the 2D Lorentz group is Abelian: see excercise IXA5.3). bNote that the dtterm doesn't contribute if we choose a gauge where ta=a 1 (i.e.,tm=e1m). Demonstrate this by evaluating in polar coordinates for a disk, and in spherical coordinates for the half-sphere. Show the result is halfthat for a whole sphere (excercise IXA5.3). Repeat the calculation for thedisk in Cartesian coordinates (so then only thedtterm contributes). REFERENCES 1 A. Einstein, Sitz. Preuss. Akad. Wiss. Berlin, Math.-phys. Kl. (1914) 1030, (1915) 778, 799, 831, 844, Ann. der Phys. 49(1916) 769: general relativity. 2E. Cartan, Lecons sur la g eometrie des espaces de Riemann (Gauthier-Villars, 1928): general relativity in terms of vierbein and (GL(D) or Lorentz) connection. 3Weyl, loc. cit. (IC): covariant derivatives on spinors. 4L. Infeld and B.L. van der Waerden, Sitz. Preuss. Akad. Wiss. Berlin, Math.-phys. Kl. (1933) 380: general relativity in two-component spinor notation. 5H. Weyl, Mat. Z. 2(1918) 384: Weyl tensor. 6D. Hilbert, Nachrichten K onigl. Ges. Wiss. G ottingen, Math.-phys. Kl. (1915) 395: found the action for Einstein's equations slightly earlier than Einstein. 7A. Palatini, Rend. Circ. Mat. Palermo 43(1919) 203: rst rst-order action for gravity (in terms of metric and Christoel symbols). A. ACTIONS 499 8H. Weyl, Sitz. Preuss. Akad. Wiss. Berlin, Math.-phys. Kl. (1918) 465, Raum
zeit
materie (Springer, 1919) p. 246 [English: Space-time-matter (Dover, 1952)]: Weyl scale. 9B. Zumino, Eective Lagrangians and broken symmetries, in Lectures on elementary particles and quantum eld theory , proc. 1970 Brandeis University Summer Institute in Theoretical Physics, eds. S. Deser, M. Grisaru, and H. Pendleton (MIT, 1970) v. 2, p.437:dilaton. 10E.S. Fradkin and V.I. Vilkovisky, Phys. Lett. 73B (1978) 209: nonlocal action. 11C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, 1970): introductory, long-winded. 12S.W. Hawking and G.F.R. Ellis, The large-scale structure of spacetime (Cambridge University, 1973):mathematical. 13R.M. Wald, General relativity (University of Chicago, 1984): intermediate between the above two. 14S. Weinberg, Gravitation and cosmology (Wiley & Sons, 1972): old-fashioned. 500 IX. GENERAL RELATIVITY ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: B. GAUGES ::::::::::::::::::::::::::::: We now consider various gauge choices for coordinate, Lorentz, and scale trans- formations. 1. Lorentz We begin with gauges that preserve global Lorentz invariance, which are useful for perturbation theory. Therefore, we look rst at perturbation by nding the kineticterm, which is sucent for nding linear gauge conditions. (It can also be derived fromgeneral principles, as will be shown in subsection XIIA5.) We expand the vierbeinabout its at value, e am=am+ham At the linearized level, local Lorentz invariance implies that only the symmetric part of the eld,1 2h(ab), appears in the curvature and the action; we will denote this by hab to simplify notation. (In other words, the linearized curvature is invariant under the linearized local Lorentz transformations, which gauge away the antisymmetric part of the eld. This is equivalent to working directly with the metric.) We then can ndthe linearized curvature, e.g., from the results of subsection IXA5 for the variation ofthe curvature, by considering variation about at space: i.e., replacing  ab!haband ra!@a. The result is Rabcd@[a@[chb]d] )Rab1 2abRhab+@a@bhc c@(a@chb)cab(hc c@c@dhcd) Since this comes from varying the action, the quadratic part of the gauge-invariant action must be SGZ 1 4[habhab+2 (@bhab)2ha ahb b+2ha a@b@chbc] This part of the action, and the linearized curvature, are invariant under the linearized gauge transformations hab=@(ab). Excercise IXB1.1 Take the Newtonian (weak-eld, nonrelativistic) limit of gravity: (1) Linearizethe action by perturbing about at space ( e am=m a+ham). Keep just the part of the pure gravity action quadratic in the perturbation, the part of the mattercoupling linear in it, and the complete at-space matter action. (2) Assumesmall velocities. Now consider the problem of a massive point particle in the B. GAUGES 501 eld of a much more massive point particle (or spherical body); in the above approximations: aShow the eect of the gravitational eld generated by the heavier particle on the lighter particle is given by the action for the lighter particle (in the gaugex 0t=) S=msZ dt(m1 2m.x2 i+mh00) bShow this eld is given by solving Laplace's equation R00
h00=T00 cShow that, with our conventions for normalizing functional dierentiation, a point mass Min D=4 generates T00=M(2)23(x))h00=M r using the usual solution to Laplace's equation for a point source. Combining these results, we see that the potential energy for the particle is U=mh00=Mm r which agrees with Newtonian gravity if we identify G=.( I f w e r e s t o r e units, this becomes G=2.) The BRST transformations (see subsection VIA4) for gravity again follow from the gauge transformations: Qeam=Cn@neamean@nCm+Cabebm QCm=Cn@nCm;Q Cab=Cn@nCab+CacCcb Q~Cm=iBm;Q ~Cab=iBab (Other forms follow from dierent parametrizations of the gauge transformations, and are equivalent to eld redenitions. For theories without spinors, we can work interms of the metric, and avoid Lorentz gauge xing.) Lorentz gauges for coordinate invariance are similar to Yang-Mills. For gravity, the gauge-xing function is f a=@bhab1 2@ahb b The BRST procedure works similarly to Yang-Mills. Looking at just the graviton kinetic term, the gauge-xed quadratic Lagrangian for gravity is then, in the Fermi- Feynman gauge, LG!LG;FF =LG+1 2(@bhab1 2@ahb b)2=1 4habhab+1 8ha ahb b 502 IX. GENERAL RELATIVITY plus ghost terms. Note that the trace part of happears with opposite sign to the traceless part. This prevents any redenition which would allow rewriting the La- grangian in the simple form 1 4habhab. However, all derivatives have been absorbed into 's, which makes the linearized eld equation a simple Klein-Gordon equation. There are various generalizations of this gauge condition to include nonlinear terms, such as the \de Donder (harmonic) gauge", which uses the gauge-xing func- tion fn=1 2@m(pggmn) For example, this allows the eld equation for a scalar to be written with only terms with both partial derivatives acting on the scalar. 2. Geodesics Consider the eld equations for coupling gravity and electromagnetism to a scalar particle: From subsection IIIB3, the action for a particle in external elds, rewritten in Hamiltonian form, is SH=Z df.xmema(x)[aAa(x)] +vHg;H =1 22+(x) where we have pulled the vout ofHfor convenience, and use the \covariant momen- tum" a=eampm+Aa(x)=eam(pm+Am) in place of pm(the canonical conjugate to xm) for covariance. All the equations of motion except the Lagrange-multiplier constraint 1 22+=0 follow from the usual Poisson-bracket relation v1. O=i[H;O] which can be evaluated by using the canonical commutation relations (following from the simpler ones for pm) i[a;xm]=eam;i [a;b]=cabcc+Fab; [x;x]=0 Thus,aacts eectively like iea+Aa, which is the covariant derivative for gravity and electromagnetism, less the Lorentz term. The.xequation is the obvious va=.xmema B. GAUGES 503 that follows from varying SHwith respect to a, while the equation of motion for  is v1.a=cabcbcFabbra Using the relation 0=Tabc=cabc+![ab]c)ca(bc)+!(bc)a=0 we nd v1.abc!bca+Fa bb+ra=0 This is the coordinate-covariant form of the Lorentz force law (plus scalar eld). With only the gravitational eects we have the covariantization of the free particle equation, Dpa.pavpbpc!bca=0 where \D" is understood as a covariantized derivative (along a worldline with metric v). It's useful to consider a continuum of particles (\dust") moving under the in u- ence of these elds, such that any two innitesimally close particles have innitesi- mally dierent velocities, and only one particle passes through any particular point in spacetime (at least within some small region of spacetime). We then can treat a (orpm) as a eld dened for all x: Choosing a point xalso chooses a curve X()f o r whichx=X()f o rs o m e ,s ow ec a nw r i t e (x)i np l a c eo f (). Specifying the eldalso determines this family of curves, since the tangent to any curve is given by theXequation of motion. Xm=vaeam (To determine the parametrization, we also specify v, and the hypersurface given by the collection of points X(0) from each curve.) Then we can express the derivative in terms of xderivatives:d d=. Xm@m=vaea which gives the manifestly covariant form of the equation of motion brba+Fabb+ra=0 For vanishing F(and thusA) and constant (=1 2m2), this equation pbrbpa=0 describes \geodesics", which are curves of extremal length, since the action is S=ms;ds2=dxmdxngmn 504 IX. GENERAL RELATIVITY for massive particles. These are the analogs of straight lines in at space. (For positive-denite metric, they are shortest lines. Because of the indenite signature of the Minkowski metric, the worldlines of massive particles are actually longest lines,while massless particles travel along lines with no length.) For some purposes we need a weaker (but equivalent) form of the geodesic equa- tion: If for some scalar fand vector n a (n
r)na=fna)pa=gna; (p
r)pa=0;f =(n
r)lng and we can integrate fto ndg. In particular, we can identify g=v1)nm=. Xm Thus, the more general geodesic equation allows arbitrary parametrization of the geodesics, while the stricter version ( f= 0) corresponds to ane parametrization (v= 1) if we still want to identify pwith. X. (Remember, as with all constrained systems, the equations of motion p
rp=0i m p l y( d=d)p2=0 ,s oa n yg e o d e s i c satisfying the stricter equation will have some xed mass along that particular curve.) Excercise IXB2.1 Show that in D=2 (one space dimension, one time) anylightlike curve is a geodesic, using the weaker form of the geodesic equation. (Find f.) This is a consequence of the fact that it is impossible to change direction in D=2without slowing down. The particle (geodesic) version of the conservation of momentum in the direction of a Killing vector is p
rp a=0)p
rK
p=0)d dK
p=0 where covariant conservation p
rhas become ordinary conservation d=d (no con- nection term) because K
pis a scalar. (See also excercise IXA2.4.) This is the same as for the conserved current Ja=KbTba(subsection IXA6). Excercise IXB2.2 Check explicitly that the current and energy-momentum tensor are covari-antly conserved for the scalar particle coupled to gravity and electromag-netism (see subsection IIIB4), using the geodesic equation of motion for thelatter. (Note: In subsection IIIB4 the 's need e's to be covariant in curved space.) B. GAUGES 505 3. Axial The denition of axial gauges in terms of the covariant derivative is the same as for Yang-Mills ( n
r=n
@). In terms of the explicit gravity elds, n
r=n
@)nmnaeam=nam a;na!abc=0 In the case of gravity, this implies that lines in the nadirection are geodesics (see previous subsection): @mna=0) (n
r)n=(n
@)n=0 To analyze the consequences of axial gauge conditions for the metric, we need a further identity: For any vector eld na, consider the action of n
ronnm=emana, treating it as a scalar; in this calculation we ignore any indirect action of ron curved indices. Then (n
@)emana=(n
r)emana=ema(n
r)na+na(n
r)ema The last term simplies for vanishing torsion, since: nnrnema=nnr[nem]a+nnrmena=nnTnma+nnrmena=rmnaenarmnn =rmnaena@mnn We thus have (n
@)emana=ema(n
r)na+@m(1 2n2 a)(enana)@mnn Applying this identity to the axial gauge condition, we nd n
r=n
@; @ mna=0) (n
@)emana=0)nmemana=a mna by choosing the appropriate constants of integration. (This amounts to xing a residual gauge invariance.) We can now determine the form of the gauge conditionon the metric: n m=nam a;nm=a mna)nmnngnm=nnnm The lightcone gauge is again useful for eliminating unphysical degrees of freedom. The linearized lightcone gauge conditions with na=a imply, for the unsymmetrized h, naeam=nam a)h+a=0 506 IX. GENERAL RELATIVITY emana=a mna)eamb mnb=na)ha+=0 and thush(+a)= 0, so we can again work with just the symmetrized h. Separating out the trace part as hij=hT ij+ijh,w h e r ehT ijis traceless, we nd for the linearized gauge-xed action L0 G1 4habhab+1 8ha ahb b1 2f2 =1 4hTijhTij1 2(h0i)2+D2 2hh0 h0ifi=@+hi+@jhTijD4 2@ih h0@+f+D4 4h=@+2h+@+@ihiD2 2@+@h+D4 4h where we have simplied some algebra by writing the gauge-invariant action as the Lorentz gauge one minus its gauge-xing terms. (There is some ambiguity in that we can shift h0iby a@ihterm, and absorb the generated terms into h0.) We see that all but hT ijare auxiliary elds (we redened hiandhby just shifting and applying@+), and can be eliminated (but watch out if there are matter couplings, when eliminating them gives Coulomb-like interactions). The temporal gauge (known also as \Gaussian normal coordinates") is used when treating time and space separately: In this case we have for the metric nm=m 0)g0m=0m An alternate way of dening the temporal gauge is to start with a spatial hypersurface, and determine the geodesics normal to this hypersurface ( g0i= 0), where the positions on this hypersurface dene xi, constant along the geodesics, and the proper times along the geodesics dene x0(g00=1), withx0= 0 at the hypersurface. The fact that these are geodesics guarantees that the hypersurfaces of xed, but nonvanishing,(proper) time are still orthogonal to the geodesics ( g 0istays zero): n
V=0; (n
r)n=0)n
[(n
r)V]=0 Equivalently, we can consider a dust of massive particles and choose an initial hyper- surface orthogonal to their (timelike) geodesics to dene x0=s=0 . T h i sc o o r d i n a t e system is thus the \rest frame" of the dust; all the information about the geometry of the space is contained in the time dependence of the spatial separation of the particles(g ij). There is still the residual coordinate ambiguity of how to assign xion the initial hypersurface. Gaussian normal coordinates thus can be useful for studying the dynamics of particles: For example, we can study a gravitational eld of distant, unknown (or B. GAUGES 507 ignored) origin (i.e., the curvature of spacetime) by watching the relative motion of two nearby particles of such a dust, neglecting the gravitational force/curvature eect acting between the two particles themselves. If the two particles start out relatively at rest at some initial time (which is well-dened only if they are close and relatively slow), then in the temporal gauge the paths of both particles are described by xedx i, independent of x0, since their geodesics are simply lines in the time ( na=a 0) direction, and the proper times of both particles are the same as the time x0.T h e n the distance between the particles at any given time is given by the magnitude ofdx mema,w i t hdx0=0a n ddxitheir innitesimal separation. Thus, since the xi's, and thusdxi, are xed, we want to study the change in ema(really just eia;e0a=a 0) with time. Using our evaluation of ( n
r)emafrom above, we nd (n
r)2ema=(n
r)rmna=[n
r;rm]na=nn[rn;rm]na=nbncRbdcaemd using the axial gauge condition n
r=n
@. For the Gaussian case na=a 0,w et h e n have ..ema=R0b0aemb (Of course, vanishing curvature implies geodesics that start parallel remain that way, because the space is then at.) By observing dierent sets of particles initially atrest with respect to each other, we can choose dierent timelike directions n,a n d determine all the curvature components from their linear combinations. Excercise IXB3.1 Let's examine some 2D examples of axial gauges in spaces with positive-denite metric: aGaussian normal coordinates need not be Cartesian in at space. Show that polar coordinates for the plane dene an axial gauge. What is the coordinate in the \n a" direction? Give the geodesic interpretation. bRepeat the above for a curved space | the (2D) sphere in spherical coordi- nates. cApply the above \equation of motion" (..e=Re) to the sphere. (See excercise IXA5.3.) Show its solution agrees with the obvious. 4. Radial Another useful gauge similar to the axial gauge is the radial gauge (\Riemann normal coordinates"), discussed for Yang-Mills in subsection VIB1. In this case we have nm=xm) (n
r)na=xm@mxna n=na 508 IX. GENERAL RELATIVITY a case of the more general form of the geodesic equation. Applying the same identity as for the axial, we again have n
r=n
@; @mna=a m) (n
@)emana=(n
@)mana)emana=a mna but now the boundary condition is already implied by the gauge condition near the origin: For any innitesimal xm=m, mema(0) =ma m)ema(0) =a m m!mab(0) = 0)!mab(0) = 0 Thus, there is no residual gauge invariance, unlike axial gauges (where the coordi- nates of the initial hypersurface need additional determination). Any reference frame satisfying these conditions at the origin is called a \local inertial frame", and is themost natural for an observer at that point in spacetime. (In at space, this yields Cartesian coordinates.) Excercise IXB4.1 We can think of Gaussian normal coordinates as dened by a dust of particleswith ane parametrization v= 1 and unit mass m=1 ,w i t h=s=x 0and xiconstant for any particle (. X=p=n). For Riemann normal coordinates we can think of particles radiating out from the origin xm= 0 in all possible directions in space and time, but then some must be antiparticles (traveling backward in time), some must be massless (for the lightlike geodesics), and some must be tachyons, with m2<0 (for the spacelike geodesics). However, as for the Gaussian case, we can still identify nm=. Xm Using the radial gauge condition, show that these can be chosen as geodesics with v=e;X ()=eX(0);p =X(0) so all particles start at the origin at =1, and their position at =0 is determined by their initial (constant) momentum. (Thus particles withproportional momenta travel the same path, but arrive at dierent points at = 0; however, in this case is neither the time x 0nor the proper time s, but just an arbitrary parameter.) As we saw in subsection VIB1, the radial gauge is related to gauge-covariant translation (in general relativity, \parallel transport") as, for any tensor , ~ (~y)=exa(y)Da (y)=e (~y); ~ym=exa(y)Eam(y)Dmym B. GAUGES 509 whereyis the \origin",  = IMIis just a Lorentz transformation, and Dis the covariant derivative acting at y: Da=Eam(y)Dm+$aI(y)MI;Dm=@ @ym;[Da;Db]=TabcDc+RabIMI As in general for coordination transformation parameters a,xanow transforms under local Lorentz transformations. (In background eld language, this \quantum eld" transforms under the \background" Lorentz transformations.) Thus, xais now a function of y; it cannot be made even covariantly constant in general: Daxb=0) 0=[Da;Db]xc=xdRabdc (For more practical reasons, if we dened it to be invariant or constant, the manip- ulations that follow would break down.) At this point we have only made a Lorentztransformation on , since it and ~ are evaluated at the same point ~ y. However, as for Yang-Mills in subsection VIB1, for the next step we want to identify x aas the new coordinate: 0(xa)=~ (~ym(ym;xa)) =exaDa (y) where 0has implicit dependence on y, since in radial gauges the choice of origin is gauge parameters that dene the gauge. (The coordinates are dened as radial with respect to the origin y.) Thus, we have made a Lorentz transformation !~ followed by a coordinate transformation ~ ! 0. We also want to dene a covariant derivative for xby r 0=(D )0=exaDaDa (y) wherer(as for Yang-Mills) contains only @a=@=@xaand notDm: ra=eab(x)@b+!aI(x)MI At this point we no longer distinguish at and curved indices, since the Lorentz gauge has been xed. We have then transformed y; ;D!x; 0;r Note that the y-coordinate tensors are the x-coordinate tensors evaluated at the origin: 0(0) = (y); (r 0)(0) = (D )(y) We can identify this as the radial gauge when x(y) satises the geodesic condition, since then (x
D)x=0)x0ex
Dx=x 510 IX. GENERAL RELATIVITY )x
r 0=x
(D )0=(x
D )0=x
D 0=x
@ 0 making use of 0(x)=ex
D (y). Unfortunately, it is somewhat dicult to continue this construction in terms of the covariant derivative, but simpler in terms of the \dual" dierential forms. We therefore dene the (Lorentz-covariantized) Lie derivative as Lx
D^ =x
D^ ;Lx
D^D=[x
D;^D] for any \tensor" (object carrying only at indices) ^ and any \covariant derivative" (object with a at vector index free, but multiplying partial derivatives and Lorentz generators) ^D. We generalize to evaluate on not only andD, but to apply the Lie derivative also as part of the transformation exp(Lx
D). For that reason, for the remainder of this section we will abbreviate Lx
Das justL.W et h e nh a v e [x
D;Da]=(Daxb)Db+xb(TbacDc+RbaIMI) Dening Lie derivatives to satisfy the usual Leibniz and distributive rules like any derivative (since we use them as innitesimal transformations), we then nd (LEam)Emb=Daxb+xcTcab;EamL$mI=xbRbaI In terms of dierential forms, dened as Ea=dymEma;$I=dym$mI D=EaDa=d+$IMI)Dxa=dxaxb$ba we then have Lxa=0;L(T;R)=x
D(T;R) LEa=DxaEbxcTcba L$I=EaxbRbaI)L (Dxa)=xbEcxdRdcba which covers all the quantities that appear in evaluating eLonEaand$I.T h e geodesic condition prevents higher derivatives of xafrom appearing in the transfor- mation law, and allows us to freely reorder all the x's to the left at the end of the calculation for identifying the coecients of the Taylor expansion, at which point we can forget that xdepends on y. Thus, these few equations for the action of L allow any transformed quantity to be evaluated straightforwardly by iteration, Taylorexpandinge Lin powers ofL. B. GAUGES 511 The important distinction between the transformation laws for DaandEais that forEathe derivatives of xappear only in the combination dx,w h i c hm a k e s changing coordinates from y(or ~y)t oxeasier. Specically, by iterating the above Lie derivatives, we nd a solution of the form E0a=eLEa=EbAba+(Dxb)Bba;$0I=eL$I=EaAaI+(Dxa)BaI whereAba,Bba,AaI,BaIare functions of xand of tensors evaluated at the \origin" ((D


DT )(y), (D


DR )(y)). For Riemann normal coordinates, we want to x y (e.g.,y= 0), so we evaluate the above at dy= 0. Furthermore, we can choose the gauge!(0) = 0 (at least for vanishing torsion), so also Dx!dx. Then the solution is Ea=dxbBba;!I=dxaBaI Thus,BbaandBaIare the inverse vierbein emaand Lorentz connection !mIfor the new coordinate system, ra=(B1)ab(@b+BbIMI) written explicitly as a Taylor expansion in xby the above method, all of whose coecients are tensors (torsions and curvatures and their derivatives) evaluated at the origin. However, we can also use these results for rst-quantization (where actions are expressed in terms of, e.g., E0a=dfor the particle) in background eld gauges, by choosingyas the background and xas the quantum coordinate (see subsection VIB1); then we keep both the dyandDxterms. Excercise IXB4.2 Find the rst few orders of this expansion. aUsing the above method, show that for vanishing torsion Ea=dxb(a b1 6xcxdRcbda+:::);!I=1 2dxaxbRbaI+::: bCheck the validity of this result by evaluating [ r;r] to this order from the rgiven by this Eand!. cUse instead the covariant-derivative method of subsection VIB1. In this case, we nd ra=ex
~D~Daex
~D+habex
~D@bex
~D wherehabis chosen to cancel all Dmterms inr, and we have dened ~Da=Eam(y)Dm+$aI(y)fMI 512 IX. GENERAL RELATIVITY where now xais \constant", so DmandfMdo not act on it. (Otherwise, in this approach, we would be stuck with tons of D


Dxterms.) In terms of the previously dened Lorentz generators, Mab=fMab+x[a@b] Findhto this order, and use it to obtain ra=(b a+1 6xcxdRcadb+:::)@b+1 2(1 2xbRbacd)Mdc restoringfMtoM. 5. Weyl scale The gauge-xed kinetic term can be simplied by including the conformal com- pensator (see subsection IXA7). The quadratic part of the gauge-invariant Lagrangianis then (=1+ 1 2) L0=1 4e1(R4D1 D2) 1 4[habhab+2(@bhab)2ha ahb b+2ha a@b@chbc]1 2(ha a@a@bhab)+1 4D1 D2 The nicest (globally Lorentz) covariant gauge comes from choosing the coordinate and scale gauge-xing functions fa=@bhab1 2@ahb b+1 2@a; f =ha a We use these to obtain the gauge-xed Lagrangian (see subsection VIB9) L=L0+1 2(@bhab1 2@ahb b+1 2@a)21 8(ha a)(hb b) =1 4habhab+1 41 D2 plus ghost terms. Now the hkinetic term is simpler. Also, remember that de- couples from conformal matter. These features of gauge xing make this formalism closely analogous to the St uckelberg formalism for the massive vector. We can also dene nonlinear versions of these gauge-xing functions, such as @m(e1=2eam)o r @m(2pggmn) for the coordinate gauge, and e1=2or2pgfor the scale. Excercise IXB5.1 Find the ghost terms for linearized gravity in the Fermi-Feynman gauge, andits simplication with the compensator. The scale gauge can also be xed in terms of the vierbein/metric alone: For example, we can x the gauge e=1 B. GAUGES 513 in which case acts simply as a renaming of e. A more unusual gauge is R=0 This is not a restriction on the geometry, since the physical Ricci scalar is eectively replaced by its scale transform R0=(D+2)=(D2)(R4D1 D2) which is scale invariant. In the gauge =1 ,R0=R, but in the gauge R= 0 it is proportional to . Excercise IXB5.2 Show that the ghosts for scale transformations propagate in the gauge R=0 : Find their contribution to the action. More general gauges are possible when matter elds appear. For example, con- sider coupling gravity, with compensator, to a physical conformal scalar .W i t h appropriate normalization of the compensator and physical scalar, the kinetic termsfor the two elds are identical except for sign: There is a manifest O(1,1) symmetry. We can take advantage of this by using a \lightcone" basis for these elds: Dening  = , the full nonlinear (in gravity) Lagrangian Lbecomes (S=R dxe1L) L=+(D1 D21 4R) The overall normalization is arbitrary, including sign, since we can rescale either eld by a constant. Many Weyl scale gauges are possible, and somewhat more transparentthan making eld redenitions on the corresponding action without compensator. Eectively, we can redene the elds  arbitrarily as long as we don't x +=to a constant, since that combination is scale invariant. (I.e., +=can be redened, but not xed.) Some of the more interesting choices are: =1')L=1 4R'(D1 D21 4R)' =e')L=1 4RD1 D2'' ='1a)L='[(1a2)D1 D21 4R]' +='; =1)L=1 4R' We can also have any of these gauge-xed Lagrangians with opposite overall sign, simply by changing the choice of either +orby a sign. The rst two choices are useful because they put the action in standard form, as the usual gravity action 514 IX. GENERAL RELATIVITY plus a physical scalar kinetic term. (Thus, coupling a massless scalar to gravity either conformally or minimally is equivalent, and the two cases are distinguished only by interactions.) In fact, the rst choice, or \temporal gauge" ++=constant ,j u s t returns us to the form without compensator, = 1. On the other hand, changing the sign ofyields the \axial gauge" +=constant , which is xing the physical scalar as = 1. The overall sign of the action changes because the physical scalar is traded for the compensator, or the corresponding part of the metric. This gaugeis closely related to the \string gauge": In our third choice above the gravity actionis invisible until the surviving scalar has been expanded about its vacuum value.The constant ais arbitrary except that it must not vanish (so that  +=is not a constant). In particular, this action appears in string theory, with the choice a=1p D1)L='(1 4R)' which eliminates explicit D-dependence. Again the scalar appears with the wrong- sign kinetic term, but Rappears with the right sign (or vice versa), because of more complicated redenitions. The sign of the changes back to the usual for jaj>1. However, forjaj= 1, it disappears completely. A similar result occurs for the last choice, or \lightcone gauge" =1 . Excercise IXB5.3 The property that distinguishes this kinetic term for a scalar coupled to grav-ity is the O(1,1) symmetry: aBefore xing the Weyl scale gauge, the continuous SO(1,1) subgroup of this symmetry is just the scaling  !1. After gauge xing, this trans- formation may change the gauge, and thus may need to be combined with a constant Weyl scale transformation to preserve the gauge. In that case the vierbein will also transform under the resulting modied SO(1,1) trans-formation. Find the SO(1,1) transformations for 'ande amin the above 4 gauges. bThere is also the \parity" transformation of this O(1,1), +$.F i n dt h e modied form of this transformation for 'andeam. Although all these choices are equivalent in perturbation theory (though the physical scalar may require a nonvanishing vacuum value), they aren't necessarilyso nonperturbatively, depending on the ranges of the various scalars. Unfortunately, nonperturbative gravity is not understood well enough (even classically) to make such distinctions, even though they may be important physically. The above considera-tions generalize straightforwardly to the case with many physical scalars, where we B. GAUGES 515 may consider symmetry groups such as O(n,1). If the physical scalars form a nonlin- earmodel, the compensator may join in to make the -model groups noncompact: Examples of this appear in supergravity and strings (see below). The appearance of a physical scalar can also aect the way scale gauges are chosen in conjunction with coordinate gauges. For example, a result similar to the one foundat the beginning of this subsection can be obtained from the (linearized) action withboth compensator and physical scalar (whereh i=0 ) , LL 01 4 choosing the same -dependent coordinate-xing term ( fa)2, but imposing the scale gauge =1p 2(ha a) The result is identical to the one given at the beginning of this subsection, except that now no scale ghosts appear: The scalar that appears as haais now physical, and no longer needs a ghost to cancel it. This is the perturbative \string gauge" for scaleinvariance, which appears automatically in covariantly gauge-xed string theory. Excercise IXB5.4 Let's investigate such gauge choices further: aStarting with the Fermi-Feynman-gauge-xed linearized gravity action of sub- section IXB1, add the physical-scalar kinetic term 1 4 . Separate the traceless and trace pieces of hab. Show that the string-gauge action (i.e, the one given at the beginning of this subsection if we ignore ghosts) follows fromsimply switching $ 1p Dha a and then identiying the new withp 2=(D2). bThe way the physical scalar of string theory appears in the gauge-invariant and gauge-xed action is slightly more clever than as described above. (Seesubsections XIB5-6 below.) The kinetic term (already in the string gauge forscale invariance) is S=Z dx( 1 4R) where the missing ehas been absorbed into  by a eld redenition. (Since  is thus not a scalar, we dene b ye1=2e1=2, since e1=2 is a scalar.) Expanding  = 1 + , the linearized gauge xing is now simply L!L+1 2(@bhab+@a)2 516 IX. GENERAL RELATIVITY (or we can use the nonlinear gauge-xing function @m(eam)). Show the result i st h es a m ea sa b o v e . REFERENCES 1T. de Donder, La gravique einsteinienne (Paris, 1921). 2J. Scherk and J.H. Schwarz, Gen. Rel. and Grav. 6(1975) 517; M. Kaku, Nucl. Phys. B91 (1975) 99; M. Goro and J.H. Schwarz, Phys. Lett. 127B (1983) 61: lightcone gauge for gravity. 3Cartan, loc. cit. (IXA), ch. X; M. Spivak, A comprehensive introduction to dierential geometry (Publish or Perish Inc., 1979) v. II, p. 299:Riemann normal coordinate expansion. 4U. M uller, C. Schubert, and A.E.M. van de Ven, A closed formula for the Riemann normal coordinate expansion, gr-qc/9712092. 5W. Siegel, Phys. Lett. 149B (1984) 162, 151B (1985) 396, 211B (1988) 55: string gauge. C. CURVED SPACES 517 ::::::::::::::::::::: ::::::::::::::::::::: ::::::::::::::::::::: C. CURVED SPACES ::::::::::::::::::::: There are some important solutions of general relativity that have no close analog in Yang-Mills. Here we consider the ones relevant to the only experimental verica-tions of this theory: Solutions outside approximately spherical matter distributions(like the Sun and Earth), and those describing the Universe itself. 1. Self-duality Plane wave solutions can be constructed for gravity in the same way as for Yang- Mills (see subsection IIIC3): A little more work (solving the torsion constraint, orusing the result of the free theory) gives r +=@+1 2xixjR+i+j(x)@xiR+i+j(x)Mj(r=@;ri=@i) whereR+i+jis an arbitrary function of x, but symmetric in ij, and the empty-space eld equations imply it is also traceless: R+i+i=0 If we want to couple Yang-Mills to gravity, then we can still write exact solutions as long as both waves are parallel; then R+i+i=2T++=1 g2tr(F+iF+i) where here g2refers to the Yang-Mills coupling. (Similarly, we can add in other elds, such as massless, neutral scalars or particles.) Excercise IXC1.1 Check that the gravitational plane wave solution satises the eld equationsand torsion constraint. Show that we can also nd more-special solutions ofthis form satisfying g mn=mn+1 g2tr(AmAn);Rmnpq =1 g2tr(FmnFpq) This has the interpretation that the \graviton" is the bound-state of two \gluons". However, it is only a kinematic eect, since the two gluons happen to be traveling in the same direction at the same speed. (We saw in subsectionVIIIA7 that a similar eect always occurs in D=2, since there only two spatialdirections exist.) 518 IX. GENERAL RELATIVITY Self-duality for Yang-Mills was discussed in subsection IIIC4. Similar remarks apply to gravity: We again impose [ra;rb]=1 2abcd[rc;rd] Self-duality again implies the eld equations, by dualizing the Bianchi identities: For gravity R[abc]d=0) 0=1 2abcdRbcde=1 4abcdbcfgRfg de=Ra e While it might appear that the self-duality condition is still second-order because solving the torsion constraint makes the Lorentz connection the derivative of the vierbein, the self-duality allows the gauge where the connection is also self-dual, and this condition eectively becomes a rst-order eld equation: Rabcd=Rcdab)Rabcd=1 2cdefRabef)!abc=1 2bcde!ade In four dimensions (2 space + 2 time), lightcone methods can again be applied (see subsection IIIC5): Now [r0;r 0]=C 1 2R0000M00 (!0 =0 ) T h ef a c tt h a t[r(0;r )0] has only an M00term poses an additional constraint; the full solution is then r0=@0;r 0=@ 0+(@0@0)@ 0+1 2(@0@0@ 0)M0 0 R00 00=i@0@0@ 0@0 (In this case, the existence of covariantly constant spinors is a consequence of self- duality.) The equation of motion that follows from the nal condition is now i(@0@0)(@ 0@ 0)=0 2. De Sitter The simplest spaces are those where the Ricci scalar is constant, and the other parts of the curvature vanish: Rabcd=kc [ad b] These are special solutions of the eld equations without matter, but with a cosmo- logical term, where there are no physical gravitons (the Weyl tensor vanishes), and thus represent the vacuum. Since there are no physical degrees of freedom, we can represent this space by just the conformal compensator: i.e. the vierbein (metric) C. CURVED SPACES 519 is just the at one up to a local Weyl scale transformation. We thus have (from subsection IXA7) Rabcd![c [a@b]@d]c [ad b](@)2=kc [ad b] where we have written the curvature as a scale transformation of at space Rabcd= 0;r=@: The space is \conformally at". Separating this equation into its ir- reducible parts with respect to the Lorentz group, the Weyl tensor part vanishesidentically, leaving 2 D(@)2=Dk; D@ a@b=ab (The latter equation isn't implied in D= 2, where the global conformal group is larger, and more general coordinate choices are possible for this solution. However,we can still use it consistently.) The latter equation can be solved easily: Looking ata6=b, we see that  is a sum of functions of one variable. Then looking at a=btells us that these functions are quadratic and have the same quadratic coecient, whilethe former equation gives k: =A+B axa+C1 2xaxa;k =2ACB2 We can choose any A,Ba,a n dCthat give the desired value of k: For example, we can choose the solution  = 1 +1 4kx2(giving the usual at-space coordinates for k=0 ) ,o r= Baxa(choosing the direction of Baas appropriate to k=B2| spacelike, lightlike, or timelike). Excercise IXC2.1 Show that after a Weyl scale transformation the action for gravity including acosmological term is (up to a sign) that of a conformal self-interacting scalarcoupled to gravity. Use this to show that the de Sitter space solution (in theR=0 gauge) yields an \instanton" for this scalar theory, and compare with theYang-Mills instanton of subsection IIIC6. Show that similar solutions existfor massless scalars in arbitrary dimensions with potentials  nfor arbitrary n(but thenk=0 ) . The geometry of this space can be understood most easily as that of a D- dimensional hyperboloid embedded in a at (D+2)-dimensional space, where we addone space and one time dimension: Again using the methods of subsections IA6 andIVA2, we now supplement the constraint y 2=0)yA=ewA; (w+;w;wa)=( 1;1 2xaxa;xa) 520 IX. GENERAL RELATIVITY with the additional constraint nAyA=1)e=1 nAwA=1 nn+1 2xaxa+naxa wherenAis a (D+2)-vector, yielding the intersection of a cone and plane. In partic- ular, forn26= 0 we can write the metric on the space whose coordinates are all but n
y: yA=(jn2j1=2;zA))z2+n2=0;ds2=dz2 which is the denition of a hyperboloid. Comparing the metric, we nd the previous result: ds2=dy2=e2dxadxa;e =1)k=n2 This gives the most general coordinate system for de Sitter space as a local scale of at space, since conformal transformations are the most general coordinate trans- formations that will just replace this scale factor with another, and they just rotate nA. The symmetry group of the D-dimensional subspace that satises these two con- straints is as big as the Poincar e group, namely SO(D,1), ISO(D 1,1), or SO(D1, 2), depending on whether n2<,= ,o r>0: The former constraint preserves the conformal group, while the latter kills a timelike, lightlike, or spacelike coordinate. Excercise IXC2.2 We can also start instead with a D+1-dimensional space, which is a naturalchoice for the symmetry group of de Sitter space: Consider the metric and constraint kd s 2=dz2=kd zadzbab+dz2 D+1; 1=z2=kzazbab+z2 D+1 Both equations have the same global symmetry group, determined by the sign ofk;k= 0, at space, can be considered as a limiting case of the others. aSolve the constraint y2=1!z(x) as in subsection IVA2 for 2=m2!(), and substitute to nd the metric in terms of x. bFind the conformal transformation on xathat relates this coordinate sys- tem to the more general one above. (Hint: Use zof the D+2-dimensional construction.) C. CURVED SPACES 521 3. Cosmology To a good approximation the universe can be described by a spacetime which is (spatially) rotationally invariant (\isotropic") with respect to a preferred time direc- tion. Furthermore, it should be (spatially) translationally invariant (\homogeneous"),so the metric should depend only on that time coordinate. This means that the 3D subspace at any xed time should be 3D spherical, at, or de Sitter space, up to an overall time-dependent scale factor: ds 2=d2+2() where  is the de Sitter metric for the 3 other dimensions for k=1;0;1( g i v e n , e.g., by the coordinates in the previous subsection.) By a simple redenition of the time coordinate, this can be put in a form which is conformal to a static space: ds2=2(t)bds2;bds2=dt2+ where by \(t)" we really mean \ ((t))", and the two time coordinates are related by d=dt )t=Z d1 ()or  =Z dt ((t)) Using previous results for 3D de Sitter space, we nd bds2has curvature ^Rijkl=kk [il j];r e s t =0 wherek=1;0;1. To a good approximation the matter in the universe can be approximated as a \dust", a collection of noninteracting particles. It should also be rotationally invariantwith respect to the preferred time direction, so the momenta of the particles should be aligned in that time direction. (Really it is this matter that denes the time direction, since it generates the curvature of spacetime.) Furthermore, the dust shouldbe translationally invariant, so the energy-momentum tensor should depend only on that time coordinate. We then can write (compare excercise IIIB4.1) T ab M=M(t)uaub;ua=a 0 whereMis just the spatial density of particles in the \rest" frame. One way to derive thedependence of Mthat generalizes straightforwardly to other cases is by using conservation laws: By considering particles all of the same mass in units m=1 , or by considering JandTfor each individual particle (since in this case we neglect interactions), we have from current conservation Ja=Mua) 0=raJa=e@me1Jaeam=(4)@0(4)(M1) 522 IX. GENERAL RELATIVITY )M=3a3 for some nonnegative constant 3 a, and using (covariant) energy-momentum conser- vation as a check, uaraub=r0ub=1@0ub=0 (geodesic ))raTab M=ubraJa+Jaraub=0 where we have used (from the result of subsection IXA7 for scaling covariant deriva- tives) br0=@0)r 0=1@0 (Note, however, that rihas Lorentz pieces, and M0iJiJ06= 0 even though Ji=0 . ) For radiation, the momenta of the photons can't be timelike (they're lightlike, of course), but we can still use rotational and translational invariance, together with the fact that the trace of the energy-momentum tensor vanishes (from scale invariance:see subsection IXA7). Then T ab R=R(t)1 3(4uaub+ab) There is no conserved current, but energy-momentum conservation alone determines 0=raTab R=4 3MuaubraR M+rb1 3R=1b 0(4 3M@0R M1 3@0R) =1b 04=3 M@04=3 MR)R=3 2b4 for some nonnegative constant3 2b. Writing the vierbein as a scale-transformation of the constant curvature space discussed above (de Sitter in spatial directions, at in other directions), the gravita- tional eld equations with matter and radiation become (using results from subsection IXA6 or 7): 6[(bra)(brb)1 2ab(br)2]+[ (abbbrabrb)+( ^Rab1 2ab^R)]2=2 (TMab+TRab)4 The only independent components of this equation are the 00-component and trace, which are, after multiplying by an appropriate power of : 1 2. 2+1 2k2=a+1 2b;.. +k=a Fork= 1, these are just energy conservation and the equation of motion for a harmonic oscillator (centered at =a). The 00 equation gave energy conservation becauseT00is the energy density. The trace equation gave the eld equation for  due to the relation of TaatoS= given earlier. These equations are easily solved: C. CURVED SPACES 523 Imposing the initial condition (0) = 0 (i.e., we set the \Big Bang", when curvatures and energy density were innite, to be t=0 )a n d. (0)>0( s o0), k=8 < :1 0 19 = ;:=a8 < :1cos t 1 2t2 cosh t19 = ;+p b8 < :sint t sinh t9 = ; The \physical" time coordinate is then =R 0dt . In general can't be expressed directly in terms of , so we use the expressions for both in terms of t. For example, fork=1a n db= 0 (just matter), we get a cycloid, which has only such a parametric expression. Explicit expressions can be found for a= 0 (just radiation): ()i st h e n a circle, parabola, or hyperbola for k=1, 0,1. Also, for k=0a n db=0 ,2=3 (vs.pfora=0 ) . For the case of pure matter ( b= 0), the energy conservation equation written in terms of the coordinate becomes, using d=d t, 1 2d d2 a =k 2 This is the same as the Newtonian equation for the radial motion of a particle under the in uence of a xed point mass (or the relative motion of 2 point particles). 4. Red shift The most obvious eect of the cosmological expansion is the cosmological \red shift". The expansion of the universe causes photons to lose energy, including thoseof the black-body radiation of the universe as well as those emitted long ago fromdistant sources. The easiest way to see this is to consider Killing vectors. Since the cosmological solutions are related to static, isotropic, homogeneous spaces by a time dependent (but space independent) scale transformation, the symmetries of thisspace are just in the spatial directions, and are basically the same as before the scaletransformation. Specically, the Killing vectors that survive the scale transformationr a=bra+(brb)Mabsatisfy 0=r(aKb)=br(a(Kb))2abK
br )Ka=1bKaforbK
br=0 We then nd for conserved momenta Kapa1pa.S i n c e t h e K's which survive are just the spatial ones, we at rst nd only the spatial components of 1paconserved, but the conservation of the time component follows from papa= 0 for photons. Thus, 524 IX. GENERAL RELATIVITY pa1.S i n c epais what an observer measures as the components of momen- tum (in his \local inertial frame", a gauge where at his location the metric is at and its rst derivative vanishes), observers measure the photon's energy, frequency, and corresponding black-body radiation (whose distribution depends only on E=T) as having time dependence 1(and wavelength as ). The spectrum of radiation emitted by a distant object is then shifted by this energy loss, so the amount of shift determines how long ago it was emitted, and thus the distance of the emitter. Excercise IXC4.1 Using this result for the dependence of the momenta of individual particles, we can now rederive the dependence of 's of the previous subsection directly from the explicit expressions for JandTof the point particle. aRederiveJandTin curved space as in subsection IIIB4 and show that Jm0 m=(p0)e(2)23(xX);Tmn=Jmpn=Jnpm bFrom Killing vectors we just saw that pa0 a0(m6=0 ) 1(m=0 ) where the massive particles are at rest ( pa=ma 0). Combine these results to nd Ja0 a3;Tab0 a0 b3(m6=0 ) 4(m=0 ) cFind the factors multiplying the 's inTabfor the two cases of dust and radiation from the explicit expression for Tmn. For the massive case (dust) all particles can be taken at rest, but for the massless case the particles travel at the speeed of light, so average over particles traveling in the three spatial directions. ( is a continuous function obtained by summing the functions of all the particles. However, for the above results it is sucient to consider each individual particle for the massive case, and 3 particles at the same point going in orthogonal directions for the massless case.) Astronomers use 3 parameters which are more directly observable: Hd=d ;qd2=d2 (d=d )2; 3H2 where=T00=M+R. Note that in our conventions spatial integrals are weighted asRdD1x=(2)D=2; thus the relation of our density to the more standard one is (see, e.g., excercise IXC4.1a) =( 2)2usual)=4 3Gusual H2 C. CURVED SPACES 525 whereG=in our conventions (but sometimes G= 1 is useful, especially for solutions describing stars and planets. See excercise IXB1.1.) The \Hubble constant" Hmeasures the expansion rate, and gives a length (time) scale. The \deceleration parameter" qis a dimensionless parameter which tells how fast the expansion rate is slowing down. The \density parameter" is another dimensionless parameter which measures matter density with respect to the amount needed to close the universe. Inthe case of pure matter, and with vanishing cosmological constant, =q. Then the \critical" value is q= 1 2,f o rw h i c h k=0 : F o rq>1 2,k= 1, while for q<1 2,k=1. In this case we also see that for a given value of Hthe critical value of the matter density isc=3 2H2. If the matter of the universe has this density, we have k=0 , and spacetime is conformally at. If it has greater density, we have k>1, and space is closed. Excercise IXC4.2 Solve forandqin terms of just a;b;k; and(but no time derivatives). In particular, show b=0)=q=( 2k a)1 a=0) 2=q=( 1k b2)1 These quantities are dicult to measure. Some recent estimates for their present values are: H1= 14(2)109yrs:; : 02<q< 2 Even the gravitational constant is not so easy to measure: Its presently accepted value is G=6:67259(85)1011m3kg1s2 which is accurate to only a few parts per 10,000, compared to the standard atomic and nuclear constants, which are known to better than 1 part per million. In \natural(Planck) units," c=G= =h=1 ,H 1=1:5(2)1061. 5. Schwarzschild All gravitational experiments outside of cosmology are based on the \Schwarz- schild solution", which describes spherical symmetry outside the region with matter. Assuming also time independence, which is a consequence of spherical symmetry (Birkho's theorem), we look for a metric of the form ds2=A2(r)dt2+B2(r)dr2+r2(d2+sin2d 2) 526 IX. GENERAL RELATIVITY (Other coordinate choices are possible, e.g. A2(r)dt2+B2(r)[dr2+r2(d2+ sin2d 2)].) The rst step is the choice of a vierbein: The simplest choice following from this metric is et=A@t;er=B@r;e=1 r@;e=1 rs i n@ (This can also be used as a starting point in place of the metric.) The next step is to nd the commutators of the e's, which tells us what !terms ther's must have to cancel these cabc's (vanishing torsion): [e;e]=r1cot  e [er;et]=B(lnA)0et [er;e]=r1Be [er;e]=r1Be)rhas M rthas Mtr rhas Mr rhas Mr)rr=B@r rt=A@t+Mtr r=r1@+Mr r=(rs i n )1@+ Mr+M where;; and depend only on r, whiledepends also on . (Their explicit forms are already clear at this point, but we'll collect the results below.) We can now determine these Lorentz connections and compute the curvatures by calculating thercommutators. Since we now use explicit functions for the vierbein and connections, we use the method described in subsection IXA2 for this situation:Using the identities [M 12;V2]=22V1; [M12;M23]=22M13 [r1;r2]=[e1+!1;e2+!2] =f[e1;e2]+(e1!2)M2(e2!1)M1g+f!1[M1;r2]!2[M2;r1]!1!2[M1;M2]g we then nd: [rt;r]=Mt)Rtt= [rt;r]= Mt)Rtt= [rt;rr]=B(lnA)0etB0Mtr+rt=[B(lnA)0]et+(2B0)Mtr )=B(lnA)0;Rtrtr=2B0 [rr;r]=B re+B0Mr+r=(B r)e+(2+B0)Mr )=B r;Rrr=(2+B0) [rr;r]=B re+B 0Mr+B0M+ r =( B r)e+( 2+B 0)Mr+( +B0)M C. CURVED SPACES 527 ) =B r;Rrr=( 2+B 0);Rr=( +B0) [r;r]=cot  re+1 r(@)M+r+ MMr =(cot  r)e+( )Mr+(2+ +1 r@)M )=cot  r;R=(2+ +1 r@);Rr=0 Collecting the results: rt=A@t+B(lnA)0Mtr rr=B@r r=1 r@+B rMr r=1 rs i n@+cot  rM+B rMrRtrtr=BA[B(A1)0]0 Rtt=Rtt=B2 r(lnA)0 Rrr=Rrr=BB0 r R=1B2 r2 Excercise IXC5.1 Find the covariant derivative for the 2-sphere in spherical coordinates ds2=d2+sin2d 2 in terms of the single SO(2) generator Mab=abMby the above methods (and not that of excercise IXA4.5). Calculate the curvature. Find the three Killing vectors. (Hint: What is the symmetry of the sphere?) Having all the curvatures, we can now calculate the Ricci tensor, which appears in the eld equations. The nonvanishing components are: Rtt=Rtrtr+2Rtt;Rrr=Rtrtr+2Rrr;R=R=Rtt+Rrr+R Vanishing of Rab1 2abRis equivalent to vanishing of Rab. In terms of these curvatures, we see it also implies Rtrtr=2Rtt=2Rrr=R These are easy to solve: First, Rtt=Rrr) (lnA)0=(lnB)0)A=B1 where we have xed the proportionality constant by requiring A;B!1a sr!1 (redening tby a constant scale transformation). Also, 2Rrr=R) (1B2)0=1 r(1B2)) 1B2=k r 528 IX. GENERAL RELATIVITY )B=r 1k r for some constant k. The last eld equation is then redundant. (As usual, the eld equations are related by the Bianchi identity.) The constant kcan be related to the nonrelativistic result by comparing at large distances. (See excercise IXB1.1.) We then ndk=2GM, so the nal result is: ds2= 12GM r dt2+ 12GM r1 dr2+r2(d2+sin2d 2) More generally, if we have some spherically symmetric, static matter distribu- tion, then the only nonvanishing components of the energy-momentum tensor will be Ttt,Trr,a n dT=T(representing energy denisty, radial pressure, and isotropic pressure), all functions of just r. Repeating the above procedure, we integrate [r(1B2)]0=2r2Ttt; [ln(AB)]0=r B2(Ttt+Trr) while the remaining equation is redundant. For example, for a spherically symmetric, static electromagnetic eld the only nonvanishing components of the eld strength are FtrandF, corresponding to electric and magnetic charges, respectively. Then the invariance of Tunder a duality transformation (see subsections IIA7, IIIA4) implies T=T)Ttt=Trr)A=B1 again, since on this Fabduality eectively replaces ( ;)$(it;r), with the ifrom Wick rotation. Local scale invariance (see subsection IXA7) then tells us Taa=0)Ttt=Trr=T=T Excercise IXC5.2 Let's rederive these results by brute force: aDeriveTabfor a general electromagnetic eld by varying its action with respect toeamorgmn. bFind each of T's components explicitly in terms of FtrandFin the case where those are the only nonvanishing components, and show they appearonly in the combination F 2 tr+F2 . As usual, these eld strengths can be found easily from the integral form of Gauss' law by integrating over a sphere: For example, for the magnetic eld magneticcharge 1 2Z dxmdxnFmn=4r2F C. CURVED SPACES 529 for theFcomponent of Fab(integrating over and), since the metric (and vierbein) forandis the same as for at space. By duality, the solution for Ftrin terms of the electric charge is the same. The result is Ttt=Q r4;Q =2(e2+g2) for electric charge eand magnetic charge g.T h e 1=r4dependence also follows from scale invariance, since the charges are dimensionless (and the matter eld equationsdecouple from AandB). (Again, since the solution does not extend to r=0 ,w e normalize by comparing F aborTabatr=1to the at-space solution.) The net eect on the Schwarzschild metric is 12GM r!12GM r+2Q r2 Our solution relates to the usual mechanics normalization of the charges (see subsec- tion VIIA3), restoring G,a s 2Q=G2(e2+g2)=G(e2 m+g2 m) Excercise IXC5.3 Let's also apply brute force to solving Maxwell's equations raFab=r[aFbc]= 0 (outside the matter). aAs a warm-up, using directly the above covariant derivatives, show that in at space Va=a rVr)raJa=r2@rr2Jr Note that the covariant derivative of a vanishing component doesn't necessar- ily vanish (just as the ordinary derivative of a function that vanishes at somepoint doesn't necessarily vanish at that point): Components of rother than r rcontain Lorentz generators that rotate other components of VtoVr. bSolve Maxwell's equations in dierential form for Fabin the above case. Use the empty-space solution to dene the normalization at innity. (Actually, in this case, the charge is well-dened in terms of the ux of the elds, asdescribed above, but gives the same result here because the space is asymp-totically at.) Excercise IXC5.4 Consider the plane wave in the coordinates ds 2=2dx+dx+L2(x) e2(x)dy2+e2(x)dz2 530 IX. GENERAL RELATIVITY Calculate the covariant derivatives and curvature tensor by the method de- scribed above (double-counting and subtracting). Show that the eld equa-tions reduce to L 00+(02)L=0 Excercise IXC5.5 Use the method applied to the Schwarzschild metric to calculate the covariantderivative and curvature tensor for the metric ds 2=dt2+2exdt dy1 2e2xdy2+dx2+dz2 Show that this metric satises the eld equations with a cosmological term for a dust at rest with respect to this time coordinate; i.e. Rmn1 2gmn(R4) =m 0n 0 where  and are both constants. Excercise IXC5.6 Use this method to calculate the covariant derivative and curvature tensor forthe cylindrically symmetric metric ds 2=A2(r)dt2+B2(r)dr2+r2d2+dz2 Assume the matter in this problem is a \perfect uid", Tab=uaub+P(ab+uaub)(u2=1) Solve the equations of motion for the gravitational eld to nd AandB,a s well as the pressure Pand particle density . What is the implied relation betweenPand? Excercise IXC5.7 Use this method to calculate the covariant derivative and curvature tensor for the following metric, corresponding to that outside a planar mass distribution: ds2=A2(z)dt2+B2(z)(dx2+dy2)+dz2 Solve Einstein's equations in empty space to nd AandB(up to some con- stants of integration). C. CURVED SPACES 531 6. Experiments All experiments (excluding cosmology) are based on the Schwarzschild metric. The rst type of experiment involves gravitational redshift, but unlike the cosmolog-ical case, the relevant reference frames of observation are not local inertial frames but the static reference frame in which the Schwarzschild metric is dened. (There are also measurements of redshift from airplanes, whose reference frame is dened with respect to the Schwarzschild one.) In this reference frame the relevant Killing vector is the one which expresses the fact that the space is static, K m@m=@[email protected] h e momentum which is measured by the observer is pa,n o tpmorpm, since the observer still uses a reference frame for which the metric at his position is at (but not its rst derivative, since he is not in free fall). (In fact, this is one of the purposes for using a vierbein, as a frame of reference.) The conserved quantity is then E=Kapa=r 12GM rbE where the energy of a particle bEis the time component of paas measured in this frame. Thus, conservation of Efor a photon gives the r-dependence of the observed energybE(and thus the frequency, which in turn determines the wavelength, since p2=0 ) . To compare with nonrelativistic mechanics, we instead evaluate Efor a massive particle in the Newtonian limit: E 1GM r (m+K)m+KGMm r giving the \conserved energy" Ein terms of the \particle energy" bE(rest massm+ kineticK), including the potential energy. The other type of experiment involves properties of geodesics, so we need to solve the geodesic equations of motion. Without loss of generality, we can choose the angular coordinates such that the initial position and direction of the particle is in the equatorial plane ==2, where it remains because of the symmetry $,a si n the nonrelativistic case. Also as in the nonrelativistic case, we can nd constants ofthe motion corresponding to the energy Eand (z-component of) angular momentum Lby using the Killing vectors K m@m=@=@t and@=@ to nd the conserved quantities Kapa=Kmgmn.xn(in the parametrization v=1 ) : Egtm.xm= 12GM r. t; Lgm.xm=r2.  532 IX. GENERAL RELATIVITY In the case where the particles come from innity, these are the initial kinetic energy and angular momentum. We have chosen an ane parametrization, which requires m2=gmn.xm.xn= 12GM r. t2+ 12GM r1.r2+r2. 2 Solving the previous equations for. tand. , this reduces to the radial equation 0=E2+.r2+ 12GM rL2 r2+m2 )1 2.r2+ GMm2 r+L2 2r2GML2 r3 =1 2(E2m2) This looks like a typical nonrelativistic Hamiltonian for \energy"1 2(E2m2) with the same terms as in the Newtonian case but with an extra r3term. (To take the nonrelativistic limit for the massive case, rst scale the ane parameter!s=m.) Since there are good coordinate systems for a \black hole" using ras a coordinate (e.g., see the following subsection: randr 00+t00, as seen from the gure for Kruskal-Szkeres), this equation can even be used to descibe a fall into a blackhole. (For example, for L= 0 we get the same cycloid solution as in cosmology and in Newtonian gravity, reaching the singularity at r= 0 in nite proper time.) Because of the r 3term in the potential, noncircular orbits are no longer closed. In particular, let's consider orbits which are close to circular. Circular orbits arefound by minimizing the potential for the r-equation: 0=dV dr=GMm2 r2L2 r3+3GML2 r4 0<d2V dr2=2GMm2 r3+3L2 r412GML2 r5 The near-circular orbits are described by small (harmonic) oscillations about this minimum, with angular frequency given by !2 r=d2V dr2=GMm2(r6GM) r3(r3GM) from solving for L2=GMm2r2=(r3GM). On the other hand, the frequency of the circular orbit itself in terms of its angular dependence is just. =L=r2, giving !2 =GMm2 r2(r3GM) This means that the perihelion (closest approach to the Sun) of an orbit, which occurs every period 2 =!rof the radial motion, results in the change of angle 2+=Z2=!r 0dd d=2 !r!=2 16GM r1=2 C. CURVED SPACES 533 )6GM r in the weak-eld approximation. This eect contributes to the measurement of the precession of the perihelion of the (elliptical) orbit of Mercury, but so do the precessionof Earth's axis, the oblateness of the Sun, and gravitational interaction with other planets. As a result, this relativistic eect contributes less than 1% to the observed precession. In particular, the solar oblateness is dicult to measure. The eects on geodesics of photons are much easier to measure, since there are no Newtonian eects. As a result, the weak eld approximation is sucient. We rst consider bending of light by the Sun: A photon comes in from innity and goes back out to innity (actually to the Earth, which we assume is much farther from the Sun than the photon's closest approach to it), and we measure what angle its trajectory was bent by. (For example, we look at the apprarent change of position in stars whenthe Sun passes in their direction during an eclipse.) Starting with the exact solution for a photon's geodesic (case m 2= 0 above), we use the equations for.rand. to nd dr d=r E2 L2r4r2+2GMr Changing variables, ub r;bL E;aGM b)d=dup 1u2+2au3 The impact parameter bL=E would be the closest approach to the Sun neglecting gravitational eects ( L=rp=bE). We now make the weak eld approximation: For asmall, ddup 1u2 1au3 1u2 =d 1asin3 cos2 (usin ) =d a cos  +1 cos  Dening=0a tr=1, the integral is a(1cos  )2 cos )+a(1cos  )2 cos  )u=sinsin +a(1cos  )2 The change in from incoming photon to outgoing photon follows from the two solutions for r=1: u=0)=0;+4a 534 IX. GENERAL RELATIVITY Therefore the deviation of from a straight line is 4 GME=L . (Mathematical note: All variable changes were those suggested by the at space case a= 0: E.g.,b=r=sin , whereis whatw o u l db ei n a ts p a c e . ) A similar experiment involves measuring the round-trip travel time for radio waves from Earth to some re ector (on another planet or an articial solar satellite), with and without the Sun near the path of the waves. Now, instead of dr=d we want, in unitsb=1 dr dt= 12a rr 11 r2+2a r3 )dtrdrp r21+2adrp r21 =dhp r21+2ac o s h1ri =dhp r2b2+2GM cosh1r bi putting the b's back. (We have neglected the gravitational eect on.r,w h i c hi s negligible compared to that on. t, since for most of the path rb.) We then integrate fromr=rminbtor=rEarth, add the integral from r=rmintor=rreflector , multiply by 2 for the round trip, and throw in a factor to convert to the proper time sof the observer (which turns out to have a negligible eect to this order in a). This result is then compared to the same measurement when both observer and re ector have revolved further about the Sun, so bchanges signicantly (but not rEarth nor rreflector ). The biggest contribution (as seen from
t=
b), forbrEarth andrreflector , comes from the cosh term: Forx1,cosh1xln(2x), so
s8GM
(lnb) 7. Black holes For physical massive bodies the Schwarzschild solution applies only outside the body, where Tab= 0. The form of the solution inside the body depends on the distribution of matter, which is determined by its dynamics. Generally the surface ofthe body is at rGM, but we can try to nd a solution corresponding to a point mass by extending the coordinates as far as possible, till the curvature components R abcdblow up. The Schwarzschild metric is singular at r=2GM.I n f a c t ,rand tswitch their roles as space and time coordinates there. There is no corresponding singularity there in the curvatures, which are r3. This unphysical singularity can be eliminated by rst making the coordinate transformation, for r>2GM, r0=Z dr 12GM r1 =r+2GM lnr 2GM1 C. CURVED SPACES 535 and then making a second coordinate transformation by rescaling the \lightcone" coordinates as r00t00=4GMe(r0t)=4GM=4GMrr 2GM1e(rt)=4GM The result is the \Kruskal-Szekeres coordinates" ds2=2GM rer=2GM(dt002+dr002)+r2(d2+sin2d 2) wherer(r00;t00) is dened by r002t002=( 4GM)2r 2GM1 er=2GM This can now be extended past r=2GM down to the physical singularity at r=0 . The complete space now looks like (plotting just r00andt00): t" r"r = 0 r = 0r = 2GM r = 2GM In this diagram lines at 45to the axes represent radial lightlike geodesics. Since nothing travels faster than light, this indicates the allowed paths of physical objects. Curves of xed rare hyperbolas: In particular, the physical singularity is the curve t002r002=( 4GM)2(r= 0), while t002r002=0(r=2GM) is the \event hori- zon" which allows things to go only one way (out from the bottom half or into the top half), and r=1is bothr00=1. Nothing can communicate between the 2 \outside worlds" of the left and right 90wedges. In particular, a star which col- lapses (\gravitational collapse") inside its \gravitational radius" 2 GM is crushed to a singularity, and the spherically symmetric approximation to this collapse must be 536 IX. GENERAL RELATIVITY represented by part of the Kruskal-Szekeres solution (outside the star) by Birkho's theorem, patched to another solution inside the star representing the contribution of the matter (energy) there to the eld equations. This means using just the topand right 90 wedges, with parts near the left edge of this modied appropriately. The top wedge is called a \black hole". (If a situation should exist described by justthe bottom and right wedges, the bottom wedge would be called a \white hole".)Similarly, stable stars are described by just the right wedge, patched to some interiorsolution. This right wedge represents the original Schwarzschild solution in the regionr>2GM where its coordinates are nonsingular. In that region lines of constant t are just \straight" radial lines in the Kruskal-Szekeres coordinate system ( r 00t00). Besides the fact that nothing can get out, another interesting feature of the black hole is that an outside observer never sees something falling in actually reach the eventhorizon: Consider an observer at xed r> 2GM using Schwarzschild coordinates, so his proper time st. Then light radiating radially from an in-falling object is received later and later, up till t=1, by the observer as the object approaches the event horizon, although it takes the object a nite amount of proper time to reachthe event horizon and the physical singularity. There are also more complicated black-hole solutions with spin and electric charge. Another interesting eect of the event horizon is the eventual decay of the black hole (\Hawking radiation"): Pair creation can result in a similar way to that inan electrostatic potential of sucient strength (see excercise IIIB5.1). Particles areemitted near the event horizon (the edge of the gravitational barrier), carrying energyo to innity, while their antiparticles fall into the singularity. There are two features of the black hole that are less than desirable: the existence of singularities indicates a breakdown in the eld equations, and the existence of eventhorizons results in an \information loss". Both these properties might be avoidablequantum mechanically: For example, quantum eects can generate curvature-squaredterms in the eective action, which modify the short-distance behavior of the theory.One might think that such short-distance eects would have an eect only at shortdistances away from regions of high curvature such as the singularity, and thus removethe singularities but not the event horizons. However, it is possible (and examples ofsuch solutions have been given) that the prevention of the creation of the singularityin stellar collapse would eventually result in a reversal of the collapse (\gravitationalbounce"): The would-be black hole solution is patched to a would-have-been whitehole by short-distance modications, resulting in an exploding star that initially re-sembled a black hole but has no true event horizon. C. CURVED SPACES 537 REFERENCES 1J. Ehlers and W. Kundt, Exact solutions of gravitational eld equations, in Gravitation: An introduction to current research , ed. L. Witten (Wiley, 1962) p. 49: gravity plane waves. 2J.F. Pleba nski, J. Math. Phys. 16(1975) 2395: reduction of self-dual metric to single component. 3Siegel, loc. cit. (IVC, 2nd ref. 17): lightcone gauge for self-dual (super)gravity. 4W. de Sitter, P r o c .K o n .N e d .A k a d .W e t . 19(1917) 1217, 20(1917) 229. 5A. Friedmann, Z. Phys. 10(1922) 377; H.P. Robertson, Astrophys. J. 82(1935) 284, 83(1936) 187, 257; A.G. Walker, Proc. Lond. Math. Soc. 42(1936) 90: cosmology in general relativity. 6Particle Data Group, loc. cit. (IC): astrophysical \constants". 7K. Schwarzschild, Sitz. Preuss. Akad. Wiss. Berlin, Math.-phys. Kl. (1916) 189. 8M.D. Kruskal, Phys. Rev. 119(1960) 1743; G. Szekeres, Publ. Mat. Debrecen 7(1960) 285. 9S.W. Hawking, Comm. Math. Phys. 43(1975) 199. 10V.P. Frolov and G.A. Vilkoviskii, Phys. Lett. 106B (1981) 307: possibility that quantum corrections to gravity eliminate black holes. 538 X. SUPERGRAVITY X. SUPERGRAVITY In the previous chapter we studied the symmetry principles behind general rel- ativity; now we add supersymmetry to the picture. Supergravity is a fundamental part of many of the applications of supersymmetry. ::::::::::::::::::::::::: ::::::::::::::::::::::::: ::::::::::::::::::::::::: A. SUPERSPACE ::::::::::::::::::::::::: We rst need to understand the \geometry" associated with local supersymmetry. 1. Covariant derivatives In subsection IVC3 we discussed superspace covariant derivatives for super Yang- Mills. Similar methods can be applied to supergravity, the theory of the graviton (spin2) and gravitino (spin 3/2). In that case we want to gauge the complete (unbroken) global symmetry of the theory: Besides the obvious Poincar e and supersymmetry, there is also the axial U(1) (\R") symmetry that transforms the spin-3/2 eld. (Thebest we might have expected is superconformal symmetry, which also has conformal boosts and scale, but which are broken by the vacuum just as in ordinary gravity, and S-supersymmetry, which is also broken because it's the square root of conformal boosts.) It is introduced in the same way as local Lorentz invariance in ordinary gravity, and acts on at spinor indices (but cancels on vector indices). We thereforewant to gauge the translations @ M(which have been generalized naturally to super- space from @mappearing in ordinary gravity to include supersymmetry), the Lorentz generatorsM,M.. of ordinary gravity, and the (second-quantized) hermitian U(1) generatorY, dened to act on the covariant derivatives as [Y;r]=1 2r; [Y;r.]=1 2r.; [Y;ra]=0 We now use the \ " to refer to hermitian conjugation without reordering, i.e., keep- ing the partial derivatives and other generators on the right. (As in ordinary gravity, transformations are not truly unitary, and covariant derivatives truly hermitian, be- cause of ordering.) Then the gauge parameter, covariant derivative, and eld strengths are expanded over these generators, as in ordinary gravity: K=KM@M+1 2KM+1 2K.. M. .+iK1Y rA=EAM@M+1 2 A M +1 2 A. . M. . +iAAY A. SUPERSPACE 539 [rA;rBg=TABCrC+1 2RAB M +1 2RAB. . M. . +iFABY (EAMis known as the \supervierbein" or \vielbein".) Alternatively, we can write the MandYterms collectively as1 2KABMBA(and similarly for the covariant derivative and eld strengths), where MABare the generators of OSp(3,1 j4), by algebraically constraining KABto contain just the appropriate pieces (and relating KabtoKin the usual way). Also, the shorthand KIMInow includes Lorentz and U(1) terms. Excercise XA1.1 Use the denition in the above commutation relations to express the torsionT ABCdirectly in terms of the structure functions CABC, Lorentz connection A , and U(1) connection AA. The constraints in supergravity are a combination of the kinds used in ordinary gravity and super Yang-Mills: those that (1) dene the vector derivative in terms ofthe spinor ones ir . =fr;r. g (2) dene the spinor (Lorentz and R) connections T =T ;(. . )=Tbb=0 and (3) allow the existence of chiral (scalar) superelds r=0)f r ;rg=0 ( Y=0 ) (The rst two constraints imply the generalization of this chirality condition to Y6=0 and chiral superelds with undotted indices, like the Yang-M ills eld strength.) Excercise XA1.2 Rewrite the rst and last set of constraints directly in terms of eld strengths. The explicit solution of all these constraints is a bit messy, but we will need only a certain subset of them to nd the prepotentials and supergravity action. Theform of the solution is a generalization of super Yang-Mills in a way similar to how general relativity generalizes ordinary Yang-Mills. In particular, just as the vierbein e a=eam@mis a generalization of the Yang-Mills vector Aato describe gauging of the translations, the generalization of the super Yang-Mills prepotential Vto supergravity isH=Hm(i)@m, which appears in an exponential eHjust asVappears aseV:T h e chirality-preserving constraints, expressed explicitly in terms of the vielbein, is fE;Eg=C E 540 X. SUPERGRAVITY If the commutator vanished like the Yang-Mills case, Ewould be partial derivatives on some complex two-dimensional subspace, the usual @up to some complex (su- per)coordinate transformation, as for din at superspace. However, the fact that their algebra still closes means they still generate translations within such a subspace, and are thus linear combinations of such partial derivatives: E= N^E; ^E=e @e ; = M(i)@M where we have separated the matrix coecient into a local complex scale (scale  U(1)) and a local Lorentz transformation N. For most purposes we will nd it convenient to x all these invariances by choosing the gauge N= )E=^E As for Yang-Mills, solution of the chirality condition introduces a new, chiral gauge invariance: e 0=eie eiK; = M(i)@M;K =KM(i)@M [@.;]@.) @.m=@.=0 where .is not chiral, since it generates terms in the transformation law of E.that can be canceled by including @..terms in the transformation law of  N...T h i s means we can use .andKto gauge = .=0) = m(i)@m) ^E=@+^Em@m where ^Em=i@ m+:::from expanding the exponentials as multiple commutators. We can again transform to a chiral representation, and work in terms of eU=e e ;eU0=eieUei; ^E.=@.; ^E=eU@eU whereUnow generalizes the constant hUi=.(i)@.used in at superspace. Also as for Yang-Mills, the usual local component transformations (now for coordinate, supersymmetry, scale, U(1), and S-supersymmetry) reappear in the chiral parameters mand . For a component analysis, we look at the linearized transformation (see excercise IVC4.3) Umi()mi1 2[hUi;+ ]m =i()m1 2.@.(+ )m+(..)+i1 2(2@.2.@..) A. SUPERSPACE 541 where for  we use as independent just the chiral parameters mand ; the nonchiral ., having already been used to gauge away U, is now xed in terms of the others as .=eU.eU to maintain U= 0. The rst term in the transformation tells us that the surviving component elds are the same as for super Yang-Mills, with \ m" as the group index: Um=eam(); m(2); .m(2);Am(22) The second term in the transformation law gives eam@amfrom mj=mj= m.T h e ncontains the rest of the gauge parameters: =;a+ib();();(2) = supersymmetry, scale +iU(1), Lorentz, S-supersymmetry The third term in the transformation law then shows scale and Lorentz gauge way pieces of the vierbein, as usual, while S-supersymmetry gauges away the trace of m. It also forces mto include .at orderto maintain the gauge; we then see that mis the gauge eld for supersymmetry, with contributions from the second and fourth terms. Finally, the fourth term also shows that Amis the gauge eld for U(1). The resulting component content is that of \conformal supergravity", which will be transformed later to ordinary supergravity through a compensator supereld. For perturbation theory, or comparison with global supersymmetry, we should expand about \ at" superspace (which is nontrivial because of nonvanishing torsion T . cin empty superspace). We then modify the chiral representation: e e =eU=ehUi=2eHehUi=2) ^E.=d.; ^E=eHdeH We now expand all derivatives over the covariant derivatives dMof global supersym- metry (constructed from hUias before) H=HM(i)dM;EA=EAMdM instead of over partial derivatives @M, which is just a change of basis. This also modies the description of the  gauge parameters: eH0=eieUei; = M(i)dM;K =KM(i)dM [d.;]d.) d+m@m=1 2fd.;[d.;Ld]g ) =d2L; .=id.L 542 X. SUPERGRAVITY in terms of a new parameter L. From this we nd the linearized transformation law H.dL.d.L Excercise XA1.3 Expand this transformation law in components, and compare with the previ-ous analysis. Besides chirality, we also need a certain combination of the other constraints: 0=T bbT . . =(1)BCBB+CiA where theAterm comes from the contribution1 2iA. . toT . . . (See excercise XA1.1.) Using the gauge E=^Ewithout loss of generality, we then nd (comparing similar manipulations in subsection IXA2) iA=E@ME1EM=E1 EE where the backwards arrow on E=EM @Mmeans all derivatives act on everything to the left (see subsection IA2), and Esdet EAM (The superdeterminant was dened in subsection IIC3.) We now need the general identity, for any function Aand rst-order dierential operatorB, Ae B=( 1
e Be B)Ae B=( 1
e B)(e BAe B)=( 1
e B)(eBAeB) =( 1
e B)(eBA) ) 1=( 1
e B)e B=( 1
e B)[eB(1
e B)] The nal result in the gauge =1(Nis trivially restored) is then iA=ET; eT=E(1
e ) This can be used to solve chirality conditions on matter elds: In this gauge, we have Y=y) 0=r.=( E.+iyA.) ) =eyTe ; @.=0 A. SUPERSPACE 543 Again as for Yang-Mills, the chiral-representation eld transforms under only the  transformations: 0=( 1
ei )yei; =i(m@m+@) Thus, scalars in the real representation become densities in the chiral representation (except for y= 0). In particular, we have for the special case y=1)0=ei )Z dx d2=0 which will prove useful later for chiral integration. For now, we note that such a chiral scalar, with y6= 0, can be seen to compensate from the @term: This term allowsUmto eat the complex \physical" scalar and spinor, xing scale, U(1), and S-supersymmetry, while the complex auxiliary scalar survives, along with coordinate, Lorentz, and supersymmetry invariance. We also note that for perturbation about at superspace we have i(1
) =@mmd=d2dL Excercise XA1.4 Show that preservation of the chirality of implies the previous chirality con- ditions on . Thus, as for nonsupersymmetric or nongravitational theories, the gauge group follows more simply from starting with matter representa- tions. We also note that in the gauge =1 ,a n da l s o = .= 0, the superdetermi- nant is simply (see subsection IIC3) E1=det(Ema) whereEmais a component of EMA(not the inverse of Eam). 2. Field strengths These constraints can be completely solved for all the eld strengths. Alterna- tively, we can impose them, together with the Bianchi identities (Jacobi identities of the covariant derivatives), to nd a smaller set of algebraically independent eld strengths, and the dierential equations that relate them. The method is analogousto the case of super Yang-Mills treated in subsection IVC3. We begin with the con- straints analogous to the Yang-M ills ones: fr ;r. g=ir . ;fr;rg=RIMI 544 X. SUPERGRAVITY (where the latter will simplify from later results). From just the latter, we nd [r(;fr;r )g]=0)R( )i1 2(F )=r(R )I=0 Using both constraints, we also have [r(;fr);r. g]+[r. ;fr;rg]=0 ) [r(;r). ]=iR. . r. 1 2Fr. i(r. RI)MI ) [r;r. ]=iCW. +1 2iR. . r. 1 4Fr. 1 2i(r. RI)MI for some operator W. =W. ArA+W. IMI. So far the excercise has been analogous to the super Yang-Mills case (where the extra \ i" in the denition of Wis due to our use of antihermitian generators, except for Y) . N o ww ei m p o s et h er e m a i n i n g constraints, which can be combined conveniently as 0=T;. . =iCW. . )W.=W.r+W.. r. +W.IMI Following again the steps for Yang-M ills, we analyze the next-higher-dimension Jacobis, beginning with 0=fr(;[r);r . ]g+[r . ;fr;rg]=iC (fr);W. g+
. 
. =1 2i(r(R ) . .)r.1 4(r(F) )r. 1 2i(r(r. R) I)MI +1 2R (. .r).+1 4iF (r ). 1 2i(r. R ())r1 4(r. F ()r) +(r . RI)MIR r. R . .r . By inspection, or applying the previous Jacobis, we see
( ). =0)
. =C (
). ;
. =1 3
.  automatically, so the only new information comes from the trace of this Jacobi, fr;W. g=i
.  Evaluatingfr;Wgin terms of its pieces, we nd R. . =F=0;R = ( )B;W.. =B. . W.=r.B+rW.;W. =1 2r(W. ) A. SUPERSPACE 545 rW. =rW. . . =0;rW.  =( (W. )+1 2ir). )B whereWis theYpart ofW(=WiY+:::). Excercise XA2.1 Show that fr;rg=BM;r[rr ]=0)r(r2+B)=1 2BrM and thusr2+Bg i v e sac h i r a ls u p e r  e l dw h e na c t i n go na n ys u p e r  e l dw i t h o u t dotted indices. For the other Jacobi of this dimension, we have 0=[r.;fr;r. g]+fr;[r. ;r.]g+fr. ;[r;r.]g =i[r.;r. ]ifr;C. .W+1 2(rB)M. .gifr. ;CW.+1 2(r.B)Mg =iC.. [ffr;Wg+1 2(r2B)M]h:c: )f=1 2fr(;W)g1 2(r2B)M;fr;Wg+fr.;W.g=0 (Here \h:c:" means \hermitian conjugate" without the reordering, which would gen- erate non-operator terms.) Evaluatingfr;Wgin terms of its pieces, and combining with the results of the previous Jacobi, we obtain the nal result: fr.;r. g=BM.. ;fr;r. g=ir .  [r.;ir . ]=C. .W1 2(rB)M.. ; [ir.;ir . ]=C. .fh:c: W=BrG. r. +1 2(r. G. )M. . +1 2W M +iWY+i1 6WM f=i1 2G(. r). 1 2(r(B+i1 3W()r)+W r 1 2(r(G). )r. (1 2r2B+BB+1 12ir W )Mi1 8[(r(. G ). )M +$] +1 2W M +1 4(r(r. G). )M. . +i1 2(r(W))Y W =1 4!r(W ) The \reduced tensors" B;Ga;W;W satisfy the \reduced Bianchi identities" Ga=Ga;r.B=r.W=r.W =0;r.G.=rBiW rW i1 3r(W )=i1 2r(.G ).;rW+r.W.=0 546 X. SUPERGRAVITY Note thatBorWmay vanish in certain gauges, for reasons to be explained in subsection XA4. Excercise XA2.2 In IXA4 we saw that integrals of total covariant derivatives vanished in curved space by virtue of the identity Tabb= 0. Show that these torsions satisfy the superpace generalization (1)BTABB=0 Excercise XA2.3 Using the expression for abcdin terms of spinor indices from subsection IIA5, show Tbcd=Gaabcd ThusGais an axial vector. Excercise XA2.4 By hermitian conjugation, nd the commutators not written explicitly above, and show the result is essentially the same as switching dotted and undotted indices (and similarly for bars), except that Ga,Y,a n dW(andW.)g e t extra minus signs. This illustrates CP invariance, and the fact that Gais an axial vector, while Yis a pseudoscalar (and similarly for W). Excercise XA2.5 Derive the Bianchi identities in the absence of constraints, in terms of thetorsions and curvatures (as follow from the Jacobi identity): r [ATBC)DT[ABjETEjC)D=R[ABC )D r[ARBC)IT[ABjEREjC)I=0 Excercise XA2.6 Show that in 4-component notation we can write Tc=i c ;Ta = aG ;G=G;rG=W This gives another way to see the result of excercise XA2.2. Show that this expression for G=(Ga;B;B) gives it an interpretation as an SO(3,3) 6- vector in SL(4) notation (see subsection IC5). A. SUPERSPACE 547 3. Compensators Just as in ordinary gravity, compensators for scale transformations can be intro- duced, but for supergravity the compensator should be a supersymmetric multiplet. The simplest choice is the chiral scalar supereld  considered earlier: Its complex\physical" scalar (scalar + ipseudoscalar) compensates local scale (the real part) and U(1) (the imaginary part), its spinor compensates local S-supersymmetry, and itsauxiliary complex scalar appears as one of the auxiliary elds of supergravity. Compensators are much more important in supergravity than in ordinary gravity: Almost any at space action can be coupled to gravity by the minimal coupling prescription | replacing derivatives with covariant ones, and throwing a factor of e 1in for the measure. In supergravity this is not the case: As we'll see in the next section, we have both integrals over all superspace, which use E1, but also integrals over chiral superspace (for integrating chiral superelds), which instead use  for the measure. The minimal coupling procedure is then: (1) Use  (and ) to make a at-superspace action superconformally invariant, (2) replace the at derivatives dA with the curved ones rA, and (3) throw in the measure factors appropriate for the integrals. (The last two steps couple conformal supergravity to a globally conformally invariant theory.) Another compensator that is commonly used is the \tensor multiplet". (This is sometimes confused in the literature with the \(complex) linear multiplet", an- other version of the scalar multiplet with no gauge elds whatsoever.) Treated as a matter multiplet, it has the same physical content as the scalar multiplet, but thepseudoscalar is replaced with a second-rank antisymmetric tensor gauge eld, B mn=@[mn] To make things simpler, let's look at at space. We rst note that this tensor is \dual" to a pseudoscalar in the sense of switching eld equations and constraints ofthe eld strength (see excercises IIB2.1 and VIIIA8.2): For the free elds, F a=@a')@[aFb]=0;Ga=1 2abcd@bBcd)@aGa=0 with the eld equations following from \self-duality" under F$G: Fa=Ga)@aFa=@[aGb]=0 Since the theory of Babmust be described in terms of Gaalone (because of gauge invariance), no renormalizable self-interactions are allowed; thus, this eld is of little 548 X. SUPERGRAVITY interest in quantum eld theory outside of supergravity. In terms of the scalar, the fact that only the eld strength Faappears in the eld equations means there is the global symmetry '= for constant parameter . This generalizes to the nonabelian symmetries of nonlin- earmodels, resulting in derivative interactions (again nonrenormalizable) but no potentials. Excercise XA3.1 Consider coupling the tensor eld to Yang-Mills: To preserve the tensor's owngauge symmetry, this coupling must be nonminimal. To produce such a cou- pling, we start with the scalar and duality transform. The coupling we choose is another 4D analog to the 2D model we considered in excercise VIIIA8.2,replacing the pseudoscalar and total derivative 1 2abFabwithtr(1 8abcdFabFcd). (In general dimensions, the dual to a scalar is a rank-D 2 antisymmetric tensor.) We start with the Lagrangian L=1 4+1 16tr(abcdFabFcd) for some coupling constant . Making use of the Chern-Simons form Babcof subsection IIIC6 to write in this action only as @a, write a rst-order form of this action and perform a duality transformation to obtain L0=1 24eH2;eHabc=1 2@[aBbc]+Babc Find the Yang-Mills gauge transformation of Bab.( H i n t :eHis gauge invari- ant.) The tensor multiplet is described by a chiral spinor gauge eld =id2dK (K=K) Duality is then described in terms of the real scalar supereld strength (in the free case) F=+) d2dF=0;G =1 2(d+d..)) d2G=0 with the eld equation F=G (Faappears at order inG,a n dBabat orderin.) Again the pseudoscalar has a global symmetry: In terms of the supereld, =i A. SUPERSPACE 549 Now we return to curved space, covariantizing the above with respect to confor- mal supergravity. We now identify the above global symmetry with the local axial U(1) (R-)symmetry of supergravity. Thus, the supereld Gdoes not compensate for this symmetry; it remains as a symmetry in actions that use this compensator. In particular, there are noR d2terms in such theories, except those that are locally superscale invariant (so the compensator decouples). The matter tensor multiplet also diers from the scalar multiplet in that it has no auxiliary elds (except for the auxiliary components of the gauge eld). 4. Scale gauges Since all the covariant derivatives are built up from the spinor part of the vielbein, we dene the local superscale transformations for the covariant derivatives by rst dening E0 =LE whereLis a real, unconstrained supereld. The constraints then imply r0 =Lr+2 (rL)M+6 (rL)Y;r0.=Lr.+2 (r. L)M. .6(r.L)Y From the anticommutator we nd ir0 .=L2(i)r.+4L(rL)r.+4L(r.L)r +1 2L2(rr. L4)M. .+1 2L2(r.rL4)M3 2L2([r;r.]L4)Y Using the commutation relations, we then can show B0=L6(r2+B)L4;W0 =L3[W12i(r2+B)rlnL] G0 .=( 2 [r;r.]+G.)L2;W0  =L3W From the way they appear in the commutators we also have that YGa=0;Y W =1 2W;Y W  =1 2W ;Y B =B From linearization, we see that BandWpick out exactly the two irreducible halves of the real scalar supereld L: The \vector multiplet" in W0 and the \scalar multiplet" inB0. (Compare the vector multiplet eld strength and chiral scalar gauge xing for the prepotential Vas described in subsections IVC4 and VIB9.) This means we can completely x the superscale gauge by the choice B=W=0 550 X. SUPERGRAVITY as the generalization of the scale gauge in ordinary gravity that xes the Ricci scalar to vanish. Excercise XA4.1 Derive the superscale transformations by use of the Bianchi identities: aUse the commutation relations of the covariant derivatives (and the solution to the Jacobi identities) to nd all the transformations above. Show they implyE0=L4E. bAn easier way is to use the reduced Bianchi identities: Determine the trans- formations of the reduced eld strengths, up to constants, using chirality,dimensional analysis, etc., and then solve for the constants by plugging into the reduced identities. We then dene the scale (and U(1)) transformations of the compensators:  0=L2;Y =1 3;r.=0 G0=L4G; YG =0 ; (r2+B)G=0;G =G where the scale weights follow from the U(1) weights (vanishing for Gby reality) by consistency with the constraints they satisfy. Excercise XA4.2 Show that a supereld can be chiral only if it has no dotted indices. Then show the relation that any such supereld has between scale and U(1) weights. All these transformations can be derived either by consistency with the con- straints, or by using the solution of the constraints: In terms of the unconstrained superelds that solve the constraints, the superscale transformation is trivial: 0=L ; N0 =N; 0= The net result, as for super Yang-Mills, is that all the supercial transformations of the constrained covariant derivatives are completely replaced with the new invariances that appear upon solving the constraints: K1andLeliminate ,KkillsN,a n d KMreduces Mto its real part UM, which transforms only under M. Excercise XA4.3 RederiveAas in subsection XA1, but in a general gauge, to nd iA=ET; eT= 2E(1
e ) Show this result gives a superscale transformation for Athat agrees with the result above. Show the explicit solution for  in terms of andTalso gives it a superscale transformation that agrees with the above. A. SUPERSPACE 551 Excercise XA4.4 Often it is easier to use the solution to the constraints than the Jacobi iden- tities: aSolve forFABin terms of A, and use the solution for Afrom subsection XA1, to derive W=i(r2+B)r(T+T) and use this to rederive the superscale transformation above. (Hint: Dene and use the chiral representation.) bFind an explicit expression for B, and use it to rederive its superscale trans- formation. (Hint: You will need to nd  rst. Since Bis a scalar, you can choose the Lorentz gauge N= .) In subsection XA1 we found that a convenient way to simultaneously x Lorentz, U(1), and scale gauges was to choose E=^E. (However, the corresponding com- ponent invariances reappeared in the chiral gauge invariances.) Compensators allow more freedom for gauge xing: For example, we can x the gauge B=W=0a s described above, or we can x to 1 the compensator or a physical matter multiplet (string gauge, as for gravity in subsection IXB5): The possibility of gauges such as =1o rG= 1 depends on the existence in the action of such elds, and not on the details of how they appear (as long as the gauge choice is consistent with the allowed vacuum values). In particular, it does not depend on the signs of their kinetic terms, which is the only thing that determines what is physical and what is a compensator. Note that either  = 1 or G= 1 completely xes the superscale gauge, in spite of the constraints on these superelds. (E.g,  = 0=1)L= 1.) This is due to the appearance of the U(1) connection: For example, before xing the scale and U(1) gauges the chirality condition on , rather than constraining , actually determinesthe spinor U(1) connection A : r=(Ei1 3A)=0)A=3iEln (But the chirality of the ratio of two chiral superelds with the same weights really xes it to be chiral; in other words, chirality of scalars makes all but one truly chiral, since the U(1) connection can be determined only once.) As a result, the scale (  = 1) and U(1) ( = = 1) gauge choice  = 1 determines W: =1)A=0)W=0 Similarly, G=1) 0=(r2+B)G=B 552 X. SUPERGRAVITY Conversely, we see that whenever one of the two eld strengths BandWis elim- inated by a superscale(/U(1)) gauge choice in terms of one of the two compensators a n dG, the other eld strength can be made superscale invariant: If we introduce the compensator by a superscale transformation (as for gravity in subsection IXA7), substituting either L4!or G in the above transformation laws, we nd ~B=()3=2(r2+B) = 1=23=2(r2+B) fW=G3=4[W+3i(r2+B)rlnG] as locally superscale invariant, where using  forfWorGfor~Byields zero. We can therefore interpret gauging away the compensators as gauging them into the eld strengths: We have a choice of either =1)W=0; ~B=B G=1)B=0;fW=W This is analogous to St uckelberg gauges (and their nonlinear generalizations): One of these two tensors (gauge elds with respect to superscale) \eats" the compensator.However, it diers from St uckelberg in that a second \gauge eld" is completely gauged away. In fact, we'll see in the next section that the pure supergravity actions constructed using either of these compensators gives the corresponding eld strength as its eldequation:  ) ~B=0  )fW=0 Thus, either compensator can be used to eliminate both BandW,o n ea sa e l d equation and the other as a gauge choice. This result is already clear at this pointfrom dimensional analysis and chirality; similarly, we must have  Um) ~Ga=0 where ~Gais the result of applying a superscale transformation to Gawith whichever of the two compensators is being used in the action. This leaves W as the on-shell eld strength. The analogy to ordinary gravity is (R;Rab1 2abR;Wabcd)$(B=W;Ga;W ) A. SUPERSPACE 553 Although the Ricci tensor must appear in Ga, superscale invariance allows the choice of gauges where the = 0 component is arbitrary: From the above we nd the linearized transformations G.4[r;r.]L; A.6[r;r.]L (For purposes of evaluating at = 0 we can neglect AjinAa.) Thus, this axial vec- tor component eld can be moved around as convenient for component expansions. In ~Ga, they appear only in their invariant combination, Ga+2 3Aain this approximation. Innonsupersymmetric gauges , we can even gauge Gaj=0 . Excercise XA4.5 Use the Bianchi identities instead of explicit superscale to track down the axial vector: aUse the relation of W (which is scale covariant) to W(the eld strength forAA)a n dGato show that it is just this combination that appears in W (as its curl, for U(1) invariance). bShow that G=1)B=0)raGa=0 Thus, in this gauge the axial vector gauge eld Aahas been gauged out of Gaj(although its eld strength may appear at higher order: the gauge G=1 doesn't x U(1)). What replaces it? (Hint: What's in G?) REFERENCES 1D.Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara, Phys. Rev. D13 (1976) 3214: supergravity. 2V.P. Akulov, D.V. Volkov, and V.A. Soroka, JETP Lett. 22(1975) 187: covariant derivatives for supergravity. 3V. Ogievetsky and E. Sokatchev, Nucl. Phys. B124 (1977) 309; V.P. Akulov, D.V. Volkov, and V.A. Soroka, Theor. Math. Phys. 31(1977) 285; S. Ferrara and B. Zumino, Nucl. Phys. B134 (1978) 301: linearized o-shell supergravity (including auxiliary elds). 4P. Breitenlohner, Phys. Lett. 67B (1977) 49; Nucl. Phys. B124 (1977) 500: o-shell supergravity in components. 5Wess and Zumino; Wess; loc. cit. (IVC, ref. 5): superspace eld equations. 6W. Siegel, Supergravity superelds without a supermetric, Harvard preprint HUTP-77/A068 (November 1977);The supereld supergravity action, Harvard preprint HUTP-77/A080 (D ecember 1977); A polynomial action for a massive, self-interacting chiral supereld coupled to super-gravity, Harvard preprint HUTP-77/A077 (D ecember 1977); 554 X. SUPERGRAVITY A derivation of the supercurrent supereld, Harvard preprint HUTP-77/A089 (D ecem- ber 1977): o-shell supergravity in superspace, compensator supereld. 7M. Kaku, P.K. Townsend, and P. van Nieuwenhuizen, Phys. Rev. D17 (1978) 3179: conformal supergravity in components. 8M.F. Sohnius and P.C. West, Phys. Lett. 105B (1981) 353: R-symmetry generator in superspace covariant derivative. 9V.I. Ogievetsky and I.V. Polubarinov, Sov. J. Nucl. Phys. 4(1967) 156; K. Hayashi, Phys. Lett. 44B (1973) 497; M. Kalb and P. Ramond, Phys. Rev. D9(1974) 2273; E. Cremmer and J. Scherk, Nucl. Phys. B72 (1974) 117: antisymmetric tensor gauge eld. 10J. Wess, Acta Phys. Austriaca 41(1975) 409: N=2 supersymmetric tensor multiplet. 11W. Siegel, Phys. Lett. 85B (1979) 333: N=1 supersymmetric tensor multiplet. 12H. Nicolai and P.K. Townsend, Phys. Lett. 98B (1981) 257; E. Bergshoe, M. de Roo, B. de Wit, and P. van Nieuwenhuizen, Nucl. Phys. B195 (1982) 97;G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105: Yang-Mills appearing as Chern-Simons contribution to tensor eld strength. 13P.S. Howe and R.W. Tucker, Phys. Lett. 80B (1978) 138: superscale transformations. 14Gates, Grisaru, Ro cek, and Siegel, loc. cit. : complete superspace treatment of N=1 supergravity. B. ACTIONS 555 ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: B. ACTIONS ::::::::::::::::::::::::::::: Now that we understand the structure of superelds in curved superspace, we an- alyze various supergravity theories through their actions. We will use several methods for nding and evaluating supergravity actions. These actions are signicantly more complicated than those we have encountered previously, and it is dicult to see alltheir features simultaneously, so for any particular application we use the method which best simplies the property we most need: (1) Superspace methods are the best for nding general actions and their symmetry properties, manifesting super- symmetry, using globally supersymmetric gauges, and performing quantum calcula- tions. (2) Component methods are useful for comparing actions and other propertiesto nonsupersymmetric theories. Such approaches sometimes make some use of super- space, but not superspace integration. (3) Compensators are useful in conjunction with either of these methods, and can extract many important features and termsin the action with little more than the results of global supersymmetry. They reveal useful broken symmetries, and are the simplest way to analyze the \superhiggs eect" (Higgs for local supersymmetry). 1. Integration The action for supergravity follows from dimensional analysis: Since the usual Einstein-Hilbert Lagrangian has dimension +2, as doesR d4, the superspace La- grangian must be dimensionless. The only covariant possibility in terms of the po- tentials (EAMand AI)i st h u s SSG=3Z dx d4E1 including a normalization factor that will prove convenient later. Introducing the compensator and local scale invariance must also make the usual action for super-gravity look like the kinetic term for the compensator multiplet, i.e., S SG;c=3Z dx d4E1 (For simplicity we will restrict ourselves for the most part to the simplest compensator, the chiral scalar.) The previous form then corresponds to the scale gauge  = 1; often the scale + U(1) gauge = 1 is more convenient. There should also be a supersymmetrization of the cosmological term. This might seem dicult, requiring explicit prepotentials. However, we know from our study of de Sitter space that the cosmological term is basically a statement about the 556 X. SUPERGRAVITY conformal compensator. Therefore, the cosmological term for supergravity, in terms of the superconformal compensator , should be the supersymmetrization of the corresponding term in ordinary gravity, a dimensionless self-interaction for a scalar. The solution of the chirality condition can be written as Y3=3) 3=3e  E in the gauge = 1. The result for the cosmological term is then Sscosmo =Z dx d2E13+h:c:=Z dx d23+h:c: independent of scale or U(1) gauge: As we saw in subsection XA1, this expression is invariant under  transformations, and the integrand itself is invariant under Kand Ltransformations. Excercise XB1.1 Although this nal result for the cosmological term in terms of is locally superscale invariant, the derivation started in the gauge = 1. Generalize the derivation, and the result in terms of , to arbitrary gauges (see excercise XA4.3). A chiral expression for SSGcan be found by similar methods: SSG=3Z dx d2E1B=3Z dx d2E1B Thus, as for super Yang-Mills, the action can be expressed as a real, chiral, or an- tichiral integral. With the compensator, SSG=3Z dx d2E1(r2+B)=3Z dx d2E1(r2+B) or, more generally, Z dx d4E1L=Z dx d2E1(r2+B)L=Z dx d2E1(r2+B)L which is just the naive covariantization of the at-space result (at least in the gauge = 1: see excercise XB1.1). Clearly this method generalizes to coupling to other multiplets, and allows bothRd2andRd4integrals to be generalized to curved su- perspace. In fact, the analysis of the compensator is much simpler than that of the conformal supergravity that couples to it to produce ordinary supergravity: Just asin ordinary gravity, where use of just the compensator allowed us to study certain in- teresting solutions in gravity, namely de Sitter space and cosomology, some properties of supergravity can be analyzed in terms of just the compensator. B. ACTIONS 557 A simple expression (as simple as the super Yang-Mills case) can be written for the supergravity action in terms of unconstrained superelds. We rst give a rst- order action, analogous to the one for Yang-M ills (subsection IVC5): In that case the action was in terms of VandAa;h e r ei ti si nt e r m so f UmandEma. Using just the constraints solved in subsection XA1, we write the action as SSG;1=3Z dx d4E1(11 4T;..) in terms of the torsion T . c. (Note the similarity to the Yang-Mills case, replacing the Chern-Simons form with the same component of the torsion.) We already evaluated everything except this torsion, which is also easily found in the gauge =1 ,N= , = .=0 : SSG;1=3Z dx d4d e t (Ema)(11 4^EamEma) =[det(Ema)]1=3(1
e  )1=3e ;f^E;bE.g=i^E.m@m In the chiral representation this simplies to SSG;1=3Z dx d4[det(Ema)]1=3(1
e U)1=3(eU)(11 4Em.i@.^Em) Excercise XB1.2 Find the algebraic eld equation for Ema. Use this to eliminate it from the action, yielding expressions for Ema,E, and the (second-order) action in terms ofUmonly. The expansion of the superspace action in terms of unconstrained superelds is needed for supergraphs, the most ecient way to do quantum calculations. We will not consider quantization here; the methods are similar to those described in subsec- tions VIB5, 9-10, and C5 for super Yang-Mills. In particular, one uses background eld methods: For example, as for super Yang-Mills, e !e Be Q) ^E!eH^EeH;eH=e Qe Q The end result is that the expansion is about background-covariant derivatives DA, e.g., br=eHDeH;r=eH( D+ IMI)eH etc. However,DAsatisfy the same constraints as the full covariant derivatives: For example, they have nonvanishing torsion T . c=i . . . T h i sd i  e r sf r o mt h ee x - pansion implied above in Umabout partial derivatives, which anticommute without torsion: For a perturbation expansion useful for quantum calculations, one must ex- pand inhababoutheami, rather than in eamitself; thus (at least) the vacuum value hUimust be separated from U. 558 X. SUPERGRAVITY 2. Ectoplasm Although all supersymmetric theories can be analyzed directly in superspace (in- cluding classical solutions, eective potentials, Feynman graphs, etc.), for comparisonto nonsupersymmetric theories it is necessary to expand superelds in components. Since all fundamental theories are described by actions, it is sucient to give a pre- scription for evaluating any action in terms of component elds, as in subsection IVC2for global supersymmetry. In locally supersymmetric theories, the vielbein needs to be expanded in terms of the prepotentials for supergraphs. We can also nd compo- nent actions by a straightforward Taylor expansion in of the prepotentials in the superspace action. However, for component expansions of classical actions, one can get by more simply by applying Bianchi identities to the covariant derivatives and dierential forms (antisymmetric tensors). It is unnecessary to know even the explicitform of the measure in terms of the vielbein or prepotentials. The fundamental idea is to think of the Lagrangian not as a scalar times a mea- sure, but more \geometrically" as an antisymmetric tensor. Although this approach does not work for the usual superspace Lagrangians because of the peculiarities offermionic integration, it can be applied to component Lagrangians integrated over 4D spacetime, treated as the bosonic subspace of superspace. We thus write the component action as S= 1 4!Z dxmdxndxpdxqLmnpq(x;) whereLMNPQ is a graded antisymmetric supereld. Of course, the action should be independent of , even though we have integrated over only x.T h i si se q u i v a l e n tt o requiring that the integral should be independent of the choice of 4D hypersurface insuperspace. We are familiar with a similar requirement for conserved charges, which are dened as integrals over 3D hypersurfaces: Treating the conserved current in terms of the 3-form dual to the vector, J mnp=pgmnpqJq)Q=1 3!Z dxmdxndxpJmnp the dual to the usual conservation law as vanishing (covariant) divergence of the vector is vanishing curl of the 3-form: d dtQ=0)@[mJnpq]=0 An important point is that neither the denition of the charge nor the conservation law requires a metric, since integration in general does not. We thus require for our supersymmetric action @ @S=0)@[MLNPQR )=0; LMNPQ =1 3!@[MNPQ ) B. ACTIONS 559 where the gauge invariance allows us to drop terms in the Lagrangian that are total derivatives (surface terms). Note that both Landare assumed to be local func- tions of the elds and their (nite-order) derivatives. (As a result, this is not the usual \cohomology", where both would be allowed to be arbitrary functions of the coordinates.) Converting the curl-free condition to at indices (see subsection IVC5 for the Chern-Simons superform), 1 4!r[ALBCDE )1 2!3!T[ABjFLFjCDE )=0; L ABCD =1 3!r[ABCD )1 (2!)2T[ABjEEjCD) The plan is then to nd LABCD , in terms of which the action can be written as S=Z dx(1 4!)mnpqEqDEpCEnBEmALABCD whereEmAis exactly the nontrivial part of the inverse vielbein EMA: EmA=(ema; m) namely the inverse vierbein and the gravitino. (If we also write m=ema a,w e can collect all emafactors into a factor of e1using thetensor.) The next step is to explicitly solve the curl-free condition on the 4-form in terms of the usual scalar superspace Lagrangian. (An alternative is to solve the Bianchis for the eld strength of the 3-form gauge eld, which is also a 4-form.) The procedureis the same as that used to solve the Bianchi identities for covariant derivatives (in subsection XA2): We start with the lowest-dimension equations and work up. The equations that include the constant (vacuum/ at-space) part of the torsion can besolved algebraically, the rest give dierential constraints. Of course, we will need to use the results of subsection XA2 for the torsions and their constraints. The result is L cd=. ;.;cdL;Lbcd=i.;bcdr.L;Labcd=abcd[(r2+3B)L+h:c:] and their complex conjugates, the rest vanishing, where Lis the usual chiral super- space Lagrangian (superpotential): r.L=0 (We use the shorthand notation a=(.), etc.) The result is thus the component expansion of the usual curved superspace action S=Z dx d2E1L+h:c: 560 X. SUPERGRAVITY Using the Bianchi identities of the covariant derivatives one can also covariantize the usual solution to the chirality condition: L=(r2+B)L which allows us to identify the action as S=Z dx d4E1(L+L) soLcan be taken real without loss of generality for general d4integrals. On the other hand, for supergravity we can take LSG=3; LSG=0)LSG=3B; LSG=0 or vice versa, and the curvature appears in terms of Rand not R.. .. (like the corresponding fwithout f.. for Yang-Mills), with half as many terms to collect (for the same nal result). In general, we thus have an expression for Sin terms of EmAand the components of LABCD , and for the latter in terms of curvatures and covariant derivatives of L, which can be evaluated by the same methods as in at space (except that the commutation relations of the covariant derivatives are more complicated). Components of a supereld are again dened by evaluating its covariant deriva- tives at= 0. However, as in the case of global supersymmetry, the value of is arbitrary, since the result for the action is independent: We therefore will gener- ally drop the \j" in component expansions of actions; any supereld then implicitly refers to the corresponding component. This independence also means that it is not necessary to make any gauge choices: These methods automatically express the action in terms of just the component elds that cannot be completely gauged away. (For example, Emjnever appears.) Excercise XB2.1 Collect all the above results: aFind the complete component expression for the most general chiral (Rd2) action in terms of covariant derivatives of the superpotential, and the super- gravity elds. bD ot h es a m ef o rar e a l(Rd4)a c t i o n . Excercise XB2.2 Evaluate the above actions for massive 3theory (for a chiral matter eld ) in terms of the components of . B. ACTIONS 561 Excercise XB2.3 Do the same for super Yang-Mills: aSolve the Bianchi identities for super Yang-Mills in curved superspace. bUse this result to evaluate the component expansion of its action. 3. Component transformations We saw in the previous subsection that in component expansions the supergravity gauge elds naturally appear as EmA, since by denition we restrict to the bosonic submanifold. Similar remarks apply to the component form of their supercoordinatetransformations (i.e., local supersymmetry), and the related component expansion of their eld strengths: In subsection IXB4, we saw that coordinate transformations (as applied to solving the radial gauge condition) were simpler for E MAbecause the derivative term on the parameter was just @MKA. We therefore begin by rewriting the gauge (coordinate) transformations in terms ofEmA. As for gravity (see subsections IXA2 and IXB4), we can choose to make the transformation laws more manifestly covariant by writing the generators in terms ofcovariant derivatives. Then, as for gravity, K=K ArA+KIMI;rA=[K;rA]) EMA=rMKAEMBKCTCBA+KIMIEMA; MI=rMKI+EMBKCRCBI (TABC;RABI)=K(TABC;RABI) using (EAM)EMB=EAMEMB.H e r erM=EMArA=@M+ MIMI,s ot h e s e transformations on EmAand mAcontain only bosonic derivatives @m, other than those implicit in the torsions and curvatures. (Similar remarks apply to only EmA and mIappearing on the right.) Since the component expansion is an expansion in , it is really only supersymmetry K=which is no longer manifest; specializing to that, the transformations on the physical gauge elds become, using the results ofsubsection XA2 for the torsions, e m.=i( m.+. m); m=rm+iem. (G. .  B) We have not yet determined the solution for the Lorentz connection !maband the transformation laws for the auxiliary elds. We will also need the relation between 562 X. SUPERGRAVITY the usual curvature and the components of the supereld strengths. All of these can be found by use of the identities (see subsection IXA2): EnCEmBTBCA=TmnA=@[mEn]A+E[mB n]BA EnDEmCRCDab=Rmnab=@[m n]ab+ [mac n]cb The application of these identities is similar to that for expansion of the action in the previous section: The separation of the factors of EmAinto bosonic and fermionic parts yields an expansion in powers of the gravitino eld. For the torsion case wherethe indexA=awe solve for ! mabin terms of the torsions (auxiliary elds and constants) and vielbein; for the torsion case A=we solve for Tab ,u s e df o rt h e transformation law of the auxiliary elds, and for the curvature case we solve forR abcd, used in the component expansion of the action, in terms of these auxiliaries, the vielbein, and the just-determined connection. The U(1) connection Amneeds no solution: It is pure (superscale) gauge, and will cancel in actions (after perhaps an appropriate redenition of Ga). For these manipulations we use the relations that TABCandRABcdhave toB,Ga,W,W , and their derivatives (as expressed by the solution to the Bianchi identities given in subsection XA2). The solution is !mbc=ema[!abc1 2(^Tbca^Ta[bc])]; ^TabceamebnTmnc=abcdGd+i [a  b]. Tab =eamebnr[m n] +i( aG .  a.  Ba$b)Ct..  +C.. t B=2 3t; G.=(t.+1 3Ct. . . )+h:c: Rab =eamebnRmn +B a( b) 2i[ a(Ct . +1 3 ( )t. ..) a. 1 6t( )a$b] where \" refers to the usual expression for pure gravity, aeam m,a n dw e have chosen the superscale gauge W= 0 for simplicity. 4. Component approach We can now take the superspace action of subsection XB1, as expanded in com- ponents by the ectoplasm method of subsection XB2, and substitute the component expansions of the eld strengths found in subsection XB3, to nd the componentaction L SG=LG+L +e1La LG=1 4e1R; L =mnpq m.1 2fen.;rpg q;La=3 8(Ga)2+3BB B. ACTIONS 563 (Note the signs are again consistent with GaandB;Bforming a 6-vector of SO(3,3), though not in the same way as in excercise XA2.6.) Here randLGare the usual covariant derivative and Einstein-Hilbert action of general relativity in terms of eand !, but!is slightly dierent from any of the connections used previously (see excercise XB4.1 below). It also diers from the !given above in that we have explicitly extracted the Gapiece (which is the sole source of !in the ectoplasm approach). An alternative to ectoplasm to determine !is to use a rst-order formalism: Rather than imposing the usual torsion constraint, we can leave the Lorentz connection asan independent eld in Rand in therinL . Eliminating the Lorentz connection by its eld equation yields a modied torsion constraint, and produces 4terms in the action. We have written L in a form manifestly symmetric with respect to integration by parts. (Alternatively, we can write  er er .) As an alternative to deriving the component action from the simpler superspace expression, we can postulate the component action directly. In the component ap- proach writing the action in components is more direct than the superspace approachby denition, but proving supersymmetry invariance is less so. This is not so true when coupling to matter, where writing component actions can also be as compli- cated as deriving them from superspace, so here we consider the simplest case, puresupergravity. We thus begin by postulating L SG=LG+L ; the rst term is obvious, while the second follows from minimal coupling for the free gravitino action, which can be derived easily by many methods (see, e.g., subsection XIIA5 below). We ignore the auxiliary elds, which are necessary for o-shell closure of the supersymmetry algebra, but not for supersymmetry invariance of the action. We write the action for gravity in a form that more resembles the gravitino action (see excercise IXA5.5): LG=1 4e1R=1 16mnpqabcdemaenbRpqcd=1 8mnpqemaenb~Rpqab =i1 8mnpqem.(Rnp.. e q. Rnpeq.) where we have switched to spinor notation for the curvature (see subsection IXA1) and used duality in both vector and spinor notation (see subsection IIA7). (Alternatively, we can regard this as the denition of the gravity action.) We then have for the variation of this part of the action (after some integration by parts) LG=i1 4mnpqem.f[Rnp.. e q. Rnpeq.][(!n.. )T pq. (!n)Tpq.]g where we have used (see excercise IIIC1.2) RmnI=r[m!n]I;Z dxrmVm=Z dx @mVm=0 564 X. SUPERGRAVITY Next, we pick the obvious transformation law for the gravitino eld as the gauge eld of supersymmetry:  m=rm The transformation laws for eand!will be derived as a by-product of the invariance proof, as will the explicit expression for !in terms of eand . Substituting this expression for  intoL , L =mnpq[rm1 2fen.;rpg q..rm1 2fen.;rpg q +1 2 m.(en.)rp q1 2 m(en.)rp q. +1 2 m.en.(!p) q1 2 men.(!p.. ) q. ] where we have integrated by parts to free the supersymmetry parameters of deriva- tives. We then use the antisymmetrization on all curved indices to collect the resultingterms into torsion and curvature as rfe;rg=fe;rrg +(re)r= 1 2fe;Rg1 2Tr The curvature terms then cancel those from LG,i fw ec h o o s ef o r einLGthe transformation law em.=i( m.+. m) Then we also substitute this expression for einL , and note that half those terms immediately drop out, since [m n]=0 by antisymmetry. The remaining terms from both LGandL then can be collected as LSG=1 8mnpqeTmn.
pq. eTmn.Tmn.i [m. n]
pq.i(![p.. )e q]. i(![p)eq].+r[p q]..r[p q] We now note that the former factor in LSGvanishes by virtue of the equation of motion from varying the connection: Rather than vanishing, the torsion now satises secondorder :eTmn.=0 We can regard eTas the \supersymmetrized torsion"; this is equivalent on shell to the result we found in the previous subsection from superspace. We can therefore quit now, since in a second-order formalism the torsion (and thus the Lorentz connection) B. ACTIONS 565 would satisfy this equation even o shell. (This approach, using the second-order formalism but not bothering to substitute the supersymmetry variation of the con- nection, is called the \1.5-order formalism".) On the other hand, we can just as easilyrecognize that in the rst-order formalism cancelation of L SGis also guaranteed by allowing vanishing of the latter factor to dene the supersymmetry variation of the (independent) connection: firstorder :
pq.=0 Thus, use of the rst-order formalism requires no more work than 1.5-order (contrary to remarks in the literature), which is really the same as second-order, and providesthe bonus of yielding the transformation law for !. However, it is useful to note that not all quantities should have their longer forms substituted at the beginning of acalculation (just as we learned in high-school algebra not to plug in numbers till theend). Excercise XB4.1 Let's complete this calculation to the bitter end, nding all the properties ofthe connection: aSolve the torsion constraint for !(see subsection IXA3). bFind the transformation law for !that follows from cancelation of the above terms o shell (i.e., without imposing the torsion constraint). cShow the above two results are consistent (modulo terms with eld equa- tions, which can be canceled by contributions from auxiliary elds) by plug-ging the expressions for eand into the variation of the result for part a and comparing with the result for part b. dCompare these results with the connection found in the previous subsection. How does the appearance of G aaect the transformation law? 5. Duality Although antisymmetric tensor gauge elds can be avoided in general, they tend to turn up in string theory, so we now look at them a little more generally, examiningtheir actions and how they relate to those for scalars. In particular, we note thata sensible action for such a tensor alone cannot be constructed that is conformallyinvariant: From the same analysis as for electromagnetism or Yang-Mills (subsectionIXA7), we see that ( F a)2does not give a scale-invariant action in four dimensions. Thus, such a eld is not suitable as a compensator for pure gravity. However, in 566 X. SUPERGRAVITY supergravity the tensor multiplet (see subsection XA3) also has an ordinary scalar, and an appropriate power of it can make the tensor's action conformal. Therefore, we now examine general duality transformations for the supersymmetric case, whichis more relevant for understanding its use in gravity. We will consider explicitly a at superspace background for simplicity, but generalization to curved superspaceby covariantization is straightforward, replacing at superspace derivatives with (su-perconformal) covariant derivatives, introducing supergravity eld strengths wherenecessary (in this case, just d 2!r2+Bfor chirality), and using the covariant integration measures. Duality transformations can be performed directly in the action by use of rst- order formulations. Starting with the general tensor multiplet action Stm=Z dx d4K(G) whereKis some function and G=d+h:c:, we write this in rst-order form as S0 tm=Z dx d4[eK(V)+VG] whereVis an unconstrained real supereld and eKis the Legendre transform of K: For this action to reduce to the previous upon applying the algebraic eld equationofV,w em u s th a v e [eK(V)+VG]j @~K(V) @V=G=K(G) The duality transformation is then performed by varying instead ofVinS0 G: Remembering that is chiral, so Z d4xd4VG =1 2Z d4xd2()d2dV+h:c: we solve the condition on Vas d2dV=0)V=+ since thinking of Vas the prepotential for a vector multiplet says that it is pure gauge. The dualized action is then S=Z dx d4eK(+) We can also reverse the procedure through another rst-order action S0 =Z dx d4[K(V)V(+)] B. ACTIONS 567 w h e r ei nt h i sc a s ev a r y i n gw i t hr e s p e c tt o implies d2V=0)V=G while varying with respect to Vgives the inverse Legendre transform [K(V)V(+)]j@K(V) @V=+=eK(+) The simplest case is the Lagrangian1 2G2: We then nd K(V)=1 2V2,eK(V)=1 2V2 so the duality is Ltm=1 2G2,L=1 2(+)2 In at space, this gives the usual free result L=, but in curved space the 1 22+h:c:part does not vanish because E1is not chiral. Consequently, this action is not the conformal one (as we already knew from the component argument above). However, the conformal one is easy to nd by starting with : Making the eld redenition !eexpresses the action in terms of +. Legendre transforming, eK(V)=eV,K(V)=V(lnV1) so the duality is L=e+,Ltm=G(lnG1) These two conformal actions for matter, when coupled to conformal supergravity, become the two \minimal" actions for supergravity, when the overall sign is changed to make the matter elds into compensators: The version with as the compensator is called \old minimal", while that with is called \new minimal". They dier only o-shell, in their choice of auxiliary elds. Note that the eld equations for the two conformal multiplets, d2=0 ( e); d2dlnG =0 reproduce the compensator part of the supergravity eld equations ~B=0a n dfW=0 described in subsection XA4. Again, the full expressions follow from the usual su- pergravitational and superscale invariances, which were used to nd ~BandfW;t h e compensator dependence is enough to identify them as the appropriate covariantiza- tions. Excercise XB5.1 We saw in subsection IVC5 for the Chern-Simons form, or XB2 for ectoplasmic 568 X. SUPERGRAVITY integrals, that dierential forms can be dened in superspace. Do the same for the tensor multiplet: aBy generalizing the bosonic case to superspace with curved indices, and then \ attening" the indices (as for the Chern-Simons superform), show that thesuper 2-form B ABwith eld strength HABC and gauge parameter Ais de- scribed by BAB=r[AB)TABCC;HABC=1 2r[ABBC)1 2T[ABjDBDjC) (Hint: Show that replacing EA!rAandCABC!TABCyields only cancel- ing connection terms.) bShow that the torsions given in subsection XA2 satisfyZ dx d4E1H;.;.=0 Note that this also implies the gauge invariance of the Chern-Simons form of the super Yang-Mills action in curved superspace. cShow that the constraints H =H. =Hc=0;H ;. ; . =iC C. . G (and complex conjugates) can be solved by B=B . =0;B .;. =iC.. ;r.=0 Bab=C.. b+Cb.. ;b=1 2r();G=1 2(r+r..) Relate the results for the gauge elds BABto those for the Yang-Mills eld strengthsFAB(subsection IVC3). dSupersymmetrize the construction of excercise XA3.1: Show one can dene a eld strength eHABC=HABC+BABC using the Chern-Simons superform BABC. The supergravity component action with the tensor multiplet compensator diers from the one of the previous subsection in that Band the longitudinal part of Ghave been replaced by the gauge eld Bmn: La!1 2mnpqGm@nBpq which is the only possibility that preserves the gauge invariances of both GandB while leaving them both auxiliary. (Their eld equations are that their eld strengths vanish.) B. ACTIONS 569 6. Superhiggs Supergravity aects spontaneous supersymmetry breaking in a simple way: From the discussion of the immediately preceding subsections, we know that supergravity can be described more simply as conformal supergravity coupled to a compensator.Simple (N=1) conformal supergravity contains no scalars: It consists of only confor- mal gravity (the traceless part of the metric), the conformal (traceless) part of the gravitino eld, and an auxiliary gauge vector. Since symmetry breaking involves giv-ing vacuum values to only scalars, we can replace supergravity by just its compensator for these purposes. For a general analysis, consider a kinetic term S K=Z dx d43eK(i;i)=3 This is the most general kinetic term with the usual number of spacetime derivatives: Any term of the form f(;;i;i) can be rewritten in this form after appropriate eld redenitions. In particular, if we start with elds with arbitrary Weyl scale weight, then this form follows after rescaling elds so only carries scale weight, since all terms in the Lagrangian must have the same scale weight, xed by (super)conformalinvariance. is then the only eld to carry U(1) weight, which is proportional to scale weight by superconformal invariance. Then appears only as , whileKis an arbitrary function of  iand i. The rst step in evaluating this action in components is to simply ignore conformal supergravity altogether, and evaluate this action as is, in terms of matter and compensator multiplets, by the methods we have considered previously for evaluating integration. The next step is to add back in some parts of conformal supergravity: (1) the conformal graviton, which can be put back in easily and uniquely using coordinate and local scale invariance; (2) the U(1) axial gauge vector, whose coupling is minimal, and thus follows directly from U(1) covariantizingthe spacetime derivatives; and (3) the conformal gravitino, whose quartic couplingscan be quite complicated, but as a practical matter we are interested in only the mass term, which is determined from the mass of the Goldstone fermion it eats, which appears in the compensator (and the kinetic term, which is the usual one).(The exponential form will prove convenient for later component analysis.) Before considering the general case, we look at the pure supergravity case under this analysis: Looking at just the bosons, we nd S SG;b=Z dxe11 2f3[(ri1 3A)]
[(r+i1 3A)]1 2R +6BBg 570 X. SUPERGRAVITY whereris the usual covariant derivative of general relativity, Ais the U(1) gauge vector, and the relative coecent of the Rterm was xed by local scale invariance (see subsection IXA7). Note that here Bis the usual auxiliary eld from ,a n di s not associated with conformal supergravity. Choosing the component U(1) and scalegaugesj= 1, this reduces to S SG;b!Z dxe1(1 4R1 6A2+3BB) Relating to Ga, we recall that if we had included it from conformal supergravity, for this compensator ~Ga=Ga+2 3Aa,s ow ec a ni d e n t i f y Aawith3 2~Ga.T h u s , t h e compensator method immediately yields the bosonic action, including auxiliary elds. Returning to the general case, the part of the action for the \physical" scalars ( j andj) then starts out as SK;ps=Z dxe1eK=31 2f3[(ri1 3A)]
[(r+i1 3A)]+[(r+i1 3A)]
(@iK)ri +[(ri1 3A)]
(@iK)ri+[(@i@jK)1 3(@iK)(@jK)](rj)
(ri)1 2Rg ignoring until the following subsection the auxiliary scalars, which are irrelevant for the kinetic term. We use the notation @i=@=@i,@i=@=@i. We then choose the U(1) and scale gauges j=eK(ij;ij)=6 where we have explicitly written the j's to emphasize that this is a nonsupersymmetric gauge choice for the component j. Finally, we eliminate Aby its algebraic eld equation. We thus obtain SK;ps!Z dxe11 2[(@i@jK)(rj)
(ri)1 2R] Except for the Rterm and covariant derivatives, this is what would follow in at superspace from the action Rdxd4K. For supersymmetry breaking, we also need the super cosmological term Sc=Z dx d2 3+h:c: for some constant . We could consider more general potentials 3ef(i)(again the power ofis xed by scale and U(1)), but then the eld redenition !ef=3 would remove it while replacing K!K+f+f. (This invariance, and the form of the \metric" on the space of elds and appearing in the action, identify Kas a \Kahler potential".) B. ACTIONS 571 The analysis for SKcan also be made by performing a duality transformation on the compensator. Following the same steps as described in the previous subsection forthe case without matter (factoring the overall 3 out of the process for convenience), we nd S K!Z dx d4[3GlnG +GK(;)] Since in this form Adecouples, the result is obvious from the at-space result. Normally any kind of symmetry breaking will generate a cosmological term, since a scalar getting a vacuum value implies the potential itself getting one, giving a termR dxe1constant . This would require adding a cosmological term to the action by hand to cancel the generated one, since within observational limits no cosmologicalconstant is observed in nature. (In any case, the constant generated would corre-spond to a subatomic length scale, whereas a realistic cosmological constant requiresa cosmological length scale, which means a constant, going as 1/length 2, of the order of 1080in subatomic units.) An exception is when the potential is at in some direc- tion: In supersymmetry energy is always positive, and the supersymmetric vacuumhas zero energy, but some potentials allow other, perhaps nonsupersymmetric, vacuuathat also have zero energy, and thus generate no cosmological constant. This avoidsthe ad hoc procedure of \ne tuning" the cosmological constant of an added term forexact cancelation (or at least to order 10 80). 7. No-scale A useful example of the superhiggs eect with a at potential is \no-scale super- gravity". This theory has an explicit super-cosmological term, but the kinetic termis such that this term does not generate a component cosmological term, but doesspontaneously break supersymmetry. The simplest example describes supergravitycoupled to a single chiral scalar multiplet. The kinetic term has an SU(1,1) symme-try, and also appears in N=4 supergravity (see subsection XC6 below). Written interms of just the compensator part of supergravity, it is S K=Z dx d43(+) whereis the compensator and is the matter. We have written it in a manifestly U(1,1) covariant form, where the U(1,1) metric is o-diagonal (0 11 0
instead of the usual diagonalized1 00 1
). For the above component analysis we redene !)SK!Z dx d43(+))K=3ln(+) 572 X. SUPERGRAVITY (Many other supereld redenitions are possible to put this in more conventional forms, such as (3 ),(3), etc.) The kinetic term for the physical scalars follows from the same analysis we applied to the CP(1) model in subsection IVA2.The only dierences here are: (1) the symmetry is U(1,1), not U(2), and (2) theconstraint on the norm of the complex 2-vector follows not from a Lagrange multipler(or a low-energy limit), but as a local scale gauge chosen to give the Einstein-Hilbertcurvature term the usual normalization. Alternatively, we can use the analysis givenin the previous subsection for the general case to nd S K;ps!Z dxe11 2 3jrj2 (+)21 2R However, to study just the supersymmetry breaking, we want to look at the \po- tential" terms: terms that involve the auxiliary scalars instead of spacetime deriva-tives. We thus now need to include the super cosmological term, which breaks theSU(1,1) invariance. Again evaluating at rst without conformal supergravity, thenputting some (all but the conformal gravitino) back in, we nd the contributions fromS KandSc Saux=Z dxe13[BB(+)+( Bb+Bb)+(B2+B2)] whereB=d2andb=d2. We then see that eliminating the auxiliaries gives nothing, so there is no potential to generate a cosmological term. However, there isstill a mass term for the gravitino: As always, S calso contains the spinor term 6(2+h:c:) where=dis the trace of the gravitino. The gravitino in this model therefore has a mass proportional to h+i1=2. SU(1,1) invariant kinetic terms also appear in superstring theory, but unlike N=4 and no-scale supergravity, the kinetic term is ( +)1=3instead of just +. (See subsection XIB6.) When applying no-scale supergravity to nature, more mattermultiplets are added, S=Z dx d 43(+ii)+Z dx d2 3ef(i=)+h:c: generalizing SU(1,1) to SU(n,1) in the rst term. (N=5 supergravity has such an SU(5,1) symmetry; see below.) Then acts as the \hidden" matter sector that doesn't directly couple to the observed matter i, but serves only to break supersymmetry. B. ACTIONS 573 REFERENCES 1W. Rarita and J. Schwinger, Phys. Rev. 60(1941) 61: action for spin 3/2. 2Freedman, van Nieuwenhuizen, and Ferrara, loc. cit. (XA): complete component formalism for supergravity, except for auxiliary elds. 3S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335: rst-order formalism for supergravity. 4A.H. Chamseddine and P.C. West, Nucl. Phys. B129 (1977) 39; P.K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 67B (1977) 439: 1.5-order formalism. 5P. van Nieuwenhuizen, Phys. Rep. 68(1981) 189: general review of supergravity. 6S.J. Gates, Jr., hep-th/9709104, Ectoplasm has no topology: the prelude, in Supersym- metries and quantum symmetries , proc., July 22-26, 1997, D ubna (Lecture notes in physics, v. 524), eds. J. Wess and E.A. Ivanov (Springer, 1999) p. 46;S.J. Gates, Jr., M.T. Grisaru, M.E. Knutt-Wehlau, and W. Siegel, hep-th/9711151,Phys. Lett. 421(1998) 203: ectoplasm. 7Wess and Zumino; Wess; loc. cit. (IVC, ref. 5): component expansions for supergravity. 8W. Siegel, loc. cit. (XA); K.S. Stelle and P.C. West, Phys. Lett. 74B (1978) 330; S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. 74B (1978) 333: old-minimal supergravity. 9Sohnius and West, loc. cit. (XA): new-minimal supergravity. 10D.V. Volkov and V.A. Soroka, JETP Lett. 18(1973) 312; S. Deser and B. Zumino, Phys. Rev. Lett. 38(1977) 1433: superhiggs. 11B. Zumino, Phys. Lett. 87B (1979) 203: Kahler in N=1 nonlinear models. 12E. Cremmer, S. Ferrara, C. Kounnas, and D.V. Nanopoulos, Phys. Lett. 133B (1983) 61: no-scale supergravity. 574 X. SUPERGRAVITY ::::::::::::::::: ::::::::::::::::: ::::::::::::::::: C. HIGHER DIMENSIONS ::::::::::::::::: A convenient method for describing extended supersymmetry in D=4 is to ap- ply dimensional reduction to supersymmetry in D >4, since (1) spinors are bigger in D>4, so even simple supersymmetry reduces to extended supersymmetry, and (2) the Lorentz group is bigger in D >4, so some 4D scalars arise as parts of higher-D vectors, etc., meaning fewer Lorentz representations in the multiplet in D >4. 1. Dirac spinors We saw in subsection IC1 that coordinate representations of orthogonal groups SO(D) could be dened in terms of self-conjugate fermions, Gab=1 2[ a; b];f a; bg=ab We now will construct explicit matrix representations of the Dirac matrices for arbi- traryD, and examine their properties. This is useful for understanding: (1) repre- sentations of internal symmetries, such as in Grand Unied Theories; (2) theories inhigher dimensions, which give simpler formulations of certain four-dimensional the- ories when the extra dimensions are eliminated, and appear in string theory; and (3) properties of spinors that are independent of D, or their dependence on D,w h i c h is useful for comparison and for perturbation in quantum eld theory. An explicit solution can be found easily by rst looking at even dimensions, and breaking up the problem into D/2=ntwo-dimensional problems. Furthermore, we can look rst at the Euclidean case (SO( D)), and solve for the other cases (SO( D +,D)) by Wick rotation. The solution for SO(2) is just two of the Pauli matrices. The general solution then comes from the direct product of the two-dimensional cases, using the third matrix to introduce appropriate \Klein factors" (see excercise IA2.3) to insure that the matrices from one two-dimensional subspace anticommute with those from another. The resulting matrices are then: 1p 2(p 23


p 23 p 2i I


I);1p 2(p 23


p 23) wherei=1;2, there are a total of nfactors, and the number ofp 23andIfactors in the rst expression ranges from 0 to n1. The last matrix can always be included to extend SO(2 n)t oS O ( 2n+1); in fact, up to normalization, it's simply the product of all the other 's. (In other words, the product of all the matrices is proportional to the identity.) C. HIGHER DIMENSIONS 575 The next step is to notice that this construction generally gives a reducible rep- resentation. Reducibility comes from two properties: (1) For SO(2 n) we really have SO(2n+1); and (2) the representation may be real. In fact, most of the interest- ing cases involve SO(2 n) (in particular, SO(3,1) for Lorentz and SO(4,2) for confor- mal in four dimensions). In that case we can call the rst (or any other) matrix (1 I


I)f o rS O ( 2n+1) \ 1", and take the rest as those for SO(2 n). Then the projection operators =1 2(1p 2 1)) 2=;+=+=0;++=1 commute with the SO(2 n) generators Gab , so they can be used to project the representation of the 's into two representations of SO(2 n). These two halves of a Dirac spinor are known as \Weyl spinors". A convenient representation of the matrices for this purpose is the one given in subsection IIA6, with the representation of the Pauli matrices used in our SU(2)/SL(2,C) discussion of subsections IIA1 and5,  1=1p 21 00 1
; 2=1p 20 11 0
; 3=1p 20 ii 0
We then can write the spinor, which has 2ncomponents (since it represents the direct product of nrepresentations of matrices, each of which has two components) as two 2n1-component spinors projected by = I ::: I;  +=1 00 0
;=0 00 1
The matrices then take the block-diagonal form 1=1p 2I 00 I
;o t h e r =0 ~ 0
We will refer to these reduced matrices (and ~), and the matrices themselves for SO(2n+1), as generalized Pauli ( ) matrices. The reality properties of the representation depend on the existence of a metric . AB(or . ABfor psuedoreality, which doesn't reduce the representation), as in our discussion of classical groups of subsection IB5. In fact, all the spinor representationsof any orthogonal group are also dening representations of another group: For less than seven dimensions, this leads to the identication of covering groups discussed in subsection IC5; for more than six dimensions, it only identies the orthogonal groupas a subgroup of this new group. (An interesting exception is SO(8), where the spinor representations are also 8-dimensional, and are the two other dening representations of SO(8).) In matrix notation, we look for a matrix C=or such that we can dene the operation of charge conjugation as !C 1 *;G !C1(G )*)G=C1G*C 576 X. SUPERGRAVITY If we like, we can also choose C=Cy=C1 without loss of generality. For a representation to be invariant under charge conju- gation (i.e., real) =C1 *)C*=C1 For our matrix representation, the matrix to look at is C=::: C2p 23 C2 C2p 23 C2 where C2=0 ii 0
independent of the representation used for the Pauli matrices. (Our representation is simplest, since then C2p 23=I,a n dC=Cy. In other representations, Cmay also need ann-dependent factor of iif we wantC=Cy.) Using properties of matrices we found in our discussion of SO(3) in subsection IIA2, such as C2*C2=,w e nd C1 *C=(1)n ; C *=(1)n(n+1)=2C1;CT=(1)n(n+1)=2C; Cy=C1 This distinguishes 8 cases, where the irreducible spinors are: SO(8m): Weyl and real SO(8m+1): real SO(8m+2): Weyl SO(8m+3): pseudoreal SO(8m+4): Weyl and pseudoreal SO(8m+5): pseudoreal SO(8m+6): Weyl SO(8m+7): real For SO(4m+2), charge conjugation does not preserve 1, and thus  . Therefore, in those cases there is no metric . ABor . ABon the irreducible spinor: The Dirac spinor consists of two irreducible spinors that are complex conjugate representations of each other. In general, a Dirac spinor has 2ncomplex components for SO(2 n)a n d SO(2n+1); the Weyl condition reduces this a factor of two for SO(2 n), as does reality where applicable. (Pseudoreality does nothing.) It is useful to know the other group metrics, if they exist. For unitarity properties we look for a metric . ABsuch that G=1Gy) =y C. HIGHER DIMENSIONS 577 (Thusg=eGsatises g1=gy.) We can also choose =1 without loss of generality. We therefore look for a metric satisfying 1 y= soG[ ; ] is antihermitian with respect to . For SO(D), we have simply =I since the hermiticity of the matrices implies that of the matrices. We then can also dene a metric to raise and lower indices in terms of these two metrics, by contractingthe dotted (or undotted) indices: In matrix notation, we then have (C T)1 T(CT) = (1)n )G=(CT)1GT(CT) For all cases except SO(4 m), this also denes the symmetry properties of the gener- alizedmatrices: They can be dened as CT for SO(2n+1), and as its diagonal blocks with respect to  for SO(4m+2); but for SO(4 m) it's o-diagonal, so the generalized matrices appearing there carry one each of the two dierent kinds of spinor indices, and thus have no symmetry. Then we rewrite the above result as (CT )T=(1)n(n1)=2(CT ) 2. Wick rotation The indenite-metric groups SO( D+,D)(D+6=06=D) can be treated by Wick rotation: giving i's toDof the a's, so the corresponding components of ab get minus signs. Since this aects Ti nt h es a m ew a ya s , the metric CTi s unchanged. In other words, CT=CT E (E=I) in terms of the Euclidean Cof the previous subsection. However, *a n d yare aected in the opposite way to and T(i's instead of i's).  then becomes (up to normalization) the product of the timelike 's, Y aa<0p 2 a which also determines the modication of C. In the above equations for yand T, we then nd a factor of 1 for each rotated dimension, coming from anticommutation 578 X. SUPERGRAVITY with each timelike in , so we redene all 's by an overall factor of iDto preserve their pseudohermiticity. This changes the normalization to Gab=(1)D1 2[ a; b];f a; bg=(1)Dab In odd dimensions the matrices are the matrices (up to multiplication by one of the metrics), while in even dimensions the matrices consist of two o-diagonal blocks of the matrices. To write actions we also need the \dual" Dirac spinor, in the sense of a Hilbert-space inner product,  = y with  as dened above. In particular, it is justp 2 0inD=1 . We then nd (e.g., using the explicit representation given above) that Chas the same properties with regard to symmetry and 1for SO(D+,D)a sf o rS O ( D+1, D1). Thus, the properties of these metrics on the irreducible (as opposed to Dirac) spinors follow easily from the Euclidean case by using CT:SO(D+;D) SO(D++D) C:SO(D+;D) SO(D+D) Then the properties of  follow from the above two in the cases where all 3 exist; the few cases where only  exists, which have Deven, follow from the next higher D (increasing D+by 1). Excercise XC2.1 Find the explicit matrices for D= 2 and 4 from the construction of the previous section, and apply this Wick rotation. Compare the results with theconventions of subsections VIIIA7 and IIA6. We can use these properties to determine that the number of real components D 0 of an irreducible spinor is D0=2[D2+f(D+D)]=2;xm o d 8 01234567 f(x) 01232321 The complete results can be summarized by the following table, showing for each case of SO( DD,D), forDmod 8 and Dmod 4, the types of irreducible spinors , the types of metrics (symmetric) and (antisymmetric) for these irreducible spinors, and the type of generalized matrices (and its symmetry, where relevant): C. HIGHER DIMENSIONS 579 D 0 1 2 3 D Euclidean Lorentz conformal  0  .  0  . 0.. . . 0  . 0  .      1.... . . . . ()()()()         2 .. . . ()()()()()()()()     3  .. . . . .  .. ()()()()  0  .  0  . 4  ..  . .  0  . 0  .      5  ..  .. . . . . [][][][]         6 . . .. [][][][][][][][]     7.. . . . ... [][][][] Excercise XC2.2 The vectors of SO( D+,D)f o r3D++D6 were expressed as tensors with two spinor indices, with the appropriate symmetry and tracelessness conditions, in subsection IC5. aShow that the spinor metrics found from the Dirac analysis are sucient to identify each of these covering groups. bShow the equivalence of each orthogonal group to its covering group by show- ing that (1) the Lie algebras have the same dimension, and (2) the deter- minant (or its square root) of this tensor gives the appropriate orthogonal 580 X. SUPERGRAVITY metric. (Hint: Do the cases D++D= 4 and 6 rst, and specialize to 3 and 5.) Excercise XC2.3 Consider the groups SO(n,n) and SO(n,n+1). Explicitly construct a realrepresentation of the matrices, demonstrating that all such spinors are real. Besides the Dirac spinors and Dirac matrices , and the irreducible spinors and Pauli matrices , it is also useful to introduce irreducible real (\Majorana") spinors and corresponding matrices . When the irreducible spinors are already real these are the same, but when the irreducible spinors are complex this real spinor is just the direct sum of the irreducible spinor and its complex conjugate, a spinor with twice asmany components. In general, these generalized Majorana spinors and matrices have many properties that are independent of the number of dimensions, but depend on the number of time dimensions: D 0 1 2 3 Euclidean Lorentz conformal  0    0     0 ()()0 [][] ForDodd, there is only one irreducible spinor, so there is a metric MorM0 to relate the two spinors listed. For D2Dtwice odd (2 mod 4), the original irreducible spinor was complex, so there is a metric representing a U(1) generator that rotates the complex spinor and its complex conjugate oppositely. (I.e., it's theidentity on the complex spinor and minus the identity for the complex conjugate.) ForD2D =3;4;5mod 8, the original spinor was pseudoreal, and this U(1) can be extended to an SU(2): Since the complex spinor and its complex conjugatetransform the same way under the orthogonal group, they can be paired as a doubletof SU(2). This doubled representation is a real representation of the orthogonal group SU(2), since the direct product of the two antisymmetric charge-conjugation matrices is symmetric (two 's under transposition). 3. Other spins Before considering supersymmetry in higher dimensions, we rst study represen- tations of the Poincar e group there. From the general analysis of section IIB, we know that general on-shell representations follow from the massless ones, which can C. HIGHER DIMENSIONS 581 be classied by their representation of the lightcone little group SO(D 2). Speci- cally, the bosons can be described as traceless tensors of a certain symmetry (labeled by a Young tableau), while the fermions can be labeled as the direct product of suchtensors with an irreducible spinor, with a tracelessness condition imposed between any vector index and the spinor index using a ormatrix. Similar methods can be used to nd the o-shell representations in terms of representations of SO(D 1,1), but without subtracting traces. (For full details, see chapter XII.) The gauge degreesof freedom can be subtracted from these Lorentz representations by dropping all lower vector indices with the value \ ", by the usual lightcone gauge condition; this tells us the number of total physical + auxiliary degrees of freedom. In practice, the only interesting massless elds in higher dimensions are: (1) the metric (graviton), (2) totally antisymmetric tensors (including scalars and vectors),(3) spin-3/2 (gravitino), desribed by vector spinor, and (4) spinors. By the methods described above, the counting of physical, auxiliary, and gauge degrees of freedom for these elds is (where D 0is the number of components of an irreducible spinor of SO(D1,1) | see the previous subsection): eld physical auxiliary gauge h(ab)1 2D(D3)D D A[a1:::an]D2 n
D2 n1
D1 n1
a1 2D0(D3)1 2D0(D+1 )D0 1 2D0 1 2D00 Excercise XC3.1 Derive all the entries in the table. For each type of eld, nd the minimum Dfor which physical degrees exist. We next consider exactly how many higher dimensions are relevant. From the previous subsection, we see that an irreducible spinor (which we use for the super- symmetry generators) has 1 component in D=2, 2 in D=3, 4 in D=4, 8 in D= 5 or6, 16 in D=7, 8, 9, or 10, 32 in D=11, etc. Since the maximal Lorentz symmetry canbe obtained by looking at the maximum D for which a certain size spinor exists, we see that the appropriate D for which an irreducible spinor reduces to N irreducible spinors (for N-extended supersymmetry) in D=4 is 582 X. SUPERGRAVITY ND 14 26 410 811 etc. From the discussion of subsection IIC5, we know that supergravity exists only for N8, and super Yang-Mills only for N 4. This means that simple supergravity (i.e., any supergravity) exists only for D 11, and simple super Yang-Mills for D 10. Since theories with massless states of spin >2 are not of physical interest (in fact, no interacting examples have been constructed), we can restrict ourselves to looking atjust D=4, 6, 10, and 11. In general, an irreducible multiplet in some D can becomereducible in lower D. However, since irreducible multiplets of supersymmetry are con- structed as the direct product of the smallest representation of supersymmetry with an arbitrary representation of the Poincar e group, this reducibility corresponds di- rectly to the reducibility of that Poincar e representation, which occurs simply because the Lorentz group gets smaller upon reduction. In particular, the smallest represen- tation of supersymmetry is itself irreducible. For the case of simple supersymmetry, this is the scalar multiplet (scalars and spinors) in D=6, the vector multiplet (superYang-Mills: vectors, spinors, and scalars) in D=10, and supergravity in D=11. Thestatement that it is the smallest multiplet in that number of dimensions is directlyrelated to the fact that it does not exist in higher dimensions. 4. Supersymmetry We rst generalize to arbitrary dimensions some denitions used earlier: To dis- cuss the properties of supersymmetry that are common to all dimensions (but one time), it's most convenient to use the Majorana form fq;qg=a pa which is consistent with the general symmetry of these matrices. The supersymmetry generators are then q=i@ @+1 2a @ @xa andqgenerates the innitesimal transformations =; xa=i1 2a  C. HIGHER DIMENSIONS 583 where (q)y=q. The covariant derivatives are d=@ @+1 2a pa and they satisfy the same algebra as supersymmetry fd;dg=a pa but with the opposite hermiticity condition ( d)y=+d. The invariant innitesimals are d;d xa+i1 2(d)a  Superelds can be expanded as either (x;)=(x)+ (x)+::: or =d; ::: giving the transformations = ; =i1 2a @a+:::; ::: Representations can be found as for D=4; we don't have twistors in general, but we can always use a lightcone frame. We rst need to dene a, which in general is independent of a (only the latter was needed to dene supersymmetry above): The analog of the Dirac anticommutation relations (which can be reconstructed if wecombine the two 's, as generalized 's, to form a generalized )i s (a  b) =ab  In the lightcone frame the momentum is just pa=a +p+withp+=1b e i n gt h es i g n of the (canonical) energy. In this frame we have the constraint q= 0. This projects away half the q's, sinceare projection operators: Using the anticommutation relations of , =) 2=;+=+=0;++=1 The equality of the sizes of the two subspaces follows from parity symmetry, $. We thus need to consider only half of the q's, namely +q. We therefore switch to a notation where we consider the truncated spinor qwith just that half of the 584 X. SUPERGRAVITY components. This \lightcone spinor" is an irreducible spinor of SO(D 2). In a Majorana basis it satises the same commutation relations as Dirac matrices, fq;qg= Sinceqhas an even number of components in D> 3, the states that represent this algebra form a Dirac spinor of SO(2n)(n> 0) that is reducible to two Weyl spinors. (These spinors should not be confused with those of SO(D 2), such as q, which is a vector of this SO(2n).) Since supersymmetry takes each of these \spinors" into the other, one spinor contains all the bosons, while the other contains all the fermions. There are an equal number of physical boson and fermion states because the two Weyl spinors are equal in size. Since SO(D 2)SO(2n), each Weyl spinor of SO(2n) is reducible with respect to SO(D 2). The only exception is D=4, where SO(D2)=SO(2n)=SO(2), and there is one bosonic state and one fermionic one. This \Dirac spinor" of SO(2n) is the smallest representation of supersymmetry. It can also be represented in terms of anticommuting coordinates, by dividing up qinto two halves, one of which is complex coordinates, the other half being both the complex and canonical conjugate (as for the fermionic harmonic oscillators of excercise IA2.3).The most general representation of supersymmetry is then the direct product of thisone with an arbitrary representation of the Poincar e group. All the results of this section can be extended to \extended supersymmetry", with supersymmetry generators q ifor an N-valued \internal" index i, as expected from our discussion of supergroups in subsection IIC4: For example, in D=4 the supergroup describing extended conformal supersymmetry, SU(2,2 jN), includes con- formal symmetry SU(2,2), internal symmetry U(N), N supersymmetries, and N S-supersymmetries. In general, the supersymmetries then satisfy the algbera fq i;qjg=ija pa The smallest representation of an extended supersymmetry follows as before, where now the complete lightcone qacts as Dirac matrices for SO(N2n). Other representa- tions are again found by direct product, now between this smallest supersymmetry representation and an arbitrary representation of both Poincar e and the internal sym- metry. For the more interesting cases, where N itself is a power of 2, the smallestrepresentation can also be derived by dimensional reduction from higher dimensions of N=1 (\simple") supersymmetry, changing the higher-dimensional algebra only by setting some components of the momentum to vanish, and noting that a spinor ofhigher dimensions reduces to many spinors, as clear from our explicit construction C. HIGHER DIMENSIONS 585 earlier. (Other representations tend to be reducible, since the Poincar er e p r e s e n t a t i o n in the direct product is reducible upon dimensional reduction.) Dimensional reduc- tion can also be dened for an action (for supersymmetric or nonsupersymmetrictheories), by again setting the derivatives with respect to the \extra" coordinates tovanish, and also restricting the integration to the reduced set of coordinates. An-other interpretation is that we expand the elds over all momentum modes in theextra coordinates, and then drop all but the zero (constant) modes. We also recall from subsection XC2 the index structure of spinors in D=6, 10, and 11, which we need to write supersymmetry covariant derivatives. We thus have,for simple supersymmetry, D=6:fd i;djg=Cjii@ D=1 0:fd;dg=a i@a D=1 1:fd;dg=a i@a where in the case of D=6 we have taken advantage of the fact that SO(5,1)=SU*(4) to eliminate vector indices, and introduced the SU(2) index ifor spinors to make them Majorana. 5. Theories We rst consider the scalar multiplet in D=6. The constraints and eld equations are given by the statement, in terms of supersymmetry covariant derivatives, thatthere are only scalars and spinors on shell, and by supersymmetry their physicalpolarizations must be equal in number. Since a spinor has 4 polarizations in D=6, we must have 4 real scalars, and thus d ijk0=Cji k0 The second SU(2) index k0is introduced again to make a spinor (this time the eld) Majorana, and performs a similar service for the scalars. This one equation is suf-cient to completely describe this multiplet on shell in the free case; interactionsrequire derivatives, so we won't consider them here. This multiplet reduces to N=2in D=4 in a very simple way: The SU(2) index on dlabels the 2 supersymmetries, and the 4-component spinor index reduces in the obvious way to SL(2,C) indices,!(;.), with appropriate 6D spinor conventions. Excercise XC5.1 Show the equations given for the 6D scalar multiplet give the complete eld 586 X. SUPERGRAVITY equations for all the components, and that only the scalars and spinors shown explicitly in that equation survive on shell. This six-dimensional theory gives a simple example of nontrivial dimensional re- duction: Assume we have a 5-dimensional theory with a nontrivial U(1) symmetry.Then we can dimensionally reduce by choosing the elds to depend on the fth coor-dinate in such a way that the fth component of the momentum of each eld is equalto a constant m(with dimensions of mass) times its U(1) charge Q: p 4=Z=mQ This is consistent at the interacting level because each term in the action satsies con- servation of the U(1) charge as well as conservation of momentum. (This is equivalentto how we introduced masses by dimensional reduction earlier for free elds, since anyfree eld can be \complexied".) This has an interesting eect on the supersymmetryalgebra: It introduces a U(1) charge Z(called \central" because it commutes with the rest of the algebra). For example, if we start with the 6D supersymmetry algebra (like the above algebra for the supersymmetry covariant derivatives), introduce thecentral charge in reducing to 5, and then do an ordinary reduction to 4 (or vice versa),the supersymmetry algebra becomes (see subsection IVC7) fq i;qj. g=j ip . ;fqi;qjg=CCijZ;fqi.;qj. g=C.. CijZ If the higher-dimensional theory was massless, then p2+Z2= 0 for the 4D theory. More generally, if the higher-dimensional theory already had masses before the central charge was introduced, then by supersymmetry it satised p2+M2 0=0 ,M2 00 (since supersymmetry always has positive potentials), while afterwards the 4D theorysatises p 2+Z2+M2 0=0)M2=M2 0+Z2Z2 whereMis the 4D mass, in terms of the higher-D mass M0. However, in general, in the absence of central charges, massive representations of supersymmetry are biggerthan massless ones (because there are twice as many independent supersymmetry generators on shell, since qis a spinor with 1 helicity for the massless case, but an SU(2) doublet for the massive). So, M 2=Z2>0 has the advantage of allowing smaller massive representations than when M2>Z2=0o rw h e n M2>Z2>0. Note that when M2=Z2, so all masses arise from the central charge, (total) mass is conserved, just as in nonrelativistic physics, although in the relativistic case the mass Zcan be negative. (Of course, its square is always positive, as is physical energy. The relation between the relativistic and nonrelativistic cases can be understood through C. HIGHER DIMENSIONS 587 dimensional reduction: See excercise IA4.5. The mass is also a central charge for the Galilean group, but there the reduction is for a lightlike dimension.) In the present case, we can choose our U(1) symmetry to be a subgroup of the extra SU(2) internal symmetry ( k0index) of the 6D scalar multiplet. Note that the algebra of the d's is modied in the same way as that of the q's. Super Yang-Mills is a bit more interesting, because interactions are easier to introduce. From the counting arguments given in subsection XC3, we see that a supersymmetric theory consisting of 1 vector and 1 spinor can exist in D=3, 4, 6,or 10. This corresponds directly with our analysis of the largest dimensions for simple supersymmetries: Dimensional reduction of a vector gives also scalars, so the condition of no scalars gives maximum dimensions. We now make an analysis similar to that of the previous subsection: By dimensional analysis for physical elds, and using single-Majorana-spinor-index notation, fr ;rg=a ira [r;ra]=aW [ra;rb]=iFab Applying the Jacobi (Bianchi) identities, we nd a(a )=0 This identity can be satised only in D=3, 4, 6, or 10. The Bianchi identities imply the eld equations for D=10. Excercise XC5.2 Multiply the identity a(a )=0b yb, and use the matrix anticom- mutation relation (a b) =ab to show that D=3, 4, 6, or 10. Similar methods can be applied to D=11 supergravity. Our component counting for general dimensions, and our helicity analysis for general extended supersymmet- ric theories in D=4 (applied to the dimensionally reduced theory), can be satised by adding to the metric (44 physical components) and gravitino (128) a third-rankantisymmetric tensor gauge eld (84) A mnp(with eld strength Fmnpq =1 6@[mAnpq]). The action for the graviton and gravitino are like those in 4D N=1, while Ahas not only the obvious quadratic term but also a \Chern-Simons term": L=e1[1 4R+ m mnprn p+1 96(Fabcd)2+ 2F+ 4] +1 4
3!(4!)2mnpqrstuvwxAmnpFqrstFuvwx 588 X. SUPERGRAVITY (There are also more-complicated fermion interaction terms than in 4D N=1.) The necessity of the last term can be shown by nding the component form of the super- symmetry transformations, or by nding the eld equations implied by the superspace formulation. 6. Reduction to D=4 We now look instead at the component formulation of higher-dimensional super Yang-Mills. This formulation is o shell except for the lack of auxiliary elds. Since the elds are just a vector and a spinor, the Lagrangian consists of just that of superYang-Mills coupled to a spinor in the adjoint representation of the Yang-Mills group. Upon dimensional reduction, the vector produces some scalars. For example, the D=10 theory has an SO(9,1) symmetry, which reduces in D=4 to the SO(3,1) SO(6) subgroup. The SO(6) symmetry of the 6 attened dimensions is the SU(4) symmetry of the N=4 supersymmetries. Under this reduction, the vector becomes 10 !(4;1) (1;6), namely a 4-vector and scalars that form a 6 of SU(4), while the spinor becomes 16!(4;4 ) ,a4 Ds p i n o rt h a ti sa l s oa4o fS U ( 4 )( l i k et h es u p e r s y m m e t r yg e n e r a t o r s ) . Although (or, or ) matrices are necessary in D=10, in D=4 we can convert to spinor notation for both SO(3,1) (=SL(2,C)) and SO(6) (=SU(4)). Thus vectors and the Minkowski metric reduce as V a!(V.;Vij); (V.)*V. =V. ; (Vij)*Vij=1 2ijklVkl ab!(CC.. ;1 4ijkl):V
W!V.W.+1 2VijWij while spinors and Pauli matrices reduce as !1p 2 i  i. ;a Va!iCVijj iV .  i jV.iC. .Vij V=!V.1 2( ii.i i.)+1 2i(Vij ij +Vij i.j.) The two terms in the 10D Lagrangian then reduce as 1 8F2!1 8F2+1 8[r;ij]
[r;ij]1 32[ij;kl][ij;kl] a [ira; ]!  i.[ir.; i]+1 2i( i[ij; j ]+ i.[ij; j.]) Excercise XC6.1 Looking at the SU(3) subgroup of SU(4), decompose the states of N=4 super Yang-Mills into those of N=3. (Use the analysis of subsection IIC5 to count states, in SU(N) representations.) Do the same to decompose N=4 into N=2 C. HIGHER DIMENSIONS 589 super Yang-Mills plus scalar multiplet, this time using the SU(2) SU(2) sub- group for which 4 !(1 2;0)(0;1 2) (i.e.,i!(i;i0)). This is another way of understanding where the second SU(2) of the scalar multiplet comes from. Excercise XC6.2 Derive the commutation relations of the N=4 Yang-Mills covariant derivativesof subsection IVC7 by dimensional reduction of those for 10D N=1 given inthe previous subsection. (Don't forget the scalars come from the componentsof the vector covariant derivative in the extra dimensions.) Dimensional reduction of (super)gravity is an example of the comparative sim- plicity of the vierbein (covariant derivative) formalism vs. the metric or even inversevierbein (dierential form) formalisms. The reason in this case is that gravity is treated like Yang-Mills theory, and gauge vectors result from reducing the graviton. This is seen most easily from comparison of the coordinate transformation laws: e am=n@neamean@nm ema=n@nema+ena@mn gmn=p@pgmn+gp(m@n)p Fixing the index m=1o nmto get the gauge transformations of an Abelian vector resulting from reduction from one extra dimension, and setting @1=0w h e na c t i n g on any eld as the denition of reduction, we see the identication (in an appropriategauge for the SO(D,1)/SO(D 1,1) generators M 1a) eam! m1 aeamAa 10 ! whereAtransforms in the usual way for a gauge vector, and is an additional scalar. A more transparent way to write this is as m@m;eaeam@m;ea=[;ea] !+1@1;ea!(ea+Aa@1; @1) which makes it clear that reduction has simply U(1)-covariantized the gauge param- eter, transformation, and eld, where @1is the U(1) generator. (Under reduction all elds are U(1) neutral, but nonabelian or charged elds can be obtained by more complicated compactications.) On the other hand, the reduction of ema,b e i n gt h e inverse ofeam,a n dgmn, being the square of that, yields nonlinear reductions, and 590 X. SUPERGRAVITY the U(1) covariantization is not manifest. (In particular, in the metric formalism the metric, and thus the U(1) vector, does not even appear in the covariant derivative,except through its derivatives.) Excercise XC6.3 Let's work out the details of this simple example, reduction of pure gravity from one extra dimension: aFind the reduction of c abcby examining the commutators of the reduced ea.(Fabcomes out directly.) Using the expression of the Lagrangian in terms of the c's from excercise IXA5.2, nd the reduced action, including a cosmological term. (Drop theR dx1. We can think of this as compactication on a circle, independence from x1yielding a constant factor upon integration, which can be absorbed.) bThe scalar appears in a funny way, seen previously in subsection IXB5. Rather than eld redenitions, it is more convenient to reintroduce local scale invari- ance (after the reduction), as in subsection IXB5, introducing the dilaton . Then make a simple redenition that replaces andwith the \canonical" elds.( T h eF2and cosmological terms then appear with powers of .) Similar results can be obtained for supergravity, but the results are more com- plicated, because the scalars (which appear for N >3) appear in nonlinear models. Furthermore, although these models can be constructed by the coset method dis- cussed in subsection IVA3, the coset space G/H is noncompact, because the group G is noncompact, although the subgroup H is compact. This is a consequence of the fact that the \compensating" scalars of the group H=U(N) (or SU(8) for N=8) appear with the wrong-sign kinetic term (as the dilaton even in ordinary gravity). Thus, con-formal supergravity is coupled to \matter" with scalars in the adjoint representation of the noncompact group G, while gauging away the compensating scalars leaves the physical scalars of the coset space G/H. A simpler analog is N=1 supergravity coupled to a scalar multiplet (see subsection XB7). This is the same as conformal supergravity coupled to the matter action , which has a symmetry G=U(1,1), while N=1 supergravity has a gauge group U(1). Including Weyl scale invariance GL(1), the physical scalars then inhabit the coset space U(1,1)/U(1) GL(1)=SU(1,1)/GL(1). In the case of extended supergravity, the group G can be found by noting that the physical scalars parametrizing G/H form the representation  ijkl(totally anti- symmetric, and complex conjugate) of the group H. We then look for the group G whose adjoint representation transforms under the H subgroup as these scalars + ad- joint of H. We can also determine G by dening group generators for G as Mijfor H, C. HIGHER DIMENSIONS 591 andMijkl(and hermitian conjugate Mijkl) for G/H, and write commutation relations consistent with covariance under H. For N=8 we also have Mijkl=1 4!ijklmnpqMmnpq (and the same for the corresponding physical scalars). The result for the coset spaceG/H is N=4:SU(4) SU(1;1)=U(4) =SU(1;1)=U(1) 5:SU(5;1)=U(5) 6:SO*(12)=U(6) 8:E 7(+7)=SU(8) whereE7(+7) is a noncompact form (Wick rotation) of the exceptional group E7. An additional complication is that the vectors represent the full H symmetry only on shell. For example, for N=2 we have a single vector, as in electromagnetism.Maxwell's equations without sources have a U(1) symmetry, \S-duality", that trans-formsf by a phase (and f.. by the opposite), that mixes the eld equations with the Bianchi identities. With sources, it mixes electric and magnetic charge, since it mixes electric and magnetic elds. So, in general we must introduce both electricand magnetic potentials for each vector. Furthermore, for N=6 the vectors appearas bothf ij andfijklmn (one extra vector). (For N=8, the two are related by an tensor, just as and .) In this version of extended supergravity, all the vectors are Abelian. There is also a version where they gauge SO(N), but that theory has acosmological constant. REFERENCES 1 Georgi, loc. cit. (IB): Dirac and irreducible spinors for SO(n). 2Kugo and Townsend, loc. cit. (IIC): Dirac and irreducible spinors for SO(n +,n). 3Siegel, loc. cit. : spinors and ;;matrices in SO(n +,n). 4L. Brink, J.H. Schwarz, and J. Scherk, Nucl. Phys. B121 (1977) 77: D=10 super Yang-Mills. 5E. Cremmer, B. Julia, and J. Scherk, Phys. Lett. 76B (1978) 409: D=11 supergravity. 6W. Nahm, Nucl. Phys. B135 (1978) 149: arbitrary free representations of supersymmetry in arbitrary dimensions. 7W. Siegel, Phys. Lett. 80B (1979) 220: higher-dimensional superspace. 8E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48, Nucl. Phys. B159 (1979) 141: N=8 supergravity. 592 X. SUPERGRAVITY 9L. Brink and P. Howe, Phys. Lett. 88B (1979) 268; W. Siegel, Nucl. Phys. B177 (1981) 325: N=8 superspace formulation of supergravity. 10E. Cremmer and S. Ferrara, Phys. Lett. 91B (1980) 61; L. Brink and P. Howe, Phys. Lett. 91B (1980) 384: superspace formulation of D=11 supergravity. 11Th. Kaluza, Sitz. Preuss. Akad. Wiss. Berlin, Math.-phys. Kl. (1921) 966; Klein, loc. cit. (IIB); H. Mandel, Z. Phys. 39(1926) 136: dimensional reduction/compactication of general relativity. A. SCATTERING 593 XI. STRINGS There are three areas of application of QCD, as dened by the region of momen- tum space they address: (1) One is perturbative QCD, which applies to large relative, \transverse" velocity of (some of) the constituents of the hadrons. In this approach such an amplitude is divided into a half consisting of high-energy, asymptotically-freepartons, which is calculated perturbatively in the gauge coupling, and a half con- sisting of low-energy, conned partons, which is nonperturbative, and therefore not calculated. (2) Another area deals with the low-energy behavior of QCD | with properties of the vacuum (e.g., broken chiral symmetry), or the lowest-mass hadrons, scattering at small relative velocities. This approach is nonperturbative with respect to the gauge coupling, and instead perturbs in derivatives, as in rst-quantized JWKB. Themethods used include instantons, lattice QCD, current algebra, dispersion relations, nonlinearmodels, and duality. This low-energy behavior really says nothing about connement, just as the low-energy states of the hydrogen atom tell us nothing about ionization. A closely related problem is that the nonperturbative information about QCD that comes from (electromagnetic-type) duality considerations, which relates \weak"coupling to \strong" coupling as g$1=g, is not really relating quark-gluon physics to hadronic physics, but is relating quark-gluon physics to monopole physics; i.e., it relates a description of weakly coupled \electric" color charges to a similar lookingtheory of weakly coupled \magnetic" color charges. Thus, the dual theory, being formally of the same type as the original, except for a relabeling of what is called \electric" and what is called \magnetic", does not give anything that looks any more like hadrons, or make it any easier to calculate. (3) The one nonperturbative approach that does deal with high (hadron) energies is string theory: It incorporates hadrons of arbitrarily high mass, and studies their scattering at high energies. It also shows that stringy (hadron-like) behavior is a characteristic of QCD coupling g1, whileg0o r1have non-stringy (parton- like) behavior: String perturbation expands in G=lng,n o tgnor 1=g; duality is the symmetryG$G. Furthermore, this Gis the coupling that denes the free string, i.e., how partons bind to form strings. The coupling that determines how hadrons couple to each other is 1 =N c, as described topologically in subsection VC9. (However, the relation of duality to the 1 =Ncexpansion is unclear, since duality has been studied so far only in relation to theories where the group is spontaneously broken to U(1), 594 XI. STRINGS so eectively Nc= 1, or with respect to instantons, which are always dened for SU(2) subgroups, so eectively Nc= 2. Thus, it has not been possible to apply such duality arguments simultaneously with a 1 =Ncanalysis. Similarly, since both these duality approaches deal only with low-energy behavior, they are dicult to relate toconnement.) Most of the research eort on string theory has been directed toward models with \critical dimension" D 4 (10, 11, or 26): Attempts to relate to the real world involve choosing a solution of minimal energy with all but 4 dimensions \compactied", and considering perturbations about that solution. (The extra dimensions cannot be com-pletely eliminated without losing renormalizability.) Unfortunately, the ambiguity inchoice of such a solution results in the loss of predictability, which is what unicationand renormalizability are all about. However, historically the true usefulness of such string theories has been for the concepts and features of eld theory they have revealed: For example, supersymmetry (sections IIC and IVC, and chapter X), the Gervais-Neveu gauge (subsection VIB4),topological (1/N) expansion (subsection VIIC4), rst-quantized BRST approach togauge theory (chapter XII), and certain simplications in one-loop amplitudes wereall discovered through studies of 10- and 26-dimensional string theory, even thoughthey all are now understood more simply through ordinary eld theory. This is dueto the fact that string theories are so complex and restrictive that they require themost powerful techniques available. Clearly such strings are useful toy models forlearning about particle eld theories, and about general properties of string theorythat might lead to generalizations to include realistic 4-dimensional string theories.(In fact, the rst paper on string theory was written in 1747 by d'Alembert, and wasthe rst appearance of the wave equation and the d'Alembertian. Thus, eld theory, quantum mechanics, and special relativity can trace their origins to string theory.) In subsections IVB1 and VIIC4 we brie y discussed how hadrons are expected to arise as strings from QCD. In this chapter we analyze the dynamics of this mecha-nism. We begin by formulating the theory in terms of strings directly. Perturbativecalculations are performed using rst-quantized path integrals. The only experimen-tal evidence for strings is as a description of hadrons; to some extent the way that QCD leads to strings can be understood with similar rst-quantized methods, based on random lattices. A. SCATTERING 595 ::::::::::::::::::::::::: ::::::::::::::::::::::::: ::::::::::::::::::::::::: A. SCATTERING ::::::::::::::::::::::::: String theories are the only known theories that exhibit (S-matrix) duality. Unlike other S-matrix approaches, they provide an explicit perturbative calculational scheme,like eld theory, and string theory can be formulated as a eld theory. Also like eldtheory, string theory has consistency conditions at the classical and quantum levels,related to gauge invariance and renormalizability. The dening concept of the string is that it is a two-dimensional object: Just as the particle is dened as a point object whose trajectory through spacetime is one-dimensional (a worldline), the string has as its trajectory a two-dimensional surface,the \worldsheet". This leads to a much simpler picture of interactions: For interact-ing particles, the geometric picture of a worldline becomes a graph, whose geometryis singular at the interaction points. For interacting strings, we have instead a world-sheet with nontrivial topology: sphere, disk, torus (doughnut), etc. For example, atree graph now looks more like a real tree, in that the branches now have thickness,and they join smoothly to the rest of the tree. There are two types of free strings:open (two ends) and closed (no boundary). Their worldsheets are a rectangle and atube (cylinder). t s st 1. Regge theory In principle there is no dierence between a fundamental state and a bound state: We can always write an action with every state represented by an independenteld. Of course, such an action might not be renormalizable, but that seems moreof a formal distinction. A more physical one is based on the qualitative propertythat bound states have radial and other excitations with related properties, whilefundamental states are more unique. Regge theory is an approach to bound states that treats them as fundamental. A family of states that are dierent excitations of the same ground state is treated as asingle entity. Although basically an approach based on fundamental properties of theS-matrix, when combined with perturbation theory it leads directly to string theory. 596 XI. STRINGS A quantitative denition of this concept follows from a generalization of a concept seen in perturbative eld theory. In amplitudes following from Feynman diagramsthe nature of intermediate states can be seen from the momentum-space behavior: Single-particle states appear as poles (in the sense of complex analysis) in some mo- mentum invariants, 1 =(p 2+m2), where this pis the sum of some of the external momenta, representing the momentum of the internal state. (Any tree graph is a simple example.) Two-particle states appear as cuts in these invariants, where the branch point represents the state where the two particles are at rest with respect to one another, and the rest of the cut corresponds to arbitrary relative velocities. (For example, a one-loop propagator correction has a branch point at p2=(m1+m2)2 for intermediate particles of masses m1andm2.) Similar remarks apply to other multi-particle states. \Analytic S-matrix theory" was an attempt to formulate par- ticle physics in terms of the S-matrix by replacing the property of locality of the action with \maximal" analyticity of the S-matrix in momentum space. (Of course, unitarity and Poincar e invariance can be described easily in terms of the S-matrix; even analogs of renormalizability can be formulated in terms of certain properties of the high-energy behavior.) Unfortunately, the most general form of such nonanalyticbehavior (poles, cuts, etc.), as discovered from analyzing Feynman diagrams, proved to be too complicated to provide a practical method for dening a theory. Since Poincar e invariance means that not only momentum is conserved but also angular momentum, a natural next step was to consider the analytic behavior in that variable as well. This behavior is seen already in nonrelativistic theories; here we will approach the concept in a language most relevant to relativistic physics. The simplest example of \Regge behavior" is the 4-point S-matrix; this is the relativistic analog of a nonrelativistic particle in a potential. (We can think of an innitely massive second particle as producing the potential, or separate center-of-mass and relative coordinates for two nite-mass particles.) Also, the Feynman diagrams that appear in the nonrelativistic problem are \ladder diagrams": The sides of the ladder represent the two scattering particles, while the rungs represent a perturbation expansion forthe potential. It can be shown that such diagrams give the leading behavior of this amplitude at high energies. Here the appropriate high energy limit is dened in terms of the Mandelstam variables (see subsection IA4); by high energy we mean, e.g.,s!1 for xedt. The high-energy behavior of ladder diagrams can be shown to be of the form (in units of an appropriate mass) A(s;t)=kg 2[(t)]s(t); (t)=a+g2b(t) A. SCATTERING 597 whereais a constant that describes the behavior of the tree graph, and b(t)i sd e - termined by the one-loop graph. (Both, and the constant k, are independent of g.) This amplitude takes a simple form under a modied type of \Sommerfeld-Watson transform": A(s;t)=IdJ 2i(J)sJf(J;t)=1X J=01 J!(s)Jf(J;t) The contour integral is taken as counterclockwise about the positive real axis to obtain the last form, where it picks up the poles of ( J) (see excercise VIIA2.3b), but can be deformed to surround the singularities of f. In this case, a pole at J=(t)i nf will reproduce our original ladder amplitude, while integrating it around the positive real axis gives f(J;t)=kg2 J(t))A (s;t)=k1X J=01 J!(s)Jg2 J(t) which shows that particles of spin Jcontribute simple poles in tto the amplitude at(t)=Jwhen(t) can be approximated as linear near that value. The spin of the intermediate particle follows from the sJfactor. (This is clear from examining a 4-point tree graph where the external lines are scalars and the internal line carries J indices, and must contract the momenta of its two ends. There are also contributionsof lower spins from traces.) Thus the \Regge trajectory" (t) determines not only the high-energy behavior of the amplitude (for negative t), but also the spins and masses of the bound states (for positive t): Looking at the graph for J=(t), there is a bound state of spin Jand massp twhenever the curve crosses an integer value ofJ. The contribution at nloops in perturbation theory to f(J;t) is a multiple pole (Ja)(n+1), which contributes to A(s;t) a term proportional to sa(lns)n. Unfortunately, for eld theories with a nite number of fundamental particles, the trajectories are rather boring, containing only a nite number of bound states. In cer- tain cases a trajectory may include one of the fundamental particles itself (\Reggeiza- tion"). Because of the usual infrared divergences, such calculations can be applied directly to S-matrix elements only for fundamental massive particles; for fundamental massless particles, as in conning theories (like QCD), these results require externallines to be o-shell, and some knowledge of the parton wave functions is needed. Regge behavior thus gives a measurable denition of connement: If the scattering amplitudes of color-singlet states (or color-singlet channels of o-shell amplitudes ofcolor-nonsinglet states) have linear trajectories, the constitutent color-nonsinglet par- ticles can be said to be \conned". On the other hand, if the Regge trajectory rises only to nite spin and then falls, as with the Higgs eect, then there is only \color 598 XI. STRINGS screening"; color-singlet states might not be observable, but we do not see the innite number of radial excitations characteristic of connement. Another possib ility is that arbitrarily high spin is reached at nite energy: This is characteristic of Coloumbbinding, and indicates that a new, \ionized" phase is reached above that energy. Experimentally, hadrons are observed to have Regge behavior with respect to both high-energy behavior and spectrum. However, those Regge trajectories are approximately linear, thus indicating an (near) innite number of bound states. Thelinearity of the trajectories can be shown to be related to the relative stability of these unstable particles (as compared to what is found in ladder approximations). This suggests a formulation of the theory of hadrons where the whole Regge trajectoryis treated as fundamental. It can be shown that in any such \Regge theory" based on a perturbation expansion where the \tree" graphs have only poles in the angular momentum J(whose accuracy is implied by the linearity of the observed trajectories), that the theory has a further property called ( s-t) \duality": This property states that the amplitude can be expressed as a sum of poles in either the sort\channel", rather than as a sum over both: A(s;t)=X nCn(t) ssn=X n~Cn(s) ttnS S= This holds even when the sets of particles exchanged in the two channels are dif- ferent, due to quantum numbers of the external states. This relation has also beenexperimentally veried (approximately). Explicit realizations of such \dual models" of the S-matrix in terms of rst- quantized systems are called \string theories". They explain the linearity of the Regge trajectories by the harmonic-oscillator structure of the string Hamiltonian, and the duality of the amplitudes by the conformal invariance (\stretchiness") of thestring worldsheet. 2. Classical mechanics We now consider string theory as derived by rst quantization. As for particles, the rst step is to study the classical mechanics, which determines the appropriate set of variables, the kinetic term of the eld theoretic action, some properties of the interactions, and some techniques useful for perturbation. Just as the simplestsuch action for the particle produces only the relatively uninteresting case of the scalar, the most obvious action for the string yields a model that is not only too simple, but quantum mechanically consistent only in 26 dimensions. However, this A. SCATTERING 599 toy model exhibits many relevant qualitative features, such as Regge behavior and duality. Later we'll consider the source of its problems by relating to four-dimensionalparticle theories. The simplest classical mechanics action for the string is a direct generalization of that for the massless scalar particle: For the Lagrangian form of this action we write S L=1 0Zd2 2pggmn1 2(@mXa)
(@nXb)ab whereXa(m) is the position in spacetime of a point at worldsheet coordinates m= (0;1)=(;),gmn(m) is the (inverse) worldsheet metric, and 0is a normalization constant, the string tension. It can also be associated with the at-space spacetime metricab; if we couple a spacetime metric, then its vacuum value can be taken as ab=0,w h e r e0is the gravitational coupling, as discussed in subsection IXA5. Excercise XIA2.1 Analyze the classical mechanics of the string by approximating by a set of discrete points, so X0()!Xn+1Xn, etc. Choose the gauge X0=. Show that the string then acts as a bunch of particles connected by springs, and nd all the usual spring properties: tension, speed of wave propagation, etc. A new feature of this action (compared to the particle's) is that it is (2D) Weyl scale invariant. This gauge invariance can be used to gauge away one component of the metric, in addition to the two that can be gauged away using 2D general coordinateinvariance. The net result is that the worldsheet metric can be completely gauged away (except for some bits at boundaries), just as for the particle. However, this same invariance prevents the addition of a worldsheet cosmological term: In the particle case, such a term was needed to introduce mass. Here, mass is introduced through the coecient 1 = 0of the (@X)2term: The same scale invariance that prevents use of a cosmological term also prevents this coecient from being absorbed into the denition of the worldsheet metric. Just as for the particle, the metric can be eliminated by its equation of motion, resulting in a more geometrical, but less useful, form of the action: In this case the equation of motion (\Virasoro constraints") (@mX)
(@nX)=1 2gmngpq(@pX)
(@qX) after taking the determinant of both sides, gives S=1 0Zd2 2p ~g; ~gmn=(@mX)
(@nX) 600 XI. STRINGS This is the area of the string in terms of the \induced" metric ~ gmn, analogously to the particle case. The induced metric measures length as usually measured in spacetime: dmdn~gmn=(dm@mX)
(dn@nX)=(dX)2 Equivalently, this action can be written in terms of the area element dXa^dXb: S=1 20Zq 1 2(dXa^dXb)2;d Xa^dXb=(d0@0X[a)(d1@1Xb]) For purposes of quantization, it's also useful to have the Hamiltonian form of the action. This also allows us to see how the Virasoro constraints generalize the Klein-Gordon equation, and then nd the BRST operator. By the usual methods of converting from Lagrangian to Hamiltonian, we nd SH=Zd2 2(. X
P+H);H=pg g111 2(0P2+01X02)+g01 g11X0
P where .=@0and0=@1. Various combinations of components of the worldsheet metric now appear explicitly as Lagrange multipliers. If we dene ^P()=1p 2(01=2P01=2X0)) [^P(+);^P()]=0 the constraints can be written as two independent sets ^P2 (). Excercise XIA2.2 Show that if we call gthe Lagrange multipliers for ^P2 (),t h e ni nc o n v e n i e n t local Lorentz and Weyl scale (but not coordinate) gauges we can write in a lightcone basis e=em@m=1p 2(@0g@1) while in another scale gauge we can write dxmem=1p 2(dx0gdx1) Excercise XIA2.3 Find the (equal- ) commutation relations [ ^P();^P()]. Show that the (semi- classical) commutation relations of the constraints ^P2 ()close. (Hint: Use the identityf(a)0(ab)=f(b)0(ab)f0(b)(ab).) Since 2D general coordinate (and even just Lorentz) invariance is no longer man- ifest, for some purposes we need to generalize this to a form that is rst-order withrespect to both andderivatives: S 1=1 0Zd2 2[(@mX)
Pm+(g)1=2gmn1 2Pm
Pn] A. SCATTERING 601 obviously reproduces SLafter eliminating Pm. Eliminating just P1gives a simpler way of deriving SH(withP0=0P). Since open strings have boundaries, the action implies boundary conditions, orig- inating from integration by parts when deriving the eld equations. In the last formof the action variation of the rst term gives, in addition to theR d 2terms (P)
@X and(X)
@Pfor the eld equations, a boundary termR dmmn(X)
Pn,w h e r e dmis a line integral along the boundary, and the mnpicks the component of Pm normal to the boundary. We thus have nmPm=0atboundaries wherenmis a vector normal to the boundary. From the constraint imposed by varying gmn, it then follows that (tmPm)2=0atboundaries wheretmis a vector tangent to the boundary (or any vector, for that matter). Since by the eld equations Pmgmn@nX, this means that the boundary is lightlike in spacetime: The ends of the string travel at the speed of light. 3. Gauges In direct analogy to the particle (subsection IIIB2), the two most useful gauges are the \conformal gauge", dened by completely xing the worldsheet metric, and the lightcone gauge, which is not manifestly globally covariant but is a complete xingof the residual gauge invariance of the conformal gauge. In the conformal gauge weset g mn=mn by using the 2 coordinate invariances and the 1 scale invariance to x the 3 compo- nents of the symmetric tensor gmn. The coordinate part of this gauge is essentially the temporal gauge g0m=0m, just as for the particle ( g00=v2=1 ) . A l s oa sf o rt h e particle, this gauge can't be xed everywhere (see also subsections IIIA5 and IIIC2),but the equation of motion from the metric is implied everywhere by imposing it at the just the boundaries in . In this gauge the equations of motion for Xare just the 2D Klein-Gordon equation, which is easy to solve in 2D lightcone coordinates: @ +@X=0)X=X(+)(+)+X()() 602 XI. STRINGS ( W eh a v eu s e d in place of for later convenience.) The constraints are then ^P2 ()(X0 ())2= 0. This directly relates to the form of 2D conformal transformations, which are innite-dimensional in D=2: ds2=2d+d)0+=f(+)(+);0=f()() The constraints are the generators of these conformal transformations. (As described in subsection IIIA5, the constraints generate the gauge transformations; the global transformations are those that preserve the temporal gauge.) For the lightcone gauge, we again x the (spacetime) +-components of the vari- ables, and solve the +-components of the equations of motion (found by varying the -components). Looking at the equations of motion rst, using the rst-order form of the action, 0=S Pm@mX++(g)1=2gmnP+n ) (g)1=2gmn=(A
B)1(mpApnqAqBmBn);Am=P+m;Bm=@mX+ (as seen, e.g., by using mnAn;Bmas a basis), and 0=S X@mP+m)d dZ d P+0=0 which identiesR d P+0as the conserved momentum p+,u pt oaf a c t o ro f2 0 (sincepis really the coecient of.xin the action, where X()=x+:::). Similarly, S=gmndetermines Pm, and thusX. We then choose as our main set of gauge conditions X+=k; P+0=k for some constant k, which explicitly determines , and determines up to a function of: An equivalent way to dene the lightcone in terms of an arbitrary spacelike coordinate 0is =k1Z 0d0P+0(0) which identies as the amount of momentum p+between that value of and=0 (at xed). We thus have that the length of the string (the range of ,n o tt h e physical length) is l=k1Z d P+0=20p+k1 We then need to x the location of = 0 as some function 0(): Since in this gauge @1P+1=0 A. SCATTERING 603 soP+1is also a function of just , we further x the gauge for by choosing P+1=0) (g)1=2gmn=mn Thus the lightcone gauge is a special case of the conformal gauge, after also xing scale gauge g=1. For the open string, this almost xes 0()a t=0 ,w h i c hw e can take as one boundary: The boundary condition for X+is now 0=n
@X+n0 since in this (and any conformal) gauge @mXmnPn. Thus the normal to the boundary must be in the direction, so the boundary is at constant . This means we have one constant left to x: =0atone boundary (open string ) This invariance was left because all our previous gauge conditions preserved global  translation. Unfortunately, there is no corresponding convenient gauge choice for the closed string, so there we leave just this one invariance. In summary, our complete set of ligtcone gauge conditions is now: gauge :X+=k; P+m=km 0; =0atone boundary (open string ) The lightcone action is now, in Hamiltonian form, Slc=Z d.xp++Zd 2h. XiPi1 2(0P2 i+01X02 i)i The only distinction between open and closed strings is the boundary condition (since closed strings by denition have no boundary). For closed strings we have only periodicity in (by denition of \closed"), while for open strings we have X0(;0) =X0(;l)=0 One consequence, as we just saw, is that closed strings have one residual gauge invari- ance in the lightcone gauge. These two strings can be made to resemble each other more closely by extending the open string to twice its length, dening Xfor negative by X(;)=X(;) This is the known as the \method of images": X(;) is identied with its mirror image in the axis,X(;). Then the two strings both satisfy periodic boundary conditions, while the open string has this one additional condition. We also choose k=20p+; =1(open) 1 2(closed ))l=  604 XI. STRINGS so the length of the closed string is 2 , while the open string has original length that has now been doubled to match the closed string. Our choice of \phase" in relating X for positive and negative for the open string automatically enforces the boundary conditionX0(;0) = 0 at one end of the string, while the condition X0(;)=0a t the other end is implied in the same way by the closed string \boundary condition"of periodicity, which can be written as X(;)=X(;). The picture is then that the open string is a closed string that has collapsed on itself, so that for half of therange ofX doubles back over the path it covered for the other half. Becausehas a nite range, Xcan always be expanded in Fourier modes in that variable; the boundary conditions slightly restrict the form of this expansion. We sawthat the equations of motion, being second-order in -derivatives, gave two modes for each initial state: a left-handed one and a right-handed one. We need to be a bitmore precise about the zero-modes (independent of ): We can separate them out as X(;)=x+2 0 lp+q 0 2[Y(+)(+)+Y()()];Z d Y ()=0 whereYcontains only nonzero-modes. (The normalization of p, conjugate to x,c o m e s from the.x
pterm in the Lagrangian.) Then xrepresents the \center of mass" of the string, and pits total momentum. Note that this implies X()aren't quite periodic: X()(+2)=X()()+20p Now the periodicity boundary conditions shared by open and closed strings imply Y()(+2)=Y()() while the extra boundary condition for the open string implies Y(+)()=Y()()=Y() allowing us to drop the subscript in that case. Thus, the closed string has twice as many modes as the open, except for the nonperiodic part, corresponding to the totalmomentum and average position. This is related to the interpretation that the openstring is a closed string with its two halves occupying the same path. This doublingalso shows up in the constraints: For the closed string we have ^P 2 (), while for the open string we can consider just ^P2 (+),s i n c e ^P2 ()()= ^P2 (+)(). In the lightcone gauge we solve these constraints for X, by integrating 0=^P2 (). X2 ()=. Xi ()2 k. X ()(. Y()+p 20p)2 A. SCATTERING 605 Excercise XIA3.1 Rederive the solution to the boundary conditions for the open string without usingX(;)=X(;) (and periodicity): The string, as originally, extends between boundaries at 0 and . This separation of zero-modes from nonzero-modes also allows us to nd the spin and mass of the string: In any conformal gauge, 0=p2+M2=1 02Z= 0d 21 2(. X2+X02))M2=1 20X Zd 2. Y2 () Jab=x[apb]+Sab=1 0Z= 0d 2X[a. Xb])Sab=X Zd 2Ya (). Yb () (using the Hermitian form of the Lorentz generators, for classical purposes), where for the open string we can replace X Z 0d 2!Z2 0d 2;Y ()!Y For the lightcone gauge we then have the gauge condition to determine X+and Virasoro constraints to determine X: Y+ ()=0; 0p+q 1 20. Y ()=1 20p+ 0pi+q 1 20. Yi ()2 Excercise XIA3.2 Consider gauge xing in the temporal gauge, replacing X+withX0.T h e classical interpretation is now simpler, since andX0can now be identied with the usual time. Everything is similar except that the Virasoro constraints can't be solved (e.g., for X1) in general without square roots. aShow that some 3D solutions (2 space, 1 time) for the open string are given by 1p 2(. Y1i. Y2)()=cein; for nonzero integer n. (Without loss of generality, we can choose creal and positive.) Find the mass (energy) and spin as M=cp 0;S12=c2 n=0 nM2 FindXexplicitly, and show it describes an \n-fold spinning rod". bShow that the above solution can be generalized to closed strings by using two suchY's, and xing the relative magnitude of the two c's. Consider the special cases where n=n+. Find the explicit masses, spins, and X's, and show that one describes another n-fold spinning rod, while the other is an \n-fold oscillating ring". 606 XI. STRINGS 4. Quantum mechanics The more interesting features of the string don't appear until quantization. In particular, we can already see at the free level the discrete mass spectrum character-istic of Regge theory, or of bound states in general. Canonical quantization is simplest in the lightcone gauge. As for particles, canon- ical quantization is convenient only in mechanics (rst quantization), not eld theory (second quantization). As can be seen from the lightcone action, the Hamiltonianis part of the constraints: For the spinless particle, we had only the constraint p 2+m2= 0, which became E=Hin the lightcone gauge X+=p+after iden- tifying the lightcone \energy" E=p+pand its Hamiltonian H=1 2(p2 i+m2). (See subsection IIIB2.) The string Hamiltonian can be rewritten conveniently in terms of ^P. Since the closed string is eectively just a doubling of the open string, we treat the open string rst. The Hamiltonian is simply H=Z d 21 2^P2 i where ^P=^P(+). Since we have chosen X+=20p+,w eh a v eE=20p+p. To identify the individual particle states, we Fourier expand the worldsheet vari- ables in. As for the particle, we can work at = 0, since all the dynamics is contained in the constraints. Equivalently, from the nonrelativistic view of the light- cone formalism, we can work in the Schr odinger picture where the dependence is in the wave function instead of the operators. We expand as ^P()=1X n=1~anein; ~a0=p 20p; ~an=~any From the canonical commutation relations for PandX, [^Pi(1);^Pj(2)] =2i0(21)ij we have [~aim;~ajn]=mm+n;0ij as well as the usual [ pi;xj]=iij, and thus can relate the modes to the usual harmonic oscillator creation and annihilation operators: ~an=pnan; ~an=pnany) [am;any]=mn for positive n. After normal ordering, we nd for the Hamiltonian H=0p2 i+N0;N =1X n=1nainyain A. SCATTERING 607 )HE=0(p2 a+M2);M2=01(N0) for some constant 0. From the expression for the mass in terms of the number operator N,w es e e that thenth oscillator ainyraises the mass-squared of the ground state j0ibyn (and similarly for multiple applications of these oscillators). For any given mass, the highest-spin state is the symmetric, traceless tensor part of multiple ai1y's acting on j0i: This describes the leading Regge trajectory, with spins j=0M2+0 Let's look rst at the rst excited level, obtained by acting on the scalar ground statej0iwith the lowest-mass oscillators ai1y. Clearly this describes a (lightcone) transverse vector, with no St uckelberg scalar for describing a massive vector. (I.e., it has only D2 components, not the D 1 necessary for a massive vector.) Thus this state describes a massless vector, so 0=1 The ground state is then a scalar tachyon with M2=01. For any given level past the rst excited level, one can check explicitly that the states coming from the various oscillators include the necessary St uckelberg elds. For example, at the second excited level, ai1yaj1ycontains a traceless, symmetric tensor and a scalar (coming from the trace), while ai2yis a vector; they combine to describe a massive tensor. The proof that this works to all mass levels is closure of the Poincar e algebra quantum mechanically: The only nontrivial commutator is [ Ji;Jj] = 0, since only Jiis higher than quadratic (cubic) in oscillators (from the form of XandPafter solving constraints), so normal-ordering ambiguities lead to more than just constant terms. For reasons to be explained in chapter XII, the algebra(ic computations) in calculatingthis commutator is the same in any rst-quantized theory as the rst-quantized BRST algebra. Of course, the proof of closure is already the same in principle because both algebras are a consequence of the constraints, the conformal algebra. Thus, anyanomaly must show up in the conformal algebra itself, which will be considered in the following subsection. The closed string works similarly to the open, but with two sets of harmonic oscillators, but with p (+)=p()=1 2p In that case we nd M2=201(N(+)+N()2) 608 XI. STRINGS whereN(+)andN()are the number operators for the two independent sets of os- cillators. In the lightcone gauge the closed string has the residual gauge invariancegenerated byR d X 0
=X ; this gives the residual constraint N(+)=N() The closed-string states are thus the direct product of two open-string states of the same mass: For example, the ground state is a scalar tachyon with M2=201, while the rst excited states are massless ones from the product of two vectors | ascalar, an antisymmetric tensor, and a symmetric, traceless tensor. The leading Reggetrajectory consists of states created with equal numbers of a i1(+)y's andai1()y's, with j=1 20M2+2 In summary, the leading trajectory for open or closed string is given by j=0M2+1  Covariant quantization of the string can be performed in several ways: One is to use the OSp methods of chapter XII, as applied to the Lorentz generators derivedfrom the lightcone analysis. Another is to use the usual BRST of subsection VIA, asapplied to gravity in subsection IXB1, treating the mechanics of the string as a 2Deld theory. For the case of the conformal gauge, introducing ghosts C mcorresponding to the gauge parameters, and antighosts Bmnpaired with the Lagrange multipliers of the gauge conditions (Nakanishi-Lautrup elds), we nd the ghost action pggmn=mn;pggmn=r(mn)gmngpqrpq )Lg=B++rC+Br+C+ where in the last step we have introduced a background \zweibein" for applications such as the background eld gauge, or geometries that do not admit the conformalgauge globally, and attened the indices on the ghosts so the tracelessness of B(which follows from that of p ggmn) can be solved explicitly. (The conformal-gauge Weyl and Lorentz gauge conditions do not involve derivatives in the transformation lawsof the gauge-xing conditions, so the ghosts are just algebraic.) A. SCATTERING 609 5. Anomaly Closure of the BRST algebra, or lightcone Poincar e algebra, is nontrivial because of the innite summations over osc illators, or equivalently because of integration over the two-dimensional \momentum" (which is quantized as mode number in the di- rection because of the nite extent of ). As usual, BRST invariance is equivalent to gauge invariance, and we can check for anomalies in the usual way, now appliedto the 2D \eld theory" corresponding to the mechanics of the string. Because ofconformal invariance the boundaries of the worldsheet are irrelevant; we can confor-mally transform them to innity; then the anomaly calculations are similar to thoseapplied to the Schwinger model in subsection VIIIA8. (In particular, see excercise VI-IIA8.1.) The gauge invariances in question are coordinate invariance and local scaleinvariance, whose preservation is the vanishing of the divergence and trace of theenergy-momentum tensor. As seen from our analysis for the Schwinger model, thisimplies that the quantum corrections to the energy-momentum tensor must them-selves vanish when the external elds are restricted to gravity only. The calculationagain involves one-loop propagator corrections; performed in position space, we getthe product of two propagators between the same two points, with various numbersof derivatives acting on either end of either propagator. The classical energy-momentum tensors can be found by various methods: A straightforward way is to use the methods of chapter IX. A simpler method is to notethat the local Lorentz group in D=2 is Abelian (SO(1,1)=GL(1)), and the coupling ofthe Lorentz connection is proportional to the \spin" (number of + minus indices, as used in the 2D spinor notation of subsections VIIIA7-8). This immediately tellsus the coupling of the \zweibein" from the action for terms linear in derivatives: iL= wr 1w)iT++=1 2 w$ @+ 1w+(w1 2)@+( w 1w) where wis any eld with wthe number of + vector indices minus (half the number of spinor indices). We have written the most general terms with one derivative. (Itis@ +to get the right Lorentz weight for T++, but also because @vanishes by the free eld equation.) The rst term is the \zweibein" term, the second is the Lorentzconnection term; the coecient of the second term is xed by considering the casesw= 0 (scalar, so all derivatives act on it) and 1 w= 0. (As a check, the second term vanishes by antisymmetry for w=1w=1=2.) For convenience we rewrite this result as iT ++=w w@+ 1w+(w1)(@+ w) 1w 610 XI. STRINGS The same result holds for Tfrom ar+action. This includes the classical string action P+
P+P+
rX+P
r+X as well as the conformal-gauge ghost action B++rC+Br+C+ (The factor of e1can be absorbed into the elds by a local scale transformation; it's irrelevant for dening T.) The calculation for the matrix element for two T++'s (i.e., for two hbackground elds) is now simple: The propagators go as 1 =(xx0), and the various directions of derivatives from the vertices give either both hitting the same propagator or one hitting each, giving either ( @x1)2=x4orx1@2x1=2x4. The matrix element is then proportional to [w2+(w1)2]+2 [ 2w(w1)] = 6(w1 2)21 2 with the usual extra minus sign in the fermionic case. (As a check, this result is consistent with bosonization: A real, bosonic scalar gives the same result as a complex, fermionic spinor.) Taking (spacetime dimension) D w=0X's and2( f e r m i o n i c ) w= 2 ghosts (actually one with w= 2 and the conjugate with w=1), this adds up to D26 = 0)D=2 6 for anomaly cancelation. Thus, we have two conditions on this string that make it unsuitable for describing mesons: unphysical intercept 0for the leading Regge trajectory (massless particles) and unphysical spacetime dimension D. Excercise XIA5.1 In the derivation of the anomaly for Xused for anomaly cancelation, the eect of theP2term was neglected. Using the result for the hT++term from P+
rX, eliminate Pfrom the action, and redo the anomaly calculation in terms of this form of SL. (Hint: Show each step in the above calculation is still correct, even though PandXaren't independent.) A. SCATTERING 611 6. Tree amplitudes Unlike particles, Feynman diagrams for strings can be treated by rst-quantized methods for arbitrary loops. The basic idea is that interacting strings are just stringswith nontrivial geometries: For example, while an open-string propagator can be described by a rectangle, an open-string tree graph can be described by a rectangle that has parallel slits cut from two opposite ends of the rectangle part-way into theinterior; this describes initial strings that join and split at their ends (interactions). This is the lightcone picture of interactions, where conservation of total p +means conservation of the sum of the lengths of the strings. This corresponds to the choicek= 1 in the language of subsection XIA3, since the worldsheet coordinates must be chosen consistently over the whole worldsheet: X +=)l=20p+ More general conformal gauges are dened by conformal transformations of this con- guration: For example, the boundary of this slit rectangle can be transformed to a single straight line by the usual methods of complex analysis, so the worldsheet be-comes simply a half-plane. (For the innite rectangle, relevant for asymptotic states, the transformation is =Pp + rln(zZr).) Then even the geometry is irrelevant; all that matters is the topology, which tells how many loops the diagram has. (See the discussion of subsection VIIC4.) First-quantized path integrals are then the easiest way to calculate arbitrary S-matrix elements in string theory. However, the calculations still can be quite com- plicated (as expected from a theory with an innite number of one-particle states), so here we will consider just the tree-level scattering of ground states, which is suf- cient to illustrate the qualitative features. We can start from a gauge where thestring is an innite strip (a rectangle of innite length but nite width), with all but two of the external states associated with points on one side of the strip, the remaining two states being at the ends at innity. This is equivalent to a picture ofa propagator in an external eld, with all but two of the external states associated with the external eld. Similar calculations are possible for particles, but give only a single graph; for strings this gives the only graph, since dierent cyclic orderings 612 XI. STRINGS are related by conformal transformations. (We are restricted to cyclic orderings by group theory, as for the 1/N expansion for particles.) This method can be used ineither the lightcone gauge or Lorentz covariant conformal gauge. In the latter case,we integrate over the relative position of the vertices on the end of the string; thisis equivalent to integratingR 1 0d eH=1=Hfor the propagators between vertices. The result is the usual particle type expression VH1V:::H1V. The ground-state vertex of external momentum kis simplyeik
X(T;0), where the vertex is positioned at the end of the string at =0 . Since the functional integral is Gaussian, all we need to evaluate it is the two- dimensional propagator, subject to the boundary condition X0= 0. As for 2D elec- trostatic problems, we simplify the problem by a conformal transformation from theinnite strip to the half-plane: z=e  where=iis the complex (Wick-rotated worldsheet) coordinate for the strip. The bottom of the strip ( = 0, the real axis) is mapped to the positive real axis, while the top of the strip ( =) is mapped to the negative real axis; the interior is mapped to the upper-half complex plane. The 2D propagator (in worldsheet coordinate space,needed for the explicit =T rdependence of the vertices) now follows from that for the whole plane by the method of images: G(z;z0)=2(zz0))G(z;z0)=1 2(lnjzz0j+lnjzz0*j) (Note that we have changed normalization from our earlier discussions of general dimensions.) Thus, the N-point amplitude is given by the path integral AN=gN2limz1!1 zN!0N1Y r=3Zzr1 zr+1dzrz2 1Z DX eS S=iNX r=1kr
X(zr)+Zd2 1 2(@X)2 where we have chosen units 2 0=1 . ( S oH=1 2p2+N1.) Since the zrare the values ofzfor the vertices, we integrate over zrfrom 0 to 1, but zr>zr+1;z 1=1;z 2=1;zN=0 While only z3;:::;zN1are integrated over, and vertices are inserted originally at only z2;:::;zN1, we have inserted two extra vertices at the ends of the string at z1andzN A. SCATTERING 613 (i.e.,=1), to excite the true ground states from k= 0 to nonzero momentum (as in ordinary quantum mechanics, where eikxj0i=jkiforPjki=kjki.) The extra factor ofz2 1was inserted to amputate the external lines of these two vertices: jki=zeik
X(z)z1 2p2+N1j0i=eik
X(z)j0i hkj=h0jz(1 2p2+N1)eik
X(z)z=h0jeik
X(z)z2 The extra factors of zcome from the conformal transformation of this \vertex operator": Conformal invariance requires that d V ()=dz V (z), and it is actually V()=zV(z) that creates an appropriately normalized state acting on j0i. As usual, we perform the functional integral by completing the square, S=Z 1 2XX+iJX!Z 1 2J1J=1 2ZZ JGJ HereJis a density in terms of krand afunction; we can thus use its integration to dene it (rather then bothering with its explicit form in terms of functions in or z):Z J
X=X rkr
X(zr))ZZ J
GJ=X r;skr
ksG(zr;zs) (using also the fact that XandGare scalars). We drop the innite terms coming from connecting a vertex to itself with a propagator: This is a \renormalization" ofthe vertex, called \normal ordering" in the operator approach. (It can also be treatedin a more careful way by taking the vertices to correspond to nite-width strings, as they would in the lightcone approach, and taking the limit where their widths vanish.) The result of the functional integral is then A N=gN2limz1!1 zN!0N1Y r=3Zzr1 zr+1dzrz2 1Y 1r<sN(zrzs)kr
ks =gN2N1Y r=3Zzr1 zr+1dzrY 2r<sN(zrzs)kr
ks where we have used the ground-state mass-shell condition k2=2f o rk1. The simplest case is the four-point function, A4=g2Z1 0dz z(s)1(1z)(t)1 (s)=1 2s+1;s =(k3+k4)2;t =(k2+k3)2 614 XI. STRINGS which we recognize as the Beta function (see subsection VIIA2) A4=g2B[(s);(t)] =g2[(s)][(t)] [(s)(t)] Duality of this amplitude follows from the fact that it can be written as a sum of poles in either the sortchannel (see excercise XIA6.1 below): A4=1X j=0[(t)+j][(t)+j1]


[(t)+1 ] j!1 j(s) where the pole at j=(s)h a sm a x i m u ms p i n j, as indicated by the residue of order tj. Regge behavior is also found in the appropriate limit (see excercise XIA6.2 below): lim s!1 tf i x e dA4=g2[(t)][(s)](t) Excercise XIA6.1 Derive the pole structure of the 4-point string amplitude by Taylor expandingthe integrand of the Beta function in z. Excercise XIA6.2 Use the Stirling approximation (see excercise VIIC2.1) to derive the Regge limit of the 4-point string amplitude. Show that the same result can beobtained directly from the integral (Beta function) representation of the am- plitude, where the main contribution comes from znear 1. Excercise XIA6.3 Show the Regge limit can be obtained from the Sommerfeld-Watson transformof subsection XIA1. (Hint: The Beta function is a sum of Regge trajectories.) On the other hand, in the correspsonding limit with xed angle (i.e., xed t=s; again using the Stirling approximation), lim s!1 f i x e dA4ef(cos  )(s) cos 1+2t s;ft slns t +u slns u This limit corresponds to that treated in perturbative QCD, with large transverse en- ergies. However, this Gaussian behavior in the momentum diers from the power-law behavior (with logarithmic corrections) found in QCD (see subsection XIC3 below). Similar methods can be used for calculating closed string diagrams: The vertices are then located anywhere on the worldshseet instead of just on the boundary, so there A. SCATTERING 615 are integrals over both zand z. Since the closed-string Hilbert space is the direct product of two open-string Hilbert spaces (except for momentum), for left- and right- handed modes, the z-zintegrands are products of two open-string integrands. There are also generalizations to strings with worldsheet fermions; the main dierences are supersymmetry and D=10 (instead of 26). == Because of the local scale invariance of string theory, dierent string graphs are distinguished by their topology rather than their geometry: Any two graphs that can be \stretched" into one another are equivalent. In particular, in any loop graph any hole or window can be pulled out so that it appears as a tadpole. The result is thatany graph is equivalent to a tree graph with insertions of some one-loop open- or closed-string tadpoles. However, this does not mean that any graph constructed with only open-string propagators and interactions can be expressed as an open-string treegraph with tadpole insertions: The one-loop open-string graph with two \half-twists" on the open-string propagators in the loop is equivalent to a tree graph with a closed- string intermediate state, as can be seen by stretching the surface, or by tracing theroutes of the boundaries. (For example, drawing this graph in a psuedo-planar way, as a at ring with external states connected to both the inner and outer edges, pulling the inner edge out of the plane reveals a closed string connecting the two edges.) This phenomenon is similar to 2D bosonization: A closed string can be represented as the \bound state" of two freeopen strings just as a massless scalar in D=2 can be represented as the bound state of two free massless spinors. = REFERENCES 1T. Regge, Nuo. Cim. 14(1959) 951, 18(1960) 947. 2R. Dolen, D. Horn, and C. Schmid, Phys. Rev. Lett. 19(1967) 402: duality. 616 XI. STRINGS 3G. Veneziano, Nuo. Cim. 57A (1968) 190; M. Suzuki, unpublished: birth of string theory (as dual models). 4K. Bardak ci and H. Ruegg, Phys. Rev. 181(1969) 1884; C.J. Goebel and B. Sakita, Phys. Rev. Lett. 22(1969) 257; Chan H.-M. and T.S. Tsun, Phys. Lett. 28B (1969) 485; Z. Koba and H.B. Nielsen, Nucl. Phys. B10 (1969) 633, 12(1969) 517: generalization of 4-point amplitude. 5M. Virasoro, Phys. Rev. D1(1970) 2933; I.M. Gelfand and D.B. Fuchs, Functs. Anal. Prilozhen 2(1968) 92: Virasoro constraints. 6D.J. Gross, A. Neveu, J. Scherk, and J.H. Schwarz, Phys. Lett. 31B (1970) 592; C. Lovelace, Phys. Lett. 34B (1971) 500; E. Cremmer and J. Scherk, Nucl. Phys. B50 (1972) 222: string loops as 1-loop tadpole insertions. 7Y. Nambu, Quark model and the factorization of the Veneziano amplitude, in Proc. of International conference on Symmetries and quark model , ed. R. Chaud, Wayne State U., June, 1969 (Gordon and Breach, 1970) p. 269;L. Susskind, Phys. Rev. D1(1970) 1182, Nuo. Cim. 69A (1970) 457; H.B. Nielsen, An almost physical interpretation of the integrand of the n-point Veneziano model, 15th international conference on high energy physics, Kiev, 1970: dual model as string, through mechanics. 8Y. Nambu, lectures at Copenhagen Symposium, 1970, unpublished; O. Hara, Prog. Theo. Phys. 46(1971) 1549; T. Goto, Prog. Theo. Phys. 46(1971) 1560; H. Noskowitz, unpublished: area action for string. 9P. Goddard, J. Goldstone, C. Rebbi, and C.B. Thorn, Nucl. Phys. B56 (1973) 109: lightcone quantization for string. 10P.A. Collins and R.W. Tucker, Phys. Lett. 64B (1976) 207; L. Brink, P. Di Vecchia, and P. Howe, Phys. Lett. 65B (1976) 471; S. Deser and B. Zumino, Phys. Lett. 65B (1976) 369: mechanics of string with worldsheet metric. 11A.M. Polyakov, Phys. Lett. 103B (1981) 207: worldsheet ghosts. 12C. Lovelace, Phys. Lett. 34B (1971) 500: discovery of D=26 requirement. 13D. Friedan, Introduction to Polyakov's string theory, in Recent advances in eld theory and statistical mechanics , eds. J.B. Zuber and R. Stora, proc. 1982 Les Houches summer school (Elsevier, 1984) p. 839: covariant vertex operators with ghosts. 14S. Mandelstam, Nucl. Phys. B64 (1973) 205: lightcone path integrals for interacting strings. 15O. Alvarez, Nucl. Phys. B216 (1983) 125: covariant path integrals for interacting strings. 16M . B .G r e e n ,J . H .S c h w a r z ,a n dE .W i t t e n , Superstring theory , 2 v. (Cambridge Uni- versity, 1987); A. SCATTERING 617 J. Polchinski, String theory , 2 v. (Cambridge University, 1998): comprehensive texts on strings. 17P.H. Frampton, Dual resonance models and string theories (World Scientic, 1986); Siegel, loc. cit. ; M. Kaku, Introduction to superstrings and M-theory , 2nd ed. (Springer-Verlag, 1999): other string texts. 18S. Fubini; G. Veneziano; V. Alessandrini, D. Amati, M. Le Bellac, and D. Olive; J.H.Schwarz; C. Rebbi; S. Mandelstam; Dual theory , ed. M. Jacob (North-Holland, 1974); J. Scherk, Rev. Mod. Phys. 47(1975) 123: old string reviews. 618 XI. STRINGS :::::::::::::::::::::::: :::::::::::::::::::::::: :::::::::::::::::::::::: B. SYMMETRIES :::::::::::::::::::::::: In the previous section we looked in detail at the simplest of the string models. In this section we examine some of the general properties of string theory, shared byall known models, but expected to apply also to more realistic strings. These featurescan be used for phenomenological applications of string theory, but also may help point to new generalizations. When string theory is used as a unied theory of gravity and other forces, the most interesting predictions are those for the \low-energy" (with respect to the Planckmass) part of the theory. Although the possible low-energy limits of known stringtheories have not all been explored, the indications are that there are only a few restrictions beyond the usual eld theoretic ones: (1) The dilaton counts the number of loops. (2) The spectrum of the closed string is given by the direct product oftwo open strings. (3) String theory has noncompact symmetries, \S-duality" and\T-duality", resulting from the amplitudes also having this direct-product structure.All these properties survive the low-energy limit. In the supersymmetric case, thelast property follows from the rst two. (However, the D=10 superstring is actuallya D=11 supermembrane nonperturbatively. Since the observed features of hadrons,as well as qualitative arguments from QCD, indicate stringy but not membrane-like behavior, we will abstract only the perturbative features of higher-dimensionalstrings.) 1. Massless spectrum The known string models all have massless particles. A string model with massless particles can be applied to hadrons only if masses are given to all these states throughthe Higgs mechanism or some other change in the vacuum. An alternative is to usesuch a model to describe fundamental massless particles (graviton, photon, gluons,neutrinos), although this would also require the usual Higgs of the Standard Modelfor generating masses for some particles (W, Z, quarks, charged leptons, Higgs). Inparticular, all known string models have a graviton, and there is no known method whereby this graviton would gain mass, so these models seem suited only for unied theories of gravity plus matter. For this purpose, the massive elds have little phe-nomenological interest. They might improve high-energy behavior, but only near thePlanck scale, which is eectively unobservable. Therefore, it is necessary to analyzethe massless subsector of such string theories to nd signs of fundamental strings innature. B. SYMMETRIES 619 The low-energy limits of the known string models follow simply from gauge in- variance once the spectrum is known, and the spectrum itself follows from simpleconsiderations. In particular, we saw in subsection XIA4 for the simplest model thatthe massless spectrum of the closed string was simply the direct product of that oftwo open strings. This is true for all known models, and is expected to be a generalfeature: It is a consequence of the fact that all free massless elds in D=2 propagateto either the left or right; the left-handed modes are associated with one open-stringfactor, the right-handed with the other. In general, the two factors may be from dierent open-string theories. The massless sector of the open string includes spin 1 and no higher. This is true for the known string models, and also is expected to be a general result, sinceotherwise the closed string would include massless states with spin higher than 2, forwhich no consistent interacting theory is known. Spin 1/2 leads to supersymmetry, as described in subsection XIB3 below; for now we limit ourselves to bosonic strings. For the spin-1 particles to obtain nontrivial self-interactions (Yang-Mills), we need to introduce internal symmetry. This is naturally introduced by associating indiceswith the ends of the open string (as if they were quarks), as applied to particleeld theory in subsection VC9 (but which originated in string theory). These \Chan- Paton factors" can also be associated with worldsheet variables that live only on string boundaries. As in the eld theory case, these indices are associated with orientationof the boundaries (arrows) only for U groups, not for SO or USp. (Exceptional groupscannot be described by Chan-Paton factors.) The bosonic string contains at least the graviton, a scalar (usually going by the misnomer of \dilaton"), and a pseudoscalar (the \axion", described by an antisym-metric-tensor gauge eld), and can contain additional vectors and scalars (if theopen strings had scalars in addition to the vector). This analysis can be performedcovariantly, but it is simpler to use a helicity or lightcone analysis. Then the helicitiesof the closed-string states are just the sums of those of the open strings: For theproduct of two vectors (the minimal case), (+11) (+11) = +2002 giving the graviton, scalar, and axion. Similarly, additional scalars for one open string give additional vectors for the closed, while additional scalars for both open stringsgive also additional scalars. Excercise XIB1.1 Make the same analysis in terms of covariant elds, both for the elds them- 620 XI. STRINGS selves and their gauge transformations. Note that the trace of the gravita- tional eld hab(determinant of the metric gmn) is missing. (It's unphysical, and can be found from the ghost sector, as explained in chapter XII.) 2. Reality and orientation In the case that the two open strings are the same, it is possible to restrict this direct product to its symmetric part. This eliminates the axion, but not the scalar.Thus, a massless scalar appears even in the simplest case. To understand this restriction better, we examine the discrete symmetries of the worldsheet. As in D=4, local, unitary, Poincar e invariant 2D eld theories are always CPT invariant. (In particular, CPT doesn't switch left- and right-handed modes,which dier in some string theories.) Thus, we can always impose invariance of the wave function/eld under the worldsheet CPT transformation worldsheet CPT : [X()] =  y[X()] where parity transforms !to preserve [0;] for the open string; for the closed string the is irrelevant because of periodicity and invariance under  translation. (As for the particle, we drop dependence. We have written only the Xcoordinate explicitly for simplicity; similar remarks apply to other coordinates, such as ghosts, with possible extra signs due to 2D Lorentz indices.) Hermitianconjugation for the open string (for the closed string the eld is not a matrix), instead of just complex conjugation (for C), simply switches the internal symmetry (Chan- Paton) factors associated with the left and right ends of the open string (matrixtransposition), as also required by parity. In particular, this implies that the matricesassociated with the Yang-Mills elds are hermitian, so the Yang-Mills group is unitary. In addition to this reality condition, if the 2D theory is also invariant under CP and T, it is also possible, though not necessarily required, to impose such a quantummechanical invariance under CP, and thus T: worldsheet T : [X()] =M*[X()]M 1, worldsheet CP : [X()] =MT[X()]M1 where the matrix \ M" is the Yang-Mills group metric for the open string (we drop the Mfor the closed string), either symmetric or antisymmetric depending on whether the group is orthogonal or symplectic; without imposing T and CP the group isjust unitary. (The T condition is the usual reality condition for the particle, as B. SYMMETRIES 621 discussed in subsection IA5.) Thus, all the classical groups are allowed (at least in the classical eld theory). Since imposing invariance of the states (not just the action) under CP and T makes it impossible to observe the left/right handedness of the worldsheet, such strings are \unoriented", as opposed to the \oriented" strings that satisfy just the CPT condition. Thus, orientability of the surface is directly related to orientability of its bo undaries (oriented for U, unoriented for SO or USp). Also, as in the particle eld theory, unorientability allows \twisted" worldsheets that are prohibited in the oriented case (because we can distinguish the \front" of the worldsheet from the \back"): This allows such exotic geometries as M obius strips and Klein bottles. Open strings produce closed ones as bound states (open and closed strings are parts of the same worldsheet with dierent boundaries); in theories of open and closed strings, they must be both oriented or both unoriented. Since the worldsheet-CP and -T switch lefty and righty modes, this invariance on the closed string results in the restriction introduced earlier, keeping only the symmetric part of the direct product. 3. Supergravity Again focusing on just the massless spectrum, we now look at the restrictions imposed by supersymmetry. The open string can also contain massless spin 1/2, but only if it is related by supersymmetry to its massless spin 1, since it leads to spin 3/2 in the closed string, and massless spin 3/2 is known to be inconsistent in an interacting theory unless related by supersymmetry to the graviton. Thus there are two possibilities for the massless sector of each open string: (1) vectors and scalars for an open bosonic string, or (2) vector multiplets (vectors, spinors, and scalars, all related by some number of supersymmetries) for an open superstring. From our analysis of subsection IIC5, there are furthermore 3 types of vector multiplets in D=4, corresponding to N=1,2, or 4 supersymmetries. This leads to four types of closed strings: (1) The bosonic string, from bosonic bosonic, was discussed in subsection XIB1. (2) The \heterotic" string, comes from bosonic super. It thus can have N=1,2, or 4 supersymmetries. (3) The \(Type II) superstring" comes from super super. The total number of its supersymmetries is the sum of those from the open strings: Depending on the type of supersymmetric open strings used, the superstring can have N=2,3,4,5,6, or 8 (in other words, anything greater than 1, since N=7 supersymmetry is equivalent to N=8, and 8 is the maximum for supergravity). (4) In the super case, if the left and right open strings are the same, we can impose symmetry as in the bosonic case (\Type I"). Then we may also include 622 XI. STRINGS the open strings in the spectrum: This symmetrization also identies the left and right supersymmetries, so then N=1,2 or 4, the same for open and closed states (so theycan be consistently coupled). The spectrum again can be analyzed by helicity: For example, for the N=1 het- erotic string, we have (1 1 21 21) (11) = (23 23 22)(1 2001 2) which is supergravity plus a scalar multiplet. As for the bosonic string, all supersym- metric closed strings include the scalar, again coming from vector vector. Excercise XIB3.1 Make the same analysis for the superstring. 4. T-duality Another symmetry of all known string models is \T-duality". It is closely related to the open open structure of closed string states, and thus expected to be a general property of string theory. We consider the simple bosonic model as an example.Including constant background elds, working in the conformal gauge for convenience,the Lagrangian is L=(@ +Xm)(@Xn)Mmn;Mmn=Gmn+Bmn where the curved indices now refer to spacetime, Gmnis the spacetime metric, and Bmnis an antisymmetric tensor gauge eld (\axion"). Writing the action in rst-order form L0=P+m@XmPm@+Xm+P+mPnMmn whereMmnis the inverse of Mmn,w ev a r yXinstead ofPto solve the eld equation @+Pm+@P+m=0)P+m=@+~Xm;Pm=@~Xm and substitute to nd the \dual" Lagrangian L00=(@+~Xm)(@~Xn)Mmn Thus the \duality transformation" from Xto~Xis an invariance of the theory, as long as we also transform the background: Xm!~Xm;Mmn!Mmn B. SYMMETRIES 623 Note that in at space ( Mmn=mn), using the Pequation of motion in L0,w eh a v e P+m=mn@+Xn;Pm=mn@Xn so duality just changes the sign of the right-handed modes ( @X=@~X) while leaving invariant the left-handed ones ( @+X=@+~X). (The treatment of the zero- modes is more tricky: We have ignored them by taking the background constant.) We can see this to lowest order in the background, since M!M1,t ol o w e s t order in perturbation about hMi=, changes the sign of the eld, corresponding to the fact that their vertex operators are linear in both left- and right-handed modes ((@+X)(@X)). However, in full nonlinearity, duality mixes the spacetime metric Gmn(X)a n da x i o n Bmn(X). This invariance can be generalized to a continuous O(D,D) symmetry by com- bining it with (global) Lorentz transformations. The above discrete symmetry is a kind of \parity" for this larger group: There are also \re ections" from performingthe duality on just one component of X m. The easiest way to see the full symmetry is in the Hamiltonian formalism, where it can be made manifest: We rst combine X0mand the canonical momentum Pminto an O(D,D) vector: ZM=(Pm;X0m) ) [ZM(1);ZN(2)] =i0(21)MN;MN=0n m m n0 where the O(D,D) metric MNis constant even in curved space. (We have abbreviated \1" for \1", etc.) The Virasoro constraints are then 1 2MNZMZN=1 2MMNZMZN=0 whereMis not only symmetric but also an element of the O(D,D) group: MMN=GmnGmpBpn BmpGpnGmnBmpGpqBqn =MNM=MP(M1)PQQN If the elds are constant in only d of the D dimensions, than the symmetry is re- duced to O(d,d); thus O(d,d) is a symmetry of the dimensionally reduced theorywith arbitrary elds. 624 XI. STRINGS 5. Dilaton We can extend the spectrum analysis o shell: The procedure (to be justied in chapter XII) includes the ghost and antighost (multiplets) for the vector (multiplet)as a doublet of the ghostly Sp(2) symmetry. The direct product of vector vector now clearly gives a traceless symmetric tensor (graviton), the corresponding trace (physical scalar), and an antisymmetric tensor (axion). In the direct product of the ghosts, the Sp(2) singlet gives the trace part of the metric tensor, which is the truedilaton. This dilaton (the determinant of the metric tensor in the nonlinear case)is required in gravity for constructing local actions (see subsection IXA7), but doesnot contain a physical degree of freedom. The physical polarizations of the gravitonare contained in the traceless (actually det=1) part of the metric, which describes the conformal part of gravity. The direct products involving ghosts also give Sp(2) nonsinglets, which are the ghosts of the massless sector of the closed string. BRSTtransformations (and thus gauge transformations) can also be obtained by this direct-product procedure. The natural coupling of background elds in the classical mechanics of the string re ects this direct-product structure, as seen in the previous subsection. This meansthat the background metric as we have dened it has as its determinant not the usualone, but that times a power of the physical scalar: It is a physical degree of freedom. T-duality mixes physical degrees of freedom with each other. If we try to construct a low-energy action for the massless elds of the bosonic string, it is not too dicult to nd a scalar invariant under T-duality to act as the Lagrangian. However, it is impossible to use the usual measureR dxp gbecauseg is not invariant under T-duality. This problem is solved by including the spacetimed i l a t o n e l d ( X): It couples to the string as S dil=Zd2 2pg1 2rln(X) where we denote the worldsheet curvature by r() (only in this subsection) to distin- guish it from the spacetime curvature R(x). (There are also boundary contributions: see excercise IXA7.3.) This term can also be expressed as a coupling to the world- sheet ghosts (according to the above arguments), allowing the worldsheet metric tobe completely xed by gauge transformations, as usual. Since there is no Xdependence of S dilfor constant dilaton eld (no @Xfactors, unlikeGandB), the constant dilaton is invariant under T-duality. Furthermore, since it couples to the worldsheet curvature, which counts the number of loops, the dilaton B. SYMMETRIES 625 must appear homogeneously in the classical action. The dilaton that appears as above in the string action transforms as a density under general coordinate transformations, allowing the construction of actions invariant under both T-duality and coordinatetransformations. The resulting spacetime action is S massless =Z dx(1 4R+1 24HabcHabc+ )  whereHabc=1 2r[aBbc]is the eld strength for the axion. T-duality determines the only arbitrary coecient, the relative weight of the andRterms. Note the absence of the factor e1, which has been absorbed into the denition of : The covariant derivative acting on , since it is a density that transforms as e1=2,a c t s asra=e1=2eae1=2. Excercise XIB5.1 Find the eld equations following from this action. Then make the eldredenition  = e 1=2e, to nd the result:  )(r)2+1 4R+1 24H2+=0  Bab)raHabc+2Habcra=0  eam)Rab=2rarb Excercise XIB5.2 This action is in the string gauge (see subsection IXB5). aMake the physical scalar explicit in the action by the eld redenition (Weyl scaling: see subsection IXA7) eam!eam leaving  and Bmnunchanged. bThe resulting scalar action can be (o-)diagonalized by further redenitions: Noting that the known string theories are dened forp D1 an (odd) integer (5 or 3), write the dimension in general as (for any D> 1,nnot necessarily integer) D=n2+1 Restoring the e1to the action, redene =e1=2(n1)=2(n+1) +(n+1)=2(n1) ; =1=(n+1) +1=(n1) 626 XI. STRINGS which also gives the scalars the canonical Weyl scale weights, to obtain the nal result for the Lagrangian L(whereS=R dxe1L) L=+(n2 n211 4R)+1 24(n+5)=(n+1) +(n5)=(n1) H2 +(n1)=(n+1) +(n+1)=(n1) cThis redenition is singular for D=2(n=1 ) . F i xt h i sb ym a k i n gt h e additional redenition !(n1)=2  and then taking the limit n!1. (We can also use redenitions equivalent in the limit, such as !(D2)=4 .) Show the result is then L!1 4+(lnR)+1 243 +2 H2+ We can no longer choose the gauge += 1, since it is now scale invariant, but we can still choose =1 . 6. Superdilaton When applied to the supersymmetric cases (superstring or heterotic string), the inclusion of ghosts in the direct-product procedure also gives the auxiliary elds. (The dilaton itself is an auxiliary eld.) For example, in the heterotic case, the direct product of the physical parts of the vector and vector multiplet give conformalsupergravity (the supersymmetrization of the traceless part of the metric) and a physical tensor multiplet (the supersymmetrization of the axion and scalar). On the other hand, the ghosts of the vector multiplet form a chiral scalar supereld; its procuct with the scalar ghost of the vector gives another chiral scalar supereld, the compensator, containing the dilaton. (See chapter X.) The two conditions of supersymmetry and that the dilaton must appear homoge- neously (quadratically after an appropriate eld redenition) are now enough to x the form of the action (except for the nonminimal heterotic cases, where the open string's scalars introduce extra vector multiplets). For convenience we redene thechiral scalar compensator as ! 2=3so that it appears quadratically in the cos- mological termRd4xd22. Thus, by dimensional analysis now has scale weight 3 2. The axial-vector eld strength of the axion appears as [ r;r.]G,s oGhas scale weight 2. (Gauge elds are Weyl scale invariant with curved indices for consistency with gauge transformations; thus Hmnphas weight 0 while Habchas weight 3.) Of course, these weights also follow from local superscale transformations, the global part B. SYMMETRIES 627 of which transforms elds as L2w(see subsection XA4). The only action quadratic in the dilaton consistent with global scale and U(1) (R) invariance is then (with implicit covariantization with respect to conformal supergravity, which makes these invariances local) S=Z dx d4G1=2+ Z dx d22+h:c: Excercise XIB6.1 Use the methods of subsection XB6 to nd all of the terms in this actioninvolving only bosonic elds. Compare to the bosonic string action of the previous subsection. Besides T-duality, string theories also have \S-duality" symmetries that are re- alized only on the eld equations, or after performing electromagnetic-type dualitytransformations on the elds: If we convert Ginto a second, physical chiral multiplet by such a duality as described in subsection XB5, the above action is converted to S=Z dx d 4()2=3(+)1=3+ Z dx d22+h:c: After the redenitions 2!; 2! the rst term becomes manifestly SU(1,1) invariant (see subsection XB7): S=Z dx d4(+)1=3+ Z dx d2+h:c: It is now the original T-duality that can be realized only on shell. Also, in this form the condition that the dilaton should appear homogeneously is obscured. (Such S- dualities were rst seen in extended supergravity theories, especially when obtained by reduction from higher dimensions, where antisymmetric tensors are often required.) Excercise XIB6.2 Apply the results of excercise XB5.1 to include vector multiplets in the above actions by replacing G!~Gin the rst action and performing duality trans- formations. (The super Yang-Mills appears in the spectrum from the product(vectorscalars) vector multiplet in the heterotic string.) This substi- tution is dictated by homogeneity in the dilaton, which prevents the usual conformalRd 2W2term. Such terms occur naturally in higher-dimensional couplings of supergravity to super Yang-Mills. Classical and quantum symmetries of mechanics formulations of particle and string theories in background elds are often used to derive equations for those back- grounds. These features are not peculiar to these theories or their formulations: They 628 XI. STRINGS are a general feature of describing a particle/eld of some (super)spin in a gauge back- ground. These equations fall into two distinct types: (1) A supersymmetric system in a gauge background of higher superspin generates constraints on the background, necessary for consistently dening the coupling (see subsections IVC4 and XA1). (2)Any gauge system in a background of the same gauge eld generates eld equations for the background (see excercise VIB8.2). For example, the classical symmetries of the superparticle always generate con- straints on its background, but give eld equations for it only if the number of super-symmetries is enough to insure its superspin is as high as that of its background (e.g.,10D N=1 in background super Yang-Mills or 11D N=1 in background supergravity).Similarly, the bosonic string generates eld equations for background gravity at thequantum mechanical level because quantization is required to reveal the masslessgraviton excited state contained in the string itself. On the other hand, the 10D superstring already generates eld equations for background supergravity classically, since the ground state of the superstring (closed if boundary conditions are ignored),the only part that is evident (semi)classically, already contains supergravity. 7. Conformal eld theory We saw in subsections VIIIA7-8 some unusual features of massless theories in D=2. Since the mechanics of the string is mathematically equivalent to 2D eld theory (as the mechanics of the particle is to 1D eld theory), we now examine such eld theories in a little more detail. In particular, since the string we studied insection XIA possessed local Weyl scale invariance on the worldsheet, we are directedto 2D conformal eld theories coupled to 2D gravity. Since for the most part we will be interested in free elds, quantization will be described most easily by the path-integral method. Although 2D eld theory alreadylooks quite dierent from the 1D eld theory of particle mechanics, free 2D masslesselds depend on only one of the two lightcone coordinates  (or are the sum of two such terms), and hence 2D conformal eld theory is similar to 1D massive eldtheory. Consequently some of the features of particle mechanics or nonrelativisticeld theory, such as the commutator, can still be useful and 2D Lorentz covariant.In particle mechanics, the (equal-time) commutator is evaluated by path-integralmethods as h[A;B](t)ilim !0hA(t+)B(t)B(t+)A(t)i B. SYMMETRIES 629 (and similarly for the anticommutator), since AandBare treated as classical func- tions when evaluating the path integral hfiZ D feiS where now \hi" refers not to just the vacuum expectation value, but incorporates arbitrary initial and nal states through the boundary conditions, or explicit wavefunctions in the path integral (see subsections VA1, XIA6). In general, this denition ofhiactually gives the time-ordered expectation value, as follows from the derivation of subsection VA1: The 's were introduced to enforce the appropriate ordering. For the rest of this subsection time ordering will be implicit in expectation values. For simplicity, we assume the worldsheet boundary is at innity; this can be achieved by conformal (coordinate) transformation ( z=e on the Euclidean world- sheet: see subsection XIA6), or we can look at just short-distance eects. The upper- half plane, for the open string, can be completed into the full plane, as describes the closed string, by extending the open string to have closed-string boundary conditions,but with half as many elds, as described in subsection XIA3: In the following we consider only elds that are left- or right-handed (functions on-shell of only  +or ,a sX()or^P()), which are always dened on the whole worldsheet (or, before conformal transformation, periodic on the cylinder). The simplest example is the 2D \spinor". From subsection VIIIA7, we have h (m) (0m)i=i (0)i(0)h1i+::: (not summed over ) or in operator notation simply  (m) (0m)=i (0)i(0)+::: where \:::" means terms that are nite in the limit !0, resulting from propagators that extend not between these two fermions, but between either one and the initialor nal wave function. Using the identity (see excercise VA3.1) i x+ii xi=2(x) we then have the usual f (); (0)g=2(0) Normally, this would taken at equal , but in 2D conformal eld theory, since the elds' time-dependence is given by their depending on just +or just, and since 630 XI. STRINGS even interacting string calculations in terms of these \free" (with respect to rst- quantization) 2D elds factorizes into two separate calculations for the left-handed elds and for the right-handed elds, we generally treat just +or justas the only argument. (After Wick rotation, this becomes just the complex variable zor just z.) This limiting procedure can be avoided when evaluating commutators of -inte- grated quantities, such as conserved charges: Since such \surface" integrals in D=2 are basically contour integrals (see excercise IIA1.2c), a commutator of the form Zd 2() (); (0) =(0) (now dropping the index) for some anticommuting function ,w h i c hb yt h ea b o v e denition is the integral over the dierence of two contours, becomes the integral over a closed contour surrounding 0, directly picking up the contribution from the pole in0(see excercise VA3.1b). One unusual consequence of 2D massless eld theory is bosonization. In subsec- tion VIIIA7 we saw from propagators that the usual massless scalar =(+)+() can be used to construct fermions as =ei(), =ei(). From the above argu- ments this implies the usual anticommutation relations. (Note that such commutators are completely quantum mechanical and not merely semiclassical: The calculation of subsection VIIIA7 involved multiloop diagrams with arbitrary numbers of propaga-tors of scalars, whereas Poisson brackets eectively use only a single propagator of a fundamental eld.) As a further check of this equivalence we now examine their conformal transfor- mations. The commutator of the scalar conformal generators T =1 2(@)2=1 2(@())2 with that scalar follows from a single propagator (see subsection VIIIA7): ()(m)()(0m)=ln[i(0)] +::: ) [@()(m)]()(0m)=1 0+::: ) [@()(1);()(2)] =i2(12) (again dropping the on, using \1" for \ 1", etc.), conrming that @is canonically conjugate to . We then nd [T(1);()(2)] =i2(12)0 ()(2); [T;()]=0 B. SYMMETRIES 631 Similarly, for the composite fermion we nd [@()(m)]eia()(0m)=ia 0eia()(0m)+::: ) [@()(1); (2)] = 2(12) (2); [@()(1); (2)] =2(12) (2) in agreement with the identication @()=  . Then the product with Tgives two factors of 1 =(0) for the most singular term, as well as a less-singular term from connecting a propagator to either @(but not both): 1 2(@)2(m)eia(0m)=1 2a 02 eia(0m)1 0 eia(0m)0 +::: w h e r ew eh a v eu s e d [ia@(m)]eia(0m)= eia(0m)0 +O(0) The nal result is thus: [T(1); (2)] =i2(21) 0 (2)i0(21) (2); [T; ]=0 These results generalize easily to general linear exponentials for multiple scalars: Including an indenite metric h0ji()(m)j()(0m)j0i=ij[iln(0)] we have eia
()(m)eib
()(0m)=[i(0)]a
beia
()(m)+ib
()(0m)+::: as the most singular contribution to the path integral, using the same method as in subsection VIIIA7, where indices are raised and lowered with ij, as usual. Further- more, we can generalize the conformal generators with a linear term, as results from a1 2Riiterm in the action: L=1 4ij(ri)
(rj)+1 2Rii)T=1 2ij(@i())(@j())i@2 i() Its commutators then generalize, for any \covariant" elds ,a s  iZd0 2(0)T(0);() =()0()+w()0()() i1 2[T(1);(2)] =(21)0(2) +w()0(21)(2) 632 XI. STRINGS wherew()are the conformal weights: w(+)+w()is the usual conformal weight, whilew(+)w()is the \spin". For the elds eia
()we then nd w()(a)=1 2a2+i
a; w ()=0 In particular, for =0w es e et h a t hasw()=1 2, while()is covariant with weightw()=0 . Excercise XIB7.1 Use the result for eiaeibto derive   =@()(up to an innite constant), dening the equal-time product as a limit, as above. Excercise XIB7.2 Thei
aterm inwis classical, since it comes from a single propagator: aDerive the above Lagrangian for a single scalar by starting with the nonlocal termR(1=)Rand applying a local Weyl scale transformation, introducing the scalar as the compensator. bFind the classical scale weight of eiafrom its local scale transformation. (Hint: In deriving the local scale transformation in part a, an exponential will be needed, so that etransforms homogeneously.) The most important use of such exponentials, outside of bosonization, is in the use of external elds (or \vertex operators"): The conformal generators (energy- momentum tensor) have conformal weight 2; requiring that background elds preserve conformal invariance implies that such vertex operators must have conformal weight1 and be local on the worldsheet: i1 2[^T(1);^T(2)] =0(21)[^T(1) + ^T(2)] ^T()=T()+fW())fWh a sw =1; [fW();fW(0)] = 0 )fW()=2(0)W(0) where we have assumed the conformal anomaly cancels (or ignored its contribution), and solved for closure of the algebra perturbatively in the background. If we write a background spacetime eld ( X(m)) as a Fourier transform, then we see that its conformal weight is proportional to k2, the square of the external momentum. Hence a vertex operator consisting of just a scalar eld produces the tachyonic ground state. Excited states are created by products of derivatives of X times elds (with spacetime Lorentz indices contracted); the derivatives add to the conformal weight, forcing k2to decrease in compensation, resulting in massless and massive (m2>0) states. B. SYMMETRIES 633 8. Triality Bosonization can also be applied to representations of groups (Lorentz or inter- nal). In particular, to obtain the correct anticommutation relations for a fermion eia
 and its conjugate eia
we require a2= 1. Their weights are then1 2
a.T h e simplest choice to obtain 2 nfermions from nscalars is, for ij=ij,i na( c o m p l e x ) null basis for SO(2n), SO(2n)vector :ai V=(1;0;0;:::);(0;1;0;:::);::: Klein factors should be included to make fermions using dierent scalars anticom- mute (see subsection IA2). This construction follows that for the Dirac matrices in subsection XC1: Each scalar corresponds to a 2D subspace of SO(2n), and each com- ponent of a D-vector aiis the corresponding eigenvalue of the 2D spin. From that construction we see that spinors should be SO(2n)spinors :ai S;S0=(1 2;1 2;:::) with independent 's, with the product of all of them equal to 1 for one Weyl spinor and1 for the other. (The conventions are slightly dierent from subsection XC1: Now we use a representation where 3is diagonal, and 1is chosen as the last .) However, the resulting operators have the correct anticommutation relations and conformal weights only for SO(8) ( n= 4), corresponding to the lightcone symmetry for the 10D superstring. This follows from the \triality" symmetry between the vector and two spinors: By simply changing the scalar-eld basis so that the a's for one of the spinors arethe basis elements, the a's for the vector and the other spinor take the same form as above for the two spinors (i.e., the above expressions for the vector and spinor a's are permuted). Similar triality constructions apply to lower dimensions by taking into account supersymmetry, which also relates a vector to a spinor. In D=6, simple supersym-metry has an internal SU(2) (R) symmetry: Thus, there is a triality relating this SU(2) to the two SU(2)'s of the lightcone's SO(4). In terms of these, the vector is the ( 1 2;1 2;0) representation, while the spinors are (1 2;0;1 2)a n d( 0;1 2;1 2). The resulting operators are given by aV=(1p 2;1p 2;0);aS=(1p 2;0;1p 2);aS0=( 0;1p 2;1p 2) To relate to the SO(8) results we use the vector to dene the basis, yielding aV=(1;0;0);(0;1;0) 634 XI. STRINGS )aS=(1 2;1 2;1p 2);(1 2;1 2;1p 2);aS0=(1 2;1 2;1p 2);(1 2;1 2;1p 2) This also follows directly from the SO(8) result by dropping the third and fourth scalars for the vector, and using only (1 =p 2) their sum for the spinors. (I.e., it represents only internal symmetry.) For D=4 the construction is even simpler: Besides the SO(2)=U(1) of the lightcone, there is a second U(1) for R symmetry. In terms ofthe complex plane dened by these two quantum numbers, there is an obvious triality for the three cube roots of 1; thus a V=(1;0);aS=(1 2;p 3 2);aS0=(1 2;p 3 2) which again also follows from SO(8), now combining its last 3 scalars. Excercise XIB8.1 We now extend the analogy to the construction of subsection XC1: aShow for general SO(2n), in analogy to the Dirac 's, that the (integral of) products of two vector fermions, antisymmetrized in the vector indices, act in the same way as the group generators, by examining their commutators with each other and with the vector and spinor operators. bShow for the triality cases that the (anti)commutator of two representations yields the third (supersymmetry). These constructions can be generalized from the lightcone to manifest Lorentz covariance by adding equal numbers of scalars of positive and negative metric (at least one of each): Their contributions to the spinors' operator product (power of) then cancel, preserving the anticommutation relations. One of the extra scalars of positive metric yields the two \longitudinal" spacetime directions to complete the SO(D2) vector and spinor representations to SO(D 1,1). The rest of the scalars come from \ghosts". Note that the spacetime metric is unrelated to the metric for the scalars: The Minkowski spacetime metric comes from Wick rotation of the scalars, as applied in subsection XC2 to the construction of subsection XC1 for Dirac spinors.For either Euclidean or Minkowski spacetime the basis is null; the only dierence is in reality. REFERENCES 1 Ramond; Neveu and Schwarz; loc. cit. (IIC): almost superstrings (fermions included). 2M.B. Green and J.H. Schwarz, Phys. Lett. 109B (1982) 444, 149B (1984) 117: superstrings. 3D.J. Gross, J.A. Harvey, E. Martinec, and R. Rohm, Phys. Rev. Lett. 54(1985) 502, Nucl. Phys. B256 (1985) 253, 267(1986) 75: heterotic string. B. SYMMETRIES 635 4W. Siegel, Phys. Lett. 134B (1984) 318; T.H. Buscher, Phys. Lett. 194B (1987) 59, 201B (1988) 466: T-duality as transformation on worldsheet. 5K. Kikkawa and M. Yamasaki, Phys. Lett. 149B (1984) 357; N. Sakai and I. Senda, Prog. Theor. Phys. 75(1986) 692; V.P. Nair, A. Shapere, A. Strominger, and F. Wilczek, Nucl. Phys. B287 (1987) 402; B. Sathiapalan, Phys. Rev. Lett. 58(1987) 1597; R. Dijkgraaf, E. Verlinde, and H. Verlinde, Comm. Math. Phys. 115(1988) 649; K.S. Narain, M.H. Sarmadi, and E. Witten, Nucl. Phys. B279 (1987) 369; P. Ginsparg, Phys. Rev. D35 (1987) 648; P. Ginsparg and C. Vafa, Nucl. Phys. B289 (1987) 414; S. Cecotti, S. Ferrara, and L. Girardello, Nucl. Phys. B308 (1988) 436; R. Brandenberger and C. Vafa, Nucl. Phys. B316 (1988) 391; A. Giveon, E. Rabinovici, and G. Veneziano, Nucl. Phys. B322 (1989) 167; A. Shapere and F. Wilczek, Nucl. Phys. B320 (1989) 669; M .D i n e ,P .H u e t ,a n dN .S e i b e r g , Nucl. Phys. B322 (1989) 301; J. Molera and B. Ovrut, Phys. Rev. D40 (1989) 1146; K.A. Meissner and G. Veneziano, Phys. Lett. 267B (1991) 33; A.A. Tseytlin and C. Vafa, hep-th/9109048, Nucl. Phys. B372 (1992) 443; M. Ro cek and E. Verlinde, hep-th/9110053, Nucl. Phys. B373 (1992) 630; J.H. Horne, G.T. Horowitz, and A.R. Steif, hep-th/9110065, Phys. Rev. Lett. 68(1992) 568;A. Sen, Phys. Lett. 271B (1991) 295; A. Giveon and M. Ro cek, hep-th/9112070, Nucl. Phys. B380 (1992) 128: T-duality as symmetry of spacetime elds. 6Siegel, loc. cit. (IXB, 1st and 2nd papers of ref. 5): coupling of (unphysical) dilaton to string (through ghosts). 7E.S. Fradkin and A.A. Tseytlin, Phys. Lett. 158B (1985) 316, Nucl. Phys. B261 (1985) 1:coupling of dilaton through worldsheet curvature. 8T. Banks, D. Nemeschansky, and A. Sen, Nucl. Phys. B277 (1986) 67: relation between above two couplings. 9Fradkin and Tseytlin, loc. cit. ; C.G. Callan, D. Friedan, E.J. Martinec, and M.J. Perry, Nucl. Phys. B262 (1985) 593: low-energy closed-string actions for massless elds. 10S. Cecotti, S. Ferrara, and M. Villasante, Int. J. Mod. Phys. A 2(1987) 1839: superspace action for 4D massless part of heterotic string. 11W. Siegel, Phys. Lett. 211B (1988) 55: massless part of heterotic string is old-minimal supergravity coupled to tensor multiplet, as follows from direct product of open strings. 12W. Siegel, hep-th/9510150, Phys. Rev. D53 (1996) 3324: massless part of string actions from direct product and dilaton homogeneity. 13Virasoro; Gelfand and Fuchs; loc. cit. (XIA); S. Fubini and G. Veneziano, Nuo. Com. 67A (1970) 29, Ann. Phys. 63(1971) 12; A. Galli, Nuo. Cim. 69A (1970) 275; J.L. Gervais, Nucl. Phys. B21 (1970) 192; J.-L. Gervais and B. Sakita, Nucl. Phys. B34 (1971) 477; 636 XI. STRINGS A. Chodos and C.B. Thorn, Nucl. Phys. B72 (1974) 509: theory of free conformal elds. 14R. Marnelius, Nucl. Phys. B211 (1983) 14: eect ofRterm on Virasoro operators. 15E. Witten, D=10 superstring theory, Fourth workshop on grand unication ,U n i v e r s i t y of Pennsylvania, Philadelphia, April 21-23, 1983, eds. H.A. Weldon, P. Langacker, andP.J. Steinhardt (Birkh auser, 1983) p. 395: vectors and spinors of SO(8) as dierent exponentials of bosons. 16W. Siegel and B. Zwiebach, Nucl. Phys. B263 (1986) 105: bosonization of ghosts (bosonic string). 17D. Friedan, E. Martinec, and S. Shenker, Phys. Lett. 160B (1985) 55, Nucl. Phys. B271 (1986) 93; V.G. Knizhnik, Phys. Lett. 160B (1985) 403: vectors and spinors of SO(9,1) as dierent exponentials of bosons. 18M. Kaku, Strings, conformal elds, and topology: an introduction (Springer-Verlag, 1991);P. Ginsparg and G. Moore, hep-th/9304011, Lectures on 2D gravity and 2D stringtheory, in Recent directions in particle theory: from superstrings and black holes to the standard model , Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 1992), Boulder, CO, June 1-26, 1992, eds. J. Harvey and J.Polchinski (World Scientic, 1993) p. 277;S.V. Ketov, Conformal eld theory (World Scientic, 1995); P. Di Francesco, P. Mathieu, and D. S enechal, Conformal eld theory (Springer, 1997). C. LATTICES 637 :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::::::::::::::::::::::::: C. LATTICES :::::::::::::::::::::::::::: Integrals are dened as limits of sums. For some cases it can be convenient to dene quantum theories on discrete spacetimes (\lattices"), perform all calculations there, and then take the limit of continuous spacetime. Two types of such lattices will be considered here: (1) Physical four-dimensional spacetime can be treated as aregular hypercubic lattice. Then the existence and uniqueness of a continuum limit where Lorentz invariance is restored must be proven. (2) In rst-quantization of particles or strings, the worldline or worldsheet can be approximated as a randomlattice. Integration over the metric of the worldline or worldsheet is then replaced with summation over lattices with dierent geometries. The continuum limit is not required by physical criteria, but only for purposes of comparison to the theory as dened in the continuum. 1. Spacetime lattice The use of a regular 4D lattice for quantizing QCD has three main advantages: (1) The lattice acts as a gauge invariant regulator for UV divergences (and, if thelattice is nite, also IR ones). (2) Gauge xing is no longer necessary, since the path integral can be performed without it. (3) Nonperturbative calculations are possible, some analytically and some numerically (if the lattice is small enough). Gauge elds are associated with translations through the covariant derivative. However, on a lattice, even a regular one, innitesimal translations are no longer possible: For example, scalar elds are dened only at vertices of the lattice. We therefore consider covariantizing nite translations, as in subsections IIIA5 and IIIC2, e kmrm=P exp iZx xkdx0
A ek
@=Ux;xkek
@ Without loss of generality, we can restrict ourselves to translations along links, from one vertex straight to an adjacent one (keeping all coordinates but one constant), and successive combinations of these. Then the gauge eld is replaced with the group elementUx;xkassociated with each link, where kis now any of the 4 orthonormal basis vectors (in Euclidean space). The gauge transformation of this representationof the gauge eld follows from either the path-ordered denition or the covariant- translation denition: e k
r(x)0=g(x)ek
rg1(x))U0 x;xk=g(x)Ux;xkg1(xk) Note that, while the gauge eld is a group element associated with a link, the gauge transformation is a group element associated with a vertex. Furthermore, the eld 638 XI. STRINGS strength can be associated with the product of these group elements of the links bounding a \plaquet": Ux;xkUxk;xkk0Uxkk0;xk0Uxk0;x=P eiH dx
A =ek
rek0
rek
rek0
r e[k
r;k0
r]1+ikak0bFab w h e r ew eh a v eu s e d eBeC=eB+C+1 2[B;C]+::: (In general, there is a geometric prescription associating a scalar with a point, a vector with a line, a second-rank antisymmetric tensor with a surface, etc.) We now dene a gauge-invariant action by looking for an expression in terms of these group elements that approximates the usual Yang-Mills action to lowest order in the lattice spacing, while involving the least number of factors of the group elements.The result is: S= 1 g2trX plaquets(Ux;xkUxk;xkk0Uxkk0;xk0Uxk0;x1) 1 g2trX plaquets1 2(ikak0bFab)21 g2trX xF2(x) (expanding the exponential as above to quadratic order, and noting that total commu- tators vanish when traced). Since our elds are now represented by group elements, we no longer need to x the gauge to make the functional path integral well dened:In contrast to the continuum case, where integrating a gauge-invariant action over gauge transformations would produce an innite factor, here such an integral at any one point is just an integral over the group space, which is nite (for compact groups, which have nite volume). The functional integration is now integration over Ufor each link, where the range of Uis the group space (which is nite, since the group is compact). Matter can also be introduced: Scalars are naturally associated with vertices, just as vectors are with links, and second-rank antisymmetric tensors with plaquets. However, fermions do not have such a natural geometric interpretation. In particular,it has been proven (the \Nielsen-Ninomiya theorem") that massless fermions can't be dened in a useful way on the lattice without \fermion doubling": There must be a multiple of 2 Dmassless fermion elds for D lattice dimensions. This is closely related to the existence of axial anomalies: The absence of an anomaly is implied by the existence of a regularization that manifestly preserves a symmetry (in this case, chiral symmetry as a consequence of the existence of lattice-regularized fermions). However, C. LATTICES 639 massless fermions can be dened as limits of massive ones (so chiral invariance is not manifest). Alternatively, nonlocal spinor kinetic operators can be found that preserve masslessness and chirality without doubling. (The nonlocality can be controlled, but at the cost of a signicantly more complicated action.) In general, fermions are more dicult to integrate over, particularly when using \numerical methods" (computers), since fermions are not numbers. In principle one can integrate out the fermions analytically to produce functional determinants in terms of bosonic elds, but nonlocality makes them hard to evaluate by iterative schemes. In practice fermion loops are usually ignored (\quenched approximation"), which corresponds to leading order in an expansion in the inverse of the number of avors, or the approximation of heavy quarks. The resulting accuracy of QCD calculations for low-energy parameters (masses of light hadrons, decay constants, etc.)is of the order of 5-10%. (Getting good numbers in nonperturbative calculations is signicantly harder than in perturbative ones. The situation is expected to improve somewhat with the advent of faster computers.) Finding scattering amplitudes, or other properties that involve high-mass hadrons, is presently beyond the scope of lattice methods. However, lattice QCD is the only method so far to obtain numbers for comparison with experiment from nonperturbative calculations with the QCD action. The spacetime lattice allows a direct nonperturbative analysis of connement. For example, consider the potential between a heavy quark-antiquark pair. The heaviness again allows us to ignore pair creation, and to treat the quarks as static. For simplicity, consider scalar quarks, as described by rst-quantization. Since we approximate the quarks as static, the only relevant term in the quark mechanics actionis the interaction termR d.x
A=R dx
A. Taking into account the nonabelian nature of the group, and ignoring the rst-quantized path integration Dx(sincexis assumed xed), the factor e Sfor the quark becomes just the path-ordered expression P(eiR dx
A) we have been considering, while for the antiquark we get the inverse expression. To get a gauge-invariant expression, we connect the paths at top andbottom, since the elds will be xed at the boundaries at t=1. (Functional integration over any gauge-eld link picks out just the singlet part of the integrand, since the integral is over the group, and nonsinglet representations can be rotated to minus themselves by an appropriate group element, canceling the contribution.) The result is a \Wilson loop" trP e iH dx
A 640 XI. STRINGS The strong-coupling expansion is applied by expanding the functional integrand eSin powers of S, which is an expansion in powers of 1 =g2, and which is also an ex- pansion in the number of plaquets. Clearly the dominant term in this expansion is theone with the fewest factors of S. To be nonvanishing, each link variable must appear in a singlet combination: The function of that link, when expanded in irreduciblegroup representations, must include a term that is proportional to the identity. Forexample, for any unitary group, this is true for the product U ij(U1)kl, where the two U's are for the same link, and the indices are the group indices; this has the constant pieceUij(U1)ji. For the case of the Wilson loop, if we assume the simplest case where the path is a rectangle, then we need at least a factor of Sfor each plaquet enclosed by the loop, so there will a UU1for each link on the boundary (one factor from the loop, one from S), as well as for each link enclosed by it (both factors from the contribution to Sfrom either side). The result for the path integral is then A=D trP eiH dx
AE eVt1 g2rt whereVis the (potential) energy ( S=R dt(V+T) in Euclidean space), tis the time separation between the top and bottom of the rectangle, and ris the spatial separation between the two sides. We thus have a linear quark-antiquark potential V(r)(lng)r so the quark-antiquark pair is conned. Unfortunately, we can get a similar result from QED, by dening the U(1) group in terms of a phase factor (so eectively the range of group integration is 2 , dening a \compact" group). The reason is that for this U(1) theory this strong couplingexpansion is not accurate. The approximation is better for nonabelian theories, butthe persistence of connement has not been proven in the continuum limit (smallcoupling). In fact, while the transition to deconnement in Abelian theories has beenfound at nite coupling, it has been proven that such a phenomenon can occur in thenonabelian theory only near zero coupling. However, the perturbative properties ofthe continuum theory show that this is exactly where one expects the appearance ofambiguities in the theory (known in lattice terminology as \nonuniversality"); thisis directly related to the ambiguities in the Borel transform discussed in subsectionVIIC3. C. LATTICES 641 2. Worldsheet lattice In string theory there are two spaces, the two-dimensional space of the world- sheet, and physical spacetime. In the previous subsection we considered approximat-ing spacetime by a lattice; in this subsection we instead approximate the worldsheet by lattices. For the spacetime of QCD we used a regular lattice, representing the xed geometry of at spacetime. In string theory we considered worldsheets of ar-bitrary geometry, described by a worldsheet metric, so our lattices should be more arbitrary; in fact, functional integration over the worldsheet metric must be replaced by summation over dierent lattices. We saw that the topological expansion of QCDin 1/N generated polyhedra analogous to the worldsheet, with 1/N acting as the string coupling. We therefore identify the Feynman diagrams themselves, with faces chosen by the 1/N expansion, as these lattices, to give a more precise correlationbetween the second-quantized path integral of QCD (and other eld theories) and the rst-quantized path integral of string theory. Presently the relation between such eld theories and string theory is not well understood, and has been described only for the bosonic string. Since the bosonicstring has only the worldsheet metric and spacetime coordinates as degrees of free- dom, it corresponds to a (N N-matrix) scalar eld theory. Since a lattice requires a scale, while conformal invariance includes scale invariance, we must break the confor- mal invariance of the worldsheet. The simplest coordinate-invariant yet scale-variant property of a space is its volume, so we add a volume (area) term to the string action.Furthermore, to describe interactions we need to include a term containing the string coupling constant. In string theory the coupling is topological, in the sense that the power of the coupling constant is counted by the \genus" of the worldsheet (the num-ber of \windows" plus twice the number of \handles"). This is the \Euler number" , counted by the integral of the worldsheet curvature (see excercise IXA7.3). Our worldsheet action thus consists of the three terms S=Zd 2 2pg1 0gmn1 2(@mX)
(@nX)++(ln)1 2R A lattice version of this action is (with g i v e ni ns u b s e c t i o nV I I C 4 ) S1=1 0X hjki1 2(XjXk)2+X j1+(ln)0 @X j1X hjki1+X J11 A wherejare vertices of the lattice, hjkiare the links, and Jare the plaquets. Excercise XIC2.1 Put the particle on a random lattice \Minkowski" worldline. (See excercise 642 XI. STRINGS VB1.2.) Show the propagator for a massless particle, written in momentum space, before taking the limit lattice spacing !0, is
=2 1eip2 Show this has unphysical poles at p2=2n= for arbitrary integer n.H o w do these results dier if the propagator is dened for Wick-rotated ? The corresponding eld theory is easily found, according to our earlier discussions, by (1) identifying the worldsheet lattice with a position-space Feynman diagram (the vertices of the lattice being those of the diagram, the links of the lattice being the Feynman propagators; see subsection VC8), and (2) using the 1/N expansion toassociate the faces of the worldsheet polyhedra with the U(N) indices of the scalar eld (see subsection VIIC4). We then can immediately identify the three terms in the string action with their counterparts in the scalar eld theory: (1) The Xterm gives the propagators, (2) the area term (which counts the vertices) gives the vertex factor (coupling constant), and (3) the curvature term (which classies the topology) gives the 1/N factors of the topological expansion. Thus, the three constants in the string action can be identiedwith the mass, coupling, and number of colors of the scalar eld theory. Explicitly, the eld theory action is S 2=Nt rZdDx (20)D=2(1 2e0=2G1 nn) w h e r ew eh a v ei d e n t i  e d G=e;1 N=; m2=2 0 and we have put an overall factor of N(associated with 1 =h)s ot h a tG(andm2)i s xed (rather than Gtimes some power of N) when the 1 =Nexpansion is performed. (The reverse can be made true by rescaling .) The unusual kinetic operator e0=2=1 m2(m2 +:::) comes from identifying the second-quantized particle propagator as it appears in the rst-quantized path integral for the string: A=ZY jdDXj (20)D=2eS1)
(x;y)=e(xy)2=20 Unlike the spacetime lattice, the worldsheet lattice preserves spacetime Poincar e symmetry, so it's not necessary to take any limits to dene a physical theory (or at C. LATTICES 643 least taking limits won't improve the physical relevance of this model). This model thus describes a sti or lumpy string. The usual continuum-worldsheet string then canbe identied with a particular limit of this more general string. Explicit calculationshave demonstrated that this lattice regularization of the worldsheet reproduces theresults of the continuum approach. These results have been limited to spacetimedimension1 because of the inconsistencies introduced by the tachyon, which is the ground state in higher dimensions. Unfortunately, this prevents study of themore interesting properties, such as scattering amplitudes and the precise form ofthe potential (we have left narbitrary in S 2), since it's superrenormalizable in D 2 regardless of its form. However, these limitations probably would not appear in a corresponding formulation of the superstring, which has no tachyons. An interesting feature of this model is the use of Gaussian propagators to get rid of the usual perturbative divergences of momentum integration. Naively, onemight suspect that such eld theories were completely nite. However, we know inthis case that the bosonic string does have divergences perturbatively in the stringcoupling, and that there are further problems unless D=26. This demonstrates that modifying a theory to x problems seen in perturbation theory does not preclude the reappearance of such diculties nonperturbatively. 3. QCD strings The main problem with known string theories is that they describe spacetimes of dimension D=10. To describe physics in D=4, it is usually assumed that 6 ofthe dimensions choose to \compactify" to submicroscopic dimensions, correspondingto length scales well below the range of present experiments. Such a solution tothe classical eld equations with minimal energy is chosen as the \vacuum", about which perturbations are performed, but nothing is known to preclude contributions to the functional integral from other vacuua, whether 4-dimensional, 10-dimensional, orelsewhere. Although strings have been chosen to describe quantum gravity becauseof their renormalizability (niteness), this advantage is lost after compactication,since the arbitrariness in choice of compactication is tantamount to the loss of pre-dictability in nonrenormalizable theories. However, there are some general featuresof string theories, such as the appearance of a dilaton-like physical scalar, that havesuggested certain 4D models (such as no-scale supergravity, which is also suggestedby extended supergravity: see subsection XB7). Furthermore, D=10 string theories have recently been discovered to be (compacti- cations of) D=11 membrane theories in disguise, where the eleventh dimension shows 644 XI. STRINGS up only nonperturbatively. Not only is using a formalism where not all of the dimen- sions are manifest a technical obstacle, but the quantum mechanics of membranessuers from several problems, including nonrenormalizability, which is what stringtheories were chosen to avoid in the rst place. This suggests that D=10 strings arenonrenormalizable at the nonperturbative level. Furthermore, even the few proper-ties of 4D theories suggested by strings are no longer required, since compacticationfrom D=11 need not proceed by way of D=10. On the other hand, renormalizabilty of theories with a nite number of elds predicts D=4, since theories in higher dimensions are all nonrenormalizable (or haveunbounded potentials:  3theory). Furthermore, both experiments with hadrons and theoretical arguments in QCD suggest the existence of an inherently 4D stringtheory. (For example, the existence of a continuum limit for conning spacetime-lattice theories requires asymptotic freedom.) The reason 10D strings avoid the D=4 limit is clear from the worldsheet lattice approach of the previous subsection: The partons that make up the string have Gaus-sian propagators, and Gaussian integrals always converge. This leads to Gaussian be-havior of xed-angle scattering (subsection XIA6), in con ict with hadronic physics,where power-law behavior is observed for partons with large transverse momenta, andis a theoretical consequence of asymptotic freedom with the usual propagators. (Infact, it is the main empirical verication of QCD.) Since nonrelativistic rst-quantization gives Gaussian propagators e x2=t,i ti sn o t surprising that the simplest strings should result in partons with Gaussian propaga-torse x2. However, the fact that rst-quantization for particles leads instead to, e.g., 1=x2propagators for massless particles in 4D position space suggests that an analogous treatment for strings should be possible. We thus attempt to follow thederivation of the previous subsection from parton to string, but starting with realis-tic parton propagators. The rst step is to exponentiate the propagator so that theexponent can be identied with a rst-quantized action. The easiest way, and thatmost analogous to the nonrelativistic case, is to use the Schwinger parametrizationof the propagator, which follows from the appearance of the worldline metric in theaction:1 1 2p2=Z1 0d ep2=2 As we saw in subsection VC8, a Feynman diagram in a scalar eld theory with nonderivative self-interactions is then written as Z dx0 idpijdijeP hiji[ijp2 ij=2i(xixj)
pij] C. LATTICES 645 In the (worldsheet) continuum limit of this expression, pbecomes a worldsheet vector, somust become a symmetric worldsheet tensor. Since on a regular square lattice (\ at" worldsheet) there are two propagators per vertex (for the two independentdirections), must be a traceless tensor. (This also explains why can't be just a scalar.) Imposing this tracelessness through a Lagrange multiplier ,w ec a nw r i t e the (Wicked rotated) continuum action as S=Zd 2 2fiPm
@mX+1 2mn(Pm
Pn+ggmn)+pg[+(ln)1 2R]g Thusacts as a kind of second worldsheet metric. However, since Schwinger param- eters are positive, mnmust be positive denite, and thus a Euclidean metric. This also implies that gmnmust be Minkowskian, to be consistent with the tracelessness condition. Note that if we set equal to a constant, and ignore the positivity condi- tion on, then eliminating by the equation of motion from varying gmnreproduces the usual string action, where we can identify 0==hi. This indicates a possible approximation scheme. The two components of that survive this tracelessness condition correspond to the two lightlike directions dened by gmn: If we use a \zweibein", dened as usual by gmn=e(m+en), to atten the indices on , then the Lagrange multiplier constraint can be solved by simply setting += 0. The action is then S=Zd2 2pg[iP
em@mX+1 2P
P++(ln)1 2R] Back on the lattice, this implies that the directions chosen by the propagators (links) on whichPis dened are lightlike. Thus, the matrix model dened by this theory should have only 4-point vertices, with the four propagators coming from any vertexforming the worldsheet lightcone at that point on the worldsheet. The eld theoryaction is thus S 2=Nt rZdDx (2)D=2(1 4G1 44) ForD= 4, this action describes an asymptotically free theory, \wrong-sign" 4 theory. Unlike conventional strings, the QCD string has critical dimension D=4 for renor- malizability. (In conventional strings all momentum integrals are Gaussian and thusconverge.) Another reason for D=4 is T-duality: T-duality interchanges the positionsof the vertices with the momenta of the loops. This is clear from our discussion ofthe classical mechanics of Feynman diagrams in subsection VC8, if we note that theprocedure we used there to translate from coordinates to loop momenta is exactly the 646 XI. STRINGS random lattice version of the T-duality transformation performed in subsection XIB4 (with ~Xas the loop momenta). Thus, invariance of a string theory under T-duality must include invariance of the propagators of the underlying eld theory under Fouriertransformation. This is trivial for conventional strings, since the Fourier transform of a Gaussian is a Gaussian. However, by dimensional analysis (or explicit evalua- tion: see excercise VIIB4.2), we see that the Fourier transform of 1 =p 2is 1=x2only in D=4: T-duality implies both D=4 and masslessness. Furthermore, we can look atinteractions by considering the simplest case: The at worldsheet is represented by a regular, at lattice. For  4theory we have the usual square lattice, which is self-dual under switching vertices with loops (T-duality). On the other hand, triangular andhexagonal lattices, corresponding to  6and3theory, are dual to each other (i.e., n is dual to2n=(n2), as follows from geometry). Thus T-duality also implies the 4 interaction. Excercise XIC3.1 Let's examine T-duality for the random lattice more carefully: aRepeat the T-duality transformation of subsection XIB4, but for the QCD string (see subsection VC8), without a background ( Mmn=mn). Show that invariance under Xm!~Xmrequires that the matrix also be replaced by its inverse, with some factors of the 2D tensor. (also transforms; you can avoid this complication by using the zweibein form of the action.) bWrite the massless scalar propagator in momentum space of arbitrary dimen- sion D as an exponential using a Schwinger parameter . Show that after T-duality | Fourier transformation combined with !1=(which leaves the exponent invariant) | a -dependent \measure" factor is introduced, except for D=4. REFERENCES 1K.G. Wilson, Phys. Rev. D10 (1974) 2445: QCD lattice. 2H.B. Nielsen and M. Ninomiya, Nucl. Phys. B185 (1981) 20, 195 (1982) 541, 193 (1981) 173, Phys. Lett. 105B (1981) 219. 3P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25 (1982) 2649; P. Hasenfratz, hep-lat/9709110, Nucl. Phys. Proc. Suppl. 63A-C (1998) 53, hep-lat/9802007, Nucl. Phys. B525 (1998) 401, P. Hasenfratz, V. Laliena, and F. Nie- dermayer, hep-lat/9801021, Phys. Lett. 427B (1998) 125; H. Neuberger, hep-lat/9707022, Phys. Lett. 417B (1998) 141, hep-lat/9801031, Phys. Lett.427B (1998) 353; M. L uscher, hep-lat/9802011, Phys. Lett. 428B (1998) 342: avoiding Nielsen-Ninomiya, while preserving chiral invariance. C. LATTICES 647 4M. Creutz, Phys. Rev. D21 (1980) 2308; E. Marinari, G. Parisi, and C. Rebbi, Phys. Rev. Lett. 47(1981) 1795: computer calculations with lattice QCD. 5E.T. Tomboulis, Phys. Rev. D25 (1982) 606, Phys. Rev. Lett. 50(1983) 885: deconnement can occur only at zero coupling. 6't Hooft, loc. cit. (VIIC, ref. 3): convergence of singularities in the complex coupling plane at zero coupling. 7H.J. Rothe, Lattice gauge theories, an introduction (World Scientic, 1992). 8H.B. Nielsen and P. Olesen, Phys. Lett. 32B (1970) 203; D.B. Fairlie and H.B. Nielsen, Nucl. Phys. B20 (1970) 637; B. Sakita and M.A. Virasoro, Phys. Rev. Lett. 24(1970) 1146: worldsheet lattice as Feynman diagrams. 9F. David, Nucl. Phys. B257 [FS14] (1985) 543; V.A. Kazakov, I.K. Kostov and A.A. Migdal, Phys. Lett. 157B (1985) 295: integration over worldsheet metric as sum over Feynman diagrams. 10M.R. Douglas and S.H. Shenker, Nucl. Phys. B335 (1990) 635; D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64(1990) 127; E. Br ezin and V.A. Kazakov, Phys. Lett. 236B (1990) 144: continuum limit for worldsheet lattice with metric and 1/N expansion. 11Veneziano, loc. cit. (XIA); V. Alessandrini, D. Amati, and B. Morel, Nuo. Cim. 7A(1971) 797; D.J. Gross and P.F. Mende, Phys. Lett. 197B (1987) 129, Nucl. Phys. B303 (1988) 407; D.J. Gross and J.L. Ma~ nes, Nucl. Phys. B326 (1989) 73: Gaussian behavior of string amplitudes. 12P.A. Collins and R.W. Tucker, Nucl. Phys. B112 (1976) 150; B. de Wit, M. L uscher, and H. Nicolai, Nucl. Phys. B305 (1988) 545: problems quantizing membranes. 13C.M. Hull and P.K. Townsend, hep-th/9410167, Nucl. Phys. B438 (1995) 109: the 11th dimension arises nonperturbatively in string theory. 14Siegel, loc. cit. (VC): QCD string with Schwinger parameters as second worldsheet metric. 15A.M. Polyakov, Nucl. Phys. B268 (1986) 406: two-metric formulation of usual string. 648 XII. MECHANICS XII. MECHANICS String theories describe particles of arbitrarily large spins: So far in this text we have concentrated on lower spins, but we can describe (at least) free gauge-invariantactions for arbitrary spins based on quantum mechanical BRST. Gauge invariance is required in eld theory to manifest Lorentz invariance. The basic problem is that a four-vector wave function cannot have the obvious Minkowskiinner product, since the time component would have a minus sign in its normalization,resulting in negative probability. In the classical action there is a gauge invariancethat allows the time component to be dropped from the action. However, such gaugesdestroy manifest Lorentz invariance, since a three-vector cannot represent Lorentztransformations in a local way. More useful gauges keep all components of the four-vector, while also introducing scalar fermionic \ghosts" to cancel the eects of thebad part of the four-vector. A certain symmetry between the bosonic and fermionicunphysical degrees of freedom is needed to enforce this cancelation: It is the eldtheoretic version of the BRST symmetry discussed in section VIA. Another complication is that gauge transformations do not allow the elimination of traces in a simple way: Although it is Lorentz covariant to constrain a tensorto vanish when a pair of its vector indices is contracted, this interferes with gaugeinvariance in interacting theories, such as gravity. A related complication is massivetheories, which can't always be described simply by adding mass terms to massless theories. There is a simple solution to all these problems, which determines the free part of the action for any theory. (Interactions are a separate problem.) This methodautomatically introduces all the correct elds, including ghosts, for any massless or massive theory. It also gives a simple universal expression for the BRST symmetry that cancels unphysical modes, as well as providing a simple proof that these modesdisappear in the lightcone gauge. The method is based on the idea of introducingextra fermionic dimensions to spacetime that are unphysical (unlike superspace forsupersymmetry), which cancel unphysical degrees of freedom associated with the timedimension. Although for most purposes the only spins of fundamental particles relevant in eld theory for are 0, 1/2, 1, 3/2 (maybe), and 2, and these few cases can be studiedseparately, in this chapter we'll analyze all free theories because: (1) The ultimatetheory of particles may require them; (2) some of the theories presently under most ac-tive investigation (such as strings and membranes) require them; (3) many observed, A. OSp(1,1j2) 649 though perhaps not fundamental, particles have higher spin; and (4) a better under- standing of eld theory can be obtained by determining exactly which properties all  e l d sh a v ei nc o m m o na sw e l la sh o wt h e yd i  e r . ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: A. OSp(1,1 2)::::::::::::::::::::::::::::: This construction involves the introduction of spacetime symmetries that are not manifest on the physical coordinates. An important analog is the conformal group inD dimensions, which acts nonlinearly on the usual D spacetime coordinates, but canbe represented linearly on D+2 coordinates, since the group is SO(D,2). As describedin subsection IA6 for spin 0, 1 space and 1 time coordinate can be eliminated, so thatSO(D,2) is still represented, but SO(D 1,1) is the largest orthogonal group that is still manifest. We have also seen that in the lightcone gauge this manifest symmetryis reduced again in the same way, leaving SO(D 2). In our case the relevant group is OSp(D,2j2), the natural generalization of the orthogonal group to D space, 2 time, and 2 anticommuting dimensions. This allows rotations between timelike and fermionicdirections, eventually resulting in their cancelation. 1. Lightcone We saw in subsection IIB3 how the single equation of motion Sab@b+w@a=0 , applied to eld strengths, universally described all spins in all dimensions, for free, massless particles. (A possible exception is the spinless case, where we need =0 , which is redundant otherwise. However, we can use the universal eld equation evenin that case if we use the vector eld strength formulation of spin 0.) One way wesolved this equation was to perform a unitary transformation. We can use the sameunitary transformation, plus the constraints, to simplify the Lorentz generators. Tofurther simplify matters, we can use the constraint = 0, solved for @,t oc h o o s e the gaugex+= 0, which is equivalent to working in the Schr odinger picture (no time dependence for operators). The procedure is thus: (1) Start with the manifest(antihermitian) representation of the Lorentz generators, J ab=x[a@b]+Sab (2) Apply the transformation J!UJU1;l n U =S+i@i @+ 650 XII. MECHANICS which eliminates the only S+iterm, inJ+i(while complicating Ji). (3) Finally, apply the constraints, which have already been transformed by this same transformation, =0 ()gaugex+=0 );Sab@b+w@a=0!S+w=Si=0 Our Lorentz generators are then J+i=xi@+;J+=x@++w; Jij=x[i@j]+Sij Ji=x@i+1 @+[1 2xi(@j)2+Sij@j+w@i] These generators satisfy the pseudo(anti)hermiticity condition Jaby(@+)12w=(@+)12wJab This means that the Hilbert-space metric needs a factor of ( @+)12w. This is related to the fact that the wterms can be eliminated by a nonunitary transformation with the appropriate power of @+: As part of step 2, we could have applied a second transformation U2=(@+)S+ with the result of eliminating all S+terms, so the redundant constraint S+=w would not have been needed, so wwould not appear. In any case, we generally choose w= 0 for bosons. We previously applied dimensional reduction to the eld equations for the eld strengths, to obtain the equations for the massive, free theories from the massless ones. The same methods can be applied to the Lorentz (or Poincar e) generators. (The Lorentz generators will be used later to nd the BRST operator, to obtain theeld equations in terms of the gauge elds, and the action. For that purpose thedimensional reduction can be performed at any stage in the derivation.) We thus nd the general result J +i=xi@+;J+=x@++w; Jij=x[i@j]+Sij Ji=x@i+1 @+f1 2xi[(@j)2m2]+Sij@j+Si1im+w@ig (The1 is still an index, and should not be confused with an inverse.) Note thatSiandS+were eliminated (after the unitary transformation) by the constraints, and that S+ijust dropped out. (In other words, S+i= 0 was the gauge choice for the constraint Si= 0.) This leaves only Sij(andSi1in the massive case), whose representation is that of the highest-weight part of the original eld strength. A. OSp(1,1j2) 651 However, we can more simply choose the representation of Sijas our starting point, since it is just the transverse part of the gauge eld; it denes the representation of thePoincar e group. We therefore have an explicit construction of the generators of the Poincar e group, for arbitrary representations, dened on just the physical degrees of freedom, given directly by the little group SO(D 2) spin generators S ij(or SO(D1) generators SijandSi1in the massive case) that identify the representation. For example, in D=4, SO(2) has just one generator, the helicity, so for any state of a given helicity we know the action of the Poincar e generators. Excercise XIIA1.1 Check that this lightcone representation of the Lorentz generators satisesthe correct commutation relations. Classical free eld theory is easy to dene in the lightcone, since solving the constraints in the lightcone formalism has picked out just the physical components, so the only remaining constraint is the Klein-Gordon equation. Thus, the kinetic term foranymassless bosonic eld is simply 1 2(1 2),w h e r e1 2=@+@+1 2(@i)2, and@is considered the time derivative. (In general, the kinetic operator for a massless boson is some second-order dierential operator, which reduces to on the physical components.) For fermions we have instead =@+, since we must then have an odd number of derivatives to avoid getting a trivial result after integration by parts. (For a boson, @=@(1 22), for a fermion @@ =@( @ )(@ )2=@( @ ) by anticommutativity. In general, the kinetic operator for a massless fermion is some rst-order dierential operator, which reduces to =@+after eliminating auxiliary elds.) This quantum mechanical representation of the Lorentz generators has a simple translation into classical eld theory, in terms of eld theory Poisson brackets. The denition of Poisson brackets in lightcone quantum eld theory follows directly fromthe action: Dening as usual the canonical momentum as (minus, in our conven- tions) the variation of the Lagrangian with respect to the time derivative @ +(=@) of the variable , we nd the fundamental bracket for bosons =@+) [(x;xi);(x0;x0i)] =i1 @+(2)D=2(xx0)D2(xix0i) (Note that the @+was essential for the antisymmetry of the bracket. We can also evaluate its inverse as an integral, so1 @+(xx0)=1 2(xx0). For fermions we have instead an anticommutator and no 1 =@+.) We then nd that any quantum mechanical group generator J(including internal symmetries) can be represented in 652 XII. MECHANICS eld theoretic form as J=ZdxdD2xi (2)D=21 2@+J=i1 2hjJi where we have used the relativistic inner product of subsection VB2, but for a lightlike hypersurface: For positive-energy solutions h1j2i=ZdxdD2xi (2)D=2 1*1 2(i)$ @+ 2 Note that the free Poincar e generators are local in this form, from cancelation of @+'s. In interacting theories, the generator Ji, as well as the translation generator P, which is also the Hamiltonian, gets additional terms higher-order in the elds. In this manner, relativistic quantum eld theory can be quantized in a way that more resembles nonrelativistic eld theory than in non-lightcone methods, since is quadratic in the usual time derivative @0. We won't consider lightcone quantum eld theory further; however, in the following sections we'll use this construction to derive free gauge theory and its covariant quantization, in a way that we'll generalizestraightforwardly to interactions. Thus, the same construction directly gives the formulation of free representations of the Poincar e group, from eld strengths to transverse elds to covariant gauge elds. 2. Algebra From the denition of the graded determinant in terms of Gaussian integrals (see subsection IIC3), we see that anticommuting coordinates act like negative dimen- sions: For example, sdet(kI)=kabforacommuting and banticommuting dimen- sions. Thus, if we add equal numbers of commuting and anticommuting dimensions, they eectively cancel. Here we'll do the same for theories with spin, which allows the restoration of manifest Lorentz covariance to lightcone theories: Adding 2 com-muting and 2 anticommuting dimensions to SO(D 2) gives OSp(D1,1j2) (see also subsection IIC3), which has an SO(D 1,1) subgroup. We have seen that quantum eld theory requires unphysical anticommuting elds to cancel the commuting unphysical elds introduced by using gauges that do noteliminate longitudinal polarizations. For example, the gauge eld for electromag- netism has only D 2 components in the lightcone gauge, but needs to keep all D components to maintain manifest Lorentz covariance; this requires 2 \ghosts" to can-cel the 2 extra components of the gauge eld. The general result, at least for bosonic gauge elds, is to produce elds that form representations of OSp(D 1,1j2), includ- ing gauge elds and ghosts. Furthermore, by adding 2 anticommuting dimensions the A. OSp(1,1j2) 653 BRST transformations that relate the ghosts to the longitudinal degrees of freedom can be introduced in a natural way, as translations in the new coordinates. The result is that OSp(D1,1j2) multiplets are automatic, and gauge xing receives a geometric interpretation. In this section we'll see that an even more natural interpretation ofthese BRST transformations is as rotations of the anticommuting coordinates, and that they not only make gauge xing to the simplest Lorentz covariant gauge triv- ial, but also give a simple derivation of the gauge invariant action itself. (The two points of view are related in that translations can be considered as part of \conformalrotations", as we saw in subsection IA6.) The basic idea is very simple: Take the lightcone representation of the Poincar e generators, found in the previous subsection, and extend the SO(D 2) indices and representations to OSp(D 1,1j2) ones (including appropriate signs for the grading). Conversely, we can begin the construction with the \conformal" group OSp(D+1,3 j2), nd the equations of motion for the \Poincar e" group OSp(D,2 j2), and solve them for the \lightcone group" OSp(D 1,1j2). So we can use the same expressions for the generators, but now the \transverse" OSp(D 1,1j2) index is i=(a;) whereais a D-component index of SO(D 1,1) andis a 2-component index of Sp(2). The OSp(D1,1j2) metric is  ij=(ab;C) Furthermore, we divide up the full OSp(D,2 j2) index as (;i)=(A;a);A =(;) We now interpret the SO(D 1,1) subgroup that acts on the aindex as the usual physical one, since the generators take the usual covariant form (because all thetransverse generators are linear). The orthogonal subgroup OSp(1,1 j2) that acts on theAindex, and leaves the aindex alone, is then interpreted as a symmetry group of the unphysical degrees of freedom, an extension of BRST. However, the generators withindices are nonlinear, since the indices are no longer independent from the rest. (The + were gauged away, the were xed by equations of motion.) As a result, they act on transverse indices in a nontrivial way. longitudinal, nonlinear SO(1,1):  transverse, manifest OSp(D 1,1j2):i= a =A: ghost OSp(1,1j2) : Lorentz SO(D1,1) 654 XII. MECHANICS Since the OSp(1,1 j2) generators act only in the unphysical directions, all phys- ical states should be singlets under this symmetry. This is clear from the original construction: We started with linear generators for OSp(D,2 j2) (and translations and dilatations for (D,2 j2) dimensions), applied the equations of motion in terms of them, and now we apply the OSp(1,1 j2) singlet condition last. If we had instead applied the OSp(1,1j2) singlet condition rst to the (D,2 j2) dimensional space, we would have gotten the usual (D 1,1j0) dimensional space, and nally applying the equations of motion would have given us the lightcone results of subsection IIB3. manifest symmetry: &eld equations (x ) .BRST singlets (x A) %add 2+2 (extend i!(a;))OSp(D,2j2) %.& SO(D1,1) =) OSp(D1,1j2) eld strengths gauge elds/BRST &%. SO(D2) lightcone Explicitly, the OSp(1,1 j2) generators are (choosing w=0 ) J+=x@+;J+=x@+;J=x(@)+S J=x@+1 @+[1 2x(m2+@@)+Sb@b+S1im+S@] while the SO(D1,1) generators take their usual manifest form Jab=x[a@b]+Sab Excercise XIIA2.1 Write the general commutation relations of OSp(D 1,1j2). Specialize to the case OSp(1,1j2), in lightcone notation. Show that this representation satises them, paying special attention to signs. (Use the OSp(D 1,1j2) commutators for theS's.) We can also add a \spin" part to the general \orbital" part of the OSp(1,1 j2) alge- bra we have already derived, in the sense that the two parts commute and separately satisfy the commutation relations: JAB!JAB+~SAB The simplest choices are to choose this spin part to be just quadratic in new, \non- minimal" coordinates and momenta. It will prove convenient to perform some trans- formations J!UJU1that make the OSp(1,1 j2) generators more similar to what A. OSp(1,1j2) 655 they were before adding the spin parts. We therefore make two consecutive transfor- mations: J!U2U1JU1 1U1 2:U1=e~S+@=@+;U 2=(@+)~S+ to returnJ+andJ+to their previous forms. In fact, these are just the OSp(1,1 j2) version of the same transformations we used in the previous subsection to remove S+i andS+. The result is J+=x@+;J+=x@+;J=x(@)+^S J=x@+1 @+[x(K+1 2@@)+Q+^S@] Here we have K=1 2(m2);^S=S+~S;Q=~S+Sb@b+S1im~S+K but more generally we can satisfy the commutation relations by requiring only that K,Q,a n d ^Sare independent of the unphysical coordinates xandxand their momenta, and satisfy that their only nontrivial commutators are fQ;Qg=2K^S; [^S;Q ]=Q(C) ; [^S;^S ]=( (^S)) 3. Action We saw in the previous subsection that physical states are singlets under the OSp(1,1j2) BRST symmetry. It was introduced in a trivial way, but became nontrivial after solving the equations of motion; on the other hand, applying the singlet condition rst reproduced the usual lightcone analysis. Reversing the order of applying the two conditions has the advantage of allowing the physical state condition to be expressed as a single equation, which can be derived from an action. There are two ways of doing this: One is to use this algebra to generalize to gauge elds the rst-quantized BRST as applied to eld theory in subsection VIA3. Because of their quantum mechanical origin, the gauge-invariant  Q actions directly give a form suitable for choosing the Fermi-Feynman gauge, where the kinetic operator issimply m2. However, this is somewhat unusual for fermions, whose simplest eld equation is rst-order. (But it is useful for supersymmetry, where bosons and fermions are treated symmetrically.) As a result, this approach gives actions for fermions withan innite number of aux iliary and ghost elds. The most convenient way to discover the usual nite-component gauge-invariant rst-order actions hidden there is by per- forming an appropriate unitary transformation, after which this action (for bosons or 656 XII. MECHANICS fermions) appears as the sum of three terms: the usual gauge-invariant action, a term giving the usual second-quantized BRST transformations, and a nonderivative termthat would be considered nonminimal under second-quantized BRST. This approachwill be described in detail in the following sections. The other way is to dene a function in the generators of the group OSp(1,1 j2), and use it as the kinetic operator for the action: S=Z dx dx d2x1 4@+(JAB) where the integration is over all the coordinates appearing in the OSp(1,1 j2) gener- ators. (The dxpart is the usual dDx=(2)D=2.)@+comes from the usual relativistic inner product; it is also a \measure" factor, which is a consequence of our usinggenerators satisfying the pseudohermiticity condition J y AB@+=@+JAB Equivalently, we could redene  !(@+)1=2,J!(@+)1=2J(@+)1=2(assuming @+6= 0, as usual in lightcone formalisms) to eliminate it and restore hermiticity. (This would only aect J+!J+1 2,J!J+@=2@+, making hermitian the terms1 2fx;@+gand1 4fx;@@g=@+.) Because of the function, this action has the gauge invariance =1 2JBAAB Thus, the eld equations and gauge invariance reduce  to states in the OSp(1,1 j2) cohomology. More explicitly, the function can be written as @+(JAB)=@+(J2)(J+)2(J+)2(J) =(x)2(x)(^S2)@+2J2 w h e r ew eh a v eu s e d J+(J+)=(J+)J+=0)(J+)=1 @+(x) (There is freedom in ordering of the original functions: Reordering of any two 's produces terms that are killed by the other 's.) The(^S2) can be interpreted as a Kroneckers0in the Sp(2) \spin" s: 1 2^S^S=4s(s+1 ) A. OSp(1,1j2) 657 (remember ^Sis antihermitian, and i^S is always integer while scan be half- integer). The rest of the explicit 's are Dirac 's in the unphysical coordinates, which can therefore be trivially integrated out, leaving: S=Z dx Lgi;Lgi=1 2Kgi; K gi=1 2(+m2+1 2QQ) whereis  evaluated at x=x=s= 0. Furthermore, the remaining gauge invariance is =s01 2Q fromJ,s i n c eJ,J+,a n dJ+have been used to gauge to s=x=x=0 , respectively. Excercise XIIA3.1 Show explicitly that this action is invariant under the OSp(1,1 j2) gauge trans- formations. (Hint: Use the same method as excercise XA2.1.) 4. Spinors As we saw in subsection VIA3, the BRST algebra for the (Dirac) spinor requires nonminimal terms. For the general case of fermions we add these terms in the general way described in subsection XIIA2, choosing them in terms of a (second) set of OSp(1,1j2) matrices: ~SAB=1 2[~ A;~ Bg;f~ A;~ B]=AB where ~ A=(;;~;i~) in the notation of subsection VIA3. In particular, we nd ^S=S1 2~ (~ ) Q=Sb@b+S1im+~ [~ +~ +1 2(m2)] The next step for general massless fermions (and similarly for the massive case) is to apply Q2=1 2SaSb@a@b(~ ~ +K)~ Sa@a1 2K w h e r ew eh a v eu s e d ~ ~ =C1 2[~ ;~ ]=1 At this point we note that the gauge invariance generated by Q, for gauge parameter =~ , includes a term ~  that allows us to choose the gauge ~ +=0 658 XII. MECHANICS One way to think of this is to treat ~ +as an anticommuting coordinate and ~ as its derivative; another way is to treat them as 2 2 matrices. Alternatively, we can unitarily transform the action to contain just the ~ term of the operator: From the discussion of 2D matrices of subsection VIIIA7 we nd, including that part of the spinor metric, ~~ =1 00 0
then acts as a projection operator. Either way, the net result is to reduce the action to, now restoring the mass, Sf=Z dx Lgi;f;Lgi;f=1 2^Kgi;f^; K gi;f=1 2~ (Sa@a+S1im) where ^iswith the ~ -dependence eliminated (the top component in the above matrix representation). Thus, ^diers from the bosonic case in that it not only depends on xaand is a representation of Sij, but is also a representation of ~ ,w h i c h appears in ^Sto denes=0 . The only type of representation we have missed in this analysis is self-dual anti- symmetric tensors. In terms of eld strengths, these satisfy Fa1:::aD=2=1 (D=2)!a1:::aD=2b1:::bD=2Fb1:::bD=2 which is consistent, with Lorentz metric, if D=2 is odd (as seen from applying the tensor twice). A similar condition holds for the gauge eld in the lightcone gauge (with a (D2)-dimensional tensor). Because of the tensor, this condition can't be described by adding extra dimensions to the lightcone. However, the direct product of two spinors contains all antisymmetric tensors, and the rank D=2o n ec a nb ep i c k e d out by an appropriate OSp invariant constraint. The self-dual part of this tensor comes from the direct product of chiral spinors. Excercise XIIA4.1 We now consider this construction in more detail: aDerive the generalization of 1to OSp matrices, anticommuting with both the fermionic and bosonic 's,f 1; Ag= 0. We can use the usual product for the fermionic 's, but obviously the bosonic ones will need something dierent. (Hint: For each pair of fermionic or bosonic 's there is a Klein factor, as in excercise IA2.3e; for the fermionic 's the exponential is equal to the usual product.) bIn twice-odd dimensions, consider the direct product of two spinors by rep- resenting the OSp spin operators as a sum in terms of the two dierent sets A. OSp(1,1j2) 659 of OSp matrices acting on the two dierent spinor indices. Dene the U(1) (O(2)) symmetry that mixes the two matrices by taking linear complex combinations of the matrices to form fermionic creation and annihilation operators, so the OSp-invariant U(1) generator is ayAaA. Show that the eigen- values of this generator pick out the dierent Lorentz representations. These can be made irreducible by including 1projections. Using explicit U(1) and (both) 1projectors in the action, show that self-dual tensors can be described. (Note: This description contains an innite number of aux iliary elds.) 5. Examples The OSp(1,1j2) method is thus an ecient method for nding gauge-invariant actions (though not so useful for gauge xing). We begin with examples of massless bosons, for which the gauge-invariant kinetic operator is Kgi=1 2(+1 2QQ);Q=Sa@a The scalar is a trivial example; the simplest nontrivial example is the massless vec- tor: In terms of the basis jiifor an OSp(D1,1j2) vector (D-vector plus 2 ghosts), normalized to hijji=ij)hajbi=ab;hji=C we can write the OSp(D 1,1j2) generators as Sij (1)=j[iihj)j The Sp(2)-singlet eld is then (dropping the jiterm) =jaiAa(x) We then have Q=(jihajjaihj)@a)Q21 2QQ=jaihbj@a@b1 2jihj )Lgi(1)=1 8(Fab)2 and for the gauge invariance =s01 2Q) =ji(x))Aa=@a 660 XII. MECHANICS A more complicated example is the graviton (massless spin 2): We write the eld, a graded symmetric, traceless OSp(D 1,1j2) tensor, in terms of the direct product of two vectors, with basis jiijji. The spin operators are thus Sij=Sij (1) I(1)+I(1) Sij (1) where the rst factor in each term acts on the rst factor in jiijji,e t c . ;I(1)is the spin-1 identity. The s= 0 part of the eld is then =jiijjihji;hi i=ha a+h =0)=(jaijbi+1 2jijiab)hab wherehabincludes its trace. The rest is straightforward algebra; we use identities such as: Q2=Q2 (1) I(1)+I(1) Q2 (1)+Q (1) Q (1) Q (1) Q (1)=(jijihajhbj+jaijbihjhj)@a@b (hjhj)(jiji)=hjihji=2 whereQ2 (1)was evaluated above, and in the last identity we used the fact that jiis anticommuting. The nal result is then Lgi(2)=1 4[habhab+2 (@bhab)2ha ahb b+2ha a@b@chbc] in agreement with subsection IXB1. The original OSp(1,1 j2) gauge invariance reduces to =(jijai+jaiji)a)hab=@(ab) Excercise XIIA5.1 Consider a (D2)-rank antisymmetric tensor (i.e., totally antisymmetric in D2 indices in D dimensions; see excercises IIB2.1 and VIIIA8.2, and sub- section XA3). aShow from a lightcone analysis that it is equivalent to a scalar. Derive the gauge-invariant action using OSp methods. Find the gauge transformations and eld strength. bFind a rst-order form for the action, (auxiliary eld)2+ (auxiliary eld)  (eld strength). Show that eliminating the gauge eld as a Lagrange multiplerresults in the action for a scalar. Show that switching between scalar and antisymmetric tensor is equivalent to switching eld equation and constraint for the eld strength. cFind the description for the massive case by dimensional reduction. A. OSp(1,1j2) 661 Excercise XIIA5.2 Consider a tensor totally symmetric in its vector indices. In the lightcone gauge, the irreducible tensor is traceless. Show that, upon covariantiza-tion, the eld appearing in the gauge-invariant action satises a double-tracelessness condition (or equivalently the elds appearing there are thetotally traceless tensor and another totally tracelsss tensor with two less in-dices). For massless fermions we saw K gi;f=1 2~ Sa@a The next step is to use the fact that arbitrary fermionic representations are con- structed by taking the -traceless piece of the direct product of a (Dirac) spinor with an irreducible bosonic representation. (Just as an irreducible bosonic representationof an orthogonal group is found by taking the direct product of vectors, choosing anappropriate symmetry, as described by the Young tableau, and requiring the tracein any two vector indices to vanish; here we also require that using a matrix to contract the spinor index with any vector index also vanishes. Of course, simplermethods can be used for SO(3,1), but we need methods that apply to all dimensions,so they can be applied to orthosymplectic groups.) We then can write S ij=Sij1 2[ i; jg) ^S=Say(a) where Sijis the part of the spin acting on just the vector indices, and we have combined and ~ into creation and annihilation operators, as in subsection VIA3: a=1p 2( +i~ );ay=1p 2( i~ ); [a;ay]=  For spin 1/2 S=0 ,s0projects to the ground-state of the osc illators, and we immediately nd Sa=  a)Lgi(1=2)=1 4^ ai@a^ w h e r ew eh a v eu s e d ~  =i1 2(ayaaay )=i(aya1) =i(N+1 ) and this \N"c o u n t st h e ayexcitation level. (Note that the Hilbert-space inner product between the spinors includes the usual factor of 0.) A less trivial case is spin 3/2: Now =jiii;Sa=  a+j[iha]j 662 XII. MECHANICS whereihas an explicit vector index, and an implicit spinor index. Then from - tracelessness (for irreducibility) we have for the Sp(2) singlets ii=0)=  aa)aa=0;=1p 2ay  aa After a little algebra, using identities such as 1 6 [a b c]= a b c+1 2(b(a c)ac b) we nd Lgi(3=2)=1 12^a [a b c]i@b^c From inspection, or from =s01 2Q, we nd the gauge invariance ^a=@a Excercise XIIA5.3 Let's now examine some massive examples: aFind the gauge-invariant actions for massive spin 2 and spin 3/2 by dimen- sional reduction of the massless cases. bNote that for the spin-2 case the part of the mass term quadratic in his proportional toh[aahb]b. More generally, we might have expected ( hab)2+ k(haa)2for arbitrary constant k, since the rst part gives mass to the physical (transverse, traceless) part of h, while the second term aects only the unphys- ical pieces. Find the St uckelberg terms generated from this generalized mass term by the linearized gauge invariance. Looking at just the terms quadratic in the St uckelberg vector, what is special about k=1, and why do other values ofkgive ghosts? (Hint: Compare gauge-xed electromagnetism.) REFERENCES 1K. Bardak ci and M.B. Halpern, Phys. Rev. 176(1968) 1686: lightcone representations of Poincar ea l g e b r a . 2M. Fierz and W. Pauli, Proc. Roy. Soc. A173 (1939) 211; S.J. Chang, Phys. Rev. 161(1967) 1308; L.P.S. Singh and C.R. Hagen, Phys. Rev. D9(1974) 898; C. Fronsdal, Phys. Rev. D18 (1978) 3624; T. Curtright, Phys. Lett. 85B (1979) 219; B. deWit and D.Z. Freedman, Phys. Rev. D21 (1980) 358; T. Curtright and P.G.O. Freund, Nucl. Phys. B172 (1980) 413; T. Curtright, Phys. Lett. 165B (1985) 304: covariant actions for totally symmetric representations of all (4D) spin. A. OSp(1,1j2) 663 3W. Siegel and B. Zwiebach, Nucl. Phys. B282 (1987) 125: lightcone representations of Poincar e algebra, and covariant actions, for arbitrary spin and dimension. 4G. Parisi and N. Sourlas, Phys. Rev. Lett. 43(1979) 744: cancelation of extra dimensions. 5R. Delbourgo and P.D. Jarvis, J. Phys. A15 (1982) 611; J. Thierry-Mieg, Nucl. Phys. B261 (1985) 55; J.A. Henderson and P.D. Jarvis, Class. and Quant. Grav. 3(1986) L61: ghosts from extra dimensions for indices. 6W. Siegel, Nucl. Phys. B284 (1987) 632: 1st-quantized BRST for fermions. 7N. Berkovits, hep-th/9607070, Phys. Lett. 388B (1996) 743: 1st-quantized BRST for self-dual tensors. 8Siegel, loc. cit. 9Fierz and Pauli, loc. cit. (ref. 2 above): mass term for spin 2. 664 XII. MECHANICS ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: B. IGL(1) ::::::::::::::::::::::::::::::: Although the OSp(1,1 j2) method is the simplest way to derive general free gauge- invariant actions, it does not yield a simple method for gauge xing, even though the Hilbert space contains exactly the right set of ghosts. We now describe a related method that is slightly less useful for nding gauge-invariant actions (it includesredundant aux iliary elds), but allows gauges to be xed easily. 1. Algebra For this method we use a subset of the OSp(1,1 j2) constraints, and show they are sucient. A simple analog is SU(2): To nd SU(2) singlets, it's sucient to look for states that are killed by both T3and the raising operator T1+iT2. This approach gives a formalism that turns out to be easier to generalize to interacting theories, as well as allowing a simple gauge-xing procedure. We rst divide up the Sp(2) indices as=(; )( n o tt ob ec o n f u s e dw i t h ). We then make a similarity transformation that simplies some of the generators (while making others more complicated): J!UJU1:U=(@+)iJ which changes the Hilbert-space metric (and corresponding hermiticity conditions) to =UyU=(1)iJ ;h ji=Z y This simplies (looking at the massless case without ~SABfor simplicity) J+ !1 @+J+ =x J+!J+iJ ;J !J J!J@++iJ @=(x@+)@+1 2x+Sa@a+S@ (We use the same conventions for raising and lowering Sp(2) indices as for SU(2) in subsection IIA4 and SL(2,C) in subsection IIA5.) These four generators form the subgroup GL(1j1) of OSp(1,1j2) (=SL(1j2)): We can write the generators as JIJ, whereI=( +;). In subsection XIIB4 we'll see that the singlets of this GL(1 j1) are the same as those of OSp(1,1 j2). On the other hand, because of the simplied form of these generators, it's easy to see how to reduce the group even further: J+ =x =0;J++iJ =x@+=0 B. IGL(1) 665 )J =ix@+i+S ;J=1 2x+Sa@a+S@ (Of course, this further reduction could have been performed even without the trans- formation.) We are now left with the group IGL(1), with just J andJas gener- ators. (Jacts as translations for the GL(1) generator J . )A l s o ,w eh a v er e d u c e d the unphysical coordinates to just x. We now simplify notation by relabeling c=x;b =@;S3=iS ;J =iJ +1;Q =J )J=cb+S3;Q =1 2c+Sa@a+Sb JandQare versions of the ghost-number and BRST operators introduced in sub- section VIA1. The net result for obtaining these IGL(1) generators from the originalOSp(1,1j2) generators can also be stated as J=iJ  j@=0;@+=1;Q =Jj@=0;@+=1 where@= 0 can be regarded as a gauge condition for the constraint x =0 ,a n d @+=1f o rx@+(@=@(ln@+)) = 0. The IGL(1) algebra is [J;Q]=Q; Q2=1 2fQ;Qg=0 Excercise XIIB1.1 Show that any IGL(1) subgroup of OSp(1,1 j2) (J=iJ ,Q=J)s a t i s - es these commutation relations. Check that this nal representation of theIGL(1) algebra satises them. 2. Inner product The new inner product can be derived by the same steps: Starting with the lightcone inner product of subsection XIIA1, we add extra dimensions to get theOSp(D,2j2) inner product. We next drop dependence on x , which will be eliminated in the IGL(1) formalism. Then we perform the transformation with ( @+)iJ = (@+)J1used to simplify the BRST operator. This acts on both elds in the inner product; applying integration by parts turns one such factor into ( @+)J. The net eect is that it cancels the @+in the Hilbert-space metric, which allows us to drop thexintegration, and it introduces a factor of ( 1)J. Rather than dening a Hilbert-space inner product, which is sesquilinear, it is slightly more convenient to dene a symplectic inner product, replacing the Hermitian 666 XII. MECHANICS conjugate of the wave function/state on the left with the transpose, in analogy to an ordinary vector inner product. The inner product is then h ji=i(1) Z dx dc T(x;c)(1)J(x;c) A Hilbert-space inner product can then be dened simply as h *ji(where ( *)T= y). We have included a sign factor corresponding to what would be obtained if the dcintegration were moved to the symmetric position between the two wave functions: By (1) we mean take = 0 in the exponent if is bosonic and 1 if is fermionic. We can make this manifest by dening (x;c)=hx;cj i)I=iZ dx dcjx;ci(1)Jhx;cj which allows the inner product to be evaluated between h jandjiby inserting this form for the \identity" I. As a result, any commuting or anticommuting constant factor \ a"c a nb em o v e d out of the inner product from the left or right in the usual way: h jai=h jia;ha ji=ah ji As a consequence of the anticommutativity of the integration measure, we have (1)h ji=(1) ++1 meaning that the statistics of the inner product is the opposite of an ordinary product; we can think of \ j"a saf e r m i o n . Because of the change in metric from the lightcone, the IGL(1) generators now satisfy JT=1J; QT=Q where the constant comes from dropping the extra coordinates, and the transpose \T" indicates integration by parts (the usual transpose in the innite-matrix repre- sentation of operators): Z O=Z (1) O(OT ) Before our transformations the generators all satised GT=G;n o wt h e ya r ep s e u - doantisymmetric with respect to the metric ( 1)J, up to the constant: h jOi=h~O ji) ~O=(1)JOT(1)J B. IGL(1) 667 ~J=1J; ~Q=Q FromJT=1Jalso follows the symmetry property of the inner product: h ji=(1)( +1)(+1)hj i This can be interpreted as antisymmetry once the anticommutativity of the \ j" (metric) is taken into account. The hermiticity conditions that follow from the change from the lightcone are Jy=1J; Qy=Q Before all generators were antihermitian; now they are pseudoantihermitian, up to a constant: ^O=(1)JOy(1)J ) ^J=1J; ^Q=Q The factor of iin the inner product compensates for the funny hermiticity of ( 1)J. We then nd the usual hermiticity condition for a vector inner product, h ji*=hyj yi 3. Action As explained in subsection VIA1, we are interested in states in the cohomology of the BRST operator Q, which means states satisfying Q=0; =iQ In particular, the physical states are states in the cohomology of Qat ghost number J= 0. However, now Q = 0 is the wave equation (as in subsection VIA3), and Q contains the usual gauge transformations (as in the OSp(1,1 j2) action of the previous section). The free gauge-invariant action for an arbitrary eld theory is then S0=(1)Z dx dc1 2TJ1Q=(1)Z dx dc1 2TQJ0 for a real column-vector eld . (Complex elds can be decomposed into their real and imaginary parts. For relating to quantum mechanics, we will usually considerthe column-vector in our OSp Hilbert-space notation.) This action gives Q=0a s the equations of motion, and has =Q as a gauge invariance, so the solutions are the cohomology of Q. The projector  J0is a Kronecker restricting  to vanishing 668 XII. MECHANICS ghost-number (and thus Q to ghost number 1, since [ J;Q]=Q). As we'll see in the next section, this projector is redundant: The states in the cohomology with nonvanishing momentum automatically have vanishing ghost-number, and the stateswith nonvanishing ghost-number are needed for gauge xing. The projection is usefulonly for eliminating states that are redundant for discussing gauge invariance; we'lldrop it for the remainder of this section. The \complete" free action is then S0=(1)Z dx dc1 2T(1)J1Q=1 2hjiQi=S0y where we have included the inner-product metric. This is just the translation of the free BRST operator from rst- to second-quantized form, as for the lightcone in subsection XIIA1. We now consider some simple examples, to see how this method reproduces the usual results. The simplest example is the scalar: As shown in subsection VIA3, Q=c1 2(m2))Z dc1 2(1)J1Q=1 21 2(m2) without restrictions to vanishing ghost number, unitary transformations, gauge xing, etc. Thus the scalar is in no way a gauge eld: The kinetic operator follows from simple kinematic considerations. The fundamental example of a gauge theory is a vector: It is the dening repre- sentation of the Lorentz group, and of the extended Lorentz group we used to dene the BRST operator. In the rest of this chapter we will see from its action most of the general properties of gauge theories: ghosts, gauge invariance, BRST transformationsof the elds, the gauge-invariant action, gauge xing, backgrounds, mass, etc. Inaddition to the equations given for this case in subsection XIIA5, we will use h j i=h ji=i The BRST and ghost-number operators are Q=1 2c+(jihajjaihj)@a+2jihjb; J =cb+i(jih j+j ihj) The eld is real; it can't be called hermitian, since it is a column vector, but each component of that vector is hermitian. (This is the same as reality, but for anti-commuting objects reality includes extra signs that are dened to be exactly thosecoming from hermitian conjugation.) Thus, = *=(j aiAaij iC+iji~C)ic(jaiAaj i~CjiC) B. IGL(1) 669 T=y=(Aahaj+iCh ji~Chj)+( Aahaj~Ch jChj)ic where we denote the \antields" (those at order c)b ya\ " . The BRST transformations of the elds can be found by comparing terms in  andQ: If we dene a second-quantized BRST operator ^Qsuch thatQ= ^Q, but ^Qacts only on the elds while Q( a su s u a l )a c t so n jiiand the coordinates ( xandc), then =jii(iic i))Q= ^Q=jii[(1)i^Qiic(1)i+1^Q i] In other words, we compare terms in  and Q, and throw in a minus sign for transformations of fermions. (So, e.g., \ ^QAa" is the coecient of jaiinQ.) Dropping t h e\^"o n Q, the result is QAa=@aC; QC =0;Q ~C=2i(~C1 2@
A) QAa=i(1 2Aa@a~C);Q C=1 2~C+@
A; Q~C=1 2C Note that although Qis hermitian, it is antihermitian with respect to the inner- product metric (1)J, as expected from our convention of using antihermitian gener- ators for spacetime symmetries. (The same extra sign for hermiticity vs. pseudoher- miticity, also because of ghosts introduced by relativistic quantum mechanics, occursfor the spatial Dirac matrices i: see subsection XC2.) As a result, our transforma- tions agree with those of subsection VIA4. However, while the rst-quantized Abelian transformations also agree with those of subsections VIA1-3, the second-quantizednonabelian transformations will have the extra idemonstrated in subsection VIA4, following from the iintroduced in the inner-product metric in the previous subsec- tion. (This minor yet annoying factor will be further discussed in section XIIC whenwe relate rst- and second-quantized BRST.) The Lagrangian then can be expanded as (after some integration by parts) L 0=Z dc1 2T(1)J1Q=i1 2(A
QACQC~CQ~C+A
QACQC~CQ~C) =1 8(Fab)2+(~C1 2@
A)2i~C1 2C+iA
@C w h e r ew eh a v eu s e dt h et r a n s p o s eo ft h e e l do nt h el e f to f Q. To nd the gauge- invariant action, we can evaluate it by keeping just the (anti)elds with vanishing ghost number ( Aaand~C), and then eliminate the remaining antields by their equa- tions of motion: L!1 8(Fab)2+(~C1 2@
A)2)Lgi=1 8(Fab)2 670 XII. MECHANICS Excercise XIIB3.1 Consider the example of the second-rank antisymmetric tensor (see excercisesIIB2.1, VIIIA8.2, and XIIA5.1, and subsection XA3): aConstruct the states by direct product of two vectors. Decompose into elds plus antields, physical plus ghost: In particular, note the Sp(2) representa-tion of each SO(D 1,1) representation. bFind the BRST transformations for all the (anti)elds. In particular, note that the tensor transforms into vector ghosts (as expected from the gauge in-variance), which themselves transform into scalar ghosts (\ghosts for ghosts"). cGraph all the states for s(of the Sp(2) S )v s .J, and indicate there how BRST relates them. dFind the gauge-invariant action from  Q. eGeneralize to arbitrary-rank antisymmetric tensors. Compare the results of excercise XIIA5.1a. 4. Solution The identity of the cohomology and the physical states can be proven most easily by making a unitary (\gauge") transformation to the \lightcone gauge": (Q;J)!U(Q;J)U1:U=e(S+i@i+S+b)=@+ which simplies Qwhile leaving Junchanged: Q!1 2cS@+;J!cb+S3 These are the usual lightcone indices of any D-vector, not to be confused with the used earlier when reducing from D+2 bosonic dimensions. Except for the extension to include theindex, this is the same transformation used in subsection IIB3 (and XIIA1). This makes the generators separable, allowing us to treat the two terms in Qand Jindependently. Specically, if we integrate the action over c, =ic )Z dc1 2T(1)J1Q=1 4T(1)S3i T(1)S31S@+ Then is just a Lagrange multiplier enforcing the algebraic constraint S=0 (ignoring@+, which we always assume is invertible in the lightcone approach), leaving B. IGL(1) 671 just the Klein-Gordon term for the part of that satises the constraint. We also have the gauge invariance =+c;  =iQ)=iS@+;  =S@+1 2 so we can shoose the gauge where is restricted (algebraically) to be in the cohomol- ogy ofS. To solve for the cohomology of Sit is sucient to consider the reducible repre- sentations formed by direct products of vectors (for bosons), or the direct products ofthese with a single Dirac spinor (for fermions), since by denition the OSp(D 1,1j2) generators S ijdon't mix dierent irreducible OSp(D 1,1j2) representations. We'll show that this cohomology restricts any reducible OSp(D 1,1j2) representation to the corresponding reducible SO(D 2) lightcone representation, and therefore restricts any irreducible OSp(D 1,1j2) representation to the irreducible SO(D 2) represen- tation from which it was derived. (Also, the irreducible representations in arbitrarydimensions are most conveniently found by such a construction, where reduction is performed by symmetrization and antisymmetrization and subtracting traces of vec- tor indices, and in the fermionic case also subtracting gamma-matrix traces and usingMajorana/Weyl projection.) For bosons, we rst consider the representation from which all the rest are con- structed, the vector. Writing the basis for the vector states as j ii,w h e r eSijjki= j[iij)k, we nd S=0)notj+i;j i =S)notji;ji This leaves only the transverse lightcone states, as advertised. For the direct product of an arbitrary number of vectors, we nd the same result: The unphysical directions are eliminated from each vector in the product. We might worry that extra states in the cohomology would arise from a can- celation of two terms, resulting from the action of the \ "a n d\" parts ofS. Specically, this could happen if we could separate out the supertraceless part of (the graded symmetric part of) the product of two OSp(1,1 j2) vectors. (For exam- ple, for an SO(n) vector we can separate the traceless part of a symmetric tensor asT ij1 nijTkk.) However, this is not possible, since for OSp(1,1 j2) str(B A)=22=0 Explicitly, we can look at the two likely candidates for extra states in the cohomology, j+ijiijij i 672 XII. MECHANICS (and their transposes). But using Sj+i=ji;Sj i=iji for these states we nd S(j+iji+ijij i)=2jiji Sij+ij i=j+ijiijij i so neither state is in the cohomology. Note that we take Sto anticommute with ji; the states in the Hilbert space are assigned statistics. (This is the simplest way to allow a direct relation between wave functions and elds.) Note that in the Lagrangian 1 4Ti TS@+the elds in that are nonzero when acted upon by Sare auxiliary, killed by the Lagrange multiplier . On the other hand, the elds that are Son something are pure gauge, and do not appear in the Sterm because Sis nilpotent, while they drop out of the  term because the elds multiplying them there are exactly the auxiliary ones that were killed by varying . This follows from the fact that a eld that is pure gauge with respect to Shas a nonvanishing inner product only with an auxiliary eld, since1=S)21=2S. Equivalently, a eld redenition ! +A can cancel any terms in where oneisSon something. For the example of the vector, we have explicitly for the transformation to the lightcone lnU =1 @+ (j+ihijjiih+j)@i+(j+ihjjih+j)b under which the Lagrangian becomes L!L0=1 4A
Ai~C1 2C+~C@+AiA@+C The lightcone gauge transformations are A+=@+; ~C=1 2 ~C=@+;  A=1 2 C=1 2 ; A+=1 2+; Ai=1 2i where \i" here refers to the transverse (D 2) components. B. IGL(1) 673 5. Spinors In general we can add nonminimal terms ~SABof subsection XIIA2: The easiest way is to add them as the last step, remembering that Qcomes from JandJfrom J3; this yields Q!1 2c+Sa@a+Sb+~S;J!cb+S3+~S3 This result can also be seen from rst-quantization of spin 1/2 (subsection VIA3). Alternatively, if we add ~Sat the beginning as in subsection XIIA2, performing the transformations given there, followed by the transformation U=(@+)iJ of subsection XIIB1, where J itself now contains ~Sterms, we again nd J+ !1 @+J+ =x ;J+!J+iJ J !J ;J!J@++iJ @ Adding a nal transformation U=e~S+b (which actually undoes part of an earlier one), we again obtain the above result. As in the previous subsection, we can also transform to the lightcone gauge, to nd Q!1 2cS@++~S When analyzing the BRST cohomology in the lightcone gauge, the eect of this nonminimal term is to replace (again ignoring the factor of @+) SAB!^SAB=SAB+~SAB (although actually only the SandS3parts are used here). As in subsection XIIA4, when treating fermions we choose the Dirac spinor rep- resentation of OSp(1,1 j2) for ~SAB. Thus, for the case of spin 1/2, where SABalso is a Dirac spinor representation, ^SABis represented by the direct product of two OSp(1,1j2) spinors. We now use the harmonic oscillator interpretation of the ghost coordinates used in subsection XIIA5 (extending it trivially to the fermionic ones),which can be applied to arbitrary OSp groups: S AB=1 2[ A; Bg; ~SAB=1 2[~ A;~ Bg 674 XII. MECHANICS aA=1p 2( A+i~ A);ayA=1p 2( Ai~ A) )faA;ayB]=AB; ^SAB=ay[AaB) By expanding about the oscillator vacuum, we nd this representation of OSp(1,1 j2) consists of the direct sum of totally (graded) antisymmetrized tensors: j0i;jAi= ayAj0i;j[AB)i=ayAayBj0i;... . But we have already treated this case for the bosons, the result being that only the singlet (vacuum) survives. Of course, the complete spin representation is given by the direct product of the representation of the unphysical variables ( A,~ A) and the physical ones, namely the transverse lightcone gamma matrices. Thus, the states in the cohomology are given by the direct product of all the lightcone states with the vacuum of the unphysical variables. To treat arbitrary fermions, generalization to direct products of the Dirac spinor with arbitrary numbersof vectors works the same way, since the spinor looks like the direct sum of parts of direct products of vectors as far as ^S ABis concerned. 6. Masses As usual masses can be added by dimensional reduction: Our complete result for application to massless and massive, bosons and fermions is then Q=1 2c(m2)+Sa@a+S1im+Sb(+~S);J =cb+S3(+~S3) with extrai's introduced implicitly by the procedure given in subsection IIB4. For example, for the vector we have (the \St uckelberg formalism") Q=1 2c(m2)+(jihajjaihj)@a+(jih1j+j1ihj)m+2jihjb J=cb+i(jih j+j ihj) Compared to the massless case treated in subsections XIIB3-4, the corresponding eld now has the extra terms !+j1iicj1i giving the action L0=Lgi+[~C1 2(@
A+m)]2i~C1 2(m2)C+iC(@
A+m) Lgi=1 8(Fab)2+1 4(mA+@)2 For the spinor Q=1 2c(m2) (@=imp 2)( )2b~ ~ B. IGL(1) 675 J=cbi  i~ ~ where in the notation of subsection VIA3, =; =i; ~ =~; ~ =i~; ~ = 7. Background elds The coupling of external elds can be treated by suitable modication of the BRST operator. In terms of self-interacting eld theories, this corresponds to writing the eld as the sum of quantum and background elds, and keeping in the action only the terms quadratic in the quantum elds, as discussed for semiclassical expansionsin subsection VA2 and for the background eld method in subsection VIB8. One interesting case is the coupling of an external vector gauge eld. Clearly the spacetime derivatives in Qmust be modied by the minimal coupling prescription @!r =@+iA, but dimensional analysis and Lorentz covariance also allow the addition of a nonminimal term proportional to F abSabto . With the appropriate coecient, the general result is (see subsection VIIIA3) QI=1 2c(iFabSba)+Sara+Sb where is now the covariant r2.(Jis unchanged.) In the case of spin 0, this modication is trivial. For spin 1/2, we substitute the graded generalization of the Dirac matrices, Sij=1 2[ i; jg, as discussed in subsection XIIA5. We then nd Q2 I= 0 xes the above coecient of the nonminimal term, the same as from squaring ara. This follows from the simple factorization of SijinQI: QI=c( ara)2 ( ara)( )2b=( ara+ b)c( brb+ b) where we have neglected the ~Sterm of subsection XIIB5, and used [ a;c]=f ;cg=0 In the spin-1 case, we nd the interesting result that Q2= 0 requires not only the above coecient for the nonminimal term (as expected from supersymmetry), but also that the background terms satisfy the free eld equation @bFab=0 . O n the other hand, for spins >1,Q2= 0 implies Fab= 0, so these spins can't couple minimally (at least in at spaces). Similar remarks apply to coupling gravity (spin2) to spins >2. 676 XII. MECHANICS Excercise XIIB7.1 Check these statements for spin 1. Compare the analogous result for back- ground elds in the eld theoretic approach from excercise VIB8.2 for Yang- Mills for both the gauge transformations of the gauge-invariant action and theeld-theoretic BRST transformations of the gauge-xed action. Excercise XIIB7.2 Show that electromagnetism can't couple minimally to spin 2 (i.e., gravitons can't have charge) by considering Q2= 0 for spin 2 (symmetric traceless OSp tensor) in an external vector eld. Another interesting feature of the spin-1 case is that we can dene a \vacuum" state j0i=j i which is in the BRST cohomology only at zero momentum (constant eld), where Qsimplies to Sbwithout background. However, this state has ghost number J=1. In fact, it corresponds to the global part of the gauge invariance of the theory: Gauge parameters satisfying Q = 0 have no eect in the free theory (where = iQ), but can act in the interacting theory: They do not contribute an inhomogeneous term to gauge transformations. However, gauge parameters of the form  + Qh a v e the same eect as , up to trivial transformations proportional to the eld equations. Thus, while the BRST cohomology at J= 0 gives the physical states, that at J=1 gives the global invariances associated with the gauge eld. Now the physical states can be derived by operating on the vacuum with appro- priate vertex operators: If we expand QIin the number of external elds, QI=Q+V+::: Q2 I=0)fQ;Vg=0 QI=i[QI;])V=i[Q;] whereis the gauge parameter for the background eld. We therefore see that V is in the \operator cohomology" of Q. The states in the cohomology of Qare then given by =Vj0i;=j0i) Q=0;=iQ We can check this explicitly, as V=i1 2c(fAa;@agFabSba)+iAaSa)Vj0i=Aajai+ic1 2(@aAa)j0i The second term gives~C=1 2@
A, in agreement with the eld equations. B. IGL(1) 677 8. Strings Another interesting example is strings. Since rst-quantization is essential in string S-matrix calculations, it's natural to associate string eld theory with quantum mechanical BRST. As usual, for massive elds this formalism automatically includes the St uckelberg elds that would have been found by dimensional reduction, as well as all the ghosts. However, the explicit expression for the BRST operator does not explicitly correspond to that obtained by dimensional reduction: Although the spin operatorsSa,S,a n dS3are quadratic in oscillators, S1is cubic (because ^P, and thusX, is quadratic in the lightcone gauge). Nevertheless, the representation on any particular irreducible Poincar e representation contained among all the string states is the same as obtained by dimensional reduction, as follows from the generality of our analysis. As for any Poincar e representation, reducible or not, all we need is the lightcone spin operators, given for the general case in subsection XIIA1. In subsection XIIA2 we saw that the OSp(1,1 j2) generators followed immediately from just a change in notation. The IGL(1) generators were then found in subsection XIIB1 by a uni- tary transformation and solving half the contraints of GL(1 j1); the net result was equivalent to applying \gauge conditions" to the original OSp(1,1 j2) generators: J=iJ j@=0;@+=1;Q =Jj@=0;@+=1 For the string, the gauge condition @+= 1 simply removes the last vestige of X+, whose oscillator modes were already eliminated by the string lightcone gauge. On the other hand, the condition @=0m a k e sX=X (+)+X ()the sum of two conformally covariant objects: With the elimination of the linear term in the expansion of X, X ()are periodic in their arguments, and have the usual mode expansion in terms of exponentials only (no linear term). As a result, the decomposition of Qas obtained from the lightcone becomes trivial (Jwas easy anyway, since it's quadratic): Relabeling the result for the string's lightcone Lorentz generators from subsection XIA3, J=iZd 20(X. XX. X) (where now the \ i" comes from using the antihermitian form of the Lorentz spin), separating. Xinto its () pieces, using their \chirality". X ()=X0 ()to convert thederivative into a derivative, integrating by parts, and applying the denition ^P()=1p 20(. XX0) 678 XII. MECHANICS toX,w eo b t a i n Q=ir 2 0Zd 2X X ()^P () In this form we can easily apply the Virasoro constraint, as solved in the lightcone, ^P2=0; ^P+=p 20p+) ^P=i1 2p 20(^Pi)2 w h e r ew eh a v ea p p l i e d @+= 1. Finally, we relabel X ()=C() for purposes of identication with the usual BRST procedure in terms of ghosts Cand antighostsB(see subsections XIA4-5); comparison of the (equal-time) commutation relations then gives the further identication [^Pi ()(1);^Pj ()(2)g=iij20(21) ) [^Pi ()(1);Xj ()(2)g=iq 0 2ij2(21) fB()(1);C()(2)g=2(21) )B()=q 2 0^P () The nal result is then Q=1 X Zd 20C()(1 2^P2 ()iC0 ()B()) in agreement with direct rst-quantization of this string in the conformal gauge, in terms of the constraints ^P2 (). The ghosts can be separated into zero- and nonzero- modes as C()=1 2c+q 0 2Y ();B ()=2bq 2 0Y0 () (For the closed string there is also an extra zero-mode in CandB, enforcing the constraint that the \+" contributions to M2equal the \".) Excercise XIIB8.1 By separating zero-modes in the string's Q, and comparing with its generic expression Q=1 2c(M2)+Sa@a+iS1M+Sb show that Sij=iX Zd 2Yi ()Y0j () B. IGL(1) 679 Si1M=1 2p 20iX Zd 2Yi ()(Y0j ())2 M2=1 20X Zd 2(Y0i ())2 Of particular interest is the massless level: As mentioned in subsection XIB5, using the fact that the Hilbert space of the closed string is the direct product of theHilbert spaces of open strings gives a simple analysis of the massless states of any closed string, since the massless states of any open string are given by a vector (multi- plet) plus perhaps some scalars in the nonsupersymmetric case. To nd the completeo-shell structure, including auxiliary elds and ghosts, we can either take the direct product of the two lightcone representations and then add 2+2 dimensions, or rst add the 2+2 dimensions and then take the direct product of the two OSp(D 1,1j2) representations. In the supersymmetric case the latter is more convenient, since the procedure of adding dimensions to superspace is not yet understood, but quantization of the vector multiplet is (at least for N=1, and probably for N=2, in D=4). For example, for the bosonic closed string we just multiply two OSp vectors, producing a tensor t ij, which we can decompose into its symmetric traceless part hij(graviton plus ghosts), antisymmetric part Bij(axion plus ghosts), and trace  (physical scalar): tijjji jii!hij=t(ij]2 D2ijtk k;Bij=t[ij); =ti i (See subsection XIIA5.) However, we know that string theory prefers to treat elds in the string gauge for Weyl invariance, where the action (kinetic term) is not diag-onalized in the elds (until coordinate invariance is xed also). This is understood from this direct product structure: The \natural" string elds are string gauge :8 < :t (ab)graviton t[ab]axion tdilaton (T-duality invariant) since duality aects only the Xmodes, while the diagonal elds (representations of OSp(D1,1j2)) are normal gauge :8 < :hab=t(ab)2 D2ab(tcc+t ) graviton Bab=t[ab] axion =taa+t physical scalar Excercise XIIB8.2 Show that the above OSp analysis is consistent with the diagonalizing eld re- denitions of the low-energy string action found in excercise XIB5.2. Explain 680 XII. MECHANICS the result in terms of the redenitions 21pg!e2; 2gmn!pggmn In the heterotic case, as mentioned in subsection XIB6, we take the product of the real prepotential plus two chiral ghosts of super Yang-Mills with the usual vector plus two scalar ghosts of bosonic Yang-Mills: (V) (AaC)=Ha(Va)() The result is a vector prepotential Hadescribing the physical supergravity and tensor multiplets (in a string gauge, G= 1), a chiral scalar compensator (\superdilaton")  appropriate for \old minimal" supergravity, rst-generation ghosts (Sp(2) doublets) Vanda, and second-generation ghosts (an Sp(2) triplet, for the tensor multiplet) (). If the vector is accompanied by scalars, the closed string also has additional vector multiplets: (V) 'I=VII Background elds in string theory can be treated similarly to the previous sub- section: Since any open string includes Yang-Mills, we again choose the Yang-M ills ghost as the vacuum state. (In the bosonic case, even the tachyon can be obtained from it, by what would have been considered an annihilation operator with respect to the tachyonic vacuum.) Again vertex operators can be considered as additions to the BRST operator, are in the BRST operator cohomology, create states (in the state co- homology) from the vacuum, etc. For strings, such operators are local functions of thestring variables: They are associated with coupling an external eld at a particular value of(and). These local, fermionic vertex operators with ghost number 1 (so they can be added to Q) can be derived from related, integrated, bosonic operators with vanishing ghost number: [Q;W]=0;W=Z W) [Q;W ]=V 0 )fQ;Vg0=0)fQ;Vg=0 (Usually it is convenient to use conformal eld theory methods for such methods, so we use the notation of subsection XIB7, which mainly means dropping 's.) ThusRV is the only part of Vin the cohomology, the rest being a BRST variation: Moving Vto a dierent value of is a gauge transformation. Thus V() has explicit  dependence, while Qdoes not: QI=Q+V() B. IGL(1) 681 corresponding to a vertex local in (as found in subsection XIB7). The operators Ware those that appear with the gauge-xed Hamiltonian: HI=H+W()=fQI;g=fQ;g+fV;g where the gauge-xing operator is =Z (B+f); 2=0 the rst term in  giving the usual1 2(p2+M2)i nH, whilefis left arbitrary (as long as 2= 0 is preserved) to allow more general gauges. Thus, either of VorWcan be derived from the other: V0=[Q;W ];W =f;Vg ()fQ;Vg=[ ;W]=0 ) In general we have i1 2[^G(1);(2)] =(21)0(2) +w0(21)(2) )i[H;]=0;H =Z ^G for any conformally covariant operator, so assuming Vis one, ^G()=fQ;(B+f)()g; ^GI()=fQI;(B+f)()g ) [Q;W ]=[Q;f;Vg]=[fQ;g;V]=[H;V ]=V0 in agreement with the previous. (Similarly, we can plug V0=[Q;W ]i n t of;V0gto show the consistency of W=f;Vg.) In the notation of subsection XIB7, ^Gare the conformal generators T,a n d ^GIare^T, while fV(0);(B+f)()g=fW()=2(0)W(0) The simplest examples come from adding a background directly to the Virasoro operators (as for gravity in subsection XIB4): Then for the bosonic string V=CW(X;P);Q =Z C(1 2^P2+iC0B); =Z B; If we assume that Whas conformal weight 1 with respect to the gauge-xed conformal (Virasoro) generators, so that Wis conformally invariant, then Vhas conformal weight 0. 682 XII. MECHANICS 9. Relation to OSp(1,1 j2) For comparison to OSp(1,1 j2), we perform a unitarity (gauge) transformation on the IGL(1) action. We rst dene an almost-inverse of S:S i n c eSannihilates states with s3=s, where we dene 1 2SS=4s(s+1 );S3jsi=2s3jsi (so thatsands3take their usual integer or half-integer values), we can dene S 1 such that S 1S=1s3;s;SS 1=1s3;s SS 1S=S;S 1SS 1=S 1 We then apply the transformation Q!Qdiag=UQU1 Q=cK+Q+Sb; lnU =c[fS 1;QgS 1SQS 1] The exponent of Uis nilpotent from the c, so it generates only a linear term, U= 1+lnU. Using the commutation relations from subsection XIIA2 [S;Q]=0; (Q)2=KS we nd Qdiag=cs3;s(K+QS 1Q)s3;s+(cbQs3;s+bcs3;sQ)+Sb Now we apply the identity [S3;A]=0)s3;sAs3;s=As0 since any matrix element between hs;s3j:::js0;s03igivess=s3=s03=s0!s= s0=0 ,a sw e l la st h ef a c t s SQ s0=[S;Q ]s0=2iQs0)S 1Qs0=1 2iQ s0 fQ;Qgs0=0 This yields the nal result Qdiag=c(K+1 4QQ)s0+(cbQs3;s+bcs3;sQ)+Sb B. IGL(1) 683 Excercise XIIB9.1 Use the commutation relations of S,a sw e l la s1 2SS=4s(s+ 1), to derive S S=4 [s(s+1 )s3(s31)] from which follows the explicit expression S 1=1 S SS =1s3;s 4(ss3)(s+s3+1 )S Integrating over cin the action Sdiag=R1 2T(1)J1Qdiag as in subsections XIIB3-4, we nd the Lagrangian Ldiag=1 2T(K+1 4QQ)s0+i T(1)S31s3;sQ1 2 T(1)S3S (Note that in a product of the form Ta state of eigenvalue s3multiplies one of eigenvalues3,s i n c e(S3)y=(S3)T=S3.) We now see that in this action only the s= 0 (physical) part of appears in the term, while only the s3=s(\minimal") part (including physical) appears in the  term. The only part of that appears in the term is the s3=s(minimal) part of (the \antields" to the corresponding ones in), while all, but only, the remaining (\nonminimal") part of appears in the term. In particular, the term is recognized as the OSp(1,1 j2) action of subsection XIIA3. The terms involving can be eliminated by 's gauge invariance and eld equation, and contains no propagating degrees of freedom (elds with equations of motion), as can be seen by the methods used to analyze the cohomology in subsection XIIB4. However, the auxiliary elds , and the ghosts in (the s6=0 part of)to which they couple, are useful in gauge xing, as we'll see in the next section. Again looking at the example of the vector: S=2jihj)S 1=1 4S =1 2j ih j Q=(jihajjaihj)@a)lnU =i1 2c(j ihaj+jaih j)@a The transformation is then (cf. subsection XIIB3) Qdiag=1 2c(jaihbj@a@b)s0+cbjihaj@abcjaihj@a+2jihjb Ldiag=1 8(Fab)2+iA
@C+~C2 The BRST transformations now simplify to QAa=@aC; QC =0;Q ~C=2i~C 684 XII. MECHANICS QAa=i1 2@bFba;Q C=@
A; Q~C=0 Excercise XIIB9.2 Find the ghosts and simplied BRST transformations for massless spin 2. REFERENCES 1M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443; S. Hwang, Phys. Rev. D28 (1983) 2614; K. Fujikawa, Phys. Rev. D25 (1982) 2584: 1st-quantized BRST (for strings). 2W. Siegel, Phys. Lett. 149B (1984) 157, 151B (1985) 391: gauge-invariant actions from 1st-quantized BRST. 3E. Witten, Nucl. Phys. B268 (1986) 253; A. Neveu, H. Nicolai, and P.C. West, Phys. Lett. 167B (1986) 307: BRST operator as kinetic operator. C. GAUGE FIXING 685 ::::::::::::::::::::::: ::::::::::::::::::::::: ::::::::::::::::::::::: C. GAUGE FIXING ::::::::::::::::::::::: Although the quantum mechanical BRST operator is clearly useful for gauge x- ing, its relation to the second-quantized BRST we applied in chapter VI is not obvious,since the latter BRST operator does not include the gauge-invariant action. Here werelate the two, and extend the former to interacting eld theories. In particular, we show how the  Q action leads directly to the gauge-xed kinetic term as simply as it led to the gauge-invariant one, without applying any transformations. 1. Antibracket In the usual Hamiltonian formalism we work in a phase space ( q;p)o nw h i c hi s dened a Poisson bracket, useful for studying symmetry properties and equations ofmotion in the classical theory, and for relating to the commutator of the quantum theory (see subsections IA1-2). We want to interpret the present case of interest as an analogous phase space, for which the elds (in  =ic ) correspond to q and the antields top. This automatically follows from the lightcone commutator of subsection XIIA1, by the same steps used to derive the IGL(1) algebra and inner product h ji=i(1) Z dx dc T(x;c)(1)J(c)(x;c) in subsections XIIB1-2: We thus dene this generalization of the Poisson bracket (see subsection IA2) in terms of the inner product as (f[];g[]) =fg;=IJ*  I J where we have expanded the column vector  over a basis in the usual way (see subsections IB1,5), =jIiI; T=IhIj hIjJi=IJ=(1)IJI; (IJ)* =JI;IKJK=J I etc., and the indices I;J(not to be confused with the ghost-number operator J(c) appearing in the denition of the inner product above) run over all indices on theeld, which determine the statistics of the corresponding component as ( 1) I.( T h u s  is always bosonic, the statistics of the elds coming always from expansion over jIi.) This generalized commutator \( ;)" is called an \antibracket" because of the unusual statistics associated with it, following from the same unusual statistics of the innerproduct. (The ordering of indices on  IJin the antibracket is the opposite of usual 686 XII. MECHANICS to take into account the extra sign factor from the intervention of the anticommuting \j" between the two 's.) Plugging in the denition of the inner product, we have more explicitly =iZ dx dc  I(x;c)JI(1)J(c) J(x;c) Note that while the inner product is dened between two functions of the coordinates, the antibracket is dened between two functionals of , fandg, which don't depend explicitly on ( x;c) (although we can specialize to cases where they depend on other values of the coordinates ( x0;c0)). Thus the orbital term cbinJ(c) acts only on the argumentcof J(x;c) (and not on g), while the spin term S3acts only on the index Jof J(x;c). We have used the fact that = is antihermitian, as follows from the fact that the graded commutator between it and  is always a commutator, since they always have opposite statistics:  I(x;c)J(x0;c0)= I(x;c)J(x0;c0)y = I(x;c);J(x0;c0) =J I(x0x)(c0c) ) y = T =  Thus, fyy = ;fyy = f;  =f   Other properties of the bracket follow directly from those of the inner product: (1)(f;g)=(1)f+g+1 (f;ga)=(f;g)a; (af;g)=a(f;g) (f;g)=(1)(f+1)(g+1)(g;f) (f;gh )=(f;g)h+(1)(f+1)gg(f;h) (1)(f+1)(h+1)(f;(g;h)) +cyc:=0 (f;g)y=(gy;fy) (for some commuting or anticommuting constant a). Thus, the bracket has the exact opposite symmetry as the inner product (as is the case with the usual brackets): It would be symmetric in its two arguments if not for theRdcthat sits eectively C. GAUGE FIXING 687 between the two arguments. Most of the properties follow from this fact, and that the signs obtained from pushing things around are determined by moving things naively while treating the \ ;" in the middle as anticommuting. Furthermore, the existence of a bracket with these properties allows the denition of a Lie derivative, LAB(A;B) Excercise XIIC1.1 Find all the usual properties of this derivative (statistics, linearity, distribu-tivity, hermiticity, algebra, etc.), and relate to the usual Lie derivative. We also have functional identities such as  I(x;c)hj i=iJ I(1)J(c) I(x;c) (I(x;c);f[]) =i(1)J(c)IJ J(x;c)f[] from which follow (I(x;c);J(x0;c0)) =i(1)J(c)IJ(c0c)(x0x)=i(1)S3IJ(c+c0)(xx0) as well as ((x;c);hj i)= (x;c); (h ji;hji)=h ji Here and  are wave functions in the same space as , but need not be taken as bosonic (or real): We can even take them as functionals of  when applying the chain rule, using the above expressions for the terms where the ='s don't act on them. Expressions quadratic in  will be used to perform the second-quantized (or just classical eld theoretic) version of linear rst-quantized transformations: OA1 2hjAi) (OA;OB)=O[A;Bg;A =( ;OA) df=Z dx dc (1)IdI If) (OA;f)=Z dx dc (1)I(AI) If whereAandBmust satisfy [(1)J(c)A]T=(1)J(c)A)A=(1)J(c)AT(1)J(c) to give nontrivial contributions when appearing symmetrically between the two fac- tors of . Corresponding group elements come from exponentiating bosonic rst- quantized generators, yielding fermionic second-quantized generators: f=(OA;f)=LOAf)f0=eLOAf 688 XII. MECHANICS (1)A=1) (1)OA=1) (OA;f)=(f;OA) Clearly, the latter relations must hold when replacing OAwith their nonlinear second- quantized generalizations. We then nd (also for bosonic A) A=( ;OA)=(OA;) where the minus sign is the usual from translating rst-quantized to second-quantized language (see subsection IC1). Expanding ( ;) incas I=IicIJ(1)S3J we nd (I(x);J(x0)) =(J(x);I(x0)) =J I(xx0) This allows us to reexpress the antibracket as =Z dx(1)I0 @  I I+  I I1 A For example, the antibrackets of the component elds for the vector are =jiii=jii[jiAjic(1)S3Ai]=(jaiAaij iC+iji~C)ic(jaiAaj i~CjiC) (Ai(x);Aj(x0)) =(Aj(x);Ai(x0)) =j i(xx0) ) (Aa;Ab)=ab;(C;C)=;(~C;~C)= where now \ i" refers to the OSp(D 1,1j2) index (and we use C=A,~C=A , C=A,~C=A ). 2. ZJBV To prove gauge independence of the path integral, it's useful to draw an anal- ogy of relativistic quantum mechanical BRST to second-quantized BRST. (We'll seebelow that this is not just an analogy, but an equivalence.) Translating the BRST quantization of subsection VIA2 into path integral language, the general Lagrangian path integral for BRST quantization in quantum physics is A=Z Dq e iS0;S0=S+fQ;g C. GAUGE FIXING 689 whereqis all coordinates, including ghosts. While Sand  depend only on q,t h e BRST operator is linear in the conjugate momenta p| It generates a coordinate transformation: [Q;qmg=iQqm)Q=(Qqm)pm Here the index \ m" includes all dependence of q, including time. (We saw a more explicit expression of this result in subsection VIA2, assuming the constraints Giare themselves linear in the physical momenta.) We also have [Q;S]=0 whereScan include not only the gauge-invariant action, but arbitrary additional gauge-invariant pieces. Such pieces can be used to construct states in the BRST cohomology from the vacuum. (This is the path-integral translation of the operator construction given in subsection VIA1.) It can also include pure BRST variations,fQ; 0g. Thus, to prove gauge independence of A, we need only prove the vanishing of its variation under innitesimal change of , Z Dq eiSfQ;g=0 But this is trivial, since Qacts as a total derivative, and [ Q;S] = 0. More generally, we require only 0=1
Qi[Q;S]=i(Qqm) @m(Qqm)@mS where@m=@=@qm,a n d\ 1
Q" means the derivatives in Qact backwards onto the 1 (and itself), as found by integration by parts. In cases we have considered (and almost always), 1
Qand [Q;S] separately vanish. More generally, since there is a 1 =h multiplying Simplicitly, nonvanishing values would require a \quantum correction" toS. We can also write this condition as eiS Q=0 Furthermore, we can write 1
Q=i@2Q @pm@qm These manipulations can be applied to the eld theory expressions Jfor group generators, as found in subsection XIIA1 for the lightcone: The BRST operator asfound from these generators is S= 1 2hjiQi;Q2=0, (S;S)=0 690 XII. MECHANICS Then a unitary transformation can be implemented by exponentiating the innitesi- mal transformation Q=[G;Q],S=1 2hji(Q)i=(G;S);G=1 2hjGi Q0=eGQeG,S0=1 2hjiQ0i=eLGS We want to implement gauge xing by performing a unitary transformation on Qand then evaluating Sat the antields =0 : A=Z D eiS0j =0;S0=eLS)Sgf=(eLS)j =0 By similar manipulations to the BRST case, we see that gauge independence means 0=Z D eiS(S;) =iZ D(eiS;) where we evaluate this expression at = 0, and we have again included arbitrary gauge-invariant pieces in S. We thus obtain gauge independence from ( S;S) = 0, or more generally (again using integration by parts) 0=Z dx(1)I2 IIeiS=Z dx(1)I2S II+i1 2(S;S) This is the approach to BRST of Zinn-Justin, Batalin, and Vilkovisky (ZJBV). Excercise XIIC2.1 Find the unitary transformation, in ZJBV language, that transforms the\untransformed" action for the massive vector (that which gives the gauge- invariant action upon dropping antields) into the action that has only the vector eld (and not the scalar) upon dropping antields. Writing the BRST transformations in this second-quantized ZJBV notation will allow us to gauge x interacting theories (found by adding interaction terms to the freeS) in a gauge-independent way. In particular, it proves the equivalence of the manifestly unitary lightcone gauge (which has no ghosts, only physical degres of free- dom) to the manifestly Lorentz covariant Fermi-Feynman gauges (where the kinetic operator is simply m2). Specically, the  Q action is already unitarily trans- formed to the Fermi-Feynman gauge: Keeping just the terms, we have for a bosonic theory SFF=Sj =0=Z dxdc1 2T(1)J1c1 2(m2) =Z dx1 2T(1)S31 2(m2) In other words, the Fermi-Feynman kinetic term is just the sum over all elds (but not antields) of a m2term (using the OSp(D 1,1j2)-invariant inner product: C. GAUGE FIXING 691 the (1)S3is just a sign, and can be absorbed by a eld redenition). This can also be seen from the result for the complete  Q action after dropping the antield terms: For example, for a vector the gauge-xed (free) action is simply (see subsection XIIB3) LFF=1 4AaAai~C1 2C (The Fermi-Feynman gauge for fermions also gives a m2kinetic term, but with an innite number of ghosts; this may be useful for supersymmetry.) Excercise XIIC2.2 Let's again consider arbitrary-rank antisymmetric tensors (see excercisesXIIA5.1 and XIIB3.1): aFind the Fermi-Feynman actions. bDo the same for the massive case. (Note: There are more elds.) On the other hand, we saw in subsection XIIB9 that a unitary transformation, and evaluation at = 0, gave the gauge-invariant OSp(1,1 j2) action in terms of just the physical elds:  0=1 2hjc[fS 1;QgS 1SQS 1]i)S0=Sdiag Sdiag=Z dx[1 2T(K+1 4QQ)s0+i T(1)S31s3;sQ1 2 T(1)S3S ] Sgi=Sdiagj =0=Z dx1 2T(K+1 4QQ)s0 In the usual gauge-xing approach, we would start with Sdiagand do the inverse transformation to obtain the Fermi-Feynman gauge: =0)SFF=(eL0Sdiag)j =0 The same  can be used in the interacting case, since the eect on the quadratic piece of the action will be the same. Thus, to apply the usual ZJBV procedure we can either start with Sdiagand apply some  6= 0 sucient to x the gauge, or we can start withS(the one we found from quantum mechanical BRST) and apply some equivalent , or no  at all (for Fermi-Feynman gauge). To compare with the lightcone gauge, we start with Slc=Z dx[1 4(1)S3i (1)S31S@+] which was itself obtained by unitary transformation from S(see subsection XIIB4), and make a further unitary transformation, of the form  =Rdx1 2A,t h a th a s 692 XII. MECHANICS the eect ! +AforAsuch that all terms in containing auxiliary elds, and thus also pure-gauge elds, are canceled. For this transformed Slcwe then have Slc;diagj =0=Z dx1 4(SAB) where the projection operator (SAB)p i c k so u tt h es i n g l e t so f SAB(A=(;)), i.e., the transverse (physical) degrees of the light cone. A similar procedure can be applied in the interacting case. For the example of the vector: lnU =i1 2c(j ih+j+j+ih j)(@+)1 Qlc=1 2cS@+!Qlc;diag =1 2cjiihijS@+ Llc!Llc;diag =1 4AiAi+~C@+AiA@+C which has just the transverse (lightcone) degrees of freedom when the antields are dropped: Llc;gf=1 4AiAi 3. BRST In subsection VIA2, we saw that BRST could be used to gauge x by adding a BRST variation to the gauge-invariant Lagrangian. In that case, physical states are those that are not only in the BRST cohomology, but also satisfy the equations ofmotion. On the other hand, for relativistic mechanics we saw that the equations of motion are rather redundant, since is unphysical, and so p 2(+m2) = 0 is already included as a constraint, and contained in the quantum mechanical BRST operator. In fact,we have seen how in the most general case of a free eld the correct spectrum is specied by just the cohomology of the quantum mechanical BRST operator. We therefore want to identify the quantum mechanical BRST cohomology condi- tion with the combination of the second-quantized BRST cohomology condition andthe wave equation. This essentially has been accomplished in subsection XIIC2 by decomposing Q diagwith respect to c, as we'll now see by some further analysis. From subsection XIIB9 we have Qdiag=c(K+1 4QQ)s0+(cbQs3;s+bcs3;sQ)+Sb where the rst term gives Lgiin terms of the physical part ( s=0 )o f, the second term gives the minimal BRST transformations in terms of the minimal (anti)elds, C. GAUGE FIXING 693 and the last term adds the nonminimal stu needed for xing to general gauges. In particular, we see that the BRST transformation of the physical elds are (s0)s0Q=s0Q(s;1=2s3;1=2) (Note that this diers somewhat from the expression found from Q, since the trans- formation to Qdiagis eectively a redenition of , adding to it a piece proportional to an operator on .) Thus the only occurrences of the physical elds in Qdiagare in the term that gives the gauge-invariant action and the term that gives their BRST transformation;the remaining terms introduce nonminimal elds, as well as account for the BRSTtransformations of the ghosts. But this is the denition of BRST: Take the classicalaction in terms of physical elds, construct the BRST transformation from the gaugetransformation that leaves the classical action invariant, add terms to the BRST operator that insure its nilpotency on these ghosts, and add nonminimal terms to allow gauge xing. We have just seen that Q diagis exactly of this structure, where gauge xing gives the desired Fermi-Feynman gauge by unitary transformation to Q (which becomes a canonical transformation in second-quantized language, using theantibracket). All that is left to see is how the ZJBV combination of the gauge-invariant action with the BRST operator is equivalent to ordinary BRST. Expanding in antields, ZJBV gives the gauge xed action as S=S 0+ m(Qm)+ nm nm)Sgf=S0+(=m)(Qm)+(=nm)(=nm) w h e r ew eh a v eu s e dt h ef a c tt h a tt h et h r e et e r m si n Qdiagcontain only the physical, minimal (\m"; including physical), and nonminimal (\ nm") elds, respectively. In the usual ZJBV and BRST formalisms, derived from BRST without antields, thereis no 2term, since this generates a BRST transformation iQnm=(S;nm) nm One instead introduces further nonminimal elds, the \Nakanishi-Lautrup elds", such that Qnm=NL and use an extended gauge-xing function ^=+nmNL 694 XII. MECHANICS Then the gauge xed action is Sgf=S0+fQ;^g=S0+(=m)(Qm)+(=nm)NL+2 NL After eliminating the NL elds by their (algebraic) equations of motion, we obtain the same result as found from ZJBV. (Of course, the NL elds can also be introduceddirectly into the ZJBV formalism, but are redundant for purposes of nding Fermi- Feynman gauges.) Consider the special case of Yang-Mills: Generalizing our results for free Yang- Mills to the interacting case, making use of the BRST transformations of subsectionVIA4, we have L ZJBV =1 8(Fab)2+~C2+iA
[r;C]CC2 The basic antibrackets are (Aa;Ab)=ab; (C;C)=; (~C;~C)= From the general relations we saw earlier, and the denition of Sin terms of Qfor the free case, we have iQ=( ;S)=(S;) Since in the above we have pulled out factors to the left of the elds, as =jiii=jii[jiAjic(1)S3Ai] we pull them out of the left of the antibracket, to obtain i(Q)I=(S;I);i (Q)I=(S;I) where as before ( Q)I, etc., means to evaluate the corresponding component of Q and introduce the corresponding signs for eectively pulling those factors to the left. We then nd the previous results for the BRST transformations of the elds (by construction), but also those of the antields: QAa=[ra;C];Q C =iC2;Q ~C=2i~C QAa=i1 2[rb;Fba]+ifC;Aag;Q C=[r
;A]+i[C;C];Q~C=0 (Remember that the funny signs of the antibracket come from its symmetry, plus treating the comma in \( ;)" as anticommuting. Note the generic terms i[C;g.) Excercise XIIC3.1 Generalize the above results for the action and BRST transformations with antields when Yang-Mills is coupled to matter. C. GAUGE FIXING 695 BRST was described in a dierent way in subsection VIA4: Here we apply quan- tum mechanical BRST, and nd it equivalent to applying the ZJBV form of BRST to second-quantization. The ZJBV action consists of the gauge-invariant action, plus the antields times the BRST transformations of the elds, plus (antield)2terms. The dierence between the BRST transformations obtained by the general methodsof subsection VIA1 as applied to second-quantization, and those found in this chap- ter by applying OSp methods to rst-quantization of relativistic systems, is that the Nakanishi-Lautrup eld is treated as a eld in the former approach and as an an-tield in the latter. The two give equivalent results: The latter uses fewer elds, but is slightly more restricted in choices of gauge; however, this restriction is avoided in practice. (More \nonminimal" elds can be added to allow more general gaugechoices in either case.) For example, the ZJBV action for the former treatment of Yang-Mills can be obtained from that for the latter by the replacement ~C 2!~CB The gauge-xed action is the canonically transformed action (with respect to the antibracket) evaluated at vanishing antields: Sgf=eLSZJBVj Consider Yang-Mills in the most common type of gauge, where some function of Ais xed. From the usual BRST approach (see subsection VIA4), or the ZJBV approach withB, we nd =trZ 1 2~C[f(A)+1 2B])Lgf=Lgi1 2B[f(A)+1 2B]1 2i~C@f @A
[r;C] while in the ZJBV approach without Bwe have =trZ 1 2~Cf(A))Lgf=Lgi+1 4f(A)21 2i~C@f @A
[r;C] which is equivalent to the previous for positive (after elimination of B). REFERENCES 1Siegel and Zwiebach, loc. cit. (XIIA). 2Siegel, loc. cit. 3J. Zinn-Justin, in Trends in elementary particle theory ,e d s .H .R o l l n i ka n dK .D i e t z (Springer-Verlag, 1975) p. 2:introduced antields, antibracket, etc. 4I.A. Batalin and G.A. Vilkovisky, Phys. Lett. 102B (1983) 27, 120B (1983) 166; Phys. Rev.D28 (1983) 2567, D30 (1984) 508; Nucl. Phys. B234 (1984) 106; J. Math. Phys. 26(1985) 172: generalized Zinn-Justin's approach. 5W. Siegel and B. Zwiebach, Nucl. Phys. B299 (1988) 206: 2nd-quantized ZJBV from 1st-quantization. 696 ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: AfterMath ::::::::::::::::::::::::::::::: Conversions to other common conventions ab!ab; a!1p 2 a;S!S g2!g2 82[via1 2(m2)!m2;dDx (2)D=2!dDx]or nonabeliang2 162 Natural (Planck) units c=h=k==1 (G=) 1 e2=82 e2 ft=2 e2 m=2 = 861:022516(39);H1=1:5(2)1061 1kg=2:59194(17)107;1m=1:096782(70)1035;1s=3:28807(21)1043 1K=3:98170(25)1033; 1GeV =4:62054(30)1020 Indices a;b;c;::: |( a t )v e c t o r i;j;k;::: | transverse (D1o rD2) vector or internal m;n;p;::: | (curved) vector or large summation A;B;C;::: | ( at) super or conformal vector I;J;K;::: |i n t e r n a l M;N;P;::: | (curved) super A;B;C;:::| conformal spinor ;; ;::: ;;;;::: | spinor (usually 2-valued) or fermionic ;|i n t e r n a l 0|t i m e 1 | mass (dimensional reduction) ;t;t| lightcone (longitudinal; transverse) ; | spinor or spacecone reference line IntegrationZ dxZdDx (2)D=2;Z dpZdDp (2)D=2 (xx0)(2)D=2D(xx0); (pp0)(2)D=2D(pp0) hxjx0i=(xx0);hpjp0i=(pp0);hxjpi=eip
x;hpjxi=eip
x;pa=i@a onshell :hpjp0i=(pp0) 2[1 2(p2+m2)] 697 LORENTZ (I,II) m2=p2=papbab=(p0)2+(p1)2+(p2)2+(p3)2=2p+p+2ptpt E=p0=p0;p=1p 2(p0p1);pt=1p 2(p2ip3) ds2=dx2=dxadxbab;pads=mdxa;pad=dxa 2-spinor C=C=C.. =0 ii 0
;p. = p+ ptpt p = C; .= . C. .; 2=1 2  =i  =i   .=( )y) ( )y= .;( 2)y= 2=1 2 . .=i .  . A[]=AA=CC A ;A [ ]=0 ;V2=2det V =V. V .   .;. =CC.. ; .;. ; . ;. =i(CC C.. C. . CC C.. C. . ) V=V. )V*=V.;V W *+WV*=(V
W)I 0123=0123=1; (V;W;X;Y )=it r(VW*XY*Y*XW*V) 4-spinor =  . ;  = y=( .); a b=1 2ab+Sab r==0r.  r.0 ;=p 2 0=0 C..  C0 ; 1=1p 2 i 0 0i. .! a a=2; aa= a=a=; aa=b= a=a
b; aa=b=c= a=c=b=a= tr(I)=4;t r (a=b=)=2a
b; tr (a=b=c=d=)=a
bc
d+a
db
ca
cb
d =ji +j.].;  = y=hj+ .[.j . =ji[. jj. ]hj; +=jihj; =j.ih.j Superspace q=i@ @1 2. p .  ; q.=i@ @.1 2p. ;(q)y=q. d=@ @+1 2. p . ; d.=@ @.+1 2p.;(d)y=d. fq;q. g=fd;d. g=p. ;Z d2=d2=1 2dd; d2d2d2=1 2d2 698 ACTIONS (III,IV,XII) L=1 2.q2g(q)+.qA(q)+U(q);S =Z dt L LH=.qp+H(q;p);S H=Z dt Hdq p A=Z D eiS ;Wick :A=Z D eS ;S0 Mechanics OSp(1,1j2): K=1 2(m2);Q=Sa@a+S1im [+~ (~ ~ +K)] ^S=S(1 2~ (~ ));1 2^S^S=4s(s+1 ) S=Z dxLgi;Lgi=1 2TKgi; K gi=1 2(+m2+1 2QQ)( ^S=0 ) IGL(1): Q=1 2c(m2)+Sa@a+S1im+Sb(+~S);J =cb+S3(+~S3) S=(1)Z dx dc1 2T(1)J1Q=1 2hjiQi;Sgi=SjJ=0 SFF=Sjb=0=Z dx1 2T(1)S31 2(m2) Quantum ChromoDynamics Giy=Gi; [Gi;Gj]=ifijkGk; (Gi )A=(Gi)AB B ra=@a+iAa=@a+iAaiGi;i[ra;rb]=Fab=FabiGi=@[aAb]+i[Aa;Ab] L=1 8g2tr FabFab+L(r; );t rD(GiGj)=ij;t rA(GiGj)=2Nij QAa=[ra;C];Q C =iC2;Q~C=iB; QB =0;Q  =iC(for =i) Sgf=SgiiQ;=trZ 1 2~C(f+1 2B))Lgf=Lgi1 2B(f+1 2B)+1 2i~C(f)j=C LMajorana = ir. .+m 2p 2(  + . .)!1 4 (m2) 1 2 f  LDirac = (ir=+mp 2) =(  .ir. +.ir.)+mp 2( + ..) SsuperQCD =Z dx 1 g2trZ d2W2+Z d4+Z d2mp 21 22+h:c: 699 FEYNMAN (V) S=S0+SI;S 0=Z 1 2K ;AN=NY i=1Z Ni  Z[] Z[']=eW[']=Z D e(S0[]+SI[+'])=expZ 1 2 '1 K ' eSI['] Eective action [](unrenormalized): (E.g.,L=1 4(m2)+1 6g3.) (A1) 1PI graphs only (plus S0). (ForW[], connected graphs only.) (A2) Momenta: label consistently with conservation, withR dpfor each loop. (A3) Propagators: 1 =Kfor each internal line. (E.g., 1 =1 2(p2+m2).) (A4) Vertices: read o of SI.( E . g . ,g) (A5) External lines: attach the appropriate (o-shell) elds andR dp,w i t h(Pp). (A6) Statistics: 1/n! for n-fold symmetry of internal/external lines (or keep just 1 of n! related graphs); 1 for fermionic loop; overall 1. Vacuum: (Renormalize before for minimal subtraction/after for MOM.) (B1) Find the minimum of the eective potential (for scalars). (B2) Shift (scalar) elds to perturb about minimum; drop constant in potential.(B3) Find resulting masses; nd wave function normalizations. T-matrix: (C1) Connected trees of (shifted, renormalized) : (A2-4) for L=0 with S!. (C2) Amputate external 0-propagators. (C3) External lines: appropriate to 0wave equation ~K =0 . ( E . g . ,1 . ) (C4) External-line statistics: No symmetry factors; 1 for fermion permutation. Probabilities Sconnected =iX p T dP=jTfij2DX pY all(2)D=2 !Y outdD1p (2)D1;P=2 (ImTii)(2)D=2Y in(2)D=2 ! dP dt=2ImTii !;dP ds=2ImTii m=2ImM;dP d=2ImTii=ImM2 d=dP v12=jTfij2DX p(2)D 12Y outdD1p (2)D=21!; =2 (ImTii)(2)D=2 12 2 12=(p1
p2)2m2 1m22=1 4[s(m1+m2)2][s(m1m2)2] d dt=1 2(2)3jTfij21 2 12;s=(p1+p2)2;t =(p1+p3)2;u =(p1+p4)2 d d =( 2)2jTfij2j~p3j3 12[1 2(sm2 3m2 4)!3m2 3!4]=( 2)2jTfij234 12s(CoM ) 700 GAUGES (VI): Gervais-Neveu LA=1 4A
AiAaAb@bAa1 4AaAbAaAb YangMills :L=LA+LC;LC=1 2i~C(@+iA)2C1 2~CC(@
A+iA2) GervaisNeveu :L=LA+1 4m2A2 Twistors hpj=p;jpi=p;[pj=p.;jp]=p. hpqi*=[qp]=[pq];hpqihrsi+hqrihpsi+hrpihqsi=0 p=p+j+i[+j+pji[j+ptji[+j+ptj+i[j;h+i=[+] = 1 P=jpi[pj)p+=hpi[p];p=h+pi[p+];pt=h+pi[p];pt=hpi[p+] Spacecone n
A=0;n =j+i[j L=L2+L3+L4 L2=A+(1 2P2)A+ +1 2P2 p L3=p pA ([A;pA]+f +; g)+p p  [A; ] L4=( [A+;pA]+f +; g)1 p2([A;pA+]+f +; g)[A+; ]1 p[A; +] A+=[p] h+pi;A=h+pi [p]; +=[p]; =h+pi ref:lines :p pA+=p+ pA=p p +=p+ p =1 Background-eld !'+;r!D +iA; F ab!Fab+D[aAb]+i[Aa;Ab] @
A!D
A; ~C@
rC!~CD2C+~CD
i[A;C] 1loop:K=1 2(iFabSba) Supergraphs (A21 2)'s: one for each vertex, with anRd4. (A30) Propagators: (VV;;; ): 1;1;mp 2d2 1 2p2;mp 2d2 1 2p21 1 2(p2+m2)4(0) (A41 2) Chiral vertex factors: d2on theend(s) of every chiral propagator, d2on the end(s), but drop any one such factor at a superpotential vertex. 701 LOOPS (VII,VIII): Gamma function (z)=Z1 0d z1e=1 ze z1Y n=11 1+z nez=n=1 zexp" z+1X n=21 n(n)(z)n# = lim n!1 lnn+nX m=11 m! =0:5772156649 :::;  (z)=1X n=11 nz (z+1 )=z(z); (z)(1z)=c s c (z);(z) (2z)=212zp (z+1 2) (n+1 )=n!; (n+1 2)=(n1 2)(n3 2):::1 2p=(2n)! n!22np lim z!1(z)r 2 zz ez ;l i m z!1(az+b)p 2(az)az+b1=2eaz B(x;y)=Z1 0dz zx1(1z)y1=Z1 0d x1(1 +)xy=(x)(y) (x+y) Regularization (a) [1 2(p2+m2)]a=Z1 0d a1e(p2+m2)=2 1=Z1 0d  X i ;i=i Z dk ek2=2=1;Z dkka:::kb (1 2k2)a=0 Z dk(a) (1 2k2)a(b) [1 2(k+p)2]b=(a+bD 2) (1 2p2)a+bD=2B(D 2a;D 2b) Schemes MS : h MS : (D 2)h;(1)h; etc: G:(1)D0=2 (2D 2)B(D 21;D 21)h;(12) (1 +)[(1)]2h; etc: Running coupling 1;2gtrZ dx1 8Fab1(ln1 )Fab;1=1 2cR(1)2s(4s21 3);cF+F=2;cA=2N MtrZ dx1 8F 1lnM2+2 1ln 1lnM2 F;2 M2=e1=1g2(g2)2=2 1 702 GRAVITY (IX,X) [Mab;Vc]=V[ab]c)1 2ab[Mba;Vc]=caVa; [Mab;Mcd]=[c [aMb]d] [M12;V2]=22V1; [M12;M23]=22M13 1 2abMba=1 2M+1 2.. M. . [M; ]= (C) )1 2[M; ]=  ; [M;M ]=( (M)) ra=ea+1 2!abcMcb;ea=eam@m;[ra;rb]=Tabcrc+1 2RabcdMdc [r1;r2]=[e1+!1;e2+!2] =f[e1;e2]+(e1!2)M2(e2!1)M1g+f!1[M1;r2]!2[M2;r1]!1!2[M1;M2]g ds2=dxmdxngmn;gmn=emaenbab S=Z dxe1L; e=det eamL=1 4R+L(r; );R =Rabab Rab1 2abR=2Tab;Rab=Rc acb; SM=Z dxe1(emaebm)Tab Supergravity rA=EAM@M+1 2 A M +1 2 A. . M. . +iAAY; [Y;r]=1 2r [rA;rBg=TABCrC+1 2RAB M +1 2RAB. . M. . +iFABY fr.;r. g=BM.. ;fr;r. g=ir .  [r.;ir . ]=C. .W1 2(rB)M.. ; [ir.;ir . ]=C. .fh:c: W=BrG. r. +1 2(r. G. )M. . +1 2W M +iWY+i1 6WM f=i1 2G(. r). 1 2(r(B+i1 3W()r)+W r 1 2(r(G). )r. (1 2r2B+BB+1 12ir W )Mi1 8[(r(. G ). )M +$] +1 2W M +1 4(r(r. G). )M. . +i1 2(r(W))Y W =1 4!r(W ) Ga=Ga;r.B=r.W=r.W =0;r.G.=rBiW rW i1 3r(W )=i1 2r(.G ).;rW+r.W.=0 Ectoplasm S=Z dx(1 4!)mnpqeqDepCenBemALABCD Lcd=. ;.;cdL;Lbcd=i.;bcdr.L;Labcd=abcd[(r2+3B)L+h:c:] r.L=0)L =(r2+B)L 703 ::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::: INDEX ::::::::::::::::::::::::::::::::: Topics are listed by subsection (or section, etc.). Abelian:::::::::::::::::::::::::::: IB2 Action:::::::::::::::::::::::::::: III-IV Adjoint representation ::::::::::: IB2,5,C2 Advanced propagator ::::::::::: VA3,B1-2 Ane parametrization :::::::::::::: IIIB2 Ane transformation :::::::::::::::: IB1 Analytic S-matrix theory :::::::::::: XIA1 Anni'l'n op.IC1,IIB3,C5,IIIA2,VIA1,3,XIA4Anomaly ::::::::::: VIIIA8,B2-4,C1,XIA5 Antibracket ::::::::::::::::::::::: XIIC1 Anticommutation :::::::::::::::::::: IA2 Anticommuting number/variable :::::: IA2 Antield:::::::::::::::::::: VIA3,XIIB3 Anti-Gervais-Neveu gauge ::::::::: VIB4-5 Antighost::::::::::::::::::::::::: VIA2 Antiparticle :::::::::::::::: IA4-5,IIIB4-5 Antisym.tensor IIB2,VIIIA8,XA3,XIIA5,B3Arnowitt-Fickler gauge ::::::::::::: IIIC2 Asymptotic freedom ::::: VIIIA3-4,C,XIC3 Atiyah-Drinfel'd-Hitchin-Manin :::::: IIIC7 Auxiliary variable :::IIIA1,5,B2,C2,4,IVC2 Axial anomaly :::::::::::: VIIIA8,B2-4,C1 Axial current ::::::::::: IVA4,VIIIA8,B2-4 Axial gauge ::::::::::: IIIC2,VIB6-8,IXB3 Axial vector ::::::::::::::::::::::::: IA5 Axion:::::::::::::::::::::::::::::: XIB Background eld ::::: VC1,3,VIIB2,XIIB7 Background-eld gauge :::VIB8,10,VIIIA3 Bag::::::::::::::::::::::::::::::: IVB1 Baryon number ::::::::::::::::: IVA4,B1 Baryon::::::::::::::::::::::::::::: IC4 Bhabha scattering :::::::::::::::::: VIC4 Bianchi id.:IIA7,IIIC1,IVC3,IXA2,XB2,C5 Big Bang:::::::::::::::::::::::::: IXC3 Birkho's theorem :::::::::::::::: IXC5,7 Bjorken scaling variable ::::::::::: VIIIC3 Black hole::::::::::::::::::::::::: IXC7 Borel summation/transform :::::::: VIIC3 Bnd.cond. IC1,IIIA5,C,VA,VIA1,IXA5,XIABnd.state VC1,VIIB3,C2,VIIIA7,B4,XIA1,6Bohr-Sommerfeld quantization ::::::: VA2Bosonization ::::::::: VIIIA7,XIA5-6,B7-8 Bracket::::::::::::::::::::: IA1-2,XIIA1 BRST::::: VIA-B,VIIA1,IXB1,XIA4-5,XII Cabibbo-Kobayashi-Maskawa matrix IVB3 Cartan metric ::::::::::::::::::::::: IB2 Cartan subalgebra ::::::::::::::::::: IB5 Casimir operator ::::::::::::::::::::: IB2 Causality::::::::::::::::: VA4,C6,VIIA1 Center-of-mass frame ::::::: IA4,VC7,VIC4 Central charge ::::::::::::::::: IVC7,XC5 Channels::::::::::::::::::::::::: VIIB6 Chan-Paton factors ::::::::::::::::: XIB1 C. conj. IA5,IIC5,IIIA4,IVA4,B1,XC1,XIB2Chern-Simons formIIIC6,IVC5,VIIIB2,XC5Chirality :::::::::::::::::::::: IIA7,IVC1 Chiral projector ::::::::::::::::::: IIA6-7 Chiral representation :::::::::::::: IVC4-5 Chiral spinor :::::::::::::::::: IIA7,IIIC4 Chiral supereld :::::::: IIC5,IVC1-2,VIB7 Chiral superspace :::::::::::::::::: IVC1 Chiral symmetry ::::: IVA4,VIIIB2-4,XIC1 Chiral theory ::::::::::::::::::::::: IIA7 Chromodynamics ::::::::: IC4,IVB1,VIIC4 Classical group :::::::::::::::::: IB4,IIC3 Classical limit ::::::::::::: IA1-2,IIA2-3,V Clebsch-Gordan-Wigner coecients :::IIA4 Closed string ::::::::::::: IVB1,VIIC4,XI Coherent state ::::::::::::::::::::::: IA2 Cohomology ::::::::::::: VIA1,3,XB2,XII Coleman-Weinberg mechanism :::::: VIIB2 Colinear divergence/particle :::::::: VIIA6 Color:::::::::::::::::::::::::: IC4,IVA4 Color decomposition/expansion VC9,VIIC4Color ordering ::::::::::::::: VC9,VIC1-3 Compact group :::::::::::::::::::: IB2,4 Compactication :::::::::::::: IIB4,XIC3 Compensator :::::::::::: IVA5,IXA7,XA3 Complex projective ::::::::::: IVA2,VIIB3 Complex representation ::::::::::::: IB1-2 Component approach :::::::::::::::: XB4 Component transformation :::::: IIC2,XB3 704 Compton scattering :::::::::::::::: VIC4 Connement ::::::::::::::::::: IVB1,XIC Conformal::::::::::::: IA6,IIB-C,IIIC5-7 Conformal anomaly ::::::::: VIIIC1,XIA5 Conformal boost ::::::::::::: IA6,IIB1,6-7 Conformal eld theory :::::::::::::: XIB7 Conformal gauge ::::::::::::::::::: XIA3 Conformally at :::::::::::::::::::: IXC2 Connected graph :::::::::::::::::::: VC2 Conservation ::::::::::::: IA1,IIIB4,IXA6 Constituent quark ::::::::::::::::::: IC4 Constraint::::::::::::::::::::::::: IIIA5 Constructive quantum eld th. IIIC4,VIIC3Continuum limit :::::::::::::::: VA1,XIC Contour integral :::::::::::::::::::: IIA1 Contracted indices ::::::::::::::::::: IA1 Contraction :::::::::::::::::::::::: IIC3 Coordinate representation :::::::::::: IC1 Coordinate transform. :IA,C2,IIIB1,3,IXA1 Coset space :::::::::::::::::::::::: IVA3 Cosmological red shift :::::::::::::: IXC4 Cosmological term ::::: IXA5-7,C2,XB1,6-7 Cosmology::::::::::::::::::::::::: IXC3 Counterterm :::::::::::::::::::::: VIIA1 Covar.deriv. IIC2,IIIA4,C1,IVC3,IXA2,XA1Covariant momentum ::::::::::: IA5,IIIC1 Covering group :::::::::::::::::::::: IC5 CP(n) model ::::::::::::::::: IVA2,VIIB3 CPT theorem ::::::::::::::::: IVB1,VC8 Creat'n op. IC1,IIB3,C5,IIIA2,VIA1,3,XIA4Critical dimension :::::::::::::::::::: XI Crossing symmetry :::::::::::: VC8,VIC4 Cross product :::::::::::::::::::::: IIA1 Cross section :::::::::::::::::: VC7,VIC4 Current::::::::::::::: IIA7,IIIA4,B4,VC3 Current quark ::::::::::::::::::::::: IC4 Curvature::::::::::::::::::::::::: IXA2 Curved indices ::::::::::::::::::::: IXA1 Cut propagator ::::::::::::::::::::: VC7 Cutting rule :::::::::::::::::::::::: VC6 Deceleration parameter :::::::::::: IXC3 De Donder gauge ::::::::::::::::::: IXB1 Deep inelastic scattering ::::::::::: VIIIC3 Dening representation ::::::::::::::: IB4Degree of freedom :::IVC3-4,VB3,XC3,XII Density::::::::::::::::: IIIB1,IXA4,XIB5 Density parameter :::::::::::::::::: IXC3 De Sitter space ::::::::::::::::::: IXC2-3 Determinant ::::::::::::: IB3,C2,5,IIA1,5 Dierential cross section ::::::::::::: VC7 Dierential form ::::::::::: IC2,IXA4,XB2 Dilatation:::::::::::::::::::: IA6,IIB1,6 Dilaton::::::::::::::::::::::: IXA7,XIB Dim.red IIB4,IIIC8,VIB6,VIIA1,VIIIA3,XCDimensional regularization :::::::::::: VII Dimensional transmutation ::::::::: VIIB3 Dirac delta:::::::::::::::::::: IIIA1,VA2 Dirac equation :::::::::::::::: IIB2,IIIA4 Dirac matrices/spinor ::::::: IC1,IIA6,XC1 Direct product/sum :::::::::::::::::: IB2 Discrete symmetry ::::::::::::::::::: IA5 Divergence:::::::::::::::::::::::::: VII Division algebra :::::::::::::::::::: IIC4 Double covering ::::::::::::::::::::: IIA3 Drell-Yan scattering :::::::::::::: VIIIC3 DualityIIA7,IIIA4,C4,IXC1,XB5,C6,XIB4,6Dual model :::::::::::::::::::::::: XIA1 Dual representation/space :::::::::::: IB1 Dun-Kemmer matrices ::::::::::::: IIB4 Dust::::::::::::::::::::::::::: IXB2,C3 Dynkin index :::::::::::::::::::::::: IB2 Ectoplasm::::::::::::::::::::::::: XB2 Eective action ::::::::::::::::::::: VC2 Eective potential ::::::::::::::::: VIIB2 Einstein-Hilbert action :::::::::::::: IXA5 Einstein summation :::::::::::::::::: IA1 Elastic scattering ::::::::::::::::::: VC7 Electromagnetism :::IA5,C4,IIA7,B5,IVB2 Energy::::::::::::::::: IA1,5,IIA5,B6,C1 Energy-momentum tensor ::IIIA4,B4,IXA6 Equivalence principle ::::::::::::::: IXA4 Euclidean space ::::::::::: IA4,IIIC4,VB4 Euler number :::::::::: VIIC4,IXA7,XIC2 Euler-Mascheroni constant ::::::::: VIIA2 Euler's theorem ::::::::::::::::::: VIIC4 Event horizon :::::::::::::::::::::: IXC7 Evolution equations ::::::::::::::: VIIIC3 Exceptional group ::::::::::::::: IB4,XC6 705 Exclusive scattering ::::::::::::::: VIIIC3 Explicit supersymmetry breaking ::::IVC6 Explicit symmetry breaking :::::::::: IVA Extended supersymmetry ::::: IVC7,XC3-6 External eld :::::::::::::: IIIA1,B3-5,VC External line :::::::::::::::::::::::: VC2 Extrinsic length :::::::::::::::::::: IIIB1 Face::::::::::::::::::::::: VIIC4,XIC2 Factorization ::::::::::::::::::::: VIIIC3 Family:::::::::::::::::::::::: IC4,IVB3 Fayet-Iliopoulos term ::::::::::::: IVC5-7 Fermi-Feynman gauge ::VIB2-3,9,IXB1,XII Fermion::::::::::::::::::::::::::::: IA2 Fermion doubling :::::::::::::::::: XIC1 Feynman diagram/graph ::::::::::::: VC2 Feynman parameter ::::::::::::::: VIIA2 Feynman tree theorem ::::::::::::::: VC6 Field strength ::::::: IIA7,B3,5,IIIC1,XA2 Fine tuning::::::::::::::::::::::::: XB6 Finite theory ::::::::::::::::::: VIIIA5-6 Finite transformation :::::::::::::::: IA3 1st-order IIIA5,C4,IVC5,IXA5,XB4-5,XIA2Flat indices :::::::::::::::::::::::: IXA1 Flat superspace :::::::::::::::::::: IVC3 Flavor:::::::::::::::::::::::: IC4,IVA-B Flavor-changing neutral currents ::::: IVB3 Foldy-Wouthuysen transform. :IIB5,VIIIB6 Four-vector:::::::::::::::::::::::::: IA4 Fragmentation function ::::::::::: VIIIC3 Functional::::::::::::::::::::: IC1,IIIA1 Functional integral :::::::::::::::::: VA1 Functional derivative ::::::::::::::: IIIA1 Fundamental representation ::::::::::: IB4 Furry's theorem :::::::::::: VIIA5,VIIIB3 Galilean boost/group :::::::::::::::: IA1 Gamma matrix ::::::::::::: IC1,IIA6,XC1 Gauge::::::: IIIB2,C1,VIB,IXB,XA4,XIA3 Gauge eld:::::::::::::::::::::: IIA7,III Gauge invar. :IIA7,IIIA5,B1,C1,IXA1,XIA2 Gauge parameter :::::::::::::::: IIA7,III Gaugino::::::::::::::::::::::::::: IVC5 Gauss-Bonet theorem ::::::::::::::: IXA7 Gaussian::IB3,IIC3,VA2,5,B1,VIIA2,XIC3General linear group ::::::::::::::::: IB4 General relativity ::::::::::::::::::::: IX General supereld :::::::::::::::::: IVC1 Generating functional :::::::::::::: VC1-2 Generator::::::::::::::::::::::::::: IA1 Genus::::::::::::::::::::::::::::: XIC2 Geodesic::::::::::::::::::::::::: IXB2-4 Gervais-Neveu gauge :::::::::::::: VIB4-5 Gervais-Neveu model ::::::::::::::: IVA6 Ghost:::::::::::::::::::::::::: VIA,XII Ghost number ::::::::::::::: VIA1,XIIB1 Glashow-Iliopoulos-Maiani mechanismIVB3Glashow-Salam-Weinberg model ::::: IVB2 GLDAP equations :::::::::::::::: VIIIC3 Global symmetry/transformation :::::::: I Glueball:::::::::: IC4,IVB1,VIIC4,VIIIB4 Gluino:::::::::::::::::::::::::::: IVC5 Gluon:::::::::::: IC4,IIIC,IVB1,VIC,VIII Goldstone boson/theorem :::::::::::: IVA Goldstone fermion ::::::::::::: IVC6,XB6 Graded commutation ::::::::::::::::: IA2 Graded group/symmetry ::::: IA2,IIC3,XII Grand Unied Theory ::::::: IVB4,VIIIA4 Graphs::::::::::::::::::::::::::::: VC2 Gravitational collapse/radius :::::::: IXC7 Gravitational red shift :::::::::::::: IXC6 Graviton:::IIIA4,C1,IVA1,IXB1,C1-2,XIB Gravity::::::::::::::::::::::::: IIIA4,IX Green function ::::::::::::::::::::: VA-B Gribov ambiguity :::::::::::::::::: VIB2 Group:::::::::::::::::::::::::::: IA3-II Group contraction :::::::::::::::::: IIC3 Group metric :::::::::::::::::::::::: IB4 G-scheme::::::::::::::::::::::::: VIIA3 Hadron::::: IC4,IVA4,B1,VIIC4,VIIIC,XI Hamiltonian :::::::::: IA1,IIIA,VA1,VIA1 Hamiltonian density :::::::::::::::: IIIA3 Hamilton-Jacobi equations ::::::::::: VA2 Hamilton's equations ::::::::::::::::: IA1 Handle:::::::::::::::::::::: VIIC4,XIC2 Hard particle ::::::::::::::::::::: VIIIC3 Harmonic gauge :::::::::::::::::::: IXB1 Helicity::::::::::::::::::::::::: IIB7,C5 Heterotic string ::::::::::::::::::: XIB3,6 706 Hidden matter :::::::::::::::::::::: XB7 Higgsino::::::::::::::::::::::::::: IVC5 Higgs mechanism ::::::::::::::::::: IVA6 Higgs scalar :::::::::::::::::::: IC4,IVA6 Higher dimensions :::::::::::::::::::: XC Hilbert-space metric ::::::: IB4,VC5,XIIA1 Hole:::::::::::::::::::::::::::::: VIIC4 Homogeneous space ::::::::::::::: IXC3-4 Hubble constant :::::::::::::::::::: IXC3 Hypermultiplet/hypersymmetry ::::: IVC7 Ideal mixing :::::::::::::::::::::::: IC4 iprescription ::::::::::::::::::: VA3,B1 Improved perturbation ::::::::::::: VIIC1 Inclusive scattering ::::::::::::::: VIIIC3 Index notation ::::::::::::::::: IB,IIA4-5 Induced metric ::::::::::::::::::::: XIA2 Inelastic scattering :::::::::::::::::: VC7 Inhomogeneous group :::::::::::::: XIIB1 Innitesimal transformation ::::::::::: IA1 Innity:::::::::::::::::::::::::::::: VII Infrared divergence :::::::::: VIIA6,VIIIB Infrared slavery :::::::::::::::::::: IVB1 Inhomogeneous transformation ::::: IA4,B1 Inner product ::::::::::::::::: IB1,XIIB2 Instanton:::::::::::::::::: IIIC6-7,VIIC2 Interaction picture :::::::::::::::::: VA4 Internal line :::::::::::::::::::::::: VC2 Intrinsic length ::::::::::::::::::::: IIIB1 Inversion:::::::::::::::::::::::::::: IA6 Irreducible representation ::::::::::::: IB2 Isospin:::::::::::::::::::::::::::: IVB2 Isotropic space ::::::::::::::::::: IXC3-4 Jacobi identity :::::::::::::::::::: IA1,3 Jet:::::::::::::::::::::::::::: VIIIC2-3 JWKB:::::::: IA1,VA2,C3,VIB1,VIIIB1,6 Kahler manifold/potential :::::::::: XB6 Killing equation/vector ::::IXA2,6,B2,C4,6 Kinoshita-Lee-Nauenberg :::::::::: VIIA6 Klein factor/transformation :IA2,VC5,XC1 Klein-Gordon equation :::::::::::::: IIB1 Kronecker delta :::::::::::::::::::::: IA1 Kruskal-Szekeres coordinates :::::::: IXC7Lab frame::::::::::::::::::::::::: VC7 Ladder diagram :::::::::::::::::::: XIA1 Lagrange multiplier ::::::::::::::: IIIA1,5 Lagrangian:::::::::::::::::::::::: IIIA1 Lagrangian density ::::::::::::::::: IIIA3 Lamb shift::::::::::::::::::::::: VIIIB6 Landau equations/singularity :::::::: VC8 Landau gauge :::::::::::::::::::: VIB2-3 Landau ghost ::::::::::::::::::::: VIIC2 Laplace transform ::::::::::::::::: VIIC3 Lattice::::::::::::::::::::::::::::: XIC Legendre transform ::::::::::::: VC3,XB5 Length::::::::::::: IA4,IIIB1,IXB2,XIA2 Lepton::::::::::::::::::::::::: IC4,IVB Levi-Civita tensor :::::::::::::::: IA5,B3 Lie algebra/bracket/derivative/group ::IA3 Light bending :::::::::::::::::::::: IXC6 Lightcone::::::::::::::::::::::::::: IA6 Lightcone basis :::::::::::::::::::::: IA4 Lightcone coordinates :::::::: IA4,6,XIIA1 Lightcone frame ::::::::::::::::::: IIB3,6 Lightcone gauge :::::::::::: IIB3,IIIB2,C2 Lightcone spinor :::::::::::::::::::: XC4 Linear multiplet :::::::::::::::::::: XA3 Link::::::::::::::::::::::::: VC2,8,XIC Little group ::::::::::::::::::::::: IIB3-4 Local inertial frame ::::::::::::::::: IXC4 Locality::::::::::::::::::::::::: IIIA1,3 Local symmetry/transformation :::::::: III Loop graph :::::::::::: IIIB5,VC,VII-VIII Lorentz connection ::::::::::::::::: IXA2 Lorentz force law ::::::::::::::::::: IXB2 Lorentz gauge :::::::::::::::: VIB2,IXB1 Lorentz transf. :::::::::::::::: IA4-5,IIA5 Lowering operator ::::::::::: IA6,B5,VIA1 Magnetic charge :::::::::::::::::: IIIC8 Magnetic moment ::::: IIIC4,VIIA3,VIIIB5 Magnetic monopole ::::::::::::::::: IIIC8 Majorana spinor :::::::::: IIA6,IIIA4,XC2 Mandelstam variables :::::: IA4,VC4,VIC4 Manifest symmetry :::::::::::::::::: IB1 Maximal helicity violation ::::::::::: VIC1 Maxwell's equations :::::::: IIA7,B2,IIIA4 Mechanics:::::::::::::::: IA,IIIB,VB,XII 707 Mellin transform :::::::::::::::::: VIIC3 Meson::::::::: IC4,IVA4,B1,VIIC4,VIIIB4 Method of images ::::::::::::::::: XIA3,6 Metric:::::::::::::: IA4,B2,4,IIIA1,IXA2 Minimal couplingIA5,IIB5,IIIA2,4,C1,IXA4Minimal subtraction ::::::::::::::: VIIA3 Minkowski metric/space :::::::::::::: IA4 Modied minimal subtraction ::::::: VIIA3 Mller scattering ::::::::::::::::::: VIC4 Momentum integral :::::::::::::: VII-VIII Momentum subtraction :::::::::::: VIIA3 Monopole:::::::::::::::::::::::::: IIIC8 Nakanishi-Lautrup :::::::: VIA2,4,XIIC3 Natural units ::::::::::::::::: IA1,4,IXC4 Negative energy :::::::::: IA4-5,IIIB4,VB1 Neutrino::::::::::::::::::::: IC4,IVB2-4 New minimal supergravity ::::::::::: XB5 Newtonian limit ::::::::::::::::: IXB1,C6 Nielsen-Ninomiya theorem :::::::::: XIC1 Nonperturbative ::::::::::::::::: VIIC,XI Nonrelativity :::::::::::::::::::::::: IA1 Nonrenormalization theorem :::::::: VIC5 Normal ordering :::::::::::: VIA1,XIA4,6 No-scale supergravity :::::::::::::::: XB7 Null basis::::::::::::::::::::::::::: IA4 Octonion:::::::::::::::::::::: IB4,IIC4 O shell:::::::::::::: IIB1,IVC1,3,7,XC3 Okubo-Zweig-Iizuka rule ::::::::::: VIIC4 Old minimal supergravity :::::::::::: XB5 1/N expansion ::::::::::::::: VIIC4,XIC2 One-particle-irreducible graph :::::::: VC2 1.5-order::::::::::::::::::::::::::: XB4 On shell::::::::::::::::::::::: IIB1,XC3 Open string ::::::: IVB1,VIIC4,VIIIA3,XI Operator cohomology ::::::::::::::: VIA1 Operator product expansion ::::::: VIIIC3 Optical theorem :::::::::: VIIA4,6,VIIIC3 O'Raifeartaigh model ::::::::::::::: IVC6 Orthonormal basis ::::::::::::::::::: IA4 Orthosymplectic group ::::::::::: IIC3,XII Orthogonal group :::::::::::::::::::: IB4 Outer product :::::::::::::::::::::: IIA1 Overlapping divergence :::::::::::: VIIB7Pair creation::IIIB5,VIIB4,5,VIIIA2-3,C2 Parallel transport :::::::::::::::::: IXB4 Parity:::::: IA5,IIA5,IVA4,B1,VIIA5,XIB2 Parke-Taylor ::::::::::::::::::::: VIC1-3 Partially conserved axial current ::::: IVA4 Particles::::::::::::::::: IA,IIIB,VB,XII Parton:::::::::::::::::::::::::::: VIIIC Parton distribution/model ::::::::: VIIIC3 Path integral ::::::::::::::::: VA1-2,C1-4 Path ordering :::::::::::::::: IIIC2,XIC1 Pati-Salam model :::::::::::::::::: IVB4 Pauli-Luba nski vector::::::::::::::: IIB7 Pauli matrices :::::::::::: IIA1,IIIA2,XC1 Pauli-Villars regularization :::::::: VIIIB2 Penrose transform :::::::::::::: IIB6-7,C5 Perfect uid ::::::::::::::::::::::: IXC5 Perihelion precession :::::::::::::::: IXC6 Perturbation ::::::::::::::::::::: V-VIII Pfaan:::::::::::::::::::::::::: IB3,C5 Phase space :::IA5,VC7,VIIA5,B4-5,XIIC1 Photino::::::::::::::::::::::::::: IVC5 Physical region ::::::::::::::::::::: VC7 Pion:::::::::::::::::::: IC4,IVA4,VIIIB4 Pion decay constant :::::::::::::::: IVA4 Planar diagram ::::::::::::: VC8-9,VIIC4 Planck units :::::::::::::::::: IA1,4,IXC4 Plane wave::::::::::: IIIC3,VB2,C7,IXC1 Plaquet::::::::::::::::::::::::::: XIC1 Poincar eg r o u p::::::::::::::::::: IA4,IIB Poisson bracket :::::::::::::::::::::: IA1 Polar vector ::::::::::::::::::::::::: IA5 Positive action ::::::::::::::::: IIIB1,VA5 Positive energy ::::::: IA4,IIC1,IIIA4,IVC2 Power counting :::::::::::::::::::: VIIA5 Prepotential ::::::::::::::::: IVC1,4,XA1 Projective lightcone :::::::::::::::::: IA6 Projective transformation ::::::: IA6,IVA2 Propagator:::::::::::::::::::::::: VA-B Propagator correction :::::: VIIB4-7,VIIIA Proper Lorentz transformation :::::::: IA5 Proper time ::::::::::::::::::::::::: IA4 Pseudogoldstone boson ::::::: IVA4,VIIIB4 Pseudohermiticity :::::::::::::::::::: IB4 Pseudoreal representation ::::::::::::: IB2 Pseudoscalar/pseudotensor ::::::::::: IA5 708 QCD:::::::::::::: IVB1,VIC3,VIIIA3,C Quantum-chromodynamic string ::::: XIC3 Quark::::::::::::::::::::::: IC4,IVA4,B Quark-gluon plasma :::::::::::::::: IVB1 Quaternion:::::::::::::::::: IB4,IIA1,C4 Quenched approximation :::::::::::: XIC1 Radial gauge :::::::::::::::: VIB1,IXB4 Radiation:::::::::::::::::::::::: IXC3-4 Raising operator ::::::::::::::: IB5,VIA1 Rapidity::::::::::::::::::::::::::: IIA5 Real representation :::::::::::::::::: IB2 Recursion:::::::::::::::::::::::::: VIC2 Red shift:::::::::::::::::::::::::: IXC4 Reducible representation :::::::::::::: IB2 Reference momentum/vector :::::::: VIB6 Re ection::::::::::::::::::::: IA5,IIA3,5 Regge behavior/theory/trajectory :::XIA1 Regularization :::::::::::: VIIA-B,VIIIB2 Relativistic Schr odinger equation ::::: IIB1 Relativity:::::::::::: IA4,IIA5-B,IIIA4,IX Renormalizability :::::::::::::::: VIIA1,5 Renormalizable gauge ::::::::::::::: VIB3 Renormalization :::::::::::::::::::: VIIA Renormalization group :::::::: VIIB3,6,C1 Renormalization mass scale ::::: VIIA3,B3 Renormalon ::::::::::::::::::::::: VIIC2 Representation ::::::::::::::::::::: IB-C Representation space ::::::::::::::::: IB1 Residual gauge invar.VIA1,B2,IXB3,XIA3-4Resummation :::::::::::::::::::::: VIIC Retarded propagator :::::::::::: VA3,B1-2 Ricci scalar/tensor ::::::::::::::::: IXA2 Riemann normal coordinates :::::::: IXB4 Riemann zeta function ::::::::::::: VIIA2 Rotation::::::::::::::::::::::: IA1,IIA2 R symmetry :::::::::::::::::::::::: XA1 Running coupling :::::::::::::::::: VIIB6 Scalar eld::::::::::::::::::::::::: IA1 Scalar multiplet :::::::::::::::::::: IVC1 Scalar potential :::::::::::::::::::: IIIA1 Scale weight :::::::::::::::::::: IA6,IIB1 Scaling variable ::::::::::::::::::: VIIIC3 Scattering matrix ::::::::::::::::: VA4,CSchwarzschild solution :::::::::::: IXC5-7 Schwinger-Dyson equations :::::::::: VC3 Schwinger model ::::::::::::::::: VIIIA8 Schwinger parameter :::::::::: VB1,VIIA2 S-duality::::::::::: IIA7,IIIC4,IXC1,XC6 Self-duality::::::::::::::::::: IIIC4,IXC1 Semiclassical expansion ::::: IIIA4,VA2,C3 Semisimple group :::::::::::::::::::: IB2 Sesquilinear ::::::::::::::::::::::::: IB4 Sigma matrices ::::::::::: IIA1,IIIA2,XC1 Sigma model ::::::::::::::::::::: IVA2-4 Simple group ::::::::::::::::::::: IA6,B2 Simple supersymmetry :::::::::::::: IVC7 Singularity:::::::::::::: VC8,VIIC3,IXC7 Slepton:::::::::::::::::::::::::::: IVC5 SL(2,C):::::::::::::::::::::::: IC5,IIA5 S-matrix::::::::::::::::::::::::: VA4,C Soft divergence/particle :::::::::::: VIIA6 Sommerfeld-Watson transform ::::::: XIA1 SO(3):::::::::::::::::::::::::: IA3,IIA2 Source:::::::::::::::::::::::: IIIA4,VC3 Spacecone::::::::::::::::::: VIB6-7,C1-3 Spacetime lattice ::::::::::::::::::: XIC1 Special groups ::::::::::::::::::::::: IB4 Special relativity :::::::: IA4,IIA5-B,IIIA4 Spin:::::::::::::::::::::::::::::::::: II Spin operators ::::::::::::::::::: IIB,XII Spinor helicity :::::::::::::::::::: VIB2,6 Spinor index :::::::::::::::::::::::::: II Spinor notation ::::::::::::::: IB5,C5,IIA Spinor representation :::::::::: IC5,XC1-2 Spin-statistics theorem :::::::::::::: VC5 Splitting function ::::::::::::::::: VIIIC3 Spontaneous symmetry breakdown :::IVA1 Squark:::::::::::::::::::::::::::: IVC5 S-supersymmetry ::::::::::::::::::: IIC4 Standard Model ::::::::::::::::: IC4,IVB Stationary phase/steepest descent ::::VA5 Stirling approximation ::::::::::::: VIIC2 Stokes' theorem :::::::::::::::::::::: IC2 String::::::::::::: IVB1,VIIC4,VIIIA3,XI String gauge ::::::: IXB5,XA4,XIB5,XIIB8 String tension :::::::::::::::::::::: XIA2 Strong interaction :::::::::::::: IC4,IVB1 Structure constants :::::::::::::::::: IA3 709 Structure function :::::::::::::::::: IXA2 Stuckelberg formalism :::::::::: IIB4,IVA5 Subtraction scheme :::::::::::::: VIIA1,3 Summation convention ::::::::::::::: IA1 Superconformal group ::::::::::::::: IIC4 Supercoordinate ::::::::::::::: IIC2,IIIB1 Superdeterminant ::::::::::::::::::: IIC3 Supercial divergence :::::::::::: VIIA1,5 Supereld:::::::::::::::::::::::::: IIC2 Supergraph:::::::::::::::::::::::: VIC5 Supergravity ::::::::::::::::::::: X,XIB3 Supergroup::::::::::::::::::::::::: IIC3 Superhelicity ::::::::::::::::::::::: IIC5 Superhiggs::::::::::::::::::::::::: XB6 Superpotential ::::::::::::::::::::: IVC2 Superrenormalizable :::::::::: VIC5,VIIA5 Superspace::::::::::::::::::::::::: IIC2 Superspacecone ::::::::::::::::: VIB7,C3 Superstring :::::::::::::::::::::: XIB3,6 SupersymmetryIIC,IVC,VIC3,5,VIIIA5-6,XSupersymmetry breaking :::::::::::: IVC6 Supertrace:::::::::::::::::::::: IA2,IIC3 Supertwistor :::::::::::::::::::::::: IIC5 Super Yang-Mills ::::::::::::::::: IVC3-5 SU(2):::::::::::::::::::::::::: IA3,IIA2 Symplectic:::::::::::::::::::::::::: IB4 Symmetry:::::::::::::::::::::::::: I-IV Symmetry breaking ::::::::::::::: IVA,C6 Tadpole::::::::::::::::: VIB8,C5,VIIB1 T-duality:::::::::::::::::::::::::: XIB4 Temporal gauge :::::::::::::::::::: IIIC2 Tension::::::::::::::::::::::::::: XIA2 Tensor multiplet ::::: XA3,B5,XIB6,XIIB8 Tensor notation :::::::::::::::::::::: IB5 Tetrad:::::::::::::::::::::::::::: IXA2 'T Hooft ansatz :::::::::::::::::::: IIIC6 Three-vector :::::::::::::::::::::::: IIA1 Time development ::::::::::::::::::: IA1 Time ordering ::::::::::::::::: IIIA5,VA1 Time reversal ::::::::::::: IA5,IIA5,XIB2 Time translation ::::::::::::::::::::: IA1 T-matrix::::::::::::::::::::::::::: VC4 Topological expansion :::::::::::::: VIIC4 Torsion:::::::::::::::::: IVC3,IXA2,XA1Tree graph:::::::::::::::::::: VC2,XIA6 Triality:::::::::::::::::::::::::::: XIB8 Triangle diagram :::::::::::::::: VIIIB2-5 Twist:::::::::::::::::::::::::::: VIIIC3 Twistor::::IIB6-7,C5,IIIC5-7,VIB6-7,C1-3 Two-by-two matrix :::::::::::::::: IIA-B Ultraviolet divergence ::::::::::: VII-VIII Unitarity:::::::::::: IB4,VA4,C5-6,VIIA4 Unitary gauge :IIIA5,B2,C2,IVA6,VIA4,B3 Unitary group ::::::::::::::::::::::: IB4 U(1) problem :::::::::::::::::::::::: IC4 Vacuum:::::::::::: IC1,IIIB4,IVA,IXA5 Vacuum bubble ::::::::::::::::::: VIIB1 Vacuum value ::::: IIC5,IVA1,VIIC3,IXA5 Van Vleck determinant :::::::::::::: VA2 Vector:::::::::::::::::::::::: IA3-4,IIA1 Vector eld:::::::::::::::::::::::::: IC2 Vector multiplet :::::::::::::::::: IVC3-4 Vector potential :::::::::::::::::::: IIA7 Vector representation ::::::::::::::::: IC5 Velocity::::::::::::::::::::::: IA1,IIIB1 Vertex::::::::::::::::::::::: VC2,VIIIB Vertex operator ::::::: XIA6,B4,7,XIIB7-8 Vielbein::::::::::::::::::::::::::: IVC3 Vierbein::::::::::::::::::::::::::: IXA2 Virasoro constraints ::::::::::::: XIA2,B4 Volume element :::::::::::::::::::: IIIB1 Ward-Takahashi identity ::::::::::: VIB8 Wave equation ::::::::::::::::::::::: IIB Wave-function renorm. ::VIB8,VIIIA3,5,B5 W boson::::::::::::::::::::::: IC4,IVB2 Weak interaction ::::::::::::: IC4,IVB2-4 Weak mixing (Weinberg) angle :::::: IVB2 Wess-Zumino gauge :::::::::::::::: IVC4 Weyl scale::::::::::::::: IXA7,B5,XA3-4 Weyl spinor :::::::::::::::::::: IC1,IIA5 Weyl tensor :::::::::::::::::::::::: IXA3 White hole::::::::::::::::::::::::: IXC7 Wick rotation :::::::::::::::: IC5,VA5,B4 Wilson loop :::::::::::::::::::::::: XIC1 Window:::::::::::::::::::::::: XIA6,C2 Wino:::::::::::::::::::::::::::::: IVC5 710 Worldline:::::::::::::::::::::::::: IIIB1 Worldsheet::::::::::::::::::::::::::: XI Worldsheet lattice :::::::::::::::::: XIC2 X-ray:::::::::::::::::::::: just kidding Yang-Mills::::::::::::::::::::::::: IIICYoung tableau ::::::::::::::::::::::: IC3 Yukawa coupling ::::::::::::::::: IVA4,B Zinn-Justin-Batalin-Vilkovisky ::::: XIIC2 Zino:::::::::::::::::::::::::::::: IVC5 Zweibein::::::::::::::::::::::: XIA5,C3 :::::::::: :::::::::: :::::::::: Comments on Warren Siegel's Fields ::::::::::: \The price is right." \Oh, Warren, what have you done to your students now?"\I can see you put a lot of work into it."\That's nice, honey."\You might want to add a reference to my paper..."\It's dierent."\Is this going to be on the exam?"\ I ' l lh a v eal o o ka ti tw h e nIg e tt h et i m e . "\Aren't there enough eld theory books already?"\So this is why you haven't written any papers lately."\Where are the jokes?"