weinberg QFT vol 1

The Quantum Theory of Fields Volume I Foundations Steven Weinberg University of Texas at Austin InThe Quantum Theory of Fields Nobel Laureate Steven Weinberg com- bines his exceptional physical insight with his gift for clear exposition to provide a self-contained, comprehensive, and up-to-date introduction to quantum field theory . Volume I introduces the foundations of quantum field theory . The development is fresh and logical throughout, with each step carefully motivated by what has gone before, and emphasizing the reasons why such a theory should describe nature . After a brief historical outline, the book begins anew with the principles about which we are most certain, relativity and quantum mechanics, and the properties of particles that follow from these principles . Quantum field theory then emerges from this as a natural consequence . The classic calculations of quantum electrodynamics are presented in a thoroughly modern way, showing the use of path integrals and dimensional regularization . The account of renormalization theory reflects the changes in our view of quantum field theory since the advent of effective field theories . The book's scope extends beyond quantum electrodynamics to ele- mentary particle physics and nuclear physics . It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research . Problems are included at the end of each chapter . A second volume will describe the modern applications of quantum field theory in today's standard model of elementary particles, and in some areas of condensed matter physics . This will be an invaluable reference work for all physicists and mathe- maticians who use quantum field theory, as well as a textbook appropriate to graduate courses . The quantum Theor yof Fi elds Volume I Foundations Published by the Press Syndicate of the Un iversity of Cambridge The Pitt Building, Trumpington Street, Cambridge C132 I R P 40 West 20th Street ,New York ,NY 100 1 1-42 1 1, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australi a Oc Steven Weinberg 199 5 Firstpublished1995 Printed in t he United States of Ame rica Acatalogue record fbr this book is avai lable,from theBritish Librar y Library ofCongress cataloguing i npublication data Weinberg, Steven, 1933- The quantum theory of fie lds / Steven Wein berg. p. cm . Includes bib liographical reference a ndindex, Contents: v.1.Foun dations . ISBN 0-52 1-55(]01- 7 1. Quantum field theory . I. Title . Qc174 .45.w45 1995 530.1'43-dc20 95-2782 CI P ISBN 0 52155{101 7 TAG To Louise Content s Sections marked with an asterisk are somewhat out of the book's main line of development and may be omitted in a first reading . PREFAC E NOTATIO N 1 HISTORI CAL INTRODUCTIO N 1.1 Relativistic Wave Mechanicsxx xxv 1 De Broglie waves ❑Schrodinger-Klein-Gordan wave equation ❑Fine structure ❑Spin❑Dirac equation ❑Negative energies 0Exclusion principle ❑Positrons D Dirac equation reconsidere d 1.2 The Birth ofQuantum Field Theory 1 5 Born, Heisenberg ,Jordan quant ized field ❑Spontaneous emission❑Anticom- mutators ❑Heisenberg -Pauli quantum field theory 0Furry - -Oppenheimer quan- tization of Dirac field oPauli -eisskopf quantization of scalar field❑Early calculations in quantum electrodynamics ❑Neutrons ❑Meson s 1.3 The Prob lem o fInfinities 31 Infinite electron energy shifts ❑Vacuum polarization ❑Scattering of light by light 0 Infrared divergences D Search for alternatives ❑Renormalization 0 Shelter Island Conference ❑Lamb shift ❑Anomalous electron magnetic moment Schwinger, Tornonaga, Feynman, Dyson formalisms o why not earlier ? Bibliograph y References39 40 2 RELAT IVISTIC QUANTUM MECHANICS 4 9 2.1 Quantum Mechan ics 4 9 Rays❑Scalar products ❑Observables ❑Probabilitie s Ix x Content s 2.2 Symmet ries 5 0 Wigner's theorem ❑Antilinear and antiunitary operators 0 Observables ❑Gro up structure ❑Representations up to a phase ❑Superselection rules ❑Lie groups ❑Structure constants ❑Abe lian symmetrie s 2.3 Quan tumLorentz Transformations 5 5 Lorentz transformations o Quantum operators D Inversion s 2.4 The Pnincare Algebra 58 .TPV and PP ❑Transformation properties ❑Commutation relations ❑Conserved and non-conserved generators ❑Finite translations and rotations ❑Inonu- Wigner contraction ❑Galilean algebr a 2.5 o ne-Pa rticle St ates 6 2 Transformation rules ❑Boosts ❑Little groups ❑Normalization ❑Massive particles ❑Massless particles ❑Helicity and polarizatio n 2.6 Space Inversion and Time-Reversal 74 Transformation of JP' and PP 0P is unitary and T is antiunitary ❑Massive particles ❑Massless particles ❑Kratners degeneracy DElectric dipole moment s 2.7 Project iveRepresentations' 81 Two-cocyles ❑Central charges ❑Simply connected groups ❑No central charges in the Lorentz group ❑Double connectivity of the Lorentz group ❑Covering groups ❑Superselection rules reconsidere d Appendix A T he SymmetryRepresentationTheorem 91 AppendixBGroup Operators and Homotopy C lasses 9 6 Appendix C Inversions and Degenerate Multiplets 100 Problem s References104 105 3 SCATTERING T HEORY 107 3.1 'In' a nd`hut' States 107 Multi-particle states ❑Wave packets ID Asymptotic conditions at early and late times❑Lippmann- Schwinger equations ❑Principal value and delta function s 3.2 The S-matrix 113 Definition of the S-matr ix,❑The T -matrix ❑Born approximation CI Unitarily ❑f the S-matri x 3.3 Symmetries of the S-Matrix 116 Lorentz invariance ❑Sufficient conditions ❑Internal symmetries 0Electric charge, strangeness, isospin, S U(3) ❑Parity conservation ❑Intrinsic parities ❑ Contents x i Pion parity ❑Parity non-conservation ❑Time-reversal invariance o Watson's theorem ❑PT non-conservation ❑C, CP, CPT❑Neutral K-mesons ❑CP non- conservatio n 3.4 Rates and Cross-Section s 13 4 Rates in a box 0 Decay rates ❑Cross-sectio ns❑Lorentz invariance ❑Phase space ❑Dalitz plot s 3.5 Perturba tion Theo ry 141 Oid-fashioned perturbation theory ❑'Tfime-dependent perturbation theory C] Time-ordered products ❑The Dyson series ❑Lormtz-invariant theories o Dis- torted wave Born approximatio n 3.6 Implications of Unitarity 147 Optical theorem ❑Diffraction peaks o C PT relations o Particle and antipartic le decay rates o Kinetic theory ❑Boltzmann H-theore m 3.7 Partial-Wave Expansions* 151 Discrete basis ❑Expansion in spherical harmonics ❑Total elastic and inelastic cross-sections O Phase shifts ❑Threshold behavior : exothermic, endothermic, and elastic reactions O Scattering length ❑High-energy elastic and inelastic scatterin g 3.8 R esonances' 15 9 Reasons for resonances : weak coupling, barriers, complexity ❑Energy- dependence ❑Unitarity ❑Brefit-signer formula ❑Unresolved resonances ❑ Phase shifts at resonance r] Ramsauer-Townsend effec t Problems References165 166 4 THE CLUSTE RDECOMPOSITION PRINCIPLE 16 9 4.1 B osons and Fer mions 170 Permutation phases o Bose and Fermi statistics ❑Normalization for identical particle s 4,2 Creatio nandAnnihil ation op erators 173 Creation operators 0 Calculating the adjoins ❑Derivation of commutation/ antic ornmutation relations El Rep resentation of general operators El Free-particle Hamiltonian ❑Lorentz transformation of creation and annihilation operators ❑ C, P, T properties of creation and annihilation operator s 4.3 C luster Decomposition a ndConnectedAmplitudes 177 Decorrelation of distant experiments ❑Connected amplitudes ❑Counting delta functions xii Content s 4.4 Structure ofthe I nteraction 182 Condition for cluster decomposition ❑Graphical analysis 0 Two-body spattering implies three-body scatterin g Problems References189 189 5 QUANTUM F IELDS AND ANTIPART ICLES 19 1 5.1 Free F ields 191 Creation and annihilation fields ❑Lorentz transformation of the coefficient func- tions❑Construction of the coefficient functions ❑Implementing cluster decom- position ❑Lorentz invariance requires causality ❑Causality requires antiparticles ❑Field equations ❑Normal orderin g 5.2 Causal Scalar Fields 201 Creation and annihilation fields ❑Satisfying causality ❑Scalar fields describe bosons ❑Antiparticles ❑P, C, T transformations ❑iro 53 Ca usalVect or Fi elds 207 Creation and annihilation fields D Spin zero or spin one ❑Vector fields describe bosons ❑Polarization vectors ❑Satisfying causality ❑Antiparticles ❑Mass zero limit❑P, C, T transformation s 5.4 Th eDirac Fo rmal ism 213 Clifford representations of the Poincare algebra ❑Transformation of Dirac matri- ces C7 Dimensionality of Dirac matrices ❑Explicit matrices ❑t5❑Pseudounitarity ❑Complex conj ugate and transpos e 5.5 Ca usalDirac F ields 219 Creation and annihilation fields ❑Dirac spinors ❑Satisfying causality ❑Dirac fields describe fermions ❑Antiparticles ❑Space inversion ❑Intrinsic parity of particle-antiparticle pairs ❑Chorge-conjugation ❑Intrinsic C-phase of particle-- antiparticle pairs ❑Majorana fermions 0Time-reversal ❑Bilinear covariants El Beta decay interaction s 5.6 General Irreducible Representations of the Homogeneous Lorentz Group' 22 9 Isomorphism with S U(2) @ SU(2) d (A, B ) representation of familiar fields ❑ Rarity-Schwinger field ❑Space inversio n 5.7 General Causal Fields' 233 Constructing the coefficient function s❑Scalar Hamiltonian densities ❑Satisfying causality 0Antiparticles ElGeneral spin-statist icsconnection ❑Equivalence of different field types DSpace inversion ❑Intrinsic parity of general particle - antiparticle pair s El Charge-conjugation ❑Intrinsic C-phase of antipart icles❑ Contents Xiii Self-charge-conjugate particles and reality relations ❑Time-reversal ❑Problems for higher spin ? 5.8 The CPT T heorem 24 4 CPT transformation of scalar, vector, and Dirac fields ❑CPT transformation of scalar interaction density ❑CPT transformation of general irreducible fields ❑ CPT invariance of Hamiltonia n 5.9 Mass lessParticle Fie lds 24 6 Constructing the coefficient functions ❑Na vector fields for helicity ±1 ❑Need for gauge invariance ❑AntisymmeEric tensor fields for helicity ±1 ❑Sums over helicity ❑constructing causal fields for helicity ±1 0 Gravitons o Spin >_3❑ General irreducible massless fields ❑Unique helicity for (A,B} field s Problems References255 256 6 THE FEYNMA NRULES 25 9 6.1 Derivation of the Rules 259 Pairings ❑Wick's theorem 0 Coordinate space rules g Combinatoric factors ❑ Sign factors C] Example s 6,2 C alculation of th e Propagator 274 Numerator polynomial ❑Feynman propagator for scalar fields ❑Dirac fields ❑General irreducible fields ❑Covariant propagators ❑Non-covariant terms in time-ordered product s 6.3 Momentum Space Rules 280 Conversion to momentum space O Feynman rules ❑Counting independent momenta ❑Examples ❑Loop suppression factor s 6.4 Off the Mass S hell 286 Currents ❑Off-shell amplitudes are exact matrix elements of Heisenberg-picture operators o Proof of the theore m Problems References290 291 7 T HE CANON ICAL FORMALISM 29 2 7.1 Canonica lVaria bles 293 Canonical commutation relations D Examples : real scalars, complex scalars, vector fields, Dirac fields LJ Free-particle Harni4tonians O Free-field Lagrangian ❑Canonical formalism for interacting fields xis Content s 7.2 The Lagrangian Formalism 298 Lagrangian equations of motion o Action ❑Lagrangian density ❑Euler- Lagrange equations ❑Reality of the action ❑From Lagrangians to Harniltvnians a Scalar fields revisited ❑From Heisenberg to interaction picture o Auxiliary fields❑Integrating by parts in the actio n 7.3 Global Sy mmetries 306 Noether's theorem ❑Explicit formula for conserved quantities ❑Explicit formula for conserved currents ❑Quantum symmetry generators ❑Ehergy-momentum tensor ❑Momentum ❑Internal symmetries O Current commutation relation s 7.4 Lorentz Invariance 31 4 Currents ❑Generators J~`V ❑Belinfante tensor ❑Lorentz invariance of S-matri x 7.5 Tran sition to Intera ction P icture: Examples 318 Scalar field with derivative coupling o Vector field 0 Dirac f ield 7.6 Constraint s and Dira cBrackets 325 Primary and secondary constraints ❑Poisson brackets ❑First and second class constraints ❑Dirac brackets ❑Example : real vector fiel d 7.7 Field Redefinitions and R edundan t Co uplings" 331 Redundant parameters ❑Field redefinitions ❑Example : real scalar fiel d Appendix Dirac Brackets from Canonical Commutators 332 Problems References 8 ELECTRODYNAMICS337 33$ 339 8.1 Gauge Invariance 339 Need for coupling to conserved current ❑Charge operator D Local symmetry El Photon action o Field equations ElGauge-invariant derivative s 8.2 Constraints and Gauge Conditions 343 Primary and secondary constraints ❑Constraints are first class ❑Gauge fixing ❑Coulomb gauge ❑Solution for A 4 8.3 Quantization in Coulomb Gauge 346 Remaining constraints are second class 0 Calculation of Dirac brackets in Coulomb gauge ❑Construction of Hamiltonian 0Coulomb interactio n 8.4 Electrodynamics i ntheInteracti onPicture 350 Free-field and interaction Hamiltonians ❑Interaction picture operators ❑Normal mode decomposition Contents x V 8.5 T hePhotonPropagator 353 Numerator polynomial 0 Separation of non-covariant terms ❑Cancellation of non-covariant term s 8.6 Feyurnan Rules for Spinor Electrodynamics 35 5 Feynman graphs ❑Vertices ❑External lines ❑Internal lines ❑Expansion in a/4n ❑Circular, linear, and elliptic polarization ❑Polarization and spin sums 8.7 Compton Scattering 362 S-rnatrix ❑Differential cross-section ❑Kinematics ❑Spin sums o Traces ❑ Klein-Nishina formula ❑Polarization by Thomson scattering ❑Total cross- sectio n 8.8 Generalization : p-farm Gauge Fields* 3 69 Motivation ❑p-farms ❑Exterior derivatives ❑Closed and exact p-forms Cl p-form gauge fields ❑Dual fields and currents in D spacetime dimensions ❑ p-form gauge fields equivalent to (D -- p - 2)-form gauge fields ❑Nothing new in four spacetime dimension s Appendix Traces 37 2 Problems References 9 PATH-INTEGRAL METHODS 9.1 T he Ge neralPath-Integral Formu la374 375 376 378 Transition amplitudes for infinitesimal intervals ❑Transition amplitudes for finite intervals ❑Interpolating functions ❑Matrix elements of time-ordered products D Equations of motio n 9.2 Transition to th e S-Matrix 38 5 Wave function of vacuum ❑ic term s 9.3 Lagr angianVersi on of th e Path-Integral Formula 389 Integrating out the `momenta' ❑Derivatively coup led scalars ONon-linear sigma model D Vector fiel d 9.4 Path-Integral Derivation of Fe ynman Rul es 395 Separatio nof free-field actio n❑Gaussian in tegratio n❑Propagators : scalar fields, vector fields, de rivative couplin g 9.5 Path Integrals for Fermions 399 Anticommuting c-numbers ❑Eigenvectors of canonical operators ❑Summing states by Berezin integration ❑Changes of variables ❑Transition amplitudes for infinitesimal intervals ❑Transition amplitudes for finite intervals ❑Derivation of Feynman rules ❑Ferrnion propagator ❑Vacuum amplitudes as determinants xvr Content s 9.6 Path-Integral Formulation of Quantum Electrodynamics 413 Path integral in Coulomb gauge o Reint roduction of a° ❑Transition to covariant gauge s 9.7 Varieties of 5tatistics* 41 8 Preparing `in' and 'out' states a Composition rules ❑Only bosons and fermions in ~ 3 dimensions ❑Anyons in two dimension s Appendix Ga ussian Mult ipleIntegra ls Problem s References420 423 423 10 NON-PERTURBATIVE METHODS 425 10.1Symmetries 42 5 Translations ❑Charge conservation ❑Ferry's theore m 10.2 Polology 428 Pole formula for general amplitudes o Derivation of the pole formula E7 Pion exchang e 10.3 Field and Mass Renormalizatian 436 LSD reduction formula ❑Rcnormalized fields 0 Propagator poles 0 Na radiative corrections in external lines ❑Counterterms in self-energy part s 10.4 Renormalized C harge and Ward identiti es 442 Charge operator ❑Electromagnetic field renormalization ❑Charge renormaliz- ation❑Ward-Takahashi identity ❑Ward identit y 10.5 Gauge Invariance 448 Transversality of multi-photon amplitudes ❑Schwinger terms ❑Gauge terms in photon propagator a Structure of photon propagator 0 Zero photon renormal- ized mass ❑Calculation of Z3❑Radiative corrections to choice of gaug e 10.6 Electromagnetic Form Factors and Magnetic Mo ment 452 Matrix elements of J O❑Form factors of Jar : spin zero ❑Form factors of J0; spin ~ D Magnetic moment of a spin ~ particle El Measuring the form factor s 10.7 TheKailer-Lehmann Representation* 457 Spectral functions ❑Causality relations ❑Spectral representation ❑Asymptotic behavior of propagators ❑Poles E Bound on field renormalization constant o Z = 0 for composite particle s 10.8 D ispersio n Relations' 462 History ❑Analytic properties of massless boson forward scattering amplitude Contents Xvii o Subtractions ❑Dispersion relation ❑Crossing symmetry ❑Pomeranchuk's theorem ❑Regge asymptotic behavior ❑Photon scatterin g Problem s References469 470 ON&- LD P RADIATIVE CORRECTIONS IN QUAN TUM ELECTRODYNAMICS 47 2 11.1 Counterterm s Field, charge, and mass renormalization ❑Lagrangian countefterm s 11.2Vacuum Polarization472 473 One-loop integral for photon self energy part ❑Feynman parameters ❑Wick rotation Q Dimensional regularization a Gauge invariance ❑Calculation of ~3 C1 Cancellation of divergences 0Vacuum polarization in charged particle scattering ❑Clehling effect ❑Muonic atom s 11.3 Anomalous Magnetic Moments and Charge Radii 485 One-loop formula for vertex function ❑Calculation of form factors o Anomalous lepton magnetic moments to order ac ❑Anomalous moon magnetic moment to order Y?ln(rxa ./m,) ElCharge radius of lepton s 11.4 Ele ctronSelf-Energy 493 One-loop formula for electron self-energy part 0 Electron mass renormalization ❑Cancellation of ultraviolet dive rgence s Appendix Assorted Integral s Problem s References497 498 498 12 GENERAL RENORMALIZATION THEORY 499 12.1 D egrees of Divergence 50 0 Superficial degree of divergence ❑Dimensional analysis ❑Renormalizability D Criterion for actual convergenc e 12,2 Cance llatio nof Divergences 5 05 Subtraction by differentiation 0Re-normalization program ❑Renvrrnalizable theories ❑Example : quantum electrodynamics ❑Overlapping divergences ❑ SPH2 renormalization prescription ❑Changing the renormalization point; 04 theor y 12.3 Is Renormalizability Necessary? 516 xviii content s Renormalizable interactions cataloged ❑No renormalizable theories of gravita- tion❑Cancellation of divergences in non-rena rrnalizable theories ❑Suppression of non-renormalizable interactions ❑Limits on new mass scales ❑Problems with highe rderivatives? C1 Detection of non-renormalizable interactions El Low-energy expansions in non-renormalizable theories ❑Example : scalar with only derivative coupling ❑Saturation or new physics? ❑Effective field theorie s 12.4 T he Floating Cutoff" 525 Wilson's approach ❑Renormalization group equation El Polchinski's theorem El Attraction to a stable surface ❑Floating cutoff vs renormalizatio n 12.5 Acci dental Sy mmetries' 529 General renarrnaiizable theory of charged leptons ❑Redefinition of the lepton fields❑Accidental conservation of lepton flavors, P, C, and T Prob lems Reference s 13 INFRARED EFFECT S 13.1Soft Photon Amplitudes53t 532 534 534 Single photon emission ❑Negligible emission from internal lines ❑Lorentz invariance implies charge conservation ❑Single graviton emission 0Lorentz invariance implies equivalence principle 0Multi-photon emission ❑Factorizatio n 13.2 Virtual Soft Photon s Effect of soft virtual photons ❑Radiative corrections on internal lines539 13.3 RealSoft Photons ; Ca ncellationofDivergences 544 Sum over helicities o Integration over energies ❑Sum over photon number ❑ Cancellation of infrared cutoff factors ❑Likewise for gravitatio n 13.4 General Infrared Divergences 548 Massless charged pa rticles❑Infrared div ergen cesin general ❑Jets❑Lee-Nau- enberg theore m 13.5 So ftPhoton Sca ttering` 553 Poles in the amplitude ❑Conservation relations ❑Universality of the low-energy limit 13.6The External Field Approximation' 556 Sums over photon vertex permutations ❑Noo-relativistic limit ❑Crossed ladder exchang e Problems 56 2 References 562 Contents x1X 14 BOUND STATES IN EXTERNAL FIELDS 56 4 14.1 The Dirac Equation 565 Dirac wave functions as field matrix elements D Anticommutators and complete- ness❑Energy eigenstates ❑Negative energy wave functions ❑Qrthanormaliza- tinn❑`Large' and `mall' components ❑Purity ❑Spin- and angle-dependence Radial wave equations ❑Energies ❑Fine structure ❑Nan-relativistic approxi- mation s 14.2 Radiative Corrections in External Fields 572 Electron propagator in an external field ❑Inhomogeneaus Dirac equation Effects of radiative corrections ❑Energy shift s 14.3 The Lamb Shift in Light Atoms 578 Separating high and low energies ❑High-energy term ❑Low-energy term 11 Effect of mass renormalization ❑Total energy shift 0 1'_0D( =~- 0 Numerical results 0 Theory vs experiment for classic Lamb shift 11Theory vs experiment for 1s energy shif t Problem s Refere nces AUTHOR INDE X SUBJECT INDEX OUTLINE OF VOLUME II 15 NON-ABELIAN GAUGE THEORIES 16 BACKGROUND FIELD METHOD S 17 RENORMALIZATION GROUP METHODS 18 OPERATOR PRODUCT EXPANSION S 19 SPONTANEOUSLY BROKEN GLOBAL SYMMETRIES 20 SPONTANEOUSLY BROKEN LOCAL . SYMMETRIES 21 ANOMALIE S 22 TOPOLOGICALLY COMPLICATED FIELDS 23SUPERCONDUCTIVITY594 596 597 602 PrefaceToVolum eI Whyanother book ❑n quantum field theory? Today the student of quantum field theory can choose from among a score of excellent books, several of them quite up-to-date . Another book will be worth while only if it offers something new in content or perspective . As to content, although this book contains a good amount of new ma- terial, I suppose the most dist inctive thing about it is its generality ; I have tried throughout to discuss matters in a context that is as general as pos- sible . This is in part because quantum field theory has found applications far removed from the scene of its old successes, quantum electrodynamics, but even more because I think that this generality will help to keep the important points from being submerged in the technicalities of specific Eheories . Of course, specific examples are frequently used to illustrate gen- eral points, examples that are chosen from contemporary particle physics or nuclear physics as well as from quantum electrodynamics . It is, however, the perspective of this book, rather than its content, that provided my chief motivation in writing it . I aim to present quantum field theory in a manner that will give the reader the clearest possible idea of why this theory takes the form it does, and why in this form it does such a good job of describing the real world . The traditional approach, since the first papers of Heisenberg and Pauli on general quantum field theory, has been to take the existence of fields for granted, relying for justification on our experience with electromagnetism, and `quantize' them --- that is, apply to various simple field theories the rules of canonical quantization or path integration . Some of this traditional approach will be found here in the historical introduction presented in Chapter 1 . This is certainly a way of getting rapidly into the subject, but it seems to me that it leaves the reflect ive reader w ith too many unanswered questions . Why should we believe in the ru les of canon ical quantization or path integration? Why should we adopt the simple field equations and Lagrangians that are found in the literature? For that matter, why have fields at all? It does not seem satisfactory to meto appeal to exper ience ; after all, our purpose in theoretical physics i s xx Preface xxi not just to describe the world as we find it, but to explain - in terms of a few fundamental principles -why the world is the way it is . The point of view of this book is that quantum field theory is the way it is because (aside from theories like string theory that have an infinite number of particle types) it is the only way to reconcile the principles of quantum mechanics (including the cluster decomposition property) with those of special relativity . This is a point of view I have held for many years, but it is also one that has become newly appropriate . We have learned in recent years to think of our successful quantum field theories, including quantum electrodynamics, as `effective field theories,' low-energy approximations to a deeper theory that may not even be a field theory, but something different like a string theory . On this basis, the reason that quantum field theories describe physics at accessible energies is that any relativistic quantum theory will look at sufficiently low energy like a quantum field theory . It is therefore important to understand the rationale for quantum field theory in terms of the principles of relativity and quantum mechanics . Also, we think differently now about some of the problems of quantum field theories, such as non-renormalizability and `triviality,' that used to bother us when we thought of these theories as truly fundamental, and the discussions here will reflect these changes . This is intended to bea book on quantum field theory for the era of effective field theories . The most immediate and certain consequences of relativity and quan- tum mechanics are the properties of particle states, so here particles come first they are introduced in Chapter 2 as ingredients in the repre- sentation of the inhomogeneous Lorentz group in the Hilbert space of quantum mechanics . Chapter 3 provides a framework for addressing the fundamental dynamical question : given a state that in the distant past looks like a certain collection of free particles, what will it look like in the future? Knowing the generator of time-translations, the Hamiltonian, we can answer this question through the perturbative expansion for the array of transition amplitudes known as the S-matrix . In Chapter 4 the princi- ple of cluster decomposition is invoked to describe how the generator of time-translations, the Hamiltonian, is to be constructed from creation and annihilation operators . Then in Chapter 5 we return to Lorentz invariance, and show that it requires these creation and annihilation operators to be grouped together in causal quantum fields . As a spin-off, we deduce the CPT theorem and the connection between spin and statistics . The formal- ism is used in Chapter 6to derive the Feynman rules for calculating the S-matrix . It is not until Chapter 7 that we come to L+agxangians and the canonical formalism . The rationale here for introducing them is not that they have proved useful elsewhere in physics (never a very satisfying explanation) xxii Preface but rather that this formalism makes it easy to choose interaction Hamil- tonians for which the S-matrix satisfies various assumed symmetries . In particular, the Lorentz invariance of the Lagrangian density ensures the existence of a set of ten operators that satisfy the algebra of the Poincare group and, as we show in Chapter 3, this is the key condition that we need to prove the Lorentz invariance of the S-matrix . Quantum electrodynam- ics finally appears in Chapter 8 . Path integration is introduced in chapter 9, and used to justify some of the hand-waving in Chapter 8 regarding the Feynman rules for quantum electrodynamics . This is a somewhat later introduction of path integrals than is fashionable these days, but it seems to me that although path integration is by far the best way of rapidly deriving Feynman rules from a given Lagrangian, it rather obscures the quantum mechanical reasons underlying these calculations . Volume I concludes with a series of chapters, 10---14, that provide an introduction to the calculation of radiative corrections, involving loop graphs, in general field theories . Here too the arrangement is a bit unusual ; we start with a chapter on non-perturbative methods, in part because the results we obtain help us to understand the necessity for field and mass renormalization, without regard to whether the theory contains infinities or not . Chapter 1 Ipresents the classic one-loop calculations of quantum chromodynamics, both as an opportunity to explain useful calculational techniques (Feynman parameters, wick rotation, dimensional and Pauli- Villars regularization), and also as a concrete example of renormalization in action . The experience gained in Chapter 11 is extended to all orders and general theories in Chapter 12, which also describes the modern view of non-renormalizability that is appropriate to effective field theories . Chapter 1 3is a digression on the special problems raised by massless particles of low energy or parallel momenta . The Dirac equation for an electron in an external electromagnetic field, which historically appeared almost at the very start of relativistic quantum mechanics, is not seen here until Chapter 14, on bound state problems, because this equation should not be viewed (as Dirac did) as a relativistic version of the Schrodinger equation, but rater as an approximation to a true relativistic quantum theory, the quantum field theory of photons and electrons . This chapter ends with a treatment of the Lamb shift, bringing the confrontation of theory and experiment up to date . The reader may feel that some of the topics treated here, especially in Chapter 3, could more properly have been left to textbooks on nuclear or elementary particle physics . So they might, but in my experience these topics are usually either not covered or covered poorly, using specific dynamical models rather than the general principles of symmetry and quantum tnechanics . I have met string theorists who have never heard of the relation between time-reversal invariance and final-state phase shifts, Preface xxiii and nuclear theorists who do not understand why resonances are governed by the Breit-Wigner formula . So in the early chapters I have tried to err on the side of inclusion rather than exclusion . Volume II will deal with the advances that have revived quantum field theory in recent years : non-Abelian gauge theories, the renormalization group, broken symmetries, anomalies, instantons, and so on . I have tried to give citations both to the classic papers in the quantum theory of fields and to useful references on topics that are mentioned but not presented in detail in this book . I did not always know who was responsible for material presented here, and the mere absence of a citation should not be taken as a claim that the material presented here is original . But some of it is . I hope that I have improved on the original literature or standard textbook treatments in several paces, as for instance in the proof that symmetry operators are either unitary or antiunitary ; the discussion of superselection rules ; the analysis of particle degeneracy associated with unconventional representations of inversions ; the use of the cluster decomposition principle ; the derivation of the reduction formula ; the derivation of the external field approximation ; and even the calculation of the Lamb shift . I have also supplied problems for each chapter except the first . Some of these problems aim simply at providing exercise in the use of techniques described in the chapter ; others are intended to suggest extensions of the results of the chapter to a wider class of theories . In teaching quantum field theory, I have found that each of the two volumes of this book provides enough material for a one-year course for graduate students . I intended that this book should be accessible to students who are familiar with non-relativistic quantum mechanics and classical electrodynamics . I assume a basic knowledge of complex analysis and matrix algebra, but topics in group theory and topology are explained where they are introduced . This is not a book for the student who wants immediately to begin cal- culating Feynman graphs in the standard model of weak, electromagnetic, and strong interactions . Nor is this a book for those who seek a higher level of mathematical rigor . Indeed, there are parts of this book whose lack of rigor will bring tears to the eyes of the mathematically inclined reader . Rather, I hope it will suit the physicists and physics students who want to understand why quantum field theory is the way it is, so that they will be ready for whatever new developments in physics may take us beyond our present understandings . ~** Much of the material in this book I learned from my interactions over Xx1V Prefac e the years with numerous other physicists, far too many to name here . But I must acknowledge my special intellectual debt to Sidney Coleman, and to my colleagues at the University of Texas : Arno Bohm, Luis Boya, Phil Candelas, Bryce DeWitt, Cecile DeWitt-Morette, Jacques Distler, Willy Fischler, Josh Feinberg, Joaquim Gomis, Vadim Kaplunovsky, Joe Polchinski, and Paul Shapiro . I owe thanks for help in the preparation of the historical introduction to Gerry Holton, Arthur Miller, and Sam Schweber . Thanks are also due to Alyce Wilson, who prepared the illustrations and typed the ITT input files until I learned how to do it, and to Terry Riley for finding countless books and articles . I am grateful to Maureen Storey and Alison Woollatt of Cambridge University Press for helping to ready this book for publication, and especially to my editor, Rufus Neal, for his friendly good advice . STEVEN WEINBER G Austin, Texas October, 1994 Notatio n Latin indices i j, k, and so on generally run over the three spatial coordi- nate labels, usually taken as 1, 2, 3 . Greek indices A, v, etc . generally run over the four spacetime coordinate labels 1, 2, 3, 0, with xO the time coordinate . Repeated indices are generally summed, unless otherwise indicated . The spacetime metric q,,,is diagonal, with elements ail = X22 = )113 _ The d'Alembertian is defined as C1 "'~?l/r~xyc?xv= V2 -a2/at2, where V2 is the Laplacian c"/eYO-xi . The `Levi--Civita tensor' is defined as the totally antisymmetric quantity withC0123 =+1. Spatial three-vectors are indicated by letters in boldface . A hat over any vector indicates the corresponding unit vector : Thus, "v = v /Jvl. A dot over any quantity denotes the time-derivative of that quantity . Dirac matrices j.are defined so that ~~~1,jti, +x',, r',u= 2q,,,, . Also, yF 5= iYaYIY 2T3} and fl= iy° . The step function O(s) has the value +1 for s >0and 0for s <0. The complex conjugate, transpose, and Hermitian adjoint of a matrix or vector Aare denoted A", AT, and A#=AMT, respectively . The Hermitian adjoins of an operator 0is denoted Qt, except where an asterisk is used to emphasize that a vector or matrix of operators is not transposed . +H .c. or +c .c. at the end of an equation indicates the addition of the Hermitia n xxv xxvi Notatio n adjoint or comp lex conjugate of the foregoing terms . A bar on a Dirac spinor u is defined by u = u tP. Except in Chapter Z, we use units with h and the speed of light taken to be unity . Throughout -e is the rationalized charge of the electron, so that the fine structure constant is ac = e2/4n ~-- 1/137. Numbers in parenthesis at the end of quoted numerical data give the uncertainty in the last digits of the quoted figure . Where not otherwise indicated, experimental data are taken from `Review of Particle Properties,' Ph ys . Rev .D50 , 1173 (1994) . Historical Introductio n Our immersion in the present state of physics makes it hard for us to understand the difficulties of physicists even a few years ago, or to profit from their experience . At the same time, a knowledge of our history is a mixed blessing - it can stand in the way of the logical reconstruction of physical theory that seems to be continually necessary . I have tried in this book to present the quantum theory of fields in a logical manner, emphasizing the deductive trail that ascends from the physical principles of special relativity and quantum mechanics . This approach necessarily draws me away from the order in which the subject in fact developed . To take one example, it is historically correct that quantum field theory grew in part out of a study of relativistic wave equations, including the Maxwell, Klein-Gordon, and Dirac equations . For this reason it is natural that courses and treatises on quantum field theory introduce these wave equations early, and give them great weight . Nevertheless, it has long seemed to me that a much better starting point is Wigner's definition of particles as representations of the inhomogeneous Lorentz group, even though this work was not published until 1939 and did not have a great impact for many years after . In this book we start with particles and get to the wave equations later . This is not to say that particles are necessarily more fundamental than fields . For many years after 1950 it was generally assumed that the laws of nature take the form of a quantum theory of fields . I start with particles in this book, not because they are more fundamental, but because what we know about particles is more certain, more directly derivable from the principles of quantum mechanics and relativity . If it turned out that some physical system could not be described by a quantum field theory, it would be a sensation ; if it turned out that the system did not obey the rules of quantum mechanics and relativity, it would b ea cataclysm . In fact, lately there has been a reaction against looking at quantum field theory as fundamental . The underlying theory might not be a theory of fields or particles, but perhaps of something quite different, like strings . 1 2 1 Histo rical Introductio n From this point of view, quantum electrodynamics and the other quantum fieldtheories of which we are so proud are mere `effective field theories,' low-energy approximations to a more fundamental theory . The reason that our field theories work so well is not that they are fundamental truths, but that any relativistic quantum theory will look like a field theory when applied to particles at sufficiently low energy . On this basis, if we want to know why quantum field theories are the way they are, we have to start with particles . But we do not want to pay the price of altogether forgetting our past . This chapter will therefore present the history of quantum field theory from earliest times to 1949, when it finally assumed its modern form . In the remainder of the book I will try to keep history from intruding on physics . One problem that I found in writing this chapter is that the history of quantum field theory is from the beginning inextricably entangled with the history of quantum mechanics itself, Thus, the reader who is familiar with the history of quantum mechanics may find some material here that he or she already knows, especially in the first section, where I discuss the early attempts to put together quantum mechanics with special relativity . In this case Ican only suggest that the reader should skip on to the less familiar parts , On the other hand, readers who have no prior familiarity with quantum field theory may find parts of this chapter too brief to be altogether clear . Iurge such readers not to worry . This chapter is not intended as a self-contained introduction to quantum field theory, and is not needed as a basis for the rest of the book . Some readers may even prefer to start with the next chapter, and come back to the history later . However, for many readers the history of quantum field theory should serve as a good introduction to quantum field theory itself . I should add that this chapter is not intended as an original work of historical scholarship . I have based it on books and articles by real historians, plus some historical reminiscences and original physics articles that Ihave read . Most of these are listed in the bibliography given at the end of this chapter, and in the list of references . The reader who wants to go more deeply into historical matters is urged to consult these listed works . A word about natation . In order to keep some of the flavor of past times, in this chapter I will show explicit factors of h and c (and even h), but in order to facilitate comparison with modern physics literature, I will use the more modern rationalized electrostatic units for charge, so that the fine structure constant a , 1/137 ise2~47rhc.In subsequent chapters I will mostly use the `natural' system ❑f units, simply setting h=c= 1. 1.1 Re lativisticWave M echa nics 3 1.1 R elativisticWave Mec hanics Wave mechanics started out as relativistic wave mechanics . Indeed, as we shall see, the founders of wave mechanics, Louis de Broglie and Erwin 5chr6dinger, took a good deal of their inspiration from special relativity . It was only later that it became generally clear that relativistic wave mechanics, in the sense of a relativistic quantum theory of a fixed number of particles, is an impossibility . Thus, despite its many successes, relativistic wave mechanics was ultimately to give way to quantum field theory . Nevertheless, relativistic wave mechanics survived as an important element in the formal apparatus of quantum field theory, and it posed a challenge to field theory, to reproduce its successes . The possibility that material particles can like photons be described in terms of waves was first suggested' in 1923 by Louis de Broglie . Apart from the analogy with radiation, the chief clue was Lorentz invariance ; if partic les are described by a wave whose phase at position x and time t is of the form 27r(ic  x - vt), and if this phase is to be Lorentz invariant, then the vector rc and the frequency v must transform like x and t, and hence like pand E . In order for this to be possible Kand v must have the same velocity dependence as pand E, and therefore must be proportional to them, with the same constant of proportionality . For photons, one had the Einstein relation E = hv, so it was natural to assume that, for material particles, Yc=Ply ,v=E1h, ( 1.1.1) just as for photons . The group velocity c lv/arc of the wave then turns out to equal the particle velocity, so wave packets just keep up with the particle they represent . By assuming that any closed orbit contains an integral number of particle wavelengths A= 1/, de Broglie was able to derive the old quantization conditions of Niels Bohr and Arnold Sommerfeld, which though quite mysterious had worked well in accounting for atomic spectra . Also, both de Broglie and Walter Elsasser2 suggested that de Broglie's wave theory could be tested by looking for interference effects in the scattering of electrons from crystals ; such effects were established a few years later by Clinton Joseph Davison and Lester H . Germer .1 However, it was still unclear how the de Broglie relations (1.1.1)should be modified for non-free particles, as for instance for an electron in a general Coulomb field . Wave mechanics was by-passed in the next step in the history of quantum mechanics, the development of matrix mechanics4 by Werner Heisenberg, Max Born, Pascual Jordan and Wolfgang Pauli in the years 1925-1926. At least part of the inspiration for matrix mechanics was the 4 1Historical Introductio n insistence that the theory should involve only observables, such as the energy levels, or emission and absorption rates . Heisenberg's 1925paper opens with the manifesto : The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities that in principle are observable,' This sort of positivism was to reemerge at various times in the history of quantum field theory, as for instance in the introduction of the S-matrix by John Wheeler and Heisenberg (see chapter 3) and in the revival of dispersion theory in the 1950s (see chapter 10), though modern quantum field theory is very far from this ideal . It would take us too far from our subject to describe matrix mechanics in any detail here . As everyone knows, wave mechanics was revived by Erwin 5chr4dinger . In his 1926 series of papers,5 the familiar non-relativistic wave equation is suggested first, and then used to rederive the results of matrix mechan- ics. Only later, in the sixth section of the fourth paper, is a relativistic wave equation offered . According to Dirac,6the history is actually quite different : Schrodinger first derived the relativistic equation, then became discouraged because it gave the wrong fine structure for hydrogen, and then some months later realized that the non-relativistic approximation to his relativistic equation was of value even if the relativistic equation itself was incorrect ! By the time that Schroainger came to publish his relativistic wave equation, it had already been independently rediscovered by Oskar Klein7 and Walter Gvrdon,s and for this reason it is usually called the `Klein-Gordon equation .' Schrodinger's relativistic wave equation was derived bynoting first that, for a `Lorentz electron' of mass m and charge e in an external vector potential A and Coulomb potential 0, the Hamiltonian Rand momentum p are related by* 0 = (H+ eo)2 - c2( p+ eA/c)2 - m2c4 . (1 .1.2) For a free particle described by a plane wave exp {2lri(K . x-vt}the de Broglie relations (1 .1.1) can be obtained by the identification s p==: hrc-). -i#V E_-by--+ih~ (1 .1.3) where h is the convenient symbol (introduced later byDirac) for h/2 7r. By an admittedly formal analogy, Schrodinger guessed that an electron in the external fields A, 0would be described by a wave function T(x, t) satisfying the equation obtained by making the same replacements i n ' This is Lorentz invariant, because the quantities A and 0have the same Lorentz transformation property as cp and E . Schradinger actually wrote Uand p in terms of partial derivatives of an action function, but this makes no difference to our present discussion . 1.1Relativistic Wave Mechanic s (1.x.2): f-m2C4~(X}r)U=[(th ~ + eo)2 -c2(-thy +~~)2 r~t C5 (1.1.4) In particular, for the stationary states of hydrogen we have A = 0 and 0==e/4n r,andyhas the time-dependence exp(-iEt/h), so (1 .1.4) become s 2 ,U= E+41zr C2hzv2 - M2C41- W(X)- (1.1.5) Solutions satisfying reasonable boundary conditions can be found for the energy values9 a2 a4 n 3 2n2 2n4 + 4z(1.1.6) where a = e2/4arhc is the `fine structure constant,' roughly 1/137 ;n is a positive-definite integer, and the orbital angular momentum in units of h, is an integer with 0 C t~ n -- 1.The u2 term gave good agreement with the gross features of the hydrogen spectrum (the Lyman, Balmer, etc. series) and, according to Dirac,6 it was this agreement that led Schr ;5dinger eventually to develop his non-relativistic wave equation . On the other hand, the ac4 term gave a fine structure in disagreement with existing accurate measurements of Friedrich Paschen .to It is instructive here to compare 5chr5dinger's result with that of Arnold Sommerfeld,"- -obtained using the rules of the old quantum theory : al 14 E= mc~1- 2n220n3 k 4) +(1.1.7) where m is the electron mass . Here k is an integer between 1 and n, which in Sommerfeld's theory is given in terms of the orbital angular momentum ehask + 1 .This gave a fine structure splitting in agreement with experiment : for instance, for n = 2 Eq .(1.1.7)gives two levels (k = 1 and k:;--2), split by the observed amount oc4mc~/32, or 4 .53 x 1 0-'eV. In contrast, Schrodinger's result (1 .1.6) gives an n = 2 fine structure splitting a4mcZ f 12, considerably larger than observed . Schrodinger correctly recognized that the source of this discrepancy was his neglect of the spin of the electron . The splitting of atomic energy levels by non-inverse-square electric fields in alkali atoms and by weak external magnetic fields (the so-called anomalous Zeeman effect) had revealed a multiplicity of states larger than could be accounted for by the Bohr-Sommerfeld theory ; this led George Uhlenbeck and Samuel Goudsmitll in 1925 to suggest that the electron has an intrinsic angular 6 1 Historical Introductio n momentum h/2 . Also, the magnitude of the Zeeman splitting .12 allowed them to estimate further that the electron has a magnetic momen t eh 2mc It was clear that the electron's spin would be coupled to its orbital angular momentum, so that Schr6dinger's relativistic equation should not be expected to give the correct fine structure splitting . Indeed, by 1927 several authors13 had been able to show that the spin-orbit coupling was able to account for the discrepancy between Sch- r6dinger's result (1.1.6) and experiment . There are really two effects here : one is a direct coupling between the magnetic moment (1 .1.8) and the magnetic field felt bythe electron as it moves through the electrostatic field of the atom ; the other is the relativistic `Thomas precession' caused (even in the absence of a magnetic moment) by the circular motion of the spinning electron .l¢ Together, these two effects were found to lift the level with total angular momentum + ~ to the energy ( 1.1.7)given by Sommerfeld for k = ~ + 1 = j + while the level with - ~ was lowered to the value given bySommerfeld for k = t= j + ~ . Thus the energy was found to depend only on n and j, but not separately on i": 2 [1a2 ~c4n 3 2n2 2n4j+ By accident Sommerfeld's theory had given the correct magnitude of the splitting in hydrogen (j + ~ like k runs over integer values from 1 to n) though it was wrong as to the assignment of orbital angular momentum values e to these various levels . In addition, the multiplicity of the fine structure levels in hydrogen was now predicted to be 2 for j=1and 2(2j + 1) for j > (corresponding to z'values j±~), in agreement with experiment . Despite these successes, there still was not a thorough relativistic theory which incorporated the electron's spin from the beginning . Such a theory was discovered in 1928 by Paul Dirac . However, he did not set out simply to make a relativistic theory of the spinning electron ; instead, he approached the problem by posing a question that would today seem very strange . At the beginning of his 1928 paper,15 he asks `why Nature should have chosen this particular model for the electron, instead of being satisfied with the paint charge .' To us today, this question is like asking why bacteria have only one cell : having spin h/2 is just one of the properties that define a particle as an electron, rather than one of the many other types of particles with various spins that are known today . However, in 1928 it was possible to believe that all matter consisted of electrons, and perhaps something similar with positive charge in the 1.1Relativistic Wave Mechanics 7 atomic nucleus . Thus, in the spirit of the times in which it was asked, Dirac's question can be restated : `Why do the fundamental constituents of matter have to have spin A/2? ' For Dirac, the key to this question was the requirement that probabilities must bepositive . It was known16 that the probability density for the non- relativistic Schrodinger equation is ~ya13, and that this satisfies a continuity equation of the for m so the space-integral of Iy)12 is time-independent . On the other hand, the only probability density p and current J, which can be formed from solutions of the relativistic Schrodinger equation and which satisfy a conservation law , are of the form 0p = 11TIm ~,~ ( #,~ t -ie~ o )W) J=NclItnw'V+ IA V>hc )(1.2.10) X1.1.11) (1.1.12) with N an arbitrary constant . It is not possible to identify p as the probability density, because (with or without an external potential 0)p does not have definite sign . To quote I]irac's reminiscencesO about this problem I remember once when I was in Copenhagen, that Bohr asked me what I was working on and I told him I was trying to get a satisfactory relativistic theory of the electron, and Bohr said But Klein and Gordon have already done that!' That answer first rather disturbed me . Bohr seemed quite satisfied by Klein's solution, but I was not because of the negative probabilities that it led to . I just kept on with it, worrying about getting a theory which would have only positive probabilities . According to George Gamow,18 Dirac found the answer to this problem on an evening in 1928 while staring into a fireplace at St John's College, Cambridge . He realized that the reason that the Klein-Gordon (or relativistic Schrodinger) equation can give negative probabilities is that the p in the conservation equation (1 . t.10)involves a time-derivative of the wave function, This in turn happens because the wave function satisfies a differential equation of second order in the time . The problem therefore 8 1Historical I ntroduction was to replace this wave equation with another one of first order in time derivatives, like the non-relativistic Schrvdinger equation . Suppose the electron wave function is a multi-component quantity WJx), which satisfies a wave equation of the form , ~~ where F is some matrix function of space derivatives . In order to have a chance at aLorentz-invariant theory, we must suppose that because the equation is linear in time-derivatives, it is also linear in space-derivatives, so that takes the form : _ -Ca 'V+ ~carnc2a~Z where oc 1, 12, a 3, and a4 are constant matrices . From (1.1.13) we can derive the second-order equatio n ~2 22 2 22 at, exi -ihrrzc3(araa + aWt)O+ m2C4 1 4~ - (The summation convention is in force here ; i and j run over the values 1, 2, 3, or X, y, z .)But this must agree with the free-field form of the relativistic Schrodinger equation (1 .1.4), which just expresses the relativistic relation between momentum and energy . Therefore, the matrices a and acs must satisfy the relations a;aj+ api = 2St j1 , (1.1.15) act(X4 + a%-xi = 0 ~ 1.l.16) oc¢ = 1, ~i .1.17) where ~jjis the Kronecker delta (unity for i= j;zero for i z~ j) and 1 is the unit matrix .Dirac found a set of 4 x 4 matrices which sat isfy these relations 0 0 ~i - 0 Z0 0 1 00f Z0 00 0 0- 0 Q1 a a o fY3 i a a 0 -1 00 -1 0 00 0°`2 1 0ad- 4 00 d 0i -1 0 0U a o 10 0 -1 00-i 0 0 0 (1.1.18) 0- 0 0 --1 1.1 Re lativistic Wave Mechanics 9 To show that this formalism is Lorentz-invariant, Dirac multiplie d Eq.(1.1.13) on the left with aq, so that it could be put in the for m where.,0 c X1, Y -ioc 42 y0-ix4(1.1.19) (I.t.20) (The creek indices p, v, etc . will now run over the values 1, 2, 3, 0,with x° = ct . Dirac used x4 =[ct, and correspondingly y4 T a¢ .) The matrices ,jAsatisfy the anticommutation relation s 1+1 0µ=v=1,2,3 ,U=v=U P V(1.1.21) Dirac noted that these anticommutation relations are Lorentz-invariant, in the sense that they are also satisfied by the matrices 11 P,,pv, where A is any Lorentz transformation . He concluded from this that 11r`,7` must be related to y Aby a similarity transformation : It follows that the wave equation is invariant if, under a Lorentz transfor- mation x4 --+AA,xu, the wave function undergoes the matrix transforma- tion ip --+5(A)V. (These matters are discussed more fully, from a rather different point of view, in Chapter 5 .) To study the behavior of electrons in an arbitrary external electromag- netic field, Dirac followed the `usual procedure' of making the replacement s ih ~ i ts ~~ eo - ihV ---> - ih~7' +eA (1.1-22) as in Eq . (1. 1.4). The wave equation (1. 1. 13) then takes the for m +er~ W--(--mihcV +eA)-ay) + mc2~c4~aat Dirac used this equation to show that in a central field, the conservation of angular momentum takes the for m [,,f , -ih rx V + ha/2] = 0 , (1 .1.24) where is the matrix di fferential operator ( 1.1.14) and a is th e4x 4 version of the spin matrix introduced earlier by Pauli1 9 0 0 t 0 00 1 0 0 0 (I.1.25) 0 100 10 I Historical Introductio n Since each component of a has eigenvalues +t, the presence of the extra term in (1124) shows that the electron has intrinsic angular momentum h/2. Dirac also iterated Eq . (1.1.23), obtaining a second-order equation, which turned out to have just the same form as the Klein-Gordon equation (1. 1.4) except for the presence on the right-hand-side of two additional terms [-ehca , B -iehcac-E]V7. 0 . 1.26) For a slowly moving electron, the first term dominates, and represents a magnetic moment in agreement with the value { 1 .1.5} found by Goudsmit and Uhlenbeck .11 As Dirac recognized, this magnetic moment, together with the relativistic nature of the theory, guaranteed that this theory would give a fine structure splitting in agreement (to order a'mc2) with that found by Heisenberg, Jordan, and Charles G . Darwin ." A little later, an `exact' formula for the hydrogen energy levels in I]irac's theory was derived by Darwin20 and Gordon21 -1/2 E = mc21 + --- - ,~~ n-j- ~+ [(i+1)` 2 11(1.1.27) The first three terms of a power series expansion in a2 agree with the approximate result 0.1.9). This theory achieved Dirac's primary aim : a relativistic formalism with positive probabilities . From (2 .1.13) we can derive a continuity equatio n with0.t + VJ=0 p=fV1l2, 1=eyatay(1.1.2s) (1.1.29) so that the positive quantity IVr12 can be interpreted as a probability density, with constant total probability f ly)12d3x . However, there was another difficulty which Dirac was not immediately able to resolve . For a given momentum p, the wave equation (1 .1. 3) has four solutions of the plane wave form W oc exp 1~(p x - E t) I (1 .1.34) Two solutions with E =+p_1c1+ rnIc4 correspond to the two spin states of an electron with Jz = ± h/2.The other two solutions have E 1.1Relativisti c Wave Mechanics 11 - V$ cf m c ~, anno obvious physical interpretation . As Dirac pointed out, this problem arises also for the relativistic Schrodinger equation : for each p, there are two solutions of the form (1 .1.30), one with positive E and one with negative E . Of course, even in classical physics, the relativistic relation E2=p2c2 + m2c¢has two solutions, E - ± pc + m 2c4 However, in classical physics we can simply assume that the only physical particles are those with positive E . Since the positive solutions have E > m c;2 and the negative ones have E < -mc2 , there is a finite gap between them, and no continuous process can take a particle from positive to negative energy . The problem of negative energies is much more troublesome in rela- tivistic quantum mechanics . As Dirac pointed out in his 1928 paper,' 5 the interaction of electrons with radiation can produce transitions in which a positive-energy electron falls into a negative-energy state, with the energy carried off by two or more photons . Why then is matter stable ? In 1930 Dirac offered a remarkable solution .22 Dirac's proposal was based on the exclusion principle, so a few words about the history of this principle are in order here . The periodic table of the elements and the systematics of X-ray spec- troscopy had together by 1924 revealed a .pa#tern in the population of atomic energy levels by electrons :23 The maximum ❑utnber N, of electrons in a shell characterized by principal quantum number n is given by twice the number of orbital states with that n ,:-1 Nn = 2 (24+1)=2n2 _ 2, 8, 18 ,.... ~=a Wolfgang Pauli24 in 1925 suggested that this pattern could be understood if IVnis the total number of possible states in the nth shell, and if in addit ion there is some mysterious `exclus ion principle' which forbids more than one electron from occupying the same state . He explained the puzzling factor 2 in (1 .1.31) as due to a 'peculiar, classically non-describable duplexity' of the electron s tates, and as we have seen this was understood a lit tle later as due to the spin of the electron, l l The exclusion princ iple answered a question that had remained obscure in the old atomic theory of Bohr and Somme rfeld : why do not al lthe electrons in heavy atoms fall down into the shell of lowest energy? Subsequently Pauli's exclusion principle was formalized by a number of authors25 as the requirement that the wave function of a multi-electron system is antisymmetric in the coordinates, orbital and sp in, of all the electrons . This principle was incorporated into statistica lmechanics by Enrico Fermi26 and Dirac,27 and for this reason particles obeying the exclusion pr inciple are gene rally called `fermions,* 12 1 Historical Introductio n just as particles like photons for which the wave function is symmetric and which obey the statistics of Bose and Einstein are called 'bosons .' The exclusion principle has played a fundamental role in the theory of metals, white dwarf and neutron stars, etc ., as well as in chemistry and atomic physics, but a discussion of these matters would take us too far afield here. Dirac's proposal was that the positive energy electrons cannot fall down into negative energy states because `all the states of negative energy are occupied except perhaps a few of small velocity .' The few vacant states, or `holes,' in the sea of negative energy electrons behave like particles with opposite quantum numbers : positive energy and positive charge . The only particle with positive charge that was known at that time was the proton, and as Dirac later recalled,27a`the whole climate of opinion at that time was against new particles' so Dirac identified his holes as protons ; in fact, the title of his 1930 article22 was `A Theory of Electrons and Protons .' The hole theory faced a number of immediate difficulties . One obvi- ous problem was raised by the infinite charge density of the ubiquitous negative-energy electrons : where is their electric field? Dirac proposed to reinterpret the charge density appearing in Maxwell's equations as `the departure from the normal state of electrification of the world .' An- other problem has to do with the huge dissimilarity between the observed masses and interactions of the electrons and protons . Dirac hoped that Coulomb interactions between electrons would somehow account for these differences but Hermann Weyl28 showed that the hole theory was in fact entirely symmetric between negative and positive charge . Finally, Dirac22 predicted the existence of an electron proton annihilation process in which a positive-energy electron meets a hole in the sea of negative-energy elec- trons and falls down into the unoccupied level, emitting a pair of gamma ray photons . By itself this would not have created difficulties for the hole theory ; it was even hoped by some that this would provide an explana- tion, then lacking, of the energy source of the stars . However, it was soon pointed out29 by Julius Robert Oppenheimer and Igor Tatum that electron-proton annihilation in atoms would take place at much too fast a rate to be consistent with the observed stability of ordinary matter . For these reasons, by 1931 Dirac had changed his mind, and decided that the holes would have to appear not as protons but as a new sort of positively charged particle, of the same mass as the electron.29a The second and third of these problems were eliminated by the discovery of the positron by Carl D . Anderson,30 who apparently did not know of this prediction by Dirac . On August 2, 1932, a peculiar cosmic ray track was observed in a Wilson cloud chamber subjected to a 15 kG magnetic field . The track was observed to curve in a direction that would be expected for a p ositively charged particle, and yet its range was at least 1.1 Relativistic Wave Mechanics 13 ten times greater than the expected range of a proton! Both the range and the specific ionization of the track were consistent with the hypothesis that this was a new particle which differs from the electron only in the sign of its charge, as would be expected for one of Dirac's holes . (This discovery had been made earlier by P .M.S. Blackett, but not immediately published by him . Anderson quotes press reports of evidence for light positive particles in cosmic ray tracks, obtained by Blackett and Giuseppe Occhialini .) Thus it appeared that Dirac was wrong only in his original identification of the hole with the proton . The discovery of the more-or-less predicted positron, together with the earlier successes of the Dirac equation in accounting for the magnetic moment of the electron and the fine structure of hydrogen, gave Dirac's theory a prestige that it has held for over six decades . However, although there seems little doubt that Dirac's theory will survive in some form in any future physical theory, there are serious reasons for being dissatisfied with its original rationale : (i) Dirac's analysis of the problem of negative probabilities in Sch- rodinger's relativistic wave equation would seem to rule out the existence of any particle of zero spin . Yet even in the 1920s particles of zero spin were known -far instance, the hydrogen atom in its ground state, and the helium nucleus, Ofcourse, it could be argued that hydrogen atoms and alpha particles are not elementary, and therefore do not need to be described bya relativistic wave equation, but it was not (and still is not) clear how the idea of elementaxity is incorporated in the formalism of relativistic quantum mechanics . Today we know of a large number of spin zero particles - it mesons, K mesons, and so on -- that are no less elementary than the proton and neutron . We also know of spin one particles -the W± and Z° - which seem as elementary as the electron or any other particle . Further, apart from effects of the strong interactions, we would today calculate the fine structure of `mesonic atoms,' consisting of a spinless negative it or K meson bound to an atomic nucleus, from the stationary solutions of the relativistic Klein- Gordon-Schrbdinger equation! Thus, it is difficult to agree that there is anything fundamentally wrong with the relativistic equation for zero spin that forced the development of the Dirac equation -the problem simply is that the electron happens to have spin h/2, not zero . (ii) As far as we now know, for every kind of particle there is an `antiparticle' with the same mass and opposite charge . (Some purely neutral particles, such as the photon, are their own antiparticles .) But how can we interpret the antiparticles of charged bosons, such as the aTImesons or Wt particles, as holes in a sea of negative energy states? For particles quantized according to the rules of Bose-Einstein statistics, 14 1 Historical In troduction there is no exclusion principle, and hence nothing to keep positive-energy particles from falling down into the negative-energy states, occupied or not. And if the hole theory does not work for bosonic antiparticles, why should we believe it for fermions? I asked Dirac in 1972 how he then felt about this point ; he told me that he did not regard bosons like the pion or Yet as `important : In a lecture271 a few years later, Dirac referred to the fact that for bosons we no longer have the picture of a vacuum with negative energy states filled up', and remarked that in this case `the whole theory becomes more complicated ." The next section will show how the development of quantum field theory made the interpretation of antiparticles as holes unnecessary, even though unfortunately it lingers on in many textbooks . To quote Julian Schwinger,3°uThe picture of an infinite sea of negative energy electrons is now best regarded as a historical curiosity, and forgotten ." (iii)One of the great successes of the Dirac theory was its correct prediction of the magnetic moment of the electron . This was particularly striking, as the magnetic moment (1.1.8)is twice as large as would be expected for the orbital motion of a charged point particle with angular momentum h/2 ; this factor of 2 had remained mysterious until Dirac's theory . However, there is really nothing in Dirac's tine of argument that leads unequivocally to this particular value for the magnetic moment . At the point where we brought electric and magnetic fields into the wave equation (1.1.23), we could just as well have added a `Pauli term'3 1 rcac4Yu l TFit, ( 1.1.32) with arbitrary coefficient K . (Here F,tu is the usual electromagnetic field strength tensor, with Fri = B3, F° 1= EZ, etc .) This term could be obtained byfirst adding a term to the free-field equations proportional to y''](0,210x,,,ax'')W, which of course equals zero, and then making the substitutions (1 .1.22) as before . A more modern approach would be simply to remark that the term (1.1.32) is consistent with all accepted invariance principles, including Lorentz invariance and gauge invariance, and so there is no reason why such a term should not beincluded in the field equations . (See Section 12 .3.) This term would give an additional contribution proportional to K to the magnetic moment of the electron, so apart from the possible demand for a purely formal simplicity, there was no reason to expect any particular value for the magnetic moment of the electron in Dirac's theory . As we shall see in this book, these problems were all eventually to be solved (or at least clarified) through the development of quantum field theory . 1.2 The Birth o f Quantum FieldTheor y 1.2 The Birth of Quantum Field Theory15 The photon is the only particle that was known as a field before it was detected as a particle . Thus it is natural that the formalism of quantum field theory should have been developed in the first instance in connection with radiation and only later applied to other particles and fields . In 1926, in one of the central papers on matrix mechanics, Born, Heisenberg, and Jordan32 applied their new methods to the free radiation field. For simplicity, they ignored the polarization of electromagnetic waves and worked in one space dimension, with coordinate x running from 0 to L ; the radiation field u(x,t) if constrained to vanish at these endpoints thus has the same behavior as the displacement of a string with ends fixed at x =0and x = L . By analogy with either the case of a string or the full electromagnetic field, the Hamiltonian was taken to have the form 1 ~L c~z~l H i2 .1O ~0+ c dx.ox(1.2.1) Inorder to reduce this expr ession to a sum of squares, the field u was expressed as a sum ofFourier components with u = 0at both x = 0and xZ u(x, t)_ q k(t) sin(COkX k=1C Wk=k7rcIL, so that H:)cE f .2q24k_1{1.2.2} (x.2.3) (1.2.4) Thus the string or field behaves like sum of independent harmonic oscilla- tors with angular frequencies wk, as had been anticipated 20 years earlier by Paul EhrenfestY a In particular, the `momentum' Pk(t) canonically conjugate to qk(t) is determined, as in particle mechanics, by the condition that if H is expressed as a function of the p s and q s, the n This yields a 'momentum' {1.2.5} 1 6 1 Historical Introductio n so the canonical commutation relations may be written 2 -2 M [4k(0,qj(0]= L[pk(t),qj(t)] L bkj (126) Also, the time-dependence of q k(t)is governed by the Hamiltonian equa- tion of motion Mr} =2 Pk(t) _ -~ ~6 ~ -~~4 k(~~- (1.2,8}~'k() The form of the matrices defined by Eqs . (1.2.6)-(1 .2.8) was already known to Born, Heisenberg, and Jordan through previous work on the harmonic oscillator . The q-matrix is given by qk(t)= ~,r~~ Ia~` exp(-iw~ .t) + a~ expo+icOkt)j ( 1.2.9) with aka time-independent matrix and a~ its Hermitian adjoint, satisfying the commutation relations ak,aj~ =0 . (1 .2.1) The rows and columns of these matrices are labelled with a set of positive integers nF, ~ .~,... one for each normal mode . The matrix elements ar e 4akln',n',...,~~,~r~ ... nkJnk,n k_2 nfn1 (1.2.12) 1*k For a single normal mode, these matrices may be written explicitly as o 0o 0 0 a o . a o 0 0 00 0 0 o 0 a o . a= 00 0 0 at= 0 0 ,f3-0 It is straightforward to check that (1 .2. 12) and (1 .2.13)do satisfy the commutation relations (1 .2.10)and (1 .2.11). The physical interpretation of a column vector with integer components n1, rte,- ... is that it represents a state with nk quanta in each normal mode k. The matrix ak orakacting on such a column vector will respectively 1.2The Birt h of Quantum Field Theory 17 lower or raise rzk by one unit, leaving all n .~ with e =~ k unchanged ; they may therefore be interpreted as operators which annihilate or create one quantum in the kth norma lmode . In particular, the vector with all nk equal to zero represents the vacuum ;it is annihilated by any ak . This interpretation is further borne out by inspection of the Hamilto- nian . Using { 1 .2.9} and (1 .x.10) in (1 .2.4) give s H 11Wk (a4ak + 12) - The Hamiltonian is then diagonal in the n-representatio n (H)ni,nz,...,ni,nz.._ hWk(nk + 2)dn1 "j. k(1.2. Z4) (1.2.15) We see that the energy of the state is just the sum of energies hu)kfor each quantum present in the state, plus an infinite zero-point energy EO= ~ ~khWk. Applied to the radiation field, this formalism justified the Bose method of counting radiation states according to the numbers rayof quanta in each normal mode . Born, Heisenberg, and Jordan used this formalism to derive an expres- sion for the r .m.s. energy fluctuations in black-body radiation . (For this purpose they actually only used the commutation relations (1.2.6)-(1 .2.7).) However, this approach was soon applied to a more urgent problem, the calculation of the rates for spontaneous emission of radiation . In order to appreciate the difficulties here, it is necessary to go back in time a bit . In one of the first papers on matrix mechanics, Born and Jordan33 had assumed in effect that an atom, in dropping from a stat e to a lower state x, would emit radiation just like a classical charged oscillator with displacemen t r(t)=r#,, exp(-27rivt) + r#,* exp(2nivt) , where(1.2.16) {1.2.17} and rg,, is the fl, ot element of the matrix associated with the electron position . The energy E of such an oscillator i s E _ 1 2m ~ rZ + (2nv)2 r~~ = 87C 2 mv2 1 rp,, 12 . (1 .2.18) A straightforward classical calculation then gives the radiated power ,and dividing bythe energy by per photon gives the rate of photon emiss ion 1brc3e2v 3 ~4~~ ~ ~) = ~ ~r~~~~  (1 .2.9) 3hc- 18 1 Historical Introductio n However, it was not at all clear why the formulas for emission of radiation by a classical dipole should be taken over in this manner in dealing with spontaneous emission . A little later a more convincing though even less direct derivation was given by Dirac .34 By considering the behavior of quantized atomic states in an oscillating classical electromagnetic field with energy density per frequency interval u at frequency (1.2.17), he was able to derive formulas for the rates uB(,x --*13) and uB(# --+oc) for absorption or induced emission 2 2 g~~ P)= B(fl ~~)"'~2 3h2 ~r {~a12(1.2.20) (Note that the expression on the right is symmetric between states ac and P, because rxp is just rg,w .) Einstein34' had already shown in 1917 that the possibility of thermal equilibrium between atoms and black-body radiation imposes a relation between the rate A (J3 -} oc) of spontaneous emission and the rates uB for induced emission or absorption : A(# -, a) =87rhu 3$(P->)c). (1 .2.21) C3 Using (1,2,20) in this relation immediately yields the Born-Jordan result (1.2.19)for the rate of spontaneous emission . Nevertheless, it still seemed unsatisfactory that thermodynamic arguments should be needed to derive formulas for processes involving a single atom . Finally, in 1927 Dirac35 was able to give a thoroughly quantum me- chanical treatment of spontaneous emission . The vector potential A(x, t) was expanded in normal modes, as in Eq. (1.2.2), and the coefficients were shown to satisfy commutation relations like (1.2.6). In consequence, each state of the free radiation field was specified by a set of integers rte,one for each normal mode, and the matrix elements of the electromagnetic interaction et  A took the form of a sum over normal modes, with matrix coefficients proportional to the matrices ak and a~ defined in Eqs .(1.2,1 ~)-10)- (1.2.13). The crucial result here is the factor ,/n -k+ 1 in Eq. (1.2.13); the probability for a transition in which the number of photons in a normal mode k rises from nk to nk+Iis proportional to the square of this factor, ❑r nk + 1. But in a radiation field with nk photons in a normal mode k, the energy density uper frequency interval i s v~ U(~'k)S~= In ~ kx hv kC3 so the rate for emission of radiation in normal mode k is proport ionalto Snhvk 1.2 The Birth of Quantum Field Theory 1 9 The first term is interpreted as the contribution of induced emission, and the second term as the contribution of spontaneous emission . Hence, without any appeal to thermodynamics, Dirac could conclude that the ratio of the rates uB for induced emission and Afor spontaneous emission is given by the Einstein relation, Eq . (1.121).Using his earlier result (1 .2.20) for B, Dirac was thus able to rederive the Born-Jordan formula" (1.2.19) for spontaneous emission rate A . A little later, similar methods were used by Dirac to give a quantum mechanical treatment of the scattering of radiation and the lifetime of excited atomic states,36and by Victor Weisskopf and Eugene Wigner to make a detailed study of spectral line shapes .Y" Dirac in his work was separating the electromagnetic potential into a radiation field A and a static Coulomb potential A°, in a manner which did not preserve the manifest Lorentz and gauge invariance of classical electrodynamics . These matters were put on a firmer foundation a little later by Enrico Fermi .16~ Many physicists in the 1930s learned their quantum electrodynamics from Fermi's 1932 review . The use of canonical commutation relations for q and p or a and a t also raised a question as to the Lorentz invariance of the quantized theory . Jordan and Pauli37 in 1928 were able to show that the commutators of fields at different spacetime points were in fact Lorentz-invariant . (These commutators are calculated in Chapter 5.)Somewhat later, Bohr and Lean RasenfelP used a number of ingenious thought experiments to show that these commutation relations express limitations on our ability to measure fields at spacetime points separated by time-like intervals . It was not long after the successful quantization of the electromagnetic field that these techniques were applied to other fields . At first this was regarded as a `second quantization' ; the fields to be quantized were the wave functions used in one-particle quantum mechanics, such as the Dirac wave function of the electron . The first step in this direction seems to have been taken in 1927 by Jordan .39 I n 1 921$ an essential element was supplied by Jordan and Vdigner .4° They recognized that the Pauli exclusion principle prevents the occupation number n kof electrons in any normal mode k (counting spin as well as position variables) from taking any values other than 0or 1 . The electron field therefore cannot be expanded as a superposition of operators satisfying the commutation relations (1 .2.10), (1.2.12), because these relations require rayto take all integer values from 0 to oc .. Instead, they proposed that the electron field should be expanded in a sum of operators a k, aksatisfying the anticommutation relation s ak C~~ + at a k = 6jk ( 1.122) Clk aj+ afar =0. (1 .123 ) The relations can be satisfied by matrices labelled bya set of integers 20 1Historical Introductio n n1, ra21... ,one for each normal mode, each integer taking just the value s zero and one : In~ 0, nk = 1, n'r~=j for j *k 1 nfk=1, nk=0,n f~=nffor j4k 0 otherwise .(1.2.24) (1.2.25) For instance, for a single normal mode we have just two rows and two columns, corresponding to the values unity and zero of n' and n ; the a and at matrices take the for m 100 -~ a 1 ~T 10a[0 Q The reader may check that (1 .2.24) and (1 .225) do satisfy the anticommu- tation relations (1 .2.2) and ( 1.2.23). The interpretation of a column vector characterized by integers n1, '12, ... is that it represents a state with nk quanta in each normal mode k, just as for bosons . The difference is, of course, that since each nk takes only the values 0and 1, there can be at most one quantum in each normal mode, as required by the Pauli exclusion principle . Again, ak destroys a quantum in normal mode k if there is one there already, and otherwise gives zero ; also,at creates a quantum in normal mode k unless there is one there already, in which case it gives zero . Much later it was shown by Fier and Pauli401 that the choice between commutation and anticommutation relations is dictated solely by the particle's spin : commutators must be used for particles with integer spin like the photon, and anticommutators for particles with half-integer spin like the plectron, (This will be shown in a different way in Chapter 5.) The theory of general quantum fields was first laid out in 1 929, in a pair of comprehensive articles by Heisenberg and Pauli .4i The starting point ❑f their work was the application of the canonical formalism to the fields themselves, rather than to the coefficients of the normal modes appearing in the fields . Heisenberg and Pauli took the Lagrangian L as the space-integral of a local function of fields and spacetime derivatives of fields ; the field equations were then determined from the principle that the action f Ldt should be stationary when the fields are varied ; and the commutation relations were determined from the assumption that the variational derivative of the Lagrangian with respect to any field's time-derivative behaves like a `momentum' conjugate to that field (except that commutation relations become anticommutation relations for fermion fields . They also went on to apply this general formalism to the electromagnetic and Dirac fields, and explored the various invariance and 1.2 The Birth ofQuantum Field Theory 2 1 conservation laws, including the conservation of charge, momentum, and energy, and Lorentz and gauge invariance . The Heisenberg-Pauli formalism is essentially the same as that described in our Chapter 7, and so for the present we can limit ourselves to a single example which will turn out to be useful later in this section . For a free complex scalar field O(x) the Lagrangian is taken a s (mc2)2 ] If we subject O(x) to an infinitesimal variation ~O(x), the Lagrangian is changed by the amoun t 6L_[d3x[ ~f6~ + ~6~t-C2vOt. V60 C2vO- v60t (?nc2)2(inc2)2 ] t6o Itisassumed in u sing the principle of stationary action that the variation in the fields should vanish on the boundaries of the spacetime region of integration . Thus, in computing the change in the action f L dt, we can immediately integrate by parts, and writ e JL C~~ = C2 ~Xjof❑-(mc)2) 0+60 ❑-(mc))Of But this must vanish for any 50and ~Ot, so 0must satisfy the familiar relativistic wave equation 0 (1 .2.28) and its adjoint . The `momenta' canonically conjugate to the fields 0and ofare given by the variational derivatives of Lwith respect to ~and which we can read off from (1.2.27)as It=-~ '=~t, (1 .2.29) art= & ~ _ ~ . (1 .2.30) These field variables satisfy the usual canonical co mmutat ion relations , 22 IHistorical Introductio n with a delta function in place of a Kronecker delt a [7r(X,o,O(Y,t)J = [7(X, 0, 0*(Y, 0] [Ir(X, t),Z(Y, ol [ 0(X, t), 0 (y, 0117~1(x, o,01(y,ol=-ih5l(x -Y), 17rt(x, 0, O(Y,010, 17t(x, t), 7rt(y, t)][7r(X,0, .g,(Y,010, [01(x, o,01(y, ol [O(X, o,01(y,o] 0(1.2.31) (1.2.32) (1.2.33) (1.2.34) The Hamiltonian here is given (just as in particle mechanics) by the `sum' of all canonical momenta times the time-derivatives of the corresponding fields, minus the Lagrangian ; H- dux [ 7r~+nf~f I-L or, using (1.2.26), (1.2.29), and (1.2.30): H=fdux[gtn + C2(VO)t(VO) + ((1-2-35) M2c ,) Otol (1.2.3) After the papers by Heisenberg and Pauli one element was still missing before quantum field theory could reach its final pre-war form : a solution to the problem of the negative-energy states . We saw in the last section that in 1930, at just about the time of the Heisenberg-Pauli papers, Dirac had proposed that the negative-energy states of the electron were all filled, but with only the holes in the negative-energy sea observable, rather than the negative-energy electrons themselves . After Dirac's idea was seemingly confirmed bythe discovery of the positron in 1932, his `hole theory' was used to calculate a number of processes to the lowest order of perturbation theory, including electron-positron pair production and scattering . At the same time, a great deal of work was put into the development of a formalism whose Lorentz invariance would be explicit . The most influential effort was the `many-time' formalism of Dirac, Vladimir Fock, and Boris Padolsky,'2 in which the state vector was represented by a wave function depending on the spacetime and spin coordinates of all electrons, positive-energy and negative-energy . In this formalism, the total number of electrons of either positive or negative energy is conserved ; for instance, production of an electron-positron pair is described as the excitation of a negative-energy electron to a positive-energy state, and the annihilation of an electron and positron is described as the corresponding deexcitation . This many-time formalism had the advantage of manifest Lorentz invariance, but it had a number of disadvantages ; In particular, there was a profound difference between the treatment of the photon, described in terms of a quantized electromagnetic field, and that of the electron and positron . Not all physicists felt this to be a disadvantage ; 1.2 The B irthof'Quantum Field Thenry . 2 3 the electron field unlike the electromagnetic field did not have a classical limit, so there were doubts about its physical significance . Also, Dirac¢2a conceived of fields as the means by which we observe particles, so that he did not expect particles and fields to be described in the same terms . Though I do not know whether it bothered anyone at the time, there was a more practical disadvantage of the many-time formalism : it would have been difficult to use it to describe a process like nuclear beta decay, in which an electron and antineutrino are created without an accompanying positron or neutrino . The successful calculation by Fermi43 of the electron energy distribution in beta decay deserves to be counted as one of the early triumphs of quantum field theory . The essential idea that was needed to demonstrate the equivalence of the Dirac hole theory with a quantum field theory of the electron was provided by Fock4" and by Wendell Furry and Oppenheimer44 in 1933 . To appreciate this idea from a more modern standpoint, suppose we try to construct an plectron field in analogy with the electromagnetic field or the Barn-Heisenberg-Jordan field (1 .2.2). Since electrons carry a charge, we would not like to mix annihilation and creation operators, so we might try to write the field a s where uk (x)eya"'kr are a complete set of orthonormal plane-wave solutions of the Dirac equation (1 .1.13){with k now labelling the three-momentum, spin, and sign of the energy) : uk _hr,UkUka (L2.35) -ihc~e - V +a4rnc- X1 .2.39} fu~~sedux =6k((1.2.40) and uk are the corresponding annihilation operators, satisfying the Jordan- Wigner anticommutation relations (1.2.22)-(1 .2.23). According to the ideas of `second quantization' or the canonical quantization procedure of Heisenberg and Pauli,41 the Hamiltonian is formed by calculating the `expectation value' of with a `wave function' replaced by the quantized field (1 .2.37) H=/d3xt.V,=~amkakak. (1 .2.41) k The trouble is, of course, that this is not a positive operator -half the c )k are negative while the operators a~ak take only the positive eigenr~alues Iand D . (See Eqs . (1.2.24) and (1 .2.25).) In order to cure this disease, Furry and Oppenheimer picked up Dirac's idea42 that the positron is the absence of a negative-energy electron ; the anticommutation relations are 24 1Historical Introductio n symmetric between creation and annihilation operators, so they defined the positron creation and annihilation operators as the corresponding annihilation and creation operators for negative-energy electron s (forWk < 0) (1 .2.42) bk = ak bk=41k where the label k on b denotes a positive-energy positron mode with momenta and spin opposite to those of the electron mode k . The Dirac field (1.2.37)may then he writte n w (x) ( +)akUxW+ ~-}b~ Uk~W) a (1 .2.43) k where (+ )and (- )indicate sums over normal modes k with (Ok>0 and c 0k<0, respectively, and Uk(x)= Uk(x)e-Similarly, using the anticommutation relations for the h s, we can rewrite the energy operator (1.2.4i) as k k where E Ois the infinite c-number k(1.2.44) (1.2.45) In order for this redefinition to be more than a mere formality, it is necessary also to specify that the physical vacuum is a state YD containing no positive-energy electrons or positron s Hence ( 1 . 2. 4 ) gives the energy of the vacuum as just E0.Ifwe measure all energies relative to the vacuum energy Eo, then the physical energy operator is hC - EO; and Eq . (1.2.44) shows that this is a positive operator . The problem of negative-energy states for a charged spin zero particle was also resolved in 1934, by Pauli and VLfeisskopf,45 in a paper written in part to challenge Dirac's picture of filled negative-energy states . Here the creation and annihilation operators satisfy commutation rather than anticommutation relations, so it is not possible to interchange the roles of these operators freely, as was the case for fermions . Instead we must return to the Heisenberg-Pauli canonical formalism4l to decide which coefficients of the various normal modes are creation or annihilation operators . Pauli and Weisskopf expanded the free charged scalar field in plane waves in a cube of spatial volume V 0 OX, t)= 1 ~ ~ q(k,t)e~k~x(1.2.48) k 1.2 The Birth of Quantu m Field Theory 2 5 with the wave numbers restricted by the periodicity condition, that the quantities kJL/27r for j = 1,2,3 should be a set of three positive or negative integers . Similarly the canonically conjugate variable (1 .2.29) was expanded as ~ ( 1.2.49) (xar)= ~~1:P(k, t)e-'kx The minus sign is put into the exponent here so that (1.2.29) now becomes : The Fourier invers ion formula give s q (k} t)= p(k,t)=1~~x O(x, t)e-~~~,~ Jd3X7qX' t)e+ik,, ,(1.2.5) (1.2.51) (1.2.52) and therefore the canonical commutation relations (1 .2.31)-(1 .2.34) yield for the qsand p s _ih [p (k, t)aq (1,Yfd3xPik-xe-il-x(1.2.53) [p(k,t),qf(I,t)J [p(k,t),p(I,t)J [p(k,t),p'(I,t) J _ [q(k,t) .q(1t) ]= [q(k .t)qt(Lt)]=0 (1.2.54) together with other relations that may be derived from these bytaking their Hermitian adjoints .Byinserting (1 .2.48) and (t .2.49) in the formula (1.2.36) for the Hamiltonian, we can also write this operator in terms of psandgs : whereH [p * (k, t)p(k, t)+ (1)2q~(k, t)q(k, t) k k2mi!2 ) 2 C~J =C~'~ + ( r,(1.2.55) (1.2.56) The time-derivatives of the p s are then given by the Hamiltonian equatio n p(k,t)_ {,~Ok t = --rO ~ q~~ (k,r) (1 .2.57) ~f() (and its adjoint), a result which in the light of Eq . (1.2.50)is just equivalent to the Klein-Gordon-Schrodinger wave equation (1.2.28). We see that, just as in the case of the 1926model of Born, Heisenberg, and Jordan,4 the free field behaves like an infinite number of coupled 26 1 Historical Introductio n harmonic oscillators . Pauli and Weisskopf could construct p and q op- erators which satisfy the commutation relations (1.2.53)-(1 .2.54)and the `equations of motion' { 1 .2.5} and (1.2-57), by introducing annihilation and creation operators a, b, a T, b'of two different kinds, corresponding to particles and antiparticles : q(kat)-i F~W k p(ka t)r~_2 where[a(k) exp(-icOkt) - h l(k) exp(ic)kt} ] [b(k)CXP(-iCOkt) + at( k) exp(+i(O kt}] [a(k), at(l)] [b(k), bt(I)i = &I , [a(k), a([)] [b(k), b(I)l = 0 , [a(k),b(l)] [a(k),bt(1)j [at(k),b(l) ] [at(k), bt(l)] 0 -(t.2.58) (1.2,5) (1_x.64) (1.161) (1.2.62) I t is straightforward to check that these operators do satisfy the desired relations Cl .2.53), (1.2.54), (1.2.50), and (1 .2.57). The field (1 .2.4$) may be written Ur~O -b~(-k) exp(-a k' X + iwkt) ] and the H amiltonian(1.2.55) ta kes the form(1.2.63) H2hwk[bt(k)b(k) + b( k)bf(k) + rx#( k)u(k) + a( k)a~{k}] or, using (1.2.60)--(1 .2.62) H=~= 1:hO)k [b*(k)b(k) + a~(k)a(k)] + EO k where Efl is theinfinite c-number k(1.2.64) (1.2,65) The existence of two different kinds of operators a and b, which appear in precisely the same way in the Hamiltonian, shows that this is a theory with two kinds of particles with the same mass . As emphasized by Pauli and Weisskopf, these two varieties can be identified as particles and the corresponding antiparticles, and if charged have opposite charges . Thus, 1.2The Birth of Quan tum Field Theory 27 as we stressed above, bosons of spin zero as well as fermions of spin 1/2 can have distinct antiparticles, which for bosons cannot be identified as holes in a sea of negative energy particles . We now can tell whether a and bor a tand btare the annihilation operators bytaking the expectation values of commutation relations in the vacuum state To . For instance, if a~ were an annihilation operator it would give zero when applied to the vacuum state, so the vacuum expectation value of (1.2.60)would giv e in conflict with the requirement that the left-hand side must be negative- definite . In this way we can conclude that it is ak and bk that are the annihilation operators, and therefor e- Ila(k)Tol 12=(TO, [a(k),af(k)jTo) = -~l (1 .2.66) This is consistent with all commutation relations . Thus, the canonical formalism forces the coefficient of the e+"'r in the field (1.2.58) to be a creation operator, as it also is in the Furry-Oppenheimer formalism for spin 1/2 . Equations (1 .2.64) and (1.2.67) now tell us that E 0is the energy of the vacuum state .If we measure all energies relative to E0, then the physical energy operator is H - Eo, and (1 .2.64) shows that this again is positive . What about the problem that served Dirac as a starting point, the problem of negative probabilities? As Dirac had recognized, the only probability density p, which can be formed from solutions of the Klein- Gardon-Schradinger free scalar wave equation (1.2.28), and which satisfies a conservation law of the form (1.1.10), must be proportional to th e quantity r'1p =21m lot0tJ{ 2.2.68} and therefore is not necessarily a positive quantity . Similarly, in the `second-quantized' theory, where 0is given by Eq . {1.2.63},pis not a positive operator . Since Ot(x) does not commute with O(x) here, we can write (1.2.68) in various forms, which differ byinfinite c-numbers ; it proves convenient to write it as : r?o a01 01 h P . t a t The space-integral of this operator is then easily calculated to b e N p dux = (at(k)a(k) - b t(k)b(k))k(1.2.69) (1_2.7a) and clearly has eigenvalues of either sign . 28 1 Historical Introductio n However, in a sense this problem appears in quantum field theory for spin 1/2 as well as spin zero . The density operator ~tyrof Dirac is indeed a positive operator, but in order to construct a physical density we ought to subtract the contribution of the filled electron states . In particular, using the plane-wave decomposition (1 .2.43), we may write the total number operator a s N=jdux tp'q) - t+) afi(k)a(k)+ 1: (-)b(k)b'(k) k k The anticommutation relations for the b s allow us to rewrite this a s N - No=EWa kak (-)bklbk (1.2.71) k k where No is the infinite constant k According to Eqs . (1.2.46) and (1 .2.47), No is the number of particles in the vacuum, so Furry and Oppenheimer reasoned that the physical number operator is N- No, and this now has both negative and positive eigenvalues, just as for spin zero . The solution to this problem provided byquantum field theory is that neither the y)❑f furry and Oppenheimer nor the 0of Pauli and Weisskopf are probability amplitudes, which would have to define conserved positive probability densities . instead, the physical Hilbert space is spanned by states defined as containing definite numbers of particles and/or antipar- ticles in each mode . If(Dn are a complete orthonormal set of such states, then a measurement of particle numbers in an arbitrary state T will yield a probability for finding the system in state (D,, given b y where ( (DM,T) is the usual Hilbert space scalar product . Hence, no question as to the possibility of negative probabilities will arise for any spin . The wave fields 0, ya, etc, are not probability amplitudes at all, but operators which create or destroy particles in the various normal modes . It would be a good thing if the misleading expression `second quantization' were permanently retired , In particular, the operators Nand N - Noof Eqs .(1.2.74) and (1.2.71) are not to be interpreted as total probabilities, but as number operators : specifically, the number of particles minus the number of antiparticles . For charged particles, the conservation of charge forces the charge operators to be proportional to these number operators, so the minus signs in (1.2.70) and- (1.2.71) allow us immediately to conclude that particles and antipar- ticles have opposite charge . In this field-theoretic formalism, interactions 1.2The Birth ofQua ntumField Theory 2 9 contri bute terms to the Hamil tonian which are of third, fourth, or higher order in field variables, and the rates of various processes are given by using these interaction operators in a time-dependent perturbation theory . The conceptual framework described in the above brief remarks will serve as the bas is for much of the work in this book. Despite its apparent advantages, quantum field theory did not imme- diately supplant hole theory ; rather, the two points of view coexisted for a while, and various combinations of field-theoretic and hole-theoretic ideas were used in calculations of physical reaction rates . This period saw a number of calculations of cross sections to lowest order in powers of e2 for various processes, such as eT + y ~ e- + ,/in1929 by Klein and NXShina .;46e++ e- --* 2,i in 1930 by Dirac; 47e- + e ~ e- +e- in 1932 b y oller ;48e- +Z --* e- + y +Z and y + Z ---~ e++ e- +Z (where Z denotes the Coulomb field of a heavy atom) in 1934 by Bethe and Heider ; 49 and e+ +e- --* e+ + e- in 1936 by Bb abha .$° (Rules for the calculation of such processes are g iven in Chapter 8, and worked out in detail the re for the case of electron-photon scattering .) These lowest-order calculations gave finite results, in reasonable agreement with the experimental data . Nevertheless, a general feeling of dissatisfaction with quantum field theory (whether o rnot in the form of hole theory) persisted throughout the 1930s . One of the reasons for this was the apparent failure of quantum electrodynamics to account fo rthe penet rating power of the charged particles in cosmic ray showers, noted in 1936 by Oppenheimer and Franklin Ca I'1SOn .50QAnother cause of dissatisfaction that turned out to be related to the first was the steady discovery of new kinds of particles and interactions . We have already mentioned the electron, photon, positron, neutrino, and, of course, the nucleus of hydrogen, the proton . Throughout the 1920s it was generally believed that heavier nuclei are composed of protons and electrons, but it was hard to see how a light particle like the electron could be confined in the nucleus . Anot her severe difficulty with this picture was pointed out in 1931 by Ehrenfest and Oppenheimer :" the ❑ucleus of ordinary nitrogen, IV14, in order to have atomic number 7 and atomic weight 14, wo uld have to be composed of 14protons and 7 electrons, and would therefore have to be a fermion, in conflict wit hthe result of molecular spectroscopy52 that N 14 is a boson . This problem (and others) were solved in 1932 with the discovery of the neutron, 53 and by Heisenberg's subsequent suggestion14 that nuclei are composed of protons and neutrons, not protons and electrons . It was clear that a strong non-e lectromagnetic force of short range would have to operate between neutrons and protons to ho ld nuclei together . After the success of the Fermi theory of beta decay, several authors54" speculated that nuclear forces m ight be explained in this theory as due to the exchange of electrons and neutrinos . A few years later, in 1935, 30 1Historical Introduction Hideki Yukawa proposed a quite different quantum field theory of the nuclear force .55 In an essentially classical calculation, he found that the interaction of a scalar field with nucleons (protons or neutrons) would produce a nucleon-nucleon potential, with a dependence on the nucleon separation r given by Y(r)x1 exp(-~ .r) (1 .2.74) instead of the 11r Coulomb potential produced by electric fields . The quantity ~was introduced as a parameter in Yukawa's scalar field equation, and when this equation was quantized, Yukawa found that it described particles of mass U f c.The observed range of the strong interactions within nuclei led Yukawa to estimate that U/c is of the order of 200 electron masses . In 1937 such `mesons' were discovered in cloud chamber experiments" bySeth Neddermeyer and Anderson and byJabez Curry Street and Edward Carl Stevenson, and it was generally believed that these were the hypothesized particles of Yukawa . The discovery of mesons reveated that the charged particles in cosmic ray showers are not all electrons, and thus cleared up the problem with these showers that had bothered Oppenheimer and Carlson . At the same time, however, it created new difficulties . Lothar Nardheim56a pointed out in 1939 that the same strong interactions by which the mesons are copiously produced at high altitudes (and which are required in Yukawa's theory) should have led to the mesons' absorption in the atmosphere, a result contradicted b ytheir copious appearance at lower altitudes . In 1947 it was shown in an experiment by Marcello Conversi, Ettore Pancini, and Oreste Piccioni57 that the mesons which predominate in cosmic rays at low altitude actually interact weakly with nucleons, and therefore could not be identified with Yukawa's particle . This puzzle was cleared up by a theoretical suggestion,58 and its subsequent experimental confirmation59 byCesare Lattes, Occhialini, and Cecil Powell -there are two kinds of mesons with slightly different masses : the heavier (now called the it meson or pion) has strong interactians and plays the role in nuclear force envisaged by Yukawa ; the lighter (now called the p meson, or muon) has only weak and electromagnetic interactions, and predominates in cosmic rays at sea level, being produced by the decay of rt mesons . In the same year, 1947, entirely new kinds of particles (now known as K mesons and hyperons) were found in cosmic rays byGeorge Rochester and Clifford Butler .60 From 1947 until the present particles have continued to be discovered in a bewildering variety, but to pursue this story would take us outside the bounds of our present survey . These discoveries showed clearly that and conceptual framework which was limited to photons, electrons, and positrons would be far too narrow to be taken seriously as 1.3The Problem of Infinities 31 a fundamental theory .But an even more important obstacle was presented by a pu rely theoretical problem --- the problem of infinities . 1.3TheProblem of I nfiniti es Quantum field theory deals with fields V7(x) that destroy and create parti- cles at a spacetime point x . Earlier experience with classical electron theory provided a warning that a point electron will have infinite electromagnetic self-mass ; this mass is e2 /6nac-2 for a surface distribution of charge with radius a, and therefore blows up for a -> U . Disappointingly this problem appeared with even greater severity in the early days of quantum field theory, and although greatly ameliorated by subsequent improvements in the theory, it remains with us to the present day . The problem of infinities in quantum field theory was apparently first noted in the 1929-3 0papers of Heisenberg and Pauli .41 Soon after, the presence of infinities was confirmed in calculations of the electromagnetic self-energy of a bound electron by Oppenheimer,61and of a free electron by Ivor Waller .62They used ordinary second-order perturbation theory, with an intermediate state consisting of an electron and a photon : for instance, the shift of the energy E,of an electron in the nth energy level of hydrogen is given b y AE, dkj < m ;k,~jH'Ira> 1zX1.3.1) where the sums and integral are over all intermediate electron states in. photon helicities ~, and photon momenta k, and H' is the term in the Hamiltonian representing the interaction of radiation and electrons . This calculation gave a self-energy that is formally infinite ; further ; if this infinity is removed by discarding all intermediate states with photon wave numbers greater than I /a, then the self-energy behaves like 11a2 as a --+0. Infinities of this sort are often called ultraviolet divergences, because they arise from intermediate states containing particles of very short wavelength . These calculations treated the electron according to the rules of the original Dirac theory, without filled negative-electron states . A few years later Weisskopf repeated the calculation of the electron self-mass in the new hole theory, with all negative-energy states full . In this case another term appears in second-order perturbation theory, which in a non-hole- theory language can bedescribed as arising from processes in which the electron in its final state first appears out of the vacuum together with a photon and a positron which then annihilate along with the initial 32 1 Historical Introductio n electron . Initially Weisskopf found a 1/a2 dependence on the photon wave-number cutoff 11a . The same calculation was being carried out (at the suggestion of Bohr) at that time by Carlson and Furry . After seeing Weisskopf's results, Furry realized that while Weisskopf had included an electrostatic term that he and Carlson had neglected, Weisskopf had made a new mistake in the calculation of the magnetic self-energy . After hearing from Furry and correcting his own error, Weisskopf found that the 1/02 terms in the total mass shift cancelled! However, despite this cancellation, an infinity remained : with a wave-number cutoff 1 f a, the self-mass was found to be63 Sac h flZem 27tM In (M CL2)1. 0 .3.2) The weakening of the cut-off dependence, to Ina as compared with the classical 1/a or the early quantum I/a 2, was mildly encouraging at the time and turned out to be of great importance later, in the development of renormalization theory . An infinity of quite a different kind was encountered in 1933, apparently first byDirac .64 He considered the effect of an external static nearly uniform charge density 8(x) on the vacuum, i .e., on the negative-energy electrons in the filled energy levels of hole theory . The Coulomb interaction between E(x) and the charge density of the negative-energy electrons produces a `vacuum polarization,' with induced charge densit y 2 =Af + $ ~11 p2.E+...(1.3.3)me The constant B is finite, and of order oc, On the other hand, A i logarithmically divergent, of order a Ina, where If a is the wave-number cutoff . Infinities also seemed to occur in a related problem, the scattering of light by light . Hans Euler, Bernard Kockel, and Heisenberg15 showed in 1935w 6that these infinities could be eliminated by using amore-ar-less ar- bitrary prescription suggested earlier by DiraC66 and Heisenberg67 . They calculated an effective Lagrangian density for the non-linear electrody- namic effects produced by virtual electron-positron pairs : 4(E2-]32) + 3CO ,nzrra~c~ E2 - B 2~2 + 7(E.g)'] +... ? (1 .3.4) valid for frequencies v <m,c2/h . Soon after, Nicholas Kemmer and Weisskopf68 presented arguments that in this case the infinities are spuri- ous, and that Eq . (1.3.4) can be derived without any subtraction prescrip- tion. One bright spot in the struggle with infinities was the successful treat- 1.3 The Problem ofInfinities 33 ment❑finfrared divergences, those that arise from the low-energy rather than the high-energy part of the range of integration . In 1937 it was shown by Felix Bloch and Arne Nordsieck68,that these infinities cancel provided one includes processes in which arbitrary numbers of low-energy photons are produced . This will be discussed in modern terms in Chapter 13. Yet another infinity turned up in a calculation by Sidney Michael Dancoff69 in 1939 of the radiative corrections to the scattering of electrons by the static Coulomb field of an atom . The calculation contained a mistake (one of the terms was omitted), but this was not realized until later.69a Throughout the 1930s, these various infinities were seen not merely as failures of specific calculations . Rather, they seemed to indicate a gap in the understanding of relativistic quantum field theory on the most fundamental level, an opinion reinforced by the problems with cosmic rays mentioned in the previous section . One of the symptoms of this uneasy pessimism was the continued exploration throughout the 1930s and 1940s of alternative formalisms . As Julian 5chwinger~9h later recalled, The preoccupation of the majority of involved physicists was not with analyzing and carefully applying the known relativistic theory of coupled electron and electromagnetic fields but with changing it .' Thus in 1938 Heisenberjo proposed the existence of a fundamental length L, analogous to the fundamental action h and fundamental velocity c . Field theory was supposed to work only for distances larger than L, so that all divergent integrals would effectively be cut ❑ff at distances L, or momenta h/L . Several specific proposals7°V were made for giving field theory a non-local structure . Some theorists began to suspect that the formalism of state-vectors and quantum fields should be replaced by one based solely ❑n observable quantities, such as the S-matrix introduced by John Archibald Wheeler7l in 1937 and HeisenberJz in 1943, whose elements are the amplitudes for various scattering processes . As we shall see, the concept of the S-matrix has now become a vital part of modern quantum field theory, and for some theorists a pure S-matrix theory became an ideal, especially as a possible solution to the problems of the strong interactions ." In yet another direction, wheeler and Richard Feynman74 in 1945 attempted to eliminate the electromagnetic field, deriving electromagnetic interactions in terms of an interaction at a distance . They were able to show that a pure retarded (or pure advanced) potential could be obtained by taking into account the interaction not only between source and test charges, but also between these charges and all the other charges in the universe . Perhaps the most radical modification of quantum mechanics suggested during this period was the introduction by Dirac75 of states of negative probability, 34 1 Hislarical Introductio n as a means of cancelling infinities in sums over states . This idea, of an `indefinite metric' in Hilbert space, has also flourished in quantum field theory, though not in the form originally suggested . Amore conservative idea for dealing with the infinities was also in the air during the 1930s . Perhaps these infinities could all b eabsorbed into a redefinition, a `renormalization' of the parameters of the theory . For instance, it was already known that in any Lorentz-invariant classical theory the electromagnetic self-energy and self-momentum of an electron must take the form of corrections to the mass of the electron ; hence the infinities in these quantities can be cancelled by a negative infinity in the `bare' non-electromagnetic mass of the electron, leaving a finite measurable `renormalized' mass . Also, Eq .(1. 3.3)shows that the vacuum polarization changes the charge of the electron, from e = f dux E, t o eTOTAL ;--f Vacuum polarization gives finite results in lowest order if observables like scattering cross-sections are expressed in terms of eTOTAL rather than e . The question was, whether all infinities in quantum field theory could be dealt with in this way . In 1936 VVeisslcopf76 suggested that this is the case, and verified that known infinities could be eliminated by renormalization of physical parameters in a variety of sample calculations . However, it was impossible with the calculational techniques then available to show that infinities could always be eliminated in this way, and DancQff's calculation69 seemed to show that they could not . Another effect of the appearance of infinities was a tendency to believe that any effect which turned out to be infinite in quantum field theory was actually not there at all . In particular, the 1928 Dirac theory had predicted complete degeneracy of the 2s1/2--2p1/2 levels of hydrogen to all orders in cc ; any attempt at a quantum electromagnetic calculation of the splitting of these two levels ran into the problem of the infinite self-energy of a bound electron ; therefore the existence of such a splitting was generally not taken seriously . Later Bethego recalled that This shift comes out infinite in all existing theories, and has therefore always been ignored .' This attitude persisted even in the late 1930s, when spectroscopic experiments77 began to indicate the presence of a 2si/2-2p1/2 splitting of order 1 000 MHO . One notable exception was Edwin Albrecht Uehling,78 who realized that the vacuum polarization effect mentioned earlier would produce a 2si/3-2p1/3 splitting ; unfortunately, as we shall see in Chapter 14, this contribution to the splitting is much smaller than 1 00(} MHz, and of the wrong sign . The gloom surrounding quantum field theory began to lift soon after World War II . On June 1-4, 1947, the Conference on the Foundations of 1.3 The Problem of Infinities 35 Quantum Mechanics at Shelter Island, NY brought theoretical physicists who had been working on the problems of quantum field theory through the 1930s together with a younger generation of theorists who had started scientific work during the war, and of crucial importance a few experimental physicists . The discussion leaders were Hans Kramers, Op- penheimer, and Weisskopf, One ❑f the experimentalists (or rather theorist turned experimentalist), Willis Lamp, described a decisive measurement79 of the 2s,/2-2p1/2 shift in hydrogen . A beam of hydrogen atoms from an oven, many in 2s and 2p states, was aimed at a detector sensitive only to atoms in excited states . The atoms in 2p states can decay very rapidly to the is ground state by one-photon (Lyman a)emission, while the 2s states decay only very slowly by two-photon emission, so in effect the detector was measuring the number of atoms in the metastable 2s state . The beam was passed through a magnetic field, which added a known Zeeman splitting to any 2s,/2-2pl/2 splitting naturally present . The beam was also exposed to a microwave- frequency electromagnetic field, with a fixed frequency v - 10 GHz . At a certain magnetic field strength the detector signal was observed to bequenched, indicating that the microwave field was producing resonant transitions from the metastable 2s state to the 2p state and thence by a rapid Lyman a emission to the ground state . The total (Zeeman plus intrinsic) 2s-2p splitting at this value of the magnetic field strength would have to be just hv, from which the intrinsic splitting could be inferred . A preliminary value of 1400MHz was announced, in agreement with the earlier spectroscopic measurements .77The impact of this discovery can be summarized in a saying that was current in Copenhagen when I was a graduate student there in 1954 : `Just because something is infinite does not mean it is zero! ' The discovery of the Lamb shift aroused intense interest among the theorists at Shelter Island, many of whom had already been working on improved formalisms for calculation in quantum electrodynamics . Kra- mers described his work on mass renormalization in the classical electro- dynamics of an extended electron, 79a which showed that the difficulties associated with the divergence of the self-energy in the limit of zero radius do not appear explicitly if the theory is reexpressed so that the mass pa- rameter in the formalism is identified with the experimental electron mass . Schwinger and Weisskopf (who had already heard rumors of Lamb's re- sult, and discussed the matter on the trip to Shelter Island) suggested that since the inclusion of intermediate states involving positrons was known to reduce the divergence in energy level shifts from 1/a2 to Ina, perhaps thedi fferences of the shifts in atomic energy levels might turn out to be finite when these intermediate states were taken into account . (In fact, in 1946, before he learned of Lamb's experiment, Weisskopf had already assigned this problem to a graduate student, Bruce French .) almost im- 36 1Historical Introductio n mediately after the conference, during a train ride to Schenectady, Hans Bethe8° carried out a non-relativistic calculation, still without including the effects of intermediate states containing positrons, but using a simple cutoff at virtual photon momenta of order m,c2 to eliminate infinities . He obtained the encouraging approximate value of 1040 MHz . Fully rela- tivistic calculations using the re-normalization idea to eliminate infinities were soon thereafter carried out by a number of other authors,81with excellent agreement with experiment . Another exciting experimental result was reported at Shelter Island by Isidor I . Rabi . Measurements in his laboratory of the hyperfine structure of hydrogen and deuterium had suggested82that the magnetic moment of the electron is larger than the Dirac value ehl2rrac bya factor of about 1 .0013, and subsequent measurements of the gyromagnetic ratios in sodium and gallium had given a precise value83 2rn[I DO 118 ± 0.00003]. Learning of these results, Gregory Breit suggested$3a that they arose from an order a radiative correction to the electron magnetic moment . At Shel- ter both Breit and Schwinger described their efforts to calculate this correction . Shortly after the conference Schwinger completed a successful calculation of the anomalous magnetic moment of the electron 84 P=:eh n 2m[ 1.001162]~mc[i+__ ] in excellent agreement with observation . This, together with Beth's calculation of the Lamb shift, at last convinced physicists of the reality of radiative corrections . The mathematical methods used in this period presented a bewilder- ing variety of concepts and formalisms . One approach developed by Schwinget'$5was based on operator methods and the action principle, and was presented by him at a conference at Pocono Manor in 1948, the successor to the Shelter Island Conference . Another Lorentz-invariant operator formalism had been developed earlier bySin-Itiro Tomonaga$6 and his co-workers in Japan, but their work was not at first known in the Vest . Tamonaga had grappled with infinities in Yukawa's meson theory in the 1930s . In 1947 he and his group were still out of the loop of scientific communication ; they learned about Lamb's experiment from an article in Newsweek . An apparently quite different approach was invented by Feynman,$7 and described briefly by him at the Pocono Conference . Instead of in- troducing quantum field operators, Feynman represented the S-matrix as a functional integral of exp ( iW),where W is the action integral for a 1.3The Problem of Infinities 3 7 set of Dirac particles interacting with a classical electromagnetic field, integrated over all Dirac particle trajectories satisfying certain initial and final conditions for t --> ±oo . One result of great practical importance that came out of Feynman's work was a set of graphical rules for calculating 5'-matrix elements to any desired order of perturbation theory . Unlike the old perturbation theory of the 1920s and 1930s, these Feynman rules automatically lumped together particle creation and antiparticle annihi- lation processes, and thereby gave results that were Lore ntz-in variant at every stage . We have already seen in Weisskopf's early calculatian63 of the electron self-energy, that it is only in such calculations, including particles and antiparticles on the same footing, that the nature of the infinities becomes transparent . Finally, in a pair of papers in 1949, Freeman I]yson$$ showed that the operator formalisms of Schwinger and Tomonaga would yield the same graphical rules that had been found by Feynman . Dyson also carried out an analysis of the infinities in general Feynman diagrams, and outlined a proof that these are always precisely the sort which could be removed by renormalization . One of the most striking results that could be inferred from Dyson's analysis was a criterion for deciding which quantum field theories are `renormalizable', in the sense that all infinities can be absorbed into a redefinition of a finite number of coupling constants and masses . In particular, an interaction like the Pauli term (1 .1.32), which would have changed the predicted magnetic moment of the electron, would spoil the renormalizability of quantum electrodynamics . With the publication of Dyson's papers, there was at last a general and systematic formalism that physicists could easily learn to use, and that would provide a common language for the subsequent applications of quantum field theory to the problems of physics . I cannot leave the infinities without taking up a puzzling aspect of this story . Oppenheimer6l in 1930 had already noticed that most of the ultraviolet divergence in the self-energy of a bound electron cancels when one takes the difference between the shifts of two atomic energy levels, and Weisskopf 3 in 1934 had found that most of the divergence in the self-energy of a free electron cancels when one includes intermediate states containing positrons . It would have been natural even in 1934 to guess that including positron intermediate states and subtracting the energy shifts of pairs of atomic states would eliminate the ultraviolet divergence in their relative energy shift ." There was even experimental evidence77 fo r In fact, this guess wou ld have been wro ng. As discusse din Section 14 .3, radia tive corrections to the electron mass affect atomic e nergylevels not on ly through a shift i nthe e lectron rest energy, which is the same i nallatomic energy leve ls, but also through a c hange inthe e lectron ki netic energy, that va riesfromone level to another. 38 1 Historical Introductio n a 2s1/2-2pt/2 energy difference of order 1000 MHz . So why didno one before 1947 attempt an numerical estimate of this energy difference ? Strictly speaking, there was one such attempt8gQ in 1939, but it focused on the wrong part of the problem, the charge radius of the proton, which has only a tiny effect on hydrogen energy levels . The calculation gave a result in rough agreement with the early experiments .77 This was a mistake, as shown in 1939 by Lamb ."' A fully relativistic calculation of the Lamb shift including positrons in intermediate states could have been attempted during the 1930s, using the ❑ld non-relativistic perturbation theory . As long as one keeps all terms up to a given order, old-fashioned non-relativistic perturbation theory gives the same results as the manifestly relativistic formalisms of Feynman, Schwiner, and Tamonaga . In fact, after Be#he's work, the first precise calculations8l of the Lamb shift in the USA by French and Weisskopf and Norman Kroll and Lamb were done in just this way, though Tomonoga's group 81 in Japan was already using covariant methods to solve this and other problems . The one missing element was confidence in renormalization as a means of dealing with infinities . As we have seen, renormalization was widely discussed in the late 1930s . But it had become accepted wisdom in the 1930s, and a point of view especially urged by Oppenheimer,"' that quantum electrodynamics could not be taken seriously at energies of more than about 100 Mel, and that the solution to its problems could be found only in really adventurous new ideas . Several things happened at Shelter Island to change this expectation . One was news that the problems concerning cosmic rays discussed in the previous section were beginning to be resolved ; Robert Marshak presented the hypothesis58 that there were two types of "meson' with similar masses ; the muons that had actually been observed, and the pions responsible for nuclear forces . More important was the fact that now there were reliable experimental values for the Lamb shift and the anomalous magnetic moment that forced physicists to think carefully about radiative corrections . Probably equally important was the fact that the conference brought together theorists who had in their own individual ways been thinking about renormalization as a solution to the problem of infinities . When the revolution came in the late 1940s, it was made by physicists who though mostly young were playing a conservative role, turning away from the search by their predecessors for a radical solution . Bibliography 39 Bibliograph y ❑S. Arama .ki, 'Development of the Renormalization Theory in Quantum Electrodynamics,' Historia 5cientiarum36,97 (1989) ;ibid. 37, 91{1989} . [Section 1 .3] D R . T. Beyer, ed .,Foundatians af' Nuclear Physics (Dover Publications, Inc., New York, 1949) . [Section 1 .2.] ❑L. Brown, `Yukawa's Prediction of the Meson,' Centauros 25,71 (1981 ). [Section 1 .2.] ❑L. M. Brown and L . Hoddeson, eds ., the B irthofParticle Physics (Cambridge University Press, Cambridge, 1983) .[Sections 1.1, 1.2, 1.3] ❑T. Y. Cao and S .S.Schweber, `The Conceptual Foundations and th e Philosophical Aspects of Renormalization Theory," Syrathese 97, 33 (1993) .[Section 1 .3.] ❑P. A. Dirac, The Development of Quantum Theory (Gordon and Breach Science Publishers, New York, 1971) .[Section 1 .1.] ❑E. Fermi, `Quantum Theory of Radiation,' Rev. Mod . Whys .4,87 (1932) . [Sections 1 .2 and 1.3.1 Q G . Gamo Thirty Years that Shook Physics (Doubleday and Co ., Garden City, New York, 1 966). [Section 1 .1.] ❑ hammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill Book Co ., New York, 19 65). [Section 1.1.1 D J. Mehra, `The Golden Age of Theoretical Physics : P.A.M. Dirac's Scientific work from 1924 to 1933,' in Aspects of Quantum Theory, ed. by A . Salam and E . P. Wigner, (Cambridge University Press, Cambridge, 1972). [Section 1 .1] d A_ I . Miller, Early Qua ntumElectrodynamics - A So urce Book (Cam- bridge University Press, Cambridge, UK, 1994) . [Sections 1 .1, 1.2, 1.3.] ❑A. Pais, Inward Bound (Clarendon Press, oxford, 1986) .[Sections 1.1, i.2, 1.3.] ❑S. S. Schweber, `Feynman and the Visualization of Space-Time Fro- cesses,' Rev. Mod .Phvs . 58, 449 (1986) .[Section 1 .3.] ❑S. S. 5chweber, `dome Chapters for a History of Quantum Field Theory : 1938-1952,' in Relativity, Groups , and Topology 11,ed. by B .S. DeWitt and R . Stora (North-Holland, Amsterdam, 1984) . [Sections 1 .1, 1.2, 1.3.] 40 1Historical Introductio n ❑S. S.Schweher, `A Short History of Shelter Island I,' in Shelter Island II, ed. by R . 3aclciw, S . Weinberg, and E . Witten (MIT Press, Cambridge, MA, 1985) . Section 1 .3.] ❑S.S. Schweber, QED and the MenTwo Made It : Dyson, Feynman, Schwinger, and Tornonaga (Princeton University Press, Princeton, 1994) . [Section 1.1, 1.2, 1.3] C3J. Schwinger, ed .,Selected Papers in Quantum E lectrodyn amics (Dover Publications Inc ., New York, 1958) . [Sections 1 .2 and 1 .3.] ❑SA.Tomonaga, in The Physicist's Conception of Nature (Reidel, Dor- drecht, 1973) . [Sections 1 .2 and 1 .3] CJ S. Weinberg, The Search for Unity : Notes for a History of Quantum Field Theory,' Daedalus, Fall 1977 .[Sections 1 .1, 1.2, 1.3.] ❑V. F. Weisskopf, `Growing Up with Field Theory : The Development of Quantum Electrodynamics in Half a Century,' 1979 Bernard Gregory Lecture at C ERN, published in L . Brown and L . Hoddeson, op.cit.. [Sections 1 .1, 1.2,1.3.] ❑G. Wentzel, `quantum Theory of Fields (Until 1947)' in Theoretical Physicsin the Twentieth C entury, ed .by M. Fierz and V . F. Weisskopf (Interscience Publishers Inc ., New fork, 1 960). [Sections 1 .2 and 1.3] D E . Whittaker, A History of the Theories of Aether and El ectricity (Hu- manities Press, New York, 1 973). [Section 1 .1.] Refere nces 1. L. de Broglie, Comptes Rendus 177, 507, 545, 630 (1923) ; Nature 112, 540 (1923) ; These de doctorat (Masson et Cie, Paris, 1924) ;Annale s de Physique 3, 22 (1925) [reprinted in English in WaveMecha nics, ed. by G . Ludwig, (Pergamon Pass, New York, 1968)} ;Phil. Mag . 47, 446 (1924) . 2. W.Eisasser, ,Nuturwiss.13, 711{1925 }. 3. C.J.Davison and L . H. te rmer,Fhys.Rev.30, 705(1927) . 4. Heisenberg, A . Phys . 33, 879 (1925) ; Born and P . Jordan , Z.fPhys .34,858(1925) ;P. A. M. Dirac, Proc . Roy . Soc .A109 , 642 (1925) ;M. Born, W . Heisenberg, and P . Jordan, Z.,f: Whys . 35, 557 (1925) ;W. Z . f. Phys . 36, 33 6 (1926) .These papers ar e reprinted in Sources of Quantum Mechanics, ed . by B . L. van de r Waerden (Dover Publications, Inc ., New York, 1968) . References 41 5. E. Schrodinger, Ann . Phys . 79, 3 61, 489 ;80,437; $1, 1 09(1926). These papers are reprinted in English, unfortunately in a somewha t abridged form, in Were Mechanics, Ref .1.Also see Collected Paper s on Wave Mechanics, trans . by J . F. Schearer and W . Dean s (Blackie and Son, London, 1928) . 6.See, e .g., P. A. M. Dirac, The Development of Quantum Theory (Gor- don and Breach, New Fork, 1971) .Also see Dzrac°s obituary of Schrodinger, Nature189, 35 5(I961), and his article in Scientific American 248, 45 (1963) . 7.0. Klein, Z.fPhys . 37, 89 5 (1926) .Also see V Fock, Z.f.Phys .38, 242 (192 6); ibid, 39, 22 6 (1926) . 8.W. Gordon, Z,fWhys . 40, 117 (1926) . 9. For the details of the calculation, see, e .g., L. I. Schiff, Quantum Mecha nics, 3rd edn, (McGraw-Hill, Inc, New York, 1968) :Section 51. 10. F. Paschen, Ann. Ph ys. 50, 90 1 (1916) .These experiments wer e actually carried out using He+ because its fine structure splitting is 16times larger than for hydrogen . The fine structure of spectra l lines was first discovered interferametrically by A . A. Michelson , Phil. Mag .31, 338 (1891);ibid., 34, 280 (1892) , 10a. A.5ommerfeld,MunehnerBerichte1915, pp .425,429;Ann.Pays. 51,1, 125 ( 1916). Also see W. Wilson,Phil.Mag. 29, 795 (1915) . 11. G. E. Uhlenbeck and S . Goudsmit, Nat urwiss . 13, 953 (1925 );Nature 117, 264 ( .1920 . The electron spin had been earlier suggested for other reasons by A . H. Compton, J.Frank .Inst.192, 145(1921) . 12. The general formula for Zeeman splitting in one-electron atoms ha d been discovered empirically by A . Lande, Z.fPhys .5, 231 (1921 ); ibid., 7, 398(1921) ; ibid ., 15, 189(1923);ibid.,19, 112 (1923) . At the time, the extra non-orbital angular momentum appearing in thi s formula was thought to bethe angular momentum of the atomi c `yore ;' A. Sammerfeld, Ann.Phys . 63, 221 (1920);ibid.,70, 32(1923) . It was only later that the extra angular momentum was recognized , as in Ref .11,to be due to the spin of the electron . 13. W. Heisenberg and P . Jordan, Z. f.Phys .37, 263 (1926) ; C. G. Darwin, Proc . Roy . Soc .A116, 227 (1927) . Darwin says that several authors did this work at about the same time, while Dirac quotes only Darwin . 42 IHistorical Introductio n 14. L. H. Thomas, Nature 117, 514 (192 6). Also see S . Weinberg, Gravitation andCosmology, {Wiley, New York, 1972)- Section 5 .1. 15. P. A. M. Dirac, .Proc . Roy . Soc .A117, 610 (1928). Also see Dirac, ibid.,A118, 351 (1928), for the application of this theory to the calculation of the Zeeman and Paschen-Back effects and the relative strengths of lines within fine-structure multiplets . 16. For the probabilistic interpretation of non-relativistic quantum me- chanics, see M . Born Z.fPhys . 37, 853 (1926) ; ibid, 38, 803 (1926) (reprinted in an abridged English version in Wage Mechanics, Ref. 41); G. Wentzel, Z.f Ph ys.40, 590 (1926) ; W.Heisenberg, Z ..f: Phys . 43, 172 (1927) . N. Bohr .Nature 121, 580 (1928) ; Naturwissenchaften 17, 483 (1929) ;Electrons et Photons - Rapports et Disc ussions du V' ConseildePhysique Solvay Gauthier-pillars, Pa ris, 1928) . 17. Conversation between Dirac and J . Mehra, March 28, 1969, quoted by Mehra in Aspects of Quantum Theory, ed.byA. Salem and E . P. signer (Cambridge University Press, Cambridge, 1972) . 18.Cr. Gamow, Thirty Years that Shook Physics, (Doubleday and Co., Garden City, NY, 1966): p. 125 . 19. W.Pauli,Z.fPhys.37, 263 (1 926);43, 601 (1927) . 20. C. G.Darwin,Proc.Roy..Sac.A118 , 654 (1928) ;ibid.,A120, 62 1 (1928) . 21.W.Gordon, z.f Phys.48,11 (1928) . 22.P. A. M. Dirac Proc . Roy . Soc .A126, 360 (1930) ; also see Ref .47. 23. E. C. Stoner, Phil .Mag .48, 719 (1924) . 24. W Pauli, Z . f.Phys .31, 765 (1925) . 25.W.Heisenberg, Z.fPhys . 3$, 411 (1926) ;ibid.,39, 499 (1926) ; P. A. Dirac, Pr oc.Roy.Soc.A112, 661 ( 1926); W Pauli, Z ..f: Phys .41, 81 (1927) ; J. C. Slater, Phys .Rev. 34,1293 (1929). 26.E.Fermi,Z. f.Phys.36,902 (1926) ;Rend .Accad.Lineei 3,145 (1926) . 27.P.A. M.Dirac,Ref. 25. 27a. P. A. M. Dirac, First R . Crane Lecture at the University of Michigan, April 17, 1989, unpublished . References 43 28. H. Weyl, The Tfzeary of Groups and Quantum Mechanics, translated from the second (1931) German edition by H . P. Robertson (Dover Publications, Inc ., New York) : Chapter IV, Section 12 . Also see P. A. M. Dirac, Proc . Roy .Soc. A133,61(193 1). 29. J. R. Oppenheimer, Whys .Rev.35,562 (193 0); I. Tatum, Z.fWhys . 62, 545 (1930) . 29a. P. A. M.Dirac, Proc . Roy . Soc .133,60(1931 ). 30. C. D. Anderson, Science 7 6,238 (1932) ; Phy? s.Rev. 43, 491 (1933) . The latter paper is reprinted in Foundations of Nu clearPhysics, ed. by R. T. Beyer (Dover Publications, Inc ., New York, 1949) . 30a. J. Schwinger, `A Report on Quantum electrodynamics,' in The Physi- cist'sConception of N ature (Reidel, Dordrecht, 1973) :p.415. 31. W. Pauli, Handbuch der P hysik (Julius Springer, Berlin, 1932-1933) ; Rev. Mod .Whys . 13, 203 (1941) . 32. Born, Heisenberg, and Jordan, Ref . 4, Section 3 . 32a. P. Ehrenfest, Phys .Z. 7, 52 8(1906). 33. Born and Jordan, Ref . 4. Unfortunately the relevant parts of this paper are not included in the reprint collection Sources of Quantum Mechanics, cited ire Ref . 4. 34. P. A. M. Dirac, Froc . Roy . inc .Al 12, 661(1926) .-Section 5 . For a more accessible derivation, see L . I. Schiff, Quantum Mechanics, 3rd eds. (McGraw-Hill Book Company, New York, 1968) :Section 44 . 34a. A. Einstein, Whys .Z.18, 121 (1917) ;reprinted in English in van der Waerden, Ref . 4. 35. P. A. Dirac, Proc . Roy .Soc.A114, 243 (1927) ;reprinted in Quantum El ectrody namics, ed . by J . Schwinger (Dover Publications, Inc., New York, 1958). 36. P. A. M. Dirac, Proc . Roy . Soc .A114, 710 (1927) . 36a. V F Weisskopf and E . signer, Z.fPays .63, 54 (1934) 36h. E. Fermi, .Lincei Rend . 9,881 (1929) ;12, 431 (1930) ;Rev.Mod .Phys . 4, 87 (1932). 37,P. Jordan and W . Pauli, Z.fPhys .47,151 (1928) . 44 1Historical Introductio n 38.N. Bohr and L . Rosenfeld, Kon. dansk . vid . Sedsk .,Mat.-Fys . Medd . III, Na .8 (1933) (translation in Selected Papers of Leon Rosenfeld, ed. by R . S. Cohen and J . Stachel (Reidel, Dordrecht, 1979)); Whys . Rev.78, 794 (1950) . 39.P.Jordan,Z. f.Phys.44,473 (1927) .Alsosee P.Jordanand0.Klein , Z. fPhys. 45, 751 (1929) ; P.Jordan,Phys.Zeit.30, 700 (1929) . 40. P. Jordan and E . Wigner, Z .. f. Phys .47, 631 (1928) . This article is reprinted in Q uantum Electrodynamics, Ref. 35. 40a. M. Fierz, Hein Phys .Acta 12, (1939) ; W Pauli, P}zys .Rev.58, 716 (1940) ; VLF Pauli and F . J. Belinfante, Physica 7, 177 (1940) . 41. W. He isenberg andWPauli ,Z.f:Phys.56,1(1929) ;ibid.,59, 168 (1930) . 42. P. A. M. DiracProc . Roy . Soc . A136, 453 (1932) ; P. A. M. Dirac W. A. Fork, and B . Padolsky, Phys .Veit. der Sowjetura[on 2, 468 (1932) ; P A . M. Dirac, Phys .Veit. der Sowjeturzion 3, 64 (1933) . The latter two art icles are reprinted in Quantum Electrodynamics, Ref. 35, pp. 29 and 3 12. Also see L . Rosenfeld, Z.fWhys . 76, 729 (1932) . 42a. P. A. M. Dirac, Prot . Roy . Soc .London A136, 453 (1932) . 43. E. Fermi, Z . }:Phys .88, 1 61 (1934) . Fermi quotes unpublished work of Pauli for the proposition that an unobserved neutral particle is emitted along with the electron in beta decay . This particle was called the neutrino to distinguish it from the recently discovered neutron . 43a. V. Fock, C. R. Leningrad 1933, p . 267. 44. W. H. Furry and J . R. Oppenheimer, Phys .Rev. 45, 245 (1934) . Thi s paper uses a density matrix formalism developed by P . A. M. Dirac , Prot . Carob .Phil.Soc. 30, 150 (1934) . Also see R . E. Peierls, Proc . Roy.Soc.146, 420 (1934) ; W. Heisenberg, Z . J: Pays .90, 209 (1934) ; L. Rosenfeld, Z .fPhys . 76, 729 (1932) . 45. W. Pauli and V . Weisskopf, He Iv. Whys . Acts 7, 709 (1934), reprinted in English translation in A . I. Miller, Early Quantum Electrodynamics (Cambridge University Press, Cambridge, 1994) . Also see W . Pauli, Ann. Inst.Henri Poincare 6,137 (193 6)- 4b.0. Klein and Y . Nishina, Z. ,f: Phys . 52, 853 (1929) ;Y. Nishina, ibid ., $69 (1929) ;also see I . Tamm, Z.fPhys . 62, 545 (030)- 47. P. A. M. Dirac, Prat .Camp .Phil. Soc .26,361(1930) . Reference s 48.C.Miller ,Ann.d.Phys.14,531,568 {1932}.45 49. H. Bethe and W . Heitter, Proc . Roy . Soc .A146, 83 (1934) ; also see G. Racah, Nuovo Cimento 11, Na . 7 (1934) ; ibid .,13, 69 (1936) . 50. H. J. Bhabha, Proc . Roy. Soc . A154, 195 (1936) . 50a. J. F. Carlson and J . R. Oppenheimer, Phys.Rev.51, 220 (1937) . 51.P. Ehrenfest and J . R. Oppenheimer, Phys . Rey .37, 333 (1931} . 52. W. Heitler and G . Herzbe rg,Naturwiss .17, 673 (1929) ; F. Rasetti, Z . f. Phys .61, 598 (1934)- 53. J. Chadwick, Prot . Roy . Soc . A13 6, 692 (1932) . Th is article is reprinted in the Foundations of Nuclear Physics, Ref. 30. 54. W. Heisenberg, Z. f. Whys . 77, 1 (1932) ; also see I . Curie-Joliet and F. Joliot, Campt . Rend .194, 273 (1932) . 54a.Forreferences, see L . M. Brown and H. Rechenberg, Hiss .Stud . in Phys .and Rio . Science, ZS, 1 (1994) . 55. H. Yukawa, Proc. Whys .-Math.Soc. (Japan) (3) 17, 48 (1935), This artic le is reprinted in The Foundations of N uclear Physics, Ref 3U . 56. S. H. Neddermeyer and C . D. Anderson, Phys .Rev.51, 884 (1937) ; J. C. Street and E . C. Stevenson, Ph ys.Rev.52, 1403 (1937) . 56a. L. Nordheirn and N . Webb, Whys .Rev.56, 494 (1939) . 57. M. Conversi, E . Pancini, and 0 . Piccioni, Pays. Rev .?i., 209L (19 47). 58. S. Sakata and T . Inoue, Prog .Them . Phys .1, 143 (1946) ; R. E. Marshak and H . A. Bettie, Fhys .Rev.77, 506 (1947) . 59. C. M. G. Latter, G . P. S. acchialini, and C . F. Powell, Nature 160, 453, 486 (1947) . 60. G. D. Rochester and C . C. Butler, Nature 160, 855 (1947) . 61. J. R. Oppenheimer, Phys . Rev .35, 45 1(1930) . 62. I. Waller, Z. fPhys . 59, 168 (1934) ;ibid.,61, 721, 837(1930) ;ibid., 62, 673 (1930) . 63. F . Weisskopf, Z.fPhys, 89, 27(1934), reprinted in English translation in Early Qua ntumElectrod ynamics, Ref . 45 ;ibid.,90, 817 (1934) . The electromagnetic self energy- is calculated only to lowest order in ct in these references ; the proof that the divergence 46 1 Historical Introdu ction is only logarithmic in all orders of perturbation theory was given by Weisskopf ;Phys . Rev .56,72(1939) . (This last article is reprinted in QuantumElectrodynamics, Ref. 35). 64. P. A. Dirac, XVII Conseil Solvay de Physique, p . 203 (1933) , reprinted in Early Qua ntum Electrodynamics, lief. 45. For subsequen t calculations based on less restrictive assumptions, see Heisenberg , Z.fPays .90, 2 09(1934) ; Sachs .Akad . Wiss. 86, 317 (1934) ; R. Serber, Whys . Rev . 43, 49 (1935) ;E. A. Uehling .Phys . Rev .48, 5 5 (1935) ;W. Pauli and M . Rosy, Phys . Rev . 49, 462 (1936) . Also se e Furry and Oppenheimer, Ref . 44 ; Peierls . Ref . 44 ; Weisskopf, Ref . 63. 65. H. Euler and B . Kockel, Naturw iss.23, 246 (1935) ;W. Heisenberg and H . puler, Z . f: Phys . 98, 714 (1936) . 66. P. A. M. DiracProc . Comb . Phil . Soc . 30, 1 50(1934) . 67. Heisenberg, Z.fPays . 90, 2 09(1934) . 68.N. Kemmer and V . F. Weisskopf ..Nature 137, 659 (1 936). 68ca. F. Bloch and A. Nordsieck, Whys .Rev. 52, 54 (1937) .Also see W. Pauli and M . Fier, Nuovo Cimento 15,167(1938), reprinted in English translation in Early Quantum Electrodynamics, Ref. 45. 69.S. M. Dancoff, Phys .Rey. 55, 959(1939) . 69a. H. Lewis, Phys . Rev .73,173 (1948) ;S.Epstein, Phys . Rev. 73, 177 (1948) . Also see J . 5chwinger, Ref . $4; Z. Koba and S . Tarnonaga, Prog . Theor .Ahys . 3/3, 29 0(1948) , 69b. J. Schwinger, in The Birth oj'Particle Physics, ed.byL. Brown and L . Hoddeson (Cambridge University Press, Cambridge, 1983) :p. 336 . 70. Heisenberg, Ann. d.Pays . 32, 20 (1938), reprinted in Englis h translation in Early Quantum Electrodynamics, Ref. 45. 70a. G. Wentzel, Z .fPhys .86,479, 635(1933) ;Z. f.Phys .87,726 (1034) ; M. Borand L . Infeld, Prac .Roy.Soc.A150. 141 (1935) ; W. Pauli, Ann . inst .Henri Pnincare 6,137(1936). 71. J. A. Wheeler, Phys . Rev .52,1107 (193 7). 72. Heisenberg, Z. f.Phys . 120, 51 3,673(1943) ;Z. Naturforsch . 1, 608 (1946) . Also see C . Miller, Kos .Dansk . Vid . Sels . Ma t.-Fys . Redd .23, No . 1 (1945) ;ibid.23, No . 19, (1946) . } References 47 73.See,e.g.,G. Chew,theS-MatrixTheory of'Strong Interactions (W . A.Benjamin ,inc.New York, 1 961). 74.J. A. Wheeler and R . P. Feynman, Rev. Mod .Phys . 17, 157 (1945), ibid., 21, 425 (1949) . For further references and a discussion of the application of action-at-awdistance theories in cosmology, see S . Weinberg G ravitation and Cosmology, (Wiley, 1972) : Section 16 .3. 75.P. A. M. Dirac, Proc . Roy . inc .A180, 1 (1942) . For a criticism, se e W. Pauli, Rev .Mod .Phys .15, 175 (1943) . For a review of classica l theories of this type, and of yet other attempts to solve the proble m ofinfinities, see R . E. Peierls in Rapports du 8'e Conseil de Physiqu e Solvay 1948 (R. Stoops, Brussels, 1950) : p. 241 . 76. V.F. We issko pf,Kon . Dan.Vid. Seg ., Mat . fys . Medd .XJV, No . 6 (1936), es peciallyp. 34 a nd pp . 5-6 . This a rticleisreprinted inQuantum Electrodynamics, Ref. 35, a ndinEnglish tr anslationin Early QuantumElectrodynamics, Ref.45. Also see W .PauliandM. Fierz, Ref. 68a ;H.A.Kramers,Ref.79a. 77.S. Pasternack, Phys .Rev. 54, 1113 (1938) . This suggestion was base d on experiments of W .V. Hous ton, Pays .Rev.51, 446 (1937) ; R. C. Williams, Phys . Rev .54, 558 (1938) . For a report of contrary data , see J . W.Drinkwate r, 0. Richardson, and W . E. Williams, Proc . Roy . Soc.174, 164 (1940) . 78. E. A. Uehling, ref : 64. 79. W. E. Lamb, Jr and R . C. Retherford, Phys .Rev. 72, 241 (1947) . This article is reprinted in Qua ntumElectrodynamics, Ref. 35. 79a. H. A. Kramers, Nuovo Cimento15,108(1938), reprinted in English translation in Early Q uantumElectrodynamics, Ref. 45 ;Ned.T. Natwink . 11, 134 (1944) ;Rapports du Se Conseil dePhysique Solvay 1945 (R . Stoops, Brussels, 1 950). 80.H. A. Bathe, Phys . Rev .72, 339 (1947) . This article is reprinted in Quantum E lectrodyna mics, Ref . 35. 81. J. B. French and V . F. Weisskopf ;Phys . Rev .75, 1240 (1949) ; N. M. Kroll and W E . Lamb, ibid., 75, 388 (1949) ;J.Schwinger, Phys . Rev .75, 89 8(1949) ; R. P. Feynman, Rev, Alfod . Phys .20, 367 (1948) ; Phys .Rev., 74,939, 1430 (1948) ;76, 749, 769 (1949) ; 80, 440 (1954) , H. Fukuda, Y . Miyarnoto, and S . Tomonaga, Prog.Theor .Phys .Rev. Mod . Fhys . 4, 47, 121 (1948) . The article by Kroll and Lamb i s reprinted in Quantum Electrodyncrrrz ics, Ref . 35. 49 1Histor ical Introductio n 82.J. E.Nafe, E.B.Nelson, and I.I.Rabi,Whys.Rev, 7 1,914 (1947) ; D. E.Nagel,R. S. Julian, a nd J .R. Zacharias, Phys. Rev .72,973 (1947) . 83.P.Kusch and H. M. Foley Phys.Rev.72,1256(1947). 83a. G. Breit, Phys .Rev. 71, 984 (1947) . Schwinger in Ref . 84 includes a corrected version of Breit's results . 84. J. Schwinger, Phys . Rev .73, 416 (1948) . This article is reprinted in Quantum Electrodynamics, Ref.35. 85. J. Schwinger, P hys. Rey ;. 74, 1439 (194$) ; ibid .,75, 651 (1949) ;ibid., 76,790(1949), ibid ., 82, 664, 314 (1951) ; chid ., 91, 7 13(1953);Proc . Nat.Acad . Sci . 37, 452 (1951) .All but the first two of these articles are reprinted in Qua ntum Electrodynamics, Ref, 35 . 86. S. Tomonaga, Prog .Theor . Whys .Rev. Mod . Phys .1, 27 (1946) ; Z- Koba, T . Tet i, and S . Tornonaga, ibid., 2, 101 (1947) ; S. Kanesaw a and S . Tomonaga, ibid ., 3, 1, 141 (1948) ; S. Tomonaga, Phys .Rev. 74, 224 (1945) ; D. Ito, Z . Kona, and S . Tomonaga, Prog .Theor . Phys . 3, 276 {1948} ; Z. Koba and S . Tamonaga, ibid ., 3, 290 (1948) . The firs t and fourth of these articles are reprinted in Quantum Electrodynamics , Ref. 35. 87. R. P. Feynman, Rev. Mod . Phys .20, 367 (1948) ;Phys.Rev.74, 939, 1430 (1948) ;ibid., 76, 749,769 (1949) ; ibid 80, 440 (1950) . All but the second and third of these articles are reprinted in Qua ntum Electrodynamics, Ref. 35. 88. F. J. Dyson, Phys .Rev. 75, 486, 1736 (1949) . These articles are reprinted inQuantum Electrodynamics, Ref. 35. 88a. H. Frohlich, W .Heitler, and B . Kahn, Prot . Roy, Soc .A171, 269 (1939) ; Phys .Rev.56, 96 1 (1933) . 88b. Vi{ E . Lamb, Jr, Phys .Rev.56, 384 (1939) ; Whys .Rev. 57, 458 (1940) . 89. Quoted by R . Serbe r, in the Birth ofParticle Physics, Ref . 69b, p. 270 . Relativistic Quantum Mechanic s The point of view of this book is that quantum field theory is the way it is because (with certain qualifications) this is the only way to reconcile quantum mechanics with special relativity . Therefore our first task is to study how symmetries like Lorentz invariance appear in a quantum setting . 2.1Quantum Mechanic s First, some good news : quantum field theory is based on the same quantum mechanics that was invented b ySchradinger, Heisenberg, Pauli, Born, and others in 1925-26, and has been used ever since in atomic, molecular, nuclear, and condensed matter physics . The reader is assumed to be already familiar with quantum mechanics ; this section provides only the briefest of summaries of quantum mechanics, in the generalized version of Dirac .1 (i)Physical states are represented by rays in Hilbert space . A Hilbert space is a kind of complex vector space ; that is, if c Dand T are vectors in the space (often called `state-vectors') then so is ~ O+j7T, for arbitrary complex numbers ~, q. It has a normw : for any pair of vectors there is a complex number {0,T}, such tha t The norm (T, T) also satisfies a positivity condition : (T, T) ~ 0, and vanishes if and only if 'Y = 0. (There are also certain technical assumptions that allow us to take limits of vectors within Hilbert space .) A ray is a We shall often use the Dirac bra-ket rtotatian : instead of (T 1,T2), we may write (1k2). 49 50 setof normalized the same ray if T' ICI=1-2Relativistic Quan tum M echanics vectors (i .e., (T, T) = 1)with T and T' belonging to = ~T, where ~ is an arbitrary complex number wit h (ii)Observables are represented by Hermitian operators . These are map- pings T -+ATof Hilbert space into itself, linear in the sense that (2.1.4) and satisfying the real ity condit ionAt = A, where for any linear operator Athe adjoint At is defined by (2.1.5) (There are also technical assumptions about the continuity of AT as a function of T .) A state represented by a ray . has a definite value i for the observable represented by an operator A if vectors T belonging to this ray are eigenvectors of A with eigenvalue Y: AT= aT for T in 4. (2.1.) An elementary theorem tells us that for A Hermitian, x is real, and eigenvectors with different as are orthogonal . (iii) If a system is in a state represented by a ray A, and an experiment is done to test whether it is in any one of the different states represented by mutually orthogonal rays 1, 2, ... (for instance, b ymeasuring one or more observables) then the probability of finding it in the state represented by n i s where T and T, are any vectors belonging to rays r and , ,, respectively . (A pair of rays is said to be orthogonal if the state-vectors from the two rays have vanishing scalar products .) Another elementary theorem gives a total probability unity ; if the state-vectors T, form a complete set . 2.2 Symmetries(11.8) A symmetry transformation is a change in our point of view that does not change the results of possible experiments .If an observer 0sees a system in a state represented bya ray ' or : fi r or R2... , then an equivalent observer 0' who looks at the samesystem will observe it in a different 2.2Symmetries 51 state, represented by a ray M' or 1 or ~ ... , respectively, but the two observers must find the same probabilitie s (This is only a necessary condition for a ray transformation to b ea symmetry ; further conditions are discussed in the next chapter .) An important theorem proved by Wigner2 in the early 1930s tells us that for any such transformation --+.1'of rays we may define an operator U on Hilbert space, such that if T is in ray then UT is in the rayfir, with U either unitary and linear {(D, UT) = ((D, T) , (2 .2.2) U~~(D+qT)TUcD +qUP (2.2.3) or else antauraitary and antifineu r ((D,UT) _ (ck,T)w , (2 .2.4) Wigner's proof omits some steps, A more complete proof is given at the end of this chapter in Appendix A . As already mentioned, the adjaint of a linear ❑perator L is defined by This condition cannot be satisfied for an antilinear operator, because in this case the right-hand side of Eq. (2.2.6) would be linear in 0, while the left-hand side is antilinear in c D. Instead, the adjoint of an antilinear operator A is defined b y With this definition, the conditions for unitar ily or an tiuriitarity both take the form Ut-U-1 (2.2.8) There is always a trivial symmetry transformation M ---), R, represente by the identity operator U=1.This operator is, of course, unitary and linear . Continuity then demands that any symmetry (like a rotation or translation or Lorentz transformation) that can be made trivial by a continuous change of some parameters (like angles or distances or velocities) must be represented by a linear unitary operator Urather than one that is antilinear and antiunitary . (Symmetries represented by antiunitary antilinear operators are less prominent in physics ; they all involve a reversal in the direction of time's flow . See Section 2 .6.) In particular, a symmetry transformation that is infinitesimally close to being trivial can be represented by a linear unitary operator that is 52 2RelativisticQuantum Mechanic s infinitesimally c lose to the ident ity: U - 1+tit (2.2.9) with ea real infinitesimal . For this to be unitary and linear, t must be Hermitian and linear, so it is a candidate for an observable . Indeed, most (and perhaps all) of the observables of physics, such as angular momentum or momentum, arise in this way from symmetry transformations . The set of symmetry transformations has certain properties that define it as a gro up.IfT,is a transformation that takes rays n into Win, and T2 is another transformation that takes n into ", then the result of per- forming both transformations is another symmetry transformation, which we write T2Tj, that takes .,, into ;'. Also, a symmetry transformation T which takes rays „into R' 'has an inverse, written T-1, which take s into fin, and there is an identity transformation, T = 1, which leaves rays unchanged . The unitary or antiunitary operators U(T) corresponding to these sym- metry transformations have properties that mirror this group structure, but with a complication due to the fact that, unlike the symmetry trans- formations themselves, the operators U(T) act on vectors in the Hilbert space, rather than on rays . IfT,takes n into ', then acting on a vector T, in the ray A,,U(Tjmust yield a vector U(TI)T,, in the ray 4", and if T2 takes this ray into Win, then acting on U(T1)T, it must yield a vector U(T2) U(TI)T, in the ray R".But U(T2Tl )'Pn is also in this ray, so these vectors can differ only by a phase 0,(T2, T2 ) U(T2)U(T,)`pn=eiO"(Tz°T1)U(T2Ti)`Yn. (2 .2.0) Furthermore, with one significant exception, the linearity (or antilinearity) ofU(T ) tells us that these phases are independent of the state T, Here is the Proof .Consider any two different vectors TA,TB, which are not proportional to each other . Then, applying Eq . (2.2.10) to the state TAB = T,4 + Tg, we have e~OAU(T2TI)TA +e'O$U( T2 TI)T$.(2.2.11) Any unitary or antiunitary operator has an inverse (its adjoint) which is also unitary or antiunitary . Multiplying (2 .2.1 1) on the left with U-~(T2?'i), we have the n the upper and lower signs referring to U(T2T1)unitary or antiunitary, respectively . Since TA and Tu are linearly independent, this is only 2.2Symmetrie s possible if eiOABedOAei0y53 (2.2.13) So as promised, the phase in Eq . (2.2.10) is independent of the state-vector 'Y., and therefore this can be written as an operator relatio n U(T2)U(TI) =e101T2,T,)U(T2T,). (2.2.14) For 0 = 0 , this would say that the U('I')furnish a representation of the group of symmetry transformations . For general phases O(T2, TI), we have what is called a projective representation, or a representation `up to a phase' . The structure of the Lie group cannot by itself tell us whether physical state-vectors furnish an ordinary or a projective representation, but as we shall see, it can tell us whether the group has any intrinsically projective representations at all . The exception to the argument that led to Eq.{2.2.14} is that it may not be possible to prepare the system in a state represented by 'FA+ Ta . For instance, it is widely believed to be impossible to prepare a system in a superposition of two states whose total angular momenta are integers and half-integers, respectively . In such cases, we say that there is a `superseiection rule' between different classes of states,3and the phases O(?'2,T1)ray depend on which of these classes of states the operators U(T'2)[1(7'1) and U{ T2, Ti) act upon . We will have more to say about these phases and projective representations in section 2.7.As we shall see there, any symmetry group with projective representations can always be enlarged (without otherwise changing its physical implications) in such a way that its representations can all be defined as non-projective, wit h 0. Until section 2 .7, we will just assume that this has been done, and take 0= 0 in Eq . (2.2.14). There is a kind of group, known as a connected Lie group, of special importance in physics . These are groups of transformations T(O) that are described by a finite set of real continuous parameters, say 0a, with each element of the group connected to the identity by a path within the group . The group multiplication law then takes the far m T(R) T (0) = T(f (0, 0)) (.2.15) with f°(Q, 0)a function of the 0s and fps . Taking $ 4 =- 0 as the coordinates of the identity, we must have {2.2. t6} As already mentioned, the transformations of such continuous groups must be represented on the physical Hilbert space by unitary (rather than antiunitary) operators U( T(9)). For a Lie group, these operators can be 54 2Relativistic Quantum Mec hanics represented in at least a finite neighborhood of the identity by a power series UT(O)= I + if~~t~,+;(]~'f~`'tb~, +...~ (2,2 .1 7) where ta,tbc=tcb, etc . are Hermitian operators independent of the Os. Suppose that the U(T (B))form an ordinary (i .e., not projective) representation of this group of transformations, i .e., U(T(O)) U(T(O)) = U(T(f( O_,O))) . (2.2.18) Let us see what this condition looks like when expanded in powers of 06and Oa . According to Eq .(2.2.16), the expansion of fO(B,B) to second order must take the for m f u(O,0)=Ou+Oa+fa bcOhOC +...(2.2.19) with real coefficients f"b{,. (The presence of any terms of order 02 or would violate Eq. {2.2.10.} Then Eq. (2.2.18) read s I+ioata + 1 610crbc +X[I + i0ata + loboctbc + 2 2 I+i (Oa +04 + fab'oboc + _ )t' The terms of order 1,0, 0, 02 , and ~'automatically match on both sides ofEq. (2.2.20), but from the 00terms we obtaia non-trivial conditio n This shows that if we are given the structure of the group, i .e., the function f(O, [7}, and hence its quadratic coefficient f 'bt.,we can calculate the second-order terms in U(T (O))from the generators t . appearing in the first-order terms . However, there is a consistency condition : the operator th, must besymmetric in b and c (because it is the second derivative of U(T(0))with respect to 6# and H ')so Eq . (2.2.21) requires tha t [tb,tcl = i C.tu , (2.2.22) where 'ab, are a set o freal constants known a sstructureconstant s Such a set of commutation relations is known as a Lie algebra. In Section 2.7 we will prove in effect that the commutation relation (2 .2.22) is the single condition needed to ensure that this process can becontinued : the complete power series for U(T(B))may be calculated from an infinite sequence of relations like Eq. (2.2.21), provided we know the first-order terms, the generators t, This does not necessarily mean that the operators U{T(O)} are uniquely determined for all 0 "if we know the t ., but it 2.3 Quantum Lorentz Transformations 55 does mean that the U(T(H))are uniquely determined in at least a finite neighborhood of the coordinates 0'-0of the identity, in such a way that Eq. (2.2.15) is satisfied if 0,0, and f {0, B}are in this neighborhood . The extension to all 01'is discussed in Section 2 .7. There is a special case of some importance, that we will encounter again and again . Suppose that the function f (0, 0)(perhaps just for some subset of the coordinates 0a} is simply additiv e This is the case for instance for translations in spacetime, or for rotations about any one fixed axis (though not for both together) . Then the coefficients f a b,in Eq .(2.2. 9) vanish, and so do the structure constants (2.2,23). The generators then all commut e [tb,tt]=0. (2.2.25) Such a group is called Abelian .In this case, it is easy to calculate U(T( B)) for all 0'. From Eqs . (2.2.18) and (2 .2.24), we have for and integer N (T(o)) = U (T 0 U Letting N --* oo, akeeping only the first-order term in (J(T(B fN)),we have the n and henceN U(T(O)) =lim [1+ 001a U(T(O)) =exp(it,,O' ) 2.3 Quantum Lorentz Transformations(2.2.26) Einstein's principle of relativity states the equivalence of certain `inertial' frames of reference . It is distinguished from the Galilean principle of rela- tivity, obeyed by Newtonian mechanics, by the transformation connecting coordinate systems in different inertial frames . If x4 are the coordinates in one inertial frame (with x', x2, x3 Cartesian space coordinates, and x° = t a time coordinate, the speed of light being set equal to unity) then in any other inertial frame, the coordinates xg must satisf y or equivalentlyr~.vdx"'dxf ~ =q,,dx1`dxv cx f~ ~xV qAVcox r~x~(2.3.1) (2,3.2) SG 2 Re lativistic Quantum Mechanic s Here q,,,is the diagonal matr ix,with element s qil = q2 2=q33 = +1, boa = -1 {2.3.3} and the summation convention is in force : we sum over any index like y and v in Eq . (2.3.2), which appears twice in the same term, once upstairs and once downstairs . These transformations have the special property that the speed of light is the same (in our units, equal to unity) in all inertial frames ;* a light wave travelling at unit speed satisfies ~dx/dtl T 1, or in other words 1,,,dxmdx'' = dx2- dt' =0,from which it follows that also r 1,,Vdx'Pdx'1= 0, and hence ldx`/dt'l ^ 1 . Any coordinate transformation xP -f x'Y that satisfies Eq .(2.3.2) is dinear3a x'l` = 11u,,xV + ap(2.3.4) with a ,'arbitrary constants ,and AO, a constant matrix satisfying the conditions q,UrAPp~c--?Ipff ~ (2,3.5) For some purposes, it is useful to write the Lorentz transformation condition in a different way . The matrix q,,,, has an inverse, written qW, which happens to have the same components : it is diagonal, with q00:=--1, n ii =q22= r1 33 = +1.Multiplying Ey, (2.3.5)with ql'W zand inserting parentheses judiciously, we hav e 'r,1Vl l,up(Av 6A1C'C'f[TT)= Ahf] _ qpA1'1CAl1pii Multiplying with the inverse of the matrix q,,k A,', then give s These transformations form a group . If we first perform a Lorentz transformation (2 .3.4), and then a second Lorentz transformation xA X",U,with then the effect is the same as the Lorentz transformation x" -} x`/ y, wit h X lip = ('A'pPAPV) JCv+(APGdp+Ceti). (2.3.7) (Note that if AP,and AP,both satisfy Eq . (2.3.5), then so doesPP11P,,so this is a Lorentz transformation . The bar is used here just to distinguis h There is a larger class of coordinate transformations, known as conformal transformatio ns, for which iu,,dx+,d .x" is proportional though generally not equal to quudxPdxw, and which therefore also leave the speed of light invariant . Conformal invariance in two dimensions has proved enormously important in string theory and statistical mechanics, but the physical relevance of these conformal transformations in four spacetime dimensions is not yet clear . 2.3Quantum L orentzTra.rasfbrmatinras 57 one Lorentz transformation from the other . The transformations T(A, a) induced on physical states therefore satisfy the composition rul e Taking the determinant of Eq. (2.3.5) give s (DetA)2=1 (2.3.9) soM, has an inverse, (1L- 1)'p which we see from Eq . (2.3.5) takes the form The inverse of the transformation T (, a)is seen from Eq.(2.3.8)to be T( --t, - -ta), and, of course, the identity transformation is T(1,0) . In accordance with the discussion in the previous section, the transfor- mations T(A, a) induce a unitary linear transformation on vectors in the physical Hilbert space T --~- U (11, aff . The operators U satisfy a composition rul e U(11, a) U (11,a)=U( A,11 u +a). (2-3.11) (As already mentioned, to avoid the appearance of a phase factor on the right-hand side of Eq.(2.3.11), it is, in general, necessary to enlarge the Lorentz group . The appropriate enlargement is described in Section 2 .7.) The whole group of transformations x'(11, a) is properly known as the anhomogeneous L orentzgroup, or Poirtcrxre group .It has a number of important subgroups . First, those transformations with a A=0❑bviously form a subgroup, with T(A, 0)T(A, 0) = T (A# 0 )> (2 .3.12) known as the ho mogeneousLorentz group. Also, we note from Eq .(2.3.9) that either DetA = + 1or DetA - ----1 ; those transformations with DetA = +1 obviously form a subgroup of either the homogeneous or the inhomogeneous Lorentz groups . Further, from the 00-components of Eqs.(2.3.5)and (2.3.6), we hav e (A°0)2 = I + 1 1'o'a = 1+ 11°jVj . (2 .3.13) with i summed over the values 1, 2, and 3 . We see that either A +1 or A°0 <_ -1. Those transformations with A°o ~!: +1 form a subgroup . Note that if All, and kl`,, are two such As, the n (11A)°()= 11°oA°p+VIA'() + 1102A20 +11°3A'p;. But Eq .(2.3.13) shows that the three-vector ( Alo,A20, ~3())has length (W) -1, and similarly the three-vector ( A°j, °2,O3)has length 58 2 Relativistic Quantum Mechanic s (A°0)2- 1, so the scalar product of these two three-vectors is bounded by Ii1°1 10 + Ao?A20 + 03A3()I ~V(A(10)2-1/(A °o)2-1 , (2.3.14) andso (AA)')o >_ A'OAO() - V(A ~O()- I ~(A(10)2 - I . I The subgroup of Lorentz transformations with Detl1 = +1 and AOO >+1 is known as the proper orthochroraous Lorentz group . Since it is not possible by a continuous change of parameters to jump from I]et 11 = +1 to DetA = -1, or from 1~Oo >+1 to 110 <-1, any Lorentz transformation that can be obtained from the identity by a continuous change of parameters must have Det 11 and A°0of the same sign as for the identity, and hence must belong to the proper orthochronous Lorentz group . Any Lorentz transformation is either proper and orthn chron ous,or may be written as the product of an element of the proper orthochronous Lorentz group with one of the discrete transformations .6311or t3~-or9P,~T , where 41is the space inversion, whose non-zero elements ar e and T is the time-reversal matrix, whose non-zero elements ar e .s7-°0= ---I, tiII=~22=sF33=I. (2.3.16) Thus the study of the whole Lorentz group reduces to the study of its proper orthochronous subgroup, plus space inversion and time-reversal . We will consider space inversion and time-reversal separately in Section 2.6. Until then, we will deal only with the homogeneous or inhomogeneous proper orthochronous Lorentz group . 2.4 The Poincare Algebra As we saw in Section 2 .2, much of the information about any Lie symmetry group is contained in properties of the group elements near the identity . For the inhomogencous Lorentz group, the identity is the transformation Alt, =P,, a}'= D, so we want to study those transformations wit h 11~'ir= r~a'+coyU , a}` = ey, (2 .4.1) both c~ y,and e" being taken infinitesimal . The Lorentz condition ( 2..5) 2.4 The 1'aincare Algebra 5 9 reads here We are here using the convention, to be used throughout this book, that indices may be lowered or raised by contraction with q,,,, or r~4 v t11160) P (0 14 P11PO(1),7p Keeping only the terms of first order in co in the Lorentz condition (2.3.5), we see that this condition now reduces to the antisymmetry of wpv An antisymmetric second-rank tensor in four dimensions has (4 x 3)/ 2 6independent components, so including the four components of el`, an inhomageneous Lorentz transformation is described by 6 + 4 = 1 0 parameters . Since U{1,0} carries any ray into itself, it must be proportional to the unit operator,* and by a choice of phase may be made equal to it . For an infinitesimal Lorentz transformation {2 .4.1}, €I (1 + w, r) must then equal 1 plus terms linear in w,,,and c.. We write this a s X1(1+(t),0= 1+; i cr~"'JP' -~ i~PPP +... (2.4.3) Here Jill and PP are ro- and e-independent operators, and the dots denote terms of higher order in w and/or e . In order for X1(1 +w,c)to be unitary, the operators P" and PPmust be Hermitia n ,1paf =.IPGFPit= Pp, (2.4.4) Since (o,,, is antisymmetric, we can take its coefficient JP' to be antisym- metric als o As we shall gee, P 1,P2 , and P1 are the components of the momentum operators, J23, J31, and J1 2are the components of the angular momentum victor, and P° is the energy operator, or Ha naihon ian.*" ' In the absence of 5upersclcction rules, the possibility that the propnrtionality factor may depend on the stale on which V(1,0) acts can be ruled out b ythe same reasoning that we used in Section 12 to rule out the possibility that the phases in projective representations of symmetry group may depend on the states on which the symmetries act . Where superselection rules apply, it may be necessary to redefine U(1 70)by phase factors that depend on the sector on which it acts . We will see that this identification of the angular-momentum generators is forced on us bythe commutation relations of the Ji "_ On the other hand . the commutation relations do not allow us to distinguish between Per and -P P, so the sign for the epP x' term in {2 .4.3) is a matter of convention . The consistency of the choice in (2 .4.3) with the usual definition of the Hamiltonian P0 is shown in Section 3 .1. 64 2 Relativistic Quantum Mechanic s We now examine the Lorentz transformation properties of JAff and P P We consider the produc t where Am, and a Oarehere theparameters of a ne wtransformation, unrelated to w and P.According to Eq .(2-3,11), the product U(A-', -A-'a) U(A , a)equals U(1, 0),so U( l1-1,-A-' a)is the inverse ofU(A, a). It follows then from (2.3.11)that U(A, a)U(I + cc,-)U-'(A, a)=U (A(I + (u)A-1, A e To first order in co and E, we have thenAa)A-'a) (2 .4.6) 2 2 -(Ae - Aco -la)~PY . (2 .4.7) Equating coefficients of wp, and c.on both sides of this equation (and using (2.3.1)), we find (2.4.8) (2.4.9) For homogeneous Lorentz transformations (with a" = 0 ), these transfor- mation rules simply say that .T" is a tensor and PP is a vector . For pure translations (with Al`, Pv), they tell us that PP is translation-invariant, but J P'is not . In particular, the change of the space-space components ofJP' under a spatial translation is just the usual change of the angular momentum under a change of the origin relative to which the angular momentum is calculated . Next, let's apply rules (2 .4.8), (2 .4.9) to a transformation that is itself infinitesimal, i .e., Au,, = 6P, +caPw and al' = e A, with infinitesimals coPV and e" unrelated to the previous w and c . Using Eq.(2.4.3), and keeping only terms of first order in oil', and -P, Eqs . (2.4.8) and (2 .4.9) now becom e (2.4.11 Equating coefficients of r~~ p,and c,,on both sides of these equations, we find the commutation rue s i[JflwyJPG] = JJVPJU ff _q,UPJu(7y76tijuv + y7awJPPa (2 .4.12} iipu JPCJ = ~~~p.7,tifrpp(2.4.13) a [PP, FPI = 0 . (2.4.14) This is the Lie algebra of the Poincare group . 2.4The Poincure Algebra 61 In quantum mechanics a special role is played by those operators that are conserved, Le.,that commute with the energy operator H = Pa. Inspection of Eqs . (2.4.13) and (2 .4.14) shows that these are the momentum three-vector P=1P1,P2,F31 and the angular-momenturn three-vector(2.4.15) (2.4.16) and, of course, the energy P° itself . The remaining generators form what is called the `vast' three-vecto r K= jj1Q,j2O,j3O1 .(2.4.17) These are not conserved, which is why we do not use the eigenvalues of K to label physical states . In a three-dimensional notation, the commutation relations (2 .4,12), (2 .4.13), (2 .4. 4) may be writte n V,.1j]= i cijkJk [itsKjj ifi}kKk [iaK,jj---iE'ijkJkx [Jr,PjJ=tCijkPk [Kz,Pj~ = i H 5Ij, [i,H] = iPj,(2.4.i8) (2.4.19) (2.4.20) (2.4.2j) (2.4.22) (2.4.23) (x,4.24) where i, j, k, etc . run over the values 1, 2, and 3, and ezjk is the totally antisymmetric quantity with e ,23=+1.The commutation relation (2,4 .18) will be recognized as that of the angular-momentum operator . The pure translations T(1, Q) form a subgroup of the inhomogeneous Lorentz group with a group multiplication rule given by (2.3.7) as (2.4.25) This is additive in the same sense as (2 .2.20, so by using (2 .4.3) and repeating the same arguments that led to (2 .2,26), we find that finite translations are represented on the physical Hilbert space by (2.4.26) In exactly the same way, we can show that a rotation R Obyan angle ~O around the direction of 0is represented on the physical Hilbert space by U(41O) = exp(i J 0). (2 .4.27) Itisinterest ingto compare the Poinca re algeb ra with the Lie alge bra of the symmetry group of Newtonian mechanics, the Galilean grou p. We 62 2Relativistic Qua ntum Mechanic s could derive this algebra by starting with the transformation rules of the Galilean group and then following the same procedure that was used here to derive the Paincare' algebra . However, since we already have Eqs. (2 .4.18)-(2 .4.24), it is easier to obtain the Galilean algebra as a low-velocity limit of the Poincare algebra, by what is known as an Inona- Wigner contraction 4r5 .For a system of particles of typical mass rn and typical velocity z', the momentum and the angular-momentum operators are expected to be of order J- 1, P-mv. On the other hand, the energy operator is H = M + W with a total mass M and non-mass energy W (kinetic plus potential) of order M -m,W - mv-'.Inspection of Eqs . (2.4.1$)-(2 .4.24) shows that these commutation relations have a limit for v << I of the for m Vs,Jil = i E iikA , [Ji, Kj] i Ei ikKk ~ [Kr,KJ = 0, [Ji, W1 Z- [Pi, W1 0, [Ki, W1 = i P i LieM]=[Pi,MI=[KisM] =[ }M] = 0 , with K of order 11v . Note that the product of a translation x --).x + a and a `boost' x --~x + vt should be the transformation x --} x + vt +a, but this is not true for the action of these operators on Hilbert space : exp(A ,v) exp(-iP  a) = exp(iMa  v/2) exp (i(K -v - P -a)). The appearance of the phase factor exp(iMa  v/2} shows that this is a projective representation, with a superselection rule forbidding the super- position of states of different mass . In this respect, the mathematics of the Poincare group is simpler than that of the Galilean group . However, there is nothing to prevent us from formally enlarging the Galilean group, by adding one mare generator to its Lie algebra, which commutes with all the other generators, and whose eigenvalues are the masses of the various states . In this case physical states provide an ordinary rather than a projective representation of the expanded symmetry group . The difference appears to be a mere matter of notation, except that with this reinterpre- tation of the Galilean group there is no need for a mass superselection rule. 2.5One-Particle state s We now consider the classification of one-particle states according to their transformation under the inhomogeneous Lorentz group . The components of the energy-momentum four-vector all commute with each other, so it is natural to express physical state-vectors in terms of 2.5One-Particle S tates 6 3 eigenvectors of the four-momentum . Introducing a label a to denote all other degrees of freedom, we thus consider state-vectors 'Y ,.,with pP'yP,Q=pJUTP..7.(2.5.1 For general states, consisting for instance of several unbound particles, the label uwould have to be allowed to include continuous as well as discrete labels . We take as part of the definition of a one-particle state, that the label cTis purely discrete, and will limit ourselves here to that case . (However, a specific bound state of two or more particles, such as the lowest state of the hydrogen atom, is to be considered as a one- particle state . It is not an e lementary particle, but the distinction between composite and elementary particles is of no relevance here .) Eqs. (2 .5.1)and (2 .4,26) tell us how the states 'F' prd transform under translations : We must now consider how these states transform under homogeneous Lorentz transformations . Using (2,4 .9), we see that the effect of operating on ,,,, with a quantum homogeneous Lorentz transformation U(A, 0)=U(A)is to produce an eigenvector of the four-momentum with eigenvalue 11 p P"U(A)Tp,, = U(A) I U-'(A)P" U(A)] Ipp, = U(A)(A . "~PP)T,,, , Hence U(A)T,,,, must be a linear combination of the state-vectors TAp,d ' rr' In general, it may be possible by using suitable linear combinations of the 'P p,,to choose the alabels in such a way that the matrix ,,,(A, p) is block-diagonal ; in other words, so that the 'Pp,, with awithin any one block bythemselves furnish a representation of the inhomogeneous Lorentz group . It is natural to identify the states of a specific particle type with the components of a representation of the inhomogeneous Lorentz group which is irreducible, in the sense that it cannot be further decomposed in this ay .* Our task now is to work out the structure of th e ' Of course, different particle species may correspond to representations that are komarphic, i .e., that have matrices C',!,(11,p )that are either idenlacal, or identical up to a similarity transforma- tion. In some cases it may he convenient to define particle types as irreducible representations of larger groups that contain the inharnogeneous proper orthochronous Lorentz group as a sub- group ; for instance, as we shall sec, for massless particles whose interactions respect the symmetry of space inversion it is customary to treat all the components of an irreducible representation of the inhomogeneous Lorentz group including space inversion as a single particle type . 64 2 Relativistic Quantum Mechanic s coefficients C,,Q(A, p)in irreducible representations of the inhomogeneous Lorentz group . For this purpose, note that the only functions of p "that are left invariant byall proper arxhochranous Lorentz transformations AP, are the invariant square p2=_ q.wpAp', and for p2:5 0, also the sign of p°, Hence, for each value of p2, and (for p2<_0)each sign of p0, we can choose a `standard' four-momentum, say ky, and express any p~` of this class a s PP = Ley(p)k'a (2.5.4) where L", is some standard Lorentz transformation that depends on p P, and also implicitly on our choice of the standard P. We can then define the states 'Y,,, of momentum p b y T A,7=__N (p ) (t')U(L(p))Tk,, , (L..5.5) where N(p) is a numerical normalization factor, to be chosen later . Up to this point, we have said nothing about how the ff labels are related for different momenta ;Eq.(2.5.5)now fills that gap . Operating on (2.5.5)with an arbitrary homogeneous Lorentz transfor- mation U(11),we now fin d The point of this last step is that the Lorentz transformation L -'(t1p) L{p} takes ktoL(p)k = p, and then to gyp, and then back to k, so it belongs to the subgroup of the homogeneous Lorentz group consisting of Lorentz transformations WP, that leave k}` invariant : WPti,kti'=V. (2.5.7) This subgroup is caI1ed the little gra up.$ For and W satisfying Eq .(2.5.7), we have U'lW lTk, ,rf Dt7Fd1r~Tk,, F (2.5.8) The coefficients D (W) furnish a representation of the little group ; that is, for and elements W, W we hav e Du'a(WW)Tk ,a~=~(WW)T k,, == U(W)U( aril)Tk,d '_ 4 1~YY ~`~6 rr~~~f~ C,~lf= ~ d1f~~~~6 r~rf~YY }T~~ r ff{~ d~Qf ' and so ~„ 2.5One-Particle States 65 In par ticula r, we may apply E q. (2.5.8) to the li ttle-grouptransformat ion W(A,p) =L-'{AP}AI ..(P) (2 .5.10) and then Eq .(2.5.6)takes the for m at or, recalling the definition (2.5.5) U(A)TP,OF = ( ) 1: D,,, (W(A, p)) TAp,' (2.5.11) (p)6r Apart from the question of normalization, the problem of determining the coefficients C,1,,in the transformation rule (2.5.3)has been reduced to the problem ❑f finding the representations of the little group . This approach, of deriving representations of a group like the inhomogeneous Lorentz group from the representations of a little group, is called the method of induced representations .6 Table 2 .1 gives a convenient choice of the standard momentum Vand the corresponding little group for the various classes of four-momenta . Of these six classes of four-momenta, only (a), (c), and (f )have an y known interpretations in terms of physical states . Not much needs to be said here about case (f) g P=0;it describes the vacuum, which is simply left invariant b y U( A) .In what follows we will consider only cases (a) and (c), which cover particles of mass M >0and mass zero, respectively . This is a good place to pause, and say something about the normal- ization of these states . By the usual orthonormalization procedure of quantum mechanics, we may choose the states with standard momentum k"to be orthonormal, in the sense tha t (The de lta function appears here because Tk,# and Tkf,¢f are eigenstates of a Hermitian operator with eigenvalues kand k', respectively .) This has the immediate consequence that the representation of the little group in Eqs. (2.5.8) and (2,5 .11) must be unitary " D'( )= D-'(W)- ( 2..13) Now, what about scalar products for arbitrary momenta? Using the unitarity of the operator U(A) in Eqs .(2.5.5)and (2.5.11), we find for th e "The little groups S O(2, 1)and SO{3, 1)for p2 > 0 and p+i = 0have no non-trivial finite- dimensional unitary representations, so if there were any stales with a given momentum pF' with p2>0 or p 9= 0 that transform non-trivially under the little group, there would have to be an infinite number of them . 66 2 Relativistic QuantumMechanic s Table 2 .1. Standard momenta and the corresponding little group for various classes of Four-rnornenta . Here K is an arbitrary positive energy, say IeV. The little groups are mostly pretty obvious : SO(3) is the ordinary rotation group in three dimensions (excluding space inversions), because rotations are the only proper orthochronous Lorentz transformations that leave at rest a particle with zero momentum, while S O(2,1)and S O(3,1)are the Lorentz groups in (2 + 1)-and (3 + l)-dimensions, respectively . The group IS O(2) is the group of Euclidean geometry, consisting of rotations and translations in two dimensions . Its appearance as the little group for p2 = 0 is explained below . (b) P2 =-M' 0, p~ 0 (C)p2 = 0, PO0 (e)p2 = N2 >0 (f) PP _0 scalar product :tandard V (0, 0, 0, M) (0, 0, 0' -M ) (0, 0, K, K) (0, 0, K3~ _K ) (0,0, Na0)Little Group SO(3) SO(3) ISO(2) ISO(2) S't](2,1 ) SO(3,1) (TPF'C7"TP,1qr) =N(p) ( U-'(L(p))TP',U'~ Tk,cr) =N(p)N*(p')D(W(L-'(p),p'))* 6'(k'-k)UU1 where k' = L-l(p)p'. Since also k = L-1(p)p, the delta function63(k - k' ) is proportional to 63(P - p').For p' = p, the little-group transformation here is trivial, W(L ^t(p), p)= 1, and so the scalar product i s It remain stowork out the proportionality factor relating r53(k- k') a nd c53(p-p'). Note that the Lorentz-invariant integral of an arbitrary scalar function f(p)over four-momenta with - p2 == M1 >_ 0 and p~ 0{i.e., 2.5One-Particle State s cases (a) o r(c))may be writte n jd4p6(P2+ M2)()(pO)f (p ) d3 P,f(P,Pl+ I) 2 ' p +267 {O(pl)is the step function : O(x) = 1 for x >_ 0, O(x) = 0 forx < 0.)We see that when integrating on the `mass shed]' p2+M2 =0,the invariant volume element i s d3PIP2 + M 2 The delta function is defined b y F(p) = F(Fr)b'(P - p' )dip dipF(p) [4~_+_M263(pl _ P) j'2-+ 2 so we see that the invariant delta function i s ~+M261(pf - P)= PW(P 1 - P)(2.5.1) (2-5-16) Since p' and p are related to k' and k respectively by a Lorentz transfor- mation, L(p), we have the n and therefore PO) 63(pl -P) (2.5.17) The normalization factor N(p) issomet imes chosen to be just IV(P) =1, but then we would need to keep track of the pOlk' factor in scalar products .Instead,I will here adopt the more usual con ventionthat N(p) = ~Vlpl l for which(2.5.1S) (2.5.19) We now consider the two cases of physical interest : particles of mass Al > 0, and particles of zero mass . GS 2Relativistic QuantumMechanic s Mass Positive-Definit e The little group here is the three-dimensional rotation group . Its unitary representations can, be broken up into a direct sum of irreducible unitary representations7 D~J~(R)of dimensionality 2j+1, with j;--p, ~, 1, These can be built up from the standard matrices for infinitesimal rotations Pl,k = Pik + O zk} with Pik = -ski infinitesimal : F (!) 23 - 31 1 2 ~) )~+~ ~ ~CJ~1d 714( ~ J(2-5-20 ) (2.5.2!) (2-5.22) witho-running over t he va lues j, j-1,... , -j . For a particle of mass M > 0 and spi nj, Eq . (2.5.11) now become s u(A)T,,,,,= rLA ~ iJ~ rd('`(l x,p))TAp,a' (2.5.23) with the little-group element W(A, p) (the signer rotation') given b yEq. (2.5.10) To calculate this rotation, we need to choose a `standard boast' L(p) which carries the four-momentum from kit -(0.0.0,NI) to p4 .This is conveniently chosen as LVP) = bi k+(7-WNpk Lzo(p) = L" t(p)P V~2iZ} where A=P111, 7_p2+M21M . It is very important that when A"V is an arbitrary three-dimensional rotation , the Wigner rotation W{A, p}is the same as for all p.To see this, note that the boost (2 .5.24) may be expressed a s L(p) = (P)B(~FI)R-i(P) where R(A)is a rotation (to bedefined in astandard way below, in Eq . 2.5One-Particle State s (2.5.47)) that takes the three-axis into the direction of p, and 1 ~ BOPI}01=ao Then for an arbitrary rotation0 0 yr00 ry~ 2 ~ 1 fk69 But the rotation R-1(p)RR (p)takes the three-axis into the direction and then into the direction gy p, and then back to the three-axi s,soit mu st bejust arotation by some angle 0around the three axi s cos 0 R(A) =R(O)sin0sin00 0 cvs0 0 0 0 1 0 D 0 1 Since R($) commutes with B PI), this now give s and hence W( , P) as was to be shown . Thus states of a moving massive particle (and , by extension, multi-particle states) have the same transformation un- der rotations as in non-relativistic quantum mechanics . This is another piece of good news the whole apparatus of spherical harmonics, Clebsch-Gordan coefficients, etc . can be carried over wholesale from non-relativistic to relativistic quantum mechanics . Masssera First, we have to work out the structure of the little group . Consider an arbitrary little-group element YY",, with YVy,ku=V, where Vis the standard four-momentum for this case, kju =(0,0,1,1 ). Acting on a time-like four-vector 0 =(D, 0, 0, 1), such a Lorentz transformation must yield afour-rector Wt whose length and scalar product with Wk = k are the same as those of t: Any four-vector that satisfies the second condition may be written 70 2 RelativisticQuantum Mechanie s and the first condition then yields the relation (2.5.25) It follows that the effect of W", on tV is the same as that of the Lorentz transformation 10 Sy V(a, 9)41 = ~ ~ (X P--~c oc (2.5.2b) This does not mean that W equals S(oc, fl), but it does mean that S-1(a, Vii) is a Lorentz transformation that leaves the time-like four- vector ( 0,0,0,1 )invariant, and is therefore a pure rotation . Also, S "y like }l,, leaves the light-like four-vector (0,0,1,1 )invariant, so S-1 (a,fl) must be a rotation by some angle 0 around the three-axi s where(2.5.27) The most genera lelement of the little group is therefore of the for mCos 0 RPk (O) sin0 4 0yin0 0 0 cos0 0 0 0 1 0 0 01 What group is this? We note that the transformations with 0= 0 or with rx = #=0form subgroups : (.5.29) (2.5.30) These subgroups are Abelian -that is, their elements all commute with each other . Furthermore, the subgroup with 0= 0 is invariant, in the sense that its elements are transformed into other elements of the same subgroup by any member of the grou p R(O)S(ac,P)R-1(0)=5'(ac cos f~+ fl sin0, -.xsin0 + fl cos H) . (1531 ) From Eqs .(2.5.29)-(2 .5.31) we can work out the product of any group elements . The reader will recognize these multiplication rules as those of the group ISM{2}, consisting of translations (by a vector {a, fl})and rotations (by an angle 9 )in two dimensions . Groups that do not have invariant Abelian subgroups have certain simple properties, and for this reason are called semi-si mple. As we have 2.5 One-Particle States 71 seen, the little group ISO(2) like the inhomageneaus Lorentz group i snot semi-simple, and this leads to interesting complications . First, let's take a look at the Lie algebra of I SD (2). For O a,#infinitesimal, the general group element is 1'!`(0, '~E, MI V= 61 1' ~ V D {l '_`oL+x -H0Cl)~i v ~ ~~00 00 From (2 .4.3), we see then that the corresponding Hilbert space operator is where A and B are the Hermitian operator s A = - J 1 3+Jl() = J 2 +K1,(2.5.32) (2.5.3) and, as before, J3 = J12. Either from (2 .4.18)-(2 .4.20), or directly from Eqs.(2.5.29) -(2-5-31), we see that these generators have the commutator s Since A and B are commuting Hermitian operators they (like the momen- tum generators of the inhomogeneous Lorentz group) can be simultane- ously diagonalized by states Tk.,,b The problem is that ifwe find one such set of non-zero eigenvalues of A,B,then we find a whole continuum . From Eq. (2.5.32 , we hav e * [R(H)]A U[R(B)] = A cos 0 -Bsin0 , *[R(O)]B U-1[R(0)] =Asin0 +Bcos B , and so, for arbitrary B , AT'k.,,b=(a cos0--bsinOMk a,b B'I'~ .0=(a sin0 + h cos [1) TOk.a,h, where 72 Massless particles freedom like 0 ; to that physical states a -- b=0:2Relativistic Quantum Mechanic s .re not observed to have any continuous degree of avoid such a continuum of states, we must require (now called Tk,,) are eigenvectors of A and B wit h ATka,, = B Tk,,T=0. {x..5,38} These states are the ndistingu ished bythe eigenva lue of the rema ining generator J3'I'k,a= orTk,a - {2.5.39} Since the momentum k is in the three-direction, a gives the component of angular momentum in the direction of motion, or helicity . We are now in a position to calculate the Lorentz transformation properties of general massless particle states . First note that by use of the general arguments of Section 2 .2>Eq.(2.5.32)generalizes for finite x and Pto and for finite 0to U(R(B)) = expOJ30) .(2.5.40) (2-5-41 ) An arbitrary element W of the little group can be put in the form (2.5.28), so that U(Y~)Tk,d = ex p(taA + i flB)exp(iBJ3)Tk,, = exp(a0(7)k I'k,,, and therefore Eq .(2.5.8) give s where 0is the angle defined by expressing W as in Eq .(2.5.28).The Lorenz transformation rule for a massless particle of arbitrary helicity is now given by Eqs . (2.5.11)and (2 .5.18) a s iJ(A)''p,Q = ~~~~ P with O(A,p) defined byexp(io-O( , p))TAp, (2.5.42) We shall see in Section 5 .9 that electromagnetic gauge invariance arises from the part of the little group parameterized by oc and fl. At this point we have not yet encountered any reason that would forbid the helicity aof a massless particle from being an arbitrary real number . As we shall see in Section 2 .7, there are topological considerations that restrict the allowed values of a to integers and half-integers, just as for massive particles . 2.5 One-Particle States 73 To calculate the little-group element (2 .5.43) for a given Aand p,(and also to enable us to calculate the effect of space or time inversion on these states in the next section) we need to fix a convention for the standard Lorentz transformation that takes us from V=(0,0, rc, rc) to p ". This may conveniently be chosen to have the for m L(p) = R (P)B(~pI1K) where B(u) is a pure boost along the three-direction- '1 0 B(u)01 00 000 0 0 Q (U 2+1)/2u(U2-1)12u(2.5.44) and R( P) is a pure rotation that carries the three-axis into the direction ofthe unit vector p. For instance, suppose we take pto have polar and azimuthal angles 9 and 0 : p= (sin 0 cos 0, si n6 sin0, cos B) (2.5.4G) Then we can take R(p) as a rotation by angle 0around the two-axis, which takes ( 0, 0,1 )into (sin 9, 0, cos 0), followed by a rotation by angle 0around the three-axis : U(R(p))= Cxp(iOJ3) exp(i M), (2.5.47) where 0 :!9 0 ~ ff, 0 21r . (Wegive U(R( P}) rather than R( P), together with a specification of the range of 0 and B, because shifting 0 or 0 by 27r would give the same rotation R( P), but a different sign for U(R( p)) when acting on half-integer spin states .) Since (2 .5.47) is a rotation, and does take the three-axis into the direction (2 .5.46), any other choice of such an R{ P} would differ from this one by at most an initial rotation around the three-axis, corresponding to a mere redefinition of the phase of the one-particle states . Note that the helicity is Lorentz-invariant ; a massless particle of a given helicity a looks the same (aside from its momentum) in all inertial frames . Indeed, we would be justified in thinking of massless particles of each different helicity as different species of particles . However, as we shall see in the next section, particles of opposite helicity are related by the symmetry ❑f space inversion . Thus, because electromagnetic and gravitational forces obey space inversion symmetry, the massless particles of helicity ±1 associated with electromagnetic phenomena are both cal led photons, and the massless particles of helicity ±2 that are believed to be associated with gravitation are both called gravito ns. On the other hand, the supposedly massless particles of helicity ±1/2 that are emitted in nuclear beta decay have no interactions (apart from gravitation) that 74 2Relativistic QuantumMechanic s respect the symmetry of space inversion, so these particles are given different names : neutrinos for helicity +1/2, and antineutrinos for helicity -1/2 . Even though the helicity of a massless particle is Lorentz-invariant, the state itself is not . In particular, because of the helicity-dependent phase factor exp(i aH)in Eq . (2.5.42), a state formed as a linear superposition of one-particle states with opposite helicities will be changed by a Lorentz transformation into a different superposition . For instance, a general one-photon state of four-momenta may be writte n where IX+I2+ IU_11=1. The generic case is one of elliptic polarization, with Ioc+Iboth non-zero and unequal .Circular polarizationis the limiting case where either Y+ or x_ vanishes, and linear polarization is the opposite extreme, with a+ = la-1 . The overall phase of ac+ and a_ has no physical significance, and for linear polarization may be adjusted so that ac_ = a+, but the relative phase is still important . Indeed, for linear polarizations with a_ = a+, the phase ofac+ may be identified as the angle between the plane of polarization and some fixed reference direction perpendicular to p . Eq . (2.5.42) shows that under a Lorentz transformation A,,, this angle rotates byan amount B(11> p ).Plane polarized gravitons can be defined in a similar way, and here Eq . (2.5.42) has the consequence that a Lorentz transformation A rotates the plane of polarization by an angle 20 (A, P) 2.6Space Inversion and Time-Reversa l We saw in Section 2 .3 that any homogeneous Lorentz transformation is either proper and orthochronous (i .e., DetA :;--+1 and AOO >+1) or else equal to a proper orthochronous transformation times either : or 9-orYg-, where 9 and JVare the space inversion and time-reversal transformations 4p-L- 0 00 00 1a a 0-1 0 0 0 11 0 ~~ o 1 ~'V= o 0 U 40 0 00 10 0-1 It used to be thought self-evident that the fundamental multiplication rule of the Poincare group 2.6Space Inversion and Time-Reversal 75 would b evalid even ifAand/or Ainvolved factors of ~P or °lor503T. In particular, it was believed that there are operators corresponding to and ~Fthemselves : such that (2.6.1) {.G.2} for any proper orthochronous Lorentz transformation 11 P,and translation aP. These transformation rules incorporate most of what is meant when we say that P or T are 'conserved' . In1956-57 it became understaod8 that this is true for P only in the approximation in which one ignores the effects of weak interactions, such as those that produce nuclear beta decay . Time-reversal survived for a while, but in 1964 there appeared indirect evidence9 that these properties of T are also only approximately satisfied . (See Section 3.3.)In what follows, we will make believe that operators P and T satisfying Sys .(2-6-1) and (2.6.2) actually exist, but it should be kept in mind that this is only an approximation . Let us apply Eqs .(2.6.1)and ( 2.6.2) in the case of an infinitesimal transformation, i .e., ~ ~ ~ alt=,U withw.,, _ -o),,, and e.both infinitesimal .Using(2.4.3),and equating coefficient sof r~),o¢ and eP i n Eq s.(2.6.1)and (2.6.2), we obtain the P and T transformation properties ofthe Poincare generator s piP{3p- 1= i4 PPY, TiPPT-' = ?' .1JU PPP.(2.6.3) (2.6.4) (2.6.5) (2.6.6) This is much like Eqs . (2.4.8) and (2 .4.9), except that we have not cancelled factors of i on both sides of these equations, because at this point we have not yet decided whether P and T are linear and unitary or antilinear an d antiunitary- The decision is an easy one . Setting p = 0in Eq . (2.6.4)gives Pi HPR t=iH, where H-P° is the energy operator . If P were antiunitary and antilin ear then it would anticommute with i,so PHP -l = -H . But then forand state Tof energy E > 0,there would have t o be another state P-1 T of 76 2 Relativistic Quantum Mechanic s energy - E < 0 . There are no states of negative energy (energy less than that of the vacuum), so we are forced to choose the other alternative- P is linear and unitary, and commutes rather than anticommutes with H . On the other hand, setting p =Din Eq . (2.6.6)yield s T i HT -1_----iH. If we supposed that Tis linear and un itary then wecould simply cancel theis,and find TAT -1 = -H, with the again disastrous conclusion that for a ny state Tof energy E there is another state T-1'I' of energy -E. To avoid this ,we are forced here to conclude that Tisantilinear and antxuraitary. Now that we have decided that P is linear and T is antilinear, we can conveniently rewrite Eqs .(2.6.3)-(2 .6.6)in terms of the generators (2.4.15)-(2.4.17) in a three-dimensional notatio n PIMP-1= -K (2 .6.8) PPP-1 = _P, (2 .6.9) TKTT' _ +K , (2 .6. 11) TPT-1 = -P, (2 .6.12) and, as shown before, PHP -1 = THT -1 = H. (2 .6.13) It is physically sensible that P should preserve the sign of J, because at least the orbital part is a vector product r x p of two vectors, both of which change sign under an inversion of the spatial coordinate system . On the other hand, T reverses J, because after time-reversal an observer will see all bodies spinning in the opposite direction . Note by the way that Eq . (2.6.10) is consistent with the angular-momentum commutation relations J x J = iJ, because T reverses not only J, but also i . The reader can easily check that Eqs .(2.6.7)-(2.6.13) are consistent with all the commutation relations (2 .4.1$)-(2 .4.24). Let us now consider what P and T do to one-particle states : P:M> O The one-particle states 'I'k, ,are defined as eigenvectors of P . H, and J 3 with eigenvalues 0, M, and a, respectively . From Eqs .(2.6.7), (2 .6.9), and (2.6.13), we see that the same must be true of the state P`F'k, ,, and therefore (barring degeneracies) these states can only differ by a phas e with a phase factor (I~ j= 1) that may or may not depend on the spin a . 2.6Space Inversion and Time-Reversal 77 To see that 1,is a-independent, we note from (2.5.8), (2 .5.20), and (2.5.21) that where j is the particle's spin . Operating on both sides with P, we fin d 170- = qa+l and so q,is actually independent of a. We therefore writ e with qa phase, known as the intrinsic parity, that depends only on the species of particie on which P acts . To get to finite momentum states, we must apply the unitary operator U(L(p)) corresponding to the `boost' (2 .5.24): TP,O' = ~Mlpl U (L(p)) Tk, a Wenotethat L{P} -' = L (P) p - (-P,p+ so using Eqs. (2.6.1)and (2.6.15), we hav e PTA, = /M/p0 U( L(F))qTk ,~ or in other word s T: M 0 From Eqs . (2,6.10),(2.6.12), and (2.6.13), we see that the effect of T on the zero-momentum one-particle state Tk,,is to yield a state wit h andso where d is a phase factor . Applying the operator T to (2 .6.14), and recalling that T anticom utes with J and i, we fin d Using Eq, (2 .6.14) again on the left, we see that the square-root factors cancel, and so -~~_ ~a+ t 78 2RelativisticQuantumMechanic s We write the solution as Cd with Canother phase that depends only on the species of particle : However, unlike the `intrinsic parity' q, the time-reversal phase ~ has no physical significance . This is because we can redefine the one-particle states by a change of phas e in such a way that the phase is eliminated from the transformation rul e In what follows we will keep the arbitrary phase Cin Eq .(2.6.17), just to keep open our options in choosing the phase of the one-particle states, but it should be kept in mind that this phase is of no real importance . To deal with states of finite momentum, we again apply the 'boost' (2.5.24). Note that YP P, ~P 2+M2 (That is, changing the sign of each element ofD',with an odd number of time-indices is the same as changing the signs of elements with an odd number of space-indices . Using Eqs . (2,6.2) and (2.5.5),we have the n P;M=0 Acting on a state Yk,,7, that is defined as an eigenvector of Psi with eigenvalue V_(0, 0,:c, K) and an eigenvector of J 3with eigenvalue a, the parity operator P yields a state with four-momentum (0, U, -rC, rc) and J 3equal to a. Thus it takes a state of helicity (the component of spin along the direction of motion) a into one of helicity - u. As mentioned earlier, this shows that the existence of a space-inversion symmetry re- quires that any species of massless particle with non-zero helicity must be accompanied with another of opposite helicity . Because P does not leave the standard momentum invariant, it is convenient to consider instead the operator U(R ; 1)P, where R2 is a rotation that also takes k to :k, conveniently chosen as a rotation by -180° around the two-axi s Since U(R2 ')reverses the sign OfJ3,we have 2.6 'pace Inversion and Time-Reversal 79 with q,a phase factor . Now, Rj-19 commutes with the Lorentz `boost' (2.5.45), and 9 commutes with the rotation R(p) which takes the three- direction into the direction of p, so by operating on (2.5.5)with P, we find for a general four-momentum p P PTk,, U(R(L, .)R2BICIU(R21)PTk}, r11R(-)R2 BP1 Tk,-ff To-q, U ( P ( K Note that R( ~)Ri is a rotation that takes the t hree-axis into the direction of ~p, but U(R( p)R2) is not quite equal to U{R(- P)). According to (2 .5.47), P (0 ± 70h) CXP (47 -O)J2 ) with azimuthal angle chosen as + -gor 7c according to whether 0~0<norit <_ 0 < 27c, so that it remains in the range of 0to 2n . The n U-1(R(-P)) U(R(P)R2) =exp *t- OW2 ) x exp ~ - x( O±71),I3)expO'Oh} exp(iBJ2) exp(-i7rJ2) exp ~ -- [(Ir - Owl) exp(-inJj) exp ~ - i(II - B )J2 But a rotation of ±180 °around the three-axis reverses the sign o fJ2, so U(R(-)R2) = U(R(--)) exp(±V9J3) (2.6.21) Also ,R(-p)B(~p~/k)isjust the standard boost L (Yp)in the direction `P=_ (-p, P°)We have then finally with the phase -no- or+ncr according to whether the two-component of p is posit ive or negative ,respectively.This peculiar change of sign in the operation of parit yfor massl essparticles of half- integer spin is due to the convention adopted in Eq .(2.5.47) for the rotation used to define massless particle states of ar bitrarymomentum .Because t he rotat ion group is not simply connected ,some dis continuity of this sort is unavoidable , T:M=U Acting on the state LI' k,,, which ha svalues k y= {0,0,K, K) and afor P" and .I3, the time-reversal operator T yields a state which has values (gAk)y = (0,10,-rc, rc) and - --aforPYand J3.Thus T does not change the helicity J -k,and by itself has nothing to say about whether massless particles of one helicity crare accompanied with others of hel icity -0. Because T l ike P does not leave the standard four-momentum k invariant , 80 2 Rel ativistic Quantum Mechanic s it is convenient to consider the generator LT(R2 1)T, where R2 is the rotation (2.6,19), which also takes k into 01k. This commutes with J3,so with a an other phase . Since H ~ l_-If commutes with the boost (2.5.45), and l commutes with the rotation R(p), operating with T on the state ( 2.5.5) gives (P)R2BIPA (2 .b.24) TTP,a = FjK~ UR Using Eq. (2.6.21), this yields finall y TTP,O- = , exp(±iza)'Py,,, . (2.6.25) Again, the top or bottom sign app lies accord ing to whether the two- component of pis positive or negative, respective ly. ~** It is interesting that the square T2 of the time-reversal operator has a very simple action on both massive and massless one-particle states . Using Eq. (2.6.18), and recalling that T is antiunitary, we see that for massive one-particle state s T'T or in other word s We get the same result for massless particles . If the two-component of p is positive then the two-component of 9pinegative, and vice-versa, so Eq.(2.6.25) give s T'Tp,,= T', exp(±i-gd)TyP,, exp(}irra)', exp(+bra}`'Fa d = exp (:~2ira )TA, . As long as ais an integer or half integer, this can be writte n T2y Pad=(-)2~~j%V P,#. ( 2.6.27) By the `spin' of a massless particle, we usually mean the absolute value of the helicity, so Eq .(2.6.27) is the same as Eq .(2.6.26). This result has an interesting consequence . When T2 acts on any state T of a system of non-interacting particles, either massive or massless, it yields a factor (-)2j or (---)21'1 for each particle . Hence if the state contains an odd number of particles of half-integer spin or helicity (plus any number of particles of integer spin or helicity), we get an overall change of sign 2.7Projective Representations 81 If we now `turn on' various interactions, this result will bepreserved, provided these interactions respect invariance under time-reversal, even if they do not respect rotational invariance . (For instance, these arguments will apply even if our system is subjected to arbitrary static gravitational and electric fields .) Now, suppose that T is an eigenstate of the Hamil- tonian . Since T commutes with the Hamiltonian, TT will also be an eigenstate of the Hamiltonian . Is it the same state? If so, then TT can differ from T only by a phase TT= CT , but the n in contradiction with Eq . (2 .6.28). We see that any energy eigenstate T satisfying Eq .(2.6.28) must be degenerate with another eigenstate of the same energy . This is known as a `framers degeneracy .'10Of course, this conclusion is trivial if the system is in a rotationally invariant environment, because the total angular-momentum j of any state of this system would have to be a half-integer, and there would therefore be 2j+ 1 = 2,4' ...degenerate states . The surprising result is that at least a two-fold degeneracy persists even if rotational invariance is perturbed byexternal fields, such as electrostatic fields, as long as these fields are invariant under T . In particular, if any particle had an electric or gravitational dipole moment then the degeneracy among its 2 j+ 1 spin states would be entirely removed in a static electric or gravitational field, so such dipole moments are forbidden bytime-reversal invariance . For the sake of completeness, it should be mentioned that P and T can have more complicated effects on multiplets of particles with the same mass . This possibility will be considered in Appendix C of this chapter . No physically relevant examples are known . 2.7 ProjectiveRepresentatiansw We now return to the possibility mentioned in Section 2 .2, that a group of symmetries may be represented projectively on physical states ; that is, the elements T, T, etc . of the symmetry group may be represented on the physical filbert space by unitary operators U(T), U {T}, etc ., whic h This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 8 2 2 Re lativisticQuantum.Mechanic s satisfy the composition rul e U(T)U(T)=exp(io(T, T)) U(T T) (2.7.1) with a real phase . (A baris used here just to d istinguish one symmetry operato rfrom ano ther.) The basic requirement that any phase 0 in Eq. (2.7.1) wou ld have to satisfy is the associativity condi tion which imposes on 0the corresponding conditio n O(x'27T1)+~(T37T2Tl)=~(T3 aT2) + ~(TjT2, T 1) Of course, any phase of the form(2.7.2) (2.7.3) will automatically satisfy Eq .(2.7.2), but a projective representation with such a phase can be replaced with an ordinary representation b yreplacing U(T) with (J(T) =_U(T)exp (ia(T)) for whic h Any set of functions O(T, T) that satisfy Eq .(2.7.2),and that differ only by functions ❑O(T, T) of the form {2 .7.3}, is called a `two-cocycle' . A trivial cocycle is one that contains the function c~ = 0, and hence consists of functions of the form (2 .7.3), which can be eliminated bya redefinition ofU(T) .We are interested here in whether a symmetry group allows any non-trivial two-cocyles ; that is, whether it may have a representation on the physical Hilbert space that is intrinsically projective, in the sense that the phase (T, T) cannot be eliminated in this way . In order to answer this question, it is useful first to consider the effect of a phase 0in E q. (2.7.1) on the commutation relations of the generators of infinitesimal transformations . When either T or Tis the identity, the phase 0must clearly vanis h O(T, 1) = 0 (1J) =0. (2 .7.4) When both T and 1' are near the identity the phase must be small . Using coordinates 0" to parameterize group elements (as in Section 2 .2), with T(O) 1, Eck . {2.7.4} tells us that the expansion of O(7`(O), ~(D)] around 0= 9 = 0 must start with terms of order H H (T(O), T(O)) =.f.bO`~Ob +.. ,(2.7.5) 2.7Projective Representations 83 where fib are real numerical constants . Inserting this expansion in the power series expansion of Eq. (2.7.1),and repeating the steps that led to (2.2.22), we now hav e where Cbtis the antisymmetric coefficient (2.?.7) The appearance of terms on the right-hand side of the commutation relation proportional to the unit element (so-called central charges) is the counterpart for the die algebra of the presence of phases in a projective representation of a group . The constants C b,as well as Cuh ,are subject to an important constraint, which follows from the Jacobi identity . Taking the commutator of (2 .7.6) with td, and adding the same expressions with b, c, d replaced with c,d, b and d, b, c, the sum of the three double-commutators on the left-hand side vanishes identically, and s o C'7hcC8ad+CacdCeah+CdhCoac 0 and also CabcCud + C'cdCub+ CadhCae = 0.(2.7.8) (2.7.9) Eq. (2.7.9)always has one obvious class of non-zero solutions for Ca b Cab=C'abOe (2.7.10) where rye is an arbitrary set of real constants . For these solutions, we can eliminate the central charges from Eq .(2.7.6)bya redefinition of the generators (2.7.1!) The new generators then satisfy the commutation relations without central charge s A given Lie algebra may or may not allow solutions of Ec than Eq .(2.7.10). We can now state the key theorem that governs the intrinsically projective representations . The phase ❑f any U(T ) of a given group can be chosen so that 0 in Eq. conditions are met :(2.7.12) (2.7.9)othe r occurrence of representation (2.7.1), if tw o (a) The generators of the group in this representation can be redefined (as in Eq . (2.7.11)), so as to eliminate all central charges from the Lie algebra . 84 2Relativistic Quantum Mechanic s (b) The group is simply connected, i .e., and two group elements may b e connected by a path lying within the group, and any two such path s may be continuously transformed into one another . (An equivalen t statement is that any loop that starts and ends at the same grou p element may be shrunk continuously to a point .) This theorem is proved in Appendix B of this chapter, which also offers comments about the case of groups that are not simply connected . It shows that there are just two (not exclusive) ways that intrinsically projective representations may arise : either algebraically, because the group is represented projectively even near the identity, or topologically, because the group is not simply connected, and hence a path from Ito T and then from T to T may not be continuously deformable into some other path from 1 to TT . In the latter case, the phase in Eq. (2.7.1) depends on the particular choice of standard paths, leading from the origin to the various group elements, that are used to define the corresponding U-operators . Let's now consider each of these possibilities in turn for the special case of the inhomogeneous Lorentz group . (A) Algebr a With central charges, the commutation relations of the generators of the inhomvgeneous Lorentz group would read (TUjp v q aJ"t 't `! a (2.7.14) (2.7.15) (2.7.1b) in place of Eqs . (2.4.12)-(2 .4.14). We see that the C s also satisfy the antisymmetry conditions CAP = -CAP .(2.7.17) (2.7.18) (2.7.19) We will now show that all these constants have additional algebraic properties that allow them to be eliminated by shifting the definitions of JPand PP by constant terms . (This corresponds to redefining the phase of the operators U( 11, a) .)To derive these properties, we apply the Jacobi 2.7 Projective Representation s identities J L85 (2.7.20) (2.7.21) (2.7.22) (The Jacobi identity involving three P s is automatically satisfied, and hence yields no further information .) Using Eqs .(2.7.13)-(2 .7.16) in Eqs.(2.7.20)--X2.7.22}, we obtain algebraic conditions on the C s = n.t Contracting Eq .(2.7.23) with qw,, give s C14, = 4,(2.7.23) (17.24) (2.7.25) (2.7.26) On the other hand, the constants C}`° "and C P°°f`u are not necessarily zero, but their algebraic structure is simple enough so that they can be eliminated by shifting the definitions of FP and JW, respectively . Contracting Eq . (2.7.24) with r lVF gi ves C~.= ~ ~~~~ C'F.'`~~ Also, contracting (2 ..7.5) withq,,gives 2(2.7.28) (2.7.29) (2.T.3U) (These expressions automatically satisfy Eqs . (2.7.24) and (2.7.25), so there is no further information to begained from the Jacobi identities .) We now see that if the C s are not zero, they can be eliminated by defining new generators P~=PA+Cy, (2.7.3) SG 2 Re lativistic Q uantum Mechanic s and the commutation relations are then what they would be for an ordinary representatio n 0~1,ppl=0.(2.7.33) (2.7.34) The commutation relations will always be taken in the form Eqs . (2.7.33)-- (2.7.35), but with tildas dropped . Incidentally, the fact that there is no central charge in the algebra of the J i"could have been immediately inferred from the fact that this algebra is of the type known as `serrYi-simple' . (Semi-simple Lie algebras are those that have no invariant Abelian' subalgebra, consisting of generators that commute with each other and whose commutators with any other generators also belong to the subalgebra .) There is a general theorem" that and central charges in semi-simple Lie algebras may always be removed by a redefinition of generators, as in Eq .(2.7-32) .On the other hand, the full Poincare algebra spanned byJ11'and PA is not semi-simple (the PY form an invariant Abelian subalgebra), and we needed a special argument to show that its central charges can also be eliminated in this way. Indeed, the non-semi-simple Galilean algebra discussed in Section 2.4 does allow a central charge, the mass M . We see that the inhotnogeneous Lorentz group satisfies the first ❑f the two conditions needed to rule out intrinsically projective representations . How about the second (B)Topolog y To explore the topology of the inhomogeneous Lorentz group, it is very convenient to represent homogeneous Lorentz transformations by 2 x 2 complex matrices . Any real four-vector VY can beused to construct an Hermitian 2 x 2 matrix V° + V , V,R-iv, vO-Y3 ) 7(2.7.3) where a-,are the usual Pauli matrices with cro - 1 . Conversely any 2 x 2 Hermitian matrix can be put in this form, and therefore defines a rea l four-vector V. The property of Hermiticity will be preserved under the transformatio n ~VAI (2.7.37) with ~.an arbitrary com plex 2 x 2 matrix .Furthermore, the covariant 2.7Projec tive Representations 97 square of the four-vector i s V11 VJU-( V1)2+ (V2)2+(VI)2- {V°}' =-Det z ; (2 .7.35) and this determinant is preserved b ythe transformation (2.7.37)provided that JDe#).I= i. (2.7.39) Each complex 2 x 2 matrix ) . satisfying Eq . (2.7.39) thus defines a real linear transformat ian of VA that leaves Eq . (2.7.38) invar iant, i.e., a homogeneous Lorentz transformation A0 .?) Further more, fortwo such matrices Aand ;,we hav e and s o However, two As that differ only byan overall phase have the same effect on:; in Eq .(2.7.37), and so correspond to the same Lorentz transformation . It is therefore convenient to adjust the phase of the As so tha t Det~= 1, (2 .7.42) which is consistent with Eq . (2.7.41). The 2 x 2 complex matrices with unit determinant form a group, known as S' L(2,C). (The SL stands for `special linear', with `special' denoting a unit determinant, while C stands for `complex' .) The group elements depend on 4 - 1 = 3 complex parameters, or 6read parameters, the same number as the Lorentz group . However, L (2, C) is not the same as the Lorentz group ; ifAis a matrix in S L(2,C),then so is - ;., and both ),and - Aproduce the same Lorentz transformation in Eq.(2.7.37). Indeed, it is easy to see that the matri x Iei9'r20 0 e-dO/z produces a Lorentz transformation A(.~(H))which is just a rotation by an angle 0around the three-axis, and hence ~ = -1 produces a rotation by an angle 27z . The Lorentz group is not the same as S L(2,C), but S$ 2Relativistic Q uantum Mechanic s rather" SL(2, C} fZ2, which is the group of complex 2 x 2 matrices with unit determinant, and with ).identified with - ),. Now, what is the topology of the Lorentz group? Bythe polar decom- position theoreml2 , any complex non-singular matrix ~may be written in the form 2=Ueh where u is u nitary and h is Hermitia n ufu=1, ht=h. Since Det u is a phase factor, and Det exp h = exp Tr h is real and positive, the condition (2 .7.42) requires bot h Det u = 1, T"rh=D. (The factor u simply provides the rotation subgroup of the Lorentz group ; ifs is unitary then Tr(uvut) =Try, soVa = 'Tr v is left invariant byA(u) .) Furthermore, this decomposition is unique, so SL(2, C)is topologically just the direct product (i .e., the set of pairs of points) of the space of all us and the space of all hs . Any Hermitian traceless 2 x 2 matrix h can be expressed as c a - ib ~ = a + ib - c with a, b, c real but otherwise unconstrained, so the space of all hs is topologically the same as ordinary three-dimensional flat space, R3 . On the other hand, any unitary 2 x 2 matrix with unit determinant can be expressed as d+ie f + igu =-,f+ig d - ie with d, e, f, g subject to the sing le non-linear constrain t d2+e2+P + g2=1, so the space SU(2) of all us is topologically the same as S 3, the three- dimensional surface of a spherical ball in flat four-dimensional space . Thus S L(2,C)is topologically the same as the direct product R3 X S 3. This is simply cannected : any curve connecting two points of R3 or S 3can be deformed into any other, and the same is true of the direct product . (Al l The group Z2 Cansists of just elements +Z and - I. In general, when we write GIH,with H an invariant subgroup of G, we mean the group G with elements g and gh identified if g -G and h e H . The subgroup Z ,is trivially invariant because its elements commute with all elements of SI.(2.C'). 2.7 Projective Representations 89 spheres S ,except the circle S1 are simply connected .) However, we are interested in SL(2, CVZ2, not SL(2, C) .Identifying Awith -- ).is the same as identifying the unitary factors u and -u (because eh is always positive), so the Lorentz group has the topology of R3 X S3/ Z2, where S ~3fZ2is the three-dimensional spherical surface with opposite points of the sphere identified . This is notsimply connected ; for instance, a path on S 3from u to u' cannot becontinuously deformed into a path on S 3from u to -u', even though these two paths link the same points of S3/Z2 .In fact, S3/Z2 isdoubly cannected ; the paths between any two points fall into two classes, depending on whether or not they involve an inversion u --+-u, and any path in one class can be deformed into another path of that class . An equivalent statement is that a double loop, that goes twice over the same path from any element back to itself, may be continuously contracted to a point . (As discussed in Appendix B, this is summarized mathematically in the statement that the fundamental group, or first homotopy group, of S31,Z2is Z2 .) Similarly, the inhomogeneous Lorentz group has the same topology as R4 x R3 X S3/Z2> and is therefore also doubly connected . Because the Lorentz group (homogeneous or inhomogeneous) is not simply connected, it does have intrinsically projective representations . However, because the double loop that goes twice from 1 to AtoAA and then back to 1 canbe contracted to a point, we must hav e [U (A) U(A) U- 1 (AA)] I=1 and hence the phase eiO4A,A1 is just a sig n U(A)U(A) = ±U(AA) . (2 .7.3) Likewise, for the inhomogeneous Lorentz grou p These `representations up to a sign' are familiar ; they are just the states of integer spin, for which the signs in Eqs . (2.7.43) and (2 .7.44) are always +1, and the states of half integer spin, for which these signs are +1or -1 according to whether the path from 1 to AtoAA and then back to 1 is or is not contractible to a point . This difference arises because a rotation ❑f 2n around the three-axis acting on a state with angular-momentum three-component a produces a phase ezi"", and thus has no effect on a state of integer spin and produces a sign change when acting on a state of half-integer spin . (These two cases correspond to the two irreducible representations of the first hamotoAY group, Z2 .) Thus Eq . (2.7.43) or Eq.(2.7.44) imposes a superselection rule : we must not mix states of integer and half-integer spin . For finite mass, the limitation to integer or half-integer spin was pre- 90 ? Relativistic Quantum Mechanic s viously derived by purely algebraic means from the well-known repre- sentations of the generators of the little group, which here are ,~us~ the angular-momentum matrices J(Pwith jinteger or half-integer . On the other hand, for zero mass the action of the little group on physical one- particle states is just a rotation around the momentum, and here there is no a lgebrai creason for a limitation to integer or half-integer helicity . There is, however, a topological reason : a rotation by an angle 47r around the momentum can be continuously deformed into no rotation at all, so the factor exp{47riu} must be unity, and hence 6 must be an integer or half-integer . Instead of working with projective representations and imposing a su- perselection rule, we can just as well expand the Lorentz group, taking it as SL(2, C) itself, instead of SL (2, C)/ Z2as before . Ordinary rotation invariance forbids transitions between states of integer and half-integer total spin, so the only difference is that now the group is simply-connected, and it therefore has only ordinary representations, not projective represen- tations, so that we cannot infer a superselection rule . This does not mean that we actually can prepare physical systems in linear combinations of states of integer and half-integer spin, but only that the observed Lorentz invariance of nature cannot be used to show that such superpositions are impossible . Similar remarks apply to any symmetry group . If its Lie algebra involves central charges, then we can always expand the algebra to include generators that commute with anything, and whose eigenvalues are the central charges, just as we did when we added a mass operator to the Lie algebra of the Galilean group at the end of Section 2 .4. The expanded Lie algebra is then, of course, free of central charges, so the part of the group near the identity has only ordinary representations, and does not require any superselection rule . Likewise, even though a Lie group G may not be simply connected, it can always be expressed as C IH, where C is a simply connected group known as the `universal covering gTOUP' of G, and H is an invariant subgroupt of C.In general, we may just as well take the symmetry group as C instead of G, because there is no difference in their consequences, except that G implies a superselection rule, while Cdoes not . In short, the issue of superselection rules is a bit of a red herring ; itmay or i tmay not be possible to prepare physic al systems inarbitrary superpositions of ' state,s, but one cannot settle the question by t The first homatapy group of C/t iisH.We have seen that the covering group of the homogeneous Lorentz group is SL(2,C' ), and the covering group of the three-dimensional rotation group is SU(2). This connection with Sf,and SU groups is special to the case of three, four, or six dimensions ; for gcneral dimensions dthe covering group of 50 (d)is given a special name, Appendix A Symmetry Representation Theorem . 91 reference to symmetry principles, because whatever one thinks the symmetry groupof nature may be, there isalways another group whose consequences areidenticalexceptfbr the absence of superselection rules . Appendix A The Symmetry Representation Theore m This appendix presents the proof of the fundamental theo rem of Wigner2 that any sy mmetry transformation can be represented on the Hilbert space of physical states by an operator that is either linear and unitary orantilin ear and antiunitary . For our present purposes ,the property of symmetry transformations on which w echiefly rely is that they are ray transformations T that preserve transition probabilit ies, in the sense that if 'I' ,and 'i' 2are state-vectors belong ing to rays iand z then any state-vectors T' and T' belonging to the transformed rays 'T' 1 and ~ ' 2 satisf y We also require that a symmetry transformation should have an inverse that preserves transition probabilities in the same sense . To start, consider some complete orthonormal set of state-vectors Tk belonging to rays A, wit h and let T~ b esome arbitrary choice of state-vectors belonging to the transformed rays TX .From Eq .(2.A.1),we hav e But (T~, '~) is automatically real and positive, so this requires that it should have the value unity, and therefor e it is easy to see that these transformed states T~ also form a complete set, for if there were and non-zero state-vector T' that was orthogonal to all of the T~, then the inverse transform of the ray to which 'V `' belongs would consist of non-zero state-vectors V for which, for all k: which is impossible since the Tkwere assumed to form a complete set . We must now establish a phase convention for the states V k. For thi s purpose, we single out one of the Tk, say 'Pt . and consider the state-vector s -,,F2 92 2Relativistic Quantum Mechan ics belonging to some ray Yk, with k =~1. Any state-vector Y k' belonging to the #ransformed ray T9k may beexpanded in the state-vectors 'Y{ , Ti= CWTi From Eq . (2.A.1) we have ICkk1= CkI1= and for I:~k and 14 1 : ckl= 0. For any given k, b yan appropriate choice of phase of the two state- vectors Y'~ and T~ we can clearly adjust the phases of the two non-zero coefficients c kkand ckl so that both coefficients are just 1 1 .From now on, the state-vectors Ti and 'Yk chosen in this way will be denotedUY'k and UTk . As we have seen, 72 However, it still remains to define UT for general state-vectors T . Now consider an arbitrary state-vector T belonging to an arbitrary ray and expand it in the Tk; k(2.A.6) Any state T' that belongs to the transformed ray T'1 may similarly be expanded in the complete orthonormal set U'I' k T'=1: CkUTk k(2.A.7) The equa lity of I (Tk,T) 12 and I (UTk,T')~2 tells us that fo rall 1c (including k = 1) : while the equalit yofI(Y'k, Y)I1 and J{UY`k,'I'`y2tellsusthat for all k :~1: ~C'k +C112=IG'k,+C,C~. TheratioofEqs. (2.A.9) and (2 .A.8) yields theformul a which with Eq .(2.A.8) also requires(2.A.9) P..1o) (2.A.i 1) Appendix A Symmetry Representation Theore m and the refore either CkICI =CLIC1, or e1 se93 (2.A.12) (2.A.13) Furthermore, we can show that the same choice must be made for each k. (This step in the proof was omitted by Wigner .) To see this, suppose that for some k, we have C k/C1 =Ck'/Ci, while for some I k, we have instead C' IfC1 = (C' IC')* .Suppose also that both ratios are complex, so that these are really different cases . (This incidentally requires that k:~1 and 101, as well as k *1.)We will show that this is impossible . Define a state-vector (D- ~ [fir +~'k+ TI] . Since all the ratios of the coefficients in this state-vector are real, we must get the same ratios in any state-vector Vbelonging to the transformed ray : where oc is a phase factor with jal = 1 . But then the equality of the transition probabilities 1{0, T}Iand l( V, T') I requires tha t rr + ~~ + ii and hence ~ +k+C C* 1 1C1 C1 2 ~ =Ck +~1 i C1 This is only possible i f or, in other wards, ifReCk Ci=ReCk(:) 1 Im ~` IM " = Q . i 1 Hence either C k/fir or C I/CI must be real for any pair k, 1, in contra- diction with our assumptions . We see then that for a given symmetry transformation T applied to a given state-vector ~~ ~ kTk, we must have either Eq.(2.A.12) for all k, or else Eq . (2.A.13)for all k . signer ruled out the second possibility, Eq.(2.A.13), because as he showed any symmetry transformation for which this possibility is realized would have to involve a reversal in the time coordinate, and in the proof he presented he was considering only symmetries like rotations that do 94 2 Relativistic Quantum Mechanic s not affect the direction of time . Here we are treating symmetries involving time-reversal on the same basis as all other symmetries, so we will have to consider that, for each symmetry T and state-vector k C'kTk, either Eq.(2.A.12) or Eq . (2.A.13) may apply . Depending on which of these alternatives is realized, we will now define UT to be the particular one ❑f the state-vectors T' belonging to the ray T with phase chosen so that either C 1= C 'orC1=C11', respectively . Then eithe r ~(~~kTk) = ck UTk k k or else k k(2.A.15) It remains to be proved that for a given symmetry transformation, we must make the same choice between Eqs . (2.A.14) and (2.A.15) for arbitrary values of the coefficients C k. Suppose that Eq . (2A14) applies for a state-vector ~ kAkT k while Eq . (2.A.15)applies for a state-vecto r BkTk. Then the invariance of transition probabilities requires tha t k k or equivalently Im(4A1) I m (B~B~ ) = 0 k1 We cannot rule out the possibility that E q.(2.A.16) may be satisfied for a pair of state-vectors k AkTkandEkBk'I'kbelonging to different rays . However, for any pair of such state-vectors, with neither Ak nor Bkallof the same phase (so that Eqs . (2.A.14) and (2 .A.15)are not the same), we can always find a third state-vector k C'kTk for which ` lM<C i)Im(Ak~~) z7L 0 k1 and also Im(C~ C j)Irr►(BZBt) *0. k1(2.A. 17) (2.A.18) if for some pair k,d both AkAf and BkDIarc complex, then choose all C's to vanish except for Ck and C+, and choose these two coefficients to have different phases .IfAkAp is complex but B' 8j is real for some pair k,l, then there must be some other pair m,n (possibly with either m or n but not both equal to k or 1) for which BmBR is complex . If also A;A,is complex, then choose all Cs to vanish except for C .',,, and Cn, and choose these two coefficients to have different phase . IfA.Artis real, then choose all Cs to vanish except for Ck, C'j,C,,, and C',,, and choose these four coefficients all to have different phases_ The case where Bk Bf is complex but AkA+ is real is handled in just the same way . Appendix A Symmetry Representation Theorem 95 As we have seen, it follows from Eq. (2.A.17)that the same choice between Eqs. (2.A.14) and (2 .A.15)must be made for ~:kAkTk and Ek C kTk, and it follows from Eq . (2.A.18)that the same choice between Eqs . (2.A.14) and ( 2.A,15) muss be made far Ek BkTk and Ek CkPk, so the same choice between Eqs . (2.A.14) and (2.A.15) must also be made for the two state- vectors EkAkPkand kBkTk with which we started . We have thus shown that for a given symmetry transformation T either all state-vectors satisfy Eq.(2.A.14) or else they all satisfy Eq.(2.A.15). It is now easy to show that as we have defined it, the quantum mechan- ical operator U is either linear and unitary or antifinear and antiunitary . First, suppose that Eq . (2.A.14) is satisfied for all state-vectors Ek CkTk Any two state-vectors 'Y and (Dmay be expanded a s T=A kTk} k(D 1 : BkT k and so, using Eq . (2.A.14), k k _a AkUtiYk + fl BkUTk . k k Using Eq . (2.A.14)again ,thisgives (2.A. 1) soUislinear. Also, wing Eqs . (2.A.2)and (2.A.3), the scalar product of the transformed states is k k, k1 k and hence (2.A.20) soUisunitary . The case of a symmetry that satisfies Eq . (2.A.15) for all state-vectors may be dealt with in much the same way . The reader can probably supply the arguments without help, but since antilinear operators may be unfamiliar, we shall give the details here anyway . Suppose that Eq. (2.A,15) is satisfied for all state-vectorsEk Ck Tk. Any two state-vectors T and c D may be expanded as before, and so : k k k k 96 2 Relativistic Quantum M echanics Using Eq .(2.A.15)again, this gives (2.A.2 1) so U is antidinear .Also, using Eqs . (2.A.2) and (2 .A.3), the sca larproduc t of the transfo rmed states is ki k and henc e soUis antiuraitary .(2.A.22) AppendixBGroupOperators a nd H omotopy Classe s In this appendix we shall prove the theorem stated in section 2 .7, that the phases of the operators U(T)for finite symmetry transformations T may be chosen so that these operators form a representation of the symmetry group, rather than a projective representation, provided (a) the generators of the group can be defined so that there are no central charges in the Lie algebra, and (b)the group is simply connected . We shall also comment on the projective representations encountered for groups that are not simply connected, and their relation to the homotopy classes of the group . To prove this theorem, let us recall the method by which we construct the operators corresponding to symmetry trans#'vrmations . As described in Section 2 .2, we introduce a set of real variables 0'to parameterize these transformations, in such a way that the transformations satisfy the composition rule {2 .2.15}; T(0)T(0)==T(f (0, 0)). We want to construct operators U(T(O));U[0] that satisfy the corre- sponding condition' U[9]U[H]--U[f(,0)]. (2.g.i) To do this, we lay down arbitrary `standard' paths 0(s) in group pa-0 rarneter space, running from the origin to each point 0, with 0$(0)=0 and0~(1) = Ba, and define ~Tr~(s) along each such path bythe differentia l Square brackets are used here to distinguish U operators constructed as functions of the group parameters from those expressed as functions of the group transformations themselves . Appendix B Group O perato rs and Hnmotvpy Classe s equation N5 s) ~U6 (5)`it,,~Io(s)h ~ (Oe(s))d0 ds with the initial condition uO(o) = 17 where 00, 6=o97 (2.B.2) {2.5.3} (2,B.4) We are eventually going to identify the operators U[0] with U ,9(1), but first we must establish some of the properties of Uo(s) . In order to check the composition rule, consider two points 01 and 02, and define a path that runs from 0 to 01 and thence to f (02, Oi ): Da(2s) 0Cs ~ 1pa(s)_ 03 2 (2 .8.5) At the end of the first segment, we are at Uy(~) = ~1g~{1}.To evaluate U,;o(s)along the second segment, we need the derivative of f '(09,(2s - 1),01). For this purpose, we use the fundamental associativity condition : .fa(.f(03,e2),0 1 ) = P(03,f(e2,01)) - (2.B.6) Matching the coefficients o f0~ in the lim it03 ~ 0 yie lds theresult : Do,(2.B.7) Along the second segment the differential equation (2 .B.2) for U, 9(s) is thus the same as the differential equation for Utz (2s -1).They satisfy different initial conditions, but Ug(s)Uil (1)also satisfies the same differential equation as U02(2s - 1),and in addition the same initial condition- at s = ~, both are unity . We therefore conclude that for Z - 1 , 91 and in particular URM -z-U020) U91(1)-(2.B.8) However, this does not say that Uo(1)satisfies the desired composition rule (2.8.1), because although the path Oy(s) runs f rom 0 a=0to0' = ,fa($2,01),ingeneralitwill not be the same a swhatever 'standard' path Df(o,,01)we have chosen t orun directly from 0' _ 0 to0'= fa(O?,01).We need to show that U0(l) isindependent of the path from 0to0in order to be abletoidentify U[9] as U,9( 1). 98 2 Relativistic Quantum M echanics For this purpose, consider the variation bU of UO(s)produced by a variation 60{s} in the path from 0 to 0 . Taking the variation of Eq . (2.B,2) gives the differential equatio n ds6U = tt abU h ~,( O)dOl +its Uh~,e(O)~Ot~ d~'+ itaUh ~~~r~D~'ds ds ds where h' ,~ = Oh'/00' .Using the Lie commutation relations (2 .2.22) (without central charges) and rearranging a bit, this give s ~s(u'u) ~(iU-'t~Uh%JO')ds ds("'b b C) .(2. B.9) However, by taking the limit 0 3,02-r 0 in the associativity condition {2.B.6}, we find for all 0: where f ~ eis the coefficient defined by (2.2.19).Antisymmetrizin~, in b and c shows that the last term in Eq . (2.B,9) vanishe s Eq. (2.B.9) thus tells us that the quantit y is constant along the path 9(,5) . It follows that Uq{1} is stationary under any infinitesimal variation of the path that leaves the endpoints Q( 0)=0 and 0(1) = 0(and UO{O} =1)fixed . But assumption (b) tells us that any path from Vi(a) = 0to0(1) = 0can be continuously deformed into any other, so we may now regard U O(1) as apath - independent function of 0 aline : In particular, since the path leads from 0to f (02, 01), we hav e UYG) = UIf(02, OA (2.R, 13) so that Eq . (2.B.8) shows that U[ B]satisfies the group multiplication law (2.B.1), as was to be proved . Now that we have constructed a non-projective representation Uff)], it remains to prove that any projective representation fJ[0] of the same group with the same representation generators t acan only differ from £IJ9]by a phase : Appen dixBGroupOperatorsand Homotopy Classes 99 so that the phase (Pin the multiplication law for 0[01 : can be removed by a simple change of phase of U[0]. To see this, consider the operato r Because U[ O] and U[ O] have the same generators, the derivative of the left-hand side with respect to 0"vanishes at 0'= 0, and s o 0= ' JU[0I_IU[01J+i0b(0)UI0I J 101 , wher e Differentiating this result with respect to 0':and antisymmetrizing in b and c gives immediately 0 -NO) 00C(0) OOC 60b A familiar theorem13tells us that in a simply connected space, this requires that Obis just a gradient of some function J~ afl(o) 040 )=aBb Thus the quantity U[#]] -ICJ[0] e00) is actually constant in 0. Setting it equal to its value at 0= 4, we see that &is just proportional to U: as cla imed above . The above analysis provides some information about the nature of the phase factors that can appear in the group multiplication law when the Lie algebra is free of central charges but the group is not simply connected . Suppose that the path from zero to 0tof (B, 0) cannot be deformed into the standard path we have chosen to go from zero to f {$, 0),or in other words, that the loop from zero to 0tof {0,0)and then back to zero is not continuously deformable to a point . Then U-'(f (0-,, 60)U((~~)U(0 1) can be a phase factor exp(1'0(02,01)) *1, but 0will be the same for all other loops into which this can be continuously deformed . The set consisting of all loops that start and end at the origin and that can be continuously deformed into a given loop is known as the hamotopy class14 of that loop ; we have thus seen that 0(02, fir) depends only on the homotopy class of 100 2 Relativistic Quantum Mechanic s the loop from zero to 0to f (#], 0) and then back to zero . The set of homotopy classes forms a group ; the `product' of the homotopy class for loops Y1 and 2' 2is the homotopy class of the loop formed by going around Y1 and then Y2 ; the `inverse' of the homo#opy class of the loo p is the homotopy class of the loop obtained by going arount! in the opposite direction ; and the `identity' is the homotopy class of loops that can be deformed into a point at the origin . This group is known as the first homotopy group or fundame ntal group o f the space in question . It is easy to see that the phase factors form a representation of this group : if going around loop Ygives a phase factor e0, and going around loo p gives a phase factor e'O, then going around both loops gives a phase factor e~Od .Hence we can catalog all the possible types of projective representations of a given group 9(with no central charges) if we know the one-dimensional representations of the first homotopy group of the parameter space of 19. groups will be discussed in greater detail in Volume II . Appendix C Inversions an d Degenerate Multiplet s It is usually assumed that the inversions T and P take one-particle states into other one-particle states of the same species, perhaps with phase factors that depend on the particle species . In Section 2.6 we noted in passing that inversions might act in a more complicated way than this on degenerate multiplets of one-particle states, a possibility that seems to have been first suggested by VVigner15 in 1964 . This appendix will explore generalized versions of the inversion operators, in which finite matrices appear in place of the inversion phases, but without making some of VVigrter's limiting assumptions . Let us start with time-reversal . Wigner limited the possible action of the inversion operators by assuming that their squares are proportional to the unit operator . Because T is antiunitary, it is easy to see that the corresponding proportionality factor for T2 can only be ±1, perhaps with different signs for subspaces separated bysuperselection rules . When the sign for T2 on the space of states with even or odd values of 2jis opposite to the sign (--1)2j found in Section 2 .6, the physical states involved must furnish representations of the operator T that are more complicated than that assumed so far . But if we are willing to admit this possibility, there does not seem to be any good reason to impose Wigner's condition that T2 is proportional to unity . It is not convincing to appeal to the structure of the extended Poincare group ; the only useful definition of any of the inversion operators is one that makes the operator exactly or Appendix C I nversions and Degenerate Multiplets 101 approximately conserved, and this may not be the definition that makes T2 proportional to the unit operator . To explore more general possibilities for time-reversal, let us assume that on a massive one-particle state it has the actio n 1)j-"1 9mnT-p,-C,M} (2 .C .1) where p,j,and ff are the particle 's momentum ,spin,and sp inz-component, and n, m are indices labell ingmembers of a degenerate multiplet of particle species . (The appearance of the factor I )J-1and the reversal of p and crare deduced in the same way as in section 2 .6.) The matrix is unknown, except that be cause T is antiunitary, 9-must beunitary . Now letus see how we can simplify this transformat ion by an ap- propriate choice of bas is for the one-particle states .Defining new states by the un itary transformation T~, ~, wefind the same transformation (2 .C.1),with the matrix l ,,,changed t o We cannot in general make 9'diagonal by such a choice of basis of the one-particle states, as we could if T were unitary . But we can instead make it block-diagonal, with the blocks either 1 x 1 phases, or 2 x 2 matrices of the form 10 eiOl2 e-°0/2 0 ' where the 0are various real phases . {Here is the prof . First, note that Eq. (2.C.2)gives(2-C-3) This is a unitary transformation, so it can be chosen to diagonalize the unitary matrix tl T*, Assuming this to have been done, and dropping primes, we have 9-=Dg-T (2.C.4) where D is a unitary diagon al matrix ,say with phases e *Ralong themain diagonal. On eimmediate consequence i sthatthe diagonal componen t ,n vanishes unless e *,=1.Furthermore ,if e`en = 1but e~~- =~ 1,then Eq.(2.C.4)tells us that 5 = =0. By listing first all rows and columns for which e*, = 1,the matrix .l is put in the for m 0 '(2.C.5) .~ _0 -4 where 4 is symmetric as well as unitary, and the diagonal elements of all vanish . Because is symmetric, it can be expressed as the exponential 102 2 Relativistic Quantum Mechanic s of a symmetric anti-Hermitian matrix, so it can be diagonalized bya transformation (2.C.2) acting only on W, with the corresponding submatrix of°llreal and hence orthogonal . It is therefore only necessary to consider the subrhatrix that connects the rows and columns for which e~O, D . For n :~in,Eq. (2.C.4)gives J'_nm and so °fnm = and also J~ = Hence =J-,,, = 0 unless ~~ O,,e'O-= 1. If we list first all rows and columns of ; with a given phase e iOE:~1, and then all rows and columns with the opposite phase, and then all rows and columns with some other phase eat' *1 not equal toe±*, and then all rows and columns with opposite phase, and so on, the matrix ' becomes of block diagonal for m where A_ar0... 0 ~22... 0 ~i 0(2..C.6) (2.C.7) Furthermore, the unitarity of ~Fand hence of requires that ~~j w'T = W~ ;=1, and hence 1 is square and unitary .Byapplying a transforma- tion {2 .C.2} with Vblock-diagonal in the same sense as and with the matrix in the ith block of form vi 0 1a with Vi and Wj unitary, the submatrices, , are subjected to the transfor- mations W ;---~V,-iWiW, *, so we can clearly choose this transformation to make Wz = 1 . This establishes a correspondence between pairs of individ- ual rows and columns within each block with phases e'0i and e-'6 .To put the matrix into block-diagonal form with 2 x 2 blocks of form (2.C-3), it is now only necessary to rearrange the rows and columns so that within the nth block we list rows and columns with phase e* alternating with the corresponding rows and columns with phase e -'Pi.) It is important to note that where eat=~1, it is not possible to choose states to diagonalize the time-reversal transformation . If we have a pair of states `Yp, ,,t on which T acts with a matrix (2.C.3),then (2.C.$) Then on an arbitrary linear combination of these states, time-reversal gives Appendix C Inversions and Degenerate Multiplers 103 For c+TpcF,+ + c_T Q,,,,_ to be transformed under T bya phase A, it is necessary tha t But combining these equations gives e±`ol2c+=1~1?c+e+T0l?rwhich is impossible unless either c+T c_ =0or eio is unity . Thus for e", i, time-reversal invariance imposes a two-fold degeneracy on these states, beyond that associated with their spin . Of course, if there is an additional `internal' symmetry operator S which subjects these states to the transformatio n +i0/2 then we can redefine the time-reversal operator as T' - S-1T,and this operator would not mix the states Tp, 17r+ with one another . It is only in the case where no such internal symmetry exists that we can attribute the doubling of particle states to time-reversal itself . Let's come back now to the question of the square of T . Repeating the transformation ( 2.C.)gives 2 ~i +1 o If we were toassume withWignerthat T2isproportional to the unit operator, then we would have to have eio= e-fo, and si ncethephaseis then real it would have to be + 1 or -1.The choice e °O_ - 1 would still require a two-fold degeneracy of one-particle states be yond that associated with their spin,and under signer's assumptions all particles would show this doubling . But there is n oreason not to take a general phase 0in Eq.(2.C.8),one that may vanish for some particles and not for others . Thus the fact that observed particles do not show the extra two-fold degeneracy does not rule out the possibility that others m ight. We may also consider the possibility of more complicated representa- tions of the parity operator P ,with with a unitary but otherwise unconstrained matrix Y. Unlike the case of time-reversal, here we may always diagonalize this matrix by a choice of basis for the states . But this choice of basis may not be the one in which time-reversal acts simply, so, in principle, P and T together can impose additional degeneracies that would not be required by P or T alone . As discussed in chapter 5, any quantum field theory is expected to respect a symmetry known as CPT, which acts on one-particle states as 104 2 Relativistic Quant um M echanics where n' denotes the antiparticle (or `charge-conjugate') of particle n . No phases or matrices are allowed in this transformation (though of course we could always introduce such phases or matrices by combining CPT with good internal symmetries .) It follows tha t (APT)'Tp,0r,n = (-1 )'iTP,-aM a (2 .C. 12) so the possibility suggested by Wigner of a sign -(-1)2j in the action of (CPT)2 does not arise in quantum field theory . To the extent that T is a good sYmmetry of some class of phenomena, so is the inversion CP =_(CPT)T-1 . For the states that transform under T in the conventional way TTpaa,nGCT- p,v-,ni (2.C.13) the GP operator also acts conventionall y The operator C; CPP -1then just interchanges particles and antipar ticles CTP,cr,,, ac Tp,rrynE'. (2.C.15) On the other hand, where T has the unconventional representation (2 .x.5), Eq.(2.C.11)gives In particular ,it is possible thatthe degeneracy indicated by the label ±may be the same as the particle -antiparticle degeneracy ,sothat the antiparticle (as defined b y CPT) of the state T± I sT:;. In th is case, CP would have the unconventional property of not interchanging particles and ant iparticles .As far as these particles are concerned , CP and T would bewhat are usually called P and CT .But this i snot merely amatter of definition ;on other part icles CP and T would still have their usual e ffect. No examples are known of particles that furn ishunconventional repre- sentations of inversions ,so these poss ibilities will not bepursued further here . From now on ,the inversions will be assumed to have the conven- tional action assumed in sect ion16. Problem s 1. Suppose that observer Csees a W-boson (spin one and mass m =~0) with momentum p in the y-direction and spin z-component 6 .A second observer ur' moves relative to the first with velocity v in the z-direction . How does Vdescribe the W state? References 105 2. Suppose that observer 0sees a photon with momentum p in the y-direction and polarization vector in the z-direction . A second observer G' moves relative to the first with velocity v in the z- direction . How does '' describe the same photon ? 3.Derive the commutation relations for the generators of the Galilean group directly from the group multiplication law (without using our results for the Lorentz group), Include the most general set of central charges that cannot be eliminated by redefinition of th e group generators . 4. Show that the operators P .Pj" and YV.WP commute with all Lorentz transformation operators U (A, a),where Wp = ~~,,pAJ°Pp'. 5. Consider physics in two space and one time dimensions, assuming invariance under a `Lorentz' group ,S O(2,1). How would you describe the spin states of a single massive particle? How do they behave under Lorentz transformations? What about the inversions P and T ? 6.As in Problem 5, consider physics in two space and one time di - mensions, assuming invariance under a `Lorentz' group x(2,1) . Ho w would you describe the spin states of a single massless particle? Ho w do they behave under Lorentz transformations? What about th e inversions P and T? Reference s 1. F. A.M. Dirac The Principles of'( uantum. Mecha nics, 4th edn (Ox- ford University Press, oxford, 1958) . 2. E. P. aligner, Gruppentheorie end ihre Anwendung auf die Quarat en- mecharaak der Atomspektren (Braunschweig, 1931) : pp. 251-3 (Englis h translation, Academic Press, Inc, New work, 1959) .For massles s particles, see also E. P. Wigner, in Theoretical P hysics (Internationa l Atomic Energy Agency, sienna, 1963) :p. 64. 3. G. C. Wick, A . S. Wightman, and E . P. signer, Pays .Rev.88,101 (1952) . 3a.See, e .g, S . Weinberg, Gravitation and Cosmology (Wiley, New work, 1972) : Section 2 .1. 4. E. Inane and E . F. Wigner, Nuovo Cimento IX, 705 (1952) . 5. E. P. Wigner, Apra . Math.40, 149 (1939) . 106 2 Relativistic Quantum mechanic s 6. G. W. Mackey, Ann. Math .55, 101 (19521) ;58,193 (1953) ;Acta. Math .99,265 (1958) ;Induced Representations of Groups and Quantum Mechanics (Benjamin, New York, 1968)- 7. See, e .g., A. R. Edmonds, Angular Momentum inQuantum Mechanics , (Princeton University Press, Princeton, 1957) :Chapter 4 ; M. E. Rose, Elementary Theory of'Angular Momentum (John Wiley & Sons , New York, 1957) :Chapter IV ; L. D. Landau and E . M. Lifshitz , Quantum Mechanics -Non Relativistic T heory, 3rd edn . (Pergamo n Press, Oxford, 1977): Section 5$ ; Wu-K] Tung, Group Theory i n Physics (World Scientific, Singapore, 1995) :Sections 7 .3 and 8.1. 8. T. D. Lee and C.N. Yang, Phys . Rev .104, 254 (1956) ;C. S. Wu et al., Phys .Rev. 105, 1413 (195 7); R. Darwin, L . Lederman, and M. Weinrich, Phy s.Rev.105, 1415 (1957) ; J. I. Friedman and V . L. Telegdi, Phys . Re v.105, 1681 (1957) . 9. J. H. Christenson, J . W. Cronin, V . L. Fitch, and R . Turlay, Phys . Rev. Letters 13,138(1964) . 10. H.A. Kra .mers, Proc .Acad . Sci .Amsterdam 33, 959 (193 0); also see F.J.Dyson, J.Math .Phys . 3, 140 (1 962), 11. V. Bargmann, Ann. Math .59,1(1954)- Theorem 7.1. 12. See, e .g., H . W. Turnbull and A. C. Aitken, An Introduction tothe Theory o fCanonical Matrices (Dover Publications, New York, 1961) : p. 194 . 13. See e . g. H. Flanders, Differential Forms (Academic Press, New dark, 1963) .Section 3 .6. 14. For an introduction to homotflpy classes and groups, see, e .g., J.G. Hocking and C . S. Young, Topology (Addison-Wesley, Reading, MA, 1961) :Chapter 4 ; C. Nash and S . yen, Topology and Geometry forPhysicists (Academic Press, London, 1983) :Chapters 3 and 5 . 15. E. P. Wigner, in Gr oupTheoretical Concepts and Methods in Elemen- tary Particle Physics, ed . by F . {Gursey (Gordon and Breach, New `fork, 1964) : p.37. Scattering Theor y The general principles of relativistic quantum mechanics described in the previous chapter have so far been applied here only to states of a single stable particle . Such one-particle states b ythemselves are not very exciting - it is only when two or more particles interact with each other that any- thing interesting can happen . But experiments do not generally follow the detailed course of events in particle interactions . Rather, the paradigmatic experiment (at least in nuclear or elementary particle physics) is one in which several particles approach each other from a macroscopically large distance, and interact in a microscopically small region, after which the products of the interaction travel out again to a macroscopically large dis- tance . The physical states before and after the collision consist of particles that are so far apart that they are effectively non-interacting, so they can be described as direct products of the one-particle states discussed in the pre- vious chapter . In such an experiment, all that is measured is the probability distribution, or `cross-sections', for transitions between the initial and final states of distant and effectively non-interacting particles . This chapter will outline the farmalism1 used for calculating these probabilities and cross- sections . 3.1`In'and `hut' State s A state consisting of several non-interacting particles may be regarded as one that transforms under the inhomogeneaus Lorentz group as a direct product of one-particle states . To label the one-particle states we use their four-momenta p ", spin z-component (or, for massless particles, helicity) a, and, since we now may be dealing with more than one species of particle, an additional discrete label r : for the particle type, which includes a specification or its mass, spin, charge, etc . The general transformation rule is 107 108 3 ScatteringTheor y Xj(APi)1(P2)'... d1d~ X IApI~a',in2 iAp2,d'rn2;,.. (3.1.1) where WA P) is the Wigner rotation (x .5.20), and DYa(YV) are the con- ventional (2j+ 1}-dimensional unitary matrices representing the three- dimensional rotation group . (This is for massive particles ; for any mass- less particle, the matrix D(14(A, p ))is replaced with 6,1a exp(idO(11,p)), GCF where 0is the angle defined by Eq. (2.5.43).) The states are normalized as in Eq .(2.5.19) (Y ,.0F,1,0;- ,T +permutations (3.1.2) with the terra `± permutations' included to take account of the possibility that it is some permutation of the particle types nl, nz, ... that are of the same species of the particle types ni, n2> .... (As discussed more fully in Chapter 4, its sign is - 1if this permutation includes an odd permutation of half-integer spin particles, and otherwise +1. This will not be important in the work of the present chapter .) We often use an abbreviated notation, letting one Greek letter, say Y, stand for the whole collection p j,ui,n1; p2,C2, n2;.... In this notation, Eq.(3.1.2)is written simpl y (TCXF~'Y,)= 6(a' - a) (3.1.3) with 6(oc' -- ar) standing for the sum of products of delta functions and Kronecker deltas appearing on the right-hand side of Eq .(3.1.2).Also, in summing over states, we writ e d L x.. ,=1:f.d3Pid3p~,....(3.1.4).. In particular, the completeness relation for states normalized as in Eq. (3.1.3) reads T=/dcx T, (T,,, T ). (3 .1.5) The transformation rule (3 .1.1) is only possible for particles that for one reason or another are not interacting . Setting AP, = P, and aP= (a, Q, 0, r), for which U(A ,a)= exp(iH T), Eq . (3,1 .1)requires among othe rD(jl) (W(A,p~))DC~Tj`a), (W(A, P2)) 3.1`fin' and`Out'States 109 things that T, be an energy eigenstat e HT, =ExTo : with an energy equal to the sum of the one-particle energies(3.1.6) =p°+p°+~-- (3 .1.7) and with no interaction terms, terms that would involve more than one particle at a time . On the other hand, the transformation rule (3 .11) does apply in scat- tering processes at times t __*boo. As explained at the beginning of this chapter, in the typical scattering experiment we start with particles at time t --,,-oa so far apart that they are not yet interacting, and end with particles at t -r +oo so far apart that they have ceased interacting . We therefore have not one but two sets of states that transform as in Eq.(3.1.1): the `i n'and `nut' states 'P,+and T,- will be found to contain theparticles described by the labelacif observations are made a t t--~-oo ort--~,+oo, respectively . Note how this definition is framed . To maintain manifest Lorentz invariance, in the formalism we are using here, state-vectors do not change with time - a state-vector kI' describes the whole spacetime history of a system of particles . (This is known as the Heisenberg picture, in distinction with the hradinger picture, where the operators are constant and the states change with time .) Thus we do notsay that T .± are the limits at t -).Tao of a time-dependent state-vector T(t) . However, implicit in the definition of the states is a choice of the inertial frame from which the observer views the system ; different observers see equivalent state-vectors, but not the same state-vector . In particular, suppose that a standard observer e) sets his or her clock so that t = 0 is at some time during the collision process, while some other observer G' at rest with respect to the first uses a clock set so that t' =0 is at a time t = T ; that is, the two observers' time coordinates are related b y t' = t - - r.Then if G~sees the system to b ein a state T, 60' will see the system in a state U(1, --T)T = exp(-iHr)T . Thus the appearance of the state long before or long after the collision (in whatever basis is used by 61)is found byapplying a time-translation operator eXp(-iHT) with r --*-oo or r -++oo, respectively .Of course, if the state is really an energy eigenstate, then it cannot be localized in time the operator exp(-iH,u) yields an inconsequential phase factor exp(-iE~,c) . Therefore, we must consider wave-packets, superpositions flac g(ac)T,, of states, with an amplitude g(cc) that is non-zero and smoothly varying over some finit e The labels `+' and `- ' for `in' and 'out' states may seem backward, but th ey seem to have become traditional . They arise from the signs in Eq . (3,1,16). 110 3Scattering Theory . range ❑E of energies . The `in' and `out' states are defined so that the superposition exp(-iH-c) da g( 2)T,± _ da e i Ex-cg(a)Tx± has the appearance of a corresponding superposition of free-particle states for r <-1/ LE or r > +1/AE , respectively . To make this concrete, suppose we can divide the time-translation gen- erator H into two terms, a free-particle Hamiltonian HO and an interaction V, in such a way that HO has eigenstates (Da that have the same appearance as the eigenstates T~ and ~'~ of the complete Hamiltonia n NOcD7 = E.101 , (3.1.9) Note that HO is assumed here to have the same spectrum as the full Hamiltonian H . This requires that the masses appearing in Ho be the physical masses that are actually measured, which are not necessarily the same as the `bars' mass terms appearing in H ; the difference if there is any must be included in the interaction V . not HO . Also, any relevant bound states in the spectrum of H should be introduced into HO as if they were elementary particles ."* The `in' and `out' states can now be defined as eigenstates of H, not Ho , which satisfy the cond ition f dae-iE,-cg(a) 'Pa due-iE,Tg(a),Da forr--*-oo or-c -> +x,respectively . Eq.(3.1.12) can be rewritten as the requirement that : . / j(3,I,11) (3.l.lZ) for r -r -oo or -r --* +oo, respectively . This is sometimes rewritten as a formula for the `in' and `out' states : (3.1.13) Alternatively, in non-rciativislic problems we can include the binding potential in HO . In the application ofthis method to 'rearrangement collisions,' where some bound states appear in the initial state but not the final state, or vice-versa, one must use a different split of H into HO and V in the initial and final states . 3. 1=rnfand;out'States where Q(r) - exp(+iH-u)exp(-iHoz) .III (3.1.14) However, it should be kept in mind that in Eq .(3.1.13) gives meaningful results only when acting on a smooth superposition of energy eigenstates . One immediate consequence of the definition (3.1.12)is that the `in' and `out' states are normalized just like the free-particle states . To see this, note that since the left-hand side of Eq. (11 .12) is obtained by letting the unitary operator exp(-iHr) act on a time-independent state, its norm is independent of time, and therefore equals the norm of its limit forz --* ~)c-, i.e., the norm of the right-hand side of Eq .(3.1.12): fdo~d#exp(-i(E, -Ep),r)g(a)g*(fl)(Tp±1 Y1X±) =Jdad#e xp(-i(Ea-E#) T)g (1) g *(#) (Op, IDX) . Since this is supposed to be true for all smooth functions g{a}, the scalar products must be equal (3.1.15) It is useful for some purposes to have an explicit though formal solution of the energy eigenvalue equation (3 .1. 1)satisfying the conditions (3 .1.12). For this purpose, write Eq .(3.1.11) a s The operator E,- H ois not invertible ; it annihilates not only the free- particle state c D,but also the continuum of other free-particle states (D#of the same energy . Since the `in' and 'out" states become just (D,,for V -i 0, we tentatively write the formal solutions as (1),plus a term proportional toV: or, expanding in a complete set of free-particle states , T#7tDp± = J E,- Ep +ie(3.1.16) (3.1.17) (3.1.15) with Ea positive infinitesimal quantity, inserted to give meaning to the reciprocal of E ,- Ho . These are known as the Lippmann-Schwinger equation5.'OWe shall use Eq. (3.1.17)at the end of the next section to give a slightly less unrigorous proof of the orthonormality of the `in' and 'out' states . 112 3 Scatt eringTheor y It remains to be shown that Eq. (3.1.17), with a +aF or -ie in the denominator, satisfies the condition (3.1.12) for an `in' or an `out' state, respectively . For this purpose, consider the superposition s `fi'g±~t}=fdri eiExtg(~c)''.x± , ( 3.1.19) I We want to show that 'fig+(t) and W,,-(t) approach (Pg{t} for t --*-oo and t -k +co, respectively . Using Eq . (3.1.1?)in Eq .(3.1.19) gives E,- E + ie,~ Let us recklessly interchange the order of integration, and consider first the integrals E,, # For t --+-co, we can close the contour of integration for the energy variable E,in the upper half-plane with a large semi-circle, with the contribution from this semi-circle killed by the factor exp(-iE,,t), which is exponentially small for t -*-ao and Im E , 0. The integral is then given by a sum over the singularities of the integral in the upper half- plane . The functions g(ac) and T#,,t may, in general, be expected to have some singularities at values of E,,with finite positive imaginary parts, but just as for the large semi-circle, their contribution is exponentially damped for t - ; -oo . (Specifically, -t must be much greater than both the time-uncertainty in the wave-packet g(a) and the duration of the collision, which respectively govern the location of the singularities of g(a) and Tfl,± in the complex E,,plane .) This leaves the singularity in (Ea - E #± ie)-', which is in the upper half-plane for .-fp- but no t We conclude then that J-fl+ vanishes for t --*-oc. In the same way, for t --~ +oo we must close the contour of integration in the lower half-plane, and so Jlp vanishes in this limit . We conclude that 'Yg±(t) approaches cDg(t) for t -> +oc, in agreement with the defining condition (3.1.12). For future use, we note a convenient representation of the factor ( E,- E#± ic)-1 in Eq . (3.1.17). In general, we can writ e (E ± i c)-~ _ ~ e+ i7r~~(E) , {3 .].22) 3.2The S-matri x where Y, E E-E2+e2 ir(V +C2)113 (3-1.23) (3. 1.24) The function (3.1.23) i sjust 1/Efor JE ~>e,and vanishes for E ~ 0, so for- --+ 0 it behaves just l ike the `pr incipal value function ' J E,which allows us to give m eaning to integrals of11E times any smooth funct ion of E, byexcluding an infinite simalinterval around E=0. The function (3.1.24)is of order eforJEJ>>e,and give sunity when integrated over allE,soin the l imite -+ 0 it behaves just li ke the familiar delta function J(E).With this underst anding, we can drop the label ein Eq .(3.1.22), and write simply 3.2 The S- matrix(3.1.25) An experimentalist generally prepares a state to have a definite particle content at t -> -oc ,and then measures what this state looks like at t --++ac. If the state is prepared to have a particle content for t -> -cc, then it is the `in' state T,+, and if it is found to have the particle content Pat t --*+oo, then it is the `out' state T#- . The probability amplitude for the transition OG -> #is thus the scalar produc t This array of complex amplitudes is known as the S- matrix .2 Ifthere were no interactions then `in' and `out' states would be the same, and then Spa would be just 6(a-f3) The rate for a reaction a~ f3 is thus proportional to ISPO:6{0c We shall see in detail in Section 3 .4 what S p,has to do with measured rates and cross-sections . Perhaps it should be stressed that `in' and 'out" states do not inhabit two different Hilbert spaces . They differ only in how they are labelled : by their appearance either at t ---r -oa or t --*+oc. Any `in' state can be expanded as a sum of `out' states, with expansion coefficients given by the S-matrix ( 3.2.1). Since Spa is the matrix connecting two sets of orthonormal states, it must be unitary . To see this in greater detail, apply the completeness relation (3.1.5)to the `opt' states, and write 114 3 Sc atteringTheor y Using (3.1-15), this gives ~ sPfla V) (3.2.2) or in brie, StS = 1 . In the same way, completeness for the `in' states gives f dfl S-~flSp* = 6(y - 2) (3.2.3) or, in other words, SSA = 1 . It is often convenient instead of dealing with the S-matrix to work with an operator S, defined to have matrix elements betweenftce-particle state s equal to the corresponding elements of the S-matrix : (3,2.4) The explicit though highly formal expression (3 .1. 3)for the `in' and `out' states yields a formula for the S-operator : where(3.2.5) ~('Us Ta)= il(z)~~(,ro) = exp(il~oT)exp(-aH(,r-,co))exp(-iHOz0) . (3.2.6) This will be used in the next section to examine the Lorentz invariance of the 5-matrix, and in Sec . 3.5 to derive a formula for the S-matrix in time-dependent perturbation theory . The methods ❑f the previous section can be used to derive a useful alternative formula for the S-matrix . Let's return to Eq. (3.1.21) for the `in' state T+, but this time take t -> +x .. We must now close the contour of integration for E,in the lower hatf-E,-plane, and although as before the singularities in T#x+ and g(a) make no contribution for t - *+ou, we now do pick up a contribution from the singular factor (E ,- E#+ ae)-' . The contour runs from E ,= ----co to E, = +oa, and then back to E, = -co on a large semi-circle in the lower half-plane, so it circles the singularity in a clockwise direction . By the method of residues, this contribution to the integral over E,is given by the value of the integrand at E,,= Efl- ie, times a factor -2in . That is, in the limit e --* 0 +, for t --*+ao the integral over a in (3 .1.2I) has the asymptotic behavio r JP+ --+-2bre-iEfi "I'dot 6(Ea - Efl)g(x)Tpo!+ " An alternative pr oof`isgiven at the end ofthis ctinn _ Note that for infin ite 'matrices,' the unitarity conditions S td=1 and SS t=Iare not equivalent . 3.2The S-matrix and hence, fo rt --~- +oo , T~ +W --*jdfl e-iE~t(D~ [g(fl) - 2in I da 6(Eo t But expanding (3 .1.19)forT9+ in a complete set of 'out' states give s T9 +( t)= dye-rE,t g(U) dpiT#-S#a. Since S fi,,contains a factor 6(E#- Ea ), this may be rewritten Tg+(t)= IdflTfl- CiEgt fdag(a)S#,,115 and, using the defining property (3.1.12) for `out' states, this has the asymptotic behavior for t ---),+o~~ Comparing this with our previous result, we fin d jda&)Sfl, =g(ji) -2i7r Idoc 6(Ea -Efl)goc)Tfl~4+ or in other words (3.2.7) This suggests a simple approximation for the S-matrix : for a weak interaction Y, we can neglect the difference between `in' and free-particle states in (3.1.18), in which case Eq . (3.2.7) give s Spa^-' ~(# - a) -2kr6 (E,-E~)((D#, V(D.x). ( 3.2.8) This is known as the Born approximation .3Higher-order terms are dis- cussed in Section 3 .5. ~** We can use the Lippmann-Schwinger equations (3 .1.16)for the `in' and `out' states to give a proof4 of the orthanorrnality of these states and the unitarity of the S-matrix, as well as E q.(3.2.7), without having to deal with limits at t -r +oc . First, by using (3.1.1 6) on either the left- or right-hand side of the matrix element (T~ , V`~'~ ) and equating the results, we find that =((Dp,V Ta±) + (T#±,V(Ep-Ho +iF)-'via±) . Summing over a complete set .,of intermediate states, this gives the 116 equation :3Scattering Theor y x([E,- E, ± i-F]-' - [Ep - E, T ie]-'). (3.2.9) To prove the orthonormality of the `in' and `out' states, divide Eq . (3.2.9) byE,- E'g± 2ic . This give s Tap+ Efl -Ea ±2ie+ Tflxt E,--E# ±2ic Tr~- -E,,+ ir Eff-Tyx± E, - E# i c The 2,-s in the denominators on the left-hand side can be replaced with Es, since the only important thing is that these are positive infinitesirnals . We see then that b(P- a) + Tflx±I (E# ---E,± ie) is unitary . With {3 .1.17}, this is just the statement that the T,± form two arthonorma] sets of state-vectors . The unitarity of the S-matrix can be proved in a similar fashion b ymultiplying (3.2.9)with 6 (E#-E,) instead of (E,, -Ep±2ie)-1 . 3.3 Symmetries o f the S-M atrix In this section we will consider both what is meant by the invariance of the S-matrix under various symmetries, and what are the conditions on the Hamiltonian that will ensure such invariance properties . Lorentz Invariance For any proper orthochronous Lorentz transformation x ---~Ax + a, we may define a unitary operator U(A, a) byspecifying that it acts as in Eq.(3.1.1)on either the `in' orthe `opt' states . When we say that a theory is Lorentz-invariant, we mean that the same operator U(A, a) acts as in (3.1.1) on both `in' and `out' states . Since the operator CT(A,a)is unitary, we may write SP =(T#-, T,+) = (U(A, a)T#-, U(A, a)T,,+) so using (3.1.1),we obtain the Lorentz invariance (actually, covariance) property of the S-matrix : for arbitrary Lorentz transformations A"V and translations am, 3.3Symmetries of the S-Matrix 'T!exp(iap(Pl' + P?+ pP1?P0 P~oP~... x D~il),(W(A,p))D(j2),(W(A,P2))''- 6~ ~~ ... I')*( W (A, p')) ... X D W (A, pf )) D(j 2 27-ra2,.. XSApl,Crlani;Ap2 a~2,n2;'-' AP1>d1,nj;~1P2,~2,M2;..,117 (3.3.1) (Primes are used to distinguish final from initial particles ; bars are used to distinguish summation variables .) In particular, since the left-hand side is independent of aju, so must be the right-hand side, and so the S-matrix vanishes unless the four-momentum is conserved . We can therefore wn'te the part of the S-matrix that represents actual interactions among the particles in the form : S13« -6(fl- a)=-7riM#,64(PA- p.,:). (3.3.2 (However, as we will see in the next chapter, the amplitude Mfl .,, itself contains terms that involve further delta function factors .) Eq.(3.3.1)should be regarded as a definition of what we mean by the Lorentz invariance of the S'-matrix, rather than a theorem, because it is only for certain special choices of Hamiltonian that there exists a unitary operator that acts as in (3.1.1)on both `in' and `out' states . We need to formulate conditions on the Hamiltonian that would ensure the Lorentz invariance of the S-matrix . For this purpose, it will be convenient to work with the operator Sdefined by Eq . (3.2.4): SPY-(oft,S(Da). As we have defined the free-particle states (D,in chapter 2, they furnish a representation of the inhamogeneaus Lorentz group, so we can always define a unitary ❑perator Uo {11,a}that induces the transformation (3.1.1) on these states : Uo(A, a)Opj,uj,nj ;p2,cr2,n2 ;- - =exp( -ia'.W1, +P5,+ - - .)) ~(APj)0(~2P- P~~o ... xd)Apj,t7;,nj;nP2,6t2,nz;...D(jl) D(j2)I,(W(A,P2) ) a 'Ia~(W(A~ P, 19, Cr Eq.(3.3.1)will thus hold if this unitary operator commutes with the 118 S-operator :3Scattering Theor y This condition can also be expressed in terms of infinitesimal Lorentz transformations . Just as in Section 2 .4, there will exist a set of Hermitian operators, a momentum PO, an angular momentum JO, and a boost generator KO, that together with HQ generate the infinitesimal version of inhomogeneous Lorentz transformations when acting on free-particle states . Eq . (3.3.1)is equivalent to the statement that the S-matrix is unaffected by such transformations, or in other words, that the S'-operator commutes with these generators : Because the operators HO, PO, J .and KOy generate infinitesimal inhomo- geneous Lorentz transformations on the (D,they automatically satisfy the commutation relations (14.18)--(2.4.24): [4,Joj] = i e ijkJo' [Jo, col = [ Po,Ho]=[Po, POj] =0>(3.3.4) (3.3.5) (3.3.6) (3.3.x) (3.3.8) (3.3.9) (3.3.10) where i, j, k, etc . run over the values 1, 2, and 3, and € ijkis the totally antisymmetric quantity with C123=+1- In the same way, we may define a set of `exact generators,' operators P, J, K that together with the full Hamiltonian H generate the transformations (3.1.1)on, say, the `in' states . (As already mentioned, what is not obvious is that the same operators generate the same transformations on the 'out' states .) The group structure tells us that these exact generators satisfy the same commutation relations : 1.1 Pil=YCIA pk [itI H1 1P11 HI 1PI, Pjj0,(3.3.11) (3.3.12) (3.3.13) (3.3.14) (3.3.15) (3.3.16) (3-3-17) 3.3Symmetries of the S-Matrix 1 19 In virtually all known field theories, the effect of interactions is to add an interaction term V to the Hamiltonian, while leaving the momentum and angular momentum unchanged : (The only known exceptions are theories with topologically twisted fields, such as those with magnetic monopoles, where the angular momentum of states depends on the interactions .) Eq . (3.3. 5)implies that the commu- tation relations (3 .3.11), (3 .3.14), and (3 .3.16)are satisfied provided that the interaction commutes with the free-particle momentum and angular- momentum operators IV, p ol= [V, col=0. (3.3.19) It is easy to see from the Lippmann-Schwinger equation (3,1.16) or equivalently from (3.1.1 3) that the operators that generate translations and rotations when acting on the `in' (and `out') states are indeed simply P0 and J0.Also we easily see that P0 and J (jcommute with the operator U(t, to) defined by Eq .(3.2.6), and hence with the S-operator U(oo, -oc ). Further, we already know that the S-operator commutes with H0, because there are energy-conservation delta functions in both terms in {3 .2.7}. This leaves just the boost generator Ko which we need to show commutes with the S-operator . On the other hand, it is not possible to set the boost generator K equal to its free-particle counterpart K4, because then Eqs . (3.3.15)and (3.3.8)would give H =H0, which is certainly not true in the presence ❑f interactions . Thus when we add an interaction V to H4, we must also add a correction W to the boost generator- K = KD+ W . (3.3.20) Of the remaining commutation relations, let us concentrate on Eq . (3,3 .17), which may now be put in the for m Byitself, the condition (3.3.21) is empty, because for any V we could always define W bygiving its matrix elements between H-eigenstates T . and T# as -(T#, {K0, V]`I`x} f (E # - E.Recall that the crucial point in the Lorentz invariance of a theory is not that there should exist a set of exact generators satisfying Eqs .(3.3.11)-(3.3.17), but rather that these operators should act the same way on `in' and `out' states ; merely finding an operator K that satisfies Eq . (3.121)is not enough .Eq.(3.3.21) does become significant if we add the requirement that matrix elements of W should be smooth functions of the energies, and in particular should not have singularities of the form (Eff-- E«)- l. We shall now show that 120 3 Scattering Theor y Eq.(3.3.21), together with an appropriate smoothness condition on W, does imply the remaining Lorentz invariance condition [Ko, S] = D . To prove this, let us consider the commutator of KO with the operator U(t, to) for finite t and to . Using Eq .(3.3. 10) and the fact that PO commutes with HO yields : [Ko,exI~(Wat)] = -t Pp exp(iHot ) while Eq . (3.3.21) (w hich is equivalentto Eq . (3.3.17)) y ields [K,ex p(tHt)] = -t Pexp(iH t) = -t Pa exp(W t) . The momentum operators then cancel in the commutator of K-0 with U, and we find : where W(t) = exp(iHot)W exp(-Wot) . (3.3.23) Ifthe matrix elements of W between Ufa-eigenstates are sufficiently smooth functions of energy, then matrix elements of W(t) between smooth super- positions of energy eigenstates vanish for t --*foo, so Eq .(3.3.22) gives in effec t as was to be shown . This is the essential result : Eq . (3.3.21)together with the smoothness condition on matrix elements of W that ensures that W(t) vanishes for t --~,±aa provides a sufficient condition for the Lorentz invariance of the S-matrix . This smoothness condition is a natural one, because it is much like the condition on matrix elements of V that is needed to make V(t) vanish for t --*±oo, as required in order to justify the very idea of an S-matrix . We can also use Eq.(3.3.22) with x = 0and to =Too to show tha t KS~(Too)= U(+oo)Ko, (3.3.25) where S~( :Foo) is according to (3.1.13) the operator that converts a free- particle state dux into the corresponding `gin' or `out' state T .Also, it follows trivially from Eqs . {3.3.18} and (3.3.19) that the same is true for the momentum and angular momentum : PSG(+00) -Q(--oo) Po, (3 .3. 6) Finally, since all 4),,, and T,-i are eigenstates of HO and H respectively with the same eigenvalue E ,we have 3.3 Sy mmetriesofthe S- Matrix 12 1 Eqs.(3.3.25)-(3 .3.28) show that with our assumptions, `in' and `out' states do transform under inhomogeneous Lorentz transformations just like the free-particle states . Also, since these are similarity transformations, we now see that the exact generators K, P, J, and H satisfy the same com- mutation relations as KO, Po, J O, and HO . This is why it turned out to be unnecessary in proving the Lorentz invariance of the S-matrix to use the other commutation relations (3.3.12), (3 .3.13), and {3 .3.15} that involve K . Internal Symmetrie s There are various symmetries, like the symmetry in nuclear physics under interchange of neutrons and protons, or the `charge-conjugation' symmetry between particles and antiparticles, that have nothing directly to do with Lorentz invarlance, and further appear the same in all inertial frames . Such a symmetry transformation T acts on the filbert space of physical states as a unitary operator U( T), that induces linear transformations on the indices labelling particle species X (.3.29) In accordance with the general discussion in Chapter 2, the U(T) must satisfy the group multiplication rul e U(T) U(T) = U(TT) , (3.3.30) where TT is the transformation obtained by first performing the trans- formation T, then some other transformation T . Acting on Eq .(3.3.29) with U(T ), we see that the matrices 2satisfy the same rul e !2(T)_9(T)= Q(TT). (3.3.31) Also, taking the scalar product of the states obtained by acting with U(T) on two different `in' states or two different 'out' states, and using the normalization condition (3.1.2),we see that -9(T) must be unitar y Finally, taking the scalar product of the states obtained byacting with U(T) on one `out' state and one `in' state shows that Qcommutes with the S-matrix, in the sense tha t XS~,i~~~y~yp~~z~2, .. Pi~iNi ;I~2~aN2 ;'.' 122 3 Scattering Theor y Again, this is a definition of what we mean bya theory being invari- ant under the internal symmetry T, because to derive Eq. (3.3.33) we still need to show that the same unitary operator U(T) will induce the transformation (3.3.29) on both `in' and `out' states . This will be the case if there is an `unperturbed' transformation operator UQ(T) that induces these transformations on free-particle states , Pl i:F3 2 N,nl(T)N22,, (T)Alai P ?a2AV2 ; ". and that commutes with both the free-particle and inte raction parts of the Hamiltonian (3.3.35) (3.3.3b) From either the Lippmanrt-Schwinger equation (3.1.17) or from (3 .1.13), we see that the operator U O(T) will induce the transformations (3 .3.29) on `in' and `out' states as well as free-particle states, so that we can derive Eq. (3.3.29) taking U(T) as UO(T) . A special case of great physical importance is that of a one-parameter Lie group, where T is a function of a single parameter 0, with (3.3.37) As shown in Section 2 .2, in this case the corresponding Hilbert-space operators must take the form (3.3.3) with Qa Hermitian operator . Likewise the matrices 5,A(T)take the for m 9,jFn(T(O)) =6,+,,exP(iqnO)_ (3.3.39) where q ,are a set of real species-dependent numbers . Here Eq . (3.3.33) simply tells us that the qs are conserved : Sfl,vanishes unless (3.3.4) The classic example of such a conservation law is that of conservation of electric charge . Also, all known processes conserve baryon number (the number of baryons, such as protons, neutrons, and hyperons, minus the number of their antiparticles) and lepton number (the number of leptons, such as electrons,, muons, rparticles, and neutrinos, minus the number of their antiparticles) but as we shall see in Volume II, these conservation laws are believed to be only very good approximations . There are other conservation laws of this type that are definitely only approximate, such 3.3Symmetries ofthe S-Matrix 123 as the conservation of the quantity known as strangeness, which was in- troduced to explain the relatively long life of a class of particles discovered by Rochester and Butlers in cosmic rays in 1947 . For instance, the mesons now called# K+ and KO are assigned strangeness + 1and the hyperons 11 O, E+, E0, Z- are assigned strangeness - 1, while the more familiar protons, neutrons, and it mesons (or pions) are taken to have strangeness zero . The conservation of strangeness in strong interactions explains why strange particles are always produced in association with one another, as in reac- tions like n+ +n --i, K+ + A~,while the relatively slow decays of strange particles into non-strange ones such a s 11O --~ p+7E-and K + -~,ir + + irc) show that the interactions that do not conserve strangeness are very weak . The classic example of a `non-Abelian' symmetry whose generators do not commute with one another is isotopic spin symmetry, which was suggested in 19 37on the basis of an experiment} that showed the existence of a strong proton-proton force similar to that between protons and neutrons . Mathematically, the group is S U(2 ), like the covering group of the group of three-dimensional rotations ; its generators are denoted ti, with i = 1, 2, 3, and satisfy a commutation relation like (2 .4.1$) [taftf] = iF ijktk. To the extent that is otopicspinsymmetry is re spected ,itrequires pa rticles to form degenerate mult iplets labelled with an integer or half-integer T and with 2 T+ 1 components distinguished by their t 3values,just like the degenerate sp in mu ltiplets required by rotational invariance .These include the nucleons p and n with T and tj T ~,- ~; the pions ir+, 7°, and ar-with T = 1 and t3 -: +1,,0, - 1 and the 11Ohyperon with T = 0 and 0 = D. These examples illustrate the relation between electric charge Q, the third component of isotopic spin t.1,thebaryon number B, and the strangeness S: This relation was originally inferred from observed selection rules, but it was interpreted by Dell-Mann and Ne'eman in 1960 to be a conse- quence of the embedding of both the isospin T and the `hypercha .r e' Y=-B + S in the Lie algebra of a larger but more badly broken non- Abelian internal symmetry, based on the non-Abelian group SU(3) .As we will see in Volume II, today both isospin and SU(3) symmetry are understood as incidental consequences of the small masses of the two or three lightest quarks in the modern theory of strong Interactions, quantum chramodynamics . Superscripts denote e lectric c h4ryes in units of the absolute value of the electronic charge, A `hypero n" is any part icle carrying non-zero strangeness and unit baryon number . 124 3 Scattering Theor y The implications of isotopic spin symmetry for reactions among strongly interacting particles can be worked out by the same familiar methods that were invented for deriving the implications of rotational invariance . In particular, for a two-body reaction A + B - + C + D ,Eq.(3.3.33) requires that the S-matrix may be put in the form (suppressing all but isospin labels) : Sfr3tD3,tA31B3 :7 'r,r_j where C}1} ,(ju;a,o-2) is the usual Clebsch-Gordan coefficient9 fo r ing a spin j with three-component or from spins j ,and j2with three- components aland 0-2, respectively ; and ST is a `reduced' S-matrix depending on T and on all the suppressed momentum and spin variables, but not on the xsospin three-components W, t ,93, tC3, tn3 .Of course, this, like all of the consequences of isotopic spin invariance, is only approx- imate, because this symmetry is not respected b yelectromagnetic (and other) interactions, as shown for instance by the fact that different mem- bers of the same isospin multiplet like p and n have different electric charges and slightly different masses . Parit y To the extent that the symmetry under the transformation x --+-x is really valid, there must exist a unitary operator P under which both `ire' and 'out" states transform as a direct product of single-particle states : where ran is the intrinsic parity of particles of species n, and P reverses the space components of p #- (This is for massive particles ; the modification for massless particles is obvious .) The parity conservation condition for the S-matrix is then : SPiCTl~~;p~a~n2 ;... ipj~1,~j;p2~2,~2;.,. ~qniq~~ -.. Just as for internal symmetries, an operator P satisfying Eq . {3.3. 1} will actually exist if the operator P0 which is defined to act this way on free-particle states commutes with V as well as Ho . The phases r l,,may be inferred either from dynamicai models or from experiment, but neither can provide a unique determination of the qs . This is because we are always free to redefine P by combining it with any conserved internal symmetry operator . For instance, if P is conserved, 3.3Symmetries of the S-Matrix 125 then so is P' - P exp(iaB + i flL + iy Q) , where B, L, and Qare respectively baryon number, lepton number, and electric charge, and a, P, and 7are arbitrary real phases ; hence either P or P' could be called the parity operator . The neutron, proton, and electron have different combinations of values for B, L, and Q, so by judicious choice of the phases a, fl, and y we can define the intrinsic parities of all three particles to be +1 . However, once we have done this the intrinsic parities of other particles like the charged pion (which can be emitted in a transition n --*p + 7-) are no longer arbitrary . Also, the intrinsic parity of any particle like the neutral pion 0which carries no conserved quantum numbers is always meaningful . The foregoing remarks help to clarify the question of whether intrinsic parities must always have the values +1.It is easy to say that space inversion P has the group multiplication lawp2= 1; however, the parity operator that is conserved may not be this one, but rather may differ from it by a phase transformation of some sort, In any case, whether or notp2::= 1, the operator P2 behaves just like an internal symmetry transformation : p2T t 2 z ...~± {~1d1M1 42ft2n2s-'- ?I1l1?7n2 P1 fflM1,P2d2M2 ;... If this internal symmetry is part of a continuous symmetry group of phase transformations, such as the group of multiplication by the phases exp(ixB + i flL + iyQ) with arbitrary values of a, fl, and y, then its inverse square root must also be a member of this group, say I p, with I~P' = I and [Ip, P] = 0. {For instance, if P2= exp(iocB + ...}, then take Ip = expo- 2 iaB + - -} .} We can then define a new parity operator P' -=PIP with P'2 = 1 . This is conserved to the same extent as P, so there is no reason why we should not call this theparity operator, in which case the intrinsic parities can only take the values ±1 . The only sort of theory in which it is not necessarily possible to define parity so that all intrinsic parities have the values +1 is one in which there is some discrete internal symmetry which is not a member of and continuous symmetry group of phase transformations .10For instance, it is a consequence of angular-motnentutn conservation that the total number Fof all particles of half-integer spin can only change byeven numbers, so the internal symmetry operator (-1)' is conserved . All known particles of half-integer spin have odd values of the sum B + L of baryon number and lepton number, so as far as we know,(-1)F=(-1)B+L. If this is true, then (-I)' is part of a continuous symmetry group, consisting of the operators exp(ioc{B + L}) with arbitrary real oc, and has an inverse square root exp(-in(B + L)/2) . In this case, ifp2 =(-1)F then P can be 126 3 Scattering T heory redefined so that all intrinsic parities are +1. However, if there were to be discovered a particle of half integer spin and an even value of B+ L (such as a so-called ajorana neutrino, with j = ~ and B + L = 0), then it would be possible to havep2without our being able to redefine the parity operator itself to have eigenvalues +1 . In this case, of course, we would have P4 = 1, so all particles would have intrinsic parities either ±1or (like the Majorana neutrino) ±i . It follows from Eq . (3.3,42 that if the product of intrinsic parities in the final state is equal to the product of intrinsic parities in the initial state, or equal to minus this product, then the S-matrix must be respectively even or odd overall in the three-momenta . For instance, it was observed" in 1951 that a pion can be absorbed by a deuteron from the ~ = 0 ground state of the nd atom, in the reaction 7r- + d ~ n + n . (As discussed in Section 3 .7, the orbital angular-momentum quantum number ~' can be used in relativistic physics in the same way as in non-relativistic wave mechanics .) The initial state has total angular-momentum j=I(the pion and deuteron having spins zero and one, respectively), so the final state must have orbital angular-momentum f .~ = Iand total neutron spin s = 1 . (The other possibilities e = 1, s -0;4=0,s = Iand /= 2, s= 1, which are allowed by angular-momentum conservation, are forbidden by the requirement that the final state be antisymmetric in the two neutrons .) Because the final state has t,' = 1, the matrix element is odd under reversal of the direction of all three-momenta, so we can conclude that the intrinsic parities of the particles in this reaction must be related by: The deuteron is known to be a bound state of a proton and neutron with even orbital angular-momentum (chiefly / = 0), and as we have seen we can take the neutron and proton to have the same intrinsic parity, sarid = q., and we can conclude that -1 ; that is, the negative pion is a pse udosca lar part icle. The n+ and 7r° have also been found to have negative parity, as would be expected from the symmetry (isospin invariance) among these three particles . The negat ive parity of the pion has some striking consequences . A spin zero particle that decays into th ree pions must have intrinsic parity 3 _ -1, because in the Lo rentz frame in which the decaying particle i s at rest, rotational invariance only allows the matrix element to depend on scalar products of the pion momenta with each other, all of which are even under reversal of all momenta . (The triple scalar product pl ' (P2 XP3) formed from the three pion momenta vanishes because pt + p2 + p3 = 0 .) For the same reason, a spin zero pa rticle that decays into two pions must have intrinsic par ity 1 2 = -~ 1 . In particular, among the strange particles discovered in the late 1940s there seemed to be two different particles of 3.3Symmetries of the S-Matrix 127 zero spin (inferred from the angular distribution of their decay products ) one, the z, was identified byits decay into three pons, and hence wa s assigned a parity - 1, while the other, the 0, was identified by its decay into two pions and was assigned a parity +I I. The trouble with all this was that as the r and 0were studied in greater detail, they seemed increasingly to have identical masses and lifetimes . After many suggested solutions of this puzzle, Lee and Yang in 1956finally cut the Gordian knot, and proposed that the r and 0are the same particle, (now known as the K± ) and that parity is simply not conserved in the weak interactions that lead to its decay . 12 As we shall see in detail in the next section of this chapter, the rate for a physical process .7 --+ P (with a P)is proportional toISXx12, with propor- tionality factors that are invariant under reversal of all three-momenta . As long as the states x and Pcontain definite numbers of particles of each type, the phase factors in Eq . (3.3.42) have no effect onISP,I2, so Eq. (3.3.42) would imply that the rate for x -##is invariant under the reversal of direction of all three-momenta . As we have seen, this is a trivial consequence of rotational invariance for the decays of a K meson into two or three pions, but it is a non-trivial restriction on rates in more complicated processes . For example, following theoretical suggestions by Lee and Yang, Wu together with a group at the National Bureau of Stan- dards measured the angular distribution of the electron in the final state of the beta decay Co6°--*Ni60+ e- +v with a polarized cobalt saurce .13 (No attempt was made in this experiment to measure the momentum of the antineutrino or nickel nucleus .) The electrons were found to be preferentially emitted in a direction opposite to that of the spin of the decaying nucleus, which would, of course, be impossible if the decay rate were invariant under a reversal of all three-momenta . A similar result was found in the decay of a positive moon (polarized in its production in the process ac+ --.>, P+ + v) into a positron, neutrino, and antineutrino. 14 In this way, it became clear that parity is indeed not conserved in the weak interactions responsible for these decays . Nevertheless, for reasons discussed in Section 12.5, parity is conserved in the strong and electro- magnetic interactions, and therefore continues to play an important part in theoretical physics . Time-Reversa l We saw in Section 2.6 that the time-reversal operator T acting on a one-particle state TP,a,n gives a state with reversed spin and momentum, times a phase "j- t )j-'.Amulti-particle state transforms as usual as a direct product of one-particle states, except that since this 128 is a time-reversal interchanged :3 scattering Theor y transfo rmation, we expect `in' and `out' states to b e TTP1¢1n1:F2d2n2...- i ~nj ~aM2~ 1~ ~Ypl-d1M1;YPZ-d1_M2;... (3.3.43) (Again, this is for massive particles, with obvious modifications being required for massless particles .) It will be convenient to abbreviate this assumption as TT~_'~'~a , (3 .3.44) where Y-indicates a reversal of sign of three-momenta and spins as well as multiplication by the phase factors shown in Eq . (3.3.43). Because T is antiunitary, we hav e so the time-reversal invariance condition for the S-matrix i s or in more detai l ,. f f' n1 ~ +r k ~ XS~aPI-C1 nI;9P~ -a2 n2 ;--- .dpi -a~ n~;:YpI2 -a2 nZ ;.,.(3.3.45} (3.3.46) (3.3.47) Note that in addition to the reversal of momenta and spins, the role of initial and final states is interchanged, as would be expected for a symmetry involving the reversal of time . The S-matnix will satisfy this transformation rule if the operator To that induces time-reversal transformations on free-particle state s Toga = Dg-cc (3.3.48) commutes not only with the free-particle Hamiltonian (which is automatic) but also with the interaction : To 'Ho To =HO, (3 .3.49) T1VTQ =Y. (3.3.50) In this case we can take T = T4, and use either (3 .1. 3)or(3.1-16) to show that time reversal transformations do act as stated in Eq.(3.3.44). For instance, ❑perating on the Lippmann-Schwinger equation (3.1.16) with T and using Eqs .(3.3.48)-(3,3.50), we hav e 0!fI 3.3Symmetries aj'the S-Matrix 12 9 with the sign of +i creversed because T is antiunitary . This is just the Lippmann-Schwinger equation for T-~ thus justifying Eq . (3.3.44), Similarly, because T is antiunitary it changes the sign of the i in the exponent of (t), so tha t again leading to Eq . (3.3.44). In contrast with the case of parity conservation, the time-reversal in- variance condition (3 .3.46) does notin general tell us that the rate for the process a --* ft is the same as for the process 9-a - + JI-fl . However, something like this is true in cases where the S-matrix takes the for m Sol! = ~~~ + ~~~~ , (3.3.51) where 5 Mis small, while 5 Phappens to have matrix element zero for some particular process of interest, though it generally has much larger matrix elements than S M. (For instance, the process might be nuclear beta decay, N -+ N' + e- +:V,with 5 Othe S-matrix produced by the strong nuclear and electromagnetic interactions alone, and S (i)the correction to the S-matrix produced by the weak interactions . Section 3 .5 shows how the use of the `distorted-wave horn approximation' leads to an S-matrix of the form (3.3.51) in cases of this type . In some cases S' Mis simply the unit operator .) To first order in S (i), the unitarity condition for the S-operator reads 1-SAS=~~~W5((,)+S(WSM+SOA5M. Using the zeroth-order relation S AT5O) = 1, this gives a reality condition for S M IfSM as we llas SM satisfies the t ime-reversal condition (3 .3.46), then t his can be put in the for m 5fl(ii) ~ f- Jdydy'S(O)S.~~~.~?S. (3.3.53) Since 5 (G)is unitary, the rates fog the processes ac --~.Pand Ta -y 9-fl are thus the same ifsummed over sets Jand of final and initial states that are complete with respect to SO.(By being 'complete' here is meant that if ST) is non-zero, and either ac or cc' are in J, then both states are inJ;and similarly for .~F.} In the simplest case we have 'complete' sets Jand .3;'consisting of just one state each ; that is, both the initial and the final states are eigenvectors of S Mwith eigenvalues ev° a nd eNa~, respectively . (The 3,and 60are called `phase shifts' ; they are real because 130 3 ScatteringTheor y S(°)is unitary .) In this case, Eq. (3.3.53) becomes simply = and it is clear that the absolute value of the S-matrix for the process Of --+ 13 is the same as for the process J_ac -> 9-fl . This is the case for instance in nuclear beta decay (in the approximation in which we ignore the relatively weak Coulomb interaction between the electron and nucleus in the final state), because both the initial and final states are eigenstates of the strong interaction S-matrix (with 6x =6# = 0 ). Thus if time-reversal invariance is respected, the differential rate for a beta decay process should be unchanged if we reverse both the momenta and the spin z-components aof all particles . This prediction was not contradicted in the 1956 experiments13 .i4 that discovered the non-conservation of parity ; for instance, time-reversal invariance is consistent with the ❑bservation that electrons from the decay Co 60 ---), Ni60+ e-+Vare preferentially emitted in a direction opposite to that of the Co6° spin . As described below, indirect evidence against time-reversal invariance did emerge in 1964, but it remains a useful approximate symmetry in weak as well as strong and electromagnetic interactions . In some cases we can use a basis of states for which Ti = !x and .lfl=ft,for which Eq . (3.3.54) read s (1) 2t(a'+sO)S~(~r)MSpa _ -e (3 .3.55) which just says that iS~,r) has the phase r~ , + 5#mod 7r . This is known asWatson:stheorem .15The phases in Eqs . (3.3.54) or (3-3-55) may be measured in processes where there is interference between different final states . For instance, in the decay of the spin 1/2 hyperon Ainto a nu- cleon and a pion, the final state can only have orbital angular-momentum e= 0 or e =I; the angular distribution of the pion relative to the A spin involves the interference between these states, and hence according to Watson's theorem depends on the difference ds -6Pof their phase shifts . PT Although the 1957 experiments on parity violation did not rule out time- reversal invariance, they did show immediately that the product PT is not conserved . If conserved this operator would have to b eantiunitary for the same reasons as for T, so in processes like nuclear beta decay its consequences would take the form of relations like E q. (3.3.54): flu zp'-Tfl 3.3Symmetries of the S-Matrix 13 1 where Y3- reverses the s igns of all spin z-components but not any mo- menta.Neglecting the final-state Coulomb interaction ,it would then follow that th erecould be no preference for the electron in the decay Co"" -*Ni6° + e - + v to be emitted in the same or the opposite direct ion to the Ca6° spin ,in contradiction with what was observed . C,CP, and CP T As already mentioned, there is an internal symmetry transformation, known as charge-conjugation, which interchanges particles and antiparti- cles. Formally, this entails the existence of a unitary operator C, whose effect on multi-particle states i s ~~a I n3 ;P2 0-2n2.1 11l~~(Y tf0i aP292 n2;,.. where n{~is the ant iparticle of particle type n,and ~ ,,is yet another phase . If th is is true forboth `in' and `out' states then the S-matrix satisfies the invariance condition s SFr~Qitd1;PyqZny;.,. P lalnf ;f~2~2~2a.,. w * ~ii2.. ~ Sp ] ] '2#2h~~ p1a1n~sp2CI~ i22:,.. 13.~.51 As with other internal symmetries, the S-matrix will satisfy this condition ifthe operator C othat is defined to act as stated in Eq. (3.3-56) on free- particle states commutes with the interaction V as well as H4 ; in this case, we take C-Co. The phases ,,are called charge-conjugation parities . Just as for the ordinary parities q,,, the ~„ are in general not uniquely defined, because for any operator Cthat is defined to satisfy Eq . (3.3,56 ), we can find another such operator with different ~ ,by multiplying C with any internal symmetry phase transformation, such as exp(iocB + i flL + iyQ);the only particles whose charge-conjugation parities are individually measurable are those completely neutral particles like the photon or the neutral pion that carry no conserved quantum numbers and are their own antiparticles . In reactions involving only completely neutral particles, Eq . (3.3. 7)tells us that the product of the charge-conjugation parities in the initial and final states must be equal ; for instance, as we shall see the photon is required by quantum electrodynamics to have charge-conjugation parity ref = -1, so the observation of the neutral pion decay no -> 2r requires that ~o = +1 ; it then follows that the process 7E°--*3yshould be forbidden, as is in fact known to be the case . For these two particles, the charge-conjugation parities are real, either +1 or --1 . Just as for ordinary parity, this will always be the case if all internal phase transformation symmetries are 1 32 3 Scattering Theor y members of continuous groups of phase transformations, because then we can redefine C by multiplying by the inverse square root of the internal symmetry equal to CZ, With the result that the new C satisfies C2 = 1 . For general reactions, Eq . (3.3.57)requires that the rate for a process equals the rate for the same process with particles replaced with their corresponding antiparticles . This was not directly contradicted by the 1957 experiments on parity non-conservation (it will, be a long time before anyone is able to study the beta decay of anticobalt), but these experiments showed that Cis not conserved in the theory of weak interactions as modified by Lee and Yang12 to take account of parity non-conservation, (As we shall see below, the observed violation of TP conservation would imply a violation of C conservation in any field theory of weak interactions, not just in the particular theory considered by Lee and Yang .) It Is understood today that Cas well as P is not conserved in the weak interactions responsible for processes like beta decay and the decay of the pion and muon, though both C and P are conserved in the strong and electromagnetic in#eractions . Although the early experiments on parity non-conservation indicated that neither C nor P are conserved in the weak interactions, they left open the possibility that their product OP iuniversally conserved . For some years it was expected (though not with complete confidence) that CP would b efound to be generally conserved . This had particularly important consequences for the properties of the neutral K mesons . In 1954 Gell- Mann and Pais" had pointed out that because the KO meson is not its own antiparticle (the ICS carries a non-zero value for the approximately conserved quantity known as strangeness) the particles with definite decay rates would be not Ii° or K°, but the linear combinations KO +KO. This was originally explained in terms of C conservation, but with Cnot conserved in the weak interactions, the argument may be equally well be based on CP conservation . If we arbitrarily define the phases in the CP operator and in the K° and K° states so tha t CPTy,o =Tko and CPTKa = ` i'Ka then we can define self-charge-conjugate one-particle state s 1 i-,L and 1 TKO [TKO -TK01 3.3Symmetries of the S- Matri x 1 33 which have CP eigenvalues +1and -1, respectively . The fastest available decay mode of these particles is into two-pion states, but CP conservation would allow this only*' for the If I, not the K2 . The K4 would thus be expected to decay only by slower modes, into three pions or into a pion, moan or electron, and neutrino . Nevertheless, it was found by Fitch and Cronin in 1964 that the long-lived neutral K-meson does have a small probability for decaying into two pions .17 The conclusion was that CP is not exactly conserved in the weak interactions, although it seems more nearly conserved than C or P individually . As we shall see in Chapter 5, there are good reasons to believe that although neither C nor CP is strictly conserved, the product CPT is exactly conserved in all interactions, at least in any quantum field theory . It is CPT that provides a precise correspondence between particles and antiparticles, and in particular it is the fact that CPT commutes with the Hamiltonian that tells us that stable particles and antiparticles have exactly the same mass . Because CPT is antiunitary, it relates the S-matrix for an arbitrary process to the S-matrix for the inverse process with all spin three-components reversed and particles replaced with antiparticles . However, in cases where the .S-matrix can bedivided into a weak term 5Mthat produces a given reaction and a strong term SO) that acts in the initial and final states, we can use the same arguments that were used above in studying the implications of T conservation to show that the rate for any process is equal to the rate for the same process with particles replaced with antiparticles and spin three-components reversed, provided that we sum over sets ❑f initial and final states that are complete with respect to S O). In particular, although the partial rates for decay of the particle into a pair of final states 1, P2 with 5 (O) 0 may differ from the partial rates for the decay of the antiparticle into the corresponding final states W019-J31 and JV-fl2 , we shall see in Section 3 .5 that (without any approximations) the total decay rate of any particle is equal to that of its antiparticle . We can now understand why the 1957 experiments on parity violation could beinterpreted in the context of the existing theory of weak interac- tions as evidence that Cconservation as well as P conservation are badly violated but OP is not . These theories were field theories, and therefore automatically conserved CPT . Since the experiments showed that PT con- servation but not T conservation is badly violated in nuclear beta decay, any theory that was consistent with these experiments and in which CPT is conserved would have to also incorporate C but nit CP non-conservation . The neutral K-mesons have spin zero, so the two-pion final state has ~ =0, and hence P = +1 . Further, C =-+1 for two ;r4s because the rya has C = +1, and also for an I= 0 7c it - slate because Cinterchanges the two pions . 1 34 3 Scattering Theor y similarly, the observation in 1964 of small violations of CPconservation in the weak interactions together with the assumed invariance of all interactions under CPT allowed the immediate inference that the weak interactions also do not exactly conserve T . This has since been verifiedls by more detailed studies of the KI-KO system, but it has so far been impossible to find other direct evidence of the failure of invariance under time-reversal . 3.4RatesandCross-Section s The S-matrix S¢,is the probability amplitude for the transition oc -~ fl, but what does this have to do with the transition rates and cross-sections measured by experimentalists? In particular, (3 .32shows that Spa has a factor 64(Pp_p'),which ensures the conservation of the total energy and momentum, so what are we to make of the factor [61(F#-- p«)]2 in the transition probability j5p,12? The proper way to approach these problems is by studying the way that experiments are actually done, using wave packets to represent particles localized far from each other before a collision, and then following the time-history of these superpositions of multi-particle states . In what follows we will instead give a quick and easy derivation of the main results, actually more a mnemonic than a derivation, with the excuse that (as far as Iknow) no interesting open problems in physics hinge on getting the fine points right regarding these matters . We consider our whole system of physical particles to be enclosed in a large box with a macroscopic volume V . For instance, we can take this box as a cube, but with points on opposite sides identified, so that the single-valuedness of the spatial wave function requires the momenta to be quantize d where the rayare integers, and L3= V . Then all three-dimensional delta- functions become (27c)3 V (27E)3 } where6P" Pis an ordinary Kronecker delta symbol, equal to one if the subscripts are equal and zero otherwise . The normalization condition (3.1.2)thus implies that the states we have been using have scalar products in a box which are not just gums of products of Kronecker deltas, but also contain a factor [V/(27r }3]1, where Nis the number of particles in 3.4 Rat esand Cross-Sections 135 the state . To calculate transition probabilities we should use states of unit norm, so let us introduce states normalized approximately for our bo x "2 with n orm TBox, TBox(3.4,4) where 6#x is a product of Kronecker deltas, one for each three-momentum, spin, and species label, p lusterms with part icles permuted . Correspond- ingly, the S-matr ix may be wr itten {N, +N,}l2foxSP,=[V/(27] (3 .4.5) where S}B` is calculated using the states (3 .4.3). Of course, if we just leave our particles in the box forever, then every possible transition will occur again and again . To calculate a meaningful transition probability we also have to put our system in a time box' . We suppose that the interaction is turned on for only a time T . One immediate consequence is that the energy-conservation delta function is replaced with 1 T/ 2 - 27z I-T/ 2 The probability that a multi-particle system, which is in a state oc befor e the interaction is turned on, is found in a state P after the interact ion is turned off, is 13 Z(,Vtc+N#)~S~,12 .c`~=[(2z)'/V] (3 .4.7) P ("Z --+ P) = ~ P I This is the probability for a transition into one specific box state P. The number of one-particle box states in a momentum-space volumed3pis VdIp/(2 n)3,because this is the number of triplets of integers nj,n?,n3 for which the momentum (3 .4.1 ) lies in the momentum-space volume d3P around p . We shall define the final-state interval d flas a product ofd3p for each final particle, so the total number of states in this range i s dA-ff = [V/(2 .)3]N 'dfl . (3.4,8) Hence the total probability for the system to wind up in a range d#of final states is ~~~P,~12dfl. (3 .4.9) dP(cx--+fl)= P(a - --.> fl)d.IV-fl= [(2)3/VJ' 136 3 Scattering Theor y We will res trict our attention throughout this sect ion to fina lstates P that are not only different (however slightly) from the initial sta te rc, but that also satisfy the more stringent condition, that no subset of the partic les in the state fl (other than the whole state itself) has prec isely the same four-momentum as some corresponding subset ❑f the particles in the state oc. (In the language to be in troduced in the next chapte r, this means that we are considering only the connected part of the S-matrix .) For such states, we may define a delta function-free matr ix element M#, , Our introduction of the box al lows us to interpret the s quares of delta functions in ~~P"'12 for fi 7~ cc a s 2 15 IV(Pfl - Pol V V 16T(Efl - E,)] 6T(E# - E.x)h(0) = 6T(Efl -E.,)T/27r , so Eq . (3.4.9) gives a differential transition probabilit y dP(ot (2n)2 [(27r)31~] '-1(T127r)Ifi.xl x &v (PO-P06r(Ep- Ex)dfl. Ifwe let V and T be very large, the delta function product here may be interpreted as an ordinary four-dimensional delta function 64(pfl - p ') In this limit, the transition probability is simply proportional to the time T during which the interaction is acting, with a coefficient that may be interpreted as a differential transition rate : = (2r~)3Ny-2 W-N, imo264(I~,-Pix)dfl, (3 .4.11) where now SP#= -2 7dP( Pfl-Px)p~! . (3.4.12) This is the master formula which is used to interpre tcalculations of S- matrix elements in terms of predictions for actual experiments . We will come back to the interpretation of the factor r]d(P' - P #)dplater in th is section . There are two cases of special importance : 1V,,,-1 Here the volume V cance ls in Eq . (3.4.11), w hich gives the transition rate for aj ingle-particle state ~c to decay into a general mu lti-particle state as 3.4Rates and Cross-Sections 137 Of course, this makes sense only if the time T during which the interaction acts is much less than the mean lifetime r,, of the particle a~, so we cannot pass to the limit T -+ 00 inJT(E,- E#). There is an unremovable width ❑E ^F 1/ T 1/T,, in this delta function, so Eq . (3.4.13) is only useful if the total decay rate is much less than any of the characteristic energies of the process . N,,=2: Here the rate (3 .4.11) is proportional to 11V, or in other words, to the density of either particle at the position of the other one . Experimentatists generally report not the transition rate per density, but the rate perflux, also known as the cross-section .The flux of either particle at the position of the other particle is defined as the product of the density 1 / V and the relative velocity u~ (D,x = ux/V. ( 3.4.14) (A general definition of u,, is given below ; for the moment we will content ourselves with specifying that if either particle is at rest then Ux is defined as the velocity of the other .) Thus the differential cross-section i s d~(a --+ fl) = dr(a ---Y P)/(D, =(2-g)'u.x i i p,1'6'(Pp- F,)dfl. (3 .4.15) Even though the cases N,- 1 and N,,= 2 are the most important, transition rates for N, 3are all measurable in principle, and some ❑f them are very important in chemistry, astrophysics, etc . (For instance, in one of the main reactions that release energy in the sun, two protons and an electron turn into a deuteron and a neutrino .) Section 3 .6presents an application of the master transition rate formula (3 .4.1 1) for general numbers N,,,of initial particles . We next take up the question of the Lorentz transformation properties of rates and cross-sections, which will help us to give a more general definition of the relative velocity u, in Eq . (3.4.15). The Lorentz transformation rule (3.3.1)for the S-matrix is complicated by the momentum- dependent matrices associated with each particle's spin . To avoid this complication, consider the absolute-value squared of (3.3.1)(after factoring out the Lorentz-invariant delta function in Eq . (3.4.12)), and sum over all spins . The unitarity of the matrices D(f)( ) (or their analogs for zero mass) then shows that, apart from the energy factors in (3.3. 1), the sum is Lorentz-invariant . That is, the quantit y I11fffl,~12 rl EH E -R$« (3.4.16) spins is a scalar function of the four-momenta of the particles in states a and # . (By rI« E and f1p Eis meant the product of all the single-particle energies p0 = p2-+m2 for the particles in the states a and f3 .) 138 3Scattering T heory We can now write the spin summed single-particle decay rate (3 .4. 13) as spinsP The factor dJ3/ j l#E may be recognized as the product of the Lorentz- invariant momentum-space volume elements (2.5.15), so it is Lorentz- invariant . So also are Rflx and64(Pfl- P.), leaving just the non-invariant factor 1 1E,where E,is the energy of the single initial particle . Our conclusion then is that the decay rate has the same Lorentz transformation property as 11E, This is, of course, just the usual special-relativistic time dilation -the faster the particle, the slower it decays . Similarly, our result (3 .4.15) for the spin-summed cross-section may be written a s spin s whereEland E2are the energies of the two particles in the initial stat e -x. It is conventional to define the cross-section to be (when summed over spins) a Lorentz-invariant function of four-momenta . The factors Rflx, 6'(pfl- p.), and dflff1pEare already Lorentz-invariant, so this means that we must define the relative velocity u, in arbitrary inertial frames so that u .,EIE2 is a scalar . We also mentioned earlier that in the Lorentz frame inwhich one particle (say, particle 1)is at rest, u, is the velocity of the other particle . This uniquely determines u" to have the value in general Lorentz frames * UU=r(P~7N ~~ M'1 M2 E IE2 (3.4.17) where p hp2and Ml, m2 are the four-momenta and mass of the two particles in the initial state y , As a bonus, we note that in the 'center-of-mass' frame, where the total three-momentum vanishes, we hav e pi= (P} Et)a and here Eq . (3.4. 17) givesP2=(`PaE2 U0,`API{Ej+E2}_pi ~ P2 El E2~E1 E2(3.4.1$) Eq.(3_4_17) makes i t obvious that ElEiua is a ycalxr .Also, w hen particle 1 is at rest, we h avep, = O. E,_ IMi,so pi -P2= -m, F ,,, and so Eq . (3.4,I7)gives 2 _ 2 which is just the ve locity ofparticle 2 . 3.4Rates and dross-Sections ] 39 as might have been expected for a relative velocity . However, in this frame tt', is not really a physical velocity ; in particular, Eq . (3.4.19) shows that for extremely relativistic particles, it can take values as large as 2 . We now turn to the interpretation of the so-called phase-space factor 64(Fo-P,,)dp, which appears in the general formula (3 .4.11)for transition rates, and also in Eqs . (3.4.13) and (3 .4.15) for decay rates and cross- sections . We here specialize to the case of the 'center- of-mass' Lorentz frame, where the total three-momentum of the initial state vanishe s p, =0 . (3 .4.19) (For N,,= 1, this is just the case of a particle decaying at rest .) If the final state consists of particles w ith momenta p', p 2, the n 54(P# --P l)dfl =3( Pi+P~+...~~(E~+E~+ ..--F )d'Fid'P2-.. (3 .4.20) where E - E, is the total energy of the initial state . Any one of the p~ integrals, say over pi, can be done trivially by just droppi ng the momentum delta functio n 64(PI;- Px )d~--*r5[Er +E2 + E)d3p2...(3A.21~ with the understanding that wherever p'appears (as in E') it must be replaced with P1= _P 2 P3-... We can similarly use the remaining delta function to eliminate any one of the remaining integrals . In the simplest case, there are just two particles in the final state . Here (3.4.21) give s In more detail, this i s 64(p#-pl:)dp->~rlpC~ +nr2+ I~~+ 21 -EIP1'I'dIpx'IdQ, (3.4.?3) where P7=P1 and dQ = sin 2 OdEldois the solid-angle differential fo r pi'. This can be simplified by using the standard formul a where f(x) is an arbitrary real function with a single simple zero at x = x O. In our case, the argument Ej + E~ - Eof the delta function in Eq . (3.4.23) 140 3Scattering T heory has a unique zero at l Pi l = k', wher e 3 212 E2 -rn'2+mr2 Ei =k'~ +2 2 = 2 1 2E 2 f 2 r2 E~- ~f2 + M~2 ~ ~ ~1 ~ ~~ 2E with derivative (VFlp ~+aj2+V1Fpj'~+m22-E) ]d1p, 11 kp;1=k' k' k' k'Ei ~~(3.4.24) (3.4.25) (3.4.26) (3A27 ) We can thus drop the delta function and the differential d lpi'lin Eq. {3.4.23}, bydividing by (3 .4.27), k' E' 12dQ (3.4.28) with the understanding that k', E~, and E~ are given everywhere b y Eqs. (3.4.24)-(3 .4.26). In particular, the differential rate (3 .4.13) for decay of cone-particle state of zero momentum and energy E into two particles is dI'(ac 2ar1c'EiE 2'IMPG, 12 and the differential cross-section for the two-body scattering process 12-} 1'2' is given by Eq . (3.4.15) a s A2 Eu, Elk where k = I Pt 1 -= I P2I The above case N# = 2 is particularly simple, but there is one rtice result for Np= 3 that is a lso worth recording . For N#= 3, Eq . (3.4.21) gives 64(P~ ~ ~~ )~fl --* d'P~d3P~ + tW 2 + Vpr22+ m'22 + lp 32+ tW32 - E) xV(~72 + P 13 Y_I We write the momentum-space volume a s d3P~d3P~ = IA~,I 2 dIP~I IF312d~P~IdQ3 dd~23dcas0z3 3.5 Perturbation Theory 14 1 where d Q3is the differential element of solid angle for p', and 023and 023are the polar and azimuthal angles of p'relative to the p'direction . The orientation of the plane spanned byp 2 and p'is specified by 2 3 and the direction of P',with the remaining angle Or fixed by the energy- conservation condition +VI V3 21 +ae, 2 =E The derivative of the argument of the delta function with respect to cos 023 is OEi IP'IIP~I 0cos 023 El1 so we can do the integral over cos 02, by just dropping the delta function and dividing by this derivativ e ~~~~fl --*Ia~IdIP~IIP~IdIP~IEid03d023. Replacing momenta with energies, this is finall y 6'(Pp-Pa)dfl-+ EiEaE3dE~dE~K13 d023  (3.4.31) But recall that the quantity (3 .4.16), obtained by summingIM0,12 over spins and multiplying with the product of energies, is a scalar function of dour-momenta . If we approximate this scalar as a constant, then Eq. (3.4.31) tells us that for a fixed initial state, the distribution of events plotted in the E~, ~~ plane is uniform . Any departure from a uniform distribution of events in this plot thus provides a useful clue to the dynamics of the decay process, including possible centrifugal barriers or resonant intermediate states . This is known as a Dalitz plot," because of its use byDalitz in 1953 to analyze the decay K+ --* 7E++ g+ +77-. 3.5 Perturbation Theor y The technique that has historically been most useful in calculating the S-matrix is perturbation theory, an expansion in powers of the interaction term Vinthe Hamiltonian H = Ho + V.Eqs.(3.2.7)and (3.1.18) give the S-matrix as 142 3Scattering Theor y where T,+ satisf ies the Lippmann-Sch finger equation (3 .1.17): +j . / E.,E~,+iE Operating on this equation with V and taking the scalar product with (D# yields an integral equation for T + Tfiu+ V#a+ f Ea-r where(3,5.1) (3.5.2) The perturbation series for T pa+ is obtained by iteration from Eq .(3.5.1) V#11V,1~~`flx+ = V{~x + fir'Ex -F+ief I dydy'V/3 -~ V7 Y 1 T/`f!a +(Ea -Er + fie)( E,-E-e, + ice )(3.5.3) The method of calculation based on Eq.(3.5.3), which dominated calculations of the S-matrix in the 1930s, is today known as oldfashioned perturbationtheory .Its obvious drawback is that the energy denominators obscure the underlying Lorentz invariance of the S-matrix . It still has some uses, however, in clarifying the way that singularities of the S-matrix arise from various intermediate states . For the most part in this book, we will rely on a rewritten version of Eq.(3.5.3), known as time-dependent perturbation theory, which has the virtue of making Lorentz invariance much more transparent, while somewhat obscuring the contribution of individual intermediate states . The easiest way to derive the time-ordered perturbation expansion is to use Eq . (3.2.5), which gives the S-operator a s where U(z, -ro) - exp (iHoz)exp(-iH (T -xo))exp{ -iHo-co) . Differentiating this formula for U(r, ro) with respect to T gives the differ- ential equation td~~U(z~~o )=V(T) U(z,TO), where(3.5.4) V(t)-exp(Hot) Vexp(-Wot). (3.5.5) 3.5Perturbation Theory 143 (Operators with this sort of time-dependence are said to be defined in the interacti onpicture, to distinguish their time-dependence from the time- dependence Op(t) =cxp(iHt)OHexp(-iHt) required in the Heisenberg picture of quantum mechanics .) Eq . (3.5.4) as well as the initial condition ~(,ro, -co)=Iis obviously satisfied b ythe solution of the integral equatio n To By iteration of this integra lequation, we obtain an expansion for U('r, zo) in powers of V J 1'0 taT O T ti t " +/ccdt, 10dt2JT(} Setting T = c and to - -oo then gives the perturbation expansion for the S-operator : LIS -i dt I V(t I) +(-i)' dt,fdtlV(tl)V(t') ): ~ . IX 0C. 00 This can also be derived directly from the old-fashioned perturbation expansion ( 3.53), by using the Fourier representation of the energy factors inEq.(3.5.3): Y,; with the understanding that such integrals are to be evaluated by inserting a convergence factor e-"' in the integrand, with e ->~ 0+ . There is a way of rewriting Eq.(3.5.8)that proves very useful in carrying out manifestly Lorentz-invariant calculations . Define the time- ordered product of any time-dependent operators as the product with factors arranged so that the one with the latest time-argument is placed leftmost, the next-latest next to the leftmost, and so on . For instance , and so on, where O(T) is the step funct ion, equal to + 1for -c > 0 and to zero fo rr < 0 . The time-ordered product of n Vs is a sum over all rz ! permutations of the Vs, each of which gives the same in tegral over all 144 3 Scattering theor y tl - - - t,,, so E q. (3.5.8) may be written S= 1+ E--~~,io dtldt2... dtn T V(ti) . V0,0 (3.5.10)r~.=1 This is sometimes known as the Dys on series .20This series can be summed if the V(t) at different times all commute ; the sum is the n S = exp _ifQ0 dt V(t)00 Ofcourse, this is not usually the cage ; in general (3 .5.10)does not even converge, and is at best an asymptotic expansion in whatever coupling constant factors appear in V . However Eq. (3.5.10) is sometimes written in the general case as S= T exp __if'00 dtY(t) with T indicating here that the expression is to be evaluated by time- ordering each term in the series expansion for the exponential . We can now readily find one large class of theories for which the 5 - matrix is manifestly Lorentz-invariant . Since the elements of the S-matrix are the matrix elements of the S-operator between free-particle states (1), Ofi , etc ., what we want is that the S-operator should commute with the operator UO{A, a} that produces Lorentz transformations on these free-particle states . Equivalently, the S-operator must commute with the generators of UO(A, a):H0, P0Ja, and K0 . To satisfy this requirement, let's try the hypothesis that V(t) is an integral over three-spac e V(t} = d3x { x, t) (3-5 .11) with (x) a scalar in the sense tha t (By equating the coefficients of a ofor infinitesimal transformations it can be checked that (x) has a time-dependence consistent with E q. (3.5.5).) Then S may be written as a sum of four-dimensional integral s ~~/'!....X, ~' [ (x1)  .. P(xn) 3.5.13) S=1 + n} . 1n_1 Everything is now manifestly Lorentz invariant, except for the time- ordering of the operator product . Now, the time-ordering of two spacetime points x1, x2 is Lorentz- invariant unless x1 - x2 is space-like, i .e., unless (XI -- x2)2 > 0 ,so the time-ordering in Eq. (3.5.13) introduces no special Lorentz frame if 3.5Perturbation Theory 145 (though not only if)the (x) all commute at space-like or light-like* separations ~{x}, (x')] = 0 fo r(x - x')2 > Q . (3 .5.14) We can use the results of Section 3 .3 to give a formal non-perturbative proof that an interaction (3 .5.11) satisfying Eqs .(3,5.12) and (3 .5.14) does lead to an S-matrix with the correct Lorentz transformation properties . For an infinitesimal boost, Eq . (3.5.12)gives i [Ko, ~f(xa t)] = tV, (x,t) + X cat(x, t) so integrating over x and setting t - 0, where[Ko, V] =[Kin, I d3 X~W (X, 0)] =[Ho, W1 , W= duxx'(x, 0).(3.5.15) (3.5.16) (3.5.17) If (as is usually the case) the matrix elements of (x, 0) between eigen- states of H4 are smooth functions of the energy eigenvalues, then the same is true of V, as is necessary for the validity of scattering theory, and also true of W, which is necessary in the proof of Lorentz invariance . The other condition for Lorentz invariance, the commutation relation (3.3.21), is also valid if and only if 0 = [W, YJ = I d'xjd3 YX [,k(x, 0), 0(y, 0)] . (3 .5.18) This condition would follow from the `causality' condition (3 .5.14), but provides a somewhat less restrictive sufficient condition for Lorentz in- variance of the S-matrix . Theories of this class are not the only ones that are Lorentz invariant, but the most general Lorentz invariant theories are not very different . In particular, there is always a commutation condition something like (3 .5. 14) that needs to be satisfied . This condition has no counterpart for non- relativistic systems, for which time-ordering is always Galilean-invariant . It is this condition that makes the c ombination of Lorentz invarianceand quant um m echanicsso restrictive . We write the condition on ?c and x' here as (x -- xr )2~0instead of (x - x')2>0,because as weshall see in Chapter 6, Lorentz invariance can be disturbed by troublesome singula rities at r X = X . 146 3 Scattering Theor y The methods described so far in this section are useful when the interaction operator V is sufficiently small . There is also a modified version of this approximation, known as the distorted-wave Born approximation, that is useful when the interaction contains two term s V = Y, +Vw~ ( 3.5.19) with V ,weak but Vsstrong . We can define TS.x±as what the `in'and `out' states would be if V ,were the whole inter action + 1 F Wecan then w rite (3 .1.16) as Tfl-ot+ ((Dg, VTc+ ) ([T,#- - (Efl - Ho - + Vw)T .x (Tp-, VwTz' ) + (T,,-, [V, - V~;(E# - Ho+ I.F)-'(V, + Vw)] T,,+ ) and SO The second term on the right-hand side is just what T lf,+ would be in the presence of strong interactions alon e (For a proof of Eq .(3.5.22), just drop V ., everywhere in the derivation of Eq.(3.5.21).) Eq .(3.5.21)is most useful when this second term vanishes : that is, when the process cc --* # cannot be produced by the strong interactions alone . (For example, in nuclear beta decay we need a weak nuclear force to turn neutrons into protons, even though we cannot ignore the presence of the strong nuclear force acting in the initial and final nuclear states .) For such processes, the matrix element (3.5.22) vanishes, soEq.(3.5.21) reads T#x+ = (T,sg-, YwT,x+ ). (3 .5.23) So far, this is all exact . However, this way of rewriting the T-matrix becomes worthwhile when V,, is so weak that we may neglect its effect on the state T,,+ in Eq .(3.5.23), and hence replace T,+ with the state '~s:,:+, which takes account only of the strong interaction V, In this approximation, Eq .(3.5.23) become s This is valid to first order in Vw,but to all orders in V . This approximation is ubiquitous in physics ; for instance, the S-matrix element for nuclear 3.6Implications of Unitarity 147 beta or gamma decay is calculated using Eq . (x..24) with V, the strong nuclear interaction and V, respectively either the weak nuclear interaction or the electromagnetic interaction, and with TSfl- and T5,+ the final and initial nuclear states . 3.6 Implications of Unitarit y The unitarity of the S-matrix imposes an interesting and useful condition relating the amplitude M,, for forward scattering in an arbitrary multi- particle state a to the total rate for all reactions in that state . Recall that in the general case, where the state #may or may not be the same as the state g, the S-matrix may be written as in (3.3.2): The unitarity condition then give s ~(,.r-Ol~ _ L~fl S rS,, =6(,, - a)"_27L1[54(p } i'x) "SIX +2,gi64{Py, - p .x}M,*f+4;r2[d13 54( P~ - P~)61(P#- P. x)J~t .p °~ Cancell ingtheterm 6(y- i) and a factor 27z64(Pf -p'),wefind that for pry =N + 2n dfl (3.6.1) This is most useful in the special case ac = j, where it reads (3.6.2) Using Eq . (3,4 .11), this can beexpressed as a formula for the total rate for all reactions produced by an initial state ain a volume V dfldT{ ~c -- ~ dflr,,,f (27L)314',,-2v1-N,,dfl P(pp_Pa)I W#j, 12 I(2,n)' rv,-zyt - NxIM M" . In particular, where oc is a two-particle state, this can be writte n IrnMx,= ---u~c,,/ 161z3 ,(3.6.3) (3. b.4) where u, is the relative velocity (3 .4.17) in state x, and d, is the total 148 3 Scattering Theor y cross-section in this state, given by (3 .4.15) a s GIX = fd#dd(a--*#)Id# = (2,g)4Ua 1 d# ( P#- P,).(3.6.5) This is usually expressed in a slightly different way, in terms of a scattering amplitude f (a -). fl}. Eq . (3.4.30) shows that the differential cross-section fortwo-body scattering in the center-of-mass frame i s dQ kE2 where k' and k are the magnitudes of the momenta in the initial and final states . We therefore define the scattering amplitude as p E kMfl4~2/k'EEEiE 2 sothat the differentia lcross-section is simpl y M If[~ ---> ft In particular, for elastic two-body scattering, we hav e 4z2EiE2 MflaE(3-6-7) (3.6.s) (3.6.9) Using (3 .4.15) for the relative velocity u, the unitarity prediction (3 .6.3) now reads Imf (a--+ a) = ~ ca . (3 .6.10) This form of the unitarity condition (3.6.3)is known as the optical thear- ern.22 There is a pretty consequence of the optical theorem that tells much about the pattern of scattering at high energy . The scattering amplitude f may be expected to be a smooth function of angle, so there must be some solid angle dSZ within which If 12 has nearly the same value (say, within a factor 2) as in the forward direction . The total cross-section is then bounded b y ~a~ I fI'M2!~ i.f(a~ ~)I 'as~ ~ ~ IIM.f(a-+o~)l2Asj. The phase of f is conventional, a nd is motivated by the wave mec hanical inte rpretatia n21 of f as the coefficient of the outgoing wave in the so lution of the time-independent Sc hrodinger equat ion.The no rmalization of f used here is slig htly unconventional for 1t7e)3,ST1C SC&X tP]'1(]g; usually f is defined so t hata ratio of fina land in itial velocities a ppears in the fo rmula for the differential cross-sectio n. 3.6ImplicationsofUnitarity 14 9 Using Eq.(3.6.10) then yields an upper bound on AQ Ail 327E 2Ik2Q.x (3.6.11) As we shall see in the next section, total cross-sections are usually expected to approach constants or grow slowly at high energy, so Eq .(3.6.11) shows that the solid angle around the forward direction within which the differential cross-section is roughly constant shrinks at least as fast as 1 /k2 for k -y oo . This increasingly narrow peak in the forward direction at high energies is known as a diffraction peak . Returning now to the general case of reactions involving arbitrary numbers of particles, we can use Eq . (3,6 .2)together with CPT invariance to say something about the relations for total interaction rates of particles and antiparticles . Because CPT is antiunitary, its conservation does not in general imply any simple relation between a process cc -~ ~ and the process with particles replaced with their antiparticles . Instead, it provides a relation between a process and the inverse process involving antiparticles we can use the same arguments that allowed us to deduce (3 .3.46) from time-reversal invariance to show that CPT invariance requires the S ;matrix to satisfy the conditio n where leg g- indicates that we must reverse all spin z-components, change all particles into their corresponding antiparticles, and multiply the matrix element by various phase factors for the particles in the initial state and by their complex conjugates for the particles in the final state . Since CPT invariance also requires particles to have the same masses as their corresponding antiparticles, the same relation holds for the coefficient of 64(Pc,- P#)in SU, In particular, when the initial and final states are the same the phase factors all cancel, and Eq . (3.6.13)says that (3.6.14) where a superscript c on n indicates the antiparticle of n . The generalized optical theorem (3 .6.2) then tells us that the total reaction rate foam an initia l state consisting of some set of particles is the same as for an initial state consisting of the eorresponding antiparticles with spi nsreversed : r'Fia 1nt;p20_2n2;...=1FP1-al n' ; F2 -Uzriz;.... (3.6.15) In particular, applying this to one-particle states, we see that the decay rate of any particle equals the decay rate of the antiparticle with reversed 150 3 Scattering Theor y spin . Rotational invariance does not allow particle decay rates to depend on the spin z-component of the decaying particle, so a special case of the general result (3 .6. 5) is that unstable particles and their corresponding antiparticles have precisely the same lifetimes . The same argument that led from the unit arity condit ion S tS= 1 to our result (3 .6.2) al so allows u sto use the other unit arity relation SSA =1 to derive the resul t Im M,, =-aCdpi64(p#- pjIxp 11. (3 .6,16) Putting this together with Eq. (3.6.2) then yields the reciprocity relatio n f dfl j I (Pfi - P .) ~mo~2:_ f~fl(34(po-P.1)I gIXp12(3.6.17) or in other words fdF( fid F 00 This result can be used in deriving some of the most important results ofkinetic theory, 23 If P,,dgisthe probability of finding the syste min a volume ho cof the space of multi-particle states (D,,,,then therate of decrease in F ., due totransition stoallother states is F , fdfl d T'(a --* P)/dpi, while the rate of increase of P xdue to tran sitions from allother states is fdpPVdT'(g --* a)fda;the rate of change of P ,is the n do,da d Itfollows immediately that f P ,doc is time-independent .(Just interchange the labelling of the integrat ion variables in the integral in the second term in Eq . (3.6.29)) On the other hand, the rate ofchange of the entropy -,f daP , In P,is ~~ d ~cP,InP,,= - docdpi(InP,+1) xPdr(P --+,7)_ Padr(a --* fl) oda df l Interchanging the labelling of the integration variables in the second term, this may be writte n -ddcc F, InP,, = dcc dfl Pfi In~fl)dr~--~ oo d t Pa da Now, for any positive probabilities P, and P p, the function Pfl ln(P# f P,,) 3.7 Partial-Wave Expansi ons satisfies the inequality** FflIn~ LP #-P a - The rate of change of the entropy is thus bounded b y -' c~a P IXIn P,> dacdf3[PP- P~]dr( -+ a) dt J di or interchanging variables of integration in the second ter m -ddoc P.In P, > da dfl P# [dIu _+a) _dI'{o(--I,#) dtf da d151 But the unitarity relation (3.6.18)tells us that the integral over ac on the right-hand side vanishes, so we may conclude that the entropy alway s increases: -dtf~CCP,, In P,, ~0,I(3.6.20) This is the `Boltzmann H-theorem .' This theorem is often derived in statistical mechanics textbooks either by using the Born approximation, for which MP,12 is symmetric in ac and J3 so that d'( -> x)/dac = dT'(oc --+#)/dfl,or by assuming time-reversal invariance, which would tell us that N1fl,I2 is unchanged if we interchange o~ and fland also reverse all momenta and spins . Of course, neither the Born approximation nor time-reversal invariance are exact, so it is a good thing that the unitarily result (3.6.1$) is all we need in order to derive the H-theorem . The increase of the entropy stops when the probability P, becomes a function only of conserved quantities such as the total energy and charge . In this case the conservation laws require dr'(#--~a)/d a to vanish unless P, = P#, so we can replace P# with P, in the first term of Eq. (3.6.19). Using Eq .(3.6.18)again then shows that in this case, P,, is time-independent . Here again, we need only the unitarity relation (3.6.18), not the Born approximation or time-reversal invarrtanc :e. 3.7 Partial-Wave Expansions * It is often convenient to work with the S-matrix in a basis of free-particle states in which all variables are discrete, except for the total momentu m The difFerenccbetwe enthe left- an dright-hand sides approaches the pos itivequantity (P, - Po'-/2,P# fo rPr - Pj~, and has a der ivative►vith respect to P.z that is positive- or negat ive- dCfiI11tCfor a ll P, >P#orP,, <Yfl, respectively , This se ction lie s somewhat a uiof the h ook'smain lin e of development, and may be omitt ed in afirst readi ng. 152 3 Scattering Theor y and energy . This is possible because the components of the momenta F1,... , pn in an n-particle state of defini te total momentum pand total energy E form a (3n - 4)-dimensional compact space ; for instance, fo r n = 2 particles in the center-of-mass frame with p= 0, this space is a two-dimensional spher ical surface . Any function on such a compact space may be expanded in a series of generalized `partial wares', such as the spher ical harmonics that are commonly used in represent ing functions on the two-sphere . We may thus define a basis for these n-particle states that apart f rom the continuous variables pand E is discre te: we label the free- particle states in such a basis as (DEON, with the index N incorporating all spin and species labels as well as whatever indices are use to label the generalized partial waves . These states may convenient ly be chosen to be normalized so that their scalar products are : (OEFp}Nr,E F N)= d( E' - E)6 3 (p'-p)6NrN, (3 .7.1) The S-operator then has mat rix eleme ntsinthis basis of the for m ((DEI p, Nf , 5 (DE p N) = 6 (E'-E)6 3(p' ---P)SN',rv(E, p), (3 .7.2) where ,S Nf,Nis a unitary matrix . Similarly, the T-operator, whose free- particle matrix elements ( (D#, T(D,,) are defined to be the quantities Tflx+ defined by Eq .(3.1.18), may be expressed in our new basis (in accordance with Eq . (3.4.12)) a s and the relation (3.2.7)is now an ordinary matrix equation : We shall use this general formalism in the following section ; for the present we will concentrate on reactions in which the initial state involves just two particles . For example, consider a state consisting of two non-identical particles of species nl, n zwith non-zero masses M1, 11ri2 and arbitrary spins Si, s2. In this case, the states may be labelled by their total momentum P= PI + P2, the energy E, the species labels n1, n2, the spin z-components U1, 62, and a pai rof integers 4,m (with Iml ~ f)that specify the depen- dence of the state on the directions of, say, pi . Al ternatively, we can form a convenient discrete basis by using Clebsch-Gordan coefficients' to combine the two spins to give a total sp in s with z-component u, and then using Clebsch-Gordan coefficients again to combine this with the orbital angular momentum ~ with three-component na to form a total angula rmomentum j with three-component o- . This gives a basis of states (DEpj,1,n with n a `channel index' labelling the two particle species' ni, n2), defined by their scalar products with the states of definite individual 3.7 Partial-Wane Expansions 153 momenta and spin three-components : ((Dp,.71 p2ff:W,(DEFj6Is n) =OPilEiEzlE)-"'P( F-Fr----pz.) x6 (E -p21 + i - p22 + M22) 6 nF,~ X 1: CS,S_'(S,Y;a1, Cr2)C(s(.1,IVaM}P)Y"' (P1). (3 .7.5) M,Y where Y,1 are the usual spherical harmanics .2' The factor (I pi ~ EtE2/E)-1/2 is inserted so that in the center-of-mass frame these states will be properly normalized : (OE'P' j' d'l's'M'> OE U jd fsn) = 63(pf)63 (Ef -E)6 j,J6a',a61Vk sbn',n. (3 .7.6) For identical particles to avoid double counting we must integrate only over half of the two-particle momentum space, so an extra factor should appear in the scalar product (3.7.6). In the center-of-mass frame the matrix elements of any momentum- conserving and rotationally-invariant operator 0must take the fir m ((bEptYCf,`Srnf , 0 4D EOW sO =(3'7'7) (The fact that this is diagonal in j and a follows from the commutation of0withj2and J 3, and the further fact that the coefficient of 6,,r is independent of afollows from the commutation of 0with ,11 ±U2. This is a special case of a general result known as the signer-Eckert theorem, 25) Applying this to the operator f whose matrix elements are the quantitie s #,,, it follows that the scattering amplitude (3.6.7) in the center-of-mass system takes the for m 2 F~ f ~1 ~~~1E2----47 C ~~k MW t7, - k'a;n', kaj - kQ2n 47t2f } Cs1s2lS7P;dI5(72)C6( j, G ; M, P 1 ~ff~ +TR+S~ ft'~iFiS~ .d }C ~~~~ kf) Y~ * ~k) ~Y~I+Sii 1,I5t1(E) . (3.T8) The differential scattering cross-section isIf12. We will take the direction of the initial momentum k to be along the three-direction, in which cas e YMk)=-5m°F4 77C .(3.7.9) 154 3Scattering T heory , Integrating 11,12 over the direction of the final momentum k' and respec- tively summing and averaging over final and initial spin three-components gives the total crass-section*` for transitions from channel n to Tj': 2 (3 .7.10) X Summing (3.7.10)over all two-body channels gives the total cross-sectio n for all elastic or inelastic two-body reactions : X L(1- Sj(E))t(l-Si (E))]rsn,"~►~. (17.11) For comparison, Eqs .(3.7.8),(3.7.9),(3.7.4), and the Clebsch-Gordan sum rules`* give the spin-averaged forward scattering amplitude a s The optical theorem (3-6. 10) then gives the total cross-section a s 27 r Utotal(+~ ;E)= 1: ~~,~~~1 + 1)(2s,-)+ 1~ (2i +!) Re [1-S ] (E)l (.~i:,rs,~. (3.7.12) j~s If only two-body channels can be reached from the channel n at energy E, then the matrix Sj(E)(or at least some subrnatrix that includes channel n) is unitary, and thu s [(1 - S}{E} )t(1 - S j(E))]_ 2 Re [1-Sj(E)](5n,l,n, (3.7.13) so(3.7.12) and (3.7.1 I) are equal . On the other hand, i fchannels involving three or more particles are open, then the difference of (3.7.12) and (3.7.11) " In deriving this result, we use standard sum rulc .0 for the Clebsch-Gordan cneflicient6 : firs t ~4 5~..%71~~ ~+ dl~¢?l~a~ r~ i,5.~d,fll,{f7~ _ ~ .ss'~+ij i r1L.d2 and the same with pri mes;then fir.,{l,(7: m,a)C'"p(J,FT;m,a)=~jjb¢ ;, Rdf} and finally 2j+ 1 CMj' (Y ;2(+ 1 a~l 3.7Part ial-Wage E xpansions 155 gives the total cross-section for producing extra particles : Uproductinn(ns E)-k2(2s 1 +17E-)(252+1)4~~ +1) .1 X[I fs, .,,, and this must be positive . The partial wave expansion is particularly useful when applied to pro- cesses where the relevant part of the S-matrix is diagonal . This Is the case, for instance, if the initial channel n contains just two spinless particles, and no other channels are open at this energy, as in a E+ - n + or r E+ - 7r scattering at energies below the threshold for producing extra pions (pro- vided one ignores weak and electromagnetic interactions) . For a pair of spinless particles we have j= f, and angular momentum conservation keeps the S-matrix diagonal . It is also possible for the S-matrix to be diagonal in certain processes involving particles with spin ; for instance in pion-nucleon scattering we can have j + or j but for a given jthese two states have opposite parity, so they cannot be connected by non-zero S-matrix elements . In any case, if for some n and Ethe S-matrix elements S NFj,,:Sr=(E, U)all vanish unless N' is the two-body state j, l, s, n, then unitarity requires tha t SiFsrN.Vs„(F)=exp [20y s~(E)]~ee6,',rj►?fn1 (3.7.15) where d j,,,,,(E ) is a real phase, commonly known as the phase s hift. This formula is also often used where the two-body part of the S-matrix is diagonal but channels containing three or more particles are also open ; in such cases the phase shift must have a positive imaginary part, to keep (3.7.14) positive . For real phase shifts, the elastic and total cross-sections are then given by Eq .(3.7.10) or Eq.(3.7. 2)as: 41r-k'(2~'1+1)(2S2 + ~)(2j+1) sing Jjl,M(E).(3.7.16) This familiar result is usually derived in non-relativistic quantum me- chanics by studying the coordinate space wave function for a particle in a potential . The derivation given here is offered both to show that the partial wave expansion applies for elastic scattering even at relativistic ve- locities, and also to emphasize that it depends on no particular dynamical assumptions, only on unitarily and invariance principles . It is also often useful to introduce phase shifts in dealing with problems where several channels are open, forming a few irreducible representations of some internal symmetry group . The classic example of such an internal symmetry is isotopic spin symmetry, for which the channel index n includes 156 3Scattering Theor y a specification of the isospins Ti, T2 of the two particles together with their three-components tl, t 2; the states in channel n may be expressed as linear combinations ❑f the tth components of irreducible representations T, with coefficients given by the familiar Clebsch-Gordan coefficients CT,Tz(T, r: tl, tz) . Suppose that for the channels and energy of interest the S'-matrix is diagonal in rfand s as well as j,T, and t . Unitarity and isospin symmetry then allow us to write the S-matrix a s S]efs' Tfe,lsrr = exp[2i bjesT(E)J6rf~6,,?i56TfTber, (3,7.17) with S jlST(E) a real phase shift, t-independent according to the Wigner- Eckart theorem . The partial cross-sections can again be calculated from Eq.(3.7.10), and the total cross-section is given by Eq . (3.7.I2) a s Orr Gtotal(tl, t2 ;E) = X 1 :(2.1+OCr,, ~~(T~tat1, r2)2 sing 6j,sT (E) .(3.7.1$} j?sTf For instance, in pion-pion scattering we have phase shifts 6~ea r(E)with T==0 or T = 2 for each even z" and T =--I for each odd ,, while for pion nucleon scattering we have phase shifts bjj+11 1T with T or T = 2 , We can gain some useful insight about the threshold behavior of the scattering amplitudes and phase shifts from considerations of analyticity that are nearly independent of and dynamical assumptions . Unless there are special c ircumstances that would produce singularities in momentum space, we would expect the matrix element M1 ',-Jkr dz nf, kar, -k d2n to be an analytic functions of the three-momenta kand k' near k -- 0 or k' = 0 or (for elastic scattering) k = k' =--0. Turning to the partial wave expansion (3 .7.$)for we note that k,Y,,(k) is a simple polynomia l function of the three-vector k, so ire order for Mk',' -k'd~ nr,ko-1 -ka~.nto be an analytic function of the three-momenta kand k' near k = 0 or lc' =[}, the coefficients 's'n'/sn or equivalently 6r1bv ., 6n'n-Sk1n',,sn must go as k~+1k'~+~ when k and/or k'goto zero . Hence for small k and/or W . it is only the lowest partial waves in t he initialand/or final state that contribute appreciably to the scattering amplitude . We have three possible cases : For instance, in the Born approximation (3.2.$), M is proportional to the Fourier transform of the coordinate space matrix elements of the interaction, and hence is analytic at zero momenta as long as these matrix elements fall off sufficiently rapidly at large separations . The chief exception is for scattering involving long-distance forces, such as the Coulomb force . 3.7Partial- Wave Expansions 157 Exothermic reaction s Here k'approache safinite value as k --+0,and in this limit beFA'sb"Fn - Sigoes as k ~+I.The cross-section (3.7. 11) in this case goes a sk2~-i, where eis here the lowest orbital angular momentum that can lead to the reaction . In the most usual case e_ 0, so the reaction cross-section goes as11k.(This is the case, for i nstance, in the absorption of slow neutrons bycomplex nuclei, orfor the annihil ation of electron-positron pairs into photons at low energy, aside from the h igher-order effects of Coulomb forces .)The reaction r ate is thecross-section time sthe flux,whichgoes like k , so the rate foran exothermic re action beh aveslike a constant for k--1,0. However ,it is the cross-section rather than the reaction rate that determines the probability of absorption when a beam crosses a given thickness of target material ,and the factor 1/k ma kes thi sprobabilit y very high for slow neutrons in an absorbing material like boron . Endothermic Reaction s Here the reaction is forbidden until k reaches a finite threshold value, where k' = 0 . Just above this threshold 61Y'6s f,v6n1„ - 5in,,,,n goes as WY f+'. The cross-section (3.7.11) in this case goes as {k'where er is here the lowest orbital angular momentum that can be produced at threshold . In the most usual case ~' - 0,so the reaction cross-section rises above threshold like k',and hence like E - Elhres~old . (This is the case, for instance, in the associated production of strange particles, or the production of electron-positron pairs in the scattering of photons .) Elastic Reaction s Here k = k', so k and k'go to zero together . (This is the case where n' =n, or where n' consists of particles in the same isotopic spin multiplets as are those in n .) In elastic scattering the partial waves with (_er = 0 are always present, so in the limit k - +0 the scattering amplitude (3 .7.8) becomes a constant : 2 (s ycrra~, a~)a5(~' (3.1.17) SQ whe re a is a constant, known as the scatteringlength, defined by the limi t so.wnr,nsn--*bnF,n + 2ika ,(rz--*~n') (3.7.20) 158 3 Scattering Theor y for k = k' -} 0 . Summing 4zlf 1' over final spins and averaging over initial spins gives the total cross-section for the transition n --*n' at k - k' = 0 : 4ar 5 The classic instance of the use of this formula is in neutron-proton scattering, where there are two scattering lengths, with the spin singlet length ao considerably larger than the spin triplet length a, . The partial wave expansion can also be used to make a crude guess about the behavior of cross-sections at high energy . With decreasing wavelengths, we may expect scattering to be described more or less classically : a particle of momentum k and orbital angular momentu m would have an impact parameter I'/k, and will therefore strike a disk of radius R if f .<kR. This can be interpreted as a statement about S-matrix elements f I > kRn ID kR n where R ,is some sort of interaction radius for channel n. For a given e>s, them are 2s + Ivalues of j, allclose enough to (.to approximate 2j+f ^-' 2/+1,so the sum over1'and s in Eq .(17.12) merely gives a factor of orde r }.S 5 The tata .l cross-section is then given for k >i/ by Eq . {3.7.12 a s clolal(nxE)---). Clsld,2n k2 In exactly the same way, Eq .(3.7.1 D)gives the elastic scattering cross- section r~(ra~~ .;E)~~R ~ (3.?.24) The difference between Eqs .(3.7.23) and (3 .7.24) gives an inelastic cross- section ark, which is what we would expect for collisions with an opaque disk of radius R, (The somewhat surprising elastic scattering cross-section 7rR,2, may be attributed to diffraction by the disk .) On the other hand, if we assume along with Eq .(3.7.22) that S~Isn/sn is complex only for impact parameters Ilk within a small range of width ❑,_<R, around (1k T R,,, then using the inequality jIm (1 -- Slisn/sO I ~ 2, the same analysis gives a bound on the real part of the forward scattering amplitude 3.8Resonances 159 The smallness of the real part of the forward scattering amplitude at high energy is confirmed by experiment . So far we have not said anything about whether the interaction radius R, itself may depend on energy . As a very crude guess, we may take R, as the distance at which the factor exp{-fir} in the Yukawa potential (1.2.74) takes a value proportional to some unknown power of E, in which case R, goes as log Efor E ---~ oc:., and the cross-sections go as (log F}2 . As it happens, it has been rigorously shown26on the basis of very general assumptions that the total cross-section can grow no faster than (log E)2 for E --*co, and in fact the observed proton-proton total cross-section rises something like (log E )' at high energies, so this rough picture of high-energy scattering does seem to have some correspondence with reality . 3.8Resonances * It often happens that the particles participating in a multi-particle collision can form an intermediate state consisting of a single unstable particle R, that eventually decays into the particles observed as the final state . If the total decay rate of R is small the cross-section exhibits a rapid variation (usually a peak), known as a resonance, at the energy of the intermediate state R . We shall see that the behavior of the cross-sections near a resonance is pretty much prescribed by the unitarily condition alone, which is a good thing since there are a number of very different mechanisms that can produce a nearly stable state : (a) The simplest possibility is that the Hamiltonian can be decomposed into two terms, a °strong' Hamiltonian HO} +Y.S, which has the particle R as an eigenstate, plus a weak perturbation V, which allows R to decay into various states, including the initial and final states a,flof our collision process . For instance there is a neutral particle, the ZO, with j = 1 and mass 9 1GeV, that would be stable in the absence of the electroweak interactions . These interactions allow the Z' to decay into electron-positron pairs, muon-antimuon pairs, etc ., but with a total decay rate that is much less than the ZO mass . In 1989 the Z° particle was seen as a resonance** in electron-positron collisions at CORN and Stanford , ' This section lies somewhat out of the honk's main ling of dcvclnpment, and may be omitted in a first reading . Incidentally, this example shows that a resonant state only needs to decay relatftrely slowly ; the L'~ lifetime is 2 .6 x i0-25seconds, which is not long enough for a ZO travelling near the speed oflight to cross an atomic nucleus, What is important is that the decay rate is 36limes smaller than the rate #t/M zof oscillation of the Z(' wave function in its rest frame . 160 3 Scattering Theor y in the reactions e+ + e- --*ZQ--).e++e-, e+ + e- -4 ZO->y+ + etc. (b) In some cases a part icle is long-lived because there is a potent ial barrier that nearly prevents its constituents from escaping .The classic example is nuclear alpha decay : it may be energetically possible for a nu- cleus to emit an alpha particle (a He 4 nucleus)but the strong electrostatic repulsion between the alpha part icle and the nucleu screate sa barrier region around the daughter nucleu s, which the alpha particle is classically forbidden to enter . The decay then c an proceed only by quantum me- chan ical barrier penetration ,and is exponentially slow .Such an unstable state shows up as a resonance in the scattering of the alpha part icle on the daughter nucleus . For instance, the lowest-energy state of the Be g nucleus is unstable against decay into two alpha part icles, and is seen as a resonance in Hey -Hey scattering . (In addition to Coulomb barriers, there are al so centrifug al barr iers that help lengthen the life of alpha- ,beta- , and ga mma-unstable nuclei of high spin .) (c) It is possible for complicated systems to benearly unstable for statistical reasons, without the presence of any potential barriers or weak interactions . For instance, an excited state of a heavy nucleus may be able to decay only if, through a statistical fluctuation, a large part of its energy is concentrated on a single neutron . This state will then show up as a resonance in the scattering of a neutron on the daughter nucleus . These mechanisms for producing long-lived states are so different that it is truly fortunate that most of the properties of resonances follow from unitarity alone, without regard to the dynamical mechanism that produces the resonance . First, let's consider the energy dependence of the matrix element for a reaction near a resonance . A wave packet f da g(cc)T .,+ exp(-iE,t) of `in' states has a time-dependence given by(3.1.19) fda g(a)T~+~ iEr = ~~ ~(~)(D,e-iE,t + f dfi 0, f d.e-iE,tg(a) T#a+ Ea-E +ic Asmentioned in Section 11, apole in the function Tp,+ i n the lower-half complex E,plane would make acontribution tothe second term that decays exponentially as t ---),oa.Specifically, apole atEx-ER-iI'f2 yields a term in the amplitude that behaves likeexpo- iERt -Ft/2), so itcorresp onds to a state whose probability decays like exp(- rt).We conclude then that a long-lived state of energy ER with a slow decay rate 3.8Resonance s I' produces a term in the scattering amplitude that varies as161 To go further, it will be convenient to adopt as a basis the orthonormal discrete multi-particle states ~~ ENdiscussed in the previous section ; p and E are the total momentum and energy, and Nis an index that takes only discrete (though infinitely many) values . In this basis, the S-matrix may be written SVWN' ,pEN=63(Pf -P)b(E'-E) rvF N(P, E). (3 .8.2) Near a resonance, we expect the center-of-mass frame amplitude 5 (0,E) (E)to have the for m YNFN(E)= 5N'N{0, E} = YarwN+N'N(3.8.3)E -TEx ~iF/2 where YO and are approximately consta nt at least over the relatively small range of energiesIE-ER ~ F. In this basis, the unitarity of the S-matrix is an ordinary matrix equatio n Y(E)tY(E)= 1 . (3.8.4) Applied to Eq .(3.$.3), this tells us that the non-resonant background S-matrix is unitary ,9110t,5"O = 1 , {3 .8.5} and also that the residue matrix , satisfies the two conditions (3.8.6) 2 2 These conditions can be put in a more transparent form by settin g The unitarity conditions on the matrix dare then simply(3.8.8) X3.5.9) Any such Hermi tian idempote nt mat rixis called a projection matrix .Such matrices can always be expressed as a sum of dyads of orthonormal vectors u(r) .54WN1N_ u 7~r 1 N~~ *3Y tY The d iscrete part of t he S-matrix is t hen r5rYN(E)= 6N,W,r -i U~~ U~~* orvF~x .(3.8.11 E - E~ ~'iI' ] 9' ~+F /2 1 62 3Scattering Theor y Each term in the sum over r can be thought of as arising from a different resonant state, all these states having the same values for ER and T . What has this to do with rates and cross-sections? For simplicity let's now ignore the non-resonant background scattering, setting ,9'0N1,v equal to4rN; we will come back to the more general case a little later . Then for the two-body discrete center-of-mass states described in the previous section, Eq .(3.8.11)reads : eiF{r'/1Sfix+,J6(5Yi(1:)=6i+f6fSrQAV "3fSbfitrYi r(r) ( r)* E-- E~ +i~'/2 u}rar"'IsI,~, u~~e'sn(3.8.12) In all cases the label r will include an index UR giving the z-component of the total angular momentum of the resonant state ; for a resonant state of total angular momentum jR, GR takes 2jR+Ivalues . If there is no other degeneracy, then r just labels the value of o -R, and ]aVsn .1RA6R,d tin where ~a~sn are a set of complex ampl itudes that (because of the Wigner - Eckart theorem) areindependent of cF.NowEq.(3.5.12.)gives the ampli- tude Sidefined by Eq .(3.7.7)as E - ER +il-'/2 Also, Eq . {3.8.10} now read s Isn(3.8.14) (3.8.15) with the dots representing the positive contribution of and states contain- ing three or more particles . As we shall see, the quantitiesI U~,,12 have the interpretation of branching ratios for the decay of the resonant state into the various accessible two-body states . Eq. (3.7.12)now gives the total cross-section for all reactions in channel n: E)n(2.1R+1) rrn where r,=r f2(3.$.16) (3.8.17) This is a version of the celebrated Breit-Wigner singe-level for mula.27 We can also use these results to calculate the cross-section for resonant scattering from an in itial two-body channel n to a final two-body channel 3.8Resonance s n'. Using Eq, (3 .8.14) in Eq .(3.7.10) gives163 (n E)F_ , ! ER)2 + 1-2/4 k2(2sl +1)(2S2 +1 ) This shows that the probabilities that the resonant state will decay into any one of the final two-body channels n' are proportional to the IF,, . According to Eq .(3.8.15), the sum of the r, (including contributions from final states containing three or more particles) is just equal to the total decay rate T`, so we can conclude that IF, is just the rate for the decay of the resonant state into channel n . We see in Eqs . (3.8.16)and ( 3.8.18)the characteristic resonant peak at energy ER, with a width (the full width at half maximum) equal to the decay rate F .(The individual T', are often called partial widths .) Since Fn<F, the total cross-section at the peak of the resonance is roughly bounded byone square wavelength, (2n/k)2 . This rule, that cross-sections at a single resonance are roughly bounded by a square wavelength, is universally applicable even in classical physics (where energy conservation plays the role played here by unitarity), as for instance in the resonant interaction of sound waves with bubbles in the sea, or gravitational waves with gravitational wave antennae . (In the latter case, the branching ratio for oscillations in any laboratory mass to lose their energy through gravitational radiation is tiny, so the cross-section even at a resonance peak is vastly less then a square wavelength.28} Incidentally, it often happens that a resonance is detected, but energy measurements are insufficiently precise to resolve its width . In this case, what is measured experimentally is the integral of the cross-section over the resonant peak . For the total cross-section (3 .8.14), this i s 2~~{2 jR + 1 ~ ', ~ 2s2+ 1(3.8.19)Gtotai(n;EWE =k2R(f2s1 + 1)fl Such experiments can reveal only the partial width for decay of the resonant state into the initial particles, not the total width or branching ratios . This formalism can also be applied when the resonant states with a given spin z-cornponent form a multiplet related by some symmetry group . For instance, to the extent that isospin symmetry is respected, for a resonance of total isospin TR the index r labelling resonant states includes a specification not only of the angular-momentum z-component GR, but also of an isospin three-component tR, taking values -?'R, -TR+ 1, ...TR In this case there is no change in the above results for the total and partial cross-sections, because each two-body channel n has definite values ti, t2 of the isospin z-components of the two particles, and hence can only couple 164 3 Scattering Theor y to the resonant state with the single value tl +t2 for 4. The partial widths r,, here depend on t Iand t2 only through factors C T,J-,(?'R, tR; t1, t2 )2 . The presence of a resonance shows itself in a characteristic behavior of phase shifts near the resonance . Returning to the general formula (3.8.11) (but still taking Yo=1), we see from Eq . r~-3.S.1U) that for each individual resonant state r, there is an eigenvector u of YNN(E) with eigenvalu e or, in other words, tanp)(E)_~r~E(x.8.20) We see that over an energy range of order IF centered around the resonant energy, the the `eigenphase' ~ (')(E) jumps from a value vn (with v a positive or negative integer) below the resonance to (v +1)7z above it . However, in order to use this result to say something about reaction rates, we need to know the eigenvectors u~), which, in general, have components with arbitrary numbers of particles with various momenta, spin, and species . These results are much more useful in those special cases when the particles in a particular channel N are forbidden (usually by conservation laws) from making a transition to any other channel . With this assumption, it is not difficult to include the effects of a non-resonant background scattering matrix Yo in the general result (3.9.11).In order for YN1Nto vanish for some particular Nand all N'* N, it is necessary that the same is true of YoN+N, and also true of u~~ for any r for which u(") ~ 0 . The unitarity requirement (3.8.5)then requires that for this N ~ONrN= exp (2iSnnr) bN+N andEq. (3.8.10) requires that u(r)*U (3)-b1V N - rs } so that there can b eonly one term r in Eq .(3-8.11)for which u~ ;}*0.In this case, Eq . (3.9.11) gives ir E - ER+ar/2 6N1 N exp(2ibN (E)) with total phase shif t 6N(E)_60N -arctanr/2 (E -ER(3.8.21) We see that over an e nergy range of o rderr entered aroun dthe resonant energy ER, t he phase shift 6N(E)Jumps fro ma val ue ~ONbelow t he Problems 165 resonance to &N + iT a bove it . For instance, as we saw in the p revious section, these assumptions are satisfied in various two-body reactions such as pion-pion and pion-nucleon scattering at energies below the th reshold for producing extra pions, with N incorporating the total and orbital angular-momenta j, 1 (with j = t for pion-pion scattering) and the total angular-momentum z-component ff, as well as the total isospin T and its three-component r . The Wigner-Eckart theorem a llows the phase shifts to depend only on j, rf, and T, not on t or if . There are famous resonances in these channels : in pion-pion scattering there is a resonance at 770 Mehl called p with j = ~ = 1, T = 1, and F = 150 Mel ;in pion-nuc leon scattering there is a resonance at 1232 Mel ca lled d with j = 2 , T = A and F_ 110ta 120MeV . Inspection of Eq . (3.7.12) or Eq . (3.7.18) shows that the total cross- sectio nreaches a peak when the resonant phase shift passes through 7r/2 (or odd-integer multiples of n/2 .) The non-resonant phase shifts are typically rather small, so as we saw earlier, Utotal will exhibit a sharp peak when the phase shift 6,r goes through 7r/2, at an energy close to ER- Hawever, it sometimes happens that the non-resonant background phase shift 6ONis near 7r/2, in which case the cross-section w illexhibit a sharp dip as the phase shift rises through it near ER, due to destructive interference between the resonance and the non-resonant background amp litude . Such dips were first observed by Ramsauer and Townsend29 In 1922, in the scattering of electrons by nob le gas atoms . Problems 1. Consider a t heory with a separable interaction ; tha t is, (cDg, V(D, )=g u~u,*, where g is a real coupling constant, and u,, is a set of complex quantities wit h Use the Lipptna nn-Schwinger eq uation (3 .1.16) to find explicit solu- tions for t he `in' and °out' states and the S-matrix . 2. Suppose that a resonance of spin one is discovered in e+-e- scatterin g at a total energy of 15 0GeV and with a cross-section (in the center - of-mass frame, averaged over initial spins, and summed over fina l spins) for elastic e+-e- scattering at the peak of the resonance equa l to 14-34 cm2 . What is the branching ratio for the decay of th e resonant state R by the mode R --i, e- +e+? What is the total 166 cross-section (in answering scattering .)fore+-e- scattering at the peak of the resonance? both questions, ignore the non-resonant backgroun d 3. Express the differential cross-section for two-body scattering in th e laboratory frame, in which one of the two particles is initially at rest , in terms of kinematic variables and the matrix element M . (Deriv e the result directly, without using the results derived in this chapte r for the differential cross-section in the center-of-mass frame .) 4. Derive the perturbation expansion (3.5.8)directly from the expansion (3.5.3)of old-fashioned perturbation theory . 5.We can define `standing wave' states T,° by a modified version of the Lippmann- ch infer equation r lil l'!IO ttO =rTti, +i~lT I ~!~ YE~ - &3Scattering Th eory Show that the matrix K p,=7r6(E#- E,) (Op,VT,O) is Hermitian . Show how to express the S-matrix in term sof the ,K-matrix . 6. Express the differential cross-section for elastic n+-proton and 7r-- proton scattering in terms of the phase shifts for states of definite total angular momentum, parity, and isospin . 7. Show that the states OEp f, defined by Eq.(3.7.5) are correctly normalized to have the scalar products (3.7.6). References 1. For more details, see M . L. Goldberger and K . M. Watson, Collision Theory (John Wiley &Sons, New York, 1964) ; R.G. Newton, Scattering Theory of Waves and Particles, 2nd edn (Springer- erlag, New York, 1982) . 1a. B. Lippmann and J . Schwinger, Phys .Rev. 79, 469 (1950 )- 2. J. A. Wheeler, P hy1,s. Rev . 52, 1107 (193 7); W. Heisenberg, Z.Phys . 120, 513, 673 (1943) . 3. M,horn, Z.Whys .. 37,863(1926); 38, 803(1926) . 4.C. Moller, Kg1.Danske V idenskah.Wat. Fys .Medd .23, Na 1 (1945) ; 22, No . 19 (194) . Rcjerences 1 67 S. G. D. Rochester and C . C. Butler, Nature160, 855 (1947) . For a historical review, see G . D. Roches ter, in Pions to Quarks - Partic le Physics in the 1950s, ed . by L . Brown, Dresden, and L . Hoddeson (Cambridge University Press, Cambridge, UK, 1989) . 6. G. Breit, E . U. Condon,, and R . S. Present, Phys.Rev.50, 825 (1036) , B. Cassen and E . U. Condon, F hys.Rev.50, 846 (1936) ; G. Breit and E . Feenberg, Whys .Rev.50, 850 (1936) . 7. M. A. Tuve, N . Heydenberg, and L . R. Hafstad, P hys. Re v. 50, 806 (1936) . 8. M.Dell-Mann, Cal . Tech . Synchotron Laboratory Report CTRL-20 (1961) ; Phyrs.Rev.125, 1067 (1962) ; Y. Ne'eman Nucl.Phys. 26, 222 (1961) . 9. See e .g. A.R.Edmonds, Ang ular Momentum in Quantum Mechan- ics(Princeton University Press, Princeton, 1957) : Chapter 3 (where Cj1j2(jM;MrM2)is denotedUi.7?.7InI,1rMi.12MA ;M.E. Rose, Elementary Theory of Ang ularMomentum (John W iley&Sons, New York, 1957) :Chapter III ( where Cj,~~(jm; mIrn2)is denoted C01 j2 i ; MI M2 M A 10. C. Feinberg and S . Weinberg,1V uvvoCimento eerie X, 1 4, 571 (1959) . 11. Chinowsky and J . Steinbrger, P hys, Rev .95, 1561 (1954) ; also see B . Ferret ti,Report oj' an International Cortf~rence on Fundamen- talParticles and Low Temperatures, Cambridge, 194 6 (The Physical Society, London, 1947) . 12. T. D. Lee and C . N. Yang, Pays .Rev.104, 254 (1956) . 13. C. S. Wu et a l., PhyS.Rev.105, 1413 (1957) . 14. R. Darwin, L .Lederman, and M . 11Veinrich, Phys . Rev .105, 1415 (1957) ; T.I. Friedman and V . L. Telegdi, Phys .Rev.105, 1681 (1957) . 15.K. M. Watson Pays . Rev .88, 1163 (1952) . 16. M.Dell-Mann and A . Pais, Ph ys.Rev.97, 1387 (1955) ; also see A . Pais and 0 . Piccioni, Phys . Rev ..100, 1487 (1955 , 17. J. H. Christenson, J . W. Cronin, V L . Fitch, and R . Turlay, Ply=s . Rev. Letters 13, 138 (1964) . 18. K. R. Schubert etal., Phys. Lett . 31B, 662 (1970) . This reference an- alyzes neutral kaon data w ithout the assumption of CPT invariance, and finds that the part of the CP-violating amplitude that conserves 168 3 Scattering Theor y CPS` and violates T has both real and imaginary parts five standard deviations from zero, while the part that conserves T and violates CPT is within a standard deviation of zero . 19. R.H.Dalit z,Phil.Mag.44,1068(1953) ;also see E .Fabri,Nuovo Cimento 1.1,479 (1954). 2Q. F.J. Dyso n,Phys.Rev.75,486,1736 (1949) . 21. See, e .g., L. I. Schiff, Quantum Mechanics, 1st edn (McGraw-Hill, New York, 1949)- Section 19. 22. Th is was first proved in classical electrodynamics . Sep, e . g., H . A. Kramers, Asti Congr . Intern . Fisici, Como, 1 927; repr inted in H . A. Kramers, Collected Scientific Papers (North-Holland, Amsterdam , 1956) . For the proof in quantum mechanics, see E . Feenberg, Whys . Rev. 40, 40 (1932) ; N. Bohr, R . E. Pelerls, and G . Placzek, Nature 144, 200 (1939) . 23.A general version of this argument was given in the late 1960s in unpublished work of C. N. Yang and P . Yang . Also see A . Aharony, in Modern Developments in Thermodynamics (Wiley, Now York, 1973) :pp. 95-114, and references therein . 24. See, e .g.,A.R. Edmonds, Angular Momentum in Quantum Mechanics , (Princeton University Press, Princeton, 1957 : Chapter 2 ; E . Rose, Elementary Theory of Angular Momentum (John Wiley & Sons , New York, 1957) : Appendix III ; L. D. Landau and E.M. Lifshitz , Quantum Mechanics -Non R elativi sticTheory, 3rd edn (Pergamo n Press, Oxford, 1977):Section 2 8. 25. E. P. Wigner, Gruppentheorie (Friedrich Vieweg and Sohn, Braun- schweig, 1931) ; C. Eckart, Rev. Mod .Phys, 2, 305 (1930) . 26. M. Froissart, Phys .Rev. 123, 1 053(1961). 27. C. Breit and E . P. Wigner, Phys .Rev.49, 519 (1936). 28. See, e .g.,S.Weinberg, Gravitation and Cosmology (Wiley, New fork, 1972) :Section 1 0.7. 29. R. Kollath, Phys . Veit . 31, 985 (1931) . The Cluste r Decomposition Principl e Up to this point we have not had much to say about the detailed structure of the Hamiltonian ❑perator H . This operator can be defined by giving all its matrix elements between states with arbitrary numbers of particles . Equivalently, as we shall show hers, any such operator may be expressed as a function of certain operators that create and destroy single particles . We saw in Chapter 1 that such creation and annihilation operators were first encountered in the canonical quantization of the electromagnetic field and other fields in the early days of quantum mechanics . They provided a natural formalism for theories in which massive particles as well as photons can be produced and destroyed, beginning in the early 1930s with Fermi's theory of beta decay . However, there is a deeper reason for constructing the Hamiltonian out of creation and annihilation operators, which goes beyond the need to quantize any pre-existing field theory like electrodynamics, and has nothing to do with whether particles can actually be produced or destroyed . The great advantage of this formalism is that if we express the Hamiltonian as a sum of products of creation and annihilation operators, with suitable non-singular coefficients, then the S-matrix will automatically satisfy a crucial physical requirement, the cluster decomposition principle,l which says in effect that distant experiments yield uncorrelated results . Indeed, it is for this reason that the formalism of creation and annihilation operators is widely used in non-relativistic quantum statistical mechanics, where the number of particles is typically fixed . In relativistic quantum theories, the cluster decomposition principle plays a crucial part in making field theory inevitable . There have been many attempts to formulate a relativistically invariant theory that would not be a local field theory, and it is indeed possible to construct theories that are not field theories and yet yield a Lorentz-invariant S-matrix for two-particle scattering,2 but such efforts have always run into trouble in sectors with more than two particles : either the three-particle S-matrix is not Lorentz-invariant, or else it violates the cluster decomposition principle . In this chapter we will first discuss the basis of states containing ar- 169 170 4 The Cluster Decomposition Principl e bitrary numbers of bosons and ferrnians, then define the creation and annihilation operators, and finally show how their use facilitates the construction of Hamiltonians that yield 5-matrices satisfying the cluster decomposition condition . 4.1 Bo sonsand Fermions The H ilbert space of physical states is spanned by states conta ining 0, 1, 2, - - - free pa rticles . These can be free-particle states, or `in' s tates, or out' states ; for definiteness we shall deal here with the free-particle states Fif7in]M2 C72n_,-.# but all our results will apply equally to `in' or `out' states . As usual, a labels spin z-components (or helicities, for massless particles) and n labels particle species . We must now go into a matter that has been passed over in chapter 3 ; the symmetry properties of these states . As far as we know, all particles are either bosons or .fermions, the difference being that a state is unchanged by the interchange of two identical bosons, and changes sign under the interchange of two identical fermions . That i s with an upper or lower sign if n is a boson or a fermion, respectively, and dots representing other particles that may be present in the state . (Equivalently, this could be stated as a condition on the `wave functions,' the coefficients of these multi-particle basis vectors in physically allowable state-vectors .) These two cases are often referred to as Bose or Fermi `statistics' . We will see in the next chapter that Bose and Fermi statistics are only possible for particles that have integer or half-integer spins, re- spectively, but we shall not need this information in the present chapter . In this section we shall offer a non-rigorous argument that all particles must be either bosons or fermions, and then set up normalization conditions for multi-boson or multi-fermion states . First note that if two particles with spins and momenta p, u and p%cr' belong to identical species n, then the state-vectors (D .,,p,... pf ufn ...and 0... p+ c{ n ... p cr n .. represent the same physical Mate ; if this were not the case then the particles would be distinguished by their order in the labelling of the state-vector, and the first listed would not be identical with the second . Since the two state-vectors are physically indistinguishable, they must belong to the same ray, and s o where i,, is a complex number of unit absolute value . We may regard this as part of the definition of what we mean by identical particles . 4.1 Bosons and Fermions 171 The crux of the matter is to decide on what the phase factor a, may depend . If`it depends o nlyon the species index n, t henweare nearly done . Interchanging the two particles in Eq.(4.1.2)again,we fin d z4). p'rr'n -..an~.,.Pdrr,..p,17,n so that a~ = 1, yielding Eq . (4.1.1) as the only two possibilities . On what else could a, depend? It might depend on the numbers and species of the other particles in the state (indicated by dots in Eqs. (4.1.1) and (4 .1.2}), but this would lead to the uncomfortable result that the symmetry of state-vectors under interchange of particles here on earth may depend on the presence of particles elsewhere in the universe . This is the sort of thing that is ruled ❑ut by the cluster decomposition principle, to be discussed later in this chapter . The phase x, cannot have any non-trivial dependence on the spins of the two particles that are interchanged, because then these spin-dependent phase factors would have to furnish a representation of the rotation group, and there are no non-trivial representations of the three-dimensional rotation group that are one-dimensional - that is, by phase factors . The phase g,, might conceivably depend on the momenta of the two particles that are interchanged, but Lorentz invariance would require o~n to depend only on the scalar p 2p; this is symmetric under interchange of particles I and 2, and therefore such dependence would not change the argument leading to the conclusion that a~ = 1 , The logical gap in the above argument is that (although our notation hides the fact) the states (Dpi cT, n,pz t7zn,--- may carry a phase factor that depends on the path through momentum space by which the momenta of the particles are brought to the values pi, p2, etc . In this case the interchange of two particles twice might change the state by a phase factor, so that ocn 1 . We will see in Section 9 .7 that this is a real possibility in two-dimensional space, but not for three or more spatial dimensions . What about interchanges of particles belonging to different species? If we like, we can avoid this question by simply agreeing from the beginning to label the state-vector by listing all photon momenta and helicities first, then all electron momenta and spin z-components, and so on through the table of elementary particle types . Alternatively, we can allow the particle labels to appear in any order, and define the state-vectors with particle labels in an arbitrary order as equal to the state-vector with particle labels in some standard order times phase factors, whose dependence on the interchange of particles of different species can be anything we like. In order to deal with symmetries like isospin invariance that relate partic les of different species, it is convenient to adopt a convention that generalizes Eq . (4.1.1): the state-vector will be taken to be symmetric 172 4 The Cluster Decomposition Principl e under interchange ❑f any bosons with each other, or any bosons with any fermions, and antisymmetric with respect to interchange of any two fermions with each other, in all cases, whether the particles are of the same species or na# .' The normalization of these states must be defined in consistency with these symmetry conditions . To save writing, we will use a label q to denote all the quantum numbers of a single particle : its momentum, p, spin z-component (or, for massless particles, helicity) (r, and species n . The N-particle states are thus labelled (Dqi. ,. qN(with N = 0 for the vacuum state (Do.)For N= 0 and =1 the question of symmetry does not arise : here we have (0n,(Do)=1 and(4.1.3) where 6(q'-q )is a product of all the delta functions and Kronecker deltas for the particle's quantum numbers , 6 (q'- q) =P(Pf- p) 5,f~bWn. (4.1.5) On the other hand, for =2 the states ~qlq~and ~'q~ are physically the same, so here wemust take (q;q2'7(Dqlq2 _6( 4i- cat ) 6(Ri-q2)± 6( R2 - 4i ) b(~'i-42)(4.1.6) the sign ± being - if both particles are fermions and + otherwise . This obviously is consistent with the above stated symmetry properties of the states . More generally , The sum here is over all permutations ~)"of the integers 1, 2, ... ,N. (For instance, in the first term in Eq. (4.1.6), ' is the identity, Y1 = 1,Y2 = 2, while in the second term Y1=2,Y2 = 1 .)Also, bgis a sign factor equal to -1 if 9/ involves an odd permutation of fermions (an odd number of fermion interchanges) and +1 otherwise . It is easy to see that Eq. (4.1.7) has the desired symmetry or antisymmetry properties under interchange of the q j, and also under interchange of the q j. In fact, by the same reasoning, the symmetry or antisyrnrnetry of the state-vector under inter- change of particles of the same species but different heliczkies or spin a-components is purely conventional, because we could have agreed from the beginning to list first the momenta of photons of helicity +1, then the momenta of all photons of helicity -1, then the momenta of all electrons of spin z-component +~, and so on . We adopt the cr~nuentfnM that the state-vector is symmetric or antisymmetric under interchange of identical bosons or fermians of diftrent bell-cities or spin z-components in order to facilitate the use of rotational invariance . 4.2 Creation and Annihilation Operators 173 4.2 Creation and Annihilation Operator s Creation and annihilation operators may be defined in terms of their effect on the normalized multi-particle states discussed in the previous section . The creation operator at(q) (or in more detail, at(p, a, n)] is defined as the operator that simply adds a particle with quantum numbers q at the front of the list of particles in the stat e at(4)(Dq,qz...q.-q)qqjq2...qN. (4 .2.1) In particular, the N-particle state can be obtained b yacting on the vacuum with Ncreation operator s It is conventional for this operator to b ecalled at (q); its adjoint, which is then called a(q), may be calculated from Eq . (4.1.7). As we shall now show, ❑(4) removes a particle from and state on which it acts, and is therefore known as an annihilation operator .In particular, when the particles q ql ,.. qNare either all bosons or all fermions, we hav e r=1 with a +1or -1 sign forbosons orfermivns, respectively .(Here is the proof . We want to calculate the scalar product of a (q )(Dqjq2 ...qNwith an arbitrary state qi...q~. Using Eq .(4.2.1),this i s We now use Eq.(4.1.7). The sum over permutations 01of1,2,...,N can be written as a sum over the integer r that is permuted into the first place, i .e. gr = 1, and over mappings ]' of the remaining integers 1,... , r - 1, r + 1, ..  ,Ninto 1,  - -,N-- 1. Furthermore, the sign factor i s with upper and lower signs for bosons and fermions, respectively . Hence, using Eq . (4.1.7) twice, N M x N _6jV,M+1q1,..qlM?0ql"'.9r-1qj,+1.,.qN r=1 174 4 The ClusterDecomposition Principl e Both sides of Eq . (4.2.3) thus have the same matrix element with any state 1D,7F,..q~, and are therefore equal, as was to be shown . As a special case of ~Ec~ . (4.2.3), we note that for both bosons and fermions, a(q) annihilates the vacuum a(q)(Do=0. (4.2.4) As defined here, the creation and annihilation operators satisfy an im- portant commutation or anticommutation relation . Applying the operator a(q')to Eq- (4 .2-1) and using Eq . (4.2.3) give s 'IV ti r=1-qr~(DqtIl..,qr-Iqr-1,..t1.ti` (The sign in the second term is (±),,+2 because cox is in the (r + 1)-th place in(N. j,.,q.k-)On the other hand, applying the operator at(q) to Eq . (4.2.3) gives a'(R)a(qr)(1)q,...qNN r=l Subtracting or adding, we have the n [a(q')a*(q) :~a1(q)a(q')1 (Vqj-qjv 6(q'-q)(Dql ...qN ' But this holds for all state scDq,..,.,,; (and may easily be seen to hold also forstates contain ingboth bosons and fermions )and therefore implie sthe operator relat ion In addition, Eq . (4.2.2) gives immediatel y and so also(42s ) (4.2.6) (4.2.7) As always, the top and bottom signs apply for bosons and fermions, respectively . According to the conventions discussed in the previous section, the creation and/or annihilation operators for particles of two different species commute if either particle is a boson, and anticomrnute if both are fermions . The above discussion could have been presented in reverse order (and in most textbooks usually is) . That is, we could have started with the commutation or anticomrnutation relations Eqs . (4.2.5)-(427), derived from the canonical quantization of some given field theory . Multi-particle 4.2 Creation and Annihilation Operators 175 states would have then been defined by Eq.(4.2.2), and their scalar products Eq . (4.1.7) derived from the commutation or anticammutation relations . In fact, as discussed in Chapter 1, such a treatment would be much closer to the way that this formalism developed historically . We have followed an unhistorical approach here because we want to free ourselves from any dependence on pre-existing field theories, and rather wish to understand why field theories are the way they are . We will now prove the fundamental theorem quoted at the beginning of this chapter : any operator 6'may be expressed as a sum of products of creation and annihilation operator s N=O?4t=o xCNkr(qi...q~vRj...q±tir) (4.2.8) That is, we want to show that the C'NAf coefficients can be chosen to give the matrix elements of this expression any desired values . We do this by mathematical induction . First, it is trivial that by choosing Coo properly, we can give ((Do, 60p) any desired value, irrespective of the values of C N,u with N>0and/or M > 0. We need only use Eq . {4_2 .4} to see that Eq. (4.2.8) has the vacuum expectation valu e Now suppose that the same is true for all matrix elements of 6between N-and M-particle states, with N L, :!~K or N~ L, M K; that is, that these matrix elements have been given some desired values by an appropriate choice of the corresponding coefficients Cvm . To see that the same is then also true of matrix elements of G between any L- and K-particle states, use Eq.(4.2.8) to evaluat e +terms involving C NMwith N<L, Nl SK or N < L, A4 K Whatever values have already been given to C NM with N<L, M K orN<L,Al K, there is clearly some choice of C LKwhich gives this matrix element any desired value . Of course, an operator need not be expressed in the form (4 .2.8), with all creation operators to the left of all annihilation operators . (This is often called the 'normal' order of the operators .) However, if the formula for some operator has the creation and annihilation operators in some other order, we can always bring the creation operators to the left of the annihilation operators by repeated use of the commutation or 176 4 The Cluster Decomposition Principl e anticommutation relations, picking up new terms from the delta function inEq. (4.2.5). For instance, consider any sort of additive operator F (like momentum, charge, etc .) for whic h F(Dql...q1V_(f (4i) +... +f(qnr))(Dq,...qN. (4 .2.9) Such an operator can be written as in Eq . (4.2.8), but using only the term with N=M- 1: In particular, the free-particle Hamiltonian is alway s Ho==f dq a~(q)a(q)E(q ) where E(q) is the single-particle energy(4.2.10) (4.2.21) We will need the transformation properties of the creation and annihila- tion operators for various symmetries, First, let's consider inhamogeneous proper orthochronous Lorentz transformations . Recall that the N-particle states have the Lorentz transformation propert y UO(A, a)(DPI011n1,P2(72n2,-. PPIP2 ,(W(Ap)D/ .( FF(LLp,)) x L6II0- al Qty... ... F1 .4IT1n],p2nQ2n2, Here PA is the three-vector part of lip, D~j~(R) is the same unitary spin-j representation of the three-dimensional rotation group as used in Section 2.5, and W(A,p) is the particular rotatio n where L(p) is the standard `boot' that takes a particle of mass m from rest to four-momentum p ". (Of curse, mand j depend on the species label n. This is all for m 0; we will return to the massless particle case in the following chapter .) Now, these states can be expressed as in Eq . (4.12) where (DDis the Lorentz-invariant vacuum state 4.3 C lusterDecomposition and Connected Amplitudes 17 7 In order that the state (4 .2.2) should transform properly, it is necessary and sufficient that the creation operator have the transformation rul e Up(l~,a)al(p ffn) U6-1 (A, OL)= 8 r(Ap)x~(Ap)G/p l x Da(W {A + P)) al (PACr ( 4.2.12) In the same way, the operators C, P, and T, that induce charge-conjugation, space inversion, and txxne-reversal transformations on free particle states' transform the creation operators as : Put(po-n)PTi = i}, at(--- pa n) , (4 .2.14) Tal(Pa n)T-1 = ~, ( -I)j-dal (- p - a n), (4.2.15) As mentioned in the previous section, although we have been dealing with operators that create and annihilate particles in free-particle states, the whole formalism can be applied to `gin' and 'out' states, in which case we would introduce operators aM and a,,ut defined in the same way b y their action on these states . These operators satisfy a Lorentz transforma- tion rule just like Eq . (4.2.12), but with the true Lorentz transformation operator U(A, oc) instead of the free-particle operator Uo( 11,a). 4.3 Cluster Decomposition and Connected Amplitude s It is one of the fundamental principles of physics (indeed, of all science) that experiments that are sufficiently separated in space have unrelated results . The probabilities for various collisions measured at Fermilab should not depend on what sort of experiments are being done at CERN at the same time, If this principle were not valid, then we could never make any predictions about any experiment without knowing everything about the universe . In S-matrix theory, the cluster decomposition principle states that if multi-particle processes al ` #I , ac2`fl2s... a aAf --* fl ..yare studied in K very distant laboratories, then the S-matrix element for the overall proces s We omit the subscri pt '0' on t hese operators, beca use in v irtual ly allcases w here C, P, and/or Tare conserved, the operators tha tinduce t hese transformations o n'in' a nd 'out' states are the same as t hose dofined by their actio non fr ee-partic le states .This is not the case forcontinuous Lorentz transformations, for w hich it is nec essary to disti nguish between t he operators U(A,a) and U60, a) . 178 4 The Cluster Decomposition Principl e factorizes .That i s,* if for all i ~ j, all❑f the particles in states ocj and /i iare at a great spatial distance from allof the particles in states uj and flj. This factorization of S- matrix elements will ensure a factorization of the corresponding transition probabilities, corresponding to uncorrelated experimental results . There is a combinatoric trick that allows us to rewrite Eq . (4.3.1) in a more transparent way . Suppose we define the connected part of the S-matrix, Sc, by the formula " sflit SC SC (4.3.2)fil-Itl V7 2 PAR T Here the sum is over all different ways of partitioning the particles in the state ac into clusters Xi, oc2, - .., and likewise a sum over all ways of partitioning the particles in the state flinto clusters fi1, fl2, , not counting as different those that merely arrange particles within a given cluster or permute whole clusters . The sign is + or -according to whether the rearrangements a -r xj,)c2 ... and #-}fl 102... involve altogether an even or an odd number of fermion interchanges, respectively . The term `connected' is used because of the interpretation of S('in terms of diagrams representing different contributions in perturbation theory, to bediscussed in the next section . This is a recursive definition . For each i and 11, the sum on the right- hand side of Eq.(4.3.2) consists of a term 5~, plus a sum I' over products of two or more SI-matrix elements, with a total number of particles in each of the states aj and #jthat is less than the number ofparticles i n ' We are here returning to the notation used in Chapter 3 ; Greek letters a or {3 stand for a collection of particles, including for each particle a specification of its momentum, spin, and species .AIR) .x,+ a-)+-  + cc .,4- is the state formed by combining all the particles in the states . x l , a2, .. ,and x .t. ,and likewise for fli + f32 + ...+fl.y. " This decomposition has been used in classical statistical mechanics by Ur5e11, Mayer, and others, and in quantum statistical mechanics by Lee and Yang and others .3 It has also been used to calculate many-body ground state energies by Goldstone 4 and Hugenholta .5In all of' these applications the purpose ofisolating the connected parts ofGreen's functions, partition functions, resolvents, etc,, is to deal with objects with a simple volume dependence . This is essentially our purpose ton, because as we shall see, the crucial property of the connected parts ofthe S-matrix is that they are proportional to a single momentum-conservation delta function, and in a box the delta function becomes a Kronecker delta times the volurnc . The cluster decomposition is also the same formal device as that used in the theory of naise6to decompose the correlation function of several random variables into its 'cumulants' ; if' the random variable receives contributions from a large number N of independent fluctuations, then each cumulant is proportional to N, 4.3 Cluster Decomposition and Connected Amplitude s the states aand # PART179 Suppose that the S`-matrix elements in this sum have already been chosen in such a way that Eq . (43 .2) is satisfied for states fl, ~c containing together fewer than, say, Nparticles . Then no matter what values are found in this way for the 5-matrix elements appearing in the sum E', we can always choose the remaining term 5 Cso that Eq . (4.3.2) is also satisfied for states a, fl containing a total of Nparticles . Thus Eq . (4.3.2) contains no information in itself ; it is merely a definition of SC. Ifthe states x and li each consist of just a single particle, say with quantum numbers q and q' respectively, then the only term on the right- hand-side of Eq . (4.3.2) is just S~ 'itself, so for one-particle state s c (Apart from possible degeneracies, the fact thatS.rqis proportional to (q' - q)follows from conservation laws . The absence of any proportion- ality factor in Eq.(4.3.3) is based on a suitable choice of the relative phase of `in' and `nut' states .) We are here assuming that single-particle states are stable, so that there are no transitions between single-particle states and any others, such as the vacuum . For transitions between two-particle states, Ey . (4.3.2) read s S11~q 7,~~~~z - ~~;~~,~~~~+ 6(qi -~I1)6(q~ --c~?±b (~fi-R?)6(q2-q1). (4.3.4) (We are here using Eq . (4.3.3).) The sign ± is - if both particles are fermions, and otherwise + . We recognize that the two delta function terms just add up to the norm (4 .1.6), so here Spa is just (S -I)#,,.But the general case is more complicated . For transitions between three-particle or four-particle states, Eq . (4.3.2) reads ~+ c`1q', q ~q~~qit~2qt ~q,q~R3q1q2q3 +5(q i - q l)S~zq~.qzq3+ permutation s +b(qi - gl)cS(q~ - q 2)6(q~ --q3)± permutations (4.3.5) ~ A technicality should be mentioned here . This argument works only if we neglect the possibility that for one or more of the connected S-matrix elements in Eq_ (4,3 .2), the states x1 and ti~ both contain no particles aIa11.We mist therefore de#2ne the cnlincued vacuum-vacuum element Soo to he zero . We do not use Eq.{43.2} for the vacuum vacuum S-matrix So .o, which in the absence of time-varying external fields is simply defined to be unity, 5 O,()= t. We will have more to say about the vacuum vacuum amplitude in the presence of external fields in Volume II . 180 4 The ClusterDecomposition Principl e and Sc S~Ii~~q'q~,Riqaq3Ra RtqzR34~ ,4i4z934a + Sc q',4~R2S~ R,R3 q4+permutations 12 3 4 +6(qi - q l)S~ ~3~~,~~~~~a ± permutation s +6(qi -q1)6(~`2 - 4 2)S3~~, q3q4+permutation s +6(qt - q l)6(q2 --~- q2) 6(q~ - q 3)6(q~ - q 4)± permutations . (4 .3.6) (Taking account of all permutations, there are a total of 1 + 9 + 6= 16 terms in Eq . (4.3.5) and 1 + 18 + 16+ 72 + 2 4=131 terms in Eq . (4.3.6). If we had not assumed that one-particle states are stable, there would be even more terms .) As explained previously, the definition of S~' recursive : we use Eq. (4.3.4) to define S Cfor two-particle states, then use this definition in Eq . (4.3.5) when we define S~,!for three-particle states, then use both of these definitions in Eq. (4.3.6) to obtain the definition of ,5~ for four-particle states, and so on . The point of this definition of the connected part of the S-matrix is that the cluster decomposition principle is equivalent to the requirement that S~ must vanish when any one or more of the particles in the states ~ and/or cc are far away in space from the others .t To see this, suppose that the particles in the states Pand a are grouped into clusters fil, P2, -{  and 11, act, ` ' ' ,and that all particles in the set a ; +flz are far from all particles in the set act + Pjfor any j i. Then if 5 pc.yvanishes if any particles in Xor ac' are far from the others, it vanishes if any particles in these states are in different clusters, so the definition (4 .3.2) yield s ~Ply___ ( 1)( ±) S #iiaii~9iyXix...X 1 : (2) ( +) S21a21S 022222X... ~(4.3.7) where 1Wis a sum over all different ways of partitioning the clusters X31 and a j into su bclusters Pji,flj2,... and ac jl, ccj2,.., But referring back to Eq. (4.3.2), this is just the desired factorization property (4 .3.1). For instance, suppose that in the four-particle reaction 1234 --+1'2`3'4', we let particles 1, 2, 1', and 2' be very far from 3,4J, and 4' . Then if SC vanishes when and particles an #and/or a are far from the others, the only terms in Eq . (4.3.6) that survive (in an even more abbreviated notation) ar e In order to give a meaning to 'far, we wil lhave to Fou rier tra nsform SC, so t hat each three- momentum label p i s rep laced wi tha spatia lcoordinate th ree-vector x . 4.3ClusterDecomposition and Connected Amplitudes 181 S1'2'3'4',1234 ~ SIC'21,12S3C1 4',34 + (61'l62f2± + (436q4 ±61'26Z'0"J4',34 63'443)S 1'2',12 61'A'1)(63'3&4'4 ± 63'443 ) Comparison with Eq. (4.3.4) shows that this is just the required factvriza- tion condition (4 .3.1) S1'2'3r4',1234 --* S 1'2',12S3'4',34 We have formulated the cluster decomposition principle in coordinate spare, as the condition that S Cvanishes if any particles in the states fl or a are far from any others . It is convenient for us to reexpress this in momentum space . The coordinate space matrix elements are defined as a Fourier transfor m Sx~...,~Jx2.,.=Jd3p'td3p 2...d~p1d3p2...SFIp 2...PiPz,.. (We are here temporarily dropping spin and species labels, which just go along with the momentum or coordinate labels .) If I SC'2'_'P1~ wereP1PP a.,. sufficiently well behaved (to be specific, if it were Lebesgue integrable) then according to the Riemann-Lebesgue theorem7 the integral (4 .3.8) would vanish when any combination of spatial coordinates goes to infinity . Now, this is certainly too strong a requirement . Translational invariance tells us that the connected part of the S-matrix, like the S-matrix itself, can only depend on differences of coordinate vectors, and therefore does not change at all if all of the xi and x') vary together, with their differences held constant . This requires that the elements of S'in a momentum basis must, like those ofS, be proportional to a three-dimensional delta function that ensures momentum conservation (and makes ISp P' ..plP2'.'~ got Lebesgue integrable), as well as the energy-conservation delta function required by scattering theory . That is, we can writ e This is no problem : the cluster decomposition principle only requires that Eq. (4.3.8) vanish when the differences among some of the xi and/or x4 become large . However, if Citself in Eq . (4.3.9) contained additional delta functions of linear combinations of the three-momenta, then this principle would not be satisfied . For instance, suppose that there were a delta func- tion in Cthat required that the sum ❑f the g ~ and - pjfor some subset of the particles vanished . Then Eq . (4.3.8) would not vary if all of the x' and 182 4 The Clu sterDecomposition Principl e x1 for the particles in that subset moved together (with constant differ- ences) away from all the other x~ and x~, in contradiction to the cluster decomposition principle . Loosely speak ing then, the cluster decomposition principle simply says that the connected part of the 5-matrix, u nlike the S-matrix itsetf ; contains just a single momentum-conservation delta Junction . In order to put this a bit more precisely, we can say that the coefficient function CPrPf,,.,,Pip_... in Eq .4.3.9 is a smoo th function of its momentum labels . But how smooth? It would be most straightforward if we could simply requ ire that C'~~yy, .,,~~n~... be analytic in all of the momenta at p' _ pz pi = P2 0 . This requirement would indeed guarantee that Sx~i s , ' ...,XiX2.., vanishes exponentially fast when any of the x and x' is very distant from any of the other x and x' . However, an exponential fall- off of S Cis not an essential part of the cluster decomposition principle, and, in fact, the requirement of analyticity is not met in all theories . Most notably, in theories with massless particles, S' can have poles at certain values of the pand p'. For instance, as we will see in Chapter 10, if a massless particle can be emitted in the transition I --+ 3 and absorbed in the transit ion 2 --* 4, then 5~ .12 will have a term proportional to 1 API - p3 )2. After Fourier transforming, such poles yield terms in x X2---,Xlx2-- . that fall off only as negative powers of coordinate differences .' There is no need to formulate the cluster decomposition principle so stringently that such behavior is ruled out . Thus the `smoothness' condition on SC should be understood to allow various poles and branch-cuts at certain values of the pand p', but not singularit ies as severe as delta functions . 4.4 Structure of the Interactio n We now ask, what sort of Hamiltonian will yield an S-matrix that sat- isfies the cluster decomposition principle? It is here that the formalism of creation and annihilation operators comes into its non . The answer is contained in the theorem that the S-matrix satisfies the cluster decompo- sition principle if (and as far as I know, only if)the Hamiltonian can be expressed as in Eq. (4.2.8): ~~ . ~ Jdqi...dqnrdql... ~~w N=o M=O x a~(q1)...a*(q N)a(RM). a(q1 ) xhNM(q i... qnr,Ri... qnf) {4.4.1} with coefficient functions hNMthat contain just a single three-dimensional momentum-conservation delta function (returning here briefly to a more 4.4Structure of't heInteraction 183 explicit notation ) xhN~w(P1(71Uzi' ' "~~~~~nN,AiHirai' ' 'P?~G'Mnm }, ( 4.4.2) where h NM contains no delta function factors . Note that Eq . (4.4.1 ) by itself has no content -- we saw in Section 4 .2 that any operator can be put in this form . It is only Eq . (4.4.1 ) combined with the requirement that hNMhas only the single delta function shown in Eq . (4.4.2) that guarantees that the S-matrix satisfies the cluster decomposition principle . The validity of this theorem in perturbation theory will become obvious when we develop the Feynman diagram formalism in Chapter 6 . The trusting reader may prefer to skip the rest of the present chapter, and move on to consider the implications of this theorem in Chapter 5 . However, the proof has some instructive features, and will help to clarify in what sense the field theory of the next chapter is inevitable . To prove this theorem, we make use of perturbation theory in its time- dependent form . (One of the advantages of time-dependent perturbation theory is that it makes the cornbinatorics underlying the cluster decom- position principle much more transparent ; if E is a sum of one-particle energies then e-aElis a product of functions of the individual energies, while [E - E .x + i e]-1 is not .) The S-matrix is given b yEq.(3.5.10)as' ~~~= , ~ r~.H)~~~ fdtl...dtn(DTfl, ''(V(ti) ' 'V(tM)cDa {4 .4.3} ,:=n where the Hamiltonian is split into a free-particle part Ha and an interac- tion V, and V(f) =_exp(Wt~ t)Vex p(--iH ot). (4 .4.4) Now, the states (D.,and c D#may be expressed as in Eq.(4.2.2) as products of creation operators acting on the vacuum c D4, and V(t) is itself a sum of products of creation and annihilation operators, so each term in the sum (4.4.3) may he written as a sum of vacuum expectation values of products of creation and annihilation operators . By using the commutation or anticommutation relations (4 .2.5) we may move each annihilation operator in turn to the right past all the creation operators . For each annihilation operator moved to the right past a creation operator we have two terms, as shown by writing Eq . (4.2.5) in the for m We are now adapting the convention that for n = 0, the time-ordered product in F .q. (4,4 .3) is taken as the unit operator, so the n =0term in the sum just yields the term ri( #- x)inS. 184 4 The Cluster Decomposition Principl e Moving other creation operators past the annihilation operator in the first term generates yet more terms . But Eq.(4,2,4) shows that any annihilation operator that moves all the way to the right and acts on (Dogives zero, so in the end all we have left is the delta functions . In this way, the vacuum expectation value of a product of creation and annihilation operators is given by a sum of different terms, each term equal to a product of delta functions and ± signs from the commutators or anticommutators . It follows that each term in Eq.(4.4.3) may be expressed as a sum of terms, each term equal to a product of delta functions and ±signs from the commutators or anticommutators and whatever factors are contributed by V(r), integrated over all the times and integrated and summed over the momenta, spins, and species in the arguments of the delta functions . Each of the terms generated in this way may be symbolized by a diagram . (This is not yet the full Feynman diagram formalism, because we are not yet going to associate numerical quantities with the ingredients in the diagrams ; we are using the diagrams here only as a way of keeping track of three-momentum delta functions .) Draw n points, called vertices, one for each V(t) operator . For each delta function produced when an annihilation operator in one of these V(t) operators moves past a creation operator in the initial state (D, draw a line coming into the diagram from below that ends at the corresponding vertex . For each delta function produced when an annihilation operator in the adjoint of the final state Opmoves past a creation operator in one of the V(t},draw a line from the corresponding vertex upwards out of the diagram . For each delta function produced when an annihilation operator in one V(t) moves past a creation operator in another V(t) draw a line between the two corresponding vertices . Finally, for each delta function produced when an annihilation operator in the adjoint of the final state moves past a creation operator in the initial state, draw a line from bottom to top, right through the diagrams . Each of the delta functions associated with one of these lines enforces the equality of the momentum arguments of the pair of creation and annihilation operators represented by the line. There is also at least one delta function contributed by each of the vertices, which enforces the conservation of the total three-momentum at the vertex . Such a diagram may be connected (every point connected to every other by a set of lines) and if not connected, it breaks up into a number of connected pieces . The V(t) operator associated with a vertex in one connected component effectively commutes with the V(t) associated with any vertex in any other connected component, because for this diagram, we are not including and terms in which an annihilation operator in one vertex destroys a particle that is produced bya creation operator in the other vertex if we did, the two vertices would be in the same connected 4.4Structure of the Interaction 185 component . Thus the matrix element in Eq . (4.4.3) can be expressed as a sum over products of contributions, one from each connected component : ((Dfl, TIV(ti) ...V(t,) I 10C,) m clusterings j=1((Dpj, Tf V(tp) ...V(tj.) I(D,,j ), :7. (4A,5 ) Here the sum is over all ways of splitting up the incoming and outgoing particles and V(t) operators into v clusters including a sum over v from Ito n) with the nj operators V (tj,)... V(t3 ,})and the subsets of initial particles a} and final particles fljall in the jth cluster .Of course, this means that n=n1 + ...+nV and also the set Lx is the union of all the particles in the subsets al, ac?, ... xV, and likewise for the final state . Some of the clusters in Eq . (4.4.5) may contain no vertices at all, i .e., nj= 0; for these factors, we must take the matrix element factor in Eq . (4.4.5) to vanish unless Pjand act are both one-particle states (in which case it is just a delta function 6(aj - flj)}, because the only connected diagrams without vertices consist of a single line running through the diagram from bottom to top . Most important, the subscript C in Eq . (4.4.5) means that we exclude any contributions corresponding to disconnected diagrams, that is, any contributions in which any V(t) operator or any initial or final particle is not connected to every other by a sequence of particle creations and annihilations . Now let us use Eq . {4.4.5} in the sum (4 .4.3). Every time variable is integrated from -ao to +cxD, so it makes no difference which of the ti, ... to are sorted out into each cluster . The sum over clusterings therefore yields a factor u ! fni ?n2? ---n,!, equal to the number of ways of sorting out n vertices into v clusters, each containing n> > n2,  .. vertice s 'CK~' dt i...dtn((DP!' Tf V(ti) V(tn))(D~~) .~ova nr i'2'n,! 00PART nl +~ y j :=1 ni+...+n, ~ n x((D#,, TIV(tj, )...V(ti,)J(D,), - The first sum here is over all ways of partitioning the particles in the initial and final states into clusters al ... cc, and X31 ...P,(including a sum over the number v of clusters) . The factor n! here cancels the 1/ref in Eq. (4.4.3), and the factor (----fi)l in the perturbation series for (4 .4.5) can be written as a product (--i)", ... so instead of summing over n and 186 4 The Cluster Decomposition Principl e then summing separately over ni, ... n,, constrained by n1 +... + rzw = n, we can simply sum independently over each ral, ... nV. This gives finall y PART j=1nf=O i X(00J,1 l Comparing this with the definition (4 .2.2) of the connected matrix elements 5fic., we see that these matrix elements are just given by the factors in the product her e «~ ~ n x S~n dt, dt, (Op, T ~ V(t 1)... V(tn)I(D,)c (4 .4.6) n=O00 (The subscript jis dropped on all the t s and n s, as these are now mere integration and summation variables .) We see that SCE is calculated bya very simple prescription : S~~ is the sum of all contributions to the S-matrix that areconnected, in the sense that we drop all terms in w hich any initial orfinal partite or any operator V(t) isnot connected to all theothers bya sequence of particle creations and annihilations .This justifies the adjective `connected' for Sc- As we have seen, momentum is conserved at each vertex and along every line, so the connected parts of the S-matrix individually conserve momentum : S~ contains a factor 5-~(pfl-p,,).What we want to prove is that SC:contains no other delta functions . We now make the assumption that the coefficient fractions hNM in the expansion (4 .4.1) of the Hamiltonian in terms of creation and annihilation operators are proportianai to a single three-dimensional delta function, that ensures momenta conservation . This is automatically true for the free-particle Hamiltonian HO~, so it is also then true separately for the interaction V . Returning to the graphical interpretation of the matrix elements that we have been using, this means that each vertex contributes one three-dimensional delta function . (The other delta functions in matrix elements V, 6simply keep the momentum of any particle that is not created or annihilated at the corresponding vertex unchanged .) Now, most of these delta functions simply go to fix the momentum of intermediate particles . The only momenta that are left unfixed bysuch delta functions are those that circulate in loops of internal lines . (Any line which if cut leaves the diagram disconnected carries a momentum that is fixed by momentum conservation as some linear combination of the momenta of the lines coming into or going out of the diagram . If the diagram has L lines that can all be cut at the same time without the diagram becoming disconnected, then we it has L independent loops, and there are L momenta that are not fixed by momentum conservation .) With V vertices, Iinternal lines, and L hoops, there are V delta functions, of which I--- I. go to fix internal momenta, leaving V - I+ L delta functions relating the momenta of incoming or outgoing particles . But a well-known topological identity" tells that for any graph consisting of C connected pieces, the numbers of vertices, internal lines, and loops are related b y V-I+ L = G , (4.4.7) Hence for a connected matrix element like Six, which arises from graphs with C = 1, we find just a single three-dimensional delta function63(P#- Px), as was to be proved . It was not important in the above argument that the time variables were integrated from -oo to +oo . Thus exactly the same arguments can be used to show that if the coefficients h N,Min the Hamiltonian contain just single delta functions, then U (t, to)can also be decomposed into connected parts, each containing a single momentum-conservation delta function factor . On the other hand, the connected part of the S-matrix also contains an energy-conservation delta function, and when we come to Feynman diagrams in Chapter 6 we shall see that SCcontains only a single energy-conservation delta function, 5(Ep- E.,), Zile U(t, to) contains no energy-conservation delta functions at all . It should be emphasized that the requirement that h NXI in Eq . (4.4.1) should have only a single three-dimensional momentum conservation delta function factor is very far from trivial, and has far-reaching implications . For instance, assume that V has non-vanishing matrix elements between two-particle states . Then Eq . (4.4.1) must contain a teem with N= Al = 2, and coefficient V'),z[PiA?,PrP2) = V P1p;-Pi p2 (4 .4.8) (We are here temporarily dropping spin and species labels .) But then the matrix element of the interaction between three-particle states i s + V?,2W1 P'2a P1 132)63(P~-p3) ± permuttion ~. (4.4.9) A graph consisting of a single vertex has V= 1, L = 0, and C = 1 . if we add V - 1 vertices with just enough internal lines to keep the graph cnnnccted, we have I = V-- 1, L = 0, and C = 1 . Any additional internal lines attached (without new vertices) to the same connected graph produce a n equal number ofloops, soI -=V+L-1 and C = 1 . [f a disconnected graph consists ofCsuch connected parts, the sums of I . V, and I . in each connected part will than satisfy E! =Y +L---C . 18$ 4 The Cluster Decomposition Principl e As mentioned at the beginning of this chapter, we might try to make a relativistic quantum theory that is not a field theory bychoosing V2,2 SO that the two-body S-matrix is Lorentz-invariant, and adjusting the rest of the Hamiltonian so that there is no scattering in states containing three or more particles . We would then have to take X3,3 to cancel the other terms in Eq. (4.4.9) V3-3(P iP'2P'3,PIPzP3) = -V2,2( PiP'2,PIPz)63(P'3-R3) + permutations . (4.4. 10) However, this would mean that each term in V3r3 contains two delta function factors (recall that V2,2(p gip ;, P1 P2) has a factor 63( P+P2 - p, p2) }and this would violate the cluster decomposition principle . Thus in a theory satisfying the cluster decomposition principle, the existence of scattering processes involving two particles makes processes involving three or more particles inevitable . When we set out to solve three-body problems in quantum theories that satisfy the cluster decomposition principle, the term V3,3in Eq . (4.4.9) gives no particular trouble, but the extra delta function in the other terms makes the Lippmann-Schwinger equation difficult to solve directly . The problem is that these delta functions make the kernel [E ,- Ep+ ie]-1 V O, of this equation not square-integrable, even after we factor out an overall momentum conservation delta function . In consequence, it cannot be approximated by a finite matrix, even one of very large rank . To solve problems involving three or more particles, it is necessary to replace the Lippmann-Schwinger equation with one that has a connected right-hand side. Such equations have been developed for the scattering of three or more particles,8,9and in non-relativistic scattering problems they can be solved recursively, but they have not turned out to be useful in relativistic theories and so will not be described in detail here . However, recasting the L,ippmann-5ch wi equation in this manner is useful in another way . Our arguments in this section have so far relied on perturbation theory . I do not know of any non-perturbative proof of the main theorem of this section, but it has been shown9that these reformulated non-perturbative dynamical equations are c onsistent with the requirement that Uc(t, ta) (and hence S C) should also contain only a single momentum-conservation delta function, as required by the cluster decomposition principle, provided that the Hamiltonian satisfies our condition that the coefficient functionshN,M each contain only a single momentum-conservation delta function . Problem s Problems189 1. Def ine generating func tionals for the S-matrix and its connected part : rv-tnrr-1 x S'qi-.-q~~,q,-..q,dpi...dqfvd~'1 ' ' ' dqm X SC Derive a formula relating F[v] and FC [u] . (You may consider the purely bosonic case .) 2. Consider an interac tion where g is a real constant and a(p) is the annihilation operator of a spinless boson of mass M O.Use perturbation theory to calculate the S-matrix element for scattering of these particles in the center- of-mass frame to order g2 . What is the corresponding differential cross-section ? 3. A coherent state (DA i s defined to be an eigenstat e of theannihilation operators a(q) with eig envalues ~(9).Construct such a state as a superposition of the mult i-particle states (Dq ,qa.,.qN. Reference s 1. The cluster decomposition principle seems to have been first stated explicitly in quantum field theory by E .H. Wichmann and J . H. Crichton, Phys . Rev. 132, 2788 (1963). 2. See, e.g.,B.Bakamijian and L.H. Thomas,Phys. Rev .92,1300 (1953) . 3. For references, see T .D.Lee and C .N. Yang, Phys . Rev. 113,1165 (1959) . 4. J. Goldstone, 'roc. Roy, Soc . London A239, 267 (1957) . 194 4 The ClusterDecomposition Principle 5. N. M. Hugenholtz, Physica 23, 481 (1957) . 6. See, e .g-, R . Kubo, J. Math .Phys.4, 174 (1963) . 7. E.C. Titchmarsh, lntroducricrn tothe theory oJ'Fourier I ntegra ls (Oxford University Press, Oxford, 1937) : Section 1 .8. 8. L. D. Faddeev, Zh,Ekxper, i Teor . Fig.39.1459 (1961) (translation Soviet Phys - JETP 12,1014 (1961)) ; DAl.Akad . Nauk . SSSR 138, 565 (1961) and 145, 30 (1962) (translations Soviet Physics - Doklady 6, 3 84 (196 1) and 7, 600 (1963)} . 9. S. Weinberg, Phys . Rev .133, B232 (1964) Quantum Fields and Antiparticle s We now have all the pieces needed to motivate the introduction of quan- tum fields .' In the course of this construction, we shall encounter some of the most remarkable and universal consequences of the union of relativity with quantum mechanics : the connection between spin and statistics, the existence of antiparticles, and various relationships between particles and antiparticles, including the celebrated APT theorem . 5.1 FreeFields We have seen in Chapter 3 that the S-matrix will be Lorentz-invariant if the interaction can be written a s V(t} = dux fix, 0, where rte' is a scalar, in the sense tha t and satisfies the additional condition :(5.x.1) (5.1.2) t_flx}, (')]=0for (x_x`)2>0. (5.1.3) As we shall see, there are more general possibilities, but none of them are very different from this . (For the present we are leaving it as an open question whether Ahere is restricted to a proper orthochronous Lorentz transformation, or can also include space inversions .) In order to facilitate also satisfying the cluster decomposition principle we are going to con- struct (x) out of creation and annihilation operators, but here we face a problem : as shown by Eq . (4.2.12), under Lorentz transformations each such operator is multiplied by a matrix that depends on the momentum carried by that operator . How can we couple such operators together to make a scalar? The solution is to build X,`(x) out of fi elds - bot h 191 192 5 Quantum Fields a ndAntiparticle s annihilation fields V e-~W and creat ion fields -(x) : PI~(X)_ d3PUAX -5 p, a} n)a(p, Q, n) ~ (5.1.4) 6M V -(X) E d3p ve(x ; P, CT' n)a (5.1.5) il an f(P,a, 0. with coefficients* u,,(x ; P, a, n) and ve(x; p,a, ra) chosen so that under Lorentz transformations each field is multiplied with aposition-inde- pendent matrix : Uo(A, a) w+(x)U~' (A, ac) _ De?( -1)T'{Ax + a .} , (5.1.6) Ua(, a)I (x)ilo'(A,a)_ De?(A-k)W?(Ax+ a) (We might, in principle, have different transformation matrices D± for the annihilation and creation fields, but as we shall see, it is always possible to choose the fields so that these matrices are the same .) By applying a second Lorentz transformation A,we find tha t sotakingA, = ( )-Iand 112= (A )-',we see that the D-matrices furnish arepresentation o f the homogeneous Lorentz group : There are many such representations, including the scalar D(A) = 1, the vector D( Li)"1= Af1,a and a host of tensor and spinor representations . These particular representations are irreducible, in the sense that it is not possible to by a choice of basis to reduce all D(A) to the same b lock- diagonal form, with two or more blocks . However, we do not require at this point that D(A) be irreducible ; in general it is a block-diagonal matrix with an arbitrary array of irreducible representations in the blocks . That is, the index ?' here includes a label that runs over the types of particle described and the irreducible representations in the different blocks, as well as another that runs ❑ver the components of the individual irreducible representations . Later we will separate these fields into irreducible fields that each describe only a single partic le species (and i ts antiparticle) and transform irreducibly under the Lorentz group . A reminder :the labels nand arun over alldilF`erent particle species and spin a-carnppnerils ,respectively- 5.1Free Fields 193 Once we have learned how to construct fields satisfying the Lorent z transformation rules (5.1.6) and (5.1.7),we will be able to construct the interaction density as I I V andthis will be a scalar in the sense ofE y. (5.1.2) if the constant coefficients gli---/, t1,..(, are chosen to be Lorentz covariant, in the sense that for a ll De,2) ?,---?N(5.1.10) g?fG, (Note that wedonot include derivatives here ,because we regard the derivatives of components of these fields as just additional sorts of field components .) The task of finding coefficients g _r, .,---~ that satisfy Eq. (5.1.14)isno different in principle (and not much more difficult in practi ce)than that of u sing Clebsch-Gordan c oefficients to couple together various representations of the three-dimensional rotation group toform rotational scalars .Later we will b eable to combine creat ion andannihilation fields so that thi sdensity also commute swith itself at space-like separations . Now ,what shall we take as the coefficient funct ions u~(x;p,a,ra) and ve(x; p,a, n)? Eq .(4.2.12)and its adjo int give the transformation rules** for the annihilation and creation operator s Uo(A, b)a(p, u,n)U~l (A, b) = exp (i(Ap) - b)~(ApPl e x D((W-'(A, p))a(PA, eF, n) UO(A, b)a~(p, ~T, n)U6-1 (A, b) = exp i(Ap) - b) ~F(ApP 1p o x1: Di" W-1 (A,p))a'(PA, Er, n)(5.1.11) (s.i.12) where j ,is the spin of particles of species n, and PA Is the three-vector part of Ap.(We have used the unitarity of the rotation matrices D~j~)to put both Eqs .(5-1.11) and (5.1.12) in the form shown here .) Also, as we saw in Section 2 .5 the volume element d3 p f p° is Lorentz-invariant, so w e This i5 for massive particles . The case of zero mass will betaken up in Section 5 .9. 194 5Quantum Fields and Antiparticle s can replace dip in Eqs . (5.1.4) and (5 .1..5} with d'(11,p)pOf (Ap)II.Putting this all together, we find Cron xexp(i(Ap)-b)D(j')(W-'(A, i~, n)Cr CT P))~POAAPr OPA , and ddY 2 x exP(-i(Ap)-b)D(j'~)*(W-1(A,P)) VpOI(Ap)O a'(PA, a , We see that in order for the fields to satisfy the Lorentz transformation rules (5.1.6)and (5.1.7), it is necessary and sufficient tha t ?.. (A rr, n)= ~pO /(Ap) O x I:D(j')(W-'(A,p))exp(+i(Ap)-b)u,,,(,v ;p,c,n ) and D,,?(A-l)v?(Ax + b;PA,0,n)= ~pO/(Ap) P x D(j)µ - t(,P))exp or somewhat more convenientlyi(Ap) - b)V((X; P,a, n) u~(Ax + h; p,,, ~7)D(j-) (W(A, p)) =VFp;/(TAP x D?,,(A) exP (i(Ap) ' b)u,,(x;P, a,n) and v?{11x +b;fin, d}D(j~)*(W(A .p)') ° (AP ~ x ~~,, (11) exp - i(Ap) - b V, .,{.x;p,6,n}(5.1.13) (5.1.14) These are the fundamental requirements that will allow us to calculate the ueand ve coefficient functions in terms of a finite number of free parameters . 5.1Free Fields 195 We will use Eqs .(5. x.13)and (5 .1.14) in three steps, considering in turn three different types of proper orthochronous Lorentz transformation : Translation s First we consider Eqs . (5.1.13) and (5 .1.14) with A = 1 and b arbitrary . We see immediately that u e(x;A, a, n) and vr(x ;p,a,n) must take the for m so the fields are Fourier transforms : andW+(x) T (2 n)-1/2dipu~(P, a,n)e~rxa(p,u,n), d,n d,M(5.1.15) (5.1.16) (5.1.17) (5.1.18) (The factors (2;T)-3/2could be absorbed into the definition of u,, and r .,j, but it is conventional to show them explicitly in these Fourier integrals .) Using Eqs .(5.1.15) and (5 .1.10, we see that Eqs . (5.1,13)and (5 .1.14) are satisfied if and only i f U?(PA, FT, n)D c(,j,,-)(W(A,p)) P" D?,, n) (5.1.19) CT and C1~ 4PA 'l{T7 f i)D~Vl7 ~*(V(A .p))Vi ~~~ fD1 F, ~~~ `,f(P7l/ ~ J, 1 .~~~ I for arbitrary homogeneous Lorentz transformations A. Boost s Next take p = 0 in Eqs .(5.1.19)and (5 .1.20), and let Abe the standard boost L(q) that takes a particle of mass m from rest to some faur- rnornentum qP. Then L(p) = 1, an d Hence in this special case, Eqs .(5.1.19) and (5 .1.20)give 196 and5Quantum Fields an d Antipa rticles v? (q. 6T n) -(mlq0)1/2D?c~~~~~) vl(0,a,n). (5.1.22) In other words, if we know the quanti ties ut(Q, a,n) and v e(0, a, n) fo rzero momentum, then for a given representation D(A) of the homogeneous Lorentz group, we know the functions u,~( p, ff, n) and ve( P, a, n) for all p . (Explicit formulas for the matrices D~,(L(q)) wil lbe given for arbitrary representations of the homogeneous Lorentz group in Section 5 .7.) Rotation s Next, take g= 0, but this time let A be a Lorentz transformation with PA = 0 ; that is, take A as a rotation R . Here obviously W(A,p) =R, and so Eqs . (5.1.19) and (5 .1.20) rea d and M or equivalentl y and IT(.1..4) 6.7{5.1.25} (5.1.2b) where J Wand are the angular-momentum matrices in the represen- tations DUO) and D(R), respectively . Any representation D(A) of the homogeneous Lorentz group obviously yields a representation of the ro- tation group when Ais restricted to rotations R ; Eqs .(5.1.25) and (5.1.26) tell us that if the field (x) is to describe particles of some particular spin j, then this representation D(R) must contain among its irreducible com- ponents the spin-j representation DOW}, with the coefficients u,,(O,o-,n) and a e(O,u, ra)simply describing how the spin-j representation of the ro- tation group is embedded in D(R) .We shall see in Section 5 .5 that each irreducible representation of the proper orthochronous Lorentz group con- tains any given irreducible representation of the rotation group at most once, so that if the fields W,'(x)and V ., (x) transform irreducibly, then they are unique up to overall scale . More generally, the number of free 5.1 Free Fields 197 parameters in the annihilation or creation fields (including their overall scales) is equal to the number of irreducible representations in the field . It is straightforward to show that coefficient functions ue(P, cT, n) and ve(ps ff ,n) given b yEqs. (5.1.x.1)and ( 5.1.22), with u~ (0,a, rah and vj { 0,a,n) satisfying Eqs .(5.1.23) and (5 .1,24), will automatically satisfy the more general requirements (5 .1.19 and ( 5.1.20). This is left as an exercise for the reader . Let us now return to the cluster decomposition principle . Inserting Eqs. (5.1.17) and (5.1.18)in Eq . {5.1.9} and integrating over x, the interac- tion Hamiltonian is 1 h 1N x afi( Pi ai ni)... af(PnrDiv Div)a(F~r~?~t nnr)...a(Pisinj) X 'NM41 Ori nl...~~ ~~n'N, pit7ln I...~M (7M nM)(5.1.27) withcoefficient functionsgiven b y X -T-NM {Pial ni... p] U[n1...~ ~ (5 .1.28) where nrm(Pi ~i n1... p~ d~ n~ p ,6i,71 ... p~runfrn~r) = (2~)I-1rv,1-~~~~ v~, or 'r r' x Xuez(plalnl},u~.tiJpMamn nr) (5.1.9) This interaction is manifestly of the form that will guarantee that the S-matrix satisfies the cluster decomposition principle : 'VNm has a single delta function factor, with a coefficient irNM that (at least for a finite number of field types) has at most branch point singularities at zero par- ticle momenta . In fact, we could turn this argument around ; any operator can be written as in Eq .(5.1.27), and the cluster decomposition principle requires that the coefficient 'VN,Wmay be written as in Eq.(5.1.28) as the product of a single momentum -conservation delta function times a smooth coefficient function . Any sufficiently smooth function (but notone con- taining additional delta functions) can be expressed as in Eq . (5.1.29).tThe cluster decomposition principle together with Lorentz invariance thusmakes itnatural that the interaction density should be constructedout of the anni- hilation and creation fi elds. For general functions the indices (and :` may have to run over an infinite range . The reasons for restricting (and G' to a finite range have to do with the principle of renorrnaliaability, discussed in Chapter 12 . 198 5 Q uantum Fields and Antipartic les Ifall we needed were to construct a scalar interaction density that satisfied the cluster decomposition principle, then we could combine an- nihilation and creation operators in arbitrary polynomials (5.1.9), with coupling coefficients g~' ...~' ,, subject only to the invariance condition (5.1. 0)(and a suitable reality condition) . However, for the Lorentz in- variance of the S-matrix it is necessary also that the interaction density satisfy the commutation condition (5.1.3). This condition is not satisfied for arbitrary functions of the creation and annihilation fields becaus e (P,a,n)ed''l(x-►')(5. i.3o) L1P1.+1W, T~-_ (Y) IT = (2~~-3 d3p u~ dpi cyan)v? I (with the sign + indicating a commutator or anticommutator ifthe parti- cles destroyed and created b ythe components y}+and yi~ . are bosons or fermions, respectively,) and in general this does not vanish even for x - y space-like . It is obviously not possible to avoid this problem by making the interaction density out of creation or annihilation fields alone, for then the interaction could not be Hermitian . The only way out of this difficulty is to combine annihilation and creation fields in linear combinations ., (X) Kt-'-(X) +A~00 (5.1.:1) with the constants K and ~and any other arbitrary constants in the fields adjusted so that for x - y space-lik e 1~~~~) , I~-'r, (Y)~+ = 14, c~~~, ~I~'~r(Y)1~ _ 0 . (5 .1.32) We will see in subsequent sections of this chapter how to do this for various irreducibly transforming fields . (Byincluding explicit constants K and ~in Eq. (5.1.31)we are leaving ourselves free to choose the overall scale of the annihilation and creation fields in any way that seems convenient .) The Hamiltonian density '(x) will satisfy the commutation condition (5.1.3) if it is constructed out of such fields and their adjoins, with an even number of any field components that destroy and create fermions . The condition (5.1.32)is often described as a causality condition, because if x -y is space-like then no signal can reach y from x, so that a measurement of Weat point x should not be able to interfere with a measurement of ~ .~~+ or w~, at point y. Such considerations of causality are plausible for the electromagnetic field, any one of whose components may be measured at a given spacetime point, as shown in a classic paper of Bohr and Rosenfeld,' However, we will be dealing here with fields like the Dirac field of the electron that do not seem in any sense measurable . The point of view taken here is that Eq. (5.1.32) is needed for the Lorentz invariance of the S-matrix, without any ancillary assumptions about measurability or causality . There is an obstacle to the construction of fields (5 .1.31) satisfying 5.1Free Fields 199 (5.1.32). It may be that the particles that are destroyed and created by these fields carry non-zero values of ❑ne or more conserved quantum numbers like the electric charge . For instance, ifparticles of species n carry a value q(ra) for the electric charge Q, the n Lea a(p, a, n)] = -q (n)a( A, a, n) , [Q, al (P, a# n)] = +q(n)a~(p, ff,n). In order that (x) should commute with the charge operator Q(or some other symmetry generator) it is necessary that it be formed out of fields that have simple commutation relations w ith Q : ~~~~"Wj _ ~q,,,Vj(x ) (5 .1.33) for then we can make (x) commute with Qby constructing it as a sum of products of fields V(,We,... and adjoints ip~[V)~2...such tha t fir, + q e2+...- q,,, -q.2-...-_0. Now, Eq. (5. i..33)is satisfied for one particular component ~)Px) of the annihilation field if and only if all particle species n that are destroyed by the field carry the same charge q(n) = q(,and it is satisfied for one particular component y)e (x) of the creation field if and only if all particle species n that are created bythe field carry the charge q(h) =-q(. We see that in order for such a theory to conserve quantum numbers like electric charge, there must be a doubling of particle species carrying non- zero values of such quantum numbers : if a particular component of the annihilation field destroys a particle of species n, then the same component of the creation field must create particles of a species n-, known as the antiparticles of the particles of species n, which have opposite values of all conserved quantum numbers . This is the reason fir antipart icles. If the representation D( A) is not irreducible, then we can adopt a basis for the fields in which D (A)breaks up into blocks along the main diagonal, such that fields that belong to different blocks do not transform into each other under Lorentz transformations . Also, Lorentz transformations have no effect on the particle species . Therefore, instead of considering one big field, including many irreducible components and many particle species, we shall from now on restrict our attention to fields that destroy only a single type of particle (dropping the label n) and create only the corresponding antiparticle, and that transform irreducibly under the Lorentz group (which as mentioned above may or may not be supposed to include space inversion), with the understanding that, in general, we shall have to consider many different such fields, some perhaps formed as the derivatives of other fields . In the following sections we are going to finish the determination of the coefficient functions u (. (p,c7)and ve(P, a), 200 5Quantum Fields and Antiparticle s fix the relative values of the constants rc and ~, and deduce the relations between the properties of particles and antiparticles first for fields that belong to the simplest irreducible representations of the Lorentz group, the scalar, vector, and Dirac spinor representations . After that we will repeat the analysis for a completely general irreducible representativn . A word about field equations . Inspection of Eqs .(5.1.31), (5 .1.17), and (5.1.18)shows that all the components of a field of definite mass m satisfy the Klein-Gordon equation : (0-M')A }=0. (5 .1.34) Some fields satisfy other field equations as well, depending on whether or not there are more field components than independent particle states . Traditionally in quantum field theory ❑ne begins with such field equations, or with the Lagrangian from which they are derived, and then uses them to derive the expansion of the fields in terms of one-particle annihilation and creation operators . In the approach followed here, we start with the particles, and derive the fields according to the dictates of Lorentz invari- ance, with the field equations arising almost incidentally as a byproduct of this construction . ~** A technicality must be mentioned here . According to the theorem proved in Section 4,4, the condition that guarantees that a theory will satisfy the cluster decomposition principle is that the interaction can be expressed as a surn of products of creation and annihilation operators, with all creation operators to the left of all annihilation operators, and with coefficients that contain only a single momentum-conservation delta function . For this reason, we should write the interaction in the `normal ordered' form V= dux (W(x)y W~(x)}: (5 .1.35) the colons indicating that the enclosed expression is to be rewritten (ignoring non-vanishing commutators or anticommutators, but including minus signs for permutations of fermionic operators) so that all creation operators stand to the left of all annihilation operators .By using the commutation or anticommutation relations ❑f the fields, any such normal- ordered function of the fields can just as well be written as a sum of ordinary products of the fields with c-number coefficients . Rewritin g in this way makes it obvious that despite the normal ordering, :. {w(x), O x}) :will commute with :. (W(y), qt(y)} when x - y is space-like, if it is constructed out of fields that satisfy Eq .(5.1.32), with even numbers of any fermianic field components . 5.2Causal Scalar Fields 5.2CausalScalarFields201 We first consider one-component annihilation and creation fields 0+(x) and 0-(x) that transform as the simplest of all representations of the Lorentz group, the scalar, with D(A) = 1. Restricted to rotations, this is just the scalar representation of the rotation group, for which I =:;-- 0, so Eqs.(5.1.25) and (5.1.26) have no solutions except for j ==0, in which case Cr,&take only the value zero . Thus a scalar field can only describe particles of zero spin . Assuming also for the moment that the field describes only a single species of particle, with no distinct antiparticle (and dropping the species label n as well as the spin label aand the field label I), the quantities ue(0an) and ve(Dun) are here just the numbers u{0} and v(D). It is conventional to adjust the overall scales of the annihilation and creation fields so that these constants both have the values {ern .]-W . Eqs .(5.1.21) and ( 5.1.22)then give simply u(p)= (2pa)-"z and v(p) = (2 Po)-"' The fields (5.1.17)and (5.1.18)are then, in the scalar case , and 0-0 -- f d'p (2n)-1/2(2pO)-1/2at(p)e-P` ;--0-11(x) .(5.2.1) (5.2.2) (5.2.3) (s.2.4) A Hamiltonian density (x) that is formed as a polynomial in 0+(x) and 0-(x) will automatically satisfy the requirement (5 .1.9), that it trans- form as a scalar . It remains to satisfy the other condition for the Lorentz invariance of the S-matrix, that (x) commute with (Y) at space-like separations x - y .If (x) were a polynomial in +(x) alone, there would be no problem . All annihilation operators commute or anticommute, so 0+(x) either commutes or anticommutes with O+(Y} for all x and y, according to whether the particle is a Cason or fermion, respectively : to,{x}, 0+(y} ]:~ = 0 . (525 ) Hence any (x) formed as a polynomial in 0+(x) (or, for fermions, any such even polynomial) will commute with -YC(y) for all x and y . The problem, of course, is that, in order to beHermitian, fix) must involve +1(x) =0-(x) as well as +(x), and 0+(x) does not commute or anticommute with 0-(Y)for general space-like separations . Using the commutation (for bosons) or anticommutation (for fermions) relations 202 5 Quantum Fields and Antiparticles (4.2.5), we have 2~r 2 21`1)1/2 which collapses to the sing le integra l where d+ is a standard function : d+(x} - 1d3~e~P~x (5 .2.7)~ (21r)3 P This is manifestly Lorentz-invariant, and therefore for space-like x it can depend only on the invariant square x2 0 . We can thus evaluate ❑+(x) for space-like x by c hoosing the coordinate system so tha t x° = 0 ~x~ = x1 . Eq. (5.2.7) then gives 1 d3P z p.xA+~~ (27r)32,, ~ /~ ~xa~ 41E p2dp sin(p x2) (27z)3 vIp ~rr~.~ F x2 .l0 2 Changing the variable of integration to u = p f rra, this is 00 ❑+{x} _ ~udusin(rx2u) (5 .2.8) 4ar x2 o u + ~ or, in terms of a standard Hanlcel function , 4iz2x2(M-~X' ) This isn't zero, so what are we to do with it? Note that even though ❑+(x) is not zero, for x2 0 it is even in x" . Instead of using only 0+(x), suppose we try to construct {x} out of a linear comb inatio n Using Eq . (5.2.6), we have then for x - y space-like 5.2Causal Scalar Fields 203 Both of these will vanish if and only if the particle is a boson (i.e., it is the top sign that applies) and rc and ).are equal in magnitud e ~~~~= P~l - We can change the relative phase Of K and Aby redefining the phases of the states so that a(p) --*eila(P), a#(p) - +e ~"at(P), and hence K rce`u , A --+ ire`~. Taking oc = ~Arg(~ ./rc), we can in this way make K and Aequal in phase, and bence equal . Redefining () to absorb the overall factor :c = A'., we have the n The interaction density (x) will commute with ' (y) at space-like sepa- rations x - y if formed as a normal-ordered polynomial in the self-adjoint scalar field O(x) . Even though the choice of the relative phase of the two terms in Eq.(5.2.10) is a matter of convention, it is a convention that once adopted must be used wherever a scalar field for this particle appears in the interaction Hamiltonian density . For instance, suppose that the interaction density involved not only the field (5.2.10), but also another scalar field for the same particl e with a an arbitrary phase . This ~, like 0,would be causal in the sense that O(x) commutes with 0(y)when x - y is space-like, but O(x) would not commute with 0(y)at space-like separations, and therefore we cannot have both of these fields appearing in the same theory . If the particles that are destroyed and created b yO(x) carry some conserved quantum number like electric charge, then (x)will conserve the quantum number if and only i feach term in fi(x) contains equal numbers of operators a(p) and a(A)' . But this is impossible if (x) is formed as a polynomial in O(x) = +(x) +O+t(x) . To put this another way, in order that (x) should commute with the charge operator Q (or some other symmetry generator) it is necessary that it be formed out of fields that have simple commutation relations with Q . This is true for 0+(x) and its adjoint, for whic h but not for the self-adjoint field (5.2.8). In order to deal with this problem, use must suppose that there are two spinless bosons, with the same mass in,but charges +q and -q, respectively . Let +(x) and 0+'(x) denote the annihilation fields for these 204 5Quantum Fields and Antiparticle s two pa rticles, so that * Define O(x) as the linear combinatio n which manifestly has the same commutator with Q as 0+(x) alon e The commutator or anticommutator of O(x) with its adjoint is then, at space-tike separation (IKI' TIAI')A+(X-v) while O(x) and 0(Y)automatically commute or anticommute with each other for all x and y because 0+ and O+'t destroy and create different particles . In deriving this result, we have tacitly assumed that the particle and antiparticle have the same mass, so that the commutators or anticorn- mutators involve the same function ❑+(x -- y ).Fermi statistics is again ruled out here, because it is not possible that O{x} should anticommute with 4t(Y) at space-like separations unless K = ~=0,in which case the fields simply vanish . So a spinless particle must be a boson . For Bose statistics, in order that a complex O(x) should commute with Ot{y}at space-like separations, it is necessary and sufficient that IKI' = I AI'x as well as for the particle and antiparticle to have the same mass . By redefining the relative phase of states of these two particles, we can again give rc and Athe same phase, in which case K = A. This common factor can again be eliminated by a redefinition of the field 0, so tha t or in more detail d3 P[a(p)el" +up-t(p)e-'P'x1 (5.2.11)((27r)3/2(2pO)1/ 2 This is the essentially unique causal scalar field . This formula can be used both for purely neutral spinless partic les that are their own antiparticles (in which case we take al( p) =a( p)), and for particles with distinct antiparticles (for which a(p) a(p)) . The label `c' denotes `c harge co njuga te'. It s hould be kept i nmin dthat a particle that ca rries no conserve dquantum numbers may or may not be its own antiparticle, with u`( p) = u( P). 5.2 Causal ScalarFields 245 For future use, we note here that the commutator of the complex scalar field with its adjoint i s where(5.2.1Z) ~2ir (5.2.13) Let's now consider the effect of the various inversion symmetries on this field . First, from the results of Section 4 .2, we can readily see that the effect of the space-inversion operator on the annihilation and creation operators is :** Pa(P)P-' = qa(_P)X5 .2.14) (5.2.15) where jj and q1 are the intrinsicparities of the part icle and antipartic le, respectively . Applying these results to the annihilation field (5 .2.3) and the charge-conjugate of the creation field (5 .2.4), and changing the variable of integrat ion from p to -p, we see tha t Pct+(x)P-1 =q*O+(fix) (5.2.16) (5.2.17) where as before 9x = (-x,x°) . We see that in general applying the space inversion to the scalar field O(x) -0+(x) + O+lfi(x) would give a different field Op=q*0++q10+Gt . Both fields are separately causal, but if 0 and Ofappear in the same interaction then we are in trouble, because in general they do not commute at space-like separations . The only way to preserve Lorentz invariance as well as parity conservation and the hermiticity of the interaction is to require that OP be proportional to 0, and hence that q C= r,*. (5.2.18) That is, the intrinsic parity r lq'of a state containing a spi nlessparticle and its antiparticle is even .We have now simpl y PO(x)P-' =n*o(fix) . (5.2.19) We are omitting the subscript 0 on inversion operators P, C, and T, because in virtually all cases where these inversions are good symmetries, the same operators induce inversion transformations on `in' and `out' states and on free-particle states 206 5Quantum Fields and Antipartic les These results also apply when the spinless particle i s its own a ntiparticl e, for which q' = q, and impl y that theintrinsic parity of such a part icleis real:rI = ± 1, Charge-conjugation can be h andledin much the sa me way .From t he results of Section 4 .2,we hav e Ca'l(P)C-i = ~'at( P), (5.2.21) where and ~` are the phases associated with the operation of charge- conjugation on one-particle states . It follows then that (5.2.22) (,2.23) In order that CO(x)C-1 should be proportional to the field Of(x) with which it commutes at space-like separations, it is evidently necessary tha t Just as for ordinary parity, the intrinsic charge-conjugation parity ~~ 'of a state consisting of a spinless particle and its antiparticle is even . We now have simpl y Again, these results apply also in the case where the particle is its own antiparticle, where In this case the charge-conjugation parity like the ordinary parity must he real, ~ = ±1 . Finally we come to time-reversal . From Section 4 .2 we have (5.2.26) Recalling that T is antiunitary, and again changing the variable of inte- gration from pto - p, we find tha t Tck+(x)T-i = C"O+(-fix ) (5.2.28) (5.2.29) In order for To(x)T-' to be simply related to the field 0at the time- reversed point -fix, we must hav e ~C= C o and then TO(x)T-' = C'(h(-fix ).(5.2.30) (5.2.31) 5.3Causal Vector Fields 5.3 Ca usalVector Fields207 We now take up the next simplest kind of field, which transforms as a four-vector, the simplest non-trivial representation of the homogeneous Lorentz group . There are massive particles, the ± and Z4, that at low energies are described bysuch fields and that play an increasing role in modern elementary particle physics, so this example is not merely of pedagogical interest . (Also, although we are here considering only massive particles, one approach to quantum electrodynamics is to describe the photon in terms of a massive vector field in the limit of very small mass) . For the moment we will suppose that only one species of particle is described by this field (dropping the species label rah ; then we shall consider the possibility that the field describes both a particle and a distinct antiparticle . In the four-vector representation of the Lorentz group, the rows and columns of the representation matrices D(A) are labelled with four-com- ponent indices u, v, etc ., wit h The annihilation and creation parts of the vector field are written : 0+"(x) =Y:(21r)-1/2 f d3 PU,(p, a) a(p,u) eip-x fd3p V"(p, u) a,(P,a)eip-x ff(5.3.1} (5.3.2) (5.3.3) The coefficient functions O(P> d) and aPtP, a) for arbitrary momentum are given in terms of those for zero momentum by Eqs . (5.1.21)and (5.1.22), which here read : UNPaU) = (M IP0)1/2L (F)A, uv (0,a)} (5.3.4) (s.3.5) (We are using the usual summation convention for spacetime indices it, v, etc.) Also, the coefficient functions at zero momentum are subject to the conditions (5.1.25) and (5 .1.26): and 111(01JW* jiV,(0, a)(5.3.6) (5.3.7) 208 5Quantum Fields and Antiparticle s The rotation generators ~'u in the four-vector representation are given by Eq . (5.3. 1)as VkY ifi jk(5.3.8) (5.3.9) with i, j, k here running over the values 1, 2, and 3 . We note in particular that f2 takes the form L i= d6i ~ } i From Eqs .(5.3.6) and ( 5.3.7) it follows then tha t Er u6~~~~)p(J))a~2u~(OrQ) and(5.3.10) (5.3.11) (x.3.12) (5.3.13) (5.3.14) X5.3.1S) Also, we recall the familiar result that ( J(})}2 Q = j(1 + 1)6~ra .From Eqs. (5.3.12)-(5 .3.15) we see that there are just two possibi lities for the spin of the particle described by the vector field : either j = 0, for which atp= 0 only u° and vo are non-zero, or else j = I (so that j(j+ 1) = 2), for which at p= 0 only the space-components uj and v` are non-zero . Let us look in a litt le more detail at each of these two possibilities . Spin Zer o By an appropriate choice of normalization of the fields, we can take the only non-vanishing component of ull(O) and vP(0) to have the conventional values uo(0)= r(m12)1/2 (The label ahere takes only the single value zero, and is therefore dropped .) Then Eqs, (5 .3.4) and (5.3.5)yield for general moment a U"(P) = ip"(?po)-1/2(5.3.16) 5.3Causa l Vector Field s and The vector annihilation and creation fields here are nothing but the derivatives of the scalar annihilation and creation fields ± for a spinless particle that were defined in the previous section : 0+1~(x)= 011`0+(X) 0-10-010-fix} . (5.3.18) It is obvious that the causal vector field for a spinless particle is also just the derivative of the causal scalar field : 01,U} = 0+,`(X) + 0-1(X) ~ 010W . (53 .19) Hence we need not explore this case any further here . Spin On e From Eqs .(5.3.6) and (5.3.7)we see immediately that the vectors ui(0,0) and vx(0, 0) for a =0 are in the 3-direction . By a suitable normalization of the fields, we can take these vectors to have the value s 0 1 0209 (5.117) (5.3.20 with four-vector components listed always in the order 1, 2, 3, 0 . To find the other components, we use Eqs .(5.3.6), (5 .3.7),and (5 .3.9) to calculate the effect of the raising and lowering operators J~1)± J~1) on u and v . This gives : 1 +i 0 0(5.3.21) Applying Eqs . (5.3.4) and (5.3.5)now yields0 0 where{5.3.22} (5.3.3) (5.3.24) 210 with I 05QuantumFieldsand Anttparticle .3 [L] 1 ,j2_ 0 0 The annihilat ion and creation fields (5 .3.2)and (5.3.3)here .are0 0 (5.3.25) 0-1'1(x)_(2n)-3/2:1d'p~ e~`(p, a)a(P, a) e'~'-x (5.3.26) V2 Thy fields 0+"(x) and 0+'(y)of course commute (or anticommute) for all xand y,but 0+ ;;(x) and 0-"(.Y)do not . Their commutator (for bosons) or anticommutator (for fermions) i s where2~)2 ~ P(5.3.27) (5.3.2$) A straightforward calculation using Eq . (5.3.25) shows that III''{D} is the projection matrix on the space orthogonal to the time-direction, and Eq. (5.3.24) then shows that HA''( P) is the projection matrix on the space orthogonal to the four-vector py 2 The commutator (or anticommutator) (5.3.27)may then be written in terms of the ❑+function defined in the previous section, a s 10"{A 0' ( Y)lT _ qg` M2 d+(x - y ). (5 .3.34) For our present purposes, the important thing about this expression is that for x -y space-like it does not vanish and iseven in x -- y . We can therefore repeat the reasoning of the previous section in seeking to construct a causal field : we form a linear combination of annihilation and creation field s for which, for x - y spacelike, & 1V ~`k~J, V `'(Y )l ~= ACAL1 1 ~17PV~ YATRx I 5.3Causal Vector Fields 211 and ~~~~IV M2 1 In order for both to van ish for space-like x-y ,it is necessary and sufficient that the spin one particles be bosons and that JrcJ=~Al.By a suitable choice of phase of the one-part icle states we can give r cand Athe same phase, so that K = ) .,and then drop the common factor r c by rede fining the o verall normalization of the field .After allthis, we find that the cau sal vector fiel d fo ramassive particle of spin one i s U'(x)= 0" `(x) +c~'~'t{x} . (5.3.31) We note that this is real : O(X) = 01(x). (5.3.32) However, if the particles it describes carry a non-zero value of some conserved quantum number Q, then we cannot construct an interaction that conserves Q out of such a field . Instead, we must suppose that there is another boson of the same mass and spin which carries an opposite value of Q, and construct the causal field a s VP(X)T +11(X)+ O+QA(x) {5 .3.33) or in more detail d3 J where the superscript c indicates operators that create the antiparticle that is charge-conjugate to the particle annihilated by O+}'(x) . This again is a causal field, but no longer real . We can also use this formula for the case of a purely neutral, spin one particle that is its own antiparticle, by simply setting ac(P) ==a(p). In either case, the commutator of a vector field with its adjoint is c~PP [V"(X),V"(Y)]= 17Yu - M2 A (x-y') (5 .3.3) where ❑(x-y)is the function ( 5.2.13). The real andcomple xfieldswehave constructed for a massive, spin one parti clesatisfyintere sting field equations .First,since p pin the exponent ial in Eq . (5.3.26) s atisfie sp2 _ -M', the field satisfie stheKle1 in-Gordon equation : (❑- MI )vU(x) = 0, (5.3.36) 212 5 Quantum Fields and Antiparticle s just as for the scalar field .In addition, since Eq. (5.3.24) shows tha t we now have another equation ajuv"( x)= 0.(5.3.37) {5.3.38} In the Limit of small mass, Eqs .(5.3.6)and (5 .3.38)arejust the equations for the potential four-vector ofelectrodynamics in what is called Lorentz gauge . However, wecannot obtain el ectrodynam ics from just any theory of massive sp in one part icles by letting the ma ss go to zero .The trouble can be seen by considering the rate of production of a sp in one particle byan interaction density =,I,,vl`,where.fi,isan arbitrary four-vector current . Squaring the matrix element and summing over the sp in z-components of the spin one particle gives a rate proportional t o <Ji, > e y(p,iT)* =< .1i,, >< .I, >" III''{P}, where p is the momentum of the emitted spin one particle, and < ,I1, > is the matrix element of the current (say, at x = 0) between theinitial and final states of all other particles .The term p gp'/m2in II P"(p) will ,in general, cause the emission rate t oblow up when m - y 0. The only way to avert this catastrophe is to suppose that <J~, > py va nishes, w hich in coordinate space isjust the statement that the current J "must be conserved, in the sen se thatO,.1P = 0.Indeed, the need for conservation of the current can be seen by simply counting states .A massive spin one particle has th ree spin states, which can be taken as the states with helicity +1, 0,and -1, wh ile any massless ,spin one particle like the photon can only have helicities + 1 and -1 :the current conservation condition just ensure sthat the helicit yzero states of the spin one particle are not emitted in the limit of zcro mass . The i nversions can be dealt with in much the same way as forthe scalar field d iscussed in the previous section .Toevaluate the effect of space inversion ,we need a formula for e ;(-p,a). UsingDIJ -P} = 91-1 P LPz(P} rV and Eq. (5.3.24), we have (5.3.39) Also, to evaluate the effect of time-reversal we need a formula for (-l)i+'e,'*(-p,-a), Using e" *( -a) = -el`(cr) and the above formula for Lf'4,(- p), we find {5.3.4} Using these results and the transformation properties of the annihila- tion and creation operators given in Section 4 .2, it is straightforward to 5.4The Dirac Formalism 213 work out the inversion transformation properties of the annihilation and creation fields . Once again we find that, in order for causal fields to be transformed into other fields with which they commute at space-like separations, it is necessary that the intrinsic space inversion, charge- conjugation, and time-reversal phases for spin one particles and their antiparticles be related by q C =I,*, (53.41) (5.3.42) C`=~*. (5 .3.43) (In particular all phases must be real if the spin one particle is its own antiparticle .) With these phase conditions satisfied, our causal vector field (5.3.34) has the inversion transformation propertie s PVi`(x)P-'_ -q'~"vy(°x} , (5 .3.44) W`(x) C-1 = ~ V"t(x), (5 .3.45) In particular, the minus sign in Eq.(5.3. 4) means that a vector field that transforms as a polar vector, with no extra phases ❑r signs accompanying the matrix YP,, describes a spin one particle with intrinsic parity q= -1 . 5.4TheDirac Formalis m Among all the representations of the homogeneous Lorentz group, there is one that plays a special role in physics . As we saw in Section 1 .1, thi s 3representation was introduced into the theory of the electron by Dirac, but as so often happens it was already known to mathernaticians,4 because it provides the basis of one of the two broad classes of representations of the rotation or Lorentz groups (actually, of their covering groups ---r see Section 2 .7)in any number of dimensions . From the point of view we are following here, the structure and properties of any quantum field are dictated by the representation of the homogeneous Lorentz group under which it transforms, so it will be natural for us to describe the Dirac formalism as it first appeared in mathematics, rather then as it was introduced by Dirac . By a representation of the homogeneous Lorentz group, we mean a set ❑f matrices D(A) satisfying the group multiplication law 214 5 Q uantumFields and Antiparticle s Just as for the unitary operators U(A), we can study the properties of these matrices by considering the infinitesimal case , for which(5.4.1) (5.4.2) (5.4.3) with Pw =-FA a set of matrices satisfying the commutation relations (2.4.12): L To find such a set of matrices, suppose we first construct matrices that satisfy the arttacommutation relation s and tentatively defin e It is elementary, using E q. (5.4.5), to show that(5.4.5) {5.4.x) O.4.7) and from this we easily see that Eq. (5.4.6) does indeed satisfy the desired commutation relation Eq . (5.4.4). We shall further assume that the matrices '/,Uare irreducible ;that is, that there is no proper subspace that is left invariant by all these matrices . Otherwise we could choose some smaller set of field components, which would transform as in Eqs . (5.4.3) and (5.4.6), with an irreducible set of -1,,s . Any set of matrices satisfying a relation like Eq . (5.4.5) (or its Euclidean analog, with j,,, replaced with a Kronecker delta) is called a Clifford algebra .The importance in mathematics of this particular representation of the homogeneous Lorentz group (or, more accurately, its covering group) arises from the fact (shown in Section 5.6)that the most general irreducible representation of the Lorentz group is either a tensor, or a spinor transforming as in Eqs . (5.4.3) and (5 .4.6), or a direct product of a spinor and a tensor . The commutation relation (5 .4.7) can be summarized by saying that y P is a vector, in the sense that Eq . (5.4.3) satisfie s In the same sense, the unit matrix is trivially a scala r D(A) 1D-'(A) -1(5.4.8) (5.4.9) 5.4 The Dirac Formalis m and Eq.(5.4.4) shows that JP11 is an antisymmetric tensor215 (5.4.10) The matrices yP can be used to construct other totally antisymmetric tensors PUT 'YLP Y r~~1ff TT~ n1 l!'~},,~YL .y}7J ~~ f ~ f(5.4.11) (5.4.12) The brackets here are a standard notation, indicating that we are to sum over all permutations of the indices within the brackets, with a plus or minus sign for even or odd permutations, respectively . For instance, Eq. (5.4.Z 1) is shorthand for I { R FP f f R f ! r { By repeated use of Eq . (5.4.5) we can write any product of ys as a sum of antisymmetrized products of ys times a product of metric tensors, so the totally antisymmetric tensors form a complete basis for the set of all matrices that can be constructed from the Dirac matrices . This formalism automatically contains a parity transformation, conven- tionally taken as iY0 Applied to the Dirac matrices, this gives(5.4.13 (We here label indices so that p runs over values 0,1, 2,....)The same similarity transformation, applied to any product of 7-Matrices, then yields just a plus or minus sign, according to whether the product contains an even or an odd number of ys with space-like indices, respectively . In particular, P,f4-1;"~ ( 5.4.15) Everything so far in this section applies in any number of spacetime dimensions and for any `metric' r} ,,,. In four spacetime dimensions, how- ever, there is a special feature, that no totally antisymmetric tensor can have more than four indices, so the sequence of tensors 1, y 0,YPI , 'ryfpl", ... terminates with the tensor (5 .4.12). Furthermore, each of these tensors transforms differently under Lorentz and/or parity transformations so 216 5 Quantum Fi eldsand Antiparticle s they are all linearly independent .' The number of linearly independent components of these tensors is ❑ne for 1, four for yP, six for +° O, four for P", and one for P'P" , or 1 6independent components in all . (The general rule is that a totally antisymmetric tensor with n indices in d di- mensions has a number of independent components equal to the binomial coefficient d!/n?(d - ra )!) There are at most v2 independent v x v matrices, so they must have at least 16 = 4 rows and columns . Dirac matrices of the minimum dimensionality are necessarily irreducible ; if reducible, the subspace left invariant by these matrices would furnish a representation of lower dimensionality . We shall therefore take the y "tobe4 x 4 matrices . (More generally, in any even number d of spacetime dimensions, one can form antisymrnetric tensors with 0,1,- -, d indices, which contain altogether a number of independent components equal t o ddf -2d ,=on!(d n)! so the y-matrices must have at least 2d/1 rows and columns . In spaces or spacetimes with odd dimensionality, the totally antisymmetric tensors of rank n and d - n can be linearly related by the conditions " [;'1 Y 02... Y Pr]~C.U1P2...~YLur+l7jur+2' *Yf'Rd)x for r = 0, 1, 2, ..., d -T 1, with EP 1A2' W totally antisymmetric, and the left- hand side taken as the unit matr ix for r = 0 . Under these con ditions there are only 211-1 independent tensors, requiring ~/-matrices of dimensionality at least 2(d-')I' .) Returning now to four spacetime dimensions, we shall choose an expl icit set of 4 x 4 y-ma trices . One very conven ient choice i s d 17 =-i 0~ ~ (5 .4.17)1 U -or Q where 1is the unit 2 x 2 matrix, and the components of er are the usua l Alternatively, these matrices can be shown to be linearly independent by noting that they form an orthogonal set, with the scalar product of two matrices defined by the trace of their product . Note that none of these matrices can vanish, because each component of each of these tensors is proportional to a product of different y-matrices, and such a product has a square equal to plus or minus the product of the corresponding squares, and hence equal to ±1. This constraint does not interfere with the inclusion of space inversion in the Dirac representation of the Lorentz group in odd-dimensional spacetirne, because here the tensor e~IP2--A' is even under inversion of space coordinates . If we don't care about space inversion, we can also construct lid-r]}2 _dimensional irreducible representations of the proper orthochronous Lorentz group in even spacetime dimensions by imposing the above condition relating antisymmetrized products of r and d-r Dirac matrices . An example is provided by the submatrices in Eqs .(5-4.19) and (54 .24) below . Pali matrice s 0 Ui=15.4The Dirac Formalism 1 0 -a Z 0 CT2217 (5.4.18) (The a; are just the 2 x 2 y-matrices in three dimensions .) It can be shown that any other irreducible set of i-matrices are related to these by a similarity transformation . From Eq . (5.4.17), we can easily calculate the Lorentz group generators (5 .4.6): C:ik2 ~io_ + ~cri 206k 0 0 Cr k 0 _(7i(5.4. I9) (5.4.20) (Here fi}kis the totally antisymmetric tensor in three dimensions, with 6123= + W We note that these are block-diagonal, so the Dirac ma~ traces provide a reducible representation of the proper orthochrvnaus Lorentz group, the direct sum of two irreducible representation with ,V = ±iCijkf'o . It is convenient to write the totally antisymmetric tensors (5 .4.11) and (5.4.12) in a somewhat simpler way . The matrix (5 .4.12) is totally antisym- rne#ric, and therefore proportional to the pseudotensor eP"q, defined as a totally antisymmetric quantity with E0123 = +1 . Setting p,a, r,qequal to 0,1,2,3, respectively, we see tha t where Ys = -iyOyIy2y3. The matrix ~s is a pseudoscalar in the sense that(5.4.21) (5.4.22) (5.4.23) (s.4.24) Similarly, vlP" must be proportional to &"n contracted with some matrix 4, and by setting p, u, z equal in turn to 0,1,2 or 0,1,3 or 0,2,3 or 1,2,3, we fin d The 16independent 4 x 4 matrices can therefore be taken as the com- ponents of the scalar 1, the vector y P, the antisymmetric tensor "fP¢, the `axial" vector,,j5y,, and the pseudoscalar y 5. It is easy to see that the matrix y5has unit square 218 5Quantum Fields and A ntipartic les and anticommutes with all YY fy5"19 =0. (5.4.27) The notation y5 is particularly appropriate, because the anticommuta- tion relations (5 .4.26) and (5 .4.27), together with Eq . (5.4.5) show tha t '75 provide a Clifford algebra in five spacetime dimensions . ~'°~T 1TY2 , Y3 For the particular 4 x 4 representation (5 .4.17) of the it-matrices, the matrix Y5 is a-~11(5.4.28) Y5= 1 This representation is convenient because it reduces V1 and ^ 5to block- diagonal farm . As we shall see, this makes it particularly useful for dealing with particles in the ultra-relativistic limit, v --- >c.(fit is not, however, the representation described in Section 1.1 that was originally introduced by Dirac, because Dirac was mostly interested in electrons in atoms where V << c, and in this case it is more convenient to adopt a representation for Which YOrather than y5 is diagonal .) The representation of the homogeneous Lorentz group we have con- structed here is not unitary, because the generators Kare not all represented byHermitian matrices . In particular, in the representation (5.4.17) we have 'j Hermitian, but Y~a is anti-Hermitian . Such reality conditions can conveniently be written in a manifestly Lorentz-invariant fashion by introducing the matrix P- iy Oof E q. (5.4.13), which in the representation (5 .4.17) takes the form 01 10 Inspec tionof Eq . (5.4.17) shows tha t and it follows then that fljp6ffl = /P17(5.4.29) (5.4.30) (5,x.3 1) Hence ,though not unitary ,thematrices D(A)satisfy the pseudounitarity relatio n Also, 75 is Hermitian an danticommutes with ffs o and it follows that -75Y,,(5-4-32 ) (5.4.33) (5.4.34) 5.5Causal Dirac Fields 219 The Dirac and related matrices al so have important symmetry proper- ties.Inspection of Eqs .(5.4.17) a nd(5.4.18) shows that j'~ is symmetric foru = 0, 2 and antisymmetric for y=1,3, so where T denotes a transpose, and rr2 0 It follows immediately that is+(K ~5w{5.4.35} (5.4.36) {x.4.37} (5.4.38) (5.4.39) These signs will prove significant when we consider the charge-conjugation properties of various currents in the next section .Of course, we can combine our results for adjoints and transposes to obtain the complex conjugates of the Dirac and allied matrices : Y5 - -~`~"r~5`~~~~ , -flWi5y~'W-1fl- 5.5 Causal Dirac Fie lds(5.4.40) (5.4.41) (5.4.42) (5.4.43) We now want to construct particle annihilation and antiparticle creation fields that transform under the Lorentz group according to the Dirac representation of this group, discussed in the previous section . In general these take the form given in Eqs .(5.1.17) and ( 5.1.18): and(5.5.x) (5.5.2) 220 5Quanturn.Fieldsand Antiparticle s with the part icle spec ieslabel omitte dhere .In order to ca lculate the coefficient functions u~( P, cr) and vr( P, a) appearing in these formulas, we must first use Eqs . (5 .1.25) and (5 .1.26) to find u eand vefor zero momentum, and then apply Eqs . (5.1.21) and (5 .1.22) to ca lculate them for arbitrary momenta, with D?,,(A) in both cases taken as the 4 x 4 Dirac representation of t he homogeneous Lorentz group discussed in the previous section . Using Eq . (5.4,19), the zero-momentum conditions (5,1 .25) and {5 .1. 6} read * and In other words, if we regard u,,±(O, u) and v, ,t(a,a)as the m ,a elements of matrices U±and Y±, we have in matrix notatio n U+J(P='au+- 2 - and(5.5.3) (5.5.4) Now, the ( 2j+1)-dimensional matrices J(}1 and ----J{ j)* and the 2 x 2 matrices ~a all provide irreducible representations of the Lie algebra of the rotation group . A general theorem of group theory known as Schur's lemma tells us that when a matrix like Ut or V± connects two such representations as in Eqs .(5.5.3) and (5 .5.4), the matrix must either vanish (a possibility of no interest here) or else be square and non-singular . Hence the Dirac field can only describe particles of spin j = ~ (so that 2j + 1 = 2) and the matrices J O/1) and-J(1~2),,must be the same as 'a up to a similarity transformation . In fact, in the standard representation (2.5.21),(2.5.22) of the rotation generators, we havej(1/2) =~a and -J(1/2)* = ~d 2662 . It follows then that U±and Vtff 2roust commute with a, and hence must be proportional to the unit matrix : We are here dropping the specie slabel n, and replacing the four-component inde x C with a pair of ind ices, one 2- valued indexm labelling the rows and columns ofthesubmatri cesin Eqs . (5.4.19) and (5 .4.20),and a second index taking valuest,labelling the r owsand columns of the supermatrix in Eqs .(5.4.19)and (5.4.20). 5.5Causal Dirac Fields In other words c+ 0 C_ 0 D c- 0 VA 1)d+ 0 d_d+ and the spinors at finite momentum ar e U(P' U) =~mlpPD (L(p)) u(O, a) V(P'U) =~m1pPD(L(p))v(0,a)221 (5.5.G) (5-5-7) It now only remains to say something about the constants ct and d±.In general, these are quite arbitrary -we could even choose c_ and d_ or c+ and d+ to be zero if we liked, so that the Dirac field would have only two non-vanishing components . The only physical principle that could tell us anything about the relative values of the ct or the d ±is the conservation of parity . We recall that under a space inversion, the particle annihilation and antiparticle creation operators undergo the transformations : and so a ,j d'p ve-p,cr)e-~"Y-V~ (p, a) or(5.5.$) (5.5.9) (5.5.10) (5.5.11) (.5.12) (5-5-13) (Since J32 = 1, we are no longer making a distinction between #and #-1.) In order that the parity operator should transform the annihilation and creation fields at the point x into something proportional to these fields 222 5Quantum Fields and Antiparticle s at fix, it is necessa ry that flu(O, a) and flv(0, a) be proportional to u(4, rr) and v(O, a), respect ively: where bu and b,are sign factors, b2 = h~ = 1.In this case, the fields have the simple space-inversion properties : (5-5-15 ) (S.S.1G) By adjusting the overall scales of the fields, we can choose the coefficient functions at zero momentum to have the form : V2 bu Z1(O,2~ = 10 bV0 U(O, -0 bu 0(5.5.17) (5.5.18) Now let's try to put together the annihilation and creation fields in a linear combinatio n that commutes or anticommutes with itself and its adjoint at space-like separations . A straightforward calculation gives (5.5.20) where (5.5.21) By using either the eigenva]ue conditions (5 .5.14) or the explicit formulas (5.5. 7)and ( 5.5.18), we find at zero momentum : 2(2n)32(2n)3 - 5.5 Ca usalDirac Field s From Eqs . (5.5.6) and (5.5.7)we have the n _N(p) = ~~D (L(p)) [I + bjIDI (L(p) ) M(p) = ' D (L(p)) [1 -~ bt;#]DI (L(p)) 2po The pseudounitarity condition (5 .4.32) yield s D (L( P)) flDI (L(p)) = fl and D (L(p)) DT (L(p)) -- D (L(p)) #D-'(L(p)) # -223 {s.5.2a} (.s.2s) We also recall that ~ T -iyo, so by using the Lorentz transformation rule (5.4.8) we hav e ,O(L(p))#D_'(L(p)) = -iL$I'(P)TI' =-iP'7I'/M' (5.5.26) Putting this together, we find' P 1 Using this in Eq.(5.5.20) yields finall y [W((X), W'(Y)l Tb,mjflA+(x -y) where A+ is the function introduced in Section 5 .2: P((5.5.27) dP ip_x(5.5.28) (s.s.29) We saw in Section 5.2that for x - y space-like ❑+(x- y) is an ev en funct ion of x -- y a nd, of course, th is implies that itsfirst de rivatives are odd functions of x - y.Hence, in order that both the der ivative and the non-derivative terms in the commutator or anticommut ator should vanish at space-like separations ,it is n ecessary and suffic ient tha t ~ II=+ 141z(5.5.30) Sometimes an extra factor p° f m is included in the Dirac spinnrs, so that m appears in place of10in the denominators of the spin sums (5 .5.27) and (5.5.28). The normalization convention used here has the advantage that it goes smoothly over to the case rn = 0. 224 and IKt'bu=±IAI'bu. Clearly Eq . (5.5.30)is only possible if we choose the bottom sign, -}- =+; that is, the particles described by a Dirac field must be fermions .It is also then necessary that ~K [2 = 1~12 and b,,=--b,. Just as for scalars we have the freedom to redefine the relative phase of the creation and annihilation operators to make the ratio rc/A real, in which case K = A, and byadjusting the overall scale and phase of the field W we may then tak e K=A= 1. (5 .5.32) Finally if we like we can replace tp with y 5W, which changes the sign of both b .and b,so we can always tak e For future use, we record here that the Dirac field is now 0wc(x)= (21r)-1/2 dip [ue( P,u)~'"a(p xa)+vc{P, a}e-t#'-xa't( P} er)] (5.5.34) while the coefficient functions at zero momentum ar e 0 -1 The spin sums are(5.5.3 1) (5.5.33) (5.5.35) P P so the anticommutator is given by Eq. (5.5.20) as(5.5.37 (5.5.3$) Now let's return to the requirement that under a space inversion the field T(x) must transform into something proportional to W{gx} . For this to be possible the phases in Eqs .(5.5.15) and (5 .5.16) must beequal, an d5Quantum Fields and Antiparticles 5.5 Causal Dirac Fields 225 so the intrinsic parities of particles and their antiparticles must be related by That is, the i ntrinsic parity r~rl` of a state consisting of a spin ~ particle and itsantiparticle isodd.It is for this reason that negative parity mesons like the p 1and .l/ can be interpreted as s-wave bound states of quark- antiquark pairs . Eqs .(5.5.15) and ( 5.5.16)now give the transformation of the causal Dirac field under space inversion a s P(x)P-'=q*flT(9X) . (5 .5.41) Before going on to the other inversions, this is a good place to mention that Eqs . (5.5.14), (5.5.33), and (5.5.26) show that u(p, a) and v(p, cr) are eigenvectors of -ip "y,,f rra with eigenvalues + 1and - 1, respectively : (i`YY + M)U( P} (-T)= 0 , (---aplyu +M)V (p}a)= 0- (5.5.42) If follows then that the field (5.5.33) satisfies the differential equatio n (710~+M.)fi(x) = 0 . (S.S.43) This is the celebrated Dirac equation for a free particle of spin ~ . From the point of view adopted here, the free-particle Dirac equation is nothing but a Lorentz-invariant record of the convention that we have used in putting together the two irreducible representations of the proper orthochronous Lorentz group to form a field that transforms simply also under space inversion . In order to work out the charge-conjugation and time-reversal properties of the Dirac field, we will need expressions for the complex-conjugates of the u and v coefficient functions . These functions are real for zero momentum, but to obtain the coefficient functions at finite momentum we have to multiply with the complex matrix D(L(p)) .From Eq.(5.4.41) we see that for general real co,,, ; and so i n particular We also note that Wmmi flu(0, cr) =-v(O,a) and - 1 Pv(O,u)= -u(0, 6 ), so (5.5.44) (5.5.45) In order for the field to transform under charge-conjugation into another field with which it commutes at space-like separations, it is necessary 226 5Quantum Fields and Antiparticle s again that the charge conjugation parities of the particle and antiparticle be related b y In this case, the field transforms as(5.5.46) W(X)C-1~-~*flyW*W (5.5.47) (We are calling the Hermitian adjoint of the field on the right-hand side w' instead of ipt to emphasize that this is still a column vector, not a row .) Although we have been distinguishing particles from their antiparticles, we have not ruled out the possibility that the two are actually identical . Such spin ~ particles are called Majorana fermions .Following the same reasoning that led to Eq . (5.5.47), the Dirac field of such a particle must satisfy the reality conditio n For Majorana fermians the intrinsic space-inversion parity must be imag- inary, q=Vii, while the charge-conjugation parity must be real, ~ = ± 1 . There is an important difference between fermions and bosons in the intrinsic charge-conjugation phase of states consisting ❑f a particle and its antiparticle . Such a state may be writte n c'a'J/d3p ' where (Dois the vacuum state . Under charge-conjugation, this state is transformed into a}a, Interchanging the variables of integration and summation and using the anticommutation of the creation operators and Eq. (5,5.46), we can rewrite this as C - -- /d3p d3p':C(P{f LT';P,a)u'(p, Q)a"(p', a')(Dq C'C' That is, the intrinsic charge-c onjugation parity of astate consisting of a partic ledescribed bya Dirac field and its antiparticle isodd, in the sense that if the wave function Z of the state is even or odd under interchange of the momenta and spins of the particle and antiparticle, then the charge-conjugation operator applied to such a state gives a sign -1 or +1, respectively . The classic example here is positronium, the bound state of an electron and a positron . The two lowest states are a pair of nearly degenerate s-wave states with total spin s = 0 and s = 1, 5.5 C ausalDirac Fields 227 known respectively as para- and ortho-positronium . The wave function for these two states is even under interchange of momenta and odd or even respectively under interchange of spin z-components, so pars- and ortho-positronium have C = +1 and C = -1, respectively . These values are dramatically confirmed in the decay modes of positronium : para- positronium decays rapidly into a pair of photons (each of which has C = -1), while ortho-positronium can only decay much more slowly into three or more photons . In the same way, single p° and w° mesons are produced as resonances in high-energy electron-positron annihilation through a one-photon intermediate state, so they must have C = -1, which is consistent with their interpretation as quark-antiquark bound states with orbital angular momentum zero and total quark spin one . Now we come to time-reversal . Recall the transformation properties of the particle annihilation and antiparticle creation operators given by Eq. (4.2.1 S) Ta( p, a)T- i Time-reversal of the field (5 .5. 4)thus gives(5.5.49) (5.5.50) In order to put this back in the form given for ip, we shall redefine the variables of integration and summation as -p and - a, so we nee d }, T) and v*(-p, -a-) in terms of z~~ (P, a) and r~~(P, a formulas for uf (-p, -- C respectively . For this purpose, we can use the fact that Ito anticommutes with Pand commutes with y5 together with our former result for D(L(P ))' to writ e Also ,Eqs. (5.4.36)and (5.5.35)-(5 .5-36) giv e SO U* -75W V(P' U)(5.5.5Z) (5.s.s2) 228 5 Quantum Fields andAntiparticle s We see thin that in order for time-reversal to take the Dirac field into something proportional to itself at the time-reversed point (with which it would anticommute at space-like separations) it is necessary that the intrinsic time-reversal phases be related b y Ce Cs and in this case(5.5.53) TW(x)T`1 -~ 'Y5`6W(-'Yx). (5 .5.54) Now let us consider how to construct scalar interaction densities out of the Dirac fields and their adjoints . As already mentioned the Dirac representation is not unitary, so WtW is not a scalar . To deal with this complication it is convenient to define a new sort of adjoint : Using the pseudounitarity condition (5 .4.32), we see that the fermion bilinears constructed with have the Lorentz transformation propert y Uo(A)Ifp(x)Mw (x)] pro'(A)= y~(Ax)D-1(A)M~(A).~p(Ax). (5.5.56) Also, under a space inversio n Taking the matrix M as 1,Yk~ f '1V,757'1 } or y5yields a bilinear fPV that transforms as a scalar, vector, tensor, axial vector, and pseudoscalar, respectively . (The terms `axial° and 'pseudo" indicate that these have space- inversion properties opposite to those of ordinary vectors and scalars : a pseudoscalar has negative parity, while the space and time components of an axial vector have positive and negative parity, respectively .) These results apply also when the two fermion fields in the bilinear refer to different particle species, except that in this case a space inversion also yields a ratio of the intrinsic parities . For instance, the original Fermi theory of beta decay involved an inter- action density proportional to 1~0~'Vn Fp,, Yuu. Later it was realized that the most general Lorentz-invariant and parity-conserving non-derivative beta decay interaction takes the form of a linear combination of products like this, with y ,,replaced with any one of the five covariant types of 4 x 4 matrices 1, y", u'', 75YP, or ys- (As discussed in chapter 2, we are defin- ing the space-inversion operator so that the proton, neutron, and electron all have parity +1 . If the neutrino is massless then its parity may also be defined as + 1, if necessary by replacing the neutrino field with Y5 W,-) When Lee and Yang7 called parity conservation into question in 1956, they expanded the list of possible non-derivative interactions to include ten terms proportional to iP-PMW n xeMlP V and also t~,,MW ,~~,MYsWV, with M running over the matrices 1, y,,fA', M '`}or Y5. 5.6 General Irreducible Representations 229 It is also of some interest to study the charge-conjugation properties ❑f these bilinears . Using Eqs . (5.5.47) and (5 .4.35)-(5 .4.39), we hav e = jp(g-1T y= ±tPMW[5.5.5$] the sign in the last expression being + for the matrices i, Mg, and y5, and -for y ,,and f~~. (The minus sign in the first line arises from Fermi statistics, We ignore a c-number anticommutator .) A boson field that interacts with the current fpMV must therefore have C = + 1for scalars, pseudoscalars, or axial vectors, and C = -1 for vectors or antisymmetric tensors . This is one way of seeing that the n° (which couples to pseu- doscalar or axial-vector nucleon currents) has C = +1, while the photon has C = -1 , 5.6 Ge neralIrreducible Representations of the Homogeneous Lore ntz G roup' We shall now generalize from the special cases of vector and Dirac fields to the case of a field that transforms according to a general irreducible representation of the homogeneous Lorentz group . All fields may be constructed as direct sums of these irreducible fields . A general representation of the proper orthochronous homogeneous Lorentz group (or, more properly, its infinitesimal part) is provided by a set of matrices ~V satisfying the same commutation relations (5 .4.4) as the generators of the grou p P,~~ pIr I _ ~ ~ ~~ q(Yu + ~ p 17vor - f av1PIt -A40117 11P) 11 (5,x.1) (Of course, ,.w= and indices on are as usual raised or lowered by contraction with qy" orq,,.)To see how to construct suc h matr ices, first divide the six independent component soff.,into two three-vectors :an angular momentum matri x f1 23 a 5a = f3l, 3 =f12 (5.6.2) and a boost I =jlU a 2_f20, ,*-3.= f 30. (S.G.3) Eq. (5 .6.1) then reads (5.6.4) * This section dies somewhat out of the book's main line of development, and may be omitted in a first reading . 230 isjl= -ic ijk k 7(5.6.5) (5-6-6) where i, j, k run over the values 1, 2, 3, and ~ iftis the totally antisymmetric quantity with C123 + 1. Eq . (5.6.4) just says that the matrices generate a representation of the rotation subgroup of the Lorentz group, and Eq.(5.6.5)just represents the fact that 4^is a three-vector . The minus sign in the right-hand side of Eq .(5.6.6)arises from the fact that -1, and plays a crucial role in what follows . It is very convenient to replace the matrices and with two decoupled spin three-vectors, writin g ~(~ L`) (5.6.8) It is easy to see that the commutation relations ( 5.6.4)-(5.6.6)are equivalent to (5-6-g) (5.6.]D) (s.6.i1) We find matrices satisfying Eqs . {5,6.9~-{5.6.11 } in the same way that we find matrices representing the spins of a pair of uncoupled particles as a direct sum . That is, we label the rows and columns of these matrices with a pair of integers and/or half-integers a, b, running over the value s a= b= and take*FA, -A+ 1, ..-,+A , B, -B +1,---,+B 6'hJa0aA) (-4)db',ab =7 bola($)(5.6.12) (5.6.13) (5.6.14) (5.6.1 S) wherej(A)and J 111)are the standard spin matrices for spins A or B: j(A) 3)u'u (jAJ ±Ij~A) a'a=a6a'~ (5.6.16) (5-6-17) There is an altern ativeforrnalism ,8 based on the fact that the spin j representation of the rotation group can be written as the symmetrized direct product of 2 jspin 1/2representations - i.e.,as a symmet ric 5U[2] ten sor w ith2jlwo-valued indices .We can t herefore write fields belongi ngto the(A,B)representation with 2A two-valued (1/2.0)indices and 2 Btwo-valued (0, 1/2) indices , thelatter written with dot sto distinguish them from theformer ,5Quantum Fields and Antiparticles 5.6 General Irreducible Representations 231 and likewise for J 01). The representation is labelled by the values of the positive integers and/or half-integers Aand B . We see that the (A, B) representation has dimensionality ( 2A+1)(2B + 1) . The finite-dimensional representations of the homogeneous Lorentz group are not unitary, because and are Hermitian, and therefore is Hermitian but is anti-Hermitian . This is because of the i i n Eqs.(5.6.7)and (5.6.8),which is required by the minus sign in (5.6.6), and hence stems from the fact that the homogeneous Lorentz group is not the same as the four-dimensional rotation group SO(4), a compact group, but instead is the non-compact group known as S O(3,1). It is only compact groups that can have finite-dimensional unitary representations (aside from representations in which the non-compact part is represented trivially, by the identity) . There is no problem in working with non-unitary representations, because the objects we are now concerned with are fields, not wave functions, and do not need to have aLorentz-invariant positive norm . In contrast, the rotation group is represented unitarily, with its genera- tors represented by the Hermitian matrice s f_~Q/ + ? (5.6.18) By the usual rules of vector addition, we can see that a field that transforms according to the (A,B)representation of the homogeneous Lorentz group has components that rotate like objects of spin j, wit h This is enough to identify the (A, B) representations with the perhaps more familiar tensors and spinors . For instance, a (0,0) field is obviously scalar, with only a single j-0 component . A ( 1,0) or (0, -1)field can only have i = +' ;these are the top {i .e., r'5=+1) and bottom 05 =-1) two components of the Dirac spinor . A ( 1, field has components with j = 1 and j = 0,corresponding to the spatial part v and time-component vo of a four-vector vy . More generally, an (A, A) field contains terms with only integer spins 2A, 2A - 1,... , 0, and corresponds to a traceless symmetric tensor of rank 2A. (Note that the number of independent components of a symmetric tensor of rank 2A in four dimensions i s and the tracelessness condition reduces this t o as expected for an (A,A)field .) One more example : a (1, 0) or { 0,1} field can only have j = 1, and corresponds to an antisymmetric tensor F"' that 232 5Quantum Fields and Antipartic les satisfies the further irreducibility `duality' condition s 2 for (1, 0) and (0,1 )fields, respectively .Of course, it is only in four dimensions that an antisymmetric two-index tensor FW can be divided into such `self-dual' and 'anti-self-dual' parts . A general tensor of rank N transforms as the direct product of N(1, z ) four-vector representations . It can therefore be decomposed (bysuitable symmetrizations and antisymmetrizations and extracting traces) into irre- ducible terms (A, B) with A= 12 ,N12 - 1,.. , and B =:; N12, N12 - 1,... In this way, we can construct any irreducible representation (A,B)for which A + B is an integer . The spin representations, for which A+Bis half an odd integer, can similarly be constructed from the direct product of these tensor representations and the Dirac representation (2, 0) D (0 , For instance, taking the direct product of the vector representatio n and the Dirac (~, 0)~(D, ~)representation gives a spinor-vector VA, that transforms according to the reducible representatio n 2 2 2 2 2 The quantity y,,WA would transform as an ordinary (~,0)~(0, Dirac field, so we can isolate the (~,1) ED (1 ,-1)representationt by requiring that ymT1` =0. This is the Rarita-Schwinger field .9 So far in this section we have ❑nly considered the representations of the proper orthochronous Lorentz group . In any representation of the Lorentz group including space inversion, there must be a matrix #which reverses the signs of tensors with odd numbers of space indices, and in particula r In terms of the matrices ( 5.6.7) and (5 .6.$), this is (S.G.20) Thus an irreducible (A,B) representation ❑f the proper orthochronous homogeneous Lorentz group does not provide a representation of the Lorentz group including space inversion unless A = B.As we have seen, these (A, A )representations are the scalar, the vector, and the symmetric traceless tensors . For A:~B, the irreducible representations of the Lorent z ~ According to Eq .(5.6.18),such a field transforms under ordinary rotations as a direct sum of two j = 3/2 and two j =1!components . The doubling is eliminated by imposing the Dirac equation [y"a,, +m]VJ' = 0, and the remaining j =-21component is eliminated by requiring that a"V +2 -Itd. With these conditions, the field describes a single particle of spin j - 3f2, 5.7General Causal Fields 233 group including space inversion are the direct sums (A, B )E)(S, A), of dimensionality 2(2A + 1) (2B+ 1) . One of these is the (~, D ) @ (0, ~ } Dirac representation discussed in Section 5 .4. The 4 x 4 matrix (5 .4.29) provides the P-matrix for this representation . Another familiar example is the (1,0)x(0,1 )representation, which as we have seen is just the antisymmetric tensor of second rank, including both self-dual and anti-self-dual parts . 5,7 General Ca usal Fie lds" We now proceed to construct causal fields that transform according to the general irreducible (A, B) representations described in the previous section . The index Iis replaced here with a pair of indices a, b, running over the ranges ( 5.6.12), {5.6.13}, so the fields are now written a s Wab(x)=(2n)-'I' 1 :Jdip [x a(p,ff)~'P'xuab(P,6) act(Pacr)e-p xvab(P} d)] (5 .7. 1) with rc and ~arbitrary constants . We are here leaving open the possibility that this particle is its own antiparticle, in which case a~(p, a)= a(p, a) . Our firs# task is to find the zero-momentum coefficient functions u ,,b(0,6) and vab(0, u).The fundamental conditions (5.1.25)-(5.1.26) onu(O,a)and v(0,cT)read here t1Rgo,dpt a.i1='-'" I fab,abuab ( 0 ,a) ir a,b - dab(() ,&)J ab a ~ ,ab Vab(0xCT)~ 0 or using Eqs . (5.6.14)-(5.6.15) E Udbb(oaoiaff= E iaa"Uab(O~a)+1: jbb~ Uab (0, a} , (5 .7.2) A b But Eq .(5.7.2)is the defining condition for the Clebsch-Gordan coeffi- cients CAR(,ju;ab)! These coefficients are defined by the requirement tha t This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 234 5Quantum Fi eldsand Antiparticle s ifTab are states that under an infinitesimal rotation transform a s aa bb then, under the same rotation, the stat e ab transforms a s Inspection of Eq .(5.7.2)shows that this requirement is satisfied by the coefficients uab(0, G), and therefore, up to a possible proportionality factor, gab{0, U} is just C AB (, jU; ab). This constant is conventionally chosen so tha t uab(D,a)=(2m)- "'CAB UU ;ab). (5.7.4) This result is unique because each irreducible (A , B) representation of the homogeneous Lorentz group contains a given spin j representation of the rotation group at most once .Similarly, inspection of Eqs . (5.6.16)-(5,6,17) shows that the complex con jugates of the angular momentum matrices are dfT (5.7-5) Therefore if we write Eq . (5.7.3) in terms of(-1)j-¢Vab(P,-9), it takes the same form as Eq . (5.7.2). With a suitable adjustment of a constant factor, the unique solution for v(0,u) is We must now perform a boost to calculate the coefficient functions for finite momentum . For a fixed direction P°P/lPl, we can write the boost (2.5.24) as a function of a paramete r9 defined b y cosh 0Z= Vp 2+ M2 / M, and write LPV(0) i n place of LP,( p),wheresink O = ~ P~Jm (5 .7.7) LVO) =6ik+(cosh 0-1)^;Fka L'()(0) = L°i(F) = P;Binh 6, Loo(O) = cosh 0. The adva ntage of this paramete rizationis that(5.7.5) {5.7.x) For infinitesimal 0, we have [L(O)]P, --+Sou + wPu, where do = w°i = p ;B and wiiT woo = 0 . Following the same reasoning that led from Eq . 5.7 General C ausal Field s (2.2.24)toEq.(2.2.26), it fol lows then tha t D (L(p)) = exp(-gi p-. '0) .235 (5.7.1Q) This is for any representation of the homogeneous Lorentz group ; for the irreducible (A,B) representations, Eqs .(5.6.7)and (5 .6.$) giv e and since .4 and are commuting matrice s D (L( P))=exp(- p- t40) exp(+ p-W) In more detail, using Eqs . (5.6.14) and (5.6.15) D (L(p))db%ah =(exp P)Q'a (exp( +P, ))b1b(5.7.11) (5.7.12) (x.7.13) Eqs. (5.7.4) and (5.7.6)then give the coefficient functions at finite momen- tum a s Uab* GT) (exp P, j(A)O)) ad (eXp( + P -j(B) 0) )W x CAB(Jo-; a'b' ) and(5.7.14) These results give the field explicitly for a given transformation type (A,B), so the field { 5.1.31} of this type is unique up to the choice of the constant factors ic and ).. it is very easy in this formalism to construct Lorentz scalar interaction densities . The (AB) representation of the homogeneous Lorentz group is just the direct product of the (A, 0)and ( 0, B) representations, so the general Lorentz transformation rules (5.1.6), (5 -1.7) read her e Uo{A)v,,b (x)a 1~} _ ~~ ° d(A-')Do~+(A-1)~~r b'(Ax). {5.7.1 G} a' b' Furthermore, Etas . (5.6.14) and (5.6.15) show that the matrix generators of the (A, D) and (0,8) representations are just the spin matrices for spin A and B, respectively . Thus we can construct scalars of the form 5.7.17$a,a4...a,,;hib2 -brtWalbi~~~ ~~~~2W... ?1(X .) Gl1ul.,.anb1f12_..,hrr bysimply taking g,,,2 ...are;b,b:...b,,as the product of a coefficient for coupling spins A1, A2, ... A„ to make a scalar and a coefficient for coupling spins B1,B2,....B,to make a scalar . (Even though we do not explicitly consider interactions involving derivatives, we will in this way obtain the most general interaction involving n fields, because the derivative of a field of 236 5Quantum Fieldsand Antiparticle s type (A, B) can always be decomposed into fields of other types without derivatives .) For instance, the most general Lorentz scalar formed from a product of three fields of transformation types (A,, BI), (A2, BA and {A3, BO is y: ( A,A2 A3 BI B2 B3 (1) (2) (3) (5.7.18) g Y: WQ1}~[p~2J~2Wp~3~j3 ayaza3 bj b2b3 ~~ ~~ ~~ ~~ ~~' ~ 3 with a single free parameter g . This is the most general three-field inter- action . (The brackets in (5.7.17) denote the Wigner "three j„ sym6ols :10 it j2 h 3 3) = 1: Cil j2(hM', M IM2)Cj3ii(00, M' M3)1M2 M~rM3 which describe the coupling of three spins to make a rotational scalar .) For the S-matrix to be Lorentz-invariant it is not enough that th e interaction density (x) be a scalar like (5.7.18); it is also necessary that fi(x) should commute with (y) at space-like separations x -- Y.To see how to satisfy this condition, consider the commutator or anticommutator of to fields for the same particle species, a field V of type (A,B),and the adjoint tit of a field of type (A,P). We fin d lvabw, V-alb(y)]:F=(21r)-'/'f d3p (2~')-'nab,44 ) x[Kke1 P'(`-Y) (5.7.19) where x(p) is the spin su m (2po)-i71ab,ab (R) uab(P} 67) uQ (P, a)_ Vab(P,UP*b{ p, a} (5 .7.20) and as usual, the top and bottom signs are for bosons and fermions, respect ively. (We allow here for different coefficients k a ndAin the t~ field .) In more detai l 7r.b,a;(P) = E 1 : E CA,5(ju; A) C ;jp(jer;7a'Y dW ~?h r ))aa' ( X exp( - J, - ) Pj( A)O~ (exp ~pj ~~~o));b (5 .7.1) The function n( P) has been calcula ted explicitly ." What concerns us here is the fact that it t urns out to be the mass-shell value of a polynomial funct ion P of pand p ° Ir"b'ab(p) ~Paba6 (PI. V1W + MI) (5.7.22) 5.7 General Caus al Fields 237 and that P is even or odd according to whether 2A + 2B is an even or odd integer p(_p,_Pa) = (_)1A +1kP(Pa P°). (5.7.23) We shall check this here for just one particular direction of p . Taking p in the three-direction, (5.7.21) gives 7tab,ab(P)_ CAR( .1a; ab) C'qp(1a; ab)exp([-a + b -& + b ]B) The Clebsch-Gordan coefficients vanish unless a = a + b and or = a + b, so we can replac e -a +b - a+ b = T2a+u+ 2b-a=2b -2a. Wecan write exp( ±O) as(pp±p3)/M,sohere X[(pp + p3)IM ] ke -P,)/Mi 2b-2a 2Q-2h(b ~ a} (a ~b) where p° = p + m2 . We see that 7r(p) can indeed be written as the mass-shell value of a polynomial F(p, pO). Also, Z b- 2a equals 2A + 2A minus an even integer, so the polynomial satisfies the reflection conditio n Any polynomial in p and p ~ + m . can be written in a form linear in p2+ m (by expressing even powers of p + Win terms ❑fp)so -n(R) can be writte n Rab,ab(p) =Pab,A(P) + 2 VP 2 +M2Qal,db(P) where P and Qare now polynomials in p alone, wit h p (_ P) = (_)2A + 2,1p(p) Q(_p) =__(_)2A+2RQ(p) .(5.7.24) (5.7.25) (5.7.26) Forx - y space-li ke, we can adopt a Lorentz frame in w hich xO = y°, a nd write E q. (5.7.19) as In order that this should vanish when x z~- y, we must have 238 5 Quantum Fields and Antiparticle s Now let us consider the special case where and i p- are the same, so in particular A = A and B = B . (It is unavoidable that such commutators or anticommutators will appear in [-*(x), (y)], because the hermiticity of the Hamiltonian requires that if (x) involves y), it also involves Wf~ In this case, Eqs . {5.7.27} give s This is possib le if and only i f ± (- 1)2A+2B- +1 (5 .7.28) and IKI2= I~I1(5.7.29) Of course, 2A +2B differs from 2jby an even integer, so Eq .(5.7.28) says that our particle isa boson or fermion accordi ngtowhether 2j is even or odd. This is the general relation between spin and statistics,12 of which we have already seen special examples for particles described byscalar, vector, or Dirac fields . Now let's return to the general case, where the fields ipand i~ may be different . Using E q.(5.7.27), and dividing both sides by JR11 we have It follows that, for any field, 2 (-)2$c rc , (5 .7.30) where c is the same factor for all fields of a g iven particle . Furthermore, Eq, (5 .7.29) shows that c is just a phase, jcj = 1 . We can therefore eliminate c for all fields by a redefinition of the relative phase of the operators a( p, cr) and a0 ( P, a), so that c = 1, and hence ~ = (-)2' K . Also, the factor K for each field type may be eliminated by a redefin ition of the over-all sca le of the field . We emerge from all this with a formula for the (A, B) field of a given particle, that is unique up to overall scal e Vaf~(_x)_ (27r)-9 f 2 Ejd3P [u(p acr)a(p} a)Bipx The different fields for a given particle do not really represent possibil- ities that are physical ly distinct . For instance, the possible fields for j = 0 are those of tPe ( A,A) because the triangle inequality ~ A - B ~ :!9 j :9 A+ B here requires A - B ).Starting with a (U, 0) scalar field 0, we can easily 5.7General Causal Field s construct such {A, A} fields from the 2Ath derivativ e where 1} here denotes the traceless part ; for instanc e 02 - 1 OXIIOXV239 (5.7.32) (Recall that a traceless symmetric tensor of rank Ntransforms according to the (N12, N/2) representation .) But Eq . (5.7.31) represents the unique causal (A, B) field for a given particle of spin j, so the (A,A)fields (5.7.3 1) for j = 0 can be nothing but linear combinations of the 2Ath derivatives (5.7.32) of a scalar field . More generally, any field (A, B) for a given particle of spin jcan be expressed as a differential operator of rank 2B acting on the fielcll3 rp,(x) of type (j, 0)(or a differential operator of rank 2A acting on the field of type (0,.1)). To see this, consider the fiel d This transforms as the direct product of the representations (B, B)and (j,4), and hence by the usual rules of vector addition, i t can be deco mposed into fields transform ing according to all theirreducible representations (A,B)with ~j - B IS A :::~j + B, or equivalently JA--BI j A+B. Since Eq .(x.7.31) represents the unique field of type (A, B) foragiven particle of spin j, it can be nothing" but the (A,B) field obtained from the derivatives (5.7.33). Now let us consider the behavior of these fields under inversions, beginn ing with s pace inversion . Using theresults of Section 4.2, the space- inversion properties of the part icle annihilat ion and antiparticle creat ion operato rs are, Pa"(p, Q)P-' = r joao~(-p,cr) ,(5.7.34) (5.7.35) where qand r1°are the intrinsic parities for the particle and antiparticle, respectively . The general causal (A,B)field (5 .7.3t)thus transforms unde r The only possible flaw in this argument would be if some ofthe (A, B) fields obtained in this way actually vanished . But in this case, the 0) field ~p, wou ld satisfy a field equation f 6{r? f r?x} (p(x) 0 and fence, for eachE T, a(dp)u,,( P, a)R 0. For the (j, Q) repre- sentation the Clebsch-Gordan coefficient C fl[ja;a- is just the Kronecke rsymbol Sid, so this would require M a(ip)Daa .(L(p)) _ O,which is impossible unless all the Mff(dp) vanish, since D(A) has an inverse D(A-r ) . The (j, 0) fields (p,(x)thus satisfy no field equation other than the Klein-Gordon equation (E] - m 2}rpff(x) = 0, and therefore none of the (A, B) fields obtained from (5,7 .30 can vanish . 240 S Qua ntum Fields and Antiparticle s the parity operator P int o ,b(x)P-' =(27r)-'/'E f dp [q*a(-p,a)eP'x u A,,b(p,a ) We want to change the integration variable from p to -p, and for this purpose we will need to evaluate u,,b(- F, or) and a,,&(- F, a). To do t his, we need only glance back at (5 .7-14) and (5 .7.15), and use the symmetry property of the Clebsch-Gordan coefficient 14 CAB( Ja;ab)= (-)"-8T~C,9A(iC;bac). Thisgives so'B(-Pr ~) =(-)A+B-jU~Q gyp, ~~, PWab(x)P-1= (27c)-3f2fd3P (_1)A+11-j x[a(p .e~xU(p,~) + c(_)2B~ct(p,~)eV(p, u)](5.7.37) (5.7.3 S) (5.7.39) 3.7.40) where, as before, Yx_ (--x, x°) . This is the causal field W~~ evaluated at fix, except that the coefficients of the annihilation and creation terms may not be the same as called for in Eq . (5.7.31). But these coefficients mu stbe the same up to an overall constant factor as in Eq . (5.7.31) because, aside from scale, Eq . (5.7.31) is the unique causal field of any type . Hence the ratio of the coefficients of the two terms in Eq . (5.7.40) must be the same as in Eq . (5.7.31) (but with Breplaced with A because this is supposed to be a (B, A) field s But A - B differs from the spin j by only an integer, so this gives(5.7.41) (5.7.42) We saw special cases of this result in Sections 5 .2, 5.3, and 5 .5, where j=0, j = 1, and j = 1, respectively . We now see that the resutt is general ; the intrinsic parity q'rJof a particle-antipart iclepair is +1 for bosons, and -I orferrnions .Using Eq . (5.7.42) in Eq . (5.7.40), our final result for space inversion i s Let's see how this applies to the Dirac field . For the top 0) and bottom (0,1)components of the Dirac field, the sign (-1 )A+~-~ is just 5.7 General C ausal Fiel ds 241 +1, so the parity operator simply takes x into -x ; reverses the top and bottom components ; and multiplies the field with q*. The reversal of the top and bottom components of the Dirac field is accomplished by the matrix Pin (5 .5.41). Now let us consider charge-conjugation . Its effect on the particle annihilation and antiparticle creation operators i s Ca(Pxa)C-1 = ~* d(Pra)y (5.7.44) (.7.45) where and ~ 'are the charge-conjugation parities of the particle and antiparticle, respectively . Applying this transformation to the field ( 5.7.31), we fin d C V~ ~(X)C-1T (27r)-'/' d'~,h(A}U) x[~ *a'(p, a)e-"' +~c(_)2Bal(p,J It is useful to compare this formula for the charge-conjugate of an (A,B) field with the adjoint of the (B, A)field for the same particle : ~a fi(~r)= (2~)-3,2fd'FUha(p,U) CT X[(_I)IA(_l )j-"a'(p, -u)e'P" +at(p, c)e-iP'XI To ca lculate the u', we use our previous resul t The Clebsch-Gordan coefficient in Eq . (5.7.14) is real, so{5.7,47} Uba _P- j~A) 0)) eXp J"IA(p, eXp( - P '3) 0)) -b,-b' ofV We use the reflection property of the Clebsch-Gordan coefficients 14 and the fact that those coefficients vanish unless a' + b' = a, to writ e U~h-a~ ~}-ff)*= ( --)a-~b-f~ dab (P,a) , (5.7,49) The field adjoint (5.7.47)is then (replacing a --+-a, b - +-b, a--*-a) W-b,-a(x) =(27r 1: f d3pab(p,(-T) X~(_)2A(_~+Ud(p, u)e'P'-x +al(p, cr)e-P"I - 242 5Quantum Fi eldsandAntiparticle s Using the sign relation(-)-2A-i= (-)2B+j, this i s -b,-a(x) =(2zEf d'P U 'ab'(P, (r) ff X[d (p, U) elp', + (_)j -a +2B at (p, - 1 ] In order that C Ta~(x)C-t should commute or anticommute with all ordinary fields at space-like separations, it is necessary that it be propor- tional to T$~t abecause this is the adjoint of the unique causal field of transformation type (B,A) . Comparing Eq .(5.7.50) with Eq .(5.7.46), we see that this is only possible if the charge-conjugation parities are related by ~*=~c, in which case(5.7.51) (5.7.52) We have already encountered the relation (5.7.51) for spins 0, 1, and in Sections 5 .2, 5.3, and 5 .5, and noted some of its implications for electron-positron and quark-antiquark states in Section 5 .5. In particular, for a particle that is its own antiparticle, Eq . (5.7.52)is satisfied without any charge-conjugation operator on the left-hand side or phase ~* on the right : Wehave already seen an example of this sort of reality cond ition for Majorana spin_'particles in Section 5.5. Finally wecome to time-reversal . Applied to particle annihilation and antiparticle creation operators ,this give s The irreducib le field (5 .7.3 1) thus has the transformation propert y a xL*a(_p, _Gr)e_ip-x + ;e(_1)2BQct{„A}-or)etp'x I(5.7.54) (5.7.55) (x.7.56) To calculate the complex conjugate of the coefficient function, we use Eq. (5-7.14) and the standard formula'4 CAB (i}or;a4b)=(_)A+8-jCAB(.1}-u;-a,-b) (5.7.57) 5.7 Ge neralCausal Fields 243 and find : Changing the variables of integrat ion and summat ion in Eq . (5.7.56) to -pand -a, we find that in order for an (A, B )field to be transformed by time-reversal into something proportional to another (A, B) field, it is necessary that ~`_C} in which case(5.7.59) ab -a , It should be mentioned that from time to time various difficulties have been reported15in the field theory of particles with spin j 3/2. Generally, these are encountered in the study of the propagation of a higher spin field in the presence of c-number external field . Depending on the details of the theory, the difficulties encountered include non-causality, inconsistency, unphysical mass states, and violation of unitarity . I will not go into details about these problems here, because it seems to me that they are not relevant to the calculational scheme described in this chapter, for the following reasons : (1)The fields V,,b(x) have been constructed here directly from the cre- ation and annihilation operators for physical particles, so no question of inconsistency or unphysical mass states can arise . These are free fields, but byincorporating them into an interaction Hamiltonian density in the interaction picture, we can use perturbation theory to calculate S-matrix elements that automatically satisfy the cluster decomposition principle . As long as the interaction Hamiltonian is Hermitian, there can be no difficulty with unitarity . Lorentz invariance is guaranteed in perturbation theory as long as we add appropriate locol but non-covariant terms in the Hamiltonian density ; though a rigorous proof is hacking, there is no reason to doubt that this is always possibly . Thus any difficulties with higher spin can only arise when we try to go beyond perturbation theory . (2) As discussed in section 13 .6, the solution of field equations in the pres- ence of a c-number background field (the context where all the problems with higher spin have been found) does go beyond perturbation theory, in that the results correspond to summing an infinite subset of terms in the perturbation series . This partial summation is justified, even for weak external fields, if the fields are sufficiently slowly varying, the smallness of energy denominators making up for the weakness of the fields . But the results obtained in this way depend on all the details of the interaction 244 5Quantum Fields and Antip articles of the high-spin particle with the external fields : not only the multipole moments of the particle but also possible terms in the interaction that are non-linear in the external fields . The problems reported15with higher spin have been encountered only for higher-spin particles that have been arbi- trarily assumed to have only very simple interactions with external fields . No one has shown that the problems persist for arbitrary interactions, and as we shall see in chapter 12, particles of higher spin are expected to have interactions of all possible types allowed by symmetry principles . (3)In fact, there are good reasons to believe that the problems with higher spin disappear if the interaction with external fields is sufficiently complicated . For one thing, there is no doubt about the existence of higher- spin particles, including various stable nuclei and hadronic resonances . If there is any problem with higher spin, it can only be for `point' particles, that is, those whose interactions with external fields are particularly simple . It should be kept in mind that the requirement of simplicity depends on the choice of which field we choose to represent the higher-spin particle . Remember that any free field types for a given particle can be expressed as a derivative operator acting on any other field type, so in the interaction picture any interaction with external fields may be written in terms of any field types we like, but interactions that are simple when expressed in terms of a field of one type may look complicated when expressed in berms of a field of another type . So the requirement of simplicity does not seem to have any objective content . (4) Also, both higher-dimensional `Kaluza--Klein' theories and sting theo- ries provide examples of consistent theories of a charged massive particles of spin two interacting with a electromagnetic background field .16 (It was found that the consistency of the theory depends on the assumption of realistic external fields that satisfy the field equations, a point generally ne- glected in earlier work .) Reformulating this work in the interaction picture, the spin two particle is represented by a (1, 1) free field, but as mentioned above, the interactions may be reexpressed in the interaction picture in terms of any field type (A, B) that contains the j = 2 representation of the rotation group . 5.8 T he CPT Theo rem We have seen that the demands of relativity combined with quantum mechanics require the existence of antiparticles . Not only is it necessary that every particle have an antiparticle (which may for a purely neutral particle be itself) ; there is a precise relation between the properties of particles and antiparticles, that can be summarized in the statement that 5.8 The CPTTheorem 245 for an appropriate choice of inversion phases, the product CPT of all the inversions isconserved .This is the celebrated CPT theorem .` As a first step in the proof, let us work out the effect of the product CPT on free fields of various types . For a scalar, vector, or Dirac field the results of Sections 5 .2,5.3, and 5.5 giv e CPT (x) [CPT]-' =C**q*o~(-x) , (5.$.1) CPT 0,,(x) [CPT]-' = (5-8-2) (Of course, the phases ~, and qdepend on the species of particle described byeach field.)We are going to choose the phases so that for all particle s Then any tensor formed from and set of scalar and vector fields and their derivatives transforms int o CPT [CPT]-1= (-)no u~...~.,,~-x). (5 .8.5) (Any complex numerical coefficient appearing in these tensors is trans- formed into its complex conjugate because CPT is antiunitary .) We can easily see that the same transformation rule applies to tensors formed from bilinear combinations of Dirac fields . Applying Eq .(5.8.3)to such a bilinear give s CRT Li p-I(X)W2()l[CPTJ-~ =W i(-X)75fl ;752(-x ) [P_1(-x)Y5My5W2(-x)] f. (5 .5.6) (Aminus sign from the anticommutation of Pand y sis cancelled by the minus sign from the anticommutation of fermionic operators .)If the bilinear is a tensor of rank n, then M is a product of n modulo 2 Dirac matrices, so Ys Ys = (°--I)'M , and the bilinear therefore satisfie s A Hermitian scalar interaction density -yf(x) must be formed fro m The original proofs of this theorem were by Luders and Pauli .17 It has been proved rigorously in axiomatic field theory,lg by using commutativity assumptions to extend the Lorentz invariance of the theory to the complex Lorentz group, then using complex Lorentz transformations to prove a reflection property of vacuum expectation values of products of fields, and then using this reflection property to infer the existence of an antiunitary operator that induces CPT transformations on the fields . 246 5Quantum Fields andAntiparticle s tensors with an even total number of spacetime indices, and therefor e CPT W()[CPT]-' = (-x ). (5 .8.7) More generally (and somewhat more easily) we can see that the same is true for Hermitian scalars formed from the fields Va~ (x) belonging to one or more of the general irreducible representations of the homogeneous Lorentz group . Putting together our results in the previous section for the effects ❑f inversions on such fields, we fin d CPTWAll (x)[CPT]-' -(_1)2BV~b t(-x}  {S .S.B) (For the Dirac field the factor (-1)2" is supplied b the matrix 75in Eq.(5.8.3).)In order to couple together a product y~~'hl'(x) y~~ b~2(x)... to form a scalar (x), it is necessary that both Al +A2+ ... and 8i +B2+ ' be integers, so (-i)2B,+2B2+... i la and so a Hermitian scalar -X,"(x) will automatically satisfy Eq . (5.8.7). From Eq. (5.8.7)it follows immediately that CPT commutes with the interaction V = f d3X, (5~,0) CPT V[CPT] -1= V . (5.8.9) Also, in any theory CPT commutes with the free-particle Hamiltonian Ho . Thus the operator CPT, which has been defined here by its operation on free-particle operators, acts on `in' and `out' states in the way described in section 3 .3. The physical consequences of this symmetry principle have already been discussed in Sections 3.3 and 3 .6. 5.9 Massless Particle Field s Up to this point we have dealt only with the fields of massive particles . For some of these fields, such as the scalar and Dirac fields discussed in Sections 5 .2 and 5 .5, there is no special problem in passing to the limit of zero mass . On the other hand, we saw in Section 5 .3 that there is a difficulty in taking the zero-mass limit of the vector field for a particle of spin one : at least one of the polarization vectors blows up in this limit . In fact, we shall see in this section that the creation and annihilation operators for physical massless particles of spin j~ 1 cannot be used to construct all of the irreducible (A, B) fields that can be constructed for finite mass . This peculiar limitation on field types will lead us naturally to the introduction of gauge invariance . Just as we did for massive particles, let us attempt to construct a general free field for a massless particle as a linear combination of the annihilation operators a(p, u)for particles of momentum pand helicity or, 5.9Massless Particle Fi elds 247 and the corresponding creation operators a`#( p, a) for the antiparticles :* L¢ +~ac, (F, Ov"* Q) e-ip'x I(5-9-1 ) where now F° = J pJ. The creation operators transform just like the one- particle states in Eq . (2.5.42) a exp irrft, A)a'(PA, COP 0 U(11)acfi(p,a)U-111} = (Ap)exp(iO(p, A)~act (PA, C) P and hence als o U{1~}r~(p,~)LI-1(1~)= ~ exp -iuft, A)) a( Pn, Cr)(5.9.2) (5-9-3) (5.9.4) where PA =gyp, and 0is the angle defined by Eqs . (2.5.43). Hence if we want the field to transform according to some representation D(A) of the homogeneous Lorentz grou p ~()ip,,(x)U-i(A) De?( -1)tp?(A x), (5 .9.5) then we must take the coefficient functions u and v to satisfy the relation s a U?(PA, CY)eXP ia0(p,A) )V(Ap)"~ v?(pA, a)exp ic0(p,A) P D?e(A)v((p, a)) = ~Ap , 1 :(5.9.6) (5.9.7) in place of Eqs .(5.1.19) and ( 5.1.20). (Again, PA =11p .) As in the massive particle case, we can satisfy these requirements by setting (in place o f We deal here with only a single species of particle, and drop the species label n . Also, rc and .i are constant coefficients to be determined by the requirement of causality wL[h some convenient choice of normalization of the coefficient functions ul and v,, . 248 5 Quantum Fields and Antipartic les Eqs.(5.1.21 and (5.1.22}) rLk, VAP, CT)rLk, - sin O 0 0 cos B 00 0 1 0 U ~ 1 _jwhere k is a standard momentum, say (0,0, k), and Y(p) is a standard Lorentz transformation that ta kes a massless particle from momentum k to moment um p. Also,in place of Eqs . (5.1.23) and (5 .1.24), the coefficient functions at the standard momentum must satisf y U?(k, a) exp iaO(k, W)) D?e(W)U((k, cr) v?(k, a)exp icrO(k, W)) D?,,(W)vl (k,u)(5.9.8) (5.9.9) (5.9.10) (5.9. 11) where WAi,isan arbitrary element of the `little group' forfour-momentum k= (k, J k,l},i.e.,an arbitrary Lorentz transformat ion that leaves this four- momentum invariant . Wecan extract t he content of Eqs . (5.9.10) and (5.9.f 1)by c onsidering separately the two kinds of little -group elements in Eq . (2.5.28).For a rotation R(9)by an angle 0 around the z-axis , given byEq.(2.5.27), cos B RAV(O)- sin0 0 0 we find from Eqs . (5.x.10) and (5.9.11) ,(k,o-)e" =ED?,(R(O))u,(k,a) u? T' v~(k, a)e-idlo D?,(R(O))v,(k, u) For combined rotations and boosts S(oc, fl) in the x 1 0 S1UV~~a0t(5.9.12) X5.9.13) Y plane, given b y -ac o~ -~ ~ Y -Y 1 +Y 5.9Massless P articleField s Eqs. (5.9.10)and(5.9.11)give (S(cx, } v?(k, a) D?,(S(a, fl))v,(k, a)249 5.9.1 T) (5.9.15) Eqs. (5.9.12)-(5.9.15) are the conditions that determine the coefficient functions u and v at the standard momentum k ; Eqs .(5.9.8)and (5.9.9) then give them at arbitrary momenta . The equations for vare just the complex conjugates .of the equations for u, so with a suitable adjustment of the constants rc and Awe may normalize the coefficient functions so that vAP, Cr} _ UA p, Cr}'- (5 .9.16) The problem is that we cannot find a u ethat satisfies Eq. (5.9.14) for general representations of the homogeneous Lorentz group, even for those representations for which it is possible to construct fields for particles of a given helicity in the case m :~ 0. To see what goes wrong here, let's try to construct the four-vector [(~, ~ )1 field for a massless particle of helicity ±1.In the four-vector representation, we have simpl y D,u,,{A } = 1LA,, It is conventional to write the coefficient function u .here in terms of a `polarization vector' e,, , so that Eq .(5.9.8)gives Also, Eqs .(5.9.12) and (5 .9.14) read here(5.9.17) (5.9.15) (5.9.19) (5.9.20) Eq. (5.9.19) req uires that (up to a co nstant which ca nbe absor bed into the coefficie nts and ) .), (5.9.21) But then Eq .(5.9.20)would require also that a +ifl--0, which is impos- sible for general real a, fl. We therefore cannot satisfy the fundamental 250 5QuantumFields and Antiparticle s requirement (5 .9.14) or (5,9.10); instead, we have here D'a, (W(O, 0C,fl))e'(k, +t)--S1'),(a,#)R~,(0)e'(k, +1) ICI We have thus come to the conclusion that no four-vector field can be constructed from the annihilation and creation operators for a particle of mass zero and helicity ±1 . Let's temporarily close our eyes to this difficulty, and go ahead anyway, using Eqs . (.9,18 and ( 5.9.21) to define a polarization vector for arbitrary momentum, and take the field a s a,(X) = fd3p(2,g)-'/'(2~')-1/ 2 ~ [ep(p,o)eXap,o)+ e,,{P,a}*e-~P"ar-l(p, -a)] . (5.9.23 We w ill come back la ter to consider how such a field can be used as an ingredientinaphysical theory . The field (5 .9.23) o fcourse satisfie s Other properties of the fie ld follow from those of the polarization vector . (We shall need these properties of the polarization vector later when we come to quantum electrodynamics .). Note that the Lorentz transformatio n (P)that takes a mass less particle momentum from kto p may be written as a `boost' (I pl) along the z-axis which takes the particle from energy Jkl to energy I PIk followed by a standardized rotation R( P) that takes the z-axis in to the d irection of p. Since e''( k,+1) is a purely spatial vector with only x and y components, it is unaffected by the boost along the z-axis, and s o In particul ar, e°(k,+1) = 0 and k -e(k,±l) = 0 so and It follows that a°{x}=0 and(5.9.25) (5.9.26) (5.9.27) (5.9.28) V - a(x) = 0. (5.9.29 5.9Massless Particle Fields 251 As we shall see in Chapter 9, these are the conditions satisfied by the vacuum vector potential of electrodynamics in what is called Coulomb or radiation gauge . The fact that ao vanishes in all Lorentz frames shows vividly that aP cannot be a four-vector . Instead, Eq . (5.9.22) shows that for a general mo- mentum pond a general Lorentz transformation A, in place of Eq . (5.9.6) we hav e so that under a general Lorentz transformation(5.9.30) (5.9.31) where Q(x,11) is a linear combination of annihilation and creation op- erators, whose precise form will not concern us here . As we will see in more detail in Chapter S, we will be able to use a field like al`(x)as an ingredient in Lorentz-invariant physical theories if the couplings of a ~'(x) are not only formally Lorentz-invariant (that is, invariant under formal Lorentz transformations under which ~~ --+AP,, a''), but are also invariant under the `gauge' transformations a,, - +aA+ aYO. This is accomplished by taking the couplings of uUtobeof the form a,,ju, where jPis a four-vector current with 0,,jp=0. Although there is no ordinary four-vector field for massless particles of belicity ±1, there is no problem in constructing an antisymmetric tensor field for such particles . From Eq. (5.9.22) and the invariance of k "under the little group we see immediate]} tha t D'u, (W(0,a,fl)) D V ,(W(0,o~,fl))(Ve`(k, +1)-ka eP(k,+1)) ==e±iO (Ve'(k, +1)-VeO(k, +1)) (5.9.32) This shows that the coefficient function that satisfies Eq. (5-9-6) for the antisymmetric tensor representation of the homogeneous Lorentz group is (with an appropriate choice of normalization ) uyw(Pi ±1) = i (27z)-1/2(2p)-312[Pue''(p, ±1) -p`'e"(p,±1) ],(5.9.33) where e Al{p,±1) is given by Eq .(5.9.25). Using this together with Eq. (5.9.23) gives the general antisymmetric tensor field for massless par- ticles of helicity ±1in the form (5.9.34) Note that this is a tensor even though & is not a four-vector, because the extra term in Eq .(5.9.31) drops out in Eq . (5.9.34). Note also that Eqs. (5.9.34), (5 .9.24), (5 .9.28), and (5 .9.29)show that f~ `satisfies the 252 5 Quantum Fields and Antiparticles vacuum Maxwell equations : OJUfPV = 0, PauV(5.9.35) (5.9.36) To calculate the commutation relations for the tensor fields we need sums over helicities of the bilinears elle".The explicit formula (5 .9. 1) gives k1kJ Gr=t1 and so,using Eq.(5.9.5), ei~~~Oej(p,ff). -~ tPiPi A straightforward calculation gives then(5.9.37) V, lV(x),Jp,,(J)~J=(27r)`3 H'U{lOV~ ~+ th ~09 ~~+ qY 6OVt+p -`fV6OP"PJ xJ d'p(2po)-' [JKJ2e ipx _1A12e-ip,x I - This clearly vanishes for x O= y° if and only i f K['= I A12(5.9.39) in which case since feu is a tensor the commutator also vanishes for all space-like separations .Eq. (5.9.39)also implies that the commutator of the aJ` vanishes at equal times, and as we shall see in Chapter 8 this is enough to yield a Lorentz-invariant S-matrix . The relative phase of the creation and annihilation operators can be adjusted so that K = A; the fields are then Hermitian if the particles are their own charge-conjugates, as is the case for the photon . Why should we want to use fields like a P(x)in constructing theories of massless particles of spin one, rather than being content with fields like fl"(x) with simple Lorentz transformation properties? The presence of the derivatives in Eq . (5.9.34) means that an interaction density constructed solely from f ,,V and its derivatives will have matrix elements that vanish more rapidly for small massless particle energy and momentum than one that uses the vector field a,, . Interactions in such a theory will have a correspondingly rapid fall-off at large distances, faster than the usual inverse-square law . This is perfectly possible, but gauge-invariant theories that use vector fields for massless spin one particles represent a more 5.9Massless Particle Fields 253 general class of theories, including those that are actually realized in nature . Parallel remarks apply to gravitons, massless particles of helicity ±2 . From the annihilation and creation operators for such particles we can construct a tensor .R~,,,, with the algebraic properties of the Riemann- Christoffel curvature tensor : antisymmetric within the pairs O,v and p,d, and symmetric between the pairs . However, in order to incorporate the usual inverse-square gravitational interactions we need to introduce a field h.,that transforms as a symmetric tensor, up to gauge transformations of the sort associated in general relativity with general coordinate trans- farmations . Thus in order to construct a theory of massless particles of helicity ±2 that incorporates long-range interactions, it is necessary for it to have a symmetry something like general covariance . As in the case of electromagnetic gauge invariance, this is achieved by coupling the field to a conserved "current' Of", now with two spacetime indices, satisfying 0128111=0. The only such conserved tensor is the energy-momentum tensor, aside from possible total derivative terms that do not affect the long-range behavior of the force praduced .'* The fields of massless parti- cles of spin j ~ 3 would have to couple to conserved tensors with three or more space#ime indices, but aside from total derivatives there are none, so high-spin massless particles cannot produce long-range forces . **~ The problems we have encountered in constructing four-vector fields for helicities ±1 or symmetric tensor fields for helicity ±2 are just special cases of a more general limitation . To see this, let's consider how to construct fields for massless particles belonging to arbitrary representations of the homogeneous Lorentz group . As we saw in section 5 .6, and representation D(A) of the homogeneous Lorentz group can be decomposed into (2A + 1)(2B +I)-dimensional representations (A,B),for which the generators of the homogeneous Lorentz group are represented b y 0'i l)Q'b",Qb - 'F iJk [(Jk1A)) ara bb'b+Vk{"))b'bbdu] V MLIKat, -i [(J(A))a'a 6h1b - (Jk'B))b'b ba' a where J Ware the angular-momentum matrices for spin j. For 0 infinites- IfBi'1-"YNis a tensor current satisfying d ~,, 8R I-";`,= 0, then fd37G Ba K... YN is a conserved quantity that transforms like a tensor of rank N - 1 . The only such conserved tensors are the scalar `charges' associated with various continuous symmetries, and the energy-momentum four-vector . The conservation of any other four-vector, or any tensor of higher rank, would forbid afl but forward collisions . 254 5 Quantum F ieldsand Antiparticle s imal, D(R(O)) = 1 +i239,so Eqs .(5,9.12)and (5.9.13) give and so uab (k, cr) and v,,b(k, rr) must vanish unless a= a+b and or = -a- b, respectively . Also, letting oc and Pbecome infinitesimal in Eq.(x.9.1 4) give s o= l 31 + O1)Lib,AfhfuCIyFJrI~,(,T) _ (-J1A) +aJ~'~u~rb( k, a) + (- .1iB)- iJ~$)}at~ U~b+(k,~)a or more si mply 1 2)ad J(B)+iJ('))hYUab'(k,a)=0 These require that u,,b(k, (7)vanishes unles s a =-A, h = + B (5.9.40) and the same is obviously also true of v,,b(k, a). Putting this together, we see that a field of type (A,B),can be formed only from the annihilation operators for a massless particle of helicity aand the creation operators for the antiparticle of helicity -a, wher e a=B- A. (5 .9.4) For instance, the (Z, 0)and ( 0,1) parts of the Dirac field for a massless particle can only destroy particles of helicity - ~ and +; respectively, and create antiparticles of felicity + z and - z, respectively . In the 'two- component' theory of the neutrino, there is only a (~ = 0) field and its adjaint, so neutrinos have helicity - and antineutrinos helicity + z in this theory . By the same methods as in Section 5 .7, it can be shown that the (j,0)and (U, j)fields for massless particles of spin j(i.e., helicity +j) commute with each other and their adjoints at space-like separations if the coefficients of the annihilation and creation terms in Eq .(5.9.1)satisfy Eq .(5.9.39). The relative phase of the annihilation and creation operators may then be adjusted so that these coefficients are equal . It is-easy to see that the fields for a massless particle of spin jof type ( A, A+j)or (B + j,B)are just the 2Ath or 2Bth derivatives of fields of type ( OJ) or (j,O), respectively, so these more general fields do not need to be considered separately here . We can now see why it was impossible to construct a vector field for massless particles of helicity ±1 . A vector field transforms according to Problems 255 the ( z,;) repre sentat ion,and hence according to Eq . (5.9. 9)can only describe helicity zero . (It is,of course, possible to construct a vector field forhelicity zero -just take the derivative 0,,0 of a massless scalar field 0.)The simplest covariant massless field for helicity ±1has the Lorentz transformation type (1, D) (D(0, 1 ) ;;that is ,it isan an tisymmetr ic tensor fi~V-Similarly, the simplest covariant massless field forhelicity ±2 has the Lorentz transformation type ( 2, 0) E)(0, 2):afourth rank tensor which l ike the Rieman n-Christoffel curvature tensor is antisymmetric w ithin each pair of indices and symmetric between the two pairs . The discussion of the inversions P, C, T given in the prev ious section can be carried over to the case of zero mass with only obviou s modifications . Problems 1. Show that if the zero-momentum coefficient functions satisfy the conditions {5 .1. 3} and (5 .1.24), then the coefficient functions (5 .1.21) and {5 .1.22} for arbitrary momentum satisfy the defining conditions Eqs. (5.1.19) and ( 5.1,20). 2. Consider a f ree field ip'(x) which annihilates and creates a self - charge-conjugate particle of spin ~ and mass rM :~ D . Show ho w to calculate the coefficient functions u'( p, a), which mult iply th e annihilation operators a(p, cr) in this field, in such a way that th e field transforms under Lorentz transformations li ke a D irac fie ld ~i( with an extra four-vector index p . What field equat ions an d algebraic and reality conditions does this field satisfy? Evaluate th e matrix Pill (p), defined (for p2= --rn2} b y What are the commutation relations of this field? How does the field transfatm under the inversions P, C, T ? I Conside ra free field h,`°(x) satisfying hP'(x) = h''y(x) and hAy(x) = 0 , which annihilates and creates a particle of spin two and mass m * 0. Show how to calculate the coefficient functions O''( P, cr), whic h multiply the annihilation operators a( p, a)in this field, in such a wa y that the field transforms under Lorentz transformations like a tensor . What field equations does this fie ld satisfy? Evaluate the functio n pYV," .(p), defined by 256 5Quantum Fields and Antiparticle s Whatare the commutati on relat ions of this field? How does the field transform under the inversions P, C ,T? 4.Show that the fields for a massless particle of sp injoftype (,A, A+}) or(B+ j, B)are the 2Ath or 2Bth derivatives of fields of type (0,j) or(f, D),respectively . 5.Work out the transformat ionproperties of fields of transformation type (, lsO)+ (0 a1)formassless particles of helicity ±junder the inversions P ,C,T. 6. Consider a generalized Dirac field W that transforms according to the (J, D) +(0,,j)representation ofthe h omogeneous Lorentz gaup . List the tensors that can be formed from products of the components ofWand Wt. Checkyour result against what we found for j = i , 7. Consider a gener alfield Wabdescribing partic lesof spin jandmass :~0,that transforms according to the (A, B) representation of the homogeneous Lorentz group .Suppose ithas an interaction Hamil- tonian of the for m where .Il'~is an external c-number current . What is the asymptotic behavior of the matrix element for emitting these particles for energy E m and definite helicity? (Assume that the Fourier transform of the current has values for different a, b that are of the same order of magnitude, and that do not depend strongly on E.) Reference s 1. The point of view adopted in this chapter was presented in a series of papers : S. Weinberg, Phys.Rev.133, B131 8(1964) ; 134, B882 (1964) ;138, B988(1965) ; 181, 1893 (1969), A similar approach has been followed in unpublished lectures by E . Wichmann . 2. N. Bohr and L . Rosenfeld, Kgl.Danske Vidensk .Selskrxb Mat .-Fys. Medd ., No .12(1933) (translation inSelected Papers of Leon Rosen- feld, ed, by R .S.Cohen and J . Stachel (Reidel, Dordrecht, 1979)) ; Phys .Rev.78,794 (1950) . 3. P. A. M. Dirac, Proc . Roy . Soc .(London) A11 7,610 (1928) . 4. E. tartan, Bull. Soc . Ma th.France 41, 53(1913) . References 257 5. See, e.g.,J. M. Jauch and F . Rohrlich, The Theory of Photons and Electrons -(Addison-Wesley, Cambridge, MA, 1955) :Appendix A2 . 6. See, e .g.,H. Georgi, Lie Algebras in Particle Physics (Benjamin/Cu- mmings, Reading, MA, 1982) : pp. 15,198 . The or iginal r eference is 1. Schur, Sitz.Preuss .Akad .,p. 406 (1905) . 7. T D . Lee and C . N. Yang, Whys . Rev.104, 254 (1956) . 8. See, e .g., B. L. van dux Waerden, Die gruppentheoretische Meth- odeinderQuantenmechanik (Springer Verlag, Berlin, 1932) ; G.Ya. Lyubarski, The Applications ofGroup theory in physics, translated byS.Dedijer (Pergarnon Press, New York, 1960) . 9. Rarity and J. Schwinger, Pays . Rev .60,61(1941) . 10. See, e .g., A.R. ,Edmonds, Angular Momen tum in Quant um Mechanics, (Princeton Un iversity P ress, Princeton, 1957} : Cha pter3. 11. S.Feinberg, Phys . Rev .181, 1893 (1969), Section V . 12. M. Fierz, Hedv .Phys.Acta 12,3 (1939) ; W. Pauli, Whys .Rev. 5$, 71 6 (1940) . Nora-perturbative proofs in axiomatic field theory were give n by G . iluders and B . Zumino, Phis .Rev.110, 1450 {195$} and N . Burgoyne, Nuov oCimento 8, 807 (1958) . Also see R . F. Streater an d A. S. Wightman, PCT, Spin &Statistics,and All That (Benjamin , New York, 1968) . 13.Fields in the (j, 0) + (0,j)representation were introduced by H . Joos, Fortschr .Phis . 10, 65 (1962) ; S. Weinberg, Ph}rs .Rev.133, B1318 (1964) . 14. A. R. Edmonds, Ref .10, or M . E. Rose, Elementary Theory of Angular Momentum (John Wiley & Sons, New York, 1957) : Chapter III . 15. G. Selo and D . Zwanziger, Phys .Rev.186, 1337 ( 1969); 18$, 2218 (1969);A. Wightman, in Proceedings of the Fifth Coral Gable s Conference onSymmetry Principles at High Energy, ed . byT. Gude - hus, G.Kaiser, and A . Perlmutter (Gordon and Breach, New York , 1969) ;B. Schroer, R . Seiler, and J . A. Swieca, Phys . Rev .D 2, 292 7 (1970); and other references quoted therein . 16. C.R.Nappi an d L. Witten, Phys.Rev.D 40 , 1095 (1989) ;P. C. Argyresand C .R.Nappi ,Phys.Lett.B224,89(1989) . 17. G.Liiders ,Kong . Dansk . Vid . Selskab, M at.-Fys.Medd.28,5(1954) ; Ann.Whys.2,1(1957) ; WPauli,Nuovo Cimento 6, 204 (1957) . When 258 5 Quantum Fields and Antiparticle s Coders first considered how the inversions are related, it was still taken for granted that P is conserved, so his theorem stated that C conservation is equivalent to T invariance . 18. R.Jost,HeIv .Phys.Acta30,409 (1957) ;F.I Dyson, Phys.Rev.110, 579 (1958) . Also s ee Streate randWightman, lief. 12. The Feynman Rule s In previous chapters the use of covariant free fields in the construction of the Hamiltonian density has been motivated bythe requirement that the S-matrix satisfy Lorentz invariance and cluster decomposition con- ditions . With the Hamiltonian density constructed in this way, it makes no difference which form of perturbation theory we use to calculate the S-matrix ; the results will automatically satisfy these invariance and clus- tering conditions in each order in the interaction density . Nevertheless, there are obvious practical advantages in using a version of perturba- tion theory in which the Lorentz invariance and cluster decomposition properties of the S-matrix are kept manifest at every stage in the calcu- lation . This was not true for the perturbation theory used in the 1930s, now known as `old-fashioned perturbation theory', described at the begin- ning of Section 3 .5. The great achievement of Feynman, Sch finger, and Tomonaga in the late 1940s was to develop perturbative techniques for calculating the S-matrix, in which Lorentz invariance and cluster decom- position properties are transparent throughout . This chapter will outline the diagrammatic calculational technique first described b yFeynman at the Poconos Conference in 1948 . Feynman was led to these diagrammatic rules in part through his development of a path-integral approach, which will be the subject of Chapter 9 . In this chapter, we shall use the ap- proach described by Dyson' in 1949, which until the 1970s was the basis of almost all analyses of perturbation theory in quantum field theory, and still provides a particularly transparent introduction to the Feynman rules . 6.1 D erivation of t he Rule s Our starting point is a formula for the S-matrix, obtained by putting together the Dyson series (3 .5.10)with expression (4 .2.2) for the free- particle states 259 260 6The Feynman Rule s SP1,7iRo~d~ti2;...Pl~lnl ;l~2ff2n2;"' 1:4 N E ! Id xi ...d XN(Oo, -a(FtcT1n1) [ Pa 1 t) x TfJOXI) ... X(XN)Jat(P1e_T1nJat(P2cr2n2) ... 4)0)N=O As a reminder :p, a, and n label particle momenta, spin, and species ; primes denote labels for particle sin the final state ; (Do i s the ,free-particle vacuum state ; aand atare annihilation and creation operators ; T in- dicates a time-ordering, which puts the (x} in an order in wh ich the arguments x° decrease from left to right;and (x) is the interaction Hamiltonian dens ity,taken as a polynomial in the fields and their adjo ints each term ibeing a product of definite numbers of fields and field adjoints of each type . The field of a particle of species n that transforms under a particular representation of the homogeneous Lorentz group (with or without space inversions) is given by ~IW = (27Z)-3/2fd3 P[u(p }Cr,n) a(p su, n) eipx Cr Here nc denotes the antiparticle o fthe species n, and exp(±ip  x) is calculated with p oset equal to pI+M,;.The coefficient functions u? and a edepend on the Lorentz transformation properties of the field and the spin of the particle it describes ; they were calculated in Chapter 5. (For instance, in the scalar field the uefor a particle of energy E is simply (2E)-1/7, while in a Dirac field u eand v,, are the normalized Dirac spinors introduced in Section 5.5.)The index ef on the field should here be understood to indicate the particle type and the representation of the Lorentz group b ywhich the field transforms, as well as including a running index labelling the components in this representation . There is no need to deal separately with interactions that involve derivatives of fields ; from our point of view, the derivative of a fleld(61.3)is just another field described by (6.1.3),with different u,, and v~ . We will here make a distinction between some particle species that we arbitrarily call `particles', for instance electrons, protons, etc ., and those we call `antiparticles', such as positrons and antiprotons . The field operators that destroy particles and create antiparticles are called simply `fields' ; their adjoins, which destroy antiparticles and create particles, are called `field adjoints' .Ofcourse, 6.1Derivation of theRules 261 some particle species like the photon and 70 are their own antiparticles ; for these the field adjoints are proportional to the fields . We now proceed to move all annihilation ❑perators to the night in Eq.(6.1.1), repeatedly using for this purpose the commutation or anti- commutation relations : +6'(P' - P)bafa6wn ( 6.1.4) a(p a n)a(p'cr'n') ±a(PVW)a(p an) (6.1.5) at(Pan)a'{p'cr'n'} ± al (P{a'nl)at(Pan) ( 6.1.6 (and likewise for antiparticles), the ± sign on the right being - if both particles n, n' are fermions, and +if either or both are bosons . Whenever an annihilation operator appears on the extreme right (or a creation operator on the extreme left), the corresponding contribution to Eq .(6.1.1) vanishes, because these operators annihilate the vacuum state : 0p'afi( pcr n) =0. The remaining contributions to Eq . (6.1.1)are those arising from the delta function terms on the right-hand side of Eq . (6.1.4), with every creation and annihilation operator in the initial or final states or in the interaction Hamiltonian density paired in this way with some other annihilation or creation operator . In this way, the contribution to Eq . (6.1.1) of a given order in each of the terms j in the polynomial _VO(tp(x),Vt(x)) is given by a sum, over all ways ❑f pairing creation and annihilation operators,2 of the integrals of products of factors, as follows : (a) Pairing of a final particle having quantum numbers p ', a', nr with a field adjoint Wt .(x) in i(x) yields a facto r [aii' 4Tfn% Wt(x)1 +== (2n)-'1'e-ie-Xu;(Pfafn') , (6.1.9) (b) Pairing of a final antiparticle having quantum numbers p',cr', rt"With a field We(x)in , f(x)yields a facto r CAI fdrn"), v CIJC~],, = (27Z)-3/2e-iP"-ve (p rffl n). (6 .1.10) (c) Pairing of an initial particle having quantum numbers p, ff,iT with a field ipe(x) in i(x) yields a facto r 1w,,(x), at(pern)]- 262 6 The Feynman Rule s (d) Pair ing of an initial antiparticle having quantum numbers p, a,n' with a field adjo int ~ ~(x)in.-a(x)yields a facto r [4x),at(p un')]+ _ ( 2,g)-'1'e`P"'U*(pan). (6 .1.12) (e) Pairing of a final particle (or antiparticle) having numbers p ', a', ra' with an initial particle (or antiparticle) having quantum numbers p, a, n yields a factor [aWcr'n'), a'(p an)],= 63(p I -War',gr6wn - (6.1.13) (f) Pairing of a field W& ) &x~ in i(x) wit ha field adjo int T rn(y) in QYPj(Y) yields a factor * O(X - Y)~Ve' W, W r'n I (Y) ] ,± 0 (Y - X) [T (Y), we- (X)] , -i (X, Y) , (6 .1.14) where y+ and ip- a rethe terms in ipthatdestroy particles a nd create antiparticles, res pectivel y ipe+(x) (2,A)-"'fd3P uAP ~n) d"'a( Pa n) Recall that O(x-y)isa step function ,equal to +1forx°> Y°and zero forx°<y°.These step functions appear in Eq . (6.1.14) because ofthe time-ordering in Eq .(6.1.1); wecan encounter a pairing of an annihilation field W+(x) in (x) with a creation field W+t( Y) in (Y)only if fi(x) was i nitially to the left of ,ma(y) in Eq . (6.1.1),i.e.,if xO YO; similarly, we encounter a pa iring of an ann ihilation field V-t( y) in (y)with a creation fieldW-(x)in (x )only if ( y)was initially to t heleftof { x} in Eq .(6.1.1),i.e., if yox°.(The +sign in the second term in(6.1.14) will be expla ined a little later .) The quantity (6.1.14)is known a sa propagator ; it is calculated inthe following section . The 5-mat rix is obtained by multiplying these factors together ,along with additional numerical factors tobe discussed below, then integrating over xl..,xv, then summing over all pair ings,and then over the numbers of interaction of each type .Before filling in all the details, it will be convenient first to describe a d iagrammatic formalism for keeping track ofallthese pairings . If the interaction A(x) is written in the normal-ordered form, as in Eq, (5_1_33), then there is no pairing of fields and field adjoints in the some interaction .Otherwise some sort of regulari7ation is needed to give meaning to d ,~,(0). 6.1Derivation Qf the Rule s pC5n {2n}312 x fal ~, x E°p- TlYP 6n) (2n)312 ~~ (d) >0 0 <x ►~,v (f)IXVQ(P Q~n f {2n} 3r2 N263 X CtPn) (2 R) 1/2 P(Tn P or nT} Y (e) -iA gm(xy) Figure 6 .1. Graphical representation of pairings of operators arising in the coordinate-space evaluation of the S-matrix . The expressions on the right are the factors that must beincluded in the coordinate-space integrand of the S-matrix for each line of the F'cynmar, diagram . The rules for calculating the S-matrix are conveniently summarized in terms of Feynrraan diagrams . (See Figure 6.1.)The diagrams consist of points called vertices, each representing one of the 'j(x), and lines, each representing the pairing of a creation with an annihilation operator . More specifically ;-r. , fcpan e X (a) The pairing of a final particle with a field a .djaint in one of the (x) 264 6 The Feynman Rule s is represented by a line runn ing from the vertex representing that '(x) upwards out of the diagram, carrying an arrow pointed up wards . (b) The pairing of a final antipart icle with a field inone of the -*'(x) is also represented by a line running from the vertex representing that "C(x) upwards out of the diagram ,but carrying anarrow pointed downwards . (Arrows are omitted throughout for particles like y,Ic0, etc . that aretheir own antiparticles .) (c) The pairing of an initial particle with a field inone of the (x)is represented hyaline running into the diagram from below, ending in the vertex representing that -yf(x), carrying an arrow pointed upwards . (d) The pairing of an initial antipart icle w ith a field in one of the (x)isalso represented by a line running into the diagram from below, ending i nthevertex representing that fi(x) ,but carryi ng anarrow pointed downward s. (e)The pa iring of a final particle or antiparticle with an initial particle or antiparticle is represented by a line running clear through the diagram from bottom to top,not touching an yvertex, with an arrow pointed upwards or downwards for particles or antiparticles, respectively . (f) The pairing of a field in-X,'7(x) with a field adjoint in -Ye(y) is represented by a line joining the vertices representing .e(x) and (y), carrying an arrow pointing from ytox. Note that arrows always point in the direction a particle is moving, and opposite to the direction an antiparticle is moving . (As mentioned above, arrows should be omitted for particles like photons that are their own antiparticles .) The arrow direction indicated in rule (f)is consistent with this convention because a field adjoint inj(y)can either create a particle destroyed b ya field in j(x), or destroy an antiparticle created by a field in j(x) . Note also that since every field or field adjoint in A"';(x) must be paired with something, the total number of lines at a vertex of type i, corresponding to a term j(x) in Eq . (6.1.2), is just equal to the total number of field or field adjoint factors in j(x).Of these lines, the number with arrows pointed into the vertex or out of it equals the number of fields or field adjoints respectively in the corresponding interaction term . To calculate the contribution to the S-matrix for a given process, of a given order Niin each of the interaction terms ;(x) in Eq . (6.1.2), we must carry out the following steps : (i) Draw all Feynman diagrams containing Njvertices of each type i, and containing a line coming into the diagrams from below for each particle or antiparticle in the initial state, and a line going upwards out of the diagram for every particle or antiparticle in the final state, together 6.1Derivation of the R ules 265 with any number of internal lines running from one vertex to another, as required to give each vertex the proper number of attached lines . The lines carry arrows as described above, each of which may point upwards or downwards . Each vertex is labelled with an interaction type i and spacetime coordinate ~' . Each internal or external line is labe lled at the end w here it runs into a vertex with a field type ~ (corresponding to the field T/(x) or Vt(x) that creates or destroys the part icle or antipar ticle at that vertex), and each external line whe re it enters or leaves the diagram is labelled with the quantum numbers p, a, ra or p', ff', no of the initial or final particle (or antiparticle) . (ii) For each vertex of type i, include a factor --i (from the (-i)N in Eq . (6.1.1)) and a factor gj (the coupling constant multiplying the product of fields in j(x)}. For each line running upwards out of the diagram, include a factor (6 .1.9) or (6 .1.10), depending on whether the arrow is po inting up or down . For each line running f rom below into the diagram, include a factor (6 .1.11) or (6 .1.12), again depend ing on the arrow direc tion. For each line running straight through the diagram include a factor (6 .1.13). For each internal line connecting two vertices include a factor (6 .1.14). (iii) Integrate the product of all these factors over the coordinates x1, X2, ... of each vertex . (iv) Add up the results obtained in this way from each Feynman diagram . The complete perturbation series for the S-matrix is obtained by adding up the contributions of each order in each interaction type, up to whatever order our strength permits . Note that we have not included the factor 11N! from Eq .(6.1.1)in these rules, because the time-ordered product in Eq. (6.1.1) is a sum over the N! permutations of x Ix2... xrv, each permutation giving the same contribution to the final result . To put this another way, a Feynman diagram with Nvertices is one of N! diagrams, which differ only in permutations of the labels on the vertices, and this yields a factor of N!which cancels the 11N! in Eq .(6.1.1).(There are exceptions to this rule, discussed below .) For this reason, henceforth we do notinclude more than one of a set of Feynman diagrams that differ only by relabelling the ver#ices . In some cases, there are additional combinataric factors or signs that must be included in the contribution of individual Feynman diagrams- (v) Suppose that an interaction ,(x) contains (among other fields and field adjoints) M factors of the sa me field . Suppose that each ❑f these fields is paired with a field adjaint in a different interaction (different for each one), or in the initial or final state . The first of these field adjoints 266 6The Feynmarz Rule s Figure 6 .2. Example of a graph requiring extra combinatoric factors in the S-matrix . For an interaction involving, say, three factors of some field (as well as other fields) we usually include a factor 113? in the interaction Hamiltonian, to cancel factors arising from sums over ways of pairing these fields with their adjoints in other interactions . But in this diagram there are two such factors of 1/3! and only 3!different pairings, so we are left with an extra factor of 1/3! . can be paired with any one of the M identical fields in -01i(x) ; the second with any one of the remaining M -1 identical fields ; and so on, yielding an extra factor of N 1!. To compensate for this, it is conventional to define the coupling constants g iso that an explicit factor 11M! appears in an y j(x) containing M identical fields (or field adjaints .) For instance, the interaction of Mth order in a scalar field O(x) would be written g OM/Mt. (More generally, one often also displays an explicit factor of 11M ?when the interaction involves a sum of Mfactors of fields from the same symmetry multiplet, or when for this or and other reason the coupling coefficient is totally symmetric or antisymmetric under permutations of M boson or fermion fields, ) However, this cancellation of ?factors is not always complete . For instance, consider a Feynman diagram in which the Al identical fields in one interaction j(x) are paired with Ai corresponding field adjoints in a single other interaction j(Y). (See Figure 6.2.)Then by fallowing the above analysis, we find only M! different pairings (since it makes no difference which of the field adjoints we call the first, second, ...), cancelling only one of the two factors of 11MT? in the two different interactions . In this case, we would have to insert an extra factor of 1/ M? by hand' into the contribution of such a Feynman diagram . Other combinatoric factors arise when some of the permutations ❑f vertices have no effect on the Feynman diagram . We noted earlier that the factor 11N! in the series (b.1.1) is usually cancelled by the sum over the N!diagrams that differ only in the labelling of the 1V vertices . However, this cancellation is incomplete when relabelling the vertices does not yield a new diagram . This happens most commonly in the calculation of vacuum-to-vacuum 5-matrix elements in a theory with a quadratic interaction where M may depend on external fields . (The physical significance of such vacuum fluctuation diagrams is discussed in detail in Volume II .)The Feynman diagram of Nth order in A is a 6.1Derivation of the Rules 2 67 ring with Ncorners . (See Figure 6.3.)There are only (N-1)!different diagrams here because a permutation of labels that moves each label to the next vertex around the ring yields the same diagram . Hence such a graph is accompanied with a facto r (N-1)? 1' (6 .1.1?) (vi) In theories involving fermion fields, the use of Eqs .(6.1.4)-(6.1.6)to move annihilation and creation operators to the right and left introduces minus signs into the contribution of various pairings . To be specific, we get a minus sign wherever the permutation of the operators in Eq. (6.1.1) that is required to put all paired operators adjacent to one another (with annihilation operators just to the left of the paired creation operators) involves an odd number of interchanges of fermion operators . (This is because to compute the contribution of a certain pairing, we can first permute all operators in Eq .(6.1.1)so that each annihilation operator is just to the left of the creation operator with which it is paired, ig- noring all commutators and anticomrnutators of unpaired operators, and then replace each product of paired operators with their commutators or antic :ommutators .} One immediate consequence is to produce the minus sign in the relative sign of the two terms in Eq . (6.1.14) for the fermion propagator . Whatever permutation puts the annihilation part W+(x) of a field in (x) just to the leftof the creation part y~+f(yt)of a field adjoint in. tr(y), the permutation that puts the annihilation part y)-~(Y) of the field adjoint just to the left of the creation part V-(x) of the field involves one extra interchange of fermian operators, yielding the minus sign in the second term of Eq. (6.1.14) for fermions . In addition, minus signs can rise in the contribution of whale Feynman diagrams . As an example, let us take up a theory in which the sole interaction of fermions takes the for m fnik where g (,,kare general constants, y((x) are a set of complex fermion fields, and ..(x) are a set of real bosonic (but not necessarily scalar) fields . (Not only quantum cle krodynarnics, but the whole `standard model' of weak, electromagnetic, and strong interactions, has fermionic interactions that can all b eput in his form .) Let us first take up the process of fermion- fermion scattering, 12 - -►1'2', to second order in A.The fermion operators in the second-order term in Eq .(6.1.1)appear in the order (with obvious abbreviations ) There are two connected diagrams to this order, corresponding to the 268 6TheFeynman R ules Figure 6 .3. An eighth-order graph for the vacuum-to-vacuum amplitude with particles interacting only with an external field . In this diagram the external field is represented by wiggly lines . There are 7! such diagrams, differing only by relabelling the vertices, and not counting as different those labellings that simp ly rotate the ring . The factor 1/8! from the Dyson formula (6 .1.1) is therefore not entirely cancelled here, leaving us with an extra factor 1/8 . pairing s and (See Figure 6 .4.) To go from (6.1.19) to(6.1.20) requires a n eve n permuta- tion of fermianic operators . (For instance, move W(x) past three operators to the right, and then move a(l') past one operator to the right .) Thus there is no extra minus sign in the contribution of the pairing (6.1.20). This in itself is not so important ; the overall sign of the S-matrix does not matter in transition rates, and in any case depends on sign conventions for the initial and final states . What is important is that the contributions of pairings ( 6.1.20)and (6.1.21) have opposite sign, as can be seen most easily by noting that the only difference between these two pairings is the interchange of two ferrnionic operators, a(1' )and a(2') . In fact, this relative minus sign is just what is required by Fermi statistics : it makes the scattering amplitude antisymmetric under the interchange of particles 1' and 2' (or Iand 2) . However, it must not be thought that all sign factors can be related in such a simple way to the antisymmetry of the final or initial states, even in the lowest order of perturbation theory . To illustrate this point, let's now consider fermion-antifermion scattering, 12' - -+1'2'x, to second order in the same interaction (6.1.15). The fermionic operators in the second-order 6.1Derivation of t heRules 1' 2' > ------------ < 122' ------------- 2269 Figure 6.4.The connected second-order diagrams forfermior ►-fermion scattering in a theory with interaction (6.1.18).Herestraight lines repr esent ferrnion s;dott ed linesare neutral bosons .Ther eisa minus signdifference in the contribution sof thesetwo diagrams ,arising from an extra interchange of fermion operators in the pairing srepre sented b ythesecond diagra m. term in Eq . (6.1.1) appear in the order : Here again there are two Feynman diagrams to this order, corresponding to the pairing s and(6.1.23) (6.1.24) (See Figure 6.5.) To go from (6.1.22)to {6 .1.23} requires an even permu- tation of fermionic operators (for instance, move ~)(x) past two operators to the left and move ipf(y) past two operators to the right) so there is no extra minus sign in the contribution of the pairing ( 6.1.23). On the other hand, to go from ( 6.1.22)to (6 . 1.24) requires an odd permutation of fermionic operators (the same as for (6.1.23), plus the interchange o f t(x) and rat(Y)) so the contribution of this pairing does come with an extra minus sign, ` Additional signs are encountered when we consider contributions of higher order . In theories ❑f the type considered here, in which the interactions of fermions all take the form (6.1.18), the fermian lines i n Actually, this sign is not wholly unrelated to the requirements of Fermi statistics . The same field can destroy a particle arld create an antiparticle, su there is a relation, known as `crossing symmetry', between processes in which initial particles or antiparticles are exchanged with final antiparticles or particles . In particular the amplitudes for the process 12' -1'2'" are related to those for the `crossed' process 12' -1'2; the two pairings (6.1.23) and (6.124) jusl correspond to the two diagrams for this process, which differ by an interchange of 1 and 2' or ]' and 2), so the antisymmetry of the scattering amplitude under interchange of initial (or final) particles naturally requires a minus sign in the relative contribution of these two pairings . However, crossing symmetry is not an ordinary symmetry it involves an analytic continuation in kinematic variables} and it is difficult to use it with any precision for general processes . 270 6 The Feyntrtan Rule s 11 2 {,t' 2 {2 Figure 6 .5. The connected second-order diagrams for fermion-antifermion scat- tering in a theory with interaction ( 6.1.18). Here straight lines represent fermions or antiferrnions, depending on the arrow direction ; dotted lines are neutral bosons . There is again a minus sign difference in the contributions of these two diagrams, arising from an extra interchange of fermion operators in the pairings represented by the second diagram , general Feynman diagrams form either chains of lines that pass through the diagram with arbitrary numbers of interactions with the boson fields, as in Figure 6 .6, or else fermianic loops, like that shown in Figure 6 .7. Consider the effect of adding a ferminnie loop with M corners to the Feynman diagram for any process . This corresponds to the pairing of fermionic operator s On the other hand, these operators appear in Eq . (6.1.2) in the orde r To go from (6 .1.26)to(6.1.25)requires an odd permutation of fermionic operators (move Q(xi) to the right past 2Nf - 1 operators) so the con- tribution of each such fermionic loop is accompanied with a minus sign . These rules yield the full ,S-matrix, including contributions from pro- cesses in which various clusters of particles interact in widely separated regions of spacetime . As discussed in Chapter 4, to calculate the part of the S-matrix that excludes such contributions, we should include only connected Feynman diagrams . In particular, this excludes lines passing, clean through the diagram without interacting, which would yield the factors (6.1.13). To make the Feynman rules perfectly clear, we will calculate the low- order contributions to the S-matrix for particle scattering in two differen t theories . 6.1Derivation qfthe Rules 271 i' 2. -► 2 ti L Figure 6 .6. The connected second-order diagrams for bosun-fcrtnion scattering in a theory with interaction ( 6.1.18). Straight lines are fermions ; dashed lines are ricutral bosons . ti 1 ,r 3y it ti Figure 6 .7. The lowest-order connected diagram for boson . bison scattering in a theory with interaction (6.1.19).Such fermion loop graphs yield an extra minus sign, arising from permutations of the paired fermion fields . Theory I Consider the theory of fermions and self- charge-con jugate bosons with interaction (6.1.18).The lowest-order connected d iagrams for fermion - boson scattering are shown in Figure 6.6.Following therules outlined in Figure 6. 1,the corresponding S-matrix element i s 'SPi'c~nj P 2~d2~z pla~ nlP2~:~~~s O'm' ki m 1fd x [e-'P' .Y4'(P2 c2n~)e`P2'xUk{Pz(TaH2) +e-'P~ xZ dk(P2 Id2In'2)~'~PZ vUk,( P2U2n2), (61 .27) 272 6The Feynman Rule s (The labels Iand 2 are used here for fermions and bosons, respectively .) For fermion-fermion scattering there are also two second-order diagrams, shown in Figure 6 .4. They yield the S-matrix elemen t SPl n1 P2'a~n~, PI ff I61 P 20-2 42 = (2 ;T)-6 ( -i)2g ►firlt#~cf9~~I~G O'm'kl m xu;~r(Pz, aZ"?) uif(i~i'aini) U~(P?C2n2~~r(PI(TI ni) X Id4 X Jd4y e-p~'xe-6Ye Pz'XeiP1-Y(-i)A k'klX-YI with the last term indicating subtraction of the preceding term with interchange of particles !' and 2' (or equivalently Iand 2) . There are no second-order graphs for boson-boson scattering in this theory ; the lowest-order graphs are of fourth order, such as that shown in Figure 6.7. More specific examples of formulas like Eqs . (6.1.27)and (6.1.28) will be given in Section 6.3, after we have had a chance to evaluate the propagators a ndgoover to momentum space . A different . It is instructive also to look at an example with a trilinear interaction in which all three fields are the same, or at least enter into the interaction in a symmetric way . Theory I f Now ta ke the interaction density to be a sum of terms that are trilinear in a set of real bosonic fields 0& } (X)= t gem, O ~'~~Om(x)On(x) (6.1.29) ~'mn with g ,,,,,,a real totally symmetric coupling coefficient . Suppose we want to consider a scattering process 12 -+1'2' to second order in this interaction . Each of the two vertices must have two of the four external lines attached to it . (The only other possibility is that one ❑f the external lines is attached to one vertex and three to the other vertex, but the vertex with three external lines attached to it would have no remaining lines to connect it to the other vertex so this would be a disconnected contribution .)The additional line required at each vertex must then just serve to connect the two vertices to each other . There are three graphs of this type, differing in whether the other external line that is attached to the same vertex as line Iis line 2 or 1' or 2' . (See Figure 6,8.) Following the rules given above, the contribution to the S'-matrix from those three 6.1Derivation of the Rule s 1r L+ t f 1 t 4 f 41 S1 f~ r t yf ti^} 1 GlF 2 ' ♦ t k~ ~r y f'r.....'...( rr 22 4~ rr 1 YR {ti J .ti 1 2273 Figure 6 .8. The connected-second order diagrams for boson-boson scattering in a theory with interaction (6 .1.29). diagramsis SRjdinj P~O2ny Picini P2a-n2 -1)2(Ga)-'~ere fF9iTtMfTTtrrd4'Xdy ~-iAe rf,rr~~~~~)U+P+1t3fTt +11lr1 . / I * ! J7 1 +"s 1 ! ! ! Y luf I I I P2 2 2x x uM(P1al ni)e'pl vUM,{P2a2n?)eIP'-y +U14149 i0I)e ip''xu~( Pjainl)etP1'X *u~!(A2a2ni)e `P' x ue( Pit7ini)e'Pt~ x F xuyyirlp(71n' )~~Ep 1YUiiilP2U 7Yd2~ ~r P2 }f 1!. (6.1.30) To be even more specific, if the bosons in this theory are spinless particles of a single species, then we write the interaction (6.1.29) in the for m = go3/3t and the S-matrix element (6-1.30) for scalar -scalar scatter ingis SPY, P'2,P1 P2 = 2 (27r)6 f 6E1 E,'EjE ! x[exp(_i(p i + P'2) ' x) Cxp(i(pl + P2)'A *exP{i (P1 - Pi )' x) exP(a(P2 - Pz) - Y) +eXP(r(P1 -Pa) ' X) eX P(i(P2 - Pi ) ' Y)1(6.1.31) 274 6 The FeynmanRules where dAx - y} is the scalar field propagator, calculated in the next section . There are no terms of third order in {x}, or of any odd order in41(x) . 6.2Calculation of the Propagato r We now turn to a calculation of the propagator {6 .1.14}, an essential ingredient in the Feynman rules that arises in the pairing of a field W&) with a field adjoint ~m (Y). Inserting Eys . {6.1.15} and (6.I.16)in Eq. (6.1.14), and using the commutation or anticommutation relations for annihilation and creation operators, we have immediatel y fGr Lr In the course of calculating commutators and anticommutators in Chapter 5, we showed that -~ ~~(Pff n)u~(Pun) _ (2/p2 +m~ Pr„ :(p, p2+rra~ ( 6.2.2) -~ u,,,(p c n)v*(p c n) + (2 ~p~tn,2, Ptm (-p, - Vp ~ m.2) (G.2.3) where P /,,,(P,cc)) is a polynomial in p and w . (Here as in Eq . (6.2.1), the top and bottom signs refer to bosonic and fermianic fields, respectively .) For instance, if V,~,{x} and ,,(y)are scalar fields O(x) and 0(y)for a particle of spin zero, then we have simpl y PW = I . If~)&) and V,,(y) are Dirac fields for a particle of spin then(6.2.4) P"M(P) =[(-kw" +MVJ ,,, , (6 .2.5) where l and m are here four-valued Dirac indices . (The matrix Pappears here because we are considering the pairing of ~~~(x} with V'~(Y) . It is absent in the pairing of V,,(x) with ~t(Y) - 1 .pt(Y)p) If W~(x) and W"(}')are vector fields V ~,(x) and Y,(y)for a particle of spin one, then 6.2Calculation of the Propagator 275 More generally, if ~~~{x} and W,„(y) are components of fields ~ ,b(x) and Vab(y) for a particle of spin j, in the irreducible ( A, B ) and ( A,B)repre- sentations of the homogeneous Lorentz group, the n [lrf]r5Fbfa PI aaexp(+ 0 - - P'))] by X exp(-Op J(,!))] aa, exp(+Op. J(")] ~;, 1) (6 .2.7) where Binh 0 = PI/m, while a, b, a, run by unit steps from -Ato+A, -B to +B, -A to + .!, and -B to +B, respectively, and likewise for the running indices a',b', ac', and R . Inserting Eqs .(6.2.2)and ( 6.2.3)in Eq .(6.2.1) yield s Y) = O(X - Y)P~,, -i-) A+(x - Y ) + 0 (Y - -X) P"'M i-f-) A,(y - x)Ux where ❑+(x) is the function introduced in Chapter 5(6.2.$) (6.2.9) in which p ois taken as + V+ m2 . To go further, we must say a bit about how to extend the definition of the polynomial P(p) .Eqs.(.2.2)and (6.2.3) only define P(p) for four- momenta on the mass shell' . i.e., with p° =± p -}- W . Any polynomial function of such four-momentum can always be taken as linear in ffl,, because any power (p°)2'' or {p°} 2v+lcan be written as (p'+ m')ti' or P°(A2 +m2), , respectively . Thus we can define a polynomial P(')(q) by the conditions tha t p, L) (P)= p (P) (for p° = p2 +rra2 ~ILI(q) = Pl °)(9) + &(')(q) (for general q") ,(6.2.10) where P((,1) are polynomials depending only on q.We can now use the relations ax ~X_ O (recall that O(x) has a unit step at x° ,and is otherwise constant) to move 276 6The Feynman Rules the derivative operators to the left of the 0functions in Eq.(6.2.8) (_ O X (X1 - Y") P 2)(-iV) A+(x- y)- A+ (y -x)], (6-2,12 ) where ❑Fis the `Feynman propagator ' However, for x° = 0 the function ❑+(x) is even in x, since a change x --* -x in Eq . (6.2.9) can be compensated by a change p-* - pin the integration variable . We can therefore drop the second term in Eq . (6.2.12), and write simply A,,.(x3 J')=P~.~) (_-i-_) Q~,-{x - y} . ( 6.2.14) f~x It will be most useful to use the expression of the Feynman propagator as a Fourier integral . The step functions in Eq .(6.2.13)have the Fourier representation' p(-+)ds(6.2.15) This can be combined with the Fourier integral (6 .2.9) for ❑+(x) . We introduce new integration variables, q=p, qO= p4 + s in the first term of Eq. (6.2.13), yieldin g 1 -~A,F(X)2~i,d3q dqoexp(iq x -iq°x°) (27z)3 V2 9+W X~ 00- ~q2 _+M2 iE) qO- ~q2 + m2-ie) Combining denominators and adopting afour-dirnensional notation, we have simply q, + rn, -i e where q2 = q2 -- (q°)2 . (In the denominator we have rep laced 2,- q+ era with e, because the o nly important thing a bout this quantity is tha tit To prove this, note that if t > 0 then the contour of integration can be clascd with a large clockwise semi-circ le in t he lower ha lf-plane, so the integral picks up a contribution of -2gi from the pole at s = -iF .Ift<0then the contour can be closed with a large cou nter-clockwise .w1t7i-circle in the upper half-plane, where the integrand is analytic, giving an integral equal to zero . 6.2Calculation of thePropagator 27 7 is a positive infinitesimal .) This shows incidentally that AFis a Green's function for the Klein-Gordon differential operator, in the sense tha t with boundary conditions specified by the --re in the denominator : as shown by Eq . (6.2.13), dF(x) for x0 ,--* hoc or x0 ---# -ac involves only positive ❑r negative frequency terms, exg(-ix0 p+ m) or exp(+ix ()p2+ m 2), respectively . Inserting Eq . (6.2.16) in Eq . (5.2.14) now gives the propagator a s fq2 +M2-ic(6.2.18) There is one obvious problem with this expression . The polynomial P(p) is Lorentz-covariant when p is on the mass shell, p2= -rn2 , but in Eq . (5.2.18) we integrate over all q 1', not restricted to the mass shell . The polynomialp(L)(q)is defined for general q1'to be linear in q ', a condition that clearly does not respect Lorentz covariance unless the polynomial is also linear in each spatial component q~ as well . We can instead always define our extension of the polynomial P (p) to general four-momenta qt, which we shall call simply P(q), in such a way that P(q) is Lorentz- covariant for general q P, in the sense tha t where At, is a general Lorentz transfarniation, and D(A) is the appropriate representation of the Lorentz group . For instance, for scalar, Dirac, and four-vector fields, these covariant extensions are obviously provided b y just replacing p 'with a general four-momentum q "in Eqs . (6.2.4),(6.2.5), and (6.2.6). For the scalar and Dirac fields, these are already linear in q so here there is no difference between P(L)(q)and P(q): p~~~(q )=P~„.(R) (scalar, Dirac fields) . (6 .2.19) On the other hand, for the vector field of a spin one particle, the 00 components of the covariant polynomial P ,ti,(q)_qf,,+ m-2 q,,q, are quadratic in q0, so here there is a difference : (6.2.20 (The extra term here is fixed bythe two conditions that it must cancel the (4o)2 term in Poo(q),and must vanish when q Pis on the mass shell .) Inserting this in Eq .(6.2.18 gives the propagator of a vector field as 278 6 The Feynman Rule s The first term is manifestly covariant, and the second term, though not covariant, is local, so it can be cancelled by adding a local non-covariant term to the Hamiltonian density . Specifically, if Ve(x) interacts with other fields through a term Y,,(x) .IP{x} in (x), then the effect of the second term in Eq . (6.2.21) is to produce an effective interactio n - i"Y"ff (x) aP(x) ] [ - a JV (x)] I - i r~z-'6~~~]. (The factors -i are the usual ones which always accompany vertices and propagators . The factor is needed because there are two ways to pair other fields with ,~~{x}, differi ng in the interchange of JPand P.)Thus the effect of the non-covariant second term in Ey, (622 1)can be cancelled byadding to (x )the non-covariant ter m A""NC(X) = -~*"ff W =I~JO(X)l(6.2.22) It is the singularity of the equal-time commutators of vector fields at zero separation that requires us to employ a wider class of interactions than those with a scalar density . A detailed non-perturbative proof of the Lorentz invariance of the S-matrix in this theory will be given in the next chapter . It should not be thought that this is solely a phenomenon associated with spins j 1 . For instance, consider the vector field associated with a particle of spinj-m 0, equal (as discussed in Chapter 5) to the derivative 0~0(x) of a scalar field . For the pairing of this field with a scala r the polynomial P(p) on the mass shell i s while the pairing of 0,*x) with O,O*(y ) yields a polynomial(6.2.23) (6.2.24) The covariant polynomials for general off-shell four-momenta q "are again obtained byjust substituting q Pfor#` in Eqs .(6.2.23) and (6 .2.24). Eq.(6.2.23) shows that P Jq) is already linear in qa, so here there is no difference between F Jg} and P~~'}(q ).However, for Eq . (6.2.24) there is a difference : A'11 0 A17 so here the propagator is q,e f q2 +tn2 A J7(6.2.25) (6.2.26) 6.2Calculation of the Propagator 279 Just as before, the non-covariant effects of the second term may be removed by adding to the interaction a non-covariant ter m X;5[J()(X)] (6.2.27) where P(x) is here the current which multiplies 0,,O(x) in the covariant part of z~(x ). It should be clear that at least for massive particles) the effects of non covariant parts of the propagator can always be cancelled in this way by adding non-covariant local terms to the Hamiltonian density . This is because the numerator P (11)(q)in the propagator must equal the covariant polynomial P~„ ,(q)when q~` is on the mass shell, so the difference between P'2(q) and P~,(q)must contain a factor q2+ m' . This factor cancels the denominator (q2 + M2 -jr)in the contribution of this difference to Eq. (6.2.18), so Eq . (.2.18)always equals a covariant term plus a term proportional to the delta function bl(x - y )or its derivatives . The effect of the latter term may be cancelled by adding to the interaction a term quadratic in the currents to which the paired fields couple, or in their derivatives . In what follows, it will be assumed tacitly that such a term has been included in the interaction, and in consequence we shall use the co variant polyn omial P4,(q)in the propagator (6,2.1$), and will thus henceforth drop the label `V . It may seem that this is a rather ad hoc procedure . Fortunately, in the canonical formalism discussed in the following chapter, the non-covariant term in the Hamiltonian density needed to cancel non-covariant terms in the propagator arises automatically . This, in fact, farms part of the motivation for introducing the canonical formalism . Before closing this section, it may be useful to mention some other defi- nitions of the propagator, equivalent to Eq .(6.2.1), that appear commonly in the literature . First, taking the vacuum expectation value of Eq.(6.1.14) gives 0 ±O(y- x)[Vm * (Y), ~),. W] + (6.2.28) (Here {AB... }()denotes the vacuum expectation value ((DO, AB---(Do)-) Both +(x) and y~-'(y)annihilate the vacuum, so only one term in each commutator or anticnmmutatar in Eq . (6.2.28)actually contributes to the propagator , -~~~~~~~Y) T ~(x-Y)~~'~ ~~~Wam~~3'}~0~~~~'-?~~~~ ►n~~Y)~'~' ~~)~o . (6.2.29) 280 6 The Feynman Rule s Also , W-t and zp+ would annihilate a vacuum state on the right , and W-and a p+f would ann ihilate the vacuum state on the left , so w+ and y- may bereplaced everywhere inEq.(6.2.9)with the complete field This is often writte n where T is a time-ordered product, whose definition is now extended" to all fields, with a minus sign for and odd permutation of fermioni c operators. 6.3 Momentum Space Rule s The Feynman rules outlined in Section 6 .1 specify how to calculate the contribution to the S-matrix of a given Nth order diagram, as the integral over Nspacetime coordinates of a product of spacetime-dependent factors . For a final particle (or antiparticle) line with momentum p' ,'leaving a vertex with spacetime coordinate x", we get a factor proportional to exp(-ip'  x), and for an initial particle line with momentum p Aentering a vertex with spacetime coordinate xy, we get a factor proportional to exp(+ip  x) . In Section 6.2we saw that the factor associated with an internal line running from y to x can be expressed as a Fourier integral, over off-shell four-momenta q ,, of an integrand proportional to exp(iq, (x - y) ). We can think of q ,'as the four-momentum flowing along the internal line in the direction of the arrow from y to x . Hence the integral over each vertex's spacetime position merely yields a facto r (27r )454 (1: p + 1 : q - 1 :pt-E q ;), (6.3,1) where Ep' and Y_p denote the total four-momentum of all the final or initial particles leaving or entering the vertex, and Zq' and q denote the total four-momentum of all the internal lines with arrows leaving or entering the vertex, respectively .Of course, in place of these integrals over xAs, we now have to do integrals over the Fourier variables q~`, one for each internal line . These considerations can be encapsulated in a new set of Feynman rule s This is not inconsistent with our previous definition of the lime-ordered product of Hamiltonian densities in Chapter 3, because the Hamiltonian density can only contain even numbers of fermionic ]'told factors . 6.3Momentum Space Rule s P'Wn' (2 (a) cy n(2312 vpP (d),PC --).~nP 312 -4 r~(2v~ ( Pa n") 4 (b) ___)r r + ~ I1 or P cyn281 (2 p ,ffn) Pan 53 p'a ne (e) NI- (q) YTS4 2 2 (f) Figure 6.9. Graphical representation of pairings of operators arising in the momentum-spare evaluation of the S-matrix, The expressions on the right are the factors that must be included in the momentum space integrand of the S-matrix for each line of the Feynman diagram . (see Figure 6 .9)for calculating contribution to the S-matrix as integrals over momentum variables : (i) Draw all Feynman diagrams of the desired order, just as described in Section 6 .1. However, instead of labelling each vertex with a spacetime coordinate, each internal line is now labelled with an off- mass-shell four-momentum, considered conventionally to flow in the 282 6TheFeynman Rule s direction of the arrow (or in either direction for neutral particle lines without arrows .) (ii)For each vertex oftype i ,include a facto r -- t{27r}4gi 64 (1:p+ 1 :q - 1: p- q') (6.3.2) with the momentum sums having the same meaning as in (6 .3.1). This delta function ensures that the four-momentum is conserved at every point in the diagram . For each external line runn ing upwards out of the diagram, include a factor {27r}-3/2u~( p'U'rt')or (2n)-_1j2 a, ( p'a'nr), for arrows pointing upwards or downwards, respectively . For each external line running from below into the diagram, include a factor (27r)-312U~( pan) or (2;T)-3/2V;(pUra), for arrows pointing upwards or downwards, respectively . For each internal line with ends labelled ~ and m, the arrow pointing from intoe, and carrying a momentum label q", include as a factor the integrand of the integral for -iA,,,,,(q ) -i(27r)J4pln7 (q)(R'+Mr-red . ( 6.3.3) A reminder : for scalars or antiscalars of four-momentum q, the us and v s are simply (2q0)-1/`, while the polynomial P(q) is unity . For Dirac spinors of four-momentum p and mass M.the us and vs are the normalized Dirac spinors described in Section 5 .5, and the polynomial P(p) is the matrix (-aT-,,pP+ M)fl . (iii) Integrate the product of all these factors over the Maur-moment carried byinternal lines, and sum over all field indices e,m, (iv) Add up the results obtained in this way from each Feynman diagram . Additional combinatoric factors and fermionic signs may need to be included, as described in parts (v) and (vi) of Section 6.1.Examples will be given at the end of this section . We have afour-monnentum integration variable for every internal line, but many of these are eliminated by the delta functions associated with vertices . Since energy and momentum are separately conserved for each connected part of a Feynman diagram, there will be C delta functions left over in a graph with C connected parts . Hence in a diagram with I internal lines and V vertices, the number of 'independent four-momenta that are not fixed b ythe delta functions is I f [ V-Q.This is clearly also the number L of independent loops : which is defined as the maximum number of internal lines that can be cut without disconnecting the diagram, because any such and only 6.3 Momentum Space Rules 28 3 such internal lines can be assigned an independent four-momentum . We can think of the independent momentum variables as characterizing the momenta that circulate in each loop . In particular a tree graph is one without loops ; after taking the delta functions into account there are no momentum-space integrals left for such graphs . For instance, in a theory with interaction (6 .1.18), the S-matrix (6.1.27) for fermion-boson scattering is given by the momentum-space Feynman rules as SpIdlr~~ P 2't7l2n2 -Pig] ni P26V12 n') ul(I)l gin VlWklm PrW,,( 9')xI d4q (-i(2;T)-4 q2 + M2 -tip +Uk(P2f6zn~}uk+(P2a2r~2)6'(P2 -Pif-~`~~~~P~- FT + 01 with labels l and 2 here denoting fermions and bosons, respectively . The momentum-space integral here is trivial, and give s Splrs'j P:iP2'OFzn2PI (Yin IP2672r12-i(27L)-264(pi+P2 ` P1P2) X 1: g r'm+kfgmikUl*,(Ai'n-ini)UI(Pi(l~~~) 0{rWOm XPm'm(Fl'- p ] ~ Ukr(P2 ~~2n2)Uk( P262rI ?) ~~1'l P1~~ +IYlm-iF (p2' -pj)2 + M;~- if In the same way, the S-matrix element (6.1.28)for fermion-fermion scat- tering in the same theory i s SPi 'ffiniP2+¢ n1p, u i ni P 2 aY n:i(27t)-264(p t+P2 -P1 - P2) Pk+k(Pir-PI ) X grn'mk' 9i'tk These results illu strate the need for a more compact notation .Wemay define a ferm inn-boon coupling matrix (6-3-7) The matrix elements (6.3.5)and (6.3.6)for fermion-boson and fermion-- 284 6The Feyrta nRules fermion scattering can then be rewritten in matrix notation a s SAifa~ni P2 'din' , , Pi0- 1 n I P2 0-2 n2 = ~(27t)-2d4(p 1+ P2 - P1-P2~ Wk 22 +(t(p1 an.)refP(P2~ - MI --~ i~-FkU(Pi~in~~ xuk(R21'-T?n2)UkF( P2ff?n2) and SPi pi'~2 W~ ,piainiP2 dz n2 -i(27Z)-26 1 lPl+P? - Pi-F2~ X57Pk'k(Pl'-P1 k'k(PI Mk(f.3.8) x(.ut(p2r,g2n~)rk'u(Plr 61 (.ut(p2 C72 n2) rkU(PiuIn1)~ where M2 and m2 are the diagonal mass matrices of the ferrnians and bosons in Eqs .(6.3.8) and (6.3.9), respectively . The general rule is that in using matrix notation, one writes coefficient functions, coupling matrices, and propagators in an order dictated by following lines backwards from the order indicated bythe arrows . In the same notation, the S-matrix for boson-boson scattering in the same theory would b egiven by a sum of one-loop diagrams, shown in Figure 6 .7: SPI'~jniQ2'azn2 , Piuini P2aa n4-' - ~ ~(PI + P 2-P1TP?) xY uZi(Pi, (-Ti,n1)uZ~{pi, a~, na)Uk,( Pi, ~1, n1)Uk,( Ra,ff2}nz) k,k2k~k2 P (q) P(q+p')~ x dq Tr I -'~z q2+ z -i~~r 1(q+pj)2+11fi2 _ie xrk, (q+ p'_pi)2 +M2 -ark2- (R-P~)2+M2- ie +... , (6 .3.10) where the ellipsis in the last line indicates terms obtained by permuting bosons 1',2`, and 2 . The minus sign at the beginning of the right-hand side is the extra minus sign associated with ferrnionic loops . Note that after elimination of delta functions there is just one momentum space 6.3 Momentum Space Rules 285 integral here, as appropriate for a diagram with one loop . We shall see how to do this sort of momentum-space integral in Chapter 11. To make this more specific, consider a theory with a Dirac spinor field W(x) of mass and a pseudoscalar field O(x) of mass m, interacting through the interaction -ig~VyW (The factor -i is inserted to make this interaction Hermitian for real coupling constants g .)Recall that the polynomial P(q) for the scalar is just unity, while for the spinor it is L-iYpq}l+M]fl. Also, the u for a scalar of energy E is (2E )-1J2, while for the spinor u is the conventionally normalized Dirac spinor discussed in Section 5.5. Eqs .(6.3.8), (6.3.9), and (6.3.10) give the lowest-order connected S- matrix elements for fermion-boson scattering, fermion-fermion scattering, and boson-boson scattering : SPIIC' P?P~~~ F~ - -~~~7~)-2g2(4E~E2)='1'6'( P1+P2-P1-P21 r f ar}~~~~1' W` POP + + (U(PI cyi)i5 -.T5 U(PI al)(P2 pl)2 + MI IE x (42 (;') ~/5 u(P' cr')) (142(F2)Y5 U(P1 60) 1 X fir; r~ ~ PiP~ = -(27r)-6 g2 (16E1 E2EiE2)-1/26¢(P1 + P2- Pi -Pad x coq Tr -,-iyuq" + M -iyu(q + plly + M hi q2 +X12 - (q+pi)2 +M2-iF +... where in the last formula the ellipsis indicates a sum over permutations of particles 2, 1', 2' . The factors J3 in the fermion propagator numerators have been used to replace uf with u . Another useful topological result expresses a sort of conservation law of lines . For the moment we can think of all internal and external lines as being created at vertices and destroyed in pairs at the centers of internal lines or when external lines leave the diagram . (This has nothing to do with the directions of the arrows carried by these lines .) Equating the 286 6 The FeynmanRules numbers of lines that are created and destroyed then give s 2I+E=njYz, {6 .3.11) where I and E are the numbers of internal and external lines, Yj are the numbers of vertices of various types labelled f, and nj are the number of lines attached to each vertex . (This also holds separately for fields of each type,) In particular, if all interactions involve the same number nj = n of fields, then this read s where V is the total number of all vertices . In this case, we can eliminate = 1) f from Eqs . (6.3.4) and (6.3.11).and find that for a connected (i .e.,C graph the number of vertices is given b y For instance for a trilinear interaction the diagrams for a scattering process (E = 4) with L =0, 1, 2 - .. has V = 2, 4, 6 ... vertices . In general, the expansion in powers of the coupling constants is an expansion in increasing numbers of loops . 6,4 O ffthe Mass S hell In the Feynman diagrams for any 5-matrix element all external lines are on the mass shell' ; that is, the four-momentum associated with an external line for a particle of mass m is constrained to satisfy p,,Pq= -m2- It is often important also to consider Feynman diagrams `off the mass shell', for which the external line energies like the energies associated with internal lines are free variables, unrelated to any three-momenta . For one thing, these arise as parts of larger Feynman diagrams ; for instance, a loop appearing as an insertion in some internal line of a diagram could be regarded as a Feynman diagram with two external lines, both off the mass shell . Of course, once we calculate the contribution of a given Feynman diagram off the mass shell, it is easy to calculate the associated S-matrix elements by going to the mass shell, taking the four-momentum p yflowing along the line into the diagram to have p O= p2 -+m2 for particles in the initial state and p° =- p! + rrt for particles in the final state, and including the appropriate external line factors (2ar)-1/zUe or (2 7E)-31'-V; for initial particles or antiparticles and (27z)-3/2u* or (2-H )-312zi~ for final particles or antiparticles . Indeed, when we come to the path integral approach in Chapter 8we shall find it easiest first to derive the Feynman 6.4OfftheMass Shell 287 rules for diagrams with all external lines off the mass shell, and then obtain ,S-matrix elements byletting the momenta associated with external lines approach their appropriate mass shells . Feynman graphs with lines off the mass shell are just a special case of a wider generalization of the Feynman rules that takes into account the effects of various possible external fields . Suppose we add a sum of terms involving external fields eJx} to the Hamiltonian, so that the interaction V(t) that is used in the Dyson series (3.5.10) for the 5-matrix is replaced with VF(t)=V (t)+ dux e,, ( x,t)n,, (x,t)  (6 .4.1) A The `currents' op(t) have the usual time-dependence of the interaction picture Oa(t)=exp{ iHot) o,, ( 0)exp{ -iHot}, (6.4.2) but are otherwise quite arbitrary operators . The S-matrix for any given transition a then becomes a functional S #,[,-] of the c-number functions r,,(t) . The Feynman rules for computing this functional are given by an obvious extension of the usual Feynman rules . In addition to the usual vertices obtained from V(t),we must include additional vertices : ifo,,(x) is a product of rya field factors, then any oa vertex with position label x must have nu lines of corresponding types attached, and makes a contribution to the position-space Feynman rules equal to -icA(x) times whatever numerical factors appear in oa(x ). It follows then that the rth variational derivative of S #,[e] with respect to F,(x),eh(y) ... ate=0is given by position space diagrams with r additional vertices, to which are attached respectively n, , ray,...internal lines, and no external fines . These vertices carry position labels x,y -- -over which we do notintegrate ; each such vertex makes a contribution equal to -i times whatever numerical factors appear in the associated current a , In particular, in the ca ::e where these currents are all single field factors, i.e., VF(t) V( i) +~~~C', 1X 7t lYl,, (X 7t) therthvariational derivative of 5 fl,,[e] with respect to e~,(x),e,,(y)... at dF= 0 is given by position space diagrams with r additional vert icescarry- ingspacetime la belsx, y - -, to each of which is attached a single internal particle line of type ~, m These can be thought of as off-shell external lines, with the difference that their contribution to the matrix element is not a coefficient function like (2 7t)-3f2ut(j},~)dP` or(27C)-3/2tL(p,(7)8-ip x but apropagator, as well as a factor -ifrom the vertex at the end of 288 6The Feynman Rule s theline. We obtain a momentum s pace Feynman diagram w ith particles in states o:and # on the mass shell plus rexternal lines of t ypee, m carrying momenta p ,p'... from the variational derivat ive 6' SP a [E l by stripping away the propagators on each of the off-shell lines and then taking the appropriate Fourier transforms and multiplying with appropriate coefficient functions ue, u~, etc . and a factor (-a)r . It is very useful for a number of purposes to recognize that there is a simple relation between the sum . of contributions from all perturbation theory diagrams for any off-shell amplitude and a matrix element, between eigenstates of the full Hamiltonian, of a time-ordered product of corre- sponding operators in the Heisenberg picture . This relation is provided by a theorem,3which states that to all orders of perturbation theory ' W 6 6 (Y)loa(x), -iob(y) -- - I ) 1 (6.4.3) where 0,,(x), etc . are the counterparts of o ,(x)in the Heisenberg pictur e n(t)ieffcCiHnr(6.4.5) and Tp+ and Tfl- are `in' and `out' eigenstates of the full Hamiltonian H, respectively . Here is the proof . From Eq .(3.5.10), we see immediately that the left-hand side of Eq . (6.43) i s 61-S [6]0C (-i)N+r dTj ANNi rXN-0 For definiteness, suppose that x° x° ...x°. Then we can denote and by-ro1--- T aN4 all zs greater than x° ; by X11 . ~12V ,all zs between and and so on ; finally denoting by z, .1 - - -crNr all is that are less than x° . For a single 0operator, this is a version of the Schwinger action principle .4 6AOffthe Mass Shell 289 Eq. (6.4.6) then becomes : 6rSjej _i)N+r IY=O J'x f x°xJ~l~z0l   - d T4N, 0 dT11.. ,dzlNI fxi .}x2E Np2V i... jlfr xa x...0a(x,)TJV(T xl)... y(-Cr N,+fix The factor Nf JN4 !Ni ?...Nx!is the number of ways of sorting Nrs into r + Isubsets, each containing No, N1 ,... IVr of these -cs. Instead o f - summing over No,N1,  , NT, subject to the condition No+N,+...+Nr N, and then summing over we can just sum independently over No, ATt,... ,IVY, setting Nwhere it appears in (-i)' equal to No + NI+ ... +N.This gives 000"(X,) (xr ~-00)(P,,) (6.4.7) where 0C N 11 U(t t) =;Y ( ')d-r,...dTNTJ V(-11) V(TN)J ? JrN=o The operator U(t', t) satisfies the differential equatio n dt, with the obvious initial conditio n This has the solutio n UW,t}=exp(iHot)exp{- iH(t'- t)} exp{- iHa0=Q-lWAt)(6.4.8) tb.4.9) (6.4.10) (6.4.11) with n given by Eq . (6.4.5). Inserting Eq . (6.4.11) in Ey . (6.4,7) and using 290 6 The Feyraman Rule s Eq. (6.4.4), we have JS [6] (_Ot (Q(CXD)(DO' 0", (X 1) 0", (X')Q(-OO)q)') (6 .4.12) In deriving this result we supposed thatx° ~ x° ~--- > x~,so we could just as well replace the product of operators on the right-hand-side with the time-ordered product : 6S[E] T0~1(xl)...pOr(.Xe)JQ(-oo )d}, {6.4.13} But now both sides are entirely symmetric (or for fermions antisymmetric) in the as and x s, so this relation holds whatever the order of the times x~... x4. Also, we saw in Section 3.1that (in the sense of Eq .(3.1.12)) Tp± = p+oc?)(DO . ( 6.4.14} Hence Eq . (6.4A) is the desired result (6 .4.3), Problems 1.Consider thetheory of a real scalar field 0,with interaction (in th e interaction pi cture)V=g fd3x O(x)'/ 3?.Calculate the connecte d S-matrix element for scalar -scalar scattering to second order in g , doing allintegrals . Use the results to calculate the differential cross - section for scalar -scalar scat tering in the center- of-mass system . 2. Consider a theory involving a neutral scalar field O(x) for a boso n B and a complex Dirac field y ){x} for a fermion F,with interactio n (in the interaction picture) V = ig.f dux TFWyMx )O(x).Draw al l the connected order-g2Feynman diagrams and calculate the cor- responding , S-matr ix elements for the processes F `'+ B --* F" +B, F+F" --* F+F`,and F+ F--*B+B (where F' is the antiparticl e ofF).Do all integrals . 3. Consider the theory of a real scalar field O{x},with interactio n V.-g f d ux O {x}4/4!.Calculate the S '-matrix forscalar -scalar scat- tering to o rder g ,and use the result to calculate the differentia l scattering cro ss-sect ion.Calcul ate the correction terms in the S - matrix for scalar-scalar scatter ingtoorder g 2,expressing the re sult as an integral over a single four-momentum, but doallx-integrals . References 291 4. What is the contribution in Feynman diagrams from the contraction of the derivative .(x)of a Dirac field with the adjoint m~}~) of the field ? 5, Use the theorem of Sect ion 6 .4 togive express ions forthevacuum expectat ion values of Heisenberg picture operators ( TO,4~(xffQ) an d (To, 7"{<D(x),(D(y')fTo) i n the theory of Problem 1 ., to orders g and grespe ctively. References 1. F. J. Dyson, Pays .Rev. 75, 486, 17 36(1949) . 2, The formal statement of this result is known as Wicks theorem ;see G. C. Wick, Whys .Rev.50,268 (1950) . 3. I do not know who first proved this theorem . It was known in the early 1950s to several theorists, iDcludi-ng ell-Mann and F E . Low . 4. J. Schwinger, Ph ys,Rev. $2, 914 ( 1951 ). The Canonical Forma lism Ever since the birth of quantum field theory in the papers of Born, Dirac, Fermi, Heisenberg, Jordan, and Pauli in the late 1920s, its development has been historically linked to the canonical formalism, so much so that it seems natural to begin any treatment of the subject today by postulating a Lagrangian and applying to it the rules of canonical quantization . This is the approach used in most books on quantum field theory . Yet historical precedent is not a very convincing reason for using this formalism . If we discovered a quantum field theory that led to a physically satisfactory S-matrix, would it bother us if it could not be derived by the canonical quantization of some Lagrangian ? To some extent this question is moot because, as we shall see in Section 7.1, all of the most familiar quantum field theories furnish canonical systems, and these can easily be put in a Lagz'angian form . However, there is no proof that every conceivable quantum field theory can be formulated in this way . And even if it can, this does not in itself explain why we should prefer to use the Lagrangian formalism as a starting point in constructing various quantum field theories . The point of the Lagrangian formalism is that it makes it easy to satisfy Lorentz invariance and other symmetries : a classical theory with a Lorentz-invariant Lagrangian density will when canonically quantized lead to a Lorentz-invariant quantum theory . That is, we shall see here that such a theory allows the construction of suitable quantum mechanical operators that satisfy the commutation relations of the Pvincare algebra, and therefore leads to a Loren tz-invariant S-matrix . This is not so trivial . We saw in the previous chapter that in theories with derivative couplings or spins j ~ 1, it is not enough to take the interaction Hamiltonian as the integral over space of a scalar interaction density ; we also need to add non-scalar terms to the interaction density to compensate for non-covariant terms in the propagators . The canonical formalism with a scalar Lagrangian density will automatically provide these extra terms . Later, when we come to non-Abelian gauge theories in volume II, this extra convenience will become a necessity ; it would b e 292 7.1 Canonical Variables 293 just about hopeless to try to guess at the form of the Hamiltonian in such theories without starting with a Lorentz-invariant and gauge-invariant Lagrangian density . 7.1 Canonical Variable s In this section we shall show that various quantum field theories that we have constructed so far satisfy the commutation rules and equations of motion of the Hamiltonian version of the canonical formalism . It is the Hamiltonian formalism that is needed to calculate the S-matrix (whether by operator or path-integral methods) but it is not always easy to choose Hamiltanians that yield a Lorentz-invariant S-matrix . In the balance of this chapter we shall take the Lagrangian version of the canonical formalism as our starting paint, and use it to derive physically satisfactory Hamiltonians . The purpose of the present section is to identify the canonical fields and their conjugates in various field theories, to tell us how to separate the free-field terms in the Lagrangian, and incidentally to reassure us that the canonical formalism is indeed applicable to physically realistic theories . We first show that the free fields constructed in chapter 5 automatically provide a system of quantum operators q'(x, t) and canonical conjugates pn(x, t) that satisfy the familiar canonical commutation or anticommuta- tion relations : [q"(x, t )}pjj(y, t)]+ = i63(x -- y)fin , (7.1.1) where the subscripts + indicate that these are commutators if either of the particles created and destroyed bythe two operators are bosons, and anticommutators if both particles are fermions . For instance, the real scalar field O(x) for aself-charge -conjugate particle of zero spin was found in Section 5.2 to obey the commutation relatio n where ❑is the functio n AW J2.]cO(27C)3 with k° = + m2 . We note that 294 7 The Canonical Formalis m (A dot denotes the derivative with respect to the time x° .) It is easy then to see that the field and its time-derivative ~ obey the equal-time commutation relations : Therefore we may define canonical variables(7.1.4) (7.1.5) (7.1.6) (7.1.7) which satisfy the canonical commutation relations (7.1.1)-(7.1.3). For the complex scalar field of a particle of spin zero with a distinct antiparticle, the commutation relations ar e We may therefore define the free-particle canonical variables as the com- plex operators 4{x,t}_ (x,t) , (7 .1.8) Equivalently, writing (01 + i02)/ wi th Ok Hermit ian for k = t,2, we have canonical variables Pk(X,0 = ~k(X, 0 , (7.1.11) and these satisfy the commutation relations (7.1.1)--.(7.1.3). For the real vector field of a particle of spin one, the commutation relations are given by section 5.3as M2 1 (We are using 0 rather than V AIfor the vector field because we want to reserve upper case letters here for the fields in the Heisenberg picture .) Here the free-particle canonical variables may be taken a s f OV°(x,t)(7.1.12) (7.1.13) with i = 1, 2, 3 . The reader may check that (7.1.12)and (7.1.13) satisfy the commutation relations (7.1.1)-(7,1 .3),The field equations (5-3-36) and 7.1Canonical Variables 295 (5.3.38) together with E q.(7.2.13)allow us to express 0 in terms of the other variables a s so ~;~° is not regarded as one of the q s. The extension of these results to complex vector field smay be handl edjust as forcomplex scalar fields . For the Dirac field of a non-a jorana spin;particle , Section 5.6 shows that the an ticammutator i s and Here it would be inconsistent to take W, and f to canonical variables, because their anticommutator does equal times . It is conventional instead to definebe independent not vanish a t (7.1.15) (7.1.1b) Itis easy then to see that (7.1.15)and (7.1.16) satisfy the canonical anticommut ation relations (7 .1.1)-(7.1.3). For any system of operators that satisfy commutation or anti comm u- tation relations like (7.1.1)-(7.1.3),we may de fine a quantum mechanical functional derivative : for an ar bitrary bo sonic functional ,F[R( t),F(t)] of qn(x, t )and p ,,(x, t) a ta fixed time t, we define ' 5F[q (t)aP(t)] .[Pn(X, t), F[q(t), p(t) ] 6F[q(t),p(t)][F[q(t), p(t)] ,qn(X, t)(7.1.1?) (7.1.18) This definition is motivated by the fact that if F[q(l), p(t)] is written with all qs to the left of all ps, then ( 7,1.17) and (7.1.18) are respectively just the left- and right-derivatives with respect to q 'and pM . That is, for a n 'We are here using a notation that will be adopted hencelorlh ;Iff(x, v) I's a function oftwo classes of variables collectively called x and y, then F[f (y)] indicates a functional that d(Lpends on the values of f [x, y] for all x at fixed y . By a bosonic functional we mean ❑ne in which each term contains only even number-, of fcrrnionic fields . 296 7 The Canonieal Formalis m arbitrary c-number' * variation 6q and 3p of the q s and ps, we hav e 6F[q(t),p(t)]= dux1:(5(x ,t)~6F[q(t), p(t)] ~qn(xa r) +6 FLq (t),P(t)]6Fn(X, t) 6 P n\xi0 For more general functionals we need the definitions (7.1.17) and (7 .1.18) to pin down various signs and equal time commutators that may appear . In particular, HO is the generator of time-translations on free-particle states in the sense tha t q'(x, t)= exp(iHot)q'( x, 0) exp(-Wat) , p"(x, t) = exp (Wot)P,(x,0)exp(--iHot), so the free-particle operators have the time-dependenc e Pn(x,0 Pn(x,0=-i IP't(x, 0, HolJHo 6qn(x, r)(7.1.19) (?.1.2U) (7.i.21) We recognize these as t he familiar dynamica lequat ions in the Hamiltonian forma lism. The free-particle Hamiltonia nis given as always b y HO d3kc~~(k,u, n) a (k,a, n) k2+MI(7.1.23) n,(7 This HO may be rewritten in terms of the q s and ps at time t . For instancy, it is easy to see that for a real scalar field, Eq .(7.1,23) is equal up to a constant term to the functiona l Ho _d 3X [~P2+~(Vq)2 + ;M2q2~ (7 .1.24) To be more precise, using (7.1.7)and the Fourier representation of the scalar field 0, we find that Eq. (7.1.24) becomes ; .Ho f A ko [a(k), a'(k) ] _ f t31C0(a(k)a(k) + 16'(k -1C)). (7 .1.25) Where q'tl and p„ are bosunic or fermionic, dqn and bph are understood to commute or anticom- mute with all fermionic operators, respectively, and to commute with all bosnnk operators . 7.1Canonical Variables 297 This is the same as Eq. (7.1.23), except for the infinite constant term . Such terms only affect the zero of energy, and have no physical significance in the absence of gravity,f . Explicit forms for HO as a functional of the q and p variables for other fields will be given in Section 7 .5. It is usual in textbooks on quantum field theory to derive Eq . (7.1.25) as a consequence of Eq . (7.1.24), which in turn is derived from a Lagrangian density . This seems to me backward, for Eq .(7.1.25) must hold ; if some assumed free-particle Lagrangian did not give Eq.(7.1.25) up to a constant term, we would conclude that it was the wrong Lagrangian . Rather, we should ask what free-field Lagrangian gives Eq .(7.1.25) for spinless particles, or more generally, gives the free-particle Hamiltonian (7.1.23). This question may be answered b ythe well-known Legendre transformation from the Hamiltonian to the Lagrangian ; the free-field Lagrangian is given b y LO [q(t), 4 [t)1::=1:f d3x p, (x,t)q'(,t)- H o} (7 . 1.26) n it being understood that pn is replaced everywhere byits expression in terms of qn and q' (and, as we shall see, perhaps some auxiliary fields as weli) . For instance, from the Hamiltonian (7 .1.24) and (7.1.7)we can derive the free-field Lagrangian for a scalar field : )2_ M1 2Lo fd'x[pq - lp' - I(Vq q f d3X la,o apo _ I M202 ] Whatever we suppose the complete Lagrangian of the scalar field may be, this is the term that must be separated out and treated as a term of zeroth order in perturbation theory . A similar exercise may be carried out for the other canonical systems described in this section, but from now on we shall content ourselves with guessing the form of the free-field Lagrangian and then confirming that it gives the correct free-particle Hamil#onian . We have seen that various free-field theories can be formulated in canonical terms . It is then a short step to show that the same is true of the interacting fields . We can introduce canonical variables in what is called the 'Heisenberg picture', defined b y Q'(x,t)=exp(iHt)q'(x,, 0) exp (-iHt), {7.1.2$} Pn(X,t)= eXp(iH r)PM(X,0)exp(-iHt), (7.129) ~ However, charges in such terms due tochanges in the boundary conditions for the fields, as for instance quantizing in the space between parallel plates rather than in infinite space, are physically significant, and have even been measured .E 298 7The Canonical Formalis m where H is the full Hamiltonian . Because this is a similarity transformation that commutes with H, the total Hamiltonian is the same functional of the Heisenberg picture operators as it was of the q s and PS: Also ,because Eqs .(7.1.28)-(7 .1.29) define a s imila ritytransformation, the Heisenberg picture operators aga in satisfy the canonical commutation or anticommutation relat ions: However, they now have the time-dependence n (X, 0 Pn(x, t)= i[P,(x,t),H] --6H 6 Q ( x,t){7.1.30) (7.1.31) (7.1.32) (7-1.33) (7.1.34) For instance, we might take the Hamiltonian for a real scalar field as the free-particle term (7-1,24) plus the integral of a scalar interaction density so that in terms of Heisenberg-picture variable s 1(VQ)2 +1M2Q2 + .yH=Jd'x ['P'+_ f (Q)](7.1-35) 2 2 2 In this case the canonical conjugate to Qis given bythe same formula as for free fields P= Q . (7.1.36) However, as we shall see, the relation between the canonical conjugates P,(x) and the field variables and their time-derivatives is in general not the same as for the free particle operators, but must be inferred from Eqs.(7.1.33) and (7 .1.34). 7.2 The Lagrangian Formalis m Having seen that various realistic theories may be cast in the canonical for- malism,, we must now face the question of how to choose the Hamiltonian . As we will see in the next section, the easiest way to enforce Lorentz in- variance and other symmetries is to choose a suitable Lagxangian and use it to derive the HarrwiXtonian . There is not much loss of generality in this ;. 7,27The Lagrangian Formalism 299 given a realistic Hamiltonian, we can generally reconstruct a Lagrangian from which it could be derived, by reversing the process that we are going to describe here of deriving Hamiltonlans from Lagrangians . (The derivation of Eq. (7.1,26) gives one example of this reconstruction .) But although we can go from Hamil#onians to Lagrangians or Lagrangians to Hamiltonian, it is easier to explore physically satisfactory theories by listing possible Lagrangians, rather than Hamiltonians . The Lagrangian is, in general, a functional' L[T(t), +(t)] of a set of generic fields V{x, t} and their time-derivatives 'Y'(x, t) . The conjugate fields II/(x, t) are defined as the variational derivatives* * F[1(X, t)a4"', (x,t) The equations of motion are W(x ,t)(7.2.1) (7.2.2) These field equations can be usefully reformulated as a variational prin- ciple . We define a functional of V(x) over all spacetime, known as the actio n Under an arbitrary variation of T(x), the change in I[T] i s ~I [~'] = dt d ux dT~(x) + ~ J~((x) Eck; [JT/(x) ~'I}(x ) Assuming that 6'P'(x) vanishes for t --i, ± oo, we may integrate by parts, and write _I IL] 61[T] = dux ~~'~[xdt +,`~ W(x) . (7.2.4)~ 6x) Wesee that the action is stationary with respect to all variations b T"'that vanish at t --*Sao if andonlyif the f ields satisfy the field equations (7.2.2). Recall that ire the notation we use for f'unctaonals, a functional like L in which we display the variable i is understood to depend on the fields V(x . t) and V(x, i), With the undisplayad variables { and x running over a ll their values at a fixed value of the displayed variable t . We use upper case ` 3's and 11 s to indicate that these are interacting rather than free fields . Because the ` f's and Ts do not in general satisfy simple commutation or anticommulation relations, we cannot give a simple definition of the functional derivatives oc :curing here as we did for functional derivatives with respect to the Qs and Ys in the previous section . [nstead, we will simply specify that the variational derivatives are what they would be for c-numbel variables, with minus signs and equal-time commutators or anticornmutatars supplied as needed to make the formulas correct quantum-mechanically . As far as I know, no -important issues hinge on the details here . 300 7The Canonical Formalis m Because the field equations are determined by the functional I [T), it is natural in trying to construct aLoren tz-invariant theory to make IM a scalar functional . In particular, since I[T] is a time-integral of L[T(t),T(t)], we guess that L should itself be a space-integral of an ordinary scalar function of T(x) and OT(ac) JOxP,known as the Lagrangian density L[T(t), T(t)]fd'x Y (T(x, t), VT(x, t), ~V(x, t) ) so that the act ion is f d4Xy(T(x),OT(x)/OW') ,(7.2.s) (?.Z.6) All field theories used in current theories of elementary particles have Lagrangians of this form . Varying V(x) by an amount 6V(x), and integrating by parts, we find a variation in L : 6L= dux ~ ~6T,,' +'f V6 `I~r a(VT ) d3XO OY ) WP, Te a(w ) so(with obviou sarguments suppressed ) 6L aY OY e "- ~W' - ' - tT O(Wf ) 6L0.Y dVe--OT". The field equations (7 .2.1)then rea d a Oxju O(O tYI/Ox'A)aY OT".+ ~&Pr Ale + O (7.2.7) (7.2.8) (7.2.9) These are known as the Euler-Lagr ange e quations.As expected ,if2 is a scalar then these equations are Lorentz-invariant . In addition to being Lore n#z-invariant ,the act ionIis required to be real.This is because w e want just as many field equations as there are fields . By break ingupany complex fields into their real and imaginary parts, we can always think of I as being a functional only of a number of re alfield s,say Nof them .If I were complex ,with independent real and imaginary parts, then the real and im aginary parts of the conditions that Ibe stationary (the Eyler -Lagrange equations) would yield 2 N field equations forNfields, too many to be satisfied except i nspecial cases . We will see in the next section that the reality of theaction also ensures 7.2The Lagrangia n Formalism 301 that the generators of various symmetry transformations are Hermitian operators . Although the Lagrangian formalism makes it easy to construct theories that will satisfy Lorentz invariance and other symmetries, to calculate the S-matrix we need a formula for the interaction Hamiltonian . In general, the Hamiltonian is given by the Legendre transformatio n H dux II'(Xat)4',-(X't) -L['Y(t), +(t)]- (7.2.10) Although Eq. (7.2.1) does not in general allow 'Pe to be expressed uniquely in terms of 'F" and III, it is easy to see that Eq .(7.2.14 has vanishing variational derivative with respect to 'I' 1for any IF satisfying Eq .(7.2.1), so in general it is a functional only of V and H~, . Its variational derivatives with respect to these variables ar e - " ~ = I d3 ynr(y,t)6T'~'(Y, 0 ' - -6L d3 6 L r (Y ,)T 6T"( } y ~F(iIH3 ~611e(X, t 6 rr (X, t)T d3y6L 5kP""(y, t) where subscripts denote the quantities held fixed in these variational derivatives . Using the defining equation (7.2.1)for III, this simplifies t o and6H 6L 6T,"(x, t) ~n 6T((X, T 6HV The equations of motion (7.2.2)are then equivalent t o 6H(7.2.11) (7.2.12) (7.2.13) It is tempting now to identify the generic field variables V and their conjugates n, with the canonical variables Qn and P, of the previous section, and impose on them the same canonical commutation relations (7.1,30)-(7 .1.32, so that Eqs .(7.2.12) and (7.2.13)are the same as the Hamiltonian equations ❑f motion (7.1.33) and (7 .1.34). This is indeed the 302 7The Canonical Formalis m case for the simple example of the real scalar field (Dwith non-derivative coupling . Consider the Lagrangian densit y rn2 20'-J(D) (7.2. 14) which can be obtained b yadding a real function - ((D) of c Dto the free-field Lagrangian density found in the previous section . The Euler- Lagrange equations here are 11 (7 .2.15) From this Lagrangian density, we calculate a canonical conjugate to V 01(b(7.2.16) which is the same as Eq .(7.1.36) if we identify d} and IT with the canonical variables Qand P.The Hamiltonian is now given by Eq. (7.2. 10) ash H=/i3.(nth_- .) ' + (V~)l + n~;2cDI + ( 7.2.17) _ dux [ I II which we recognize as the Hamiltonian (7 .I.35). This little exercise should not be regarded as another derivation of this Hamiltonian, but rather as a validation of the Lagrangian (7 .2.14) as a possible theory of scalar fields . Matters are not always so simple . We have already seen in the previous section that there are field variables, such as the time component of a vector field or the Hermitian conjugate of a Dirac field, that are not canonical field variables Q "and do not have canonical conjugates ; yet Lorentz invariance dictates that these must appear in the Lagrangtans for the vector and Dirac fields . From the point of view of the Lagrangian formalism, the special char- acter of field variables like the time component of a vector field or the Hermitian conjugate of a Dirac field arises from the fact that although they appear in the Lagrangian, their time-derivatives do not . We shall denote the field variables T' whose time-derivatives do not appear in the Lagrangian as Cr;the remaining independent field variables are th e t We do not include a free constant factor in the term - 50,00RD, because any such constant if positive can be absorbed into the normalization of Q} . As we shall see, a negative constant here would lead to a Hamiltonian that is not bounded below- The constant m is known as the bare mass . The most general Lagrangian that satisfies the principle ofrennrmaliaability (discussed in Chapter 12) is of this form, with .*'(<D) a quartic polynomial in <~. In order for if to be interpreted as an energy, it should be bounded below . The positivity of the first two terms shows that we guessed correctly as to the sign in the first term in Eq.(72A4) . The remaining condition is that _ 11m'-0= + ((D) must be bounded below as a function vl' (D . 7.2The Lagrangian Formalis m canonical varia bles Q' .TheQ' have canonica lconjugates303 (7.2.18) andsatisfy the commutation relation s(7.1.30)-(7.1,32), but there are no canonic al con jugates fortheC'. Becau se6L16& =0,the Hamilton ian (7.2.10)is in genera l H =fd3x P,Q~ - L[Q(t), 0(t) ,C(t)], (7.2.19) but this is not yet useful until we express the Cr and 4in terms of the Qs and P s . The equations of motion of the Cr involve only fields and their first time-derivative s 0_6LEQ(t),Q(t), C(t)](7.2.20)6Cr(x, t) In the simple cases to be discussed in this chapter, these equations together with Eq . (7.2.18)can be solved to give the C'and0110in terms of the Q s and P s . Section 7 .6shows how in such cases one can avoid the task of actually solving for the Cr andT. In gauge theories like electrodynamics other methods must be used : either choosing a particular gauge, as in chapter 8, or the more modern covariant methods to be discussed in Volume 11 . Once we have derived a Hamiltonian as a functional of the Heisenberg picture Q s and P s, to use perturbation theory we must make a transition to the interaction picture . The Hamiltonian is time-independent, so it can be written in terms of the Pn and Q 'at t =Q, which are equal to the corresponding operators p ,and q 'in the interaction picture at t --0.The Hamiltonian derived in this way may then be expressed in terms of the q s and p s of the interaction picture, and split into two parts, a suitable free-particle term Ho and an interaction V . Finally, the time-dependence equations (7.1.21) and (7.1.22) and the commutation or anticommutation relations (7 .1.1)-(7.1.2) are used to express the q s and ps in V (t) as linear combinations of annihilation and creation operators . We shall present a number of examples of this procedure in Section 7,5; for the moment we will give only one example of the simplest type, the scalar field with Hamiltonian (7.2.17).We split H into a free-particle term and an interactio n H=H4+ Y ~pd))2+m2cp~~ Ho =fd'x1zFI2+ 2 2 V= d3x {(D).(7.2.21) (7.2.22) (7.2.23) 3(}4 7 The Canonical Formalism Here (Dand II are taken at the same time t, and H is independent of t, though HO and V usually are not . We now pass to the interaction representation . Taking t =Din Eqs.(7.2.22)and (7 .2.23), we can simply replace 0, Il with the inter- action picture variables , 7r, since they are defined by Eqs . (7.1.28)and (7.1.29)to be equal at that time . To calculate the interaction Y(t) in the interaction picture, we apply the similarity transformation (3.5.5) fd'x -ye (O(x, t)). The same transformation applied to HO leaves it constant : HO exp(iHot)HOexp(-Wo(x)} d'x[ 1-n'(x, t) +1,(VO(X, o)' ] .{ The relation between 7cand ~is dictated by Eq.(7.1.21) JHO (X, !(7.2.24) (7.2.25) {7.2.2b) (This happens to bethe same relation as in Eq. (7.2.16), but as we shall see this is not to be expected in general .) Also, the equation of motion for 0is dictated by Eq .(7.1.22): 60(X, t) which together with Eq . (7.2.26)yields the field equation s The general real solution may be expressed a s O(x)=(27r )-3/2 Jd3P(2PD)-1{2[eu1a(p)+elpxulw ](7.2.27) (7.2.28) with p° = p2 -+m2 understood, and a(p) some as-yet-unknown operator function of p . Eq.(7.2.26) then gives the canonical conjugate a s [e'a(p ) In order to get the desired commutation relations, [O(X, o' F(Y' o] {x,o,0LY' ol [2T(X, 0 ,n(y, 0)=0, 5(7.2.30) (7.2.31) (7.x.32) {7.x.33) 7.2The Lagra ngian Formalis m we must take the as to satisfy the familiar commutation relation s [a(p), a*(pf)] [a(p) . a(Pf)]=0.305 (7.2.34) (7.2.35) Also, we have already shown in the previous section that using thes e ~expansions in Eq . (7.2.25) gives the usual formula (4 .2.i 1) for the free - particle Hamiltonian, up to an inconsequential additive constant . A s remarked before, these results should not be regarded so much as a n alternative derivation of Eqs . (7.2.29), (7 .2.34), and (7 .2.35) (which wer e obtained in Chapter 5 on quite other grounds) but rather as a validatio n of the first two terms of Eq . (7.2.14) as the correct free-particle Lagrangia n for a real scalar field . We can now proceed to use perturbation theory t o calculate the S-matrix, taking (7 .2.24) as V(t), with the field O{x} given b y Eq. (7.2.29). The procedures illustrated here will be carried out for examples tha t are more complicated and more interesting in Section 7 .5. In considering the various possible Lagrangian densities for physical theories it is common to apply integration by parts, treating as equivalent and Lagrangian densities that differ only b ytotal derivatives It is obvious that such total derivative terms do not contribute to the action and hence do not affect the field equations . It is also obvious that a space-derivative term V - in the Lagrangian density does not contribute to the Lagrangian and hence does not affect the quantum theory defined bythe Lagrangian .¶ What is less obvious and worth noting here is that a time-derivative ao ° in the Lagrangian density also does not affect the quantum structure of the theory . To see this, let's first consider the effect of adding a term to the Lagrangian of the more general for m AL(t)= fd'x Dn,x M01 ~"(X, 0, (7.2.36) where D is an arbitrary n- and x-dependent functional of the values ofQ at a given time . This changes the formula for the conjugate variables P(t) as functionals of Q(t) and (t) by the amoun t ❑P,(x,t) _6❑L(t) 6 n(yx t) = Dn,x[Q(t)] . (7 .2.37) It follows that there is no change in the Ham iltonian as expressed as a ~This is under the usual assumption, that the fields vanish at infinity . These resells do not necessary apply when we allow fields or different topology, as discussed in Volume il . 306 7The Canonical Formalis m functional of the Q(t) and Q(t) : fdux❑P,,{x, t}Q '(x, t) y- ❑L(t)=0. Hence also there is no change in the Hamiltonian as expressed as a functional of the old canonical variables Q "and P,, . However, the Hamil- tonian is notthe same functional of the new canonical variables (3 1and P„+FPM as it was of the Q "and PM, and in a theory described by the new Uagrangian Y + ❑ it is the new canonical variables Q" and P,, + ❑P„ rather than the Q'and P, that would satisfy the canonical commutation relations . The commutators of the Q 'with each other and of the Q" with the Pm are given by the usual canonical relations, but the commutators of the P, with each other are no w _ _~bD ►~,~[Q(t)]+tbDmy~« tt)~ ('x.2.39) In general this doesn't vanish , but if the added term in the Lagrangian is a total time-derivative Q_ then D in Eq. (7.2.36)is of the special for m Qn\ T(7.2.44) (7.2.41) In this case the commutator (7.2.39)vanishes, so the variables Q'and Pn satisfy the usual commutation relations . We have seen that a change of the form (7 .2.36)in the Lagrangian does not change the form of the Hamiltonian as a functional of the Q 'and P, and since, as we have now shown,, the commutation relations of these variables are also unchanged, the addition to the Lagrangian of the term (7.2.36)has no effect on the quantum structure of the theory . Different Lagrangian densities ❑btained from each other by partial integration may therefore be regarded as equivalent in quantum as well as classical field theory . 7.3 Global sy mmetries We now come to the real point of the Lagrangian formalism, that it provides a natural framework for the quantum mechanical implementation 7.3 Global Symmetries 307 of symmetry principles . This is because the dynamica lequations in the Lagrangian formalism take the form of a variat ional principle, the principle of stationary action . Consider and infinitesimal transformation of the fields 'I'"'(x) --o, ' I'"(x) +i~ ~{x ) that leaves the action (7.2.3) invariant : R [T ]0 = J1 = IefdxJT"(x) {(7.3.i) (7.3.2) (With ea constant, such symmetries are known as global symmetries . In general, 3w"' depends on the fields and their derivatives at x .)Of course Eq.(7.3.2)is automatically satisfied for all infinitesimal variations of the fields if the fields satisfy the dynamical equations ; by an infinitesimal symmetry transformation we mean one that leaves the action invariant even when the dynamical equations are not satisfied . If we now consider the same transformation with ean arbitrary function of position in spacetime : 'Y~(x) --o, 'i'~(x) + i e(x).~F~(x), (7 .3.3) then, in general, the variation of the action wil lnot van ish, but it will have tobe of the form 61 duxP(x) he(x) (7-3-4)0xP in order that it should vanish when e(x) is constant . If we now take the fields inI [T] to satisfy the field equations then I is stationary with respect to arbitrary field variations that vanish at large spacetime distances, including variations of the form (7 .3.3), so in this case (7 .3.4) should vanish . Integrating by parts, we see that P(x) must satisfy a conservation law- it follows immediately tha t whereOxA dF0=dt F=_/d3xJ0(7.3.5) (7.3.6) (7.3,7) There is one such conserved current J Pand one constant of the motion Ffor each independent infinitesimal symmetry transformation . This rep- resents a general feature of the canonical formalism, often referred to as Noether's theorem : sym metries imply conservation laws . Many symmetry transformations leave the Lagrangian and not just the action invariant . This is the case, for instance, for translations and 308 7 The Canonical Formalism rotations in space and also isospin transformations and other internal symmetry transformations, though not for general Lorentz transforma- tions . When the Lagrangian is invariant we can go further, and write an explicit formula for the conserved quantities F . Consider a field variation (7.3.3)in which c(x) depends on t but not x . In this case the variation in the action is 61=1 dt dux[5L[1(t)41(tJ1) .(() J~Y t(~€,) ,(6(t)'F"'(X' t ) The requirement that the Lagrangian be invariant under this transforma- tion when cis a constant yield s () W( x, t)_dt so for general fields (whether or not the field equations are satisfied) the variation in the action i s 61=idt/d3x '40 (x,t). (7-3-10)qj"(Xat) Comparing this with Eq .(73.4)gives F=-i d 'x6LCT(r)a4'~tfl_F"'(X,t). (7.111)bV(xta) Using the symmetry condition (7.3.9), the reader can easily check that this F is indeed time-independent for any fields that satisfy the dynamical equations (7 .2.2). Other symmetry #ransformations such as isospin rotations leave not only the action and the Lagrangian invariant but also the Lagrangian density . In such cases we can go even further, and write an explicit formula for the current .IP{x} . Writing the action as in Eq . (7.2.6)as the integral of the Lagrangian density, its variation under the transformation (7.3.3) with a general infinitesimal parameter e(x) i s 14 Ml'(x ) + (7 .3.12) The invariance of the Lagrangian density when e is a constant requires 7.3Global Symmetries 309 that so for arbitrary fields the variation of the action i s 31['T'] =idxOY(~'(x),~ju TO) alu--W . (7 .3,14 Comparison with Eq . (7.3.4) shows tha t J'U '~' x~`~ l ' (7 .3.15) ~(~ ~ ) Using the symmetry cond ition (7.3.13), i tis easy to see d irectly that O".l" vanishes when the fields satisfy the puler -Lagrange equations (7.2.9).Note also that the integral of the time component of thecurrent (7.3.15) has the previously de rived value (7.3.11). So far everything we have said would appl yto classical a swell as quan- tum mechanical field theories . The quantum propert ies of the conserved quan titiesF are most eas ilyseen for symmetries of the Lagrangian (not necessarily the Lagrangian density) that transform the canonical fields Qn(x,t)(that is,those of the Vwho se time der ivatives appear in the Lagrangian) into x-dependent funct ionals of themselves at the same time . For such transformat ions, we hav e n(x, t)=.Fn [Q(t) ;x]. (7.3. 16) As we shall see, infinitesimal spatial translations and rotations as well as all infinitesimal internal symmetry transformations are of the form (7.3.1), (7.3.16), with . "' a linear functional of the Q ', but we will not need to assume here that the symmetry is linear . For all such symmetries the operator F is not only conserved ; it also acts in quantum mechanics as a generator of this symmetry . To see this, note first that when V is a canonical field Q1, the functional derivative 6L15V is equal to the canonical conjugate P,,, while when V is an auxiliary field C ', this functional derivative vanishes ; hence we may rewrite Eq . (7.3.11) in the for m F =-i/d3xPx4t) .-'cxt) =-ifd3xfd3vPx,t) .P1[Q(t),x] (7.3.17) Tocalculate the commutator (not anticommutator) o f Fwith acanonical field QI(x,t)at an arbitrary t ime t, we can invoke Eq .(7.3.6)to evaluate F as a functional of the Qs and P s at the time t, and then use the equal-time 310 7 The Canonical Formalis m canonical commutation re lations (7 .1.30)-(7 .1.32) to obtain* (7.3.15) It is in this sense that F is the generator of the transformation with Eq. (7.3.16). Eq- (7.3.17)and the canonical commutation rules give als o Where F 1is linear, Eq . (7.3.19) tells us that P„ transforms contragrediently toQ'. As a first example, consider the symmetry transformation of spacetime translation : This is of the form (7.3.1),with four independent parameters cyand four corresponding transformation function s In consequence we have four independent conserved currents, convention- ally grouped together in the energy-momentum tensor TP, : a4"TPV=0 (7.3.22) from which we can derive time-independent quantities as the spatial integrals of the time components of the translation 'currents' (not to be confused with the canonical conjugate field variables P,(x, t) ) P, = d 'x T °ti, d dtP,_0.(7.3.23) (7.3.24) The Lagrangian is invariant under spatial translations, so in accordance with the above general results we can conclude that the spatial components of P, take the form P- - dux Pn(x, t)VQ"(x, r). (7 .3.25) Using the equal-time commutation relations (7.1.30)-(7.1,32), we also find the commutator of this operator with the canonical fields an d We are here assuming that for C]" bosonic or 3ermionic the variation F" is also respectively bosonic or fermionic, so that F is bosonie . The only exceptions are certain symmetries known as %upersymmetries, for which F is fermionic and (7.3.18) is an anticommutator if Q 1is also Ferrnionic . 7.3 Global Symmetrie s conjugates :311 (7.3.27) It follows that for any function of Qs a ndPs that does not also depend explicitly on x, we have 1P,27(x)]=ivs(x) . (73,28 ) These results show that the operator P is indeed the generator of space translations . In contrast, time-translations do not leave the Lagrangian L(t) invariant, However, we already know the generator of time-translations ; it is the Hamiltonian P° =-H, which as we know satisfies the commutation relatio n for any function o fHeisenberg picture operators . If we further assume that the Lagrang ianisthe integral of a Lagrangian dens ity, then wemay also obtain an explicit formula for the energy - mome ntum te nsorT.However ,the Lagrangian dens ityY(x)is not invariant under spa cetime translat ions, sowe cannot use Eq .(7.3.151 here . Instead, note that the change in the action under a spacetime-dependent translation T'~(x) --+T~(x+ c(x)) _ T"{x} + e(x)r?,,T"(x) (7.3.30) is illy 61 [LI']=Jdux ~, e~`~~p `'' + 0(0~ec,.[Epr?~'I~4] {7.3.31} L The Eule r-Lagrange eq uations (7.2.9)show that the terms proportional to e add up to E-uvA, so Y CJU + OY OyT1O,,FP((7xp 0 (0 v) Integrating by parts, we see that this takes the form of Eq . (7.3.4) U = - d uxT''.a,,eju with `currents' VP 6 ~Y 02C~T~ f1~~ V(7.3.32) (?.3.33) (7.3.34) As a check, we may note that the spatial components of Eq . (7.3.23) are the same as our prev ious formula (7 .3.25) for P, while for p = 0 312 7 TheCanonical Formalis m Eq. {7.3.23} gives the usual formula for the Hamiltonian : H = -PO d'x PnQn- , n(7.3.35) (A warning : the tensor TI" obtained by raising the second index in Eq.(7.3.34)is not in general symmetric, and therefore cannot be used as the right-hand side of the field equations of general relativity . The correct energy-momentum tensor to use as the source of the gravitational field is the symmetric tensor OM introduced in the next section .) In many theories there are also one or more symmetry principles that state the invariance of the action, under a set of linear coordinate- independent transformations of the canonical field s together with a set of suitable transformations on any auxiliary field s Cr(x) __+C''(x) + i,_a (z.)r,,CS(x). (7 .3.37) Here t,, and r,, are sets of Hermitian matrices furnishing some representa- tions of the Lie alge bra of the symmetry g roup, and we sum over repeated group indices a, b, etc . (For instance, in electrodynamics there is such a symmetry, fo rwhich the one matrix tn,, is d iagonal, wit hthe cha rges carriedby each field on the main diagonal .) From any such symmetry, we can infer the existence of another set of conserved currents .1~ whose time components are the densities of a set of time-independent operators T,,~ d 3xJa . (7.3.39) When the Lagrangian as well as the action is invariant under the trans- formation (7.3.36), Eq .(7.3.11)provides an explicit formula for the T~, : Ta =-i ,}~3xP,,( Y,t)lral'm'(x rt) The equal-time commutation relations here giv e [T"'Qjl(x)l [T11,P"(X)j=-(ta)%QNX) , +(t a)MnPm(X)(7.3.40) (7.3.41) (7.3.42) (Where r,, is diagonal, this tells us that Qn and P,, respectively lower and raise the value of T,,by an amount equal to the nth diagonal element oft,,.) Using these results, we can calculate the commutator of 7'Q with 7.3 Global Symmetrie s another generator Tb313 [Ta,Tbl-=iId'x [`P.(ta)ten(tb)'lrQ' + n (th)'k(ta)'mQ']. (7.3.43) Thus, if the matrices t, form a Lie algebra with structure constants fufiC, [ta, thl-= if abctea then so do the quantum operators Ta(?.3.44) (7.3.45) This confirms that the quantities (7 .3,40) are correctly normalized to qualify as generators of the symmetry group . Where the Lagrangian is the integral of a Lagrangian density which is invariant under (7.3.36) and (7.3.37) we can go further, and use Eq .(73.15) to provide an explicit formula for the currents associated with these global symrhetries ; O(OCr /~xP)(1a)rsC'S(7.3.46) As an illustration, suppose we have two real scalar fields of equal mass, with Lagrangian density 2a)2 2q)2 _ , (7.3.47) This is invariant under a linear transformation like (7.3.36): so there is a conserved current (7 3,46):602 = +601 , The explicit formula (7 .3. 6) for the current can be used to derive other useful commutation relations . In particular, since the Lagrangian density does not involve time-derivatives of the auxiliary fields, we hav e J°=-iPn(tU )11MQM . (7 .3.4$) We can then derive the equal-time commutators of general fields not only with the symmetry generators T, but also with the densities Ja Ijal (X, t), Qn (y, t)]=_ 6 1(X_ y)(t a)n?nQm (,, t) , [j,O7(X, t) ,pm(y, t)j 3 (X_Y)(tay, m pjj(X, t)(7-3.49) (7.3.50) If the auxiliary fields are constructed as local functions of the Ps and Qs in such a way that they transform according to a representation of the 314 7 The Canonical Formalism symmetry algebra with generators -ca, then als o We often summarize Eqs .(7.3.49) and ( 7.3.51)in the single commutation relation ~J,O(x, t)# T"( Y, t)]--63(x - Y)(ta)"j, T"' (xa t) . (7.3.52) Commutation relations like (7.3.49)-(7 .3.51) will be used in Chapter 10 to derive relations called ward identities for matrix elements invoicing the current P. 7,4 Lo rentz Invariance We are now going to show that the Lorentz invariance of the Lagrangian density implies the Lorentz invariance of the S-matrix . Consider an infinitesimal Lorentz transformatio n According to the analysis of the previous section, the invariance of the action under such transformations tells us immediately that there are a set of conserved `currents' ., PP' : OP MPP '' = 0, (7.4.3) (7.4.4) one current for each independent component of The integrals of the time-components of these `currents' then provide us with a set of time-independent tensors : jyw =d3Xauv} dPV=0,dt(7.4.5) (7.4.6) The J} "will turn out to be the generators of the homogeneous Lorentz group . We would like to have an explicit formula for the tensor but Lorentz transformations act on the coordinates and hence cannot leave the Lagrangian density invariant, so we cannot immediately use the results of the previous section . However, translation invariance allows us to formulate Lorentz invariance as a symmetry of the Lagrangian density 7.4Lorentz Invariance 3 15 under a set of transformations on the fields and field-derivatives alone . The fields undergo the matrix transformatio n OV Y'6T' V MV nTm (7.4.7) where .,are a set of matrices satisfying the algebra of the homogeneous Lorentz grou p For example, for a scala rfield we have 60= 0, so fj, ,. = 0, while for an irreducible f ield of type (A, B) we hav e where .4and are spin matrices for spin A and B, respectively . We specially note that for a covariant vector field, 6V,,= co,AVA, so her e The derivative of a field that transforms as in Eq. (7.4.7)transforms like another such field, but with an extra vector index (7.4.9) The Lagrangian density is assumed to be invariant under the combined transformations Eqs .(7.4.7)and (7 .4.9), s o OT/ 2(OxT 4K Setting the coefficient of 01' equal to zero give s a=~'} Vp ti)fm ~"~ + i ~ G~ ~~ ~ ~►r~ L ~ }~ 2 OOK~ij~ )~d BEY' iii F C I 1 0 a20(0,V ) Using the Euler-Lagrange equations (7 .2.9), and our formula (7.3.34) for the energy momentum tensor T}tV, we may write this a s ., i' mTm K12 V )1 This immediately suggests the definition of a new energy-momentum 316 7 The Canonical Formalis m tensor, known as the Belinfantetensor :2 (3)1v=TJuv-_OK[- mT2 O(K ~(~`u''n ("~` )t~`Y't' (7.4.11) O(~) O(a,V ) The quantity in square brackets is manifestly antisymmetric in Pand K, so 811v satisfies the same conservation law as T~11 , ayQyV = 4 . (7 .4.12) For the same reason, when we set y_0in Eq . (7.4.f 1)the index K runs over space components only, so the derivative term here drops out when we integrate over all spac e where P° = H . Thus OPI can be regarded as the energy-momentum tensor, just as well as T.However, Eq.(7.4.10) tells us that, unlike Till', the Selinfante tensor 0,1' is not only conserved but also sy mmetric: E)Juv =EYP . (7 .4.14) It is Q PIrather than T"' that acts as a source of the gravitational field .3 In consequence of the symmetry of OP'', we may construct one more conserved tensor density .,lu'= xPWw - x'' WA . (7 .4.15) This is conserved, in the sense tha t Thus Lorentz invariance allows us to define one more time-independent tensor .]JUV = J°;4''d3x = Jd'x(xi`@°` - XVa°P). (7.4.17) The rotation generator J k=eijk.T412 is not only time-independent, bu t also has no explicit time-dependence, so it commutes with the Hamiltonian [H, J] = 0 . (7 .4.18) Also, applying Eq. (7.3.28) to the function e4', we hav e 2 2 ax -iCzjkJd'x0°' 7.4 Lorentz Invarianc e and therefore317 (7.4.19) On the other hand, the `boost' generator Kk - .Tko, though time-independ- ent, does explicitly involve the time coordinat e Kk = fd'x(x'9" -x"Ook) , or more explicit ly K--tP+ dux x 0"(x, t) . (7 .4.2 0) Since this is a constant, we have 0 = lk = - P+ i[H, K], and the refore Also, applying E q. (7.3.28 again give s [PS, Kk] = ifdux xk a0 xio°° =-iJjkfduxOw and therefore(7.4.21) (7.4.22) For any reasonable Lagrangian density, the operator (7 .4.20) will be 'smooth" in the sense used in Section 3 .3, i.e., the interaction terms in elHo` f dux x 0°°(x, 0)e-t1111 vanish' for t --} ±ac . (Note that the interaction terms ineiHotf dux 8C'0(x, 0)e`0r must vanish for t ---),hoc in order to allow the introduction of `in' and `out' states and the S-matrix .) With this smoothness assumption and the commutation relation (7 .4. 1) in hand, we can repeat the arguments of Section 3.3, and conclude that the S-matrix is Lorentz-invariant . The same arguments were also used in Section 3 .3 to verify that the remaining commutation relations of the Lorentz group, those of the P`, with each other, take the proper form . This can also be shown directly for the commutators of the rotation generators, which here take the for m jjj = (.'OjTe -,00jTe -i(f'j)' ,Trn) (7 .4.23)OT Since the Lagrangian density does not depend on the time-derivatives of the auxiliary fields, and the rotation generators do not mix canonical an d When we say that some interaction-picture operator vanishes for r --;,+x~we mean that its matrix elements between states that are smooth superpositions of energy eigenstates vanish in this limit, 318 7 The Canonical Formalis m auxiliary fields, this can also be written as a sum over canonical fields alone : P = f d'x P . (x~ OjQn-XiajQ11 - i(J'j) " w V) . (7-4.24) It follows immediately then from the canonical commutation relations that [.1'~, P,,{x}1-= i(x ;0i-xj00Pn(x)+V~}niMPnf(x). (7.4.26) These results can be used to derive the usual commutation relations of the 0 with each other and other generators .w` If there are no auxiliary fields then the same arguments may be applied to the `boost' generators to complete the demonstration that the P11 and Jill'satisfy the commutation relations of the inhomogeneous Lorentz group . However the `boost' matrices 0will in general mix canonical and auxiliary fields such as the components W and VIof a vector field), so the direct proof of the commutation relations of the 0with each other has to be given on a case by case basis . Fortunately, this is not needed for the proof of the Lorentz invariance of the S-matrix given in Section 3 .3. 7.5 Tran sition to Interaction Picture:Exampl es At the end of Section 7.2 we showed how to use the Lagrangian of a simple scalar field theory to derive the structure of the interaction and the free fields it contains in the interaction picture . We will now turn to somewhat more complicated and revealing examples . Scalar F ield,Derivative Coupl ing First let 's co nsider a neutral scalar field ,but now with derivative coupling . We take the Lagrangian a s _ - 2 ~~PWND -~m,(DI- Jpr)YcI~ - ~ '(~~, ( 7.5.1) where JAis either a c-number external current (unrelated to currents P introduced earlier), or a functional of various fields other than c D(in which case terms involving these other fields need to be added to (7.5.1)). Th e Also, s ince JU commutes w ith H and P„Q", itcommut es wi th L.The commutat orofPiwith the auxiliary fie ldsmustthushe consistentwiththe ro tational invarianceof L. 7.5 Transition to Interaction Pictur e canonical conjugate to (Dis no w F1F~y 77- and the Hamiltonian i s H=f Idux [III - .:P] Xx[n(R+j°)+ z {v(D)2 ( n+,r°)2 Collecting terms, we can write this a s H = Ho+V, Ho = J dux [ in' + ~{V~}2 + ~rn2 d)2], V d3X[njO+J .V(D+ i(jO)2+ .e((D)319 (7.5.2 (7-5-3) (7. 5.4) (7.5.s) As explained in Section 7 .2, we can pass to the interaction picture by simply replacing f]and 4) with -aand 0(and likewise for any fields in the current P, though we will not bother to indicate this explicitly ) HQf (VO(X,2 + ~ M 01 (X,t) l +[J,,(X, t)] 2 +A-,OWX, O)l -(7.5.6) (7.5.7) The free-particle Hamiltonian is just the same as Eq. (7.2.25), and leads as in Section 7 .2to Eqs . (7.2.26)-(7.2.35). Indeed, whatever the total Hamiltonian may be, we mu sttake ( 7.5.6)as the part we split off' and call the free-particle part, with the remainder called the interaction, because as we have seen it is this form of the free-particle Hamiltonian that leads to the correct expansion (7.2.29)of the scalar field in terms of creation and annihilation operators that satisfy the commutation relations (7 .2.34), (7.2.35).The last step is to replace gin the interaction Hamiltonian with its value 4 in the interaction picture (not its value ~-,T )in the Heisenberg picture) : V W _fdux[J1(x .t)OU O(X, 0 + ~ [J°x , ~~l, + ~~~x, t) (7.5.8) 320 7 The Canonical Formalis m The extra non-invariant term in Eq .(7.5.8)is just what we saw in Section 6.2is needed to cancel a non-invariant term in the propagator of 00. Vector Field, Spin on e similar results are obtained in the canonical quantization of the vector field V ,,for a particle of spin one . Let's here keep an open mind, and write the Lagrangian density in a fairly general for m 2 ~xc~4VV c~~~~''~ ~~3OuV~,c~VVJ`- ;MZV, Vy--JMV,, (7.5.9) where ac, and m 2 are so far arbitrary constants, an dJ.iseither a c- number external current, or an operator depending on fields other than V,U, in which case additional terms involving these fields must be added toY.ThEuler-Lagrange field equations for V,, rea d Taking the divergence gives (7.S.lfl) This is the equation for an ordinary scalar field with mass m 2/(ac + and source adtil(a + M. We want to describe a theory containing only particles of spin one, not spin zero, so to avoid the appearance of 0) .V~ as an independently propagating scalar field, we take ac = -J3, in which case alV'can be expressed in terms of an external current or other fields, as -al,I1/mz .The constant oc can be absorbed in the definition of Y,,,so we can take oc = -P= 1, and therefor e 1FMv P` m,V1, Y" - ill V P where(7-5-11 ) (7.5-12) The derivative of the Lagrangian with respect to the time-derivative of the vector field is -Fog oVU(7.5.13) This is non-vanishing for ,u a spatial index i, so the W are canonical fields, with conjugates II`= F i4=Vz + di V Q (7.5.14) On the other hand P" _ 0, so P"does not appear in the Lagrangian, and VO is therefore an auxiliary field . This causes no serious difficulty- the fact that 0 /0VO vanishes means that the field equation for Vo involves no second time-derivatives, and can therefore be used as a constraint that 7.5Transition toInteraction Picture 32 1 eliminates a field variable . Specifically, the puler-Lagrange equation for v-0is ~, Fa =Fn2 Va+ jo or usingEq.(7.5.14)(7.s.15) Nov let us calculate the Hamiltonian H = f d3x (II ~ -- Y)for this theory . Eq . (7.5.14) allows us to write V in terms of H and A V_ -VV°+II=II- pip ' fI--i°), so H~ dux III + m-2(V.II){V.II-j°) 2 2 2 Again, we split this up into afree-particle term HO and interaction V : H=Ho+V r (7.5.17) and pass to the interaction picture by replacing the Heisen berg-picture quantit ies V and IIwith their interaction-picture counterpar ts v and it (and, though not shown explicitly, likewise for whatever fields and conjugates are present in J P) 2 ~D = ~~ ~ ~[12+~~~(V. n)2+4(vXY)2+ Y2 2 V= dux J v - -2j°V. n+2(j)2(7.5.19) The relation between aand v is then 6Ho (v , and the `field equation' i s Since V° is not an independent field variable, it is not related by a similarity transformation to any interaction-picture object A Instead, we can invent a quantity z'° = rra-2v .X. (7.5.22) 322 7 The Canonical Formalis m Eq. (7.5.20)then all owsus to write aas ?z=v+Vv0. (7.5.23) Inserting this in Eqs . (7.5.22)and (7.5.21) gives our field equations in the form These can be combined in the covariant for m Taking the divergence give s and hence(7.5.24) (7.5.25) (7.S.2b) A rea lvector f ieldsatisfy ing Eqs . (7.5.25) and (7 .5.26) can be exp ressed as aFourier t ransform VA(x) =(27C)-1/2 ~d3p (2po)-1/2 fe,(P, u)a(P, a)eipX (7.5.27) where p° = /p2 + m ;the e tt(p,u)forc = +1, 0, --1 are three independent vectors satisfying Ppe~`(P,U)=0 and normalized so tha t e" (P}u)e"(Pfa)=q~zti+ PpP V /M 2(7.5.28) (7.5.29) and the a( p, a) are operator coefficients . It is straightforward using Eqs. (7.5.23), (7 .5,27), and (7 .5.29) to calculate that v and itsatisfy the correct commutation relation s [v(x.t),vi(X,t) I=17r'(x, t), iri (x, t)]=0, (7 .5.30) provided thata(p, a) and aT(P, cr) satisfy the commut ation rel ations ~a(p,0-),at(Pr~Cr;)] = 6,(P, - Oju'cr [a(P, cr), a(p', (T')]= 0.(7.5.31) (7.5.32) 7.5 Transition toInteraction Picture 323 We already know that the vector field for a spin one particle must take the form (7 .5.27), so our derivation of these results serves to verify that Eq. (7.5.18) gives the correct free-particle Hamiltonian for a massive particle of spin one . It is easy to check also that Eq . (7.5.18)may be written (up to a constant term) in the standard form of a free-particle energy, as Edf d'P P~ al(p,a)a(Py a) . Finally, using Eq .(7.5.22) in Eq . {7.5,19 )yields the interaction in the interaction pictur e 1i 2m I - The extra non-invariant term in Eq .(7.5.33) is just what we found in Chapter 6is needed to cancel anon-invariant term in the propagator of the vector field . Dirac field, Spin One Hal f For the Dirac field of a particle ❑f spin 1/2, we tentatively take the Lagrangian as (7.5.34) with a real function of kY and T, This is not real, but the action is, becaus e Hence the field equations obtained byrequiring the action to be stationary with respect to Tare the adjoints of those obtained by requiring the action to be stationary with respect to P . as necessary if we are to avoid having too many field eyuatians . The canonical conjugate to 'I' i s a(7.5.35) so we should not regard T as a field like T, but rather as proportional to the canonical conjugate of T . The Hamiltonian is . H= dux [FIT - ]= ~ We write this a s where Ho= VI IH=Ho+ V . d3x 1-17017 ' V + m]T, dax,Yr('I',`If).(7.5.3G) (7.5.37) (7.5.38) 324 7TheCanonical Formalism We now pass to the interaction picture . Since Eq .(7.5.35) does not involve the time, the similarity transformation (7.1.28), (7 .1.29) yields immediately it=-FPY° - {7 .5.39} Likewise, HO and V(t) can be calculated by replacing 'Y and H with W and a in Eqs .(7.5.37) and (7 .5.38). This gives the equation of motio n bHo ormore neatly(7.5.40) (7.5.41) (The other equation ❑f motion, it=-6H~bip, yields just the adjoint of this one .) Any field satisfying Eq . (7.5.41) can be written as a Fourier transform (7.5.42) where p° = p2-+m2 ; a(P, cr) and bt( p, ff) are operator coefficients ; and u(p, ± ~ ) are the two independent solutions o f and likewis e normal izedso tha t Q 20° 2JO In order to obtain the desired anticommutator s ~Va(X,0,'~#(Y'01+=[Va(X,t), al, (Y't)]+(Y )", P(7.5.43) (7.5.44) (7.5.45) (7.5.46) (7.5.47) (7.5.48) The matrix iy%, has eigenvalues ±m, so lu~ and tai must be proportional to the projection matrices +m)/2m and (i,~upp + m) f Zm, respectively . The proportionality factor may be adjusted up to a sign by absorbing it in the definition of u and v . The overall sign is determined by positivity ; Tr Iuuf3 = I u}u and Tr It, F#=Y-v{Vmuss be positive . 7.6Constraints and Dirac Bracket s we must adopt the anticommutation relation s [a(p, cr), af (P cr~ )] + [a(p.tT), QTRa+ [a(p,o-), b(p U += [b(p, (7), b~(p I I a 0)]+ = [b(p, a-), b(p I ,CTId + ={a(p.cr)'V(p{' ffId +PW - PM"r~, =d , (7.5.50)325 (7.5.49) and their adjoints . These agree with the results obtained in Chapter 5, thus verifying that (7.5.37) is the correct free-particle Hamiltonian for spin In terms of the as and bs, this Hamiltonian i s HO d'p p' (a~(p, a)a(p, a) - b(p, a)O(p, a)) (7 .5.51) We can rewrite this as a more conventional free-particle Hamiltonian, plus another infinite c-number " HO= E f d~~p P' [al(p, a)a(p,a) + b~(p, u)b(p, a)- P(p - p)] . (7,5 .52) CT The c-number term in Eq .(7.5.52) is only important if we worry about gravitational phenomena ; otherwise here, as for the scalar field, we can throw it away, since it only affects the zero of energy with respect to which all energies are measured . With this understanding, HO is a positive operator, just as for bosons . 7.6 Constraints and Dirac Br ackets The chief obstacle to deriving the Hamiltonian from the Lagrangian is the occurrence of constraints . The standard analysis of this problem is that of Dirac,5 whose terminology we will follow here . Dirac's analysis is not really needed for the simple theories discussed in this chapter, where it is easy to identify the unconstrained canonical variables . We shall use the theory of a real massive vector field for illustration here, returning to Dirac's approach in the next chapter, where it will be actually useful . Primary constraints are either imposed on the system (as when in the next chapter we choose a gauge for the electromagnetic field) or arise from the structure of the Lagrangian itself . For an example of the latter type, consider the Lagrangian (7.5.11) of a massive vector field Vil interactin g Note the nega tive sig nof the c- number terra . The con jectured symmetry icn own as su persy mrnetry4 connects t he numbers of boson and fermion f ields, in such a way that the c-nutttbers in HOall cance l. 326 with a current J where7 The Canonical Formalism (7.G.1) {7.b.2) Suppose we try to treat all four components of V" on the same basis . We should then define the conjugates -Foy0(OOAP ) We immediately find the primary constrain t no=0.(7.6.3) (7.6.4) More generally, we encounter primary constraints whenever the equations rl? = 6LJ60OT" cannot b esolved to give all the OQV (at least locally) in terms of He and V . This will be the case if and only if the matrix 61L/b(0()T1)b(OQT') has vanishing determinant, Such Lagrangians are called irregular . Then there are secondary constraints, which arise from the requirement that the primary constraints be consistent with the equations of motion . For the massive vector field, this is just the Euler-Lagrange equation (7.5.16) for V ° Here we are finished, but in other theories we might encounter further constraints by requiring consistency of the secondary constraints with the field equations, and so on . The distinction between primary, secondary, etc, constraints is not important ; we will treat them all together here . There is another distinction between certain types of constraint that is more important . The constraints we have found for the massive vector field are of a type known as second class, for which there is a universal prescription for the commutation relations . To explain the distinction between first and second class constraints, and the prescription used to deal with second class constraints, it is useful first to recall the definition of the Poisson brackets of classical mechanics . Consider any Lagrangian L (T,'i')that depends on a set of variables T'(t) and their time -derivatives 'PI{t} . (The Lagrangians of quantum field theory are a special case, with the index a running over all pairs of ~ and x.) We can define canonical conjugates for all of these variables b y . III ~ C- q~a 7.6Constraints and Dirac Brackets 327 The Its and Ys will in general not be independent variables, but may instead be related b yvarious constraint equations, both primary and secondary . The Poisson bracket is then defined b y CAaB]p =_OAc~B O BOA MO UTa OTa 0 .IIa(7.6.7) with the constraints ignored in calculating the derivatives with respect to Ta and 1-1a, In particular, we always have [ T",IIb]p_6~. (Here and below all fields are taken at the same time, and time arguments are everywhere dropped .) These brackets have the same algebraic Properties as commutators : including the Jacobi identity(7.6.$) (7.6.9) (7.6.1o) If we could adopt the usual commutation relations [ T6, III,] =ids II~, 1=0,then the commutator of any two functions of the 'I's and Its would b ejust [A, B] = a[A, B] p.But the constraints do not always allow this . The constraints may in general be expressed in the form yy = 0, where the yN are a set of functions of the Ts and TI s . Because we are including secondary constraints along with the primary constraints, the set of all the constraints is necessarily consistent with the equations of motion A=[A,H]pa and therefor e when the constraint equations X?v = 0are satisfied . We call a constraint first class if its Poisson bracket with all the other constraints vanishes when (af ter calculating the Poisson brackets) we im- pose the constraints . We shall see a simple example of such a constraint in the quantization of the electromagnetic field in the next chapter, where the first class constraint arises from a symmetry of the action, electromagnetic gauge invariance . In fact, the set of first class constraints XN=0is always associated with a group of symmetries, under which an arbitrary quantity A undergoes the infinitesimal transformatio n (In field theory these are local transformations, because the index Ncon- 328 7The Ca nonical Formalis m tains a spacetime coordinate .) Eq . (7.6,11)shows that this transformation leaves the Hamiltonian invariant, and for first class constraints it also re- spects all other constraints . Such first class constraints can be eliminated by a choice of gauge, or treated by gauge-invariant methods described in Volume II . After all of the first class constraints have been eliminated, the re- maining constraint equations YN= 0 are such that no linear combination EIr UNLXN,ZnfrI P of the Poisson brackets of these constraints with each other vanishes . It follows that the matrix of the Poisson brackets of the remaining constraints is non-singular : DetC =~0, wher e constraints of this sort are called second class. Note that there must always be an even number of second class constraints, because an antisymmetric matrix of odd dimensionality necessarily has vanishing determinant . As we have seen, in the case of the massive real vector field the constraints are Y,tx =X2x = 0 ,CNAI =[XJr,XMI P(7.G. 13) (7.6.14) (7-6.15) where X2x=e1II;{x} - rra,Ve(x) - .I°{x} . (7.6.16) The Poisson bracket of these constraints i s and, of course, Ci x, ~ ~ -C2x,2y = 0 .(7.b.17) (7.6.18) This `matrix' is obviously non-singular, so the constraints (7.6.15) are second class . Dirac suggested that when all constraints are second class, the comrnu- tation relations will be given by [A7 $}=i[A,BID, (7 .6.19) where [A, B] D is a generalization of the Poisson bracket known as the Dirac bracket : (Here Nand M are compound indices including the position in space, 7.6Constraints andDirac Brackets 329 taking values like 1, x and 2, x in the vector field example .) He noted that the Dirac bracket like the Poisson bracket satisfies the same algebraic relations as the commutator s and also the re lations(T6.21) (7.6.22) (7.6.23) (7.s.24) which make the commutation relations (7.6.19) consistent with the con- straints XN= 0. Also, the Dirac brackets are unchanged if we replace the XNwith any functions X~ for which the equations Z = 0and XN=0 define the same submanifold of phase space . But all these agreeable prop- erties do not prove that the commutators are actually given by Eq . (7.6.19) in terms of the Dirac brackets . This issue is illuminated if not settled by a powerful theorem proved by Maskawa and iVakajima .6 They showed that for and set of canonical variables T', III governed by second class constraints, it is always possible by a canonical transformation* to construct two sets of variables Q", ~2r and their respective conjugates P,, i,such that the constraints read 91= ,= D. Using these coordinates to calculate Poisson brackets, and redefining the constraint functions as ~1 , = ~ ~, ~2r = 9,,we hav e ~,7T r 1 r C1r,1s=Prr~2'1 P= 0 , and for any functions A, B cA LA,J~2r]F =al This C-matrix has inverse C-1 = -C, so the Dirac brackets (7.6.20) are Recall that by a canonical transformation, we mean a transformation from a set of phase space coordinates ' I'p,17A to some other phase space coordinates 17Y", fl,, such that ['FO, IIblr = S db and ['7y°, q'b]p - [III, I lb)p = 0, the Poisson brackets being calculated in terms of the T° and 1IQ. It follows that the Poisson brackets for any functions A, Bare the same whether calculated in terms of `I'° and fIQ or in terms of T1 and fia . It also follows that if q j0and fla satisfy the Hamiltonian equations of maTion, then so do ~PQ and 1%, with the carne Hamiltonian . The Lagrangian is changed by a canonical transformation, but only by a time-derivative, which does not affect the action . 330 here7TheCanonical Formalis m [AxBID = [A, B] P+[A, Z 1r]Y[y?r,BjP - [A,X2rjP[Y1r,Bj P OA aB cB OA C~ l~ r~r 0,J Tf~Ir OAOB OBnA (7.G.25) aQn apr, 0, onaPn In other words, t he Dirac bracket is equal to the Poisson b racketcalculated in termsof'thereduced set of unconstrained canonical 'variables QI, P, If we assume that these unconstrained variables satisfy the canonical commutation relations, then the commutators of general operators A,B are given by Eq. (7.6.19)in terms of the Dirac brackets,* ' We now return to the massive vector field, to see how it can be quantized using Dirac brackets . This is a case where it is easy to express the constrained variables V° and CIQ in terms of the unconstrained onest Vj and III ; we have simply III = 0, and V° is given by Eq .(7.6.5). From Eqs.(7.6.17) and (7.6.18), we see that CN,where has the invers e (C- l)lx.ly_ (C -1)2x.2v= o .(7,6.26) (7.6.27) Therefore the Dirac prescription (7-6.19), (7.6.20)yields the equal-time commutator s CA,BI_i[A,B]p +im d3,([A,Ho (z] p[,,FjjZ)_M2VO(Z)-JO(z),Bjp-A" B (?.6.25) By definition, we have (7.6.29) Henc e It is still an open question whether we should adopt canonical commutation relations for the unconstrained variables Q .P,,constructed by the Maskawd-Nakajima canonical transformation . Ultimately, the test ofsuch canonical commutation relations is their consistency with the free-field commutation relations derived in Chapter 5, but to apply this test we need to know what the Q" and Pry are . In the Appendix to this chapter we display two large classes of theories in which we can identify a set of unconstrained Qs and Ps, such that the Dirac commutation relations (7.6,19)follow from the ordinary canonical commutation relations of the Q5 and Ps . We shall also show that in these cases, the Hamiltonian defined in terms of the unconstrained 'Ys and Rs may be written just as well in terms of the constrained variables , This is a special case of the theories discussed in Part A of the Appendix . 7.7 Field Redefi nitionsand Redundant Couplings 331 (7.6.3) These are indeed just the commutation relations that we would find by assuming that the unconstrained variables satisfy the usual canoni- cal commutation relations [V`(x),IIj(Y)] = i6~b3(x - Y), [[~~(x), Vf(y)] (III{x}, II,.(Y)] = 0, and using the constraints to evaluate the commutators involving IIO~ and VQ . 7.7 Field Redefinitions and Redundant Couplings * Observables like masses and S-matrix elements are independent of some of the coupling parameters in any action, known as the redundant pa- rameters . This is because changes in these parameters can be un- done by simply redefining the field variables . A continuous redefini- tion of the fields, such as an infinitesimal local transformation V(x) Te(x)+cF({tI'(x), opT (x), ' - -), clearly cannot affect and observable of the theory," though, of course, it would change the values of matrix elements of the fields themselves . How can we tell whether some variation in the parameters of a the- ory can be cancelled by a field redefinition? A continuous local field redefinition will produce a change in the action of the for m } bV{x So any change 6gd in the coupling parameters gj> for which the change in the action is of the for m ( ETft= iflgjf () may be compensated by a field redefinitio n This section lies somewhat out of the book's main linc of development, and may be Omitted -in a first reading . For instance, the theorem of Section 10 .2 shows that as long as we multi-ply b ythe correct field renormalication constants . S-matrix elements can be obtained from the vacuum expectation value of a time-ordered product of any operators that have non-vanishing matrix elements between the vacuum and the one-particle states of the particles participating in the reaction . 332 7 TheCanonical Formalis m and therefore can have no effect on any observables . In other words, a couplingparameter isredundant if the change in the action when we nary this parameter vanishes w hen we use the _ field equations H/6V' =0, For example, suppose we write the Lagrangian density of a scalar field theory in the form A 24 The constant Z is an redundant coupling, becaus e Zfd4x(EJIJI-_m2 _(D ~gZ~3) f:= and this vanishes when we use the field equatio n On the other hand, neither the bare mass m nor the coupling g are redundant, and no function of inand g is redundant . In this example, the field redefinition needed to compensate for a change ofZis a simple rescaling, in which F is proportional to (D. (For this reason Zis called a field renormalization constant .) This is the most general field transformation that leaves the general form of this action invariant . But for the more general actions considered in sections 12 .3 and 12 .4, with arbitrary numbers of fields and derivatives, we would have to consider non-linear as well as linear field redefinitions, and an infinite subset of the parameters of the theory would be redundant . Appendix Dirac Brackets from Canonical Commutator s In this Appendix we shall show, in theories of two types, that the for- mula giving commutators as Dirac brackets times i follows from the usual canonical commutation relations for a reduced set of variables . A Suppose (as i n the case of a massive vector field V~) that the quantu m variables T'and II ,appearing in the Lagrangian L may be divided into twoclasses:*one setQ1of independent canonicalvariables (likeVi(x)) with independent canonical c onjugates f'n =aL/001;and another set ` We are again using a compact notation, in which labels like a, n, and r include a space coordinate x as well as discrete indices . Repeated labels are summed and integrated . All quantum variables are understood to be evaluated at the same time, with the common time argument dropped everywhere . The quantities Y are the same as the Crintroduced in Section 7,2, Appendix Dirac brackets 33 3 2r(x)(like V° )whose time- derivatives donotappear in the Lagrang ian. The primary constra ints are the cond itions Xjr = 0, wher e fir =Yr (7 .A.1) are the variables conjugate to the A'. The secondary constraints arise from the equations of motion 0=OL/0221 for the Y; we suppose that these constraints can `solved' i .e., they may be written in the form /z, = 0, with X2, in the for m (An example is provided by Eq . (7.6.5), which gives YO in terms of the independent Ps (here, the FIj) and Qs .) We assume that the independent Qs and Ps satisfy the usual canonical commutation rules : The constraint X2r = 0yields the commutators involving 22 ; OQn 12r, S1 = ir'ts, where I -'11is the Po isson bracke t FPS=VWS]P, and, of course, all commutators involving ,vanish : Pp,, T J==~rsPn]-Vr, -~)81 = L ra sl_0.(7.A.4) (7.A.5) (7.A.6) (7.A.7) Now let us compare these commutators with the Dirac brackets . The Poisson brackets of the constraint functions ar e Ciy,is° ~1";Gi_dP--0, Clr,2s = - C25,t!' = XIr, ;(z,slP(7.A.8) (7.A,9) (7.A.in) (In the example of the massive vector field I 'r' vanishes ,but the discussion here will apply also for non-vanishing rr~.)It is easy to see that the C-matrix has the invers e (C-1}1r, ls = rrs (c- 1)2r,2s=0 (7.x.11) S 334 7 The Canonical Formalis m Also, the Poisson brackets of any function A with the constraint functions are ALA, %1r]P= IA O-A 0,A}B G ~_B 0,A 0211 'J "Hence the Dirac bracket i s [A,BID =[AfB]p- OA 0 Now, if A and B are both functions only of the independent canonical variables Q'and Pry, then c'~i Jc~,2r = O.Blajr -0,so the Dirac bracket is equal to the Poisson bracket . In particular , Where A is ' and B is a function of Qs and Ps, it is only the fifth term on the right-hand side of Eq .(7.A.12)that contributes . In particula r or r f1T" Where both A and B are l .s, we have only the fourth ter m [r ,A n = Frs - (7.A.] 5) Finally, where A is ~~r and B is anything, we have only the first and third terms, which cancel : Comparison of Eds .(7.A.13)-(7.A.16)with Eqs .(7.A.3)-(7,A,7) shows that in all cases, the commutators are equal tothe Dirac bra cketstimes 1 .This is❑nly to b eexpected ,becau se as remarked inSect ion 7 .6 all D irac brackets involv ing the constraint functions vanish,so the Dirac brackets involving Y and/or ~_3ASare g iven by using the constraint equations to express rand/orY-,in terms of the independent Qs and Ps. B Next consider the case where the constraints take the form of conditions zi,('i') = 0 on the V, which can be solved by expressing them in terms of a smaller set of unconstrained variables Q', and an equal number of separate conditions )(24 I)- {} on the III,, which can be solved b y expressing then,, in terms of a smaller set of unconstrained P . (We will seean example in the next chapter, where the constraints on the T" are Appendix Dirac Brackets 33 5 gauge fixing conditions that are used to eliminate first class constraints, and the constraints on the IIQ are secondary constraints arising from the consistency of the first class constraints with the field equations .) We assume that the unconstrained variables satisfy the usual canonical commutation relations [Q ',P,,,] =tam, [[fin, Q'] -= JP,, P .]= 0 . The constrained and unconstrained momenta are related b y it follows tha t or,in other word s,OT' ar7Tu OT'O►ID.(7.A.17) (7.A.18) Now, the constraint Xi,(T) = 0 is satisfied for T' = T" (Q)forallQ,so O-ZI rOT' (~ i70.O.1160 Furthermore, the vectors ( V,)b=JY1,/OT6 form a complete set perpen- dicular to all the vectors (U,)~ , Y61C' Q'',because if there were some other vector V bwith Vb(U,)h =0for all n, then there would be additional constraints on the 'I'a . Hence Eq . (7.A.18)implies tha t O,~Iri~~ + icy r?Tb(7.A.20) with some unknown coefficients c~ . To determine these coefficients, we make use of the other constraint, that .72,M) = 0 . It follows tha t 0 = ~, i[` ~`~, III,]OX 2,M) P.III Using E q. {7.A.20}, we have the n 0, X2rJI1 a Ox1s~~~0;(2r(FI) aria ins O'i'bc'II, We recognize the factor multiplying ca as the Poisson bracke t OV a.rib(7.A.21) Also, sine ~ls depends only on the T and Zz, depends only on the U . these are the only non-vanishing Poisson brackets of constraints, so 336 7 The Canonical Formalis m Thus Eq .(7.A.21)may be writte n 0'XN III-CSC'is,-N,(7.A.22) with Nrunning over all constraint functions . For second class constraints, this has the unique solutio n 3 r7 II. aII, Using this in Eq . (7.A.20) shows tha t -an" OTh{7.A.23} (7.A.24) The Poisson brackets of T' and III with the constraint functions ar e [Xlr, IMPOxir -f~~~OX2,[~``~a X2 Jp~aII' [X2r,r1ir]P - 0 ,(7.A.25) so the quantity in brackets on the right-hand side of Eq.(7.A.24) is the Dirac bracket [Ta, nfi] = tff'a f1b]r) {7.A.26} as was to be shown . Also, we can easi ly see that because C'-1 has no 11 or 22 components, the other Dirac brackets ar e so triviall y In addition to commutation rules, we also need an explicit formula for the Hamiltonian . The usual canonical formalism tells us to tak e the sum running over independent canonical var iables .Intheories of both types consi dered in this Appendix, this Hamilton ian may be w ritten in terms of the con strain ed var iables a s For theories of type A, th is is t rivial; the sum over a runs over values n, for which T' =Q'i and fl,, - P, are the independent canonical variables, Problems 337 together with values r, for which II, . T9,= 0. For theories of type B, we note that Eq . (7.A.17) gives h PMn =]Fjh fJ~~ ~~ T11 h which again y ields Eq ,(7.A.30). Problem s 1. Consider the theory of a set of real scalar fields (D", with Lagrangia n density ' =-1 O ~~~'OY~'f,~„z(~), where f ,,(O) is an arbitrar y real matrix function of the field . (This is called the non-linear cr- model .)Carry out the canonical quantization of this theory . Deriv e the interaction V [O(t),(t)] in the interaction picture . 2. Consider a theory of real scalar fields Vand Dirac fields V, wit h Lagrangian density Y= O+ 1, where YOis the usual free-fiel d Lagrangian density, and 1 is an interaction term involving Vand Ti, but not their derivatives . Derive an explicit expression for th e symmetric energy-momentum tensor O y'. 3. In the theory described in Problem 2, suppose that the Lagrangian density is invariant under a global infinitesimal symmetry 60" = ieEm Om Vn, 6kYi = a eT_j T'jkI'J. . Derive an explicit expression for the conserved current associated with this symmetry . 4. Consider the theory of a complex scalar field c Dand a real vector field VP, with Lagrangian densit y where D,,=0j,- ig V,, and F.,- c7~ ,V, - 0,V,,,and , is an arbitrary function . Carry out the canonical quantization of this theory . Derive the interaction in the interaction picture . 5. In the theory of Problem 4, derive expressions for the symmetric energy-momentum tensor O Y'and for the conserved current associ- ated with the symmetry under 60= ic(D, 6V ~` = 0 . 6.Prove that the Dirac bracket satisfies the Jacobi identity (7.6.23). 7. Prove that the Dirac bracket is independent of the choice of con straint functions XNused to describe a given submanifold of phase space . 338 7 The Canonieal Formalis m References 1. H. B. G. Casimir, Proc . K. Ned .Akad .Wet.51,635 (1948) ; M. J. Spaarnay, Nature 1 80, 334 (1957) . 2. F. Belinfante, Physica 6, 8$7(1939 ); also see L . Rosenfeld, femoires deI'Academie Roy . Belgique 6,30(1930) . 3. See, e .g.,S. Weinberg, Gravitation an dCosmology (Wiley, New York, 1972) :Chapter 12 . 4. See, e .g.,J. Wess and J . Bagger, S upersymmetry and S uperg ravaty, (Princeton University Press, Princeton, 1983), and original references quoted therein . 5. P. A. M. Dirac Lectures on Q uantumMechanics (Yeshiva University, New York, 1964) . Also see P . A. M. Dirac Can.J. Math .2,129(1950) ; Prod .Roy.Soc. London, ser. A, 246, 32 6 (19 5 8) - P. G. Bergmann, Heiv . Phys .Acta Suppl. IV, 79 (1956) . 6.T. Maskawa . and H . Nakajima, Prog . Theor. Phys . 56, 1295 (197 6). I am grateful to J . Feinberg for bringing this reference to my attention . Electrod ynami cs The original approach to quantum electrodynamics was to take for granted Maxwell's classical theory of electromagnetism, and quantize it . It will probably not surprise the reader that this book will follow a different path . We shall first infer the need for a principle of gauge invariance from the peculiar difficulties that arise in formulating a quantum theory of massless particles with spin, and then deduce the main features of electrodynamics from the gauge invariance principle . After that we shall follow a more conventional modern approach, in which one takes gauge invariance as the starting point and uses it to deduce the existence of a vector potential describing massless particles of unit spin . It is too soon to tell which of these two alternatives corresponds to the logical order of nature . Most theorists have tended to take gauge invariance as a starting point, but in modern string theories the argument runs the other way ; one first notices a state of mass zero and unit spin among the normal modes of a string, and then from that deduces the gauge invariance of the effective field theory that describes such particles . At any rate, as we shall see, using either approach one is led to the quantized version of Maxwell's theory, still the paradigmatic example of a successful quantum field theory . 8.1 Ga ugeInvariance Let's start by recalling the problems encountered in constructing covariant free fields for a massless particle of helicity ±1. We saw in Section 5.9 that there is no difficulty in constructing an antisymmetric tensor free field f ft ,(x)for such particles . This field can be expressed in terms of the four-potential a,,(x), given by Eq .(5.9.23), through the familiar relatio n fl,,,(x) = o .a,,(x) - c, .a,,(x) . (8.1.1) However, Eq . (5.9.23 shows that the a,,(x) transforms as a four-vecto r 339 340 8 Electrodynamic s only up to a gauge transformatio n Uo(A)a , (x)Uo-i(11,) = A,,'ay(Ax) + o,Q( x, A). (5 .1.2) There is, in fact ,no way to construct a true four-vector as a linear comb ination of the crea tionand annih ilation ope ratorsfor h elicity ± 1. This is one way of underst anding the presence of singular itiesat m = 0in the prop agator of a ma ssive vector fi eld APv(x, Y) =(2-W,.d4,!q-(x-y) ~9V + 4 1'~fv /mz q2 + m2 - ie ' which prevent us from dealing with massless particles of helici#y ±1 by simply passing #a the limit rra--*0of the theory of a massive particle of spin one . VI{e could avoid these problems by demanding that all interactions involve only* F~,,(x) ~_r7,,A,( x} - a,,A,,(x) and its derivatives, not Ap(x), but this is not the mast general possibility, and not tie one realized in nature . Instead of laanishing A,,{x} from the action, we shall require instead that the part of the action IM far matter and its interaction with radiation be invariant under tie general gauge transformatio n (atleastwhenthe ma tter fi elds satisfy the fie ldequa tions sa t hatthe ex tra term inEy.(8.L2) shou ld have no effec t.The ch angein the mat teraction underthe transfor matians (8.1.3maybe written 6!,_ dux6A1~ X)aue(x). (8 .1.4) Hence the Lorentz invariance of Imrequires tha t ~~~ Op6A AW= 0. (8.x.5) This is trivially true if I minvolves only F,,4, (x) and its derivatives, along with matter fields_ In this cas e 6IM= 20,,b1m 6A"(X) a F,,v(x) Bit if IM involves Ajx}itself then Eq . (5.1.5)is anon-trivia] constraint on the theory . Now, what sort of theory will provide conserved currents to which we can couple the field AP(x)? Vie saw in Section 7 .3 that infinitesimal interna l We now use Au and F,,V for the electrornagnelic potential vector and the field strength tensor because these are interacting fields . 8.1 Gauge Invariance 341 symmetnes of the action imply the existence of conserved currents . In particular, if the transformation* * leaves the matter action invariant for a constant c, then for general infinitesimal functions E{x} the change in the matter action must take the form 51M = -fd4x ,I~`(x)aye(x). (8 .1.7) When the matter fields satisfy their field equations, the matter action is stationary with respect to any variation of the T4, so in this case (8 .1.7) must vanish, and hence OPP = 0. (8 .1.8) In particular, we saw in section 7 .3 that if I mis the integral of a function Ym afV and r?},V, then the conserved current is given by t and this generates the transformations (8-1.6)in the sense tha t [Q,T'(x )] =----RrT"'(x ) , (8.1.9) where Q is the time-independent charge operato r Q =Id'x.l°. (8.1.10) ❑Ve can therefore construct a Lorentz-invariant theory by coupling the vector field 4to the conserved current P, in the sense that 6Im16A,,(x) is taken to be proportional to P(x). Any constant of proportionality may be absorbed into the definition of the overall scale of the charges q 1, so we may simply set these quantities equal : 61M= P {x}6A,U(X) The conservation of electric charge only allows us to fix the values of all charges in terms of the value of any one of them, conventionally taken t o k3ecausc the field transformation rnalrix is taken now to be diagonal it is not convenient here to use the summation convention for sums over field indices, so there is no sum over C in Eq.(8.1.6). Here V is understood to run over all independent fields other Than A.We use a capital psi to indicate that these are Heisenberg-picture fields, whose time-dependence includes the effects of interactions .Ofcourse, this `Yris not to be confused with astate-vector or wave function . 342 8Electrodynamic s be the electron charge, denoted -e . It is Eq .(8, 1.11) that gives a definite rrieaning$ to the value of e . The requirement (8 .1.11 may be restated as a principle of invariance :-i° the matter action is invariant under the joint transformation s JA~(x )=a.,e(x) , (8.1.12 6`~~(-x)= ie(x)R,, `Y,,(x) (8.1.13) A symmetry of this type with an arbitrary function e(x) is called a local symm .etryF, or a gauge invariance of the second kind . A symmetry under a transformation with cconstant is called a global symmetry, or a gauge invariance of the first kind . Several exact local symmetries are now known, but the only purely global symmetries appear to be accidents enforced by other principles . (See Section 12 .5.) We have not yet said anything about the action for photons themselves . As a guess, we can take this to be the same as for massive vector fields, but with m . = 0 : If ~ -~d 4x F11vF''~V - (8.1.14) This is the same as the action used in classical clectradynamics, but its real justification is that it is (up to a constant) the unique gauge-invariant functional that is quadratic in F,,ti,, without higher derivatives . Also, as we will see in the next section, it leads to a consistent quantum theory . If there are any terms in the action of with higher derivatives and/or of higher order in Feu they can be lumped into what we have called the matter action . Using Eqs .(8.1.11) and (8 .1.14), the field equation for electromagnetism now read s o= Crr +rte]=~~~~'v + F . (8.1.15)5A,, We recognize these as the usual inhomogeneous Maxwell equations, with current P.There are also other, homogeneous, Maxwell equation s which follow directly from the definition F.,="I,Av-- cruAp. In the above discussion, we have started with the existence of massless spin one particles, and have been led to infer the invariance of the matter action under a local gauge transformation (8 .1.12), (8,1.13). As usually presented, the derivation runs in the opposite direction . That is, one start s Of course, Eq . (8. 1.I 1) fixes the definition of e only after we have de lined how we are normal izing A,,(x) .The question of electromagnetic fie ld normalization is taken up in Section 10 .4. 8.2 Co nstrai nts and Gauge Conditions 343 with a global internal symmetr y and asks what must be done to promote this to a local symmetr y If the Lagrange density Y depended only on fields V(x) and not on their derivatives then it would make no difference whether e is constant or not ; invariance with Econstant would imply invariance with e a function of spacetime position . But all realistic Lagrangians do involve field derivatives, and here we have the problem that derivatives of fields transform differently from fields themselves : ~~11V( x) = ie(x)qer',,V(x) + iq~V(x)c?,,-F{x}. ($ .1.19 In order to cancel the second term here, we `invent' a vector field AJ!(x) with transformation rul e and require that the Lagrangian density depend on c~,,V and A,,only in the combination D~~~ 0}jTt -iq(A$jTt- which transforms just like V A matter Lagrangian density Y' M(T, Dom) that is formed only out of V and D .V will be invariant under the transformations (8.1,18 ),(8.1.20), with c(x) an arbitrary function, if it is invariant with Ea constant . With the Lagrangian of this form, we hav e dIM M c which is the same as Eq .($.i.ll). (We could also include F,,, and its derivatives in Ym.)From this point of view, the masslessness of the particles described by A Pis a consequence of gauge invariance rather than an assumption : a term -; rn2APAPin the Lagrangi an density would violate gauge invariance . 8.2Constraints and Gauge Condition s There are aspects of electrodynamics that stand in the way of quantizing the theory as we did for various theories of massive particles in the 344 8 Electrodynamic s previous chapter, As usual, we may define the canonical conjugates to the electromagnetic vector potential by rip OY O(OaAM) Quantization by the usua lrules would give(8.2.1 But this is not possible here, because Apt and II'' are subject to several constraints . The first constraint arises from the fact that the Lagrangian density is independent* of the time-derivative of A O, and therefor e r1°(x) -- o . (8 .2.2 This is called a primary constraint, because it follows directly from the structure of the Lagrangian . There is also a secondary constraint here, which follows from the field equation for the quantity fixed by the primary constraint :** 1. O Y OFjQ r1Aa(8.2.3) the time-derivative term dropping out because Fm =0. Even though the matter Lagrangian may generally depend on AO,the charge density depends only on the canonical matter fieldst Qn and their canonical conjugates P,~ JI)=_ i qe'T''= ---i ~ PnRn Qn- 5 .2.4)E Hence Eq . (8 .2.3)is a functional relation among canonical variables . Both Eq . (8.2.2) and Eq .(8.2.3) are inconsistent with the usual as- sumptions that [A,,(x,t),IU(y,t)] = t6~th3(x - y) and [Q'(x, t),nV(y,t)] _ lpn(xa01[IV1Y'01=0. We encountered a similar problem in the theory of the massive vector field . In that case we found two equivalent ways of dealing with it : either bythe method of Dirac brackets or, more directly, b ytreating onl y For Yp=-FuVF P'14,we have = -F UP,which vanishes for it = 0 because FO' is antisymmetric . For matter L..agrangians YM that involve only ' f"~ and D .V, the prescription (8.121) tells us that .PMdoes not depend on any der ivatives of any A.Even if the matter LagKa ngiandepends also on F~,V, r7~~q f~(c3, .4u) w illbe aga in anlisym metric ina and tip, an d therefore wi llvanish for ki= v = O . As usua l, i, j, etc .run over the va lues 1, 2, 3 , Due to ex haustion of alphabetic resources, I have had to adopt a notation here that is different from that of the previous chapter . The symbo ls Qn and P„ are now reserved for t he canonica l matter fields and their canonical conjugates, 1`CfipCCT1VC ly, whi le the canonica lelectromagnetic fields a nd canonical conjugates are A, and fl;. 8.2 Constraints andGauge Conditions 345 Ajand n, as canonical variables, solving the analog of Eq .($.2.3)to calculate A° in terms of these variables . It is clear that here we cannot use Dirac brackets ; the constraint functions Xhere are II° and 0 ;IIi - J 0 (as compared with an - m IA° - J ')and these obviously have vanishing Poisson brackets . In Dirac's terminology, the constraints (8.2.2)and ($ .2.3) arefirst class . Nor can we eliminate A 0as a dynamical variable by solving for it in terms of the other variables . Instead of giving A° for all time, Eq.(8.2.3)is a mere initial condition ; if Eq .(8.2.3)is satisfied at one time, then it is satisfied for all times, because (using the field equations for the other fields A'), we hav e ~Y 0Fi6 o, +a! aj~ ~ 0-Fjj-ai~~-~0i" and the current conservation condition then give s 000` r~F- J° = 0. z4(8.15) It should not be surprising that we still have four components of Au with only three field equations, because this theory has a local gauge symmetry that makes it, in principle, impossible to infer the values of the fields at arbitrary times from their values and rates of change at any one time . Given any solution A,,(x, t) of the field equations, we can always find another solution Ai,(x, t) +0~,c(x, t) with the same value and time- derivative at t = 0 (by choosing eso that its first and second derivatives vanish there) but which differs from Ap(x, t) at later times . Because of this partial arbitrariness of A~t(x, t), it is not possible to apply the canonical quantization procedure directly to A ,,(or, as for finite mass, toA). Of the various approaches to this difficulty, two are particularly useful . One is the modem method of gauge-invariant quantization, to be discussed in Volume II . The other, which will be followed here, is to exploit the gauge invariance of the theory, to `change a gauge' . That is, we make a finite gauge transformatio n A,,(x) -+A,jx) +0,,).(x), Tt(x) --~exp(iq1;.(X))T,,'(X ) to impose a condition on A,,(x)that will allow us to apply the methods of canonical quantization . There are various gauges that have been found useful in various applicatians :f Here 0 is any comp lex scalar field with q ~-0; this gauge condit ion is used w hen the gauge symmetry is spontaneous ly broken by a non-vanishing vacuum expectation va lue of (P . 346 8 Electrodynamic s Lorentz (or Landau) gauge : oOAP=0 Coulomb gauge : V -A = 0 Temporal gauge : A° = 0 Axial gauge : A '=0 Unitarity gauge : rea l The canonical quantization procedure works most easily in the axial or Coulomb gauge, but of course Coulomb gauge keeps manifest rotation invariance in a way that axial gauge does not, so we will adopt Coulomb gauge here .'- To check that this is possible, note that if AP does not satisfy the Coulomb gauge condition, then the gauge-transformed field A '+ ~P~will, provided we choose ~so that V2~ _ -V -A. From now on, we assume that this transformation has been made, so tha t VA=O . (8 26) It will be convenient henceforth to limit ourselves to theories in which the matter Lagrangian Y m may depend on matter fields and their time- derivatives and also on A'but not on derivatives of AP.(The standard theories of the electrodynamics of scalar and Dirac fields have Lagrangians of this type .) Then the only term in the Lagrangian that depends on F,,, is the kinematic term - 'F,,.,F,'', and the constraint equation (8.2.3)read s O-i fro = J°. (8.2.7) Together with the Coulomb gauge condition (8 .2.6), this yie lds - V2A0 ,Ja ?(8.2.8 which can be solved to giv e A°(x, t) = dayJO(Y't)(8.2.9)47rlx -- Y l The remaining degrees of freedom are Ai, with i = Z, 2, 3, subject to the gauge condition V -A = 0 . As mentioned earlier, the charge density depends only on the canonical matter fields Q" and their canonical conjugates iP,,, so Eq.($.2.9) represents an explicit solution for the auxiliary field A 0. 8.3 Quantization inCoulomb Gauge There is still an impediment to the canonical quantization of electrody- namics in the Coulomb gauge . Even after we use Eq .(8.2.9) to eliminate AO(and III) from the list of canonical variables, we cannot apply the usual 8.3 Quantization in Coulomb Gauge 34 7 canonical commutation relations to A 'and Iii, because there are two re- maining constraints on these variables .* One of them is the Coulomb gauge condition Xlx=OiAl(X)_0. (8.3.1) The Other is the secondary constraint Eq . ($.2.3), which requires tha t (8.3.2) Neither constraint is consistent wi th the usual commutation re lations [Ai{x},I Ij (y)] = i6zjS3(x - y), because operating on the right-hand side with either )/0x' or 0/0y} does not give zero . These constraints are of a type known as second class, for which there is a universal prescription for the commutation relations, discussed in section 7 .6. Note that the constraint functions have the Poisson bracket s Czx,iY=[i~1x,XIYlP_0 7 C2x,2yLY,2xa x2}lP 0 where here, for any functionals U and V , JU JV dV 6U lu,V]p d3X[ bAj(x)Mli(x)bAi(x)bHi(x)(8.3.3) The `matrix' C N,Wis non-singular, which identifies these as second class constraints . Also, the field variables A 'may be expressed in terms of independent canonical variables, which may, for instance, be taken as Q1x = Al{x}, Q2x = A2(x), with A3given by the solution of Eq . (5.3.1): A'(x)_ - ~ ds[firA'(x% x2,S) +e2A2(X 1,X2,S)] Using Eq . (8.3.2), the canonical conjugates IIi to A 'may likewise be expressed in terms of the canonical conjugates P,,, and A2, to Q1, ,and Q2x. In such cases, Part B of the Appendix to the previous chapter tells us that if the independent variables Qlx, Qzx, P1,and Fix satisfy the usual canonical commutation relations, then the commutators of the constrained variables and their canonical conjugates are given (aside from a factor i) by the corresponding Dirac brackets (7.6.20).This prescription has the great advantage that we do not have to do use explicit expressions for the dependent variables in terms of the independent ones . In this section i, j, etc, run over the va lues 1,2,3 . We continue the practice of taking all operators at the same time, and omitting the time argument . 348 8Electrodynamic s To calculate the Dirac brackets, we note that the matrix C has the inverse d~k e lk'(x-r) (27z)3k21 47zlx - Y I (8.3.4) Also,the non-vanishing Poisson bracket s of the A 'and Hi w iththe constraint func tionsare and RIO), x2ylp = +f~~~Y) Hence according to Eqs. (7.6.19)and (7.6.20),the eq ual-time commutators are r'21 iOxi0x'(4x -YJ [A"(x), A"(y)] = [1-li(x), Hj(y)] -0.(8.3-5) Note that these are consistent with the Coulomb gauge conditions (8.3.1) and (8.3.2),as is guaranteed by the general properties of the Dirac bracket . Now, what is H in electrodynamics? For the class of theories discussed in the previous section where only the kinematic term -:1fd3xF1, F"' in the Lagrangian depends on A, varying the Lagrangian with respect to A without worrying about the constraint V -A = 0 give s bAi()P_XJ But with A constrained by the condition V -A = 0, variational derivatives with respect to A are not really well defined . If the variation of L under a change Ain A is 6L = f duxy.6A, then since ❑- A = 0, we also have 6L- f dux [ , 4 0 + VJV] Afor any scalar function ~flx) . Thus all we can conclude from inspection of the Lagrangian is that H equals A .(x)+VA°{x} plus the gradient of some scalar . This ambiguity is removed by condition (8.3.2), which requires that V  H = --J° =V2A Because V  A = 0,we conclude that Eq . (5.3.6)does indeed give the correct formula for III . Although the commutation relations (8.3.5) are reasonably simple, we must face the complication that IT does not commute with matter fields and their canonical conjugates . If F is any functional of these matter degrees o ffreedom, then its Dirac bracket with A vanishes, but its Dirac 8.3Quantization in Coulomb Gauge 34 9 bracket with H i s [FkH(Z)I D = -f~3 xd 3y[F,%2x]P47r lx -yjU lye~(41 P = - j d3x X Y [F, Jo(x)] p 47rIx 1-yI Inorder to facilitate the t ransition to the inte raction picture, instead of expressing the Hamiltonian in terms of A and H, we shall write it in terms of A and IIl, where IIl is the solenoidal part of H. IIl = II-PA°=A, (8 .3.7) for w hich LF,Tll(z)]vanishes. By using thefactsthat II1(x) commu tes with Il(y) - [I-L(y) = VA°(y) and that vjA°(x)commutes with OjA°( y), it is easy to see that 71(x) satisfiesthesame commut ation rel ations (8.3.5) asC[(x), andalsothe simpl e constrain t v-n1=o. (8.3.8) Now we need to construct a Hamiltonian . According to the general results of the Appendix to Chapter 7, we can apply the usual relation between the Hamiltonian and Lagrangian using the constrained variables A and IIL, without first having explicitly to write the Hamiltonian in terms of the unconstrained Qs and P s . In electrodynamics, this give s jd'x~IIL jA~ +Pnon - Y1 1 (83 .9) where, as mentioned earlier, Tand P,, are to be understood as the matter canonical fie lds and their canonical conjugates . (We can use IIL in place of 17in Eq . (8.3.9) because V  A = 0 .) To be specific, consider a theory with a Lagrangian density of the for m Y ~FpvFyy + JvA" + smatter (8.3.10) where .I,,,is a current that does not involve AP,and maitc,- is the La- grangian for whatever other fields do appear in P, aside from their electromagnetic interactions, which are given explicitly by the term ,I,,A4 in Eq .(8.3.10).(The electrodynamics of spin particles has a Lagrangian of this form, but the electrodynamics of spinless particles is more compli- cated .) Replacing A everywhere with II-L, this gives a Hamiltonian (8.3.9) of the for m H_ d3xLni + ~(❑XA)2-;(IIl + VA4)2-.IA+ ,]OAOI+HM } 350 8Electrodynamic s where Hm is the Hamiltonian for matter fields, excluding their electro- magnetic interactions HM= X x ~ P,, On -Ymaiter). Using the solution (8.2.9)for A°, this i s H dux ~ 1JT~ + !(V x A)2 - J-A+ZJ°Ar'] + H M The term ~.IOA' may look pecul iar, but this is nothing but the familiar Coulomb energy Vc0u] = z ~ duxJOAo ?l fd3YJ11(X)J11(Y) 47L I X -($.3.12) The reader can verify, using the commutation relations (8 .3.5), that the rate of change of any operator function F of A and Ris given by iF =[F, H], as it should be . 8.4 E lectrodynamics in theInteraction Pictur e We now break up the Hamiltonian (8.3.11) into a free-particle term Ho and an interaction V H = Ho+V, H,, = f d-'x L V = -jdx J'A+ VCou j+ smatter ,(5.4.x) (8A .2) (8.4.3) where Hmattero and Vmatler are the free-particle and interaction terms in Hmatxera and Vcqu] is the Coulomb interaction (8.3.12).The total Hamiltonian (8 .4.1) is time-independent, so Eqs . (8.4.2) and (8 .4.3) can be evaluated at any time we like (as long as both are evaluated at the same time), in particular at t = 0. As in Chapter ?, the transition to the interaction picture is made by applying the similarity transformatio n V(t) = exp(iHot) V[A, Ill,Q,P]t=o exp{- Wot} V [*0, 140,q(t), A t}] (8 .4.4) where P here denotes the canonical conjugates to the matter fields Q, and any operator a(x, t) in the interaction picture is related to its value O(x, U ) 8.4Electrodynamics in the Interaction Pictur e in the Heisenberg picture at t - 0 by so that i o(x, t) = [o(x, t),fin]351 (8.4.5) (5.4.6) (We are dropping the subscript ,L on n(x) .) Since Eq . ($.4.5) is a similarity transformation, the equal-time commutation relations are the same as is the Heisenberg picture : 02 1[ax, t), az'(y, t)] = i 6ji63(x -Y)+Ox~Oxf41r Ix -yJ(8.4.7) ~a ~ (x, t), ai (y, t)]=0, (8 .4.8) and likewise for the matter fields and their conjugates . For the same reason, the constraints ($.2.6) and (8.3.8)still appl y V -a=0 , Pn=Q .(8.4.10) (8.4.11) To establish the relation between it and it, we must use Eq . (8.4.6) to evaluate a : i1.Ej(x5t)-[aZ lx7tJ}HV] C72: f d3y[3 .x-y)+7z f( y, t)c;xYx}47zlx-y l We can replace 0510xf in the second term with -r~10yf,integrate by parts, and use Eq. (8.4.11), yielding a=It (8.4.12) just as in the Heisenberg picture . The field equation is likewise determined by ifzi (x,t)=[-icj (x,t), moo] =-if d'y [6,jj'(x - y) + x(VxVx a{y, t})} ,a2 i Ox'Ox1 47c 1- YJ which (using Eys . ($.4.10)and (8 .4.12) just yields the usual wave equatio n 11a=;:= 0. (8A 13) 352 8Electrodynamic s since A Ois not an independent Heisenberg-picture field variable, but rather a functional (8.2.9)of the matter fields and their canonical conjugates that vanishes in the limit of zero charges, we do not introduce any corresponding operator ao in the interaction picture, but rather tak e ao=0. ( 8,4.14) The most general real solution of Eqs . (8.4.10), (8 .4.13), and (8,4 .14) may bewritte n alu(x) =(21r)-'/'d3P E PP eNP, a)op, cr) +e-'P"e'*(p, a) at(p,cr) (8A.15) where p O-JpJ; eg(p, cr) are any two independent 'polarization vectors' satisfying P -e*a) = 0 (8.4.16) e°(A}G) T0, (8.4.17) and a(p, a)are a pair of operator coefficients, with d a two-valued index . By adjusting the normalization of u(p, a), we can normalize the e il(p,a) so that the completeness relation read s ea*a}ei(p7 CO" = 6ij - Pi P jIIPI, (8.4.15) For instance, we could take the e{ p, rr} to be the same polarization vectors that we encountered in Section 5 .9: 1/ el'(P, ±1 );--= R(P)0(8.4. 19) where R(p )isa standard rotation that carries the three-axis into the direction of p. Using Eqs .(8.4.18) and (8.4.12), we can easily see that the commutation relation s (8.4.7)-(8.4.9) are satisfied if (and in fact only if) theoperator coefficients in Eq . (8.4.15) satisfy (8.4.20) (8.4.21) As remarked before for massive particles, this result should be regarded not so much as an alternative derivation of Eqs . (8.4.20) and (8 .4.21), but rather as a verification that Eq . (8.4.2) gives the correct Hamiltonian for free massless particles of helicity ±1 . In the same spirit one can also use Eqs . (8.4.12) and (8 .4.15) in Eq. (8.4.2) to calculate the free-photon 8.5 The Photon Propagator 35 3 Hamiltonian Ho id3plp'[a(p,u), af(p, a] + ff d'p E p' (a'(p, u)a(p, a)+ W(p - p)) (8.4.22) CT which (aside from an inconsequential infinite c-number term) is just what we should expect . Finally, we record that the interaction (8 .4.4) in the interactian picture is ~(O.- -d 3Xj,,(x, t) aP(x at)+VCoul (t)+Vrnatier (t)} (8.4.23) where in terms of the current Jin the Heisenberg pictur e j,,(x, t) = exp( Wot) J,,(x, 0) exp(-iHat) , ($ .4.24) while Vc,,,](t) is the Coulomb ter m VCOU)(t}= exp(iHot)Vco„i exp(-Wo t ) = fd 3,day ~~~x, t} j°(y, ~) (8.4.25) J 4nIx - yI and Vmatter~t) is the non -elect ramagnefic part of the matter field interaction in the interaction picture : Vrnatter(t) _ CxP(iHO0 smatter CW(-iHpO . (8 .4.26) We have written jl-ap instead of j -a in Eq. ($.4.23), but these are equal because am has been defined to have ao=0. 8.5The Photon Propagato r The general Feynman rules described in Chapter 6dictate that an in- ternal photon line in a Feynman diagram contributes a factor to the corresponding term in the 5-matrix, given by the propagator : -iAg,.(x-y)=((DVAcx T {a ,, (x),a, (Y)} (DvAc) (8 .5.1) where T as usual denotes atime-ardered product . Inserting our formula (5.4.15) for the electromagnetic potential then yield s - iAMV (x `W' Y) _d ~ P~,,( p}[e18(x -y) +e~''fy--x}O(Y - x)](2n)2Ipl(S.s.2) 354 where8Electrodynamic s =+a(8.53) and pY in the exponentials is taken with p° = pl- We recall from Eqs. (5.4.18) and (8 .4.17) that rPij(P)=61]- F Fj ~A1(8.5.4) As we saw in Chapter 6, the theta functions in Eq. (8.5.2)may be expressed as integrals over an independent time-component q Qof an off-shell four- momentum q P, so that Eq .(8.5.2)may be rewritte n A,,, (x -y)=(27r )-4f dq q2 - 1-6eiq-(X-V). (8-5.5) Thus in using the Feynman rules in momentum space, the contribution of an internal photon line carrying four-momentum qthat runs between vertices where the photon is created and destroyed by fields a" and a'is -i f'1,( 9) (27r)4 q2 - iF F(8.5.6) It will he very useful (though apparently perverse) to rewrite Eq . (8.5.4) as where n,' - (0, 0, 0, 1) is a fixed time-like vector, q' as usual is q2-(q0)2, but qO is here entire ly arbitrary . We shall choose q° in Eq . (8.5.7) to be given by four-momentum conservation : it is the difference of the matter p° s flowing in and out of the vertex where the photon line is created . The terms proportional to q,, and/or q, then do not contribute to the S-matrix, because the factors q,, or q, act like derivatives 0.and o,,, and the photon fields aA and a, are coupled to currents jF` and ,jV that satisfy the conservation condition 0,j" = O .W The term p roportional to n,nV contains a factor q2 that cancels the q2 in the denominator of the propagator, yielding a term that is the same as would be produced by a term in the action : -i ~ J dux day[-i~}A{x }] [-i.!°(Y)] d" eiq-(x-y) .f(2-g)" I CI1 "1-his argument as given here is little better than hand-waving . The result has been justified by a detailed analysis of Feynman diagrams,3 but the easiest way to treat this problem is by path-integral methods, as discussed in Section 9 .6. 8.6 Feynman Rules far SpinorElectrodynamics 355 The integral over q 0here yields a delta function in time, so this is equivalent to a correction to the interaction Hamiltonian V(t),of the farm 1J d3, J d3yAX, t) j11(Y' t) ~ 47rlx- YJ This is just right to cancel the Coulomb interaction (8 .4.25). Our result is that the photon propagator can be taken effectively as the covariant quantity q2 - iF with the Coulomb interaction dropped from now on . We see that the apparent violation of Lorentz invariance in the instantaneous Coulomb interaction is cancelled by another apparent violation of Lorentz invari- ance, that as noted in Section 5 .9the fields a"(x) are not four-vectors, and therefore have a non-covariant propagator . From a practical point of view, the important point is that in the momentum space Feynman rules, the contribution of an internal photon line is simply given b y -i 4'UV (2tr)a q' - i e and the Coulomb interaction is dropped . 8.6 Fey nman Ru les for Spin or E lectrodynamics(8.5.9) We are now in a position to state the Feynman rules for calculating the S-matrix in quantum electrodynamics . For definiteness, we will consider the electrodynamics of a single species of spin-1particles of charge q=-e and mass m . We will call these fermions electrons, but the same formalism applies to moons and other such particles . The simplest gauge- and Lorentz-invariant Lagrangian for this theory is " The electric current four-vector is then simpl y J~ _c'AP-i e'I`yPT($.6.1) (8.b.2) In Chapter 12 we wil ldiscuss reasons why more complicated leans are excluded from the Lagrangian density . 356 8Electrodynamic s The interaction (8.4.23) in the interaction picture is her e (There is no Vrr,atter here .) As we have seen, the Coulomb term V Co„I(r) just serves to cancel a part ❑f the photon propagator that is non-covariant and local in time . Following the general results ❑f Section 6.3, we can state the momentum space Feynman rules for the connected part of the S-matrix in this theory as follows : (i) Draw all Feynman d iagrams with up to some given number of vertices . The d iagrams consist of electron lines carrying arrows and photon lines without arrows, with the lines joined at vertices, at each of which there is one incoming and one outgoing electron line and one photon line . There is one external line coming into the diagram from below or going upwards out of the diagram for each particle in the initial or fina lstates, respec- tively ; electrons are represented by external lines carrying arrows pointing upwards into or out of the d iagram, while positrons are represented by lines carrying arrows pointing downwards into or out of the diagram . `here are also as many internal lines as are needed to give each verte x the required number of attached lines . Each internal line is labelled with an off-mass-shell four-momentum flowing in a def inite direction along the line (taken conventionally to flow along the direction of the arrow for electron lines . Each exte rnal line is labelled w ith the momentum and spin z-component o rfelicity of the electron or photon in the initial and final states . {ii} Associate factors with the components of the diagram as follows : Vertice s Label each vertex with a four-component Dirac index ix at the electron line with its arrow coming into the vertex, a Dirac index flat the electron line with its arrow going out of the vertex, and a spacetime index 1u at the photon line . For each such vertex, include a facto r where k and k' are the electron four-momenta entering and leaving the vertex, and q is the photon four-momentum entering the vertex (or minus the photon momentum leaving the vertex) . 8.6Feynman Rules for Spinor Electrodynamics 3 57 Exte rnal lines : Label each external line with the three-momentum p and spin z-component or helicity aof the particle in the initial or final state . For each line for an electron in the final state running out of a vertex carrying a Dirac label ~ on this line, include a factar* DP(p,a) (21r)1/2 For each line for a positron in the final state running into a vertex carrying a Dirac label a on this line, include a facto r L'x(p,(-T)(8.6.6)(27r)3/ 2 For each line for an electron in the initial state running into a vertex carrying a Dirac label at on this line, include a facto r (2,g)31' For each line for a positron in the initial state running out of a vertex carrying a Dirac label #❑n this line, include a facto r 00(~, CT)(8.6.8)(2,r)3/' The us and vs are the four-component spinors discussed in Section S .S. For each line for a photon in the final state connected to a vertex carrying a spacetime label pon this line, include a facto r (27r)3/2/i For each line for a photon in the initial state connected to a vertex carrying a spacetime label yon this line, include a facto r (2 n)1/272~jp The eP are the photon polarization four-vectors described in the previous section . Internal lines : For each internal electron line carrying a four momentum kand running from a vertex carrying a Dirac label Pto another vertex carrying a Dira c A matrix 0has been extracted from the interaction in (8 .6.4), so that u and vappear instead of ufi and VT' 358 8Electrodynamic s label x, include a factor (27y)4 k2 +rri2- ie (We are here using the very convenient `Dirac slash' notation ; for any four-vector 0`, ~ denotes For each internal photon line carrying a four-momentum q that runs between two vertices carrying spacetime labels yand v include a factor -i qsiV(8,6.12){27r}4q'--ie (iii) Integrate the product of all these factors over the four-momenta carried by the internal lines, and sum over all Dirac and spacetime indices . (iv) Add up the results obtained in this way from each Feynman diagram . Additional combinatoric factors and fermionic signs may need to be included, as described in parts (v) and (vi) of Section 6.1. The difficulty of evaluating Feynman diagrams increases rapidly with the number of internal lines and vertices, so it is important to have some idea of what numerical factors tend to suppress the contributions of the more complicated diagrams . We shall estimate these numerical factors including not only the factors of the electronic charge e associated with vertices, but also the factors of 2 and itfrom vertices, propagators, and momentum space integrals . Consider a connected Feynman diagram with V vertices, I internal lines, Eexternal lines, and L loops . These quantities are not independent, but are subject to relations already used in Section 6.3: There is a factor e(27r)4 from each vertex, a factor (2 ;r )-4 from each internal line, and a four-dimensional momentum space integral for each loop . The volume element in four-dimensional Euclidean space in terms of a radius parameter K is a E2 n2 dre2, so each loop contributes a factor 7r2 . Thus the diagram will contain a factor e2 L (2Z)4Ve v(27, )T417r2L= (21r)4gE'-2167E2 The number E of external lines is fixed for a given process, so we see that the expansion parameter that governs the suppression of Feynman graphs for each additional loop i s e ~X 16~2 - ~ =5.81x10-~. 8.6Teynmun Rules for Spinor Electrodynamics 359 Fortunately this is small enough that good accuracy can usually be ob- tained from Feynman diagrams with at most a few loops . We must say a little more about the spin states of photons and electrons in realistic experiments, where not every particle in the initial and final states has a definite known helicity or spin z-component . This consid- eration is especially important for photons, which in practice are often characterized by a state of transverse or elliptical polarization rather than helicity . As we saw in the previous section, for photons of helicity ± 1, the polarization vectors are 1/.,/2- e(p,±l) _ R(P)±i 1 0 0 where 2i( p) is the standard rotation that takes the z-axis to the pdirection . These are not the only possible photon states ; in general, a photon state can be a linear combination of helicity states `gy'p,± l ~X+T p,+l+ oc-T pa1 which is properly normalized i f 1,7+12+1~~-12 = 1 .(8.6.13) (8.6.14) To calculate the S-matrix element for absorbing or emitting such a photon, we simply replace e,,,(p,±l )in the Feynman rules wit h eu(P)=x+ ey(p, +1) + U_ el'(p, -1) . (8 .6.15) The polarization vectors for definite helicity satisfy the normalization condition e,t(P,A') e~`(pa and therefore in general(8.6.16) (8.6.17) The two extreme cases are circular polarization, for which o(_ = 0or ac+ = 0,and linear polariza tion, for which 1 ,,c+l =Ja-l = 1/,12- . For linear polarization, by an adjustment of the overall phase of the state {$ .6.13}, we can make x+ and 7_ complex conjugates, so that they can be expressed as cc+ = exp~+iO)/,,,[2_ , (8.6.15) 36(] 8Electrodynamic s Then in the Feynman rules we should use a polarization vecto r cas r~ elt(P)=R(P) sin 0 0(8.6.19) That is, 0 is the azimuthal ang le of the photon polarization in the p lane perpendicular to p. Note that the photon pola rization vector here is real, which is only possible for linear polarization . In between the extremes of circular and linear polarization are the states of elliptic polarization, for which la+l and lac_I are non-zero and unequal . More generally, an initial photon may be prepared in a statistical mixture of spin states . In the most general case, an initial photon may have any number of possible polarization vectors e~')( p), each with probability P,The rate for absorbing such a photon in a given process will then be of the for m where pis the density matrix(8.6.20) (8.6.21) Since p is obviously a Hermitian positive matrix of unit trace (because Y:,.P,=1)with puo=Poi= 0 and p, ,,,p"=PUpe= 0, it may be written as s=1,2 where e ,,(p; s) are the two orthonormal eigenvectors of p wit h ea (p; s)= eu ( P; s)pJ` =0 and A'S are the corresponding eigenvalues, wit h s=1,2 We may then wr itethe rate fo rthe photon absorption process as(8.6.2) (8.6.23) (8.6.24) Thus any statistical mixture of initial photon states is always equivalent to having just two orthonormal polarizations e,(p ;s)with probabilities A, In particular, if we know nothing whatever about the initial photon polarization, then the two probabilities As for the polarization vectors 8.6 F'eynman Rules for Spinor Electrodynamics 361 ev(p;s)are equal, so that ).1= A2 = ~, and the density matrix (and hence the absorption rate) is an average over initial polarization s Pq ( fp (8 .6.25) .c=1,2 Fortunately, this result does not depend on the particular pair of polar- ization vectors ei(p; s) over which we average ; for unpolarized photons we can average the absorption rate over any pair of orthonormal polarization vectors . Similarly, if we make no attempt to measure the polarization of a photon in the final state, then the rate may be calculated by summing over any pair of orthonarmal final photon polarization vectors . The same remarks apply to electrons and positrons ; if (as is usually the case) we make no attempt to prepare an electron or positron so that some spin states are more likely than others, then the rate is to be calculated by a verag ing over any two orthonormal initial spin states, such as those with spin z-component a = ±21; if we make no attempt to measure a final electron's or positron's spin state, then we must sum the rate over any two orthonormal initial spin states, such as those with spin z-component a- ±~ . Such sums may be performed using the relations (5 .5.37) and (5.5.38): -iP+m Cr P x0 or PXfl where e T p + rn . For instance, if the initial state contains an electron with momentum p and spin z-component a, and a positron with momentum p' and spin z-component d', then the S-matrix element for the process will be of the form ( T,(P',a') ,# up(p, a)} . Hence if neither electron nor positron spins are observed, the rate will be proportional t o C"d ITr2pa(_14,+rn) } Techniques for the calculation of such traces are described in the Appendix to this chapter . 362 8Electrodynamic s re f?6-~ f t k, e~., P,CY-~+tk,e Figure 8 .1. The two lowest-order Feynrnan diagrams for Compton scattering . Straight lines are electrons ; wavy lines are photons . 8.7Compton Scatterin g As an example of the methods described in this chapter, we shall consider here the scattering of a photon by an electron (or other particle of spin and charge -e), to lowest order in e.We label the initial and final photon momenta and polarization vectors by V, eMand 0`,e"',where k° =kI and k4f = jkf y. Also, the initial and final electron momenta and spin z-camponents are labelled p ", a and pry, a', where p° = p 2-+m Z and p'()- + rrz2, with m the electron mass . The lowest order Feynman diagrams for this process are shown in Figure S .1. Using the rules outlined in the previous section, the corresponding S-matrix element i s (27T)3/2 (27)3~22k0+(27E)3/2(27r)3 ~2 2 0 +in * J'd4q 2 ic L7r ] ~ IX f ?([e(2i4y~4(q - p' -k{)j[e(2r)4y~4(q- p - k)] [e(2)4y4.q_-)r r f).0 8.7 C omptonScattering 363 Performing the (trivia]) q-integral, collecting factors of i and fin, and rewriting the resu lt in matrix notation, we have more simpl y -iA54(p,+ k' - p - k) ~.1)~"'(-i( j+ + 2n)2 2k~'  2k O (p+k)2 -f- ~~ ~ ( (p -kI)2 + Vn2U(p, a) (8.7.2) (Here means e~y,, not (~)*. Also, we drop the -ie, because the denominators here do not vanish .) Becausep2 =-M2 and k2 = 0= 0, the denominators can b esimplifie d (p+k)-'gym` -2p - k , (8.7.3) Also, the `Feynman amplitude' M is defined in general by Eq . (3.3.2), which (because some scattering is assumed to take place) here read s so 2 ii(P'U') ~*'[ - 4 j + 0) + rn]~lp - kIk-OkO' - ~[ -i(j - g) + m]~*'lp - k' I "(p, cr)(8.7.5) (8.7.6) The differential cross-section is given in terms of M by Eq . (3.4.15), which here reads AT= (2ar)aU-1IM1264(p{ + k' - p - k}d3p'd3V. (8.7.7) Since one of the particles here is massless, Eq . (3.4.17) for the initial velocity gives u=p' ki p°k° . ( 8.7.$) To go further, it will be convenient to adopt a specific coordinate frame . Since electrons in atoms move non-relativistically, the laboratory frame for high-energy (X ray or gamma ray) photon-electron scattering experiments is usually (though not always) one in which the initial electron can be taken to be at rest, We will adopt this frame here, so tha t p=0, p°=rn . The velocity (8.7.8)is then simply(8.7.9) uT1. ( 8.7.10) 364 8Electrodynamic s To save writing, we denote the photon energies b y co' = k C'=I W1=-p-k' /M . (8.7.12) The three-momentum delta function in Eq .(8.7.7)just serves to eliminate the differential dip', setting p' = k -W.This leaves the remaining energy delta functio n Mp'O+0- p° - 0°)=6(/_ k'}2+ m.2 m - cck (8 .7.1 3) This fixes co' to satisf y where fi is the angle between kand V.Squaring both sides and cancelling02 terms gives* f M {o = (0m + c )~ 1 ~- cos 6) = r~'~{B). ~~ .?. t4} The energy delta function ($.7.13) can be writte n ~afcap - 2(9a)' cos 0+ ro'2 + m .2+ cj']/00) , I(a)'-cOcosB )f P'0 +1 1 10f M[[] Also, the differential d3k' can be writte n A' =(0' 2dco'dSZ, ($.7.16) where d Ois the solid angle into which the final photon is scattered . The final delta function in Eq .(8.7.15)just serves to eliminate the differential dco' in Eq .(8.7.16), leaving us with a differential cross-sectio n do-={?z}a1MIz ~00"'dn (8.7.17)mcD with p 'fl=m+co - co', a ndco'given b yEq. (8.7.14). Equivalently, there is an increase in wavelengt h 1 1 1-cose rt~t W M The veri fication of this formula in the scatte ring of X rays by elect rons by A .H. Compton in 1922 --3played a key role in con firming Einstein's 19(] 5proposal o fa quantum of light, wh ich soon after C'n mpton's experiments came to be known as the photon . 8.7Compton Scattering 36 5 Usually we do not measure the spin z-component of the initial or final electron . In such cases, we must sum over a'and average over a,or in other words take half the sum over aand a': da(p+k,e --.).p'+k',er} ~Eda(p,a+k, e --+ p',a'+k',e'). To calc ulate thi s,weusethe standardformula Ua (P, 6)fig(p, a) faxi4M)x~ (8.7.19) ff and likew ise for the s um over c' . Itfollows that for an arbit rary 4x 4 matrix A 5(Pr, u')Au( p,6)12E(R( p', d')Au (p, d))(u{P5cT}flA1fiu(p',cr')) 6d f ~,a' = Tr A(_i~ ~M #Atfl~ ~ M(8.7.20) P p(')} Recalling that flyf,#= Eq .(8.7.6)gives no w e4E ~12 ~(2n)6~~i p0p 'O,QfO(8.7.21) p ~k p k pk p k (Recall again that ~` means e ;y,, not (e,,y")", and likewise for ~ * .)We work in a'gauge' in whic h e,p=e'-p=e'-p=of*-p= 0 (8 .7.22) such as for instance Coulomb gauge in the laboratory frame, where e° = e'O = 0 and p= 0. This implies tha t and likewise for ~r*, ~', and (. Eq. (8.7.21 can therefore be written in 366 the greatly simplified form8Electrodynamic s 4 p1 g, '* CT af kJ~.Y1 ~y4( i~)6 Wli /'Y{~Li7 1 p fY P+Vt a p'k P'V(8.7.23) The trace of any product of an odd number of gamma matrices vanishes, so this breaks up into terms of zeroth and second order in m : 2 =e4 T1 TZ T3 a 'a,64(2g)1ft)w1p1P'1 P T4 lrl2t 1 riYry t 2 Yll2t3 Y1z2 t 4 where Tj Tr T~i T r Tr~ T4 Tr tjTr t4= Tr(8.7.2s) (8.7.2G) (8.7.27) (8.7.28) (8.7.29) (8.7.30) (8.7.31) X8.7.32) The Appendix to this chapter shows how to calculate any trace Tr~00V~...~ as a sum of products of scalar products of the four-vectors a, b, c, d, .... I n general, traces of products of 6or 8 gamma matrices like the tk orTkwould be given by a sum of 15 or 105 terms, respectively, but fortunately here most scalar products vanish ; in addition to Eq .(8.7.22), we also have k -k =k'-k' = 0 . (Furthermore, e  e* =e'- e'* -1.)To simplify the calculation further, let us specialize to the case of linearpolarization, where eP and e+P are real . Dropping the asterisks in Eqs . (8.7.25)-(8 .7.32), we have then T1 = T r ~f g ~i~~~fij 8.7 Compton Scattering 3 67 Since e~'p,, = 0and e~'ef, = 1, we hav e so T,=-Tr I~' gig ~'ij Also, kPk,, =0, so and hence T1 = -2p - k Tr ~ ~f ~ 'i Using Eq . (.A.6),thisis It is convenient to make the substitution s So T~ ~ - 16 p  k(e'k)2 + Sp - k p  k`. (8.7.33) A similar though more lengthy) calc ulationgives T2=?'3 =-8(e  k')2(p ' k) + 16(e. e')2p  k'p -k + 8(e-er)2k-k'pra2 T4 = -lb p -k'(ek')2 +8(P 'k)(p k) (8 .7.35) Com bining all these terms in Eq . {8.7.24} gives 1 =ea8(k.k')' ~~ G4(2~)bc )cc~'p°p'° (k' p)(k{' p)+ 32{e -e'}~ (5.7.35} 6, d r Allthis applies in any Lorentz frame . In the laboratory frame, we have the special results CL] f 1~~ 368 8 Electrodynamic s Combining Eq .(8.7.38) with Eq . (8.7.17),the laboratory frame cross- section is 64ac m vO x + 2+4(e-ef)2] eJ' OJ(8.7.39) This is the celebrated formula derived (using old-fashioned perturbation theory) by O . Klein and Y . Nishina4 in 1929 . As discussed in section $ .f, if the incoming photon is (as usual) not prepared in a state with any particular polarization, then we must average over two orthonormal values of e . This average give s 2 2 and the differential cross-section is the n ~ ~ d a(pau+k,e~p',o+ k,e')64~2rra2cn2 [WI+ ~ - 2(k ' e) 1 . (8.7.40) We see that the scattered photon is preferentially polarized in a direction perpendicular to the incident as well as the final photon direction, i .e., perpendicular to the plane in which the scattering takes place . This is a well-known result, responsible among other things for the polarization of light from eclipsing binary stays ." To calculate the cross-section for experiments in which the final photon polarization is not measured, we must surn Eq .($.7.44) over e', usin g This gives e4 2 f L where 0is the angle between k and W.In the non-relativistic case, w << rn, Thelight from one of the stars is po larized when it is scattered by free electrons in the outer atmosphere of the ot her, coo ler, star w hen both are a long the sa meline of sight . This polarization is normally undetectable beca use it cance ls when the astronomer adds up light from a llparts of the sta r's disk . The polarization has been observed in eclipsing bi nary stars at times w hen t he cooler star blocks the light from jus tone side of the hotter sta r. S.&Generalization p -form Gauge Fields 369 Eq. (8.7.41) gives a da =e ^~~D M2(1 + co S2B) , (8.7.42) The solid angle integral is sinB dB -16[[1+cos2e]dQ=f2df1+cos21,3 giving a total cross-section for cry <m: UT= 6~~ m2(8.7.43) This is often written UT = 8grQ/3, where ra = e2 J4 7cm = 2.818 x 10-13 Cm is known as the c lassica l electron radius .Expression (5 .7.4)is called the Thomson cross-section, after J . J. Thomson, the discoverer of the electron . Eqs .(8.7.42) and (8,7.43) were originally derived using classical mechanics and electrodynamics, by calculating the xeradlatian of light by a non-relativistic point charge in a plane wave electromagnetic field . 8.8 Generalization: p4arm Gauge Field s` The ant isymmetric field strength tensor F ,,Vof electromagnet ism is a special ca se of a g enera lclass of tensor sof speci alimportance in physics and mathematics . A p f orrn is an antisymmetri ccovariant tensor of rank p.From a p-form tY,,Pz,...UPone may construct a (p + 1)-farm called the exterior derivative" dt by taking the derivative and then antisymmetrizing with respect to all indices : JairP2~Z3_A P+1 ~Z2t U1u3_'_,UP+i~...+ (-1)PO PP+1tA1u 2...AP with square brackets indicating antisymmetrization with respect to the in- dices within the brackets . Because derivatives commute, repeated exterior derivatives vanish d(dt) = 0. (8.8.2) A p-form whose exterior derivative vanishes is called closed, while a p- form that is itself an exterior derivative is called exact .From Eq . (8.8.2) ' This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . Exterio rderivatives and p-forms p lay a special role in ge neralrelativity, i npart because the exte riorderivative of a tenso rtransforms like a tensor even thou ghit is calculated using ordinary rather than covariant derivatives .5 370 8Electrodynamic s it follows that any exact p-form is closed ; a famous theorem6 of Foincare states that in a simply connected region, any closed p-form is exacta For instance, the homogeneous Maxwell equations (8.1.16) tell us that the electromagnetic field strength two-form F ,,,, is closed ; Pairtcare's theorem then shows that it is also exact, so that it can be written as an exterior derivative, i .e., as F,,v = apAw - o .A..Again using Eq . ($.8.2), we see that the two-form F,,,, is invariant if A,,is changed by an exterior derivative, i.e., by a gauge transformation 6A,,= The formalism of p-forms and exterior derivatives makes it natural t o consider the possibility of massless particles described by p-form field gauge Melds with an invariance under gauge transformation s 6A== d S2 or in more detail(8.8.3) where is an arbitrary (p -1)-form . From such a p-form gauge field, we can construct a gauge-invariant field strength tensor given b y F = d A or in detail(8.8.4) (Alternatively, we c an start with a (p+ I)-form F, and from an assumed condition dF=0infer the existence of a p-form Awith F- dA .)By analogy with electrodynamics, w e might expect the Lagrangian density for A to take the form 2(p 1) where J is an antisyrnrnetric tensor current (either a c ;-cumber, or a function of fields other than A) that in order to make the action gauge- invariant must satisfy the conservation conditio n 0(8.8.7) t In multiply connected spaces closed forms are not necessarily exact ; although it is possible to write a closed p-form as an exterior derivative locally, this cannot in general be done smoothly throughout the space . The set of closed p-forms, modulo exact p-forms, makes up what is called the p th de Rham cohomology group of the space . There is a deep relation between the de Rham cohorrlplogy groups of a space and its topology, which will be discussed further in Volume li . $ We are speaking loosely in calling Aug-u ,, a p-form, because for F = dA to be a tensor it is only necessary for A to be a tensor up to a gauge transformation . In fact, we have already seen that in four spacelime dimensions, it is not possible to construct afour-veckof field from the creation and annihilation operators of physical massless particles of helicity ±1, so we have to deal with an AP(x) that according to Eq.(8.1,2) transforms as a four-vector only up to a gauge transformation . 8.8Generalization p form Gauge Field s The Euler -Lagrange equations arethen371 (8.8.8) These p-fflrm gauge fields play an important role in theories with more than four spacetime dimensions . For instance, in the simplest string theories in 26 spacetime dimensions, there is a normal mode of the string represented at low energies by a two-form gauge field A.But in four spacetime dimensions, p-forms offer no new possibilities . To see this, note first that in Dspacetime dimensions there are no antisymmetric tensors with more than D indices, so in general we must take p + 1 ~ D . Like any other (p + l)-form with p + 1 ~ D, the field strength F may be expressed in terms of a dual (D - p -I)-form 30F, as (8.5.9) Likewise, the p-form current Jmay be expressed in terms of a dual (D- p)-form current J, as 'ijuF-,.,.~4ft.(8.8.10) The field equation (8.8.8)and conservation condition (8.8.7)then read simply &F=f, df=4 (8.8.11) Because the dual current is closed, it may be written in terms of a (D- p - 1)-form Yas Eqs. ($.8.11)and (8.8.12) tell us that the difference of ~F and 9' is closed, and therefore according to PQincare's theorem, may be written a s F=52 +d o, (8.8.13) with 0a(D- p - 2)-form . There is an exception for the case p = D - 1, where JF and Yare zero-forms, Le,scalars, and the condition d F=cat simply tells us that ~F and Ydifl(r only bya constant . In this case the gauge field describes no degrees )f freedom at all . We may therefore concern ourselves only with the cases p ~ D - 2 . For p :::~D-2, the homogeneous `Maxwell' equations dF - 0 rea d which with Eq. (8.8.13) yields the field equation for 0(8.8.14) (8.8.15) This is invariant under a new set of gauge transformations 0 -+ 0 + dw, except that where D - p - 2 = 0 the gauge transformation that leaves F 372 8Electrodynamic s invariant is 0--o-0+ c, where c is an arbitrary constant .Wesee that inD spacetime dimensions, thetheory of a p -form gauge field A with current J isequivalent to the theory oJ a (D - P - 2)-form ga ugefield 0, wit h curren t We can now understand why p-form gauge fields offer no new possibili- ties in four spacetime dimensions . As we have seen, we need only consider the cases p ~ D - 2, or p = Q, 1, or 2 . A zero-form gauge field is a scalar 5, for which Eq . (8.8.5)reads F,, = 0,, 5, and the field equations ($ .8.5) read simply OS = -J . The gauge invariance here is invariance under a shift S--~S+c, with c a constant . This is just the theory of a massless scalar field with only derivative interactions . A one-form gauge field is a four-vector AI(x) coupled to a conserved four-vector current, just as in electrodynamics . Finally, according to the general result quoted above, a two-form gauge field in four spacetime dimensions is equivalent to a zero-form gauge field, which as we have seen is equivalent to a massless derivatively coupled scalar field . Appendix T races In calculating S-matrix elements and transition rates for processes in- volving particles of spin 1, we often encounter traces of products of Dirac gamma matrices . It will therefore be useful to give formulas for these traces that can be used in all such calculations . For products of even numbers of gamma matrices, the trace is given by TrfY, E Yea... yuax 1—:4E 6p11 17 pairedMs  (8.A.1) pairings pai rs Here the sum is over all different ways of pairing the indices { .i1, -  - {~ 2iv. A pairing can be regarded as a permutation of the integers 1, 2,-   2N into some order P1, P2,- - P(2N) , in which we pair upr with µ P2,MP3 With Yp4, and so on . Permuting pairs or permuting ps within a pair yields the same pairing, so the number of different pairings i s (2N) ?/N!2N_ (211- 1 )(2N- 3)... 1 = (2N ._1 ) ! !, (8 .A.2) We can avoid summing over equivalent pairings by requiring tha t P1<P2,P3<P4,... ,P (2N - 1)< P - {2N} (8 .A.3) and P1 < P3<PS<... (8 .A,4) With this convention, the factor bpis +1 or -1 according to whether the pairing involves an even or odd permutation of indices . The product in Eq . Appendix Traces 37 3 (8.4.1) is over all N pairs, the nth pair contributing a factorq0P(2n_I) Pp,(2n ) For instance (writing y, v,p,ff,... in place of Pl, µ2> P3, [4, ...), for N= 1, 2, and 3 we have ` Tr{Y,,Y,} = 4q'Uu Tr jy,,y,,ypy,, I =4[qp, qpa- - q'UPq"C' + qyqqP] I(S.A.S) (8.A.6) Tr {YpY,,TpyaYk?'n~=4 q'VV11pa111Cq- q,V07pk qaj+ q'UuqPqq6K + 11PF~~1'Kq(Tq - q fUptj4nfjrrx + i7Kq wPtjx~Tqpdj ~wqPq +~ P6MVO pK-q'Urc1Vpqaq+ *+P11'7V ff1fPTC + ' t'Uq'fV1C'fPQJ.q'aK~V':Mpq-~'UKq V1qPv+qPn'7vAaK (S.A.7) For an odd number of gamma matrices, the result is much simple r TrIppi"lug... y92N+I I_0 (8.A.8) The proof of Eq . (8.A.1) is by ma thematical in ductio n. First note tha t so'fi'r~ 7,,YV l = 4q,,,, in agreement with Eq . (8.A.1).Next , suppose that Eq. (8.A.1)istrue for N ~M- 1. We then hav e Tr{yA,Y~U2... 7~12M 2quIPzTrfY,,3... yu2 .'WI-Tr U.2YA,YP3... yYzM 2'1ul jja Tr~7,,,,...ypa,►f~ 2'7u~~j.jTrfY,U2Yea... yPM ~Trt7427 u3YuiYN... i42 M 2171UIP2 Tr~,/,,3... ypamI-2ql,,p3 Tr~i'uz Yu¢... 7~i2M +2q,,,,a Tr l1.2Y113Yes '- -Yu2u } -... +2?IPiMz.uTr{Y1,z... yPanr-iI-Tr~7,12.. yAzM YuI All commutators have zero trace, so the last term subtracted here is the same as the left-hand side, and s o Tr~^~~ IYM... f'MM q p1~z Tr{Yu~... yp2m If we assume that Eq .(8.A.1) correctly gives the trace of any product of2N- 2 Dirac matrices, then Eq .(8.A.9)shows that Eq .($.A.1)also correctly gives the trace of any product of 2 NDirac matrices . The easiest way to see that the trace of an odd number of Dirac matrices ❑anishes is to note that -y.is related to y ,,bya similarity transformation, -t,, = ysy ,(ys)yl. Traces are unaffected by such similarit y There are now computer programs? available for the calculation of traces of products of large numbers of Dirac matrices . 374 8Electrodynamic s transformations, so the trace of an odd number of Dirac matrices is equal to minus itself, and hence vanishes . One occasionally encounters another class of traces, of the for m TrIYsYjt, 7~42... y1jn} This van ishes for odd n for the same reason as given above for traces without a y 5.Italso vanishes for n -0 and n=2: fi't' i fj51=0 (8 .A.10) (To see this just recall that 's = iYoY17273, and note that there is no way of pairing the indices in Tr 17o,'MY3; or in Tr {yoT11/2fey°pYv ; so that the spacetime indices in each pair are equal .) For n = 4 it is possible to pair the indices in Try^1or'rY2Y3ToYY7P"1di so that the spacetime indices in each pair are equal, but only if y, v,p,aare some permutation of 0, 1, 2, 3 . Furthermore this trace must be odd under permutations of p, v, p, a since gamma matrices with different indices anticommute . Thus the trace Tr ~Y5Y,ayvY .Yd } must be proportional to the totally antisymmetric tensor 61tvf,cr . The constant of proportionality may be worked out by letting ,u, v, p, a take the values 0, 1, 2, 3, and recalling that 6o123=-1, In this way we find Tr ~ ~~s ~'{,i'}Yp ~' .J = 4ic~V pj. (8 .A.12) The trace of products of y 5with six, eight, or more Dirac matrices may be calculated by the same methods used above to verify Eq .(8.A.1), Problems 1. Calculate the differential and total cross-sections for the process e+e - -_?.eµ- to lowest order in e . Assume that electron and muffin spins are not observed . Use the simplest Lagrangian for the electrodynamics of electrons and moans . 2. Carry out the canonical quantization of the theory of a charged scalar field 0and its interaction with electromagnetism, with Lagrangian density : where D~d~ d~ -igA~d} , F~V= r'PA4,- r~VAY - Use Coulomb gauge . Express the Hamiltonian in terms of the fields A,0, and Vand their canonical conjugates . Evaluate the interaction Rafarences 37 S V(t)in the interaction-picture in terms of the interaction picture fields and their derivatives . 3. Use the results of Problem 2 to calculate the differential and total cross-sections for photon scattering by a massive charged scalar particle to lowest order in e . 4. Write a gauge-invariant Lagrangian for a charged massive vector field interacting with the electromagnetic field . 5. Calculate the differential cross-section for electron-electron scattering to lowest order in e . Assume that final and initial spins are not measured . References 1. See, e .g., M . B. Green, J . H. Schwarz, and E . Witten, Superstring Theory (Cambridge University Press, Cambridge, 1987) :Section 2 .2. 1a. V. Fock, Z. f PhyS. 39, 226 (1927) ; H. Weyl, Z. Phys .56, 33 0 (1929) . The term `gauge invariance' derives from an analogy wit h earlier speculations about scale invariance by H . Weyl, in Raum , Veit, Materie, 3rd ed . Springer-Verlag, Ber lin, 1920) . Also see F . London, Z.f.Phy°s.42, 375 (1927) . This history has been reviewe d by C . N. Yang, talk at City College (unpublished) . 2. The use of Coulomb gauge in electrodynamics was strongly advo - cated by Schwinger on pretty much the same grounds as here : tha t we ought not to introduce photons with helicities other than ±1 . See J . Schwinger, Phys..Rev. 78, 1439 (1948) ;127, 324 (1962) ; Nuov o Cime nto 30, 278 (1963) . 3. R. F. Feynman, Phyts.Rev. lUl, 769 (1949) : Sect ion 8 . 4. 0. Klein and Y . Nishina, Z. f: Phys. 52, 853 (1929) ; Y. Nishina, shad ., 869 (1929) ; also see I . Tamm, Z../: Phys. 62, 545 (1930) . 5. See, e .g., S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972) : Section 4 .11. 6. For a readable general introduction to the geometry and topology of p-farms, see H . Flanders, Differential Forms (Academic P ress, New York, 1963) . 7. T. West, Comp ut. Pays .Commun .77, 286 (1993) . Path-Integral Method s In Chapters 7 and 8 we applied the canonical quantization operator for- malism to derive the Feynman rules for a variety of theories . In many cases, such as the scalar field with derivative coupling or the vector field with zero or non-zero mass, the procedure though straightforward was rather awkward . The interaction Hamiltonian tumed out to contain a covariant term, equal to the negative of the interaction term in the La- grangian, plus aeon-covariant term, which served to cancel non-covariant terms in the propagator . I n the case of electrodynamics this non-covariant term (the Coulomb energy) turned out to be not even spatially local, though it is local in time . Yet the final results are quite simple : the Feyn- tnan rules are just those we should obtain with covariant propagators, and using the negative of the interaction term in the Lagrangian to calculate vertex contributions . The awkwardness in obtaining these simple results, which was bad enough for the theories considered in Chapters 7 and 8, becomes unbearable for more complicated theories, like the non-Abelian gauge theories to be discussed in volume II, and also general relativity . One would very much prefer a method of calculation that goes directly from the Lagrangian to the Feynman rules in their final, Lorentz-covariant form . Fortunately, such a method does exist . It is provided by the path- integral approach to quantum mechanics . This was first presented in the context of non-relativistic quantum mechanics in Feynman's Princeton Ph . D. thesis,1as a means of working directly with a Lagrangian rather than a Hamiltonian . In this respect, it was inspired by earlier work of Dirac .' The path-integral approach played a part (along with inspired guesswork) in Feynman's later derivation of his diagrammatic rules . However, although Feynman diagrams became widely used in the 1950s, most physicists (including myself) tended to derive them using the operator methods of Schwinger and Tomonaga, which were shown by Dyson in 1949 to lead to the same diagrammatic rules that had been obtained by Feynman by his own methods . The path-integral approach was revived in the late 1960s, when Faddee v 376 Path-integral M ethods 37 7 and Popov 4 and De Witt5 showed how to apply it to non-Abelian gauge theories and general relativity . For most theorists, the turning point came in 1971, when 't Hooft6 used path-integral methods to derive the Feynman rules for spontaneously broken gauge theories (discussed in Volume iI ), including in particular the theory of weak and electromagnetic interactions, in a gauge that made the high energy behavior of these theories transparent . Soon after, as also discussed in Volume II, it was discovered that the path-integral method allows us to take account of contributions to the S'-matrix that have an essential singularity at zero coupling constant and therefore cannot bediscovered in any finite order of perturbation theory . Since then, the path-integral methods described here have become an indispensable part of the equipment of all physicists who make use of quantum field theory . At this point the reader may bewondering why if the path-integral method is so convenient we bothered in chapter 7 to introduce the canonical formalism . Indeed, Feynman seems at first to have thought of his path-integral approach as a substitute for the ordinary canonical formulation of quantum mechanics . There are two reasons for starting with the canonical formalism . The first is a point of principle : although the path-integral formalism provides us with manifestly Lorentz-invariant diagrammatic rules, it does not make clear why the 5-matrix calculated in this way is unitary . As far as I know, the only way to show that the bath-integral formalism yields a unitary S-matrix is to use it to reconstruct the canonical formalism, in which unitarity is obvious . There is a kind of conservation of trouble here ; we can use the canonical approach, in which unitarity is obvious and Lorentz invariance obscure, or the path- integral approach, which is manifestly Lorentz-invariant but far from manifestly unitary . Since the path-integral approach is here derived from the canonical approach, we know that the two approaches yield the same S-matrix, so that the S-matrix must indeed beboth Lorentz-invariant and unitary . The second reason for introducing the canonical formalism first is more practical : there are important theories in which the simplest ver- sion of the Feynmari path-integral method, in which propagators and interaction vertices are taken directly from the Lagrangian, is simply wrong . One example is the non-linear Q-model, with Lagrangian density_T = -1gk~~r~ )~,u0'0'* . In such theories, using the naive Feynman rules derived directly from the Lagrangian density would yield an S-matrix that is not only wrong but even non-unitary, and that also depends on the way in which we define the scalar field .7 In this chapter we shall derive the path-integral formalism from the canonical formalism, and in this way we will see what additional sorts of vertices are needed to supplement the simplest version of the Feynman path-integral method . 3 78 9 Path-integral Method s 9.1 The General Path-Integral Formul a We start with a general quantum mechanical system, with Hermitian operator `coordinates' Q,, and conjugate `momenta' Pb, satisfying the canonical commutation relations * [C3a,Pb] =i ~ab , ~~~~QbJ = [Pa, PbJ = 0. (9 .1.2) (We shall restrict ourselves in this and the next three sections to bosonic operators, which satisfy commutation rather than anticommutation rela- tions . Our results will be generalized to include fermianic operators in Section 9 .5.) In a field theory, the index a consists of a position x and a discrete Lorentz and species index m, and we conventionally writ e P"',= Pm W (9.1.4) Also, the Kronecker delta in Eq .(9.1.1)is interpreted in a field theory a s 6x.mY,n~_63(X- Y)6,,►,- (9.1.5) However, for the present it will be convenient to use the more compact notation of Eqs .(9.1.1) and (9 .1.2). These are `Schrodinger-picture' oper- ators, taken at a fixed time (say, t =0). The time-dependent operators in the Heisenberg picture will be considered a little later . Since the Qa all commute, we can find a simultaneous eigenstate 1q}, with elgenvalues q,,: (We are using lower case qs and ps here to denote eigenvalues rather than to denote operators in the interaction picture as in Chapter 7, but since we will not be using the interaction picture in this chapter no confusion should arise .) The eigenvectors can be taken to beorthonormal , so that the completeness relation read s t=JHdqa ~q~ ~ql 11(9.1.7) (9.1.8) We are tacitly assuming here that any first class constraints are eliminated by choosing a gauge, and any remaining second class constraints are `salved" by writing the constrained degrees of freedom in terms of the unconstrained Q ,,and P,,, as in Section 7 .5. The direct application c7f path-iniegral methods lo constrained systems is described by Faddecv .9 9.1TheGeneral Path-I ntegra l Formula 379 Similarly, we can find a complete oxthonormal set of eigenstates of the PU= palP~= FaP) u(9.1.9) (9,1,10 ) (9.1.11) As usual, it follows from Eq. (9.1.1)that these two complete sets of eigenstates have the scalar product" (.l.12) In the Heisenberg picture, the Q and P operators are given a time- dependence Q,,(t)=exp(iHt) Q,,exp(-aHt ) P,(t) exp(tHt)F,,exp{ -iHt} ,(9.1.13) (9.1.14) where H is the total Hamiltonian . These have eigenstates jq ; t~ and gyp ;t} Pa{t)IP;0=Pal~; 031 given by 1q; t~= exp(ifi t)lq~ Ip; t} = e xp(ixt)Ip)(9.1.15) (9.1.16) (Note that ~q ;t)is the eigenstate of Qa(t) with eigenvalue q,and not the result of letting the state 1q)evolve for a time t . This is why its time- dependence is given by a factor exp(M) rather than exp(--tHt)-) These states obviously satisfy the completeness and orthonormality condition s ~~qc,I4; t~ ~q;tI -~,(9.1.19) (9.1.20) (9.1.21) (9.1.22) The proof' l'ctillows the same Iines as in tic quantum mechanics of point part icles. From Eq . (9.1. I we gee that P1, a,cic as -ie'je'gy, on wave functions in a q-basis . The right-hand s ide of Eq . (9,1 .121 is then seen to be the w&ve function i nthis basis of an eigens iate of P . The factor ~1 f ~+'2rr is fixedby the norma li7,atinn requirement, Eq . (9.1.10}. 380 and also9 Path-Integral Method s (q; t1P}t~=11 1 exp(iq aPa) a ~~(9.1.23) If, by measurements at time t, we find that our system is in a definite state 1q; t}, then the probability amplitude for measurements at time t' to give a state Iq'; t`} is the scalar product (q'; tflq; t). Our central dynamical problem is to calculate this scalar product . This is easy when t' and t are infinitesimally close, say t' = z + dz and t =,r . Using Eq .(9.1.17), we hav e The Hamiltonian H is given as a function H (,P), but since (9 .1.13) and (9.1. 14) are similarity transformations, and H commutes with itself, it can equally well be written as the same function of Q(t) and P(t ) H=_H(Q,P)=eW'H(Q, p)e-iHt=--H (Q(t), P (t)) . (9 .1.25) This function can be written in various different forms, with different constant coefficients, by using the commutation relations (9.1.1)and (9 .1.2) to move the Qs and Ps past each other . It will beconvenient to adopt a standard form, in which all Qs appear to the leftof all Ps . For instance, given a term in the Hamiltonian of form PaQbP,,we would rewrite it as P aQbPc= Qb P~Pe- i rS abP,. With this convention, the Qa(t)s in the Hamiltonian in Eq . (9. Z.24) may be replacedl with their eigenvalues q.'. To deal with the P(t), we use Eq.(9.1.23) to expand ~q ;-c} in P-eigenstates Ip;-r), and fin d (q';z+dTIqa-r~ { q';-r I exp ~ - ilY(Q(T), P(r )}dT)IPA T) a with each pa integrated from -00 to+00. Now let's return to the more general case of a finite time-interval . To calculate ~ q'; tr 14 > t), with t <t', we break up the time-interval from t to t' into steps t, T1, T2, ...TN,t', with (9.t.27) ~ This is only possible because with dz infinitcsirnal, exp(- iH dr)is linear in H . 9.1The General Guth-I ntegra l Formul a and sum over a complete set of states Jq ; Tk} at each time Tk:381 (qr; tfl q;~~ = Jdql...dqN1qr- trlqN; -cv~ (qN>rNI qlV-1r TN-1~...~qli-riIq3t). (9.1.28) Inserting Eq . {9.1.2}, this become s (q'; (I q; t)=ILII II udqJ[,A ~ ~ dYIC,Q/L7r -~ a TV+i X exp d (qk, a-qk-1,a)Pk-1,a -H(qka Pk-1 )d'r k=1 a 11 where(9.1.2) (9.1.30) Our result, Eq . (9.1.29), can be put in a much more elega nt fora. Define smooth interpolating functions, q(z) and p(r), such tha t qd(rk) = qk.a Pa(rk) = P k,a  (9.1.31) In the limit dz -~, 0 ( i .e.,N--+ oa), the argument of the exponen tialin Eq. (9.1.29) becomes just an integra lover x N+J 1:1 (qk ,a-qk-1,a)Pk- La -H(qk ,Pk-1)d k=1 N+t E4a(Tk)Va(Tk)-H(q(Tk),P(rk)) d r + 0(dT2) k~l a t' E4a(T)Pa(0-H(q(r),P(T)) dr. a Further, wemay define integrals over thefunctions q(r),p(T) by 27r d~~U ~C}Q k,b, C ~ ~,p r,b 27t Eq.(9.1.29) then becomes a constrained path integra l (q';t'lq; fldqa(T) fldPb(T) f,a i, b271 x exp t d~ 4a(T)Pa(z)-H( 9~(r),P(~)) (9.1.34) '~ This is called a path integra l, because we integrate over all paths that take q(T) from q at T = t to q' at r = t', as well as over all p (r). The 382 9Path-Integral Method s great advantage of writ ing matrix elements in this way is that ,as shown insection 9.3, the path integral sare easy to calculate when expanded in powers of the coupling constants in H. The path-integral formal ismallows us to calculate not only transition probability amplitudes like (q';t'jq;t),but also the matrix elements be- tween states { q';t'land 1q,t)oftime-ordered products of general operators C{P(t),Q(t)).Itwill be convenient to define these operators with (unlike H)allPs.moved to the d eft and all Qs to the right .Then b yinserting any such operator e:'[P(t), Q( t)]inEq.(9.1.26), we have ~q';,r+d-ciC(P (t), Q(t)) ~q;,r) ffj dp, X4;-[1eXp(_iH(Q('r),P(,r))dr ) 2n,,exp[-_iH(P)dr + ~ ~ (~~ ~ q,,)P,,C(F,q). (9.1.35) In order to calculate the matrix element of a product CAPOA Q01)) G` ll (P .t), Q(tB)) ...of operators with tA t J3>...a eve can insert the L--operators between the appropriate states in Eq .(9.1.2$), and use Eq .(9.1.35).For instance, if the time tA falls between -rkand zk+l, then insert (-rA(P(t,,), Q(r,)) between {q k+r;4+11and lqk;Tk). Note that inEq. (9.1.28) each successive sum over states is at a later time, so this is only possible because of our assumption that tA >'tfl>.... Following the same steps as before, we now find the general path-integral formul a (q', 60A (P(tA, Q(W) 6"'-B (P (W, Q(tB)) ...1q, t) 11 dq .(,r)IT'P' (-0 6111A (POA qh) OB (P(tB), q (tB)) - - - T,a T' h fir:(1f )= Ra l~ x exp i r d1c 4~(-r)P~(-r) -H(q(T), P(T) (9.1.36) This resu lt is only valid if the times are ordered, wit h However, nothing on the right-hand side of Eq.(9.1.36)refers to the order of time-arguments . Hence if we are presented with a path rtntegra] like the right-hand side of Eq. (9.1.36), with t A, t13,... in arbitrary order (all between t and t', with t <t'), then this path integral will equal a matrix element like the left-hand side of Eq .(9.1.36), but with the operators arranged in order (from left to right) of decreasing time . That is, for 9.1The General Path-Integral Form ula tA, tB,... in arbitrary order, we hav e (q', t'l T (6A(P(tA),Q(tA)), 611(P(tB),Q(tB)),'- I 1q,t ) rldq,(T) 11 'P' CA (AW, q(W) OR (P(tB), q(tB)) ... 2a qu(tr)-qu383 r.' xexp i dr ~Ia(`~)I~~(z) - H(q(Thpr)) (9. 1.38) .~ where T denotes the usual time-ordered product . It should perhaps be stressed that the c-number functions qa{r), pa(y) in Eq .(9.1.38) are mere variables of integration, and in particular are not constrained to obey the equations of motion of classical Harniltonian dynamics jqa(T)(9. x.39) (9.1.40) (For this reason, the Hamiltonian aH(q (z),P(T)) in Eq . (9.1,35)isnotcon- stant in T .) Nevertheless, there is a limited sense in which path integrals do respect these equations of motion . Suppose that one of the functions in Eq .(9.1.38), say G' A(pt.41, R(tA)), happens to be the left-hand side of either Eq .(9.1.39) or Eq . (9.1,40) . We note that (for t <tA t') aPa (W aH(R(tA), P(tA)) Oqa(Wexp (iI[qp]) T i exp (u[q,p]), 6p"( -tA) exp (iiqp]) = i~ Cxp (i1[qp]),qa(tA) where tI is the argument of the exponential in Eq .(9.1.38): I [q, p] d-r (-c)p.(-r)-H (q(z), p(T)) As long as t Adoes not approach t or t', the integrations over q ,,(tA)and P,(t,4)are unconstrained, and so with reasonable assumptions about the convergence ❑f these integrals, the integral of such variational derivatives must vanish . Hence the path integral (9.1.38) vanishes if CA(P, q) is taken to be the left-hand side of either of the equations of motion (9 .1.39) or (9.1.40). This simple rule applies only if the integration variables qa(W, P,(tA) are independent of any of the variables q~ (W, P,,(ta), etc . appearing in any of the other functions 61,R, Cc, etc . in Eq .(9.1.38), and hence 384 9 Path-Integral Method s only if we prohibit tA from approaching tB,tc, etc . as well as t or tf . When to approaches, say, t$, the path integral will be found to involve a non-zero term proportional to MtA-tB} or its derivatives . These delta functions are the same as would be found in the operator formalism from time derivatives of the step functions implicit in the definition of the time-ordered product . In evaluating the path integrals (9 .1.34) and (9.1.38), we only need to know the classical Hamiltonian, the c-number function H(q,p) .If we were to define a theory by the path integrals, the question would naturally arise, which of many possible quantum mechanical Hamiltonians H( Q, P) (differing is the order of Qs and Ps) governed the quantum theory that corresponds to these path integrals . Our derivation has provided an answer : the quantum Hamiltonian is to be taken with all Qs on the left, all Ps on the right . But it would be a mistake to give this prescription too much significance . There are a great many ways of interpreting the measure fldqa(T)II dp h(z) appearing in path integrals like (9 .1.34) or(9.1.38).Our prescription, of putting all Qs to the left of all Ps, is appropriate only if the measure is interpreted according to Eqs .(91.31)- (9.1,33) .Other measures would lead to other prescriptions for operator ordering . The question is not an urgent one, because different prescriptions for ordering operators in the Hamiltonian just correspond to different choices of the constants that appear as coefficients of the various terms in the Hamiltonian, and we generally formulate theories with these constants left as arbitrary parameters anyway . It is difficult to use the general path integral in Eq .(9.1.38) for numerical calculations or as a source of rigorous theorems . For these purposes it is better to use the path-integral method to calculate amplitudes in Euclidean space, where t is replaced with an imaginary quantity -ix4, and the argument of the exponential in Eq .(9.1.38) is a negative real quantity . In this way, instead of jagged paths producing rapid oscillations of the integrand from one path to another, all jagged paths are exponentially suppressed . Though we shall not go into it here, quantum field theory may be formulated from the beginning in terms of Feynman amplitudes in Euclidean spacetime .84 Under certain plausible assumptions, it is possible to reconstruct the Feynman amplitudes in Minkowskian spacetime from their Euclidean counterparts,8bBut we may as well stick to the Minkowski space formulation of the path integral if we are only going to use it to calculate Feynman amplitudes in perturbation theory . 9.2Transition to the S-Matri x 9.2 Tran sition to th e S-Matrix385 As already mentioned, we can easily convert the general quantum me- chanical results of section 9 .1 to a notation appropriate to quantum field theory, by letting the index a run over points x in space and over a spin-and-species index in,and replacing Q,(t) and fa(t) with Q,,(x,t)and P,,(x, t), respectively . Eq.(9-1.38) then reads * T (CA [P 4A OtA , &B [P (tB), Q0 01 1 q, t) dp,(xf-r) (X~ q .(x) q,"(x, q'(x)M1dq mt x} ~) 11 2 1rz,x,m r,x'm ]C CAA [is.tA,q .tA )J 6, D[p(tB)q(tB)]... c' x exp i dr dux q~(x, ~)Pm(x, t )it(9.2.1) -H[q(,r), P(,O] 11. However, in field theory Eq . (9.11) is not exactly what we want . Exper- imentalists do not measure probability amplitudes for transitions between eigenstates ~q', tf land 1q, r~ of the quantum field Q, but rather S-matrix elements, the probability amplitudes for transitions between states that at t-} -oo or t --4 . +ca contain definite numbers of particles of various types . These are called `in' and `out' states, lac, in) and fl,out), where oc and fl denote sets of particles characterized by the various particles' momenta, spin a-component (or helicity), and species . To calculate a matrix element of a time-ordered product (perhaps empty) between such states, we need to multiply Eq.(9.2.1) by the `wive functions" (fl,autlq', t'~ and ~q, tf a, in} at any fixed times t and t', taken for convenience here to be -oo and +oc, respectively, and then perform an integral over the `arguments' qm(x) and q ;,( ) of these wave functions . But instead of constraining the path integral over q,,(x,,r) bythe condition s and then integrating over q'(x) and q~(x), we can just as well do an unconstrained integral over q,,(x, T) (and also over p,,(x, T)), and set the arguments of the wave functions equal to the values given by Eq. {9.2.2} We are now writing H and the Os with square brackets, to remind us that H[q(t ).p(t)] and 6'[p(t), 4(t}] are fu vtivMais of q,,(x, i) and p,(x, t) at a fixed time f. 386 9Path-integral method s (flsout~T{&'.4[P(tA) ~~(W], G$ [P 00, Q(t$ )],...I X}ink f dq m(~,,c) 11 (dp m(x, -r)/2n ) i,x,m a,x,m *CA [P(tA ), q (tA )] C"B [At B ), q(tB N k x {ft, nut~q(+oo );+o~} (q(-cam) ; -aa lx, in} . (9.2.3) Incidentally, this result leads immediately** to Eq . (6.4.3), a theorem that we use repeatedly to relate sums of off-shell Feynman graphs to matrix elements of Heisenberg-picture operators between exact energy eigenstates . It is necessary now to consider how to calculate the wave functions appearing as the final pair of factors in Eq .(9.2.3). Let's first consider the simplest and most important case, the vacuum . (We saw in Section 6 .4 that S-matrix elements may beeasily calculated from the vacuum expectation values of time-ordered products .) We assume as usual that for t ---~±oc, matrix elements may be calculated as if there were no interactions . The `in' and `out' vacua may thus be defined by the conditions .2.4} Qo„c(P, a,r~) k~TAC, out =(] (9 where a ;n and aa,,r are the operators appearing in the coefficients of exp(ip -x - i Et} in the plane-wave expansion of the operator Q,,,(x, t) at t-~,-oc and t -~+oo, respectively . For instance, for the real scalar field of a neutral spinless particle, we have in effec t r~+raacb(x,t) -~ (27c)-3/2d3p (2E)-il' Iuo~A(P)~~~ _x +H.c.] (9.2.5) Iiis only necessary to vo le that, for a Hamilton ianH[Pft}, Q(t}] +1 : f d3x EA (x,t)CA(x,t), the S-ma tri)L i~i g iven by E y. (9,2 .3) a s outla, In), = j 11 dq.(x, r) 11 (dp .(x. T j / 2 2z) r,x,m r,x ,m ~ x exp i fdx x, p(xt - H [q(tLr() ] f d3X C-AX, 00AW, 0 A x ou Lq (+(f-!) 5 ~,Y-)F(q -oo Iayifl) The left-ha nd side of Eq- (6-4 .3) is the derivative of this expression with respect to Fa, eb„ etc ., at e=0, which yie lds the right-hand s ide of Eq . (9.2.3), and us ing Eq . (9,2 .3) again then immediately give5 the right-hand side c) I` F.q. (6.4.3). 9.2Transition to the S-Matrix . ~1(271)-3/2 d3 F(E/2)112 ~aOut(P)erp~h -~I.c~I387 (9.2.6) where p o = E- p +W,and we h ere use the conventional q) and 1-1 rather than Q and P for a scalar field , and drop the unnecessary labels m,Q, n.Inverting the Fourier transforms and taking a linear combination of the result ing express ions, we hav e e'Et Oul (27r)3/2 02 rM(9.2.7) As mentioned in Section 9 .1, the 'momentum' II(x, t) acts on wave- functions in a 0-basis as the variational derivative -ib/jO(x, t), so in this basis the conditions (9 .2.4) rea d o =Jd3x ~~'"'6+E(P)O(x)] (O(x) ; +ocvac, ~us .(9,~..s) [50(x ) The analogous ordinary d ifferential equation has a well-known Gaussia n solution, so let's try a Gaussian ansatz here : (O(X)a +a ;IVA, ~~t 4~' exp - ~ d~xdlY r'(x7 Y)O(x)O(Y) (9.2.9) with kernel and constant 1V to be determined . Substituting this in Eq. (9.2.8), we see that the functional differential equation for the vacuum wave functional is satisfied if for all 0 0=fdux e -dP'x d3y'(,6,(x, y)~(y) - E ( P)O(x) or, in other words, if d'xe-gyp x(qxsy) = E(P)8-*y, The solution is easily found by inverting the Fourier transform(9.2.11) (9.112) (Recall that E(p) = pZ_+m2) .This is actually the most useful repr esen- tation forthekerne l ', but wemay note inpassing that forx =~- y,4'may also be wr itten in terms of a Hankel function of negative orde r M d ( ' &(X} Y) -2~ r dr r(!K_i(i .nr)) (9.2.13)z 388 9Path-I ntegra l Method s wherer ~ I x-yl. The constant ~V'in Eq .(9.2.9) may be formally ob tained from the normalization condition forthe vacuum state ,but w ewill not need this result . According to Eq .(9.2.9), in calculating vacuum expectation values in the theory of a scalar field ,the product of t helast two factors in Eq . (9.2.3) is (VAC, outs ( ) ; +oa} (4(-' .-i5);-r-GIVAC, in } [O(X, +OC)O(Y, +00 ) +O(X~ _COO(Y' _C)0)D (9 .2,14) = 1'4~- 12 explef d'xd3Ydr9(x, y)o(x, T)O(y,,r)e-1jTj(_ 2 f 7or- ) 11 Where e is a positive infinitesimal . To obtain the final expression, we have used the fact that for any reasonably smooth function f (T) , 00 d-cf ( r)e-fITI, (9.2.15) ,f(+oc} +f(-co) - Ali m e1_)C 0+ f- Inserting Eq . (9.2,I4} in Eq . (9,2 .3) now give s KVAC, aut IT(CA[r1t4itAJ .COB [II0B),OOR }J,-I+VAC,in) IA-11 j fj dO(x, -r) fj(d'g(x, T)/27E)6A [70A O(W ] T,x z, x XCB[m(tB) , 0( tB)].. eXpi f d ZIf d3JC ~( Xr_r),g( x' T) ---H [O(T)} 7Z(T)] + ~ i 6r~3x day c (xa Y)e-67 O(X, z~~{Y~ z} (9 .2.1 6) We shall see in Section 9 .4 that the whole effect of the last term in the argument of the exponential in Eq . (9.2.16) is to provide the -ie in the denominator of the scalar field propagator in momentum space, [p2+ m2 _ ie]-1. We will not go into the corresponding details for fields of general spin, but will simply state that in genera l VAC, out~T iID.4[FA{tAb Q[W],C'B [PB (tH)rQ(tB)] ,...~ ~VAC,in ~P►~ ~, ~ )1'+'-'i1 ri dq,,(x,,r)j [ 1127rCA [POA q(tA)l z.x,m T, x,m c~a XOB [P(tB), q(tB)j exp i d-c d3XE 4 .(X"r)PM(X' -0 M -H [q(r .), p(r)] + ic terms (9 .2.17) 9.3Lagrangian V ersion of tie bath- Integra l Formula 389 where the `ic terms' just have the effect of putting the correct -ie in the denominators of all propagators . This is a good place to mention that field-independent factors in Eq.(9.2.17), like the constant 1~/V12, are not important . This is because such factors contribute also to the matrix element (VAC, outs AC, in) . In calculating the connected part of vacuum expectation values of time- ordered products (or the S-matrix) we eliminate the contribution of discon- nected vacuum fluctuation subgraphs by dividing by (VAC, outIAC, in), and any constant factors in the vacuum expectation values cancel in this ratio . We could go on and calculate matrix elements between multi-particle states, b yinserting the appropriate `wave functionals' in Eq .(9.2.3).These can be calculated by applying the adjoints of annihilation operators such as (9 .2.7 to the vacuum state ; just as for the harmonic oscilla- tor, these wave functionals turn out to be Hermite polynomials in the field times the vacuum Gaussian . We do not need to work all this out here, because as shown in Section 6 .4, the vacuum expectation val- ues (9 .2.17) are all we need in order to be able to calculate S-matrix elements , 9.3 Lagrangian Ve rsionofthe Pat h-Integral Formul a The integrand in the exponential in Eqs .(9.1.38) or(9.2.17) looks like the Lagrangian L associated with the Hamiltonian H . This appear- ance is somewhat misleading because here the `rnornenta' Pa(t) or p,,(x, t ) are independent variables, not yet related to q jt) or q,(x, t) or their derivatives . However, there is a large and important class of theories in which the integral over the `momenta' can be done by just replac- ing them with the values dictated by the canonical formalism, in which case the integrand in the exponential in the path integrals really is the Lagrangian . These are theories with a Hamiltonian that is quadratic in the `momenta' - in the language of field theor y J"M +Efd'x B .,[Q]P.(x)+C [Q] (9-3-1) n with a `matrix' A that is real, symmetric, positive, and non-singular . The 390 9Path-Integra l Method s argument of the exponential in Eq. (9.2.17) is then quadratic in the ps : fdTif d3_XP', (X, 04,;(X,-0-H[q (-r), p(T)] ; dux d 'ydzdr't;/,xn.gym[4]Pn(~,'r)Pm(YJ') nm where .5/xXn,r'y}ri[q] Axn,}'nl[q(T)J~(-c-Zr) fxn[q]&xn~~(-01 -4n ~~, -0 '[q]= f d TC Lq (T)l(9.3.2) (9.3.3) (9.3.4) (9.3.5) Now, in general the integral of the exponential of a quadratic expression like (9 .3.2)will be proportional to the exponential evaluated at the sta- tionary point of its argument . For a finite number of real variables this formula read s ~~' S Sr 5 _ (Det [i,Q//27r] exp iIYV_3r~,5~r - i 's5 -a (9.3.6) ~t s where ~ is the stationary poin t ~s= - 4v-I1,sr r x(9.3.7) (For a proof of this formula, see the Appendix to this chapter . Hence, as long as the G~'A, Lea, etc. inEq.(9.2.17) are independent of the ps, for such a Hamiltonian we can evaluate the path integral over the ps in E q. (9.2.17) b y setting these variables at the stationary point of the quadratic expression in the argument of the exponential . But the variational derivative of this quadratic i s b Lf_! --~ d-Td3x4,(x, -r)F,(x,,c)- H[9'(r),p(c)] +icterms P, (X, T) (X,-0 - 6 H [q(-r), p(,r) ] P,,(X, ) (The it- terms depend only on the qs .) Thus the stationary `point' p„(x,t) where this vanishes is just the value of p,(x, t) dictated by the canonical 9.3 Lagrangian Version qf the Path-integral Formula 391 formula 6I~[q(~),P(~)] qn~x' ~) = a~~{~~ ~) (9.3.8) P-p with pn(x, t) set equal to this value, the argument of the exponential in Eq. (9.117)isthe ordinary Lagrangia n L [q~~~, (T)]=/3x qn(X, T) Pn(X,t)-~ ~~(,c), ~(-r)] (9 .3.9) and we can write Eq .(9.2.17) as (VAC, o utT~OA [Q(tA)],((' .B[Q4$}],...~VAC . in 1/2 A'-11f fl dq,,(x,'r) (Det[2inv/ [q] i,x,n X CA[q(tA COB [q(tB)] ... Or-{L[q(v)jj(t)]+ ie ter ms (9 .3.10) x expif' (We have combined the 1/27r factors in the integrals over the p ,with the determinant coming from Eq, {9,3 .6},) This is the desired Lagrangian form of the path-integral formula . In deriving Eq .(93.10), it was necessary to assume that the operators GA, CB, . ,. were independent of the canonical `rnomenta' . This is not as restrictive as it may seem . For instance, in a scalar field theory for which the canonical conjugate to (Dis IT = 4), it is possible to calculate the matrix element of a time-ordered product of operators, one of which is 6(t), by taking the difference of matrix elements in which this operator is replaced with (D(t + dz) and cD(z), and then dividing bydz, with di--1'0. Equivalently, as long as t is not equal to any of the other time-arguments of the operator in Eq .(9.3.10), we can simply differentiate Eq .(9.3.14) with respect to t . The one serious remaining complication in E q.(9.3.10) is the deter- minant of -2/[q]. If ~ vl[g] is field-independent, then this is no problem ; we have already noted that overall constants make no contribution to the connected parts of vacuum expectation values, in which we divide by a vacuum-vacuum amplitude proportional to the same constant factor . This is the case for instance for the theory of a set of scalar fields din with non-derivative coupling to each other and/or derivative coupling to external currents J ,The Lagr2tngian density here is 392 9Path-integral Method s An obvious extension of the results of Section 7 .5 from one to sev- eral derivatively coupled scalars shows that this Lagrangian implies the Hamiltonian H = d ux1:[~i1~ + ~ (V~n)2 n +in,vin + JnOnn+t (in0}2+]d3x V(chi). (The (Dn are taken to be real scalars, but complex scalars can be accom- modated by separating them into real and imaginary parts .) In general there is a non-trivial term that is linear in the III,, but the coefficient of the quadratic term is a constant, just the unit `matrix' . f in Eq .(9.3.10 is here afield -independent The factor (Det[4ic1[q]])1 2 constant, and therefore is without effect . However, matters are not always so simple . As a second example, let us consider the so-called nan-linear or-model, with Lagrangian densit y 2Ecla(DnO (Dm 16nm + UnOD)l - V((D) 1d1ii A straightforward calculation gives the Hamiltonian a s 1 = f12)II Ri)nm Here ~ ~/is the field-dependent quantity i nm In cases of this sort, the determinant may be reexpressed as a contribution to the effective Lagrangian, using the relation Det,d = exp It In ,71 . By replacing the continuum of spacetime positions with a discrete lattice of points surrounded byseparate regions of very small spacetime volume Q, we may interpret the delta function in as64(x - y) - S2-16"Y, so that (In ~~}►~.x,rrry  =Jx,v ~ In (1 + ~((D(x))) - 1  In 0L n m with the logarithm of a matrix defined now by its power series expansion z 3 2 3 To evaluate the trace, we note that 1:,,...;--Q-'~ d4x,.. The determi- 9.3 Lagranglun Version of thePath-Integral Formul a ❑ant factor here is the n Det oc ex p - Q-1 d4X#rInI I + U((D(x))] ,393 where `tr' is to be understood as the trace in an ordinary matrix sense . The constant of proportionality (which arises from the - In 0term) is field-independent and therefore of no present interest . We can regard this determinant as providing a correction to the effective Lagrangian densit y AY !,?'Q-1tr I n I I I + U((D(x))] The factor Sl-1 may be written as an ultrav iolet divergent integra l f2-1=64(x - x)= (21r)-4dip-1 We shall not show this here, but the extra terms in the Feynman diagrams for this theory contributed by d.,Pcould also have been derived in the canonical formalism by taking account of the equal-time-commutator terms in the propagator of time-derivatives of the scalar field .7 Ignoring this correction would lead to a spurious dependence of the S-matrix on the way that the scalar field is defined, and would also be inconsistent with any symmetries of the Lagrangian under transformations of the scalar fields . Even where the factor (Detd)-1~2 in the path-integral formula (9.3.10) is field-independent, the Lagrangian in this formula may not be the same as the one with which we started . As an example, let's consider the theory of a set of real vector fields, with Lagrangian densit y 1~aApi, - O,An~j) (-PA MA~j A' ft +MHAnAnin AnAl-1 where the currents J nPare either externally produced c-number quantities or depend on other fields (in which case terms describing these ❑ther fields are to be added to the Lagrangian) . By a simple extension of the results of Section 7.5, we see that the Hamiltonian i s 3-, 1 : 11 1 1 7 2m22 fl 2 again with the understanding that other terms must be added involving any fields that appear in J.Here the coefficient of the quadratic term is 394 9Path-Integral Method s somewhat more complicated than in our first example : but it i sfield- independent ,so that the factor (Det.4}-1~2has no effect . On the other hand,the La grang ian (9 .3.9) ishere not the one with which we started ;it is expressed entirely in terms of A and its spacetime derivatives, with no dependence on any time-component A° .For this reason, the Lorentz invariance of Eq. (9.3.10) is far from obvious . To remedy this , we may reintroduce the auxiliary field .Suppose weadd to the Hamiltonian a ter m ❑H = - ~~ rn ~ dux~A"- rr~,~'❑-11r,+mn J4 and integrate over the AD as well as the A,, and H . This can only introduce a field-independent overall factor, since dH is a quadratic in A° ( with afield- independent coefficient in the term of second order in AO) whose stationary value vanishes . However, suppose that we now integrate over the IIIbefore integrating over the AO.The Hamiltonian in the path integral (9 .2.17) is here replaced wit h ,U2(V x A,,)2+1M2An2H+AH=fd'_xE~f n ~M~(A°)' + Jn'An- J~ A° + An dil~I. This is still quadratic in II, with a f ield-independent (and somewhat simpler) coefficient of the quadratic term, so the integral over the H,s can be done by just replacing II, with its value at the stat ionary point of the functional E,, f d'xII' A, - H - ❑H 17~ - A, + VA ° WithII„ eliminated in this way, En f d3x III  A . -H-❑H is just the Lorentz-invariant Lagrangian with which we started . In order to take account of the possible need to introduce auxiliary fields like A0 ., from now on we shall write the path-in tegral formula after elimination of the canonical conjugates in terms of fields w,, that include both canonical fields qn and auxiliary fields c, ; VAC,aut T oc fl d~i, (x, r) CA [W(tA)l 61B ~V)(tB)] ... ti,x,n if (x.3.11) x exp i ~ d r IL ['W'(r), ~) (~)] + ie ter ms 9.4Path-I ntegralDerivation ofFeynman Rules 395 it now being understood that L includes any terms arising from a possible field-dependent factor (Det ,9/)-l/ ' 9.4 Path-Integra l Derivation of Feynman Rul es We are now ready to use the path-integral formalism to derive the Feyn- man rules in a wide class of theories . We will concentrate here on the vacuum expectation values of time-ordered products of field operators (and their adjoints), {SAC, gut J TI'~`~.~(X,~), T,,, (XB)... ~ ~ VAS, ink ~~~$ ~~~~~ ...} - {SAC, autjVAC,in} (9.4.1 } from which S-matrix elements may be obtained (as shown in Section 6 .4) by stripping off the final propagators associated with each field, replacing them with the coefficient functions that multiply creation or annihilation operators in the corresponding free fields, and summing over the indices on these coefficient functions . For the simpler theories whose Hamiltonian is quadratic in the III, Ey.(9-3.11) gives I-IdWt(X)]T1A(XA)WfB(XB)_ e'llq)]~.~(9.4.2) where I [y7jis the actio n I[W]= dT ~ L [it, (c), ~,,(r)]+ietermsI(9.4.3) with L now including any terms that may arise from afield-dependent determinant in Eq .(9.3.10). Let us now suppose that the Lagrangian is the integral of a Lagrangian density, consisting of a quadratic term Owhich would be present in the absence of interactions, plus a Lagrangian interaction density I'1 : L[V(r), ~i (T)] d'x [Y~ (w (5~,z),aplp('~' 7)) X 0 X 'r))+YI(W( '-C),"Utto, (9.4.4) That is, the action (9 .4.3)is 396 9Path-Integral Method s I [V7] = 1o[0 +[1 IV] , 1010 = d'x Yo (Y)(A e"m W + icterms , I I lip] ;--dx i(ip(x), 0, q) (X)) -(9.4.5) (9.4.6) (9.4.7) Since c,and the `ie terms' are quadratic in the fie lds, we may always write 10 in the generalized quadratic for m lolwj 1. f(9.4.8) For instance, for a real scalar field of mass m, thunperturbed Lagrangian is YO 10 00 IMW and the ie terms in 10are given byEq. (9.2.16), as so here(9.4.9) liefdtfdux dux' (x, x') 0(xTt)0(x', t} (9.4.10) (X_ 1 t i (We are now dropping the factor e £Il in the if term, since it produces a correction of higher order in c.) To deal with interactions, we will expand the exponential in powers of 11, (I, IV ].N=O{9.4. 12} and then expand Ii in powers of the fields . The general integrals that we encounter in the numerator and the denominator of Eq . (9.4.2) are of the form (flui.+)x) eifa[v)l(X1) W,(X2)...,(9.4.13) where the field factors W, (xi )M~2{x2}, etc . arise from I1 [tp] and/or from the field factors Y7( :, (xA) etc . originally present in the numerator of Eq.(9.4.2). With [()IV,,] of the form (9 .4.8), the integral (9 .4.13) is of the same form as the integral evaluated in the Appendix to this chapter, with the discrete index s replaced with the pair of labels e, x. We can therefore 9.4 Path-Integral Derivation qf'Feynman Rule s use Eqs . (9.A.12) and (9 .A.15), which give her e ,..(.xlx-)...)a z[Det()f" 27rpairings o f fields397 111 pairedfieldspairs (9.4.14) This just amounts to the coordinate-space Feynman rules for calculating the numerator of Eq . (9.4.2) in their covariant form : we expand in the interaction 11, and then sum over the ways of pairing the fields in the I1 s with each other and with the fields ZPe ,,(xA etc ., with the contribution of each pairing being given by the spacetime integral of the product of the coefficients of the fields in I, [W] and the product of the 'propagators' -iA, where ❑r.iry( Xi,X2)=(9:-1)r1x1,e2-x2. (9.4.15) (The factor [Det(i /2-g)]-1/2 in Eq. (9.4.14) actually represents the con- tribution of graphs with unlimited numbers of single loops unattached to any other lines, but in any case this factor cancels in the ratio (9 .4.2).) It remains to calculate the propagators (9 .4.15). We interpret Eq. (9.4.15) as an integral equatio n Jd4x .29e1X1/2x2A1r2MX2 ,X3)= 64(XI- ~ 3A,,6. (9.4.16) In the absence of external fields, tran slation invariance will make ~' neces sarilyafunction only 0f xl - x2 . whichcanbewritten a sa Fourier integral L~~L,~~J,t2X2 =_(2-g)-4 f The solutionof Eq . (9.4.1 G} isthen JPI~2(p) (9.4.18) where - 1is the ordinary inverse of the matrix , As we will see, the ic terms have the effect of making the inverse well-defined for all real values ❑f p. We have thus reduced the problem of calculating the propagator to that of taking the inverse of a finite matrix . First consider a massive scalar field, for which the kernel 9takes the form (9 .4.11). We can write this as a Fourier integra l I (P ) I so the propagator i s d(x, y )- (2~ r)-4d4peip-(x-y )(p2+M2- icE(P))l 398 9 Path-Integral Method s We recognize this as the same scalar propagator previously obtained by operator methods . (The difference between e and e E(p) is immaterial, since both are just positive infinitesimals .) For a second example, consider a real massive vector field . The unper- turbed Lagrangian i s 4 2 We can again write IO[V] in the form (9 .4.8), with kerne l r~) px,rr}' ~ ~p~ ~ i +M2qpT1 64(x -Y) + iE'terms (27E)-4 Pedp.(-r-y)1?7PCP 2 =ppPa+M2~1r.,~+i,- terms ] We will not bother to show it here, but the `+ ic terms' here take the simple form -1EE{P)~p, The vector field propagator is then given by simply inverting the 4x4 matrix in the integran d d4pdip-(x-u) pppt 7Apa(xj) = (21r fP ( p (Terms proportional to e are dropped in the numerator . They are impor- tant in the denominator in defining how the integrand is to be treated near the mass shell, p2= _M 2.) This is the same as the propagator derived by operator methods, except that the non-covariant terms proportional to6(x4 - yc} )are now absent . These non-covariant terms were previously needed to cancel non-covariant terms in the interaction Hamiltonian, but the vertex contributions in the Feynman rules are now obtained directly by inspection of the covariant Lagrangian, and no such cancellation is needed . Theories with derivative coupling are equally simple . The factor arising from the pairing of a field derivative c'~,We(x) with any other field V),& } (perhaps itself a field derivative) i s [iidqv.(x)] (~) W.(y)cif [V~ ~ (x)] eillyl lx.~ WAX) VMW) - (9A .19)OXY Such propagators have no non-covariant pieces . For instance, for a real scalar field, the pairing of c 7.0with 0,0 gives a momentum space propagator k ,,kV1(k'- + M2 - ie) . Also, as we saw in the previous section, vertices in the theory of a scalar field with derivative couplings to other fields may be read off from the Lagrangian, and are separately covariant . 9.5Path Integrals for Fermion s 9.5 Path Integrals for Fermions399 We now turn to the problem of extending the path-integral formalism to cover theories containing fermions as well as bosons . It would be easy to proceed in a purely formal way, by analogy with the bosonic case, with the justification that this gives the `right' Feynman rules . Instead, we will here derive the path-integral formalism for fermyons directly from the principles of quantum mechanics, as we did for bosans .9 As before, we will start with a general quantum mechanical system, with 'coordinates' Q ,and canonical conjugate `momenta' P, but now satisfying anticomrnutation rather than commutation relations : {Qa.Pi}=ib,ha JC)05 Q4 = JP ., PbJ = 0(9J.1) (9..) (These are Schrodinger-picture operators, or in other words Heisenberg- picture operators at time t = 0.)Later we will replace the discrete index a with a spatial position x and a field index m . We wish first to construct a complete basis for the states ari which the Qs and Ps act . Note that for any given a,we hav e It follows that there will always be a `ket' state I(])annihilated by all Q,,- al0) =0, and a `bra' state { 01annihilated (from the right) by all P,,: APO= 0.(9.5.4) (9.5.5) For instance, we can tak e 0)~(HQa) I.f), (0 10C (g I (Pa where f )and ~g lare and kets and bras for which these expressions do not vanish . (They cannot vanish for all I f )and (gIunless the operators jla Q,,and fl,, P, vanish, which we assume not to be the case .) These states satisfy Eqs . (9.5.4) and (9.5.5)by virtue of Eq .(9.5,3). They are not in general unique, because there may be other bosonic degrees of freedom that distinguish the various possible 10~ and {0 1, but for simplicity we will limit ourselves here to the case where the only degrees of freedom are those described by the fermionic operators Q ,and Pu, and will assume that the states satisfying Eqs . (9..4) and (9.5.5)are unique up to constant factors, which we choose so that (010) =1. (9 .5.6) SOD 9Path-Integral M ethod s (Note that this normalization convention could not be imposed if we had defined ~ 01as the left-eigenstate of the Q awith eigenvalue zero, because in this case (O {QP b~1O} would vanish, which with Eq .(9.5.1)would imply that (010~ = 0.) As we saw in Section 7 .5, in the Dirac theory Q,,is not Hermitian, but instead has an adjoint - W, in which case ~0 1can be regarded as simply the adjoint of ~D} . However, there are fermianic operators (such as the `ghost' fields to be introduced in Volume II) for which P,, is unrelated to the adjoint of Q ,In what follows we will not need to assume anything about the adjoints of Q ,,or P,,, or about any relation between ~~ )and (01. A complete basis for the states of this system is provided by 10)and the states (antisymmetric in indices a, b, ...) aib,... ~ - PaPb... A (9 .5.7) with any number of different Ps acting on 10}, That is, the result of acting on these states with any operator function of the Ps and Qs can be written as a linear combination of the same set of states . In par#icular, if an index a is unequal to any of the indices appearing in ~ b, c, _~, the n Qui b7~,...~ - 0 , (9 .5.8) pal?t .~...~ ~I a, bac...~ - (9 .5.9) On the other hand, if a is equal to one of the indices in the sequence, b, c, - - , we can always rewrite the state (possibly changing its sign) so that a is the first of these indices, in which case we hav e Qal a, b, e ,...~ = a {b,e...~ (9.5.10) (9.5.11) Simiiariy, we may define a complete dual basis, consisting of (01and the states (also antisymmetric in the indices) (9.5.12) Using Eqs .(9.5.4)-(9.5.6)and the anticommutation relation (x .5.1), we see that the scalar products of these states take the value s 0if {c, d,.,.1 ~a, b,.,.1 1 ifc=u, d=b, etc.(9.s.13) where ~   1here denotes the set of indices within the brackets, irrespective of order . In deriving the Feynman rules, we would like to be able to rewrite sums over intermediate states like (9 .5.7) as integrals over eigenstates of the Q ,,or the P, However, it is not possible for these operators to have 9.5 Path Integrals for Fermions 401 eigenvalues (other than zero) in the usual sense . Suppose we try tofind a state ~q}that s atisfies (for alla) Qujq~= qulqf From Eq.(9.5.2) we see that(9.5.14) (9-5-15) which is impossible for ordinary numbers . However, nothing can stop us from introducing an algebra of `variables' (known as Grassmann variables) qa, which act like c-numbers as far as the physical Hilbert space is concerned, but which still satisfy the anticammutation relations (9.5.15). We will require further tha t tqaaqb}={qa ,Qh~= fRaaF h}=d, (9.5.16) where q and q 'denote an ytwo `value s'of the se variables. We can now construct eigenstates ~q}satis fying Eq .(9.5.1): ~q~ = exp - iEp aqa (9.5.17) with the exponential defined as usual by its power series expansion . (To verify Eq . (9.5.14), use the fact that all P ,,q,,commute with one another and have zero square, so that h:~a b¢a -t ~ ~~~b A=0 bra as required b yEq. (9.5.14).) We can also define left-eigenstates (qj~not the adjoints of 1q)), as a u u a (9.5.18) where fl,is the product in whatever order we take as standard . By the same argument as for Eq . (9.5.14), we see that 402 9Path-Integral Method s These eigenstates have the scalar produc t 0fl 1 1 1 (QO (1+iPb(qb- qb)) Moving each Qa to the right (starting with the rightmost) yields factors i2(q,a- q}, which we move to the right out of the scalar product, s o (9.5.20) We shall see that Eq. (9.5.20) plays the role of a delta function in integrals over the qs . In the same way, we can construct right- and left-eigenstates of the P, , P,Ip~=Pa 1F)a (PIPa = (PIPa r(9.5.21) (9.5.22) where the paarelikeq,, anticommuting c-numbers (taken for convenience to anticommute with the q,, and all fermionic operators as well as each other), and Ip~ = exp i Q aI~~ riPbI0)a exp(_i>I:PaQO with scalar product (now derived by moving the Ps to the left) ~PrIP~= 11(pa-Pa)(9.5.23) (9.5.25) The scalar products of these two sorts of eigenstate with each other ar e a =(iIexP(_iaPa)) (qfr1p.Wa (4 (flexP-_iaPa)) ~~~(llQa) rlPa 1 0) a andso ~qIp) = Irv eXP - ~quPa)= Irv eXp (i .Pqa) (9 .5.26 9.5Path Integrals.farFermions 403 where XN is a phase that depends only on the num berNofQ,,operator 's: Somewhat more simply, we a lso fin d It is easy to see that the states 1q)are in a sense a complete set (and so also are the lp~.)From the definitions (9.5.17), we see that the state la, b, ...} in the general basis is (up to a phase) just the coefficient of the product q ,,qb... in an expansion of 1q} in a sum of products of qs . Therefore we can write any state If)in the for m If)-fOIq)O+ falq)a+I:Jablq)ab+' R ash(9.5.27) where the f s are numerical coefficients, and a subscript a,b, on1q) denotes the coefficient of gaqh ... in I q) In summing over states, it will be very convenient to introduce a sort of integration over fermionic variables, known as Berezin integrutinn,10 that is designed to pick out the coefficients of such products of anticommuting c-numbers . For any set of such variables ~n (either ps or qs or both together), the most general function f (~ )(either a c-number or a state- vector like 1q)} can be put in the for m f G) + ter ms with fewer ~ factors (9.5,2 8) and the integral over the ~sis defined simply b y d~n (9.5.29) with the tilde in E q. (9.5.29) indicating that we use the convenient con- vention that the differentials are written in an order opposite to that of the product of integration variables in Eq .(9.5.28).Since this product is antisymmetric under the interchange of any two ~ s, the integral is likewise antisymmetric under the interchange of any two d s, so these `differentials' effectively anticommut e Also, the coefficient c may itself depend on other unintegrated c-number variables that anticommute with the ~ s over which we integrate, in which case it is important to standardize the definition of cby moving all ~s to the left of c before integrating over them, as we have done in Eq .(9.5.28). 404 9 Path-Integral Method s For instance, the most general function of a pair of anticommuting c-numbers ~1 and ~2 takes the for m because the squares and all higher powers of jand 2 vanish . This function has the integral s fd'i.f (fir, s 2)=~2Ciz + el, d~af (~I> ~2) jd~2 4 1 f (~ 1, (';2)= C12- Note that the multiple integral is the same as a repeated integral : jd~2 [f d~j f(~I, ~2)] 1 a result that can easily b eextended to integra]s over any number of fermionic variables . (It was in order to obtain this result without extra sign factors that we took the product of differentials in Eq . (9.5.29) to be in the opposite order to the product of variables in Eq .(9.5.28).)Indeed, we could have first defined the integral over a single anticommuting c-number ~1, and then defined multiple integrals in the usual way by iteration . The most general function of anticommuting c-numbers is linear in any one of them ,f(~1, ~2a...)=b( t2...)+pl4~2..) (because i = 0), and its integral over ~ jis defined a s 'id~jf(~I, ~2,-- I= CG2, - - -) - Repeating this process leads to the same multiple integral as defined by Eqs.(9.5.28) and (9.5.29). This definition of integration shares some other properties with multiple integrals (from -oa to +oo) over ordinary real variables, but there are significant differences . Obviously, Berezin integration is linear, in the sense that I ( - - - rld~,) [f w + ooi d- .) f (~) + f (11d~n g(~) (9 .5.3 1) n n and also f ( - fjd~n) d f (ol (9.5.32) where a(~') is any function (including a constant) of any anticommuting c-numbers ~ ;Mover which we are not integrating . However, linearity with 9.5 Path Integra lsfir F 'ermions 40 5 respect to left-multiplication is not so obvious . If we are integrating over v variables, then since is assumed to anticommute with all fin, we hav e a((-)%') 11~n ri a(V) (n ) = (n and so d v~')f a(~ 1)j fj d~n f (0 (9.5.33) It is therefore very convenient (though not strictly necessary) to take the differentials d~, to anticommute with all anticommuting variables (including the fin) ; in which case Eq .(9.5.33)reads more simpl y a(~') fjd~nf(D=aW)fd'n ( ?I C) f (0(9-5.34) X9.5.35) Another similarity with ordinary integration is that, for an arbitrary anticommuting c-numberVindependent of ~ , I(9.5,36) since shifting ~ by a constant only affects the terms in fwith fewer than the total number of ~-variables . On the other hand, consider a change of variables (9.5.37) where is an arbitrary non-singular matrix of ordinary numbers . The product of the new variables is n ~ ~1: ((ll9nmrtin )„ But H, ~.,here is just the same as the product (in the original order) fin fin, except for a sign c [m] whi chis+1or - 1 according to whether the permutation n--> mnis an even or odd per mutation ❑f the original order : H Ynm,) e lm] H ~n = (Det - 11 11!I1?22... n n 1 t This applies whatever order we take for the ,, as long as we take the ~n' in the same order, It follows that the coefficient of Hn ~ ;~ in and function 406 9Path-Integral M ethod s f(~)is just (Det } 'times the coefficient of a statement we write as j(~a )~f=(met,s2)-'I ~d~n f . (9.53s) This is the usual rule for changing variables of integration, except that (Det Y) appears to the power -1 instead of +1 . We shall use Eq .(9.5.38) and the linearity properties (9.5.31),(9.5.32), and (9.5.35) later to evaluate the integrals encountered in deriving the Feynman rules for theories with fermions . We can now use this definition of integration to write the completeness condition as a formula for an integral over eigenvalues . As already men- tioned, any state If }can be expanded in a series of the states I0},1a),I a, h}, etc. and these states are (up to a phase) the coefficients of the products 1,qu,quqb, etc . in the Q-eigenstate 1q}. According to the definition of integration here, we can pick out the coefficient of any product q bq,qd... in the state Iq} byintegrating the product of q} with all q ,,with a not equal to b, c,d, -...Thus, by choosing a function f (q) as a suitable sum of such products of q s, we can write any state If } as an integral : (ri dq .) (9.5.39) (We can move Iq)to the left of the differentials without any sign changes because the exponential in Eq.{9.5.17} used to define 1q} involves only even numbers of fermionic quantities .) To find the function f (q) for a given state-vector If }, take the scalar product of Eq . (9.5.39) with some bra {R' I(with q' any fixed Q-eigenvalue) . According to Eye .(9.5.35) and (9.5.20), this is ~q'jf (II(qa - q,, fldqbI ) )f(q) a )(b Moving every factor (R,,- q')to the right past every differential dq byields a sign factor where Nis now the total number ofq,, variables, so = (_)NI ( - (q'If ) 11 dqh (qu - q' .)) f (q) M a We can rewrite f(q)as f (q'+ (q-q')) and expand in powers of q --q'.All terms beyond the lowest order vanish when multiplied with the product 9.5Path Integrals for Fermions fl(q.-4')aso ~~~~ =(iia - qu)f (q)(J]j(qi"q) L7 it407 (9.5.40) which partly justifies our earlier remark that Eq .(9.5.20) plays the role of a delta function for integrals over the qs . Using Eq.{9.5.30, we now hav e f 11 dqh JI(qa- qO f (q' h ) ( a The term in the integr and proportional to JJryahascoefficient ,f(q'),so according to our definition of integration (q'If) = (-)Nf(qf).Inserting this back in Eq. (9.5.39) gives our completene ss relat ion if) = (_)Nf 1q) dqb ~qjf) b orasan operator equation J'lq) (fi -dqa (q ~ In exactly the same way, we can also show tha t fP)ljdpa~pIa(9.5.41) (9.s.42) We are now in a position to calculate transition matrix elements . As before, we define time-dependent operator s Q,(r)exp(iHt) Q,, exp(-iHt) Pi(t) exp (iHt)F,exp(-iHt) and their right- and left-eigenstates(9.5.43) (9.5.44) Iqa~~=- exp(iHt)+q} , IPA t)= exp( iHt)~P~ a (9 .5.45) {qatj=(qlexp(-iHt) (P70=(Aexp(-iHt). ( 9.5.46) The scalar product between q-eigenstates defined at infinitesimally close times is the n Now insert Eq . (9.5.42) to the left of the operator exp(-iHdT) . It is convenient here to define the Hamiltonian operator H(P, Q) with all P s to the left of all Qs, so that (for &infinitesimal ) (Aexp(-iH(P sQ)dr) l 9~= (Fjq}exp(-W(P, q)dT) . 408 9 Path-Integral Method s (We could move the c-number H(p,q)to either side of the matrix element without any sign changes because each terrain the Hamiltonian is assumed to contain an even number of fermionic operators .) This give s .~ ~q% r + dTlq ; Pq'jp~ dpa ~pj exp(-iHdr)jq ) (pjq~ exp iH(p, q)d-c) Using Eqs . (9.5.26) and (9.5.27), and noting that the products paq . and p.q. commute with all anticommuting c-numbers, we fin d f (_ !-qa)-iH(p, q)d r The rest of the derivation follows the same lines as in section 9 .1. To calculate the matrix element (R'; t' Q A(P (W) QB(P(tB), Q(tB))...1q; t)of a product of operators (with t' >tA>tB>...>t), divide the time-interval from t to t' into a large number of very close time steps ; at eachtime step insert the completeness relation (9 .5.41); use Eq .(9.5. 7) to evaluate the resulting matrix elements (with C'A>(91?, etc, inserted where appropriate) ; move alldifferentials to the left (this introduces no sign changes, because at each step we have an equal n umber of dps and dq s );and then introduce functions q a(t)and pa(t) that interpolate between the values of q ,,and p ,, at each step .Wethen fin d (q~;t' I Tf(nA(P(tA),Q(tA)), 1T"B(P(tB), Q00) , - - -I lq;t) r} i} ~ti f ,(qRlj1-qil,q{Flj'1-q{F X &4 ( l(tA) 7q(tA))C I](l(t lJ)7q(t lJ))... L+ x exp id-c - (9 .5.45) fa The symbol There denotes the ordinary product if the times are in the order originally assumed, t A>t$>- - -.However, the right-hand side istotall ysymmetric in the (except for minus signs where anticommuting c-number s are interchanged) so this formula holds for general times (between t and t'),provided T is interpreted as the time- ordered product, with an overall minus sign if t ime-order ing the operators involves an odd nu mberof permutations of fermionic operators . Up to th is point wehave kept track of theoverall p hase factor ( A^'ZN 9.5Path Integrals for Fermions 409 But in fact these phases contribute only to the vacuum-vacuum transition amplitude, and hence will not be of importance to us . The transition to quantum field theory follows along the same lines as described for Masonic fields in Section 9 .2. The vacuum expectation value of a time-ordered product of operators is given by a formula dust like Eq. (9.2.17). (VACI out ~ TICA[P(tA Q(tA)], C-B [P(tB), Q40], "'I IVAC, ID ) OCI [1I dq,, (x, T)] [ 11 dp,,, (x, -c)] ('01A~p(tA), q (ti)] z,x,m z,x,m [p(tfl .q(tB) m H[q('r), p(T)J+ ie terms (9.5.49) where the proportionality constant is the same for all operators CA, OB, etc., and the `fe terms' again arise from the wave function of the vacuum . As before, we have replaced each discrete index like a with a space position x and a field index m . We are also dropping the tilde on the product of differentials, since it only affects the constant phase in the path integral . A major difference between the fermionic and bosonic cases is that here we will not want to integrate out the ps before the qs . Indeed, in the standard model of electroweak interactions (and in other theories, such as the older Fermi theory of beta decay) the canonical conjugates p,,, are auxiliary fields unrelated to the q,,, and the Lagrangian is linear in the q., so that the quantity f dux E.pmqm-H in Eq.(9.5.49) as it stands is the Lagrangian L Each term in the Hamiltonian for a fermionic field that carries a non-vanishing quantum number (like the electron field in quantum electrodynamics) generally contains an equal number of ps (proportional to q) and res . In particular, the free-particle term HO in the Hamiltonian is bilinear in pand q, so tha t dr d'x P~(~,,r)4,(xa z) - Ho [q(t) .p(r)] +fe terms . "~ rr ¢ = - ~ c~x day na,,11~ pm(x) R n(y) (9.5,50) Mn with 2 some numerical `matrix' . The interaction Hamiltonian V = H- HO is a sum of products of equal numbers of fermianic q s and ps (with coefficients that may depend on bosonic fields) so when we expand Eq. (9.5.49) in powers of the V we encounter a sum of fermionic integrals 410 9 Path- Integral Methods ofthe for m '3rn041 M2m2...n,VM.v(X I rY1, X2, Y2...XN, Y N) fjdq i1i(X, r)J T.x,m X [11 dp .(X, T)] q., (x 1)pn,(yj )qm2 (X2) Pn2 (Y2) qMN (XN) PnN' (YN) T,X,M x exp ~ - i dux day nYP~~~~ q,,(1') ~ (9.5.51) one such term for each possible set of vertices in the Feynman diagram, with coefficients contributed by each vertex given b yi times the coefficient of the product of fields in the corresponding term in the interaction . To calculate this sort of integral, first consider a generating function for all these integral : g) =_ I [ 11 dq,,(x, r) dp,,,(x, -r)] X,T,M x exp (-iE f d'xd'y~Ymx,ny pm(x) q,(y) mn --i d4X PM(X) fin W - i dy g , (y) qn(y)), (9.5.52) where f,,(x) and g jY) are arbitrary anticommuting c-number functions . We shift to new variables of integratio n qn(Y)= 4M (Y) +fdux(.9-1)ny,tnxf?n(x) ,n Using the translation invariance condition (9.5,36 ), we then fin d J(J',g) = exp (i d4xd4}'fir:-t)ny,,ve gn( Y)f.(x))►HN x,r,m ,~ Mn The integral is a constant (i .e., independent of the functions f and g ) which can be shown using Eq .(9.5.38) to be proportional to Det ~2 . Of more importance to us is the first factor . Expanding this factor in powers 9.5 Path Integrats for Fermions 411 of gf'and comparing with the direct expansion of Eq .(9.5.52), we see tha t pairing r ~ -i-~pairedmx,nypairings pairs(9.5.54) with a proportionality constant that is independent of the x, y, m ,or n, and also independent of the number of these variables . The sum is over all different ways of pairing ps with qs, not counting as different pairings that only differ in the order of the pairs . I n other words, we sum over the N!permutations either of the ps or the q s . The sign factor 6pair2ng is +1 if this permutation is even ; -1 if it is vdd . This sign factor and sum over pairings are just the same as we encoun- tered in our earlier derivation of the Feynman rules, with the sum over pairings corresponding to the sum over ways of connecting the lines asso- ciated with vertices in the Feynman diagrams, and the factors ( playing the role of the propagator for the pairing of q,,(x) with p ,(Y). In the Dirac formalism for spin -1, the free-particle action i s dr fd'xE Pm(Y-, T)4M (X, T) J71 f4X q)(X) ryp -HO [q(T), P(T)] I where in the usual notation the canonical variables here are(9.5.55) with m a four-valued Dirac index . Comparing this with Eq .(9.5.50), we find here 0 mx,~ry~ YGXPMn (y°FiTu k~j + in - i e]) etk(x~y). (9.5.57)irk . ~ M n {2~} (Though we shall not work it out in detail, the i eterm here arises in much the same way as for the scalar field in Section 9.2.)The propagator is the n )?nn just as we found in the operator formalism . The extra factor - YO arises because this propagator is the vacuum expectation value of '{l~r~ j a -not~'l~~~l~Js ~'~rl~l~  ~~~m4~~ , - l~l~'I ^ 7 As one example of a problem that is easier to solve by path-integral than by operator methods, let us calculate the field dependence of the 412 9 Path-I ntegra l Method s vacuum vacuum amplitude for a Dirac field that interacts only with an external fiield . Take the Lagrangian as (9.5.59) where r(x) is an x-dependent matrix representing the interaction of the fermion with the external field . According to Eq . (9.5.49), the vacuum persistence amplitude in the presence of this external field i s {VAC, out sVAS, i n)r- oc[ H dq,,,(x,r)][ Tj d P. (X}T)] r,x,m r,x, m X expijdlx pT .111[~,oo~' +m + IF - ic] qj (9.5.60) with a proportionality constant that is independent of I,{x) . We write this as VAC, ou tIVAG, i n~r, cc C11 d 4,,(x}-c)] [rjdF.(X}T)] T,x,m z,x,m x ex p- i~ duxd'yPn,(?~) qn(Y) [rlntx,ny (9 .5.61) Mn where a 4 T,XT I)Mn To evaluate this, we change the variables of integration qn(x) to(9.5.62) (9.5.63) The remaining integral is now T'-independent, so the whole dependence of the vacuum persistence amplitude is contained in the determinant arising according to Eq . (9.5.35)from the change of variables : {AC, outIVAC, in) r- oc Det-*-[F] . To recover the results of perturbation theory, let us writ e and expand in powers of ~[f] .Eq.(9.5.64) gives the n (VAC, autIVAC ,in)r oc Det (.[1 + ~'-1 [I"]])(9.5.64) (9.s.6s) (9.5.66) (9.5.67) 9.6Path-Integral Formulation qfQuantum Electrodynamics 413 This is Just what we should expect from the Feynman rules : the con- tributions from internal dines and vertices in this theory are -6 1~R' an d the trace of the product of nfactors of -~-1 [r] thus corre- sponds to a loop with n vertices connected by n internal lines ; 1 In is the usual combinatoric factor associated with such loops (see Section 6 .1); the sign factor is (-1 )'+1 rather than (-1)'because an extra minus sign is associated with fermion loops ; and the sum over n appears as the argu- ment of an exponential because the vacuum persistence amplitude receives contributions from graphs with any number of disconnected loops . The F-independent factor Det -9is less easy to derive from the Feynman rules ; it represents the contribution of any number of fermion loops that carry no vertices . More to the point, a formula like Eq . (9.5.64) allows us to derive non- perturbative results by using topological theorems to derive information about the eigenvalues of kernels like A'[F]. This will be pursued further in Volume II . 9.6 Path-IntegralFormulation o f Qu antumElectrodynam ics The path-integral approach to quantum field theory really comes into its own when applied to gauge theories of massless spin one particles, such as quantum electrodynamics . The derivation of the Feynman rules for quantum electrodynamics in the previous chapter involved a fair amount of hand-waving, in arguing that the terms in the photon propagator AP''(g) proportional to q Yor qV could be dropped, and that the purely time-like terms would just cancel the Coulomb term in the Hamiltonian, so that the effective photon propagator could be taken as qPV1q2. To give a real justification of this result using the methods of Chapter 8 would involve us in a complicated analysis of Feynman diagrams . But as we shall now see, the path-integral approach yields the desired form of the photon propagator, without ever having to think about the details of Feynman diagrams . In Chapter 8 we found that in Coulomb gauge, the Hamiltonian for the interaction of photons with charged particles takes the for m H[A} U1} ...~ Hm+ dux ~ zII_L 2 + 1(V x A )' -A - J~+ Vc;ou ] (9.6.1) Here A is the vector potential, subject to the coulomb gauge conditio n while IIL is the solenoidal part of its canonical conjugate, satisfying the 414 same co nstraint9Path- Integral Method s V_II1=0. (9.6.3) Also, HM is the matter Hamiltonian and Vc,,,,1 is the Coulomb energ y VcoUi(0= 2 dux d 'y Jo(X,t)j"(Y,t) 47rlx - Y! - (9 .6.4) Just as for any other Hamiltonian system, we can calculate vacuum expectation values of time-ordered products as path integrals " x,i a,i _x/ Xexp id4x[n.a-;n'- ~ (Vxa)2+aJ+~° ~1 x[llo(v .ax)] 6([Hv ..x](9.6.5) where yaj(x) are generic matter fields . In writing Eq .(9,6. 5)in terms of a matter Lagrangian density, we are assuming that HM is local and either linear in the matter ors (as in spinor electrodynamics) or quadratic with field-independent coefficients (as in scalar electrodynamics) . We have inserted delta functions** in Eq .(9.6.5)to enforce the constraints ( 9.6.2) and (9.6.3). The argument of the exponential in Eq .(9.6.5)is evidently quadratic in the independent components of n (say, aland n2), with field-independent coefficients in the term of second order in n . Thus, according to Eq- (9.A,9), the integral over ir can be done (up to a constant factor) by setting aequal to the stationary point of the argument of the exponential , ` Note that YE(x) is the interpolating c-Wernher field for the quantum operator R L, whose com- mutation relations with each other and with A are the same as those of R, but which unlike i 1 commutes with all canonical matter variables . This is not strictly accurate .I fwe take the canonical variables to be, spy7 ut, u~ and ~1, ~2, with a3 and n3 regarded as functianals of these variables given by Lqs .(9.5.2)and (9 .6.3), then we should insert the delta function s 4 (al(x)+0z' -1(a In I fWl+6z1R2[x) f1 XddtVCoul Howeve r, this differsfrom theproduct of delta func tionsinF.,q.(9,6.5)only by a factor lletc? which a lthough infiniteis field-independent and he nce ca ncels in ratios like (9.4.1). 9.6 Path- Integral Formulation of Quantu m Electrodynamic s It = a: x.i x, &" X CA (rl'B exp i dux ['2 - ~(Vxa)2 +aj i dt V(.+ ieterms [llv .ax.]))_~415 (9.6G) To bring out the essential covariance of this result, we use a trick . Intro- duce a new variable of integration ao(x), and replace the Coulomb term - f d t VC,,j in the action wit h fd4X[__a0(xj(x)+21 (Va°w)2(9.6.7) Since (9 .6.7) is quadratic in a°, the integral over a' can be done (up to a constant factor) by setting u°(x) equal to the stationary point of (9 .6.7), i.e., to the solution o f or in❑ther words, to aa (X} t)Y d3 Yjc~ (Y~ t~ 47rx-yl(9.6.8) Using this in Eq .(9.6.7 just gives the Coulomb action -J'dt V~,,,,1.Hence we can rewrite the argu ment of the exponential in Eq . (9.6.6)as [j,2_ 1 11jI 0)2 - ¢ f jl,, f1u''+a,, jO+Y + total derivative s with ~',,,, =OUaL, -- 0, a,,, and integrate over a° as well as over a and matter fields . That is, the path integral (9 .6,6)is no w 1 ))VAC dap(x) A"(W X OA ((",g...eXP(i I [a, y 1) 6 (V-aw) where Iis the original actio n I [a, Y'] _ d 4x [ --- ~.~~ftti~,fI"+a,~,7~` + M] + ic term s.OA9 ) (9.6.10) Now everything is manifestly Lorentz- and gauge-invariant, except for the final product of delta functions which enforce the Coulomb gauge 416 9 Path-Integral Method s condition .`* To make further progress, we shall use a simple version of a trick4t1 that in Volume II will be used to treat the more difficult case of non-Abelian gauge theories . For simplicity, we shall deal here with the case where the operators GA[A,TI,0,3 [A, TI,...as well as the action I[a,W] and measure [jj dal[jj d] are gauge-invariant . First, replace the field variables of integration a,,(x) and W(x) everywhere in Eq .(9.6.9)with the new variable s WAN =_exp (i q,,,,A(x)) w W(9.6.11) (9.6.12) with arbitrary finite fi(x) . This step is mathematical triviality, like changing an integral f T,f(x)dx to readf!"_',f (Y)dy, and does not require use of the postulated gauge invariance of the theory . Next, use gauge invariance to replace apn(x) and t~1,,n( .x) in the action, measure, and 61- functions with the original fields a,,(x) and ip?(x), respectively . Eq .(9.6.9) then becomes 60,[A,,T], [A,qq, I) VAC_ x,Y x,C Xexp(i I [a, Wj) fj 5(V-a(x)+V'A(x)) (9-6,13) X Now, the function A(x) was chosen at random, so despite appearances the right-hand side of Eq .(9.6.13) cannot depend on this function . We shall exploit this fact to put the path integral in a much more convenient form . Multiply Eq. (9.6.13) by the functiona l B [11, a]= ex p- lis dux (00a'--❑'A z (9.6.14) (where a is an arbitrary constant), and integrate over A(x) . By shifting the integration variable la(x), and noting the actual A-independence of (9.6.13), we see that the effect is simply to multiply Eq .(9.6,13)with the field-independent constant exp - ~2 A) 2) [Hciix](9.6.15) Note that now o"(_r)is no t e u al to the value (9_6_8), butis an independentvariable o fintegration- Wt w illnot i ntegrate over (x) f irst, w hich would l ead backtoEq. (9.6.6),butinsteadwill t reat itintandem with a(x]. 9.6Path-Integral Formulation of 'Quantum Electrodynamics 41 7 This factor cancels out in the connected part of the vacuum expectation value ,and thus has nophysical effect .But {9 .6.13} is only A-independent after we i nteg rate over a y(x) and tt7(x).Wecanjust as well integrate over A(x) before we i ntegrate over &`(x)and zki{x},in w hich case the factor 11,6 (V  a(x)+ V'A in E q. (9-6,13) is replaced wit h dA(x) exp-'la ex(Ooa0 -v2 A)2) 6(V  a(x) + V'1 1 Ocexp i oc d4x (09 },ali)2 ) 1 (9.6.16) ~ where again means proportional with a field-independent factor . Dropping constant factors, Eq . (9.6.9)now become s X,p X ,JV I (9.6.17) where lei~~~ , Y)l [a, W1 - HOC (0 ~Q~`)2 C~4~C. (9.6.18) This is now manifestly Lorentz-invariant . We consider the new term in (9.6.18)as a contribution to the unper- turbed part of the action, whose photonic part now read s Io[CI]= dxL-~((Cx,, -f~'al') ~" 7)2 + iE' teI' iY1SJ y~ L (9.6.19) where 2 2 9wc,VY ~x~c_ (1 -~~ ox~ c~ ~, 64(X-Y)+ ie terms Yp Y The photon propagator is then found immediately byinverting the 4 x 4 matrix in the integrand of E q.(9.6.20) ~av Axx by = {2rt}-4jd4qtl~"_ i~ + ~q(9-6.21) q (q2 - i,-)2 We are free to choose aas seems most convenient . Two common choices are ac .- 1, which yields the propagator in FeynnEangauge : ~.u A~~yman = lf2n)-4 L~4~' (9.~.22) ~ ~~ 418 9 Path-Integra l Method s or oc = oo, in which case the factor (9 .6.14) acts as a delta function, and we obtain the propagator in Landau gauge (often also called Lorentz gauge) : PXjY ,f lq2 - ic (q2 -ic)2 Prac tical calculations are made far more convenient by worki ng with s uch manifestly Lore ntz-invariant in teractions and propagato rs. 9.7 Var ieties ofStatistics * We can now take up a question raised in Chapter 4 : what are the possibilities for the change of state-vectors when we interchange identical particles ? For this purpose, we will consider the preparation of the initial or final states in a scattering process . Suppose that a set of indistinguishable particles in either of these states is brought to a particular configuration with momenta p7, p2, etc . from a standard configuration with momenta k1, k2, etc ., by some sort of slowly varying external fields, keeping the particles far enough apart in the process to justify the use of non-relativistic quantum mechanics . (Spin indices are not shown explicitly here ; they should be understood to accompany momentum labels .) To calculate the amplitude for this process we can use the path-integral method,"` taking the qs and ps of Section 9 .1 as particle positions and momenta, rather than fields and their canonical conjugates . These always satisfy canon ical commutation rather than anticommutation relations, whether or not the particles are bosons or fermions or something else, so at this point we are not committing ourselves to any particular statistics . The path-integra lformula (9 .1.34) gives an amplitude ( AI,p2,... Iki,k2, .)D as an integral over paths in which one particle is brought continuously from momentum k, to momentum p1, another identical particle is brought continuously from momentum k ,)to momentum p2, and so on . The subscript "U' indicates that this is the amplitude we would calculate for distinguishable particles . In particular, th is amplitude is symmetr ic under permutations of the ps and simultaneous permutations of the ks, but has no particular symmetry under separate permutations of the ps or ks . Bu t This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . Here I am following the discussion of Laidlaw and C.De itl,11 except that they apply the path-integral method to the whole scattering process, rather than just the preparation of initial or final slates, to a relativistic theory the possibility of particle creation and annihilation makes it necessary to apply the path-integral method tofields rather than particle orbits . For us, this is not a problem, because we limit such calculations to sulficiertily early or late times, when the particles participating in a scattering process are all far apart . 9.7 Varieties rrf Statistics 419 if the particles are really indistinguishable, then there are othe rpaths that are topologically distinct, but that yield the same final configuration . For space dimensionality d L 3, the only such pathst are those that take kl,k2,... into some non-tr ivial permutation of pi,R2, ' ' -- Hence the true amplitude should be writte n (Pz,A2,... k 1a k,,...~QP~PYI>pJA2}... I ki>kz,--'}n (9 .x.1) the sum running over all Nt permutations of the Nindistinguishable particles in the state, and C a set of complex constants . These amplitudes must satisfy a composition rule appropriate for indistinguishable particles : 1 ~ *42,- kj,k2,- f d'ql d'q2 ... ~Pl, P2, lql, q2 , Using Eq . (9-7-1), this is the requiremen t CP PiaP;Yza... k I ,k2, .) D =C: Fd3qid 3q2.. Applying a permutation 29"to both the initial and final states in the first amplitude on the right ,this i s X(Py+r:~F1,py'r~+2,' Iq :y,+>>q9W+2,..)n (9orr~, q, `2, ki, k2,...}D But the amplitudes { P1,p2, '.. lkl,k2,...}D satisfy the compos ition rule for distinguishable particle s (PI, P2, Jkl,k,), -)D= f d3q, d3q2 ... * 7 V ... Iql~ (125 ... )D x ~q1, q2,'.. Ik1,k2t...~D s (9 .7.3) sothe composition rule for the physical amplitudes may be writte n r~Af ~'~pr~err ~ This is expressed formally in the statement that the first hmmotopy group of Configtiration space in d ~ 3 is the permutation group .12 By 'configuration space' for N distinguishable particles is meant the space of N d-vectors, excluding d-vectors that coincide with (or are within an arbitrary limiting distance of) each other, and identifying configurations that differ only by a permutation of the vectors . 420 9Path-Integral Method s which will be satisfied if and only i f Q T r.T ffCY rQ + (9.7.4) That is, the coefficients Cymust furnish aone-dimenaianal representation of the permutation group . But the permutation group has only two such representations : one is the identity, with C, y= +1 for all permutations, and the other is the alternating representation, with C q = +1 or Cg = -1 according to whether _~?,is an even or odd permutation . These two possibilities correspond to Bose or Fermi statistics, respectively J The nice feature of this argument is that it makes it clear why the case of two space dimensions is an exception . In this case there i a much richer variety of topologically distinct paths,1 For instance, a path in which one particle circles another a definite number of times cannot be deformed into a path where it does not . In consequence, in two space dimensions it is possible to have anyoras,is particles with more general permutation properties than just Fermi or Bose statistics .8a Appendix Ga ussian Mu ltipleIntegra ls We wish first to calculate the multiple integral, over a finite number of real variables ~ ,, of the exponential of a general quadratic function o f rld~fiexP rs r(9,x.1) (9.A.2) where K, .,, Lr, and M are arbitrary constants, except that the matrix K is required to be symmetric and non-singular . For this purpose, we begin by considering the case where If s, L ,and M are all real, with ors also positive . The result in the general case can then be obtained by analytic continuation . Any real symmetric matrix can be diagonalized by an orthogonal matrix . Therefore, there is a matrix 9'with transposeyT = 9`1 such tha t ( 5rTK,9" =~rsKr - (9.A.3) 'T'here has been much discussion in the literature of possibilities other than Rose or Fermi statistics, often under the label parasratistics . It has been shown" that parastatislics theories in d ~ 3 space dimensions are equivalent to theories in which all particles arc ordinary fermions or bosons, but carrying an extra quantum number, so that wave functions could have unusual properties under permutations of momenta and spins . This is expressed in the statement that the first hornatapy group of configuration space in two space dimensions is not the permulation group, buy a larger group known as the braid group. 14 Appendix Gaussian MultipleInteg rals 421 Because K is assumed positive and non-singular, the eigenvalues Kr are positive-definite . We can use the matrix Yto perform a change of variables : Y{(9.A.4) The Jacobian JDet Yjof this transformation is unity, so the multiple integral (9.A.1)is now given by a product of ordinary integrals : e-' ~ d~'exp e -M-Mexp~ 12K , But the determinant and the reciprocal of Eq. (9.A.3)give DetK =riKp ri ~ - I so Eq . (9.A.5) may be w ritten Det ~ i'S(9.A.5) (9.A.6) Eq.(9.A.1)defines a function of K,.,,L,., and M that is analytic in Kr;in a finite region around the surface where Krs is real and positive, where the integral converges, and for such K, is analytic everywhere in L, and M. Since (9.A.6)equals {9 .A.1} for K, L, ., and M all real, with K,S also positive, Eq . (9.A.6) provides an analytic continuation of Eq.(9.A.1) to the whole complex plane, with a cut required bythe square root . The sign of the square root is fixed b ythis analytic continuation . In field theory Kra is actually imaginary, except for a small real part due to the `ic term' . It is useful to express Eq . (9.A.6)in terms of the stationary point of the function (9.A.2): ~r - - EsK-' L s as(9,A.7) (9.A.8) ()y12exP{_Q()} 271 This is the result to remember : Gaussian i ntegra lscan be evaluated up to adeterminant factor bysetting the integra tionvariableequal tothe point where the argument of the e xponentia lis stario nury. 422 9Path-Integral Method s Wenext wish to use this result to c alculate the integrals I~.l...r2,ro fld~r6- riLIZ... ~r?N ex ~]{-~~ rS10) (9.A. 21: K r~~r ~ 5 (Integrals of this sort with an odd number of ~-factors in the integrand obviously vanish .) From the power-series expression of exp ~- ~ L r~r) in Eq .(9.A.1),we have the sum rul e N=O rirv-rz,% r - 11 &';rexp - Lr~r'"'_Krs~r~5 ~ r r r s K ()]'12 2n 2rs Det(K)I (Y L,L,K _' . (9 . A. 11) N=0 + 'N Comparing thecoefficients of L, .,L,2 ,.. L,,,❑nboth sides, weseethat I,',,2---,2Nmust b e proportional to a sum of products of elements of K-1, which symmetry requires totake the for m pairin&s of ri ,..r:n'(K-')paired indices pairs(9.A.12) Here the sum is over all ways of pairing the indices r'1   r ZN, with two pairings being considered the same if they differ only by the order of the pairs, or bythe order of indices within a pair . To calculate the constant factor Cr , awe note that the number v Nof terms in the sum over pairings in Eq.(9.A.12) is equal to the number (2N)? of permutations of the indices, divided by the number of ways N! of permuting index pairs and by the number 2' of permutations within index pair s (2N) f v~ = N!2N Therefore, Eq . (9.A.12) give s Lrj Lr,Lr2,Jrj r2---r2,V ~ VNCN (E rS(9.A. 13) {9.A.14} Compar ing this with Eq . (9.A. 11) shows that the factors (2N) ! a nd 1V!2N are cancelled by VN,leaving us with K Cr=Det 2 (9.A.15) Reference s For instance , Iri r2=1a(K-l)rjr2 [(K_1.)rjr2(K_l)r3r 4 +(K-l)rjt.j(K-1)r2r4+(K-')r1r4(K_1)rzr31r and so on, where 10 is the integral with no indice s ~ r r s )]-112= {Det(-) ] 2a Problems423 (9.A.16) (9.A.17) (9.A. 18) 1. Consider a non-relativistic particle of mass in,moving along the x-axis in a potential V(x) =fnc_)'x2/2 . Use path-integral methods to find the probability that if' the particle is at x, at time ti, then it is between xand x+dxat time t . 2. Find the wave function in field space of a state consisting of a single spinless particle of mass in D .Use the result to derive the Feynman rules for emission or absorption of such a particle . 3.Find the wave function in field space of the vacuum in the theory of a neutral vector field of mass m zf- 0 . Use the result to derive the form of the ie terms in the propagator of this field . 4. The Lagrangian density of the free spin 3/2 Rarita-Schwinger field Y±pis ap}fir + 3~ -MW ip V- Use path-integral methods to find the propagator of this field . References I.R, P . Feynman, The Prin cipleof Least Action in Quantum Mechanic s (Princeton University, 1942 ; University Microfilms Publication No . 2948, inn Arbor) . Also see R . P. Feynman and A. R. Hibbs, Quantu m Mecha nicsand Path Integrals (McGraw-Hill, New York, 1965) .For a general reference, see J . Cxlimrn and A . Jaffe, Quantum P hysics- A Functional I ntegra l Point of View, 2nd edn (Springer-Verlag, Ne w York, 1987) . 424 9Path-Integral Method s Z. P. A. M. Dirac, Fhys . Zeits.Sowjetuniora 3,62 (1933) . 3.R.P.Feynman,Rev. Mod .Phys.20,367(1948);Pays.Rev.,74,939, 1430 (1948) ; 76, 749, 769 (1949) ; 80, 440 (1950) . 4. L. D. Faddeev and N . Popov, Pays . Lett .B25, 29 (1967) . Also see R. P. Feynman, Ac ta Phys .Pol. 24, 697 (1963) ; S. Mandelstam, Phys . Rev.175, 1580, 1604 (1968) . 5.B.De Witt,Phy,s.Rev. Lett .12, 742 (1964) . 6. G. 't Hooft, IUuc l. Pays .B35, 167 (1971) . 7.I. S. Gerstein, R .Jackiw, B . W. Lee, and S .Weinberg, Phys .Rev.D3, 2486 (1971) . S. L. D. Faddeev, Teor . Mat . Fizika, 1, 3 (1969) ; translation in Theor . Math .Phys .1, 1 (1970) . 8a. J. Schwinger, Prac, Nat . Acad . Sci.44, 956 (1958) . 8b. K. Osterwalder and R . Schrader, Pays . Rev .Lett.29, 1423 (1972) ; Commun . Math . Phys .31, 83 (1973) ;Commun . Math . Phys .42,281 (1975 ). The ester alder-Schrader axioms require smoothness, Eu - clidean covariance, `reflection positivity,' permutation symmetry, an d cluster decomposition . 9. This section was largely inspired by discussions with J . Polchinski . 10. F. A.Berezin, the Method of Second Quantization (Academic Press, New York, 1966) . 11. M. G. G. Laidlaw and C . M. De Witt, Whys .Rev. D 3, f375 (1970) . 12. J. M. Leinaas and J . 1Vlyrheim, Nuovo Cimenra 3 7B, 1 (1977) . 13. Y. Ohnuki and S . Kamefuchi, Phys .Rev.170, 1279 (1968) ; Ann.Phys . 51, 337 (1969) ; K. Driihl, R . Haag, and J . E. Roberts, C'vmmun .Math . Phys .1$, 204 (1970) . 14. The braid group wa s Papersof E . Artin, ed. Reading, MA, 1965) .introduced b yE. Artie . See The Collected by S . Lang and J . E. Tate (Addison-Wesley , 15. F. Wilczek, Fhys . Rev . Lett. 49, 957 (198 2); K, Fredenhagen, M . R. GaberdieL and S. M. Riiger, Cambridge preprint ATP-94-90 (1994) . Also see J . M. Leinaas and J . Myrheim, Ref . 12. 10 Non-Pert urbative Method s We are now going to begin our study of higher-order contributions to physical processes, corresponding to Feynman diagrams involving one or more loops . It will be very useful in this work to have available a method of deriving results valid to all orders in perturbation theory (and in some cases beyond perturbation theory) . In this chapter we will exploit the field equations and commutation relations of the interacting fields in the Heisenberg picture for this purpose . The essential bridge between the Heisenberg picture and the Feynman diagrams of perturbation theory is provided bythe theorem proved in Section G .4: the sum of all diagrams for a process oc --+(3 with extra vertices inserted corresponding to operators a,(x), oh(y), etc. is given by the matrix element of the time-ordered product of the corresponding Heisenberg-picture operator s (T~-,TI -i0aW, -WAY) ... )Tu+) As a special case, where the operators D ,(x),Ob(x), etc . are elementary particle fields, this matrix element equals the sum of all Feynman dia- grams with incoming lines on the mass shell corresponding to the state x, outgoing lines on the mass shell corresponding to the state fl, and lines off the mass shell (including propagators) corresponding to the operators 0a(x),D&}, etc. After exploring some of the non-perturbative results that can be obtained in this way we will be in a good position to take up the perturbative calculation of radiative corrections . 10.1Symmetrie s one obvious but important use of the theorem quoted above is to extend the application of symmetry principles from 5-matrix elements, where all external lines have four-momenta on the mass shell, to parts of Feynman diagrams, with some or all external lines off the mass shell . For instance, consider the symmetry of spacetime translational invari- ance . This symmetry has as a consequence the existence of a Hermitia n 425 426 1 0Nan-Perturbative Method s four-vector operator PP, with the property that, for any local function O(x) of field operators and their canonical conjugates , [P,.(X), 0 wl =iOW - (10.1.1) (See Eqs .(7.3.28) and (7.3.29)-)Also, the states iand #are usually chosen to be eigenstates of the four-momentum : p~'Tj+ = T~c4 , PI'TflT = p PTO_ . (10 .1.2) It follows that for any set of local functions O(x), Ob{x}, etc . of fields and/or fiel dderivative s -Past)(q'1- ,Tf 0,,(x,), O b(X2)... JT~+) (T~-, [P,, TjO,,(x,), Ob(.X2) + '' 1] qVl+) ~,~ + fix ; ~...(i'- T~0,(x,),Ob(X2),...Ta+ j This has the solutio n (Tfl-, T I OAXO' Ob(XI), - - - Ixya+ ) =exp(i(Ffl- P.) 'X) F .b...(xi - Xza...~a where x is any sort of average spacetime coordinate(10.1.3) (10.1.4) and F depends only ❑n differences among the xs . (In particular, a vacuum expectation value can depend only on the coordinate differences .) We can Fourier transform Eq . (10 .1.4) by integrating separately over xA and the coordinate differences, with the result tha t fd'xj d4X2 -'' (To-, TIOa(X1),Oh (X2), - - - I kPU -1) x exp(-iki  x1 -iJc2 x2 -  ~ ~} oc 64(P# - Pa-kl - k? -- ' . -)  (10.1.6) We saw in Section 6 .4 that the matrix element of the time-ordered product is given b yapplying the usual coordinate-space Feynman rules to the sum of all graphs with incoming particles corresponding to particles in x, outgoing particles in fl, and external lines that simply terminate in vertices at x1,x2, .... The Fourier transform (10.1.6) is correspondingly given byapplying the momentum-space Feynman rules to the same sum of Feynman diagrams, with off-shell external lines carrying four-momenta kl, k2, ... into the diagrams . Eq .(10,1.6) is then just the statement that this sum of Feynman graphs conserves four-momentum . The result is obvious in perturbation theory, because four-momentum is conserved at 10.1Symmetries 427 every vertex, so it is not surprising to see the same result emerging without having to rely on perturbation theory . With somewhat more effort, one can use the Lorentz transformation properties of the Heisenberg-picture fields and the `in' and `out' states to show that the sum of all graphs with a given set of on- and ofd-stye]] lines satisfies the same Lorentz transformation conditions as the lowest-order terms . Similar arguments apply to the conservation of internal quantum num- bers, like electric charge . As shown in section 7 .3, a field or other operator 0,,(x)that destroys a charge q,,(or creates a charge -qu) will satisf y in the Heisenberg and interaction pictures alike . Also, if the free-particle states a and 0have charges q ,and q fl, then so do the corresponding `in' and `out' states . We then hav e (qp - qx) (T#-, T~O,(x), Oh(y), - - Ta 1 = (Tp -, [Q, T10.(x),Ob (y), - - - T,+) = -(qu + qh + - - -)(Tp-,T10OW, 04y), - - - I Tx+). Thus the amplitude (Tp -,TjD,(x),Oh(Y), vanishes unless charge is conserved (10.1.7) A somewhat less trivial example is provided by the symmetry of charge- conjugation invariance . As we saw in Chapter 5, there is an operator C that interchanges electron and positron operator s with ~ a phase factor . For the free electron field W{x}, this give s where f3C is a 4 x 4 matrix, which (for the Dirac matrix representation we have been using, with y5 diagonal) takes the far m 0 0 0 1 C0 0 ---1 0 0 ,--1 0 0 1 0 0 0 Appliedtothe free- particle electric curre nt inspinorelectrodynamics,this 428 10Non-Perturbative Method s gives If C is to be conserved in elect rodynarnics, it must then also be defined to anticommute with the free photon fiel d C(&')C=1= --&' In theories like electrodynamics for which C commutes with the interaction as well as H0, it also commutes with the similarity transformation Q(t) between the Heisenberg and interaction pictures, and so it anticommutes with the electric current of the interacting field s C(W^11~`T) C-1 = -TyuT { 10.1.8} and t he electromagnetic field i n the He isenberg-picture C(Ail)C-1 - - Au. {10.1.9} It follows then that the vacuum expectation value of the time-ordered product of any odd number of electromagnetic currents and/or fields vanishes . Therefore the sum of all Feynman graphs with an odd number of external photon lines (off or on the photon mass shell) and no other external lines vanishes . This result is known as Furry :s theorem .'It can be proved perturbatively by noting that a graph consisting of electron loops e, to each of which are attached ne photon lines, must have numbers I and E of internal and external photon lines related by an analog of Eq .(6.3.11): 21 -}- E ~- n e Hence if E is odd at least one of the loops must have attached an odd number of photon lines . For any such loop there is a cancellation between the two diagrams in which the electron arrows circulate around the loop in opposite directions . Hence Furry's theorem is a somewhat less trivial consequence of a symmetry principle than translation or Lorentz invariance ; it is not true of individual diagrams, but rather of certain sums of diagrams . Figure 1 0.1 illustrates the application of Furry's theorem that was historically most important, its use to show that the scattering of a photon by an external electromagnetic field receives no contributions of first order (or any odd order) in the external field . 10.2Pololog y one of the most important uses of the non-perturbative methods described in this chapter is to clarify the pole structure of Feynman amplitudes as 1 0.2 Polology 429 Figure 10 .1. The lowest-order diagrams for the scattering of a photon by an electromagnetic field . Here straight lines represent virtual electrons ; wavy lines represent real and virtual photons ; and the double line represents a heavy particle like an atomic nucleus that serves as a source of an electromagnetic field . The contributions of these two diagrams cancel, as required bycharge-conjugation invariance . functions of the momenta carried by external lines . Often the S-matrix for a physical process can be well approximated bythe contribution of a single pole . Also, an understanding of this pole structure will help us later in dealing with radiative corrections to particle propagators . Consider the momentum-space amplitud e Id4xi ...d'x, e-~q`xl ...e-iq,,-x, .~ TfAI(xi) ...A,(x,,)~) 4 430 10Non-Perturbative Method s The As are Heisenberg-picture operators of arbitrary Lorentz type, and ~... }~ denotes the expectation value in the true vacuum 'Yo+ =To- = To . As discussed in Section 6,4, if A1, - - ' A ,are ordinary fields appearing in the Lagrangian, then (10 .2.1) is a sum of the terms calculated using the ordinary Feynman rules, for all graphs with external lines corresponding to the fields Al,, A, carrying off-shell four-momenta qt ... qn into the graph . However, we will not be limited to this case ; the A imay be arbitrary local functions of fields and field derivatives . We are interested in poles of Gat certain values of the invariant squares of the total four-momenta carried by various subsets of the external lines . To b edefinite, let's consider G as a function of q', wher e with 1 r ::~n-1 . We will show that G has a pole at q2 = - M2, where m is the mass of any one-particle state that has non-vanishing matrix elements with the states At ...1q~ 4fo and A ,+l... A,,To,and that the residue at this pole is given b y -2 i +W(27Z)~~ a(q1 q2 ~Yi1~' -- Zc+. ,. ~ ~ r~~ where the Ms are defined by * Xi...~~~e-qI'xE , ..e-!qr',Yr(To, TIAl(xl)...Ar(Xr))TP,U = (2n)' r54 (41+' - -+ q, - P)ojp,~(q2... qr), (10 .2.4) f d4Xr+1 ,, ,d4x,e-qr+j'~Cr+1 ...e-iq,IX, x(Tp,aaTfAr+l(Xr+1)-..An(Xn)TQ with p P =_ p2+ m2), and the sum is over all spin (or other) states of the particle of mass rn. Before proceeding to the proof, it will help to clarify the significance o f Reca llthat i n the absence of lime-varying exter nal fields, there is no dis tinction betwee n`in' and out' one-particle states, so thatTp,ff -4 _ T p,M = " ~Pff 1 0.2 Polology . (10.2.3)if we wn'te it in the somewhat long-winded for m G(q, - -fin) --+ A431 1,2x(2-n)464(q, + + q, - k)(27r)"" (2Vk 2+M2) M01k,A2 q,)~ -i i X (27r)4k-2-+ M2 iE 7 x(2V'k2 +M2)112 Mk,n,O(R,+z... qn) (10.2.5) This is just what we should expect from a Feynman diagram with a single internal line for a particle of mass pn connecting the first r and the last n - r external lines ." However, it is not necessary that the particle of mass m correspond to a field that appears in the Lagrangian of the theory . Eqs.(10.2.3) and (1 0.2.6)apply even if this particle is a bound state of the so-called elementary particles whose fields do appear in the Lagrangian . In this case, the pole arises not from single Feynman diagrams, like Figure 10.2, but rather from infinite sums of diagrams, such as the one shown in Figure 1 0.3. This is the first place where the methods of this chapter take us beyond results that could be derived as properties of each order of perturbation theory . Now to the proof . Among then? possible orderings of the times x ~ ... xo in Eq . (10 .2.1), there are n! /r!{n- r)! for which the first r of the x9 are all larger than the last n -- r . Isolating the contribution of this part of the volume of integration in Eq . (10 .2.1), we hav e X0 min [x° - -.x°] - max [x°+1... x~~ X (To, T~Al(xl) ...A,(X,)l T fAr+l(xr+l) ...An(X0 I TO) + OT, (1 0.2.7) where `OT' denotes the other terms arising from different time-orderings . We can evaluate the matrix element here byinserting a complete set of intermediate states between time-ordered products . Among these ma y See Figure 10 .2. The factors (2ar)1{1[2(1c 2+nz2)] 1,'2 just serve to remove kinematic factors associated with the mass m external line in 111{ll,, and Mk ' , a, Also, the sum over d of the product of coefficient-function factors from these two matrix elements yields the numerator of the propagator associated with the internal line in Figure 10 .2. 432 10Non-PerturbativeMethod s rr+1 Figure 10 .2. A Feynman diagram with the pole structure (10.2.6).Here the line carrying a momentum k represents an elementary particle, one whose field appears in the Lagrangian , rr4-1 +2 Figure 10 .3. Figure 10 .3. A Feynman diagram of the class whose sum has the pole structure (1 0.2.6). Here the pole is due to a composite particle, a bound state of two elementary particles . The elementary particles are represented by straight lines, and interact by the exchange of particles represented by wavy lines . be the single-particle state Tl,,, of a definite species of mass m . Further isolating the contribution of these one-particle intermediate states, we have (mm II[xo...xrl-max [x,-+i...X01) d3p J ET (T0,TIA00 +OT,--A. (x,) I TPG) (Tp,, T I Ar+l(xr+l) ...An(Xn)jTo) (10.18) where `QT' now denotes other terms, here arising not only from other time- orderings, but also from other intermediate states . It will be convenien t1 1 2 1n 1 0.2 Pololog Y to shift variables of integration, so tha t and use the results of the previous section to write433 lpo~ TIAi(xi) ... A,(X,)JTp,,y ) =eip'-XI(To, TfA,(O)A2(Y2) ...A, (y,) (10.2.9) (Tp,jT5 TfAr+l (xr+l)...An (x, ) IPO ) = e-iP'Xr+1('i' P aq TfAr+l(0)...Arz(.Yn)lT p) (10 .2.10) Also, the argument ❑f the theta function become s min [x° ... x°] - max [x°+1 ... x°] = x°-X0~i + min [Oy~...y01 -max [ 0y +1... yn~ We also insert the Fourier representation (6.2.15)of the step functio n Z ~ dwe-tf 0 (r)2~i (,-cca + i c The integrals over x iand x ,.+7now justyield delta functions : X eiR2'Y :...e-igr'Yre-aRr+2 '}'r+2...C!qn'}`n fx ~ exp ~ -~c~ [rt~in[D y~...y~]- max[Uy ~t... y~1 2~i ~ W+ae X d 3p (xP,, T f A, (0) ...A,(Y),)) TP'17) x(Tp,,, T~ A,+, (0) - -- A, (y .)ITo) )4 0 ~p 2+M2+(t) x(27z 6'(++I+-+q,+p)6(qr+l + - - - +qn+ + OT (10 .2.11) We are interested here only in the pole that arises from the vanishing ❑f the denominator ci +ie, so for our present purposes we can set t he factor exp(-fco [train -max]) equal to unity . The integrals over both pand a) are 434 1 0Non-Perturbative Method s now t rivial, and yield the pole -1 +M2 +iE]G(ql ...qn)--*i(2n)'~'(qj + +qrj)Iqo ~q xE Moj_q,5(q2 . qn)Mq,5jo(4r+2 - --qn) + - - ~ (10 .2.2) OF where now q-q1+...+qr=-fir+l-'--... -qna MOj,, .,(q2...q,,)=j d'Y2 ...dlyr e-,qZ,y~...e-'qI'Y I X(To, TfAi(O)A2(Y2)-An(Yr)I TQ,or) , (10 .2.13) x T IA r+l(~ )Ar+2(Yr+2 )...An(Y►~~ITo) (10 .2.14) and the final `  ' in Eq . { 10 .2.12} denotes terms that do not exhibit this particular pole . (The `ether terms' arising from other single-particle states produce poles in q at different positions, while those arising from multi- particle states produce branch points in q, and those arising from other time-orderings produce poles and branch cuts in other variables .) Using Eqs.(1.2.9)and (10 .2.10), it is easy to see that these Ms are the same as defined by Eqs . (10 .2.4) and (10.2.5). Also, near the pole we can writ e I -q° - 007-M7 + i 6 -2 q2-+m 2 (We again redefine e by a positive factor 2/q2 +m2,which is permissible since -stands for any positive infinitesimal .) Eq . (10 .2.12) is thus the same as the desired result (10.2.3). This result has a classic application to the theory of nuclear forces . Let (D,,,(x) be any real field or combination of fields (for instance, proportional to aquark-arttiquark bilinear qp 5,c,q)that has a non-vanishing matrix element between a one-pion state of isospin a and the vacuum, normalized so tha t The matrix element of c% between one-nucleon states with four-momenta p,p' then has a pole at(p- p'}2-m2 . which 1sospin and Lorentz invariance (including space inversion invariance) dictate must take the 10.2PDlolOgY form" {N ',(7r, P 'lIJ4) IN,a, p) - >i(2,g)-3'G,,xY57aU) f)2 + -2(p - P Mir435 where u and u' are the initial and final nucleon spinor coefficient functions, including the nucleon wave functions in isospin space, and T, with a = 1,2,3are the 2 x 2 Pauli isospin matrices . The constant G . is known as the pion-nucleon coupling constant. This pole is not actually in the physical region for the matrix element (10.2.16), for which (p -p')2 >0,but it can be reached by analytic extension of this matrix element, for instance by considering the off-shell matrix elemen t where Nand N' are appropriate components of a field operator or product of field operators with non-vanishing matrix elements between one-nucleon states and the vacuum . The theorem proved above in this section shows then that exchange of a pion in the scattering of two nucleons with initial four-momenta pi, p2, and final four-momenta Pay P~ yields a pole at (P,- Pi)I_(p2- p~)1 --~ -n~~ G2 Snrinr2,nr,rv ,~ ~~(Pi + P2 - P1 - Pz )- K (p, - p')2 21 71 x(27r)-'( filly 9,~~~,)x(27z)-' u~ ySTau2). (10.2.17) (The easiest way to get the phases and numerical factors right in such formulas is to use Feynman diagrams ; our theorem just says that the pole structure is the same as would be found in a field theory in which the Lagrangian involved an elementary pion field .) Again, this pion pole is not actually in the physical region for scattering of nucleons on the mass shell, for which Cpl - pr )2>_0, but it can be reached by analytic extension of the S-matrix element, for instance by considering the off-shell matri x Lorentz and isospin invariance requires this matrix to take the form (u' I' ro u), where C' is a 4 x 4 matrix for which the bilinear { y' f-W} transforms as a pseudoscalar . Like any 4 x 4 matrix, F can be expanded as a sum of terms proportional to the Dirac matrices 1, jI„ [y+„ yw], ys~ ., and yc . The coefficients must be respectively pseudoscalar, pseudavectar, pseudotensor, vetlor, and scalar . Out of the two momenta p and p' it is possible to construct no pse ►ldo5calars or pseudovectors ; just one pseudntcnsnr, proportional to e}"'apppQ;two independent vectors, proportional to p~ or p "; and a scalar proportional to unity, in each case with a proportionality factor depending on the only independent scalar variable, (p - p' )2,By using the momentum-space Dirac equations for u and ti, it is easy to see that the tensor and pseudovector matrices in F give contributions proportional to y5, 436 element10 Nora-Pe rturbati veMethod s /d4xi dx2 d'x~d4x~ e~~Pi~'e-iP2~~1 e'~'i-x1e'P;-x, Although this pole is not in the physical region for nucleon-nucleon scattering, the pion mass is small enough so that the pole is quite near the physical region, and under some circumstances may dominate the scattering amplitude, as for instance for large e'in the partial wave expansion . Interpreted in coordinate space, a pole like this at Cpl -- p'1)2=(p2- p~)2 -m1 implies a force of range 1/r~z,~ . For instance, in Yukawa's original theory of nuclear force the exchange of mesons (then assumed scalar rather than pseudoscalar) produced a local potential of the form exp(-rrt,r)/47rr, which in the first horn approximation yields an S-matrix for non-relativistic nucleon scattering Proportional to the Fourier trans- form : d3X1d3 X2d3xId3x~ e-'x I -pI ~,-ix2 P2ex1'' P~FeiX2 F'P: r exp -~TitjEIXj -X2fX1 P X 63(XI-Xt f )63(X1 -- 41Ejxj- x 2l 1 1)2 + M 2 T he factor 1JL(Pi -Pi')2 +M2] is just the non-relativistic limit of the prop- agatorI/[(pl --p,1)2 + M~] in (1 0.2.17). (In (10.2.17) the energy transfer p° - p'~° for IP11 ~~ MN and IPiI <MN equals [p12 -ply]/2rr~ N, which is negligible compared with the magnitude IPt - p' l of the momentum trans- fer.) When Yukawa's theory was first proposed, it was generally supposed that this sort of momentum-dependence arises from the appearance of a meson field in the theory . It was not until the 1950s that it became generally understood that the existence of a pole at (p1 - pi )2 --*-rri2 follows from the existence of a pion particle and has nothing to do with whether this is an elementary particle with its own field in the Lagrangian . 10.3 Fie ldandMass Renormalizatio n We will now use a special case of the result of the previous section to clarify the treatment of radiative corrections in the internal and external line of general processes . The special case that concerns us here is the one in which the four- 10.3 Field and Mass Renormalization 437 momentum of a single external line approaches the mass shell . (In the notation of the previous section, this corresponds to taking r = 1 .)We will consider a function x (To, TI 6f(x1),A2(X2),-- ~T,,) (10.3.1) where Cj(x) is a Heisenberg-picture operator, with the Lorentz transfor- mation properties of some sort of free field W( belonging to an irreducible representation of the homogeneous Lorentz group (or the Lorentz group including space inversion for theories that conserve parity), as labelled by the subscript /', and A2, A3,etc. are arbitrary Heisenberg-picture oper- ators ;. Suppose there is a one-particle state T,,1iQ that has non-vanishing matrix elements with the states 61.To and with A2A3...To.Then ac- cording to the theorem proved in the previous section, G fhas a pole at q i = _M2, wit h Ge (q rq2-..~ ~-2i 9j~ + (2n)' (i' C~r(Q) ~',,,,) X f d4X2 ...e-iq2'-X2(Tql,u T I A2(X2) ...ITO) We use Lorentz invariance to wr ite (To,ef(O)TC11,C) =(21r)-1/2Nue(qj, a),(10.3.2) (10.3.3) where u~(9, u) is (aside from the factor (2n )-3/2)the coefficient function* appearing in the free field W1 with the same Lorentz transformation properties as 0/, and Nis a constant . (It was in order to obtain Eq .(10.3.3) with a single free constant Nthat we had to assume that Ce transforms irreducibly .) We also define a 'truncated" matrix element M,, by jdX2---e-~q2'-X2 ...(Tql,u TIA2(X2) ... ITO) Eq. (10 .3.2) the n reads, for q2 --* -r n G1 ---),=2f qt } m q~ + m2 - ie(10.3.4 (ZU.3.5) According to Eqs . (6.2.2) and (6.2.18), the quantity multiplying ,,,+ in (10.3.5)is the momentum space matrix propagator -id,,,p(gl) for the fre e For instance, for a conventionally normalized free scalar field, W . (yi,a) =[2 qi +m~1 -1/2 438 1 0 Non-Perturbative Method s field with the Lorentz transformation properties of 6~,(or at least its limiting behavior for q~ --+-rn2), so (10.3.5)allows us to identify Me as the sum of all graphs with external lines carrying momenta iql, qz .. corresponding to the operators 6(, A2,..., but with the final propagator for the01line stripped away . Eq . (10 .3.4) is then just the usual prescription for how to calculate the matrix element for emission of a particle from the sum of Feynman diagrams : strip away the particle propagator, and contract with the usual external line factor (2n)-3}2U* . The only discrepancy with the usual Feynman rules is the factor N. The above theorem is a famous result due to Lehmann, Symanzik, and Zimmerman,3 known as the reduction formula, which we have proved here bya somewhat different method that has allowed us easily to generalize this result to the case of arbitrary spin . One important aspect of this result is that it applies to and sort of operator ; G,,need not be some field that actually appears in the Lagrangian, and the particle it creates may be a bound state composed of those particles whose fields do occur in the Lagrangian . It provides an important lesson even where 6,,is some field 'PI in the Lagrangian : if we are to use the usual Feynman pules to calculate S-matrix elements, then we should first redefine the normalization of the fields by a factor 1/N, so that (with apologies for the multiple use of the symbol T) : To, T~(O)Tq,a) = (27r)-"' u,,(q,a) (10.3.6) A field normalized as in Eq. (10.3.6)is called a renvrrraulized .field . The field renormaliztion constant Nshows up in another place . Sup- pose that there is just one of the operators A2, A 3,. ,.in Eq .(10.3.1),and take it to be the adjoint of a member of the same field multiplet as Cr~ . Then Eq . (10 .3.2)read s fdxi d4 X2e"I" e-42' x2 (TO, TJ0e,(x1)(1Y'*'(X2)JY0) q2 + M2- iE (O)T41,U) x f d 4 X2e-~q2,x,(Tqp7,0T,(0)To -2d2 , ~'+--M 2 This is just the usual behavior of a propagator (the sum of all graphs with two external lines) near its pole, except for the factor IN12. According to Eq.(10.3.6), this factor is absent in the propagator of the renormalized field Te . Thus a renormalized field is one whose propagator has the same I0.3Field and Ma ssRenorrtsaliaataon 439 beha viornear its pole as fir afree field, and the renormalized mass isdefineddgby the position of the pole . To see how this works in practice, consider the theory of a real self- interacting scalar field 013, the subscript B being added here to remind us that so far this is a `}pare' (i .e., unrenormalized) field . The Lagrangian density is taken as usual a s ;0~~~~~`d~$ M2 (D' -VB{(DB) . (10.3.7) In general there would be no reason to expect that the field cbB would satisfycondition (10 .3.6),nor that the pole i nq2 would be at -m$, so let usintroduce arenormalized field and mas s ID=Z-1~2 (DB , M 2=m~ +Jrra2(103.8) (10.3.9) with Zto be chosen so that 0 does satisfy Eq .(10.3.6), an d6rn.2chosen sothat the pole of the propagator is at q2=_rra2.(The use of the symbol Zin this context hasbecome conventional ;there is a di fferent Zfor each field in the Lagrangian .)The Lagrangian dens ity(10.3.7)may then be rewritten Y=o +-rj, 40 2(D2'010110- i m where(10.3.10) (103. 11) (10.3.12) In calculating the corrections to the complete momentum space propagator of the renormalized scalar field, conventionally called S(q)> it is convenient to consider separately the one-particle-irreducible graphs : those connected graphs (excluding a graph consisting of a single scalar line) that cannot be disconnected by cutting through any one internal scalar line . An example is shown in Figure 10 .4. It is conventional to write the sum of all such graphs, with the two external line propagator factors -i(27c)-4 (q2+m2 - ie)-] omitted, as t(27r)4II*(q2), with the asterisk to remind us that these are one-particle-irreducible graphs . Then the corrections to the complete propagator are given bya sum of chains of one, two, or more of these one-particle-irreducible subgraphs connected with the usual uncorrected 440 10 Non-Perturbative method s (a) (b) Figure 10 .4. Figure 10 .4. Diagrams that (a) are, or (b) are not, one-particle irreducible . These diagrams are drawn for a theory with some sort of quadrilinear interaction, like the theory of a scalar field 0with interaction proportional to 0'. propagator factors : F27 j~(27r)4q2+ 1 + (4 q2+ - ic[i(2)4n*(q2)] ( 2n)4q2+m2 ~2 +(2)lq2+ 2[i(2)4H {gr~~(27)4 q2+ m2 ie x[i(27r)4rr(q2)]2 --+- (103 .13)[()4q+ or more simpl y A' (q) =[q2 +rri2-ie]-a +[q2 +rra2--- ie]-1f1*( 42)[qz+M2- i,,]-1 (10.3.14) Summing the geometric series, this give s A'(q)-[q2+rri2,-II*(~~) _iel-i(10.3.15) In calculating II', we encounter a tree graph arising from a single insertion of vertices corresponding to t he terms in Eq . (10.3.I2) proportional to 0,0090 and 02, pl us aterm FI~oap arising from loop graphs li kethat in Figure 10 .4(a) = 11'{q)== _(Zv`t)L~2 + M,]+Z~m.2 +IILOOP(q2). ( 10.116) The condition that m .2 is the true mass of the particle is that the pole of the propagator should be at q2= --rn2, so tha t II*(-rn2) = 0 , (10 .3.17) Also, the condition that the pole of the propagator at q 2 = -M2 should 1 0.3Field and Mass RenorFrtalizatio n have a unit residue (like the uncorrected propagator) is tha t d q2=_M 2 These conditions allow us to evaluate Z and brn2 ; ~bM _ -r ILOOP(o) , z = 1 + d ~--2TTLOO~ ~4 ~)~ q2_-M2441 (10.3.18) (10.3.19) (103.20) This incidentally shows t hat ~ 6m2 and Z-I are given by a series of terms containing one ormore coupling constant factors, justifying the treatment of the first two terms inEq.(10.3.12) as part of the interact ion 1. In actual calculations it is simplest j ust to say that from the loop terms RiJa()p(q) we must subtract a first-order polynomial in q2with coefficients chosen so that the difference satisfie s Eqs. (10.3.17)and (10.3.18).Aswe shall see, this subtraction proce dure incidentally cancels the infinities that arise from the momentu mspaceintegrals i nIIL00P. However, as this discuss ion should make clear, the renormalization of masses and fieldshas nothing directly todowith thepresence of infinities, and would be neeessary even in atheoryin wh ichall m om.entumspace integrals were convergent . An important con sequence of the cond itions ( 10.3.17)and (1 0.3.1.8)is that it isnotnecessary to include radiative corrections in external lines on the mass shell .That is , Similar remarks apply to particles of arbitrary spin . For instance, for the `bare' Dirac field the Lagrangian i s We introduce reno rmalized fie lds and massesVs(Ta) - (10.3.22) nz =MB +6M . (10.3.24) (The subscript 2 on Z2 is conventionally used to distinguish the renor- malization constant of a fermion field .) The Lagrangian density is then rewritten Y=4+ Yi, (10.3.25) 442 10Non-Ferturbatiae M ethods Leti(27t )4E*( k) be the sum of all connected graphs, w ith❑ne fermian line coming in with four-momentum k and one going out with the same four- momentum, that ca nnot be disconnected by cutt ing through any single internal fermi online, and with external l ine propagator factors -i(27r)-4 and [i +m- ic]-t o mitted .{Lorentz invariance is being used to justify writing 1* a s an ordinary function of the Lorentz scalar matrix ko"j" .) Then the complete fermion propagator i s S'(k)=[i+m-itF]-i+[ik+rri- +rrx- (1Q.3.28) In calculating V(9 ) we take into account the tree graphs from the terms in Eq.(10.3.27) proportional to ' P_ ' and TT as well as loop contributions : VW--(Z2 - 1}[i 9+ m ]+Z26M +1L00P(9 ). {1 .4.3.29} The condition that the complete propagator has a pole at 1c2=_M2 with the same residue as the uncorrected propagator is then tha t Y" (IM) = 0, ay-*(9) aYLam=D, and hence ~ k(10.3.30) Z2ayLOOP(9)(14.3.31) Just as for scalars, the vanishing of [i 1 + mj-lY*(g) in the limit g---3 im tells us that radiative corrections may be ignored in external fermion lines . Corresponding results for the photon propagator will be derived in Section 10.5. 10.4 R enormalizedCharge a ndWard I dentities The use of the commutation and conservation relations of Heisenberg- picture operators allows us to make a connection between the charges (or other similar quantities) in the Lagrangian density and the properties of physical states . Recall that the invariance of the Lagrangian density 10.4Renormalized Charge and Ward Identifies 443 with respect to global gauge transformations T~ ~ exP(iq,,oc)Tc (with 2 an arbitrary constant phase) implies the existence of a curren t satisfying the conservat ion condition(10.4.1) (14.4.2) This implies that the space-integral of the time component of J Fis time- independent ~ ~Q= Lea Hl=0, where Q =-I d3XjO(10.4.3) (10.4.4) (There is a very important possible exception here, that the integral {10 .4.4 may not exist if there are long-range forces due to massless scalars in the system . We will return to this point when we consider broken symmetries in Volume 11.} Also, since it is a space-integral, Q is manifestly translation- invarian t and since Pis afour-vector, Q is invarian#, with respect to homogeneous Lorentz transformation s It follows that Q acting on the true vacuum To must be another Lorentz- invariant state of zero energy and momentum, and hence (assuming no vacuum degeneracy) must be proportional to Toitself . But the propor- tionality constant must vanish, because Lorentz invariance requires that (TO, J ,,To)vanish . Henc e Also, Q acting on any one-particle state 'I'p,,7,n must be another state with the same energy, momentum, and Lorentz transformation properties, and thus (assuming no degeneracy of one-particle states) must be proportional to the same one-particle stat e The Lorentz invariance of Qensures that the eigenvalue q(,) is independent ofpand u, depending only on the species of the particle . This eigenvalue is what is known as the electric charge (or whatever other quantum number 444 10Nan-PerturbutiveMethod s of which Pmay be the current) of the one-particle state .To re late this to theq,,,parameters in the Lagrangian ,we note that the canonical commutation relations giv e ~Ja(x, t), Te [Y} t)~ = - q,,Te,( Y,r)b'(x - y) , (10.4.9) or integrating over x : [Q,Te(y)j = -qIT((Y) (1Q.4,10) The same is true of any local function F(y) of the fields and field derivatives and their adjoints, containing definite numbers of each : [Q, F(y)] = -qvF(y) , (10.4.11) where q Fis the sum of the q ,for all fields and field derivatives in F(Y), minus the sum of the q~ for all field adjoints and their derivatives . Taking the matrix element of this equation between a one-particle state and the vacuum, and using Eqs . (10 .4.7) and (10 .4.8), we hav e (TO, fly)Tp~a,n) (qF- q(n))=0. Hence we must have %J-qF as long as (To, F(y)qIP'T'n) =~0(10.4.12) (10.4.14) As we saw in the previous section, Eq . (1x .4.1 4) is the condition that assures that momentum space Greens functions involving F have poles corresponding to the one-particle state 'Fp,,,q . For a one-particle state corresponding to one of the fields in the Lagrangian we could take F=tiYe, in which case q F= q?, but our results heTe apply to general one-particle states, whether or not their fields appear in the Lagrangian . This almost, but not quite, tells us that despite all the possible high-order graphs that affect the emission and absorption of photons by charged particles, the physical electric charge is just equal to a parameter q e appearing in the Lagrangian (or to a sum of such parameters, like q F) The qualification that has to be added here is that the requirement, that the Lagrangian be invariant under the transformations T( -y exP(iq~a)T,,, does nothing to fix the aver-alb scale of the quantities q~- The physical electric charges are those that determine the response of matter fields to a given renormalized electromagnetic field 14P . That is, the scale of the q e is fixed by requiring that the renormalized electromagnetic field appears in the matter 1_ .agrangian min the linear combinations [ 0y- iqeAI]Te, 1 0.4 Renormalized Charge and Ward Identities 445 sothat the current JP is JA_ ~ {1 0.4.15} But AP and q eare not the same as the `dare electromagnetic field' A$and `bare charges' q l3,,that appear in the Lagrangian when we write it in its simplest form (10.4. 16) The renormalized electromagnetic field (defined to have a complete prop- agator whose pole at p2 = 0 has unit residue) is conventionally written in terms of AB as so in order for the charge q tto characterize the response of the charged particles to a given renormalized electromagnetic field, we should define the renormalized charges by qe=VZ3qB1. (10.4.18) We see that the physical electric charge q of any particle is just propor- tional to a parameter qB related to those appearing in the Lagrangian, with a proportionality constant Z, 112 that is the same for all particles . This helps us to understand how a particle like the proton, that is surrounded by a cloud of virtual mesons and other strongly interacting particles, can have the same charge as the positron, whose interactions are all much wreaker . It is only necessary to assume that for some reason the charges q1yin the Lagrangian are equal and opposite for the electron and for those particles (two u quarks and one d quark) that make up the proton ; the effect of higher-order corrections then appears solely in the common factorZ31/2, In order for charge renormalization to arise only from radiative cor- rections to the photon propagator, there must be cancellations among the great variety of other radiative corrections to the propagators and electro- magnetic vertices of the charged particles . We can see a little more deeply into the nature of these cancellations bymaking use of the celebrated relations between these charged particle propagators and vertices known as the Wardidentities . For instance, consider the greens function for an electric current P(x) together with a Heisenberg-picture Dirac field T,,(y) of charge q and its covariant adjoint T,,,(a) . We define the electromagnetic vertex function FY 446 10Non-Perturbative Method s Figure 1 0.5. Diagrams for the first corrections to the electron propagator and vertex function in quantum electrodynamics . Here straight lines are electrons ; wavy lines are photons . of the charged particle b y cox day dze'p-x e-zk-y e+V-,~To, TI P(x) T,, {Y}Tm(4)~ ~ ) 1 64(p + k (10.4.19) iqSna,(k)FnP,m,(k , where - I,~ (~C)d~(~L - C~V d'Z(i', T IT n()'Om12}ITD)e4iky efi(= (10.4.20) According to the theorem of Section 6 .4, E q. (10 .4.20) gives the sum of all Feynman graphs with one incoming and one outgoing fermion line, i.e., the complete Dirac propagator . Also, Eq . (10 .4.19) gives the sum of all such graphs with an extra photon line attached, so FP is the sum of "vertex" graphs with one incoming Dirac line, one outgoing Dirac line, and one photon line, but with the complete Dirac external line propagators and the bare photon external line propagator stripped away . To make the normalization of S' and FAperfectly clear, we mention that in the limit of no interactions, these functions take the value s The one-loop diagrams that provide corrections to these limiting values are shown in Figure 10 .5. 10.4 Renorpraalized ChargeandWard Identitie s We can derive a relation between I'j' and S' by use of the identit y a.T~P(x)Tjy)Tm(z)~ = Tf (7,,JY(x)T,(y) T-,(z) ) C1XV447 [J0ix , +6(x" -?) T IIP.(Y)ljo(x), qj,,(z)]1(10.4.21) where the deita functions arise from time-derivatives of step functions . The conservation condition (10 .4.2) tells us that the first term vanishes, while the second and third terms can be calculated using the commutation relations (10 .4.9), which here giv e IJO(x, t), Tn(y, t)] = -qT,,(y, t)6'(x - y) and its adjoint [J"(x, 0, Tn(y, 01=qTn(Y, t)6'(Y- -Y) Eq. (10 .4.21) then reads(10.4.22) 0.TfJl'(x)T,, (y)T. -q64(Xy)TIT,, (y)T ..(z) G JCS + q64(X-z) TfTyj(y)qjrn(z) (10 .4.24) Inserting this in the Fourier transform (10 .4.19) give s or in other word s This is known as the generalized Ward identity, first derived (bythese meth- 4The original Ward identity, derived earlier b yWard' ods) by Takahashi . from a study of perturbation theory, can be obtained from Eq . (10A25) byletting t approach k . In this limit, Eq . (10 .4.25) give s Oky The ferm ion propaga tor is related to the se lf-ene rgy insertion E*(*) by Eq. (10 .3.28) so Eq . (10 .4.26) may be written akp 448 10Non-Perturbative Method s Fora renormalized Dirac field, Eqs . (10.3.31) and (1 D .4.27) tell us that on the mass shell fik r'(k, k)U k=;:irk k' "Uk > (I0.A.28) where [iy ,kj'+ m]u k=[i^~pV + rrt]u~ = 0.Thus the renormalization of the fermion field ensures that the radiative corrections to the vertex function F.cancel when a fermion on the mass shell interacts with an electromagnetic field with zero momentum transfer, as is the case when we set out to measure the fermion's electric charge . If we had not used a renormalized fermian field then the corrections to the vertex function would have just cancelled the corrections due to radiative corrections to the external fermion lines, leaving the electric charge again unchanged , 10.5Gauge Invarianc e The conservation of electric charge may be used to prove a useful result for the quantities x (T, -,TfJju(x),J"(x)- - ,IT.+) ~ (10 .5. 1) In theories like spinor electrodynamics in which the electromagnetic in- teraction is linear in the field A m, this is the matrix element for emission (and/or absorption) of on- or off-shell photons having four-momenta q, q`, etc . (and/or -q, -q', etc .), with external line photon coefficient func- tions or propagators omitted, in an arbitrary transition a ---> A Our result is that Eq.(10.5.1) vanishes when contracted with any one of the photon four-momenta ... =0. (10 .5.2) Since M is defined symmetrically with respect to the photon lines, it will be sufficient to show the vanishing of the first of these quantities . For this purpose, note that by an integration b ypart s qpMAU...(q, qo".') f d 4Xfd4Xt... 07 x e-iq-Xe-iq? .Xt... (Tb-, a.TfP(x), JP'(x') ...IT,,+) (10,53)axp The electric current JP(x) is conserved, but this does not immediately imply that Eq . (10 .5.3) vanishes, because we still have to take account 1 0.5Gauge Invaria nce 44 9 of the x°-dependence contained in the theta functions that appear in the definition of the time-ordered product . For instance, for just two current s T {J(x)J"(v} } so, taking account of the conservation of JA(x ) aTfP(x)J_'(y) I =6(x" -yO)J"(x)J`(y) - 6(y) -x")J'(y)JO(x)xy = 6(X ° - Y°) [Jo(x),PWI . (10.5.4) With more than two currents, we get an equal-time commutator like this (inside the time-ordered product) for each current aside from PI(x) itself . To evaluate this commutator, we recall that (as shown in the previous section) for any product F of field operators and their adjoints and/or derivatives [Jo(X-,t), F(-, t)] = -qF F(X-", t)6-1(5~ - where q Fis the sum of the q es for the fields and field derivatives in F, minus the sum of the q(s for the field adjoints and their derivatives . For the electric current, q jis zero ;P(y) is itself an electrically neutral operator . It follows that [JU(.-+,t),J"(Y, t)] =0 (10 .5.5) andtherefore E q. (10,5 .4) vanishes, so t hatEq. (10.5.3) give s qM~~~...(q,qf, ) =0 (10.5,6) as was to be proved . There is an important qualification here . In deriving Eq .(10.5.5) we should take into account the fact that a pi-oduct of fields at the same spacetime point y like the current operator J` (Y)can only be properly defined through some regularization procedure that deals with the infinities in such products . In many cases it turns out that there are non-vanishing contributions to the commutator of J4 (X, t)with the regulated current Y{Y,t), known as Schwinger terms .'Where the current includes terms arising from a charged scalar field (D, there are additional regulator- independent Schwinger terms involving V(D . However, all these Schwinger terms are cancelled in multi-photon amplitudes by the contribution of additional interactions that are quadratic in the electromagnetic field, either arising from the regulator procedure (if gauge-invariant) or, as for charged scalars, directly from terms in the Lagrangian . We will be dealing mostly with charged spinor fields, and will use a regularization procedure (dimensionai regularization) that does not lead to Schwinger terms, so 450 10 Non-Perturbative Method s in what follows we will ignore this issue and continue to use the naive commutation relation (10.5.5). The same argument yields a result like Eq. (10 .5.2)even if other particles besides photons are off the mass shell, provided that all charged particles are taken on the mass shell, i .e., kept to the states Tp- and tI',+ . Otherwise, the left-hand side of Eq . (10.5.2)receives contributions from non-vanishing equal-time commutators, such as those we encountered in the derivation of the Ward identity in the previous section . One consequence of Eq .(10.5.2)is that S-matrix elements are unaffected if we change any photon propagator ApA ) by ❑'UV(q)-~ ~PV(q)+au 4v + qj~flw (1 0.5.7) or if we change any photon polarization vector b y where k o- k j, and rx~{, fl ., and c are entirely arbitrary (not necessarily constants, and not necessarily the same for all propagators or polarization vectors .) This is (somewhat loosely) called the gauge invariance of the S-matrix . To prove this result it is only necessary to display the explicit dependence of the S-matrix on photon polarization vectors and propagator s 14 X1AW2...v1v~---P1P2,.. T1 'T2,. . -kl, -kl r{,k2 (10.5.9) where MPI" .` is the matr ix element (10 .5.1) calculated in t he absence of electromagnetic interactions .* The invariance of Eq . (10 .5.9) under the `gauge transformations' (10 .5.7) and (10 .5.8) follows immediately from the conservat ion conditions (10 .5.2). (In Section 9 .6 we used the path- integral fo rmalism to p rove a special case of this theorem, that vacuum expectat ion values of time-ordered products of gauge-invariant operators are independent of the constant oc in the propaga tor (9 .6.21).) This result is not as elementary as it looks, as A applies not to individual diagrams, but only to sums of diagrams in which the current ve rtices are inserted in allpossible places in the diagrams . There is a particu larly important application of Eq . (10 .5.2) to the calculation of the photon p ropagator . The complete photon propagator , The states a and hare the same as x and P, but with photons deleted . Note that the argulnen#s of M are all taken to be incoming four-momenta, which is why we have to insert various signs for some of the arguments of M in Eq. (10-5-9 ). 10.5Gauge Invarianc e conventionally called ❑~~,(q), takes the form451 (10.5.10) where P'isproportional to the matrix element (10.5.1)with t wo currents and a and both the vacuum state ,and❑.,is the bare photon propagator, written here in a general I ..orentz-invariant gauge a s Z-i~ From Eq .(10.5.2) we have here q 1'1,,,{q} = 0, so that (10.5.12) On the other hand, just as we did for scalar and spinor fields in section 10.3,we may express the complete photon propagator in terms of a sum II* (q)of graphs with two external photon lines that (unlike M) are ❑ne-photon-irreducibl e or in other word s Then in order to sati sfy Eq . (10 .5.12), we must have(10.5.14) (10.5.15) This together with Lorentz invariance tells us that FI*(q) must take the form Then Eq .(10.5.13) yields a complete propagator of the for m 11,Uv - ~ (R')R~jqVlq' where(10.5.17) (10.5.18) ,,A) receives contributions only from one-photon- NQUF, because n" irreducible graphs, it is expected not to have and pole at q2 =0. (There is an important exception in the case of broken gauge symmetry, to be discussed in Volume II .) In particular, the absence of poles at q2=Din the q ,,q,, term in CI ;,,(q) tells us that the function -g(q2) in Eq . {10.5.16} also has no such pole, and so the pole in the complete propagator (10.5.17) is 452 10 Non-Perturbative Method s still at q2 = 0, indicating that radiative corrections do not give the photon a mass . For a renormalized electromagnetic field, radiative corrections should also not alter the gauge-invariant part of the residue of the photon pole inF_q.(10.5.17), so This condition leads to a determination of the electromagnetic field renor- malization constant Z3 . Recall that when expressed in terms of the renor- malized field (10 .4.17), the electrodynamic Lagrangian takes the for m The function ar(q2) in the one-photon-irreducible amp litude is the n where 7ELpOF is t he contr ibution of loop diag rams.Itfollows t hat Z3 1 + E LQOPO) - (10.5.21) In practice, we just calculate the loop contributions and subtract a constant in order to make n(0) vanish . Incidentally, Eq . (10 .5.18) shows that for q ' * 0 the gauge term in the photon propagator is altered by radiative corrections . The one exception is the case of Landau gauge, for which Ifor all q2. 10.6 E lectromagnetic Form Factors and Magnetic Momen t Suppose that we want to c alculat ethe scattering of a particle byan external electromagnetic field (orby the electromagnetic field of another particle), to first order in thi selectromagnetic field ,but to allorders in all other interactions (including electromagnetic) of our p article.For th is purpose, weneed to know the sung ,of the contributions of all Feynman diagrams with one incoming and one outgoing particle line, both onthe mass shell, plus a photon line ,which may be on or off the mass shell . Accord ing to the theorem of Section 6.4,this sum is g iven by the on e- particle matrix element of the electromagnetic current ,1 P(x).Let us see what governs the general form of th ismatr ixelement . Acco rding to spacetitne translation invariance ,the one-p article matrix element of the electromagnetic current takes t he form (XFP",,,, J"(x)TP,,) = exp(i(p - p') - x) (Tp,,,eJ"(O)Tp,,, The cur rentconservatio ncondi tionOW}' = 0 then require s V Al (TV ,c,,P(4)'gy'p d)= 0.(10.G.?] (10.x.2) 10.EElectromagnetic Form Factor s Also, setting kt= 0 and integrating over all x give s (Tpra,',QTp,a= (2 7C)W(p --p')(TPf,df, JII(O)T p, Using Eq . (10.4.8), this gives (TP'CF?' P(O)TPC) = (21r)-'q 66Y6 I453 (10.6.3) where q is the particle charge . We also have at our disposal the constraints on the current matrix elements imposed by Lorentz invariance . To explore these, we will limit ourselves to the simplest cases : spin zero and spin A . The analysis presented here provides an example of techniques that are useful for other currents, such as those of the semi-leptonlc weak interactions . Spin Zer o For spin zero, Lorentz invariance requires the one-particle matrix element of the current to take the general for m (TV, JP(O) p) = q(21r) l'l2p"}-1f2(2P0)-if2~ (p'~P)} (10 .6.4)1 where p aand p'o are the mass-shelf energi es (pa = /p2 + m),andt(p',p) is a four-vector function of the two four -vectors p "'and p y. (We have extracted a factor of the charge q of the particle from ffor future convenience .)Obviously, the most general such four-vector function takes theform of a linear combination of p"'and p ", or equiv alently of p r}`+p" and p FP - e , with scalar coe fficients . But the scalars p 2 and pry are fixed at the values p'- = P'2= -M2 ,sothescalar variables that can b eformed from eand p "'can be taken as functions only of p p', or equivalently o f Thus the function A(p', p)must take the form(10.6.5) (1o.6.6) The fact that J Pis Hermitian implies that f~`(p',p)` = f"(p,p'), so that both F(k 2)and H (k')are real . Now (P' -P)'(Pr+P) vanishes, while (p ' - p) 2 =0 is not generally zero, so the condition of current conservation is simpl y H(k')=0. (10.6.7) Also, setting p' = p and p = 0 in Eq . (10 .6.4), and comparing with Eq . (10.6.3), we find that F(a) =1 . (10.6.8) The function F(k2) is called the electromagnetic form factor of the particle . 454 10Non-Perturbative Method s Spin For spin 1, Lorentz invariance requires the one-particle matrix element of the current to take the general for m (TP%U,j- P'(O)Tp,q)=iq(27r)-3 jj(PI, t7,)F,,(P,, P)U(p, a)(10.6.9) where F" is afour-vector 4 x 4 matrix function of p',p', and a nd u is the usual Dirac coefficient function . We have extracted a factor iq to make the normalization of FP the same as in the previous section . Just as for any 4 x 4 matrix, we may expand Fil in the 16covariant matrices 1, Y O, CYp,i'aa a-5y,.,and y5- The most general four-vector I,,` can therefore b ewritten as a linear combination o f 1 p 1`,p'P Y11aP11 PFD p 1`iP,pry~f [lip, Y al LEI"*A Lf}lt, ill' Lj, ~fJt'kl> L T5: NON E with the coefficient of each term a function of the only scalar variable in the problem, the quantity (10.6.5).This can be greatly simplified by using the Dirac equations satisfied by u and U. In consequence, we can drop* all but the first three entries : p",p"', and y1'. We conclude that, on the fermion mass shell, Pmay be expressed as a linear combination of y P, pil, and p`P, which we choose to write a s PIP-- (p + p')"G(k') + H(k')] u(p, cr) .(10.6.10)2m 2m This is obvious for the terms pu J,0' ~ , e J', and p"'0',which may be replaced respectively with imP+`, imp'p, imps, and irxp'u, which are the same as terms already on our 1isl . Also, we can write 01=2y' j - {;°",0? = zy 10- 2p ", which may be replaced with 2we - 2p+`, a linear combination of ierrns already on out list, The same applies to [y O, J'] . Also , which may be replaced with 2rn2+2p  p' = -0. Hence the terms [ ~, yi']p+` and [ ~ . Or]A'A give nothing new . Finally, to deal with the last term we may use the relatio n 6( Contracting this with p,and p¢ and then moving all 0factors to the right and Jr factors to the left again gives a linear combination of p ", p'~, and 514 . 10.6Electromagnetic Farm Factor s The hermiticity of JP(O) implies tha t455 sothat F(k'`),G(k2), and H( k 2)must allbereal function sof k2. The conserv ation condition (10.6.2)is auto matically satisfied bythe first two terms inEq. (10 .6.10), becaus e (p, - P)'UY1' = -1[o i,+M)-(i i+M)] and On the other hand ,(p' -p)2 does not in general van ish,so current conservation requires the third term to vanis h H(k') = 0 . Also, letting p' -+p in Eqs .(10.6.9)and (10.6.10), we fin d (TP'fT4'U(0)TP'CT) ==iq(2n)-3fi(p, c') ~'/ P F(0)- py G (0) U (p, 6) Using the identity ~ ,P, ij+ m} = 2m TY + 2ipP, we also hav e Recall also thatM and therefor e (tp P,dr,J"(0)TP i¢= '7(2n)-3(p11Y'))6 drd[F(O) + G(O)] (10.6.13) Comparing this with Eq .(10.6.3) yields the normalization conditio n F(O)+G{O} =1. (10.6.14) It may be useful to note that the electromagnetic vertex matrix FA is commonly written in terms of two other matrices, a s fi(p',c7')F1'(p% p)u(p, a)=Ti(p',a')[Y;'Fl(k') + p), F2(k')] u(p,c) (10,6 .15) As already mentioned, we may rewrite the matrix appearing in the second term in terms of those used in defining F(k2 )and G{k'}: = T'(P I(TI) Pi i'Y" + i if 7111 11 - i + i2 il ilI U(P' 0- ) a(Pf , d)[i(p," + I~ ')+2rnyuI u(p, u) . (10.6.16) 456 10Non-Perturbative Method s Comparing Eq .(10.G,1 S) with Eq. (10.6.10), we fin d The normalization condition (10 .6-14) now reads(10.6.17) (10.6.18) In order to eva luate the magnetic moment of our particle in terms of its form factors, let as consider the spatial part of the vertex function in the case of small momenta, j p, lprl << in.For this purpose, it is usefu lto use Eq . (10,6 .16) to rewrite Eq . (10 .6.10) (with H = 0) in a third form : -i_ 2LY1, Y, i[P`[(p + p`} 1jF(k2) + G(k') ) -p)vF(k')] u(p, cr) .(10.6,19) For zero momenta the matrix elements of the commutators of Dirac matrices are given by (5 .4. 19) and (5 .4.20) a s u(0, a') [y',,jJ] u(O, cr) = 4ieijk (J~ I ) ) ,d where P') = ~ais the angular momentum matrix for spin Hence to first order in the small momenta , F)U(P,C) ~ M L(P -p') x J(3)j,,r,7 F(O). (10 .6.20) In a very weak time-independent external vector potential A(x) the matrix element of the interaction Hamiltonian H' = - f dux J(x) - A(x) between one-particle states of small momentum is therefor e L P+,Q' f H~ a ~1,ff)m(L~~~~Jdx eA ()L(p - p{)x `~~i}Jffrf! m(27[)3f dI _,, . B(X) (10 .6.21) whe reB= V x A is the magnetic field . Hence in the limit of a slowly varying weak magnetic field, the matrix elemen tof the interaction Hamiltonian i s The magnetic moment p for an arbitrary particle of general spin jis defined by the statement that the matrix element of the interaction of the particle with a weak static slowly varying magnetic field is 10.7 The Kal[erz-Lehmann Representation 45 7 Hence Eq . {10.6,22} gives the magnetic moment of a particle of charge q, mass m, and spin 1as: This contains as a special case the celebrated Dirac results y= q/2m for a spin particle without radiative corrections . We mention without proof that the form factors F(k2 ) and G(k2) of the proton may be measured for k2 >0 by comparison of experimental data for electron-proton scattering with the Rosenbluth formula$ for the laboratory frame differential cross-section : dcT e4cos'(0/2)f +2EOsin'{0 f2}A2 4(4ie)2E~ sing(BJ2) M X(flk) + G(k'))2+ k 2 22F2(V)tan2(0/2) +G2(k)4m ( )I , where EQ is the energy of the incident electron (taken here with E~ >M,); 0 is the scattering angle ; and (0/2)k~ =4E~ sin2 I + (2 EOfm.) sin2(0f 2 ) 10.7 T he W 1en--Le hmannRepresentation` We saw in Section 10 .2 that the presence of one-particle intermediate states leads to poles in Fourier transforms of matrix elements of time- ordered products, like (10 .2.1). Multi-particle intermediate states lead to more complicated singularities, which are difficult to describe in general . But in the special case of a vacuum expectation value involving just two operator, we have a convenient representation that explicitly displays the analytic structure of the Fourier transform . In particular, this representa- tion may be used for propagators, where the two operators are the fields of elementary particles . When combined with the positivity requirements of quantum mechanics, this representation yields interesting bounds on the asymptotic behavior of propagators and the magnitude of renormalization constants . Consider a complex scalar Heisenberg-picture operator fi(x), which may or may not be an elementary particle field . The vacuum expectatio n This section lies somewhat out of the book's main line of devefupment, and may be omitted in a first reading . 458 10Non-Perturbative M ethod s value of a product (D(x)V(y) may be expressed a s ~d'(x)(Dt(Y)~o = E#(D(x) ln) (nI(V(1')~O) , (10 .7.1) n where the sum runs over any complete set of states . (Here the sum over n includes integrals over continuous labels as well as sums over discrete labels .) Choosing these states as eigenstates of the momentum four-vector PP, translational invariance tells us tha t andso(10.7.2) (10.7.3) It is convenient to rewrite this in terms of a spectr alfunetion. Note that the sum n 64(p - p,)I(0 (D( O)En}I' is a scalar function of the four-vector pP, and therefore may depend only on p2and (for p' <0)on the step function O(p'). In fact, the intermediate states in Eq .(10.7.3)all have p2<_0 and po>0, so this sum takes the for m n with p( -p')=0 for p' U.(The factor (2n)-'isextracte dfrom pfor future conven ience .)The spectral funct ionp(-p 2) = 0 is clearly real and positive .With this definition , we can rewrite Eq . (10.7.3)as = (2a E)-300 d4P fa Xku) op, + P). (10.7.5) Interchanging the order of integration over p "and u 2 , this may be ex- pressed as ~(D(x)4)'(y)~o =fody'P(P2)A+(x - y; y') where Q+ is the familiar functio n In just the same way, we can show tha t x2(1U.b) (10.7-7) 10.7TheKillen-Lehmann Representatio n with a second spectral function P(y2 )defined by459 (10.7.9) We now make use of the causality requirement, that the commutator [(D(x), c W(Y)] must vanish for space-like separations x - Y.The vacuum expectation value of the commutator is Y; JU2) X.JU2)[(D(x), V(y)]) 0 o2 (p(p') A+(x A+(y5 ) - (10.7.10) As noted in Section 5.2, for x - y space-like the function d+(x - y)doe not vanish, but It does become even .In order for (14 .7.1 Q)to vanish for arbitrary space-like separations, it is thus necessary tha t This is a special case of the CPT theorem, proved here without the use of perturbation theory ; for whatever states with p 2 =;;-_-P2 have the quantum Numbers of the operator (D, there must be corresponding states with p2=,-µZ that have the quantum numbers of the operator V . Using Eq .(10.7.11), the vacuum expectation of the time-ordered product is 'U2 -00 0 where ❑F(x --y"U2)is the Feynman propagator for a spinless particle o f mass p (10.7.13) Harrowing the notation introduced in Section 1 0.3 for complete propaga- tors, we introduce the momentum space functio n iS(P) =_ fd4Xcxp [-ip -(x- y)] (Tf 4)(X)ADI(Y) P0 - Recall that P2 +p2 - 1 6 This yields our spectral representation : 00P(A2) du2 P+P2-k-(ZD.7.14) (10.7.15) (14.7.1 6) One immediate consequence of this result and the positivity of p (p2) is 460 1 0Non- Perturbatfve Methods that d'(p )cannot vanish for ~p~1 --~,oo faster" than the bare propagator 11(p2+ rra2- t,-).From time to time the suggestion is made to include higher derivative terms in the unperturbed Lagrangian, which would make the propagator vanish faster than 11p2 forjp2l-+ao, but the spectral representation shows that this would necessarily entail a departure from the positivity postulates of quantum mechanics . We can use the spectral representation together with equal-time commu- tation relations to derive an interesting sum rule for the spectral function . If O(x) is a conventionally normalized (not renormalized) canonical field operator, the n We note thata(D(x, t)}(Dt{Y} t)at aXO- --ibl{x - y} . (10.7.17) so the spectral representation (10.7.10)and the commutation relations (10.7.17)together tell us that ~(p)do'=i. 4 This implies that for lp21 -~--~ cry, the momentum space propagator (10.7.16) of the unrenormalized fields has the free-field asymptotic behavio r This result is only meaningful within a suitable scheme for regulating ultraviolet divergences ; in perturbation theory the unrenormalized fields have infinite matrix elements, and their propagator is ill-defined . Now consider the possibility that there is a one-particle state 1k)of mass m with a non-vanishing matrix element with the state { 010(0). Lorent z In fact, it is not even certain that ❑'(p) vanishes for jp~j --~ oo at all, even though this would seem to follow from the spectral representation . The problem arises from the interchange of the integrals over p}' and y2 . What is certain is that A'(p) is an analytic function of -p2 with a discontinuity across the positive real axis -p 2 = ju'- given by np(g2), as can be shown by the methods of the next section . From this, it follows that tl+ (p)is given bya dispersion relation with spectral function p(p2) and possible subtractions : A'(p) = P(p of (P ~ + P01) " P, + P, -ic- where n is a posit ive integer, ya is an arb itrary positive constant, and p(p2) is a 4-dependent polynomial in p2 of o rder n - l that is a bsent for n =0. 10.7 The Kdllen-Lehmann Representatio n invariance requires this matrix element to take the for m --112 (010(0)1k) = (27z)-3/2(2vrk2+M2)N461 {10.7.19} where Nis a constant . According to the general results of Section 14 .3, the propagator d' (P)of the unrenormalized fields should have a pole at p2__+-nay with residue Z = j N12 > D. That is, (10.7.20) where rr(p2) ~0is the contribution of multi-particle states . Together with Eq.(10.7.18), this has the consequence tha t andso0 Z 1 (10.7.22) with the equality reached only for a free particle, for which (0I(D(x) has no matrix elements with multi-particle states . Because Z is positive, Eq . (1 .7.21) can also be regarded as providing an upper bound on the coupling of the field 4)to multi-particle states : 00 0 with the equality reached for Z=0. The limit Z = 0has an interesting interpretation as a condition for a particle to be composite rather than elementary . r° In this context, a `composite' particle may be understood to be one whose field does not appear in the Lagrarigian . Consider such a particle, say a neutral particle of spin zero, and suppose that its quantum numbers allow it to bedestroyed by an operator F(T) constructed out of other fields . We can freely introduce a field (Dfor this particle by adding a term to the Lagrangian density of the forms A =((D-- F(T))', because the path integral over 0can be done by setting it equal to the stationary point c D= F(T), at which ❑ = D . But suppose instead we writ e ~,O~`d~ - ~ r~2~2 is the usual free- AY =AY O + del, where A YO - - ~ o ~ field Lagrangian, and treat AY, =AY -❑oas an interaction . A term ~0UcDO~'Din the interaction is nothing new . We encountered such a term in Eq .(10.3.12), multiplied by a factor (l - Z) ; the only new thing is that here Z = D . Instead of adjusting Z to satisfy the field renormalization condition II"(0) = 0,here we must regard this as a condition on th e t This is known in condensed matter physics as a `Hubbard-Stratonavich transformation' .11 It will be used to introduce fields for pairs aFefectrons in our discussion of superconductivity in volume [1. 462 1 0Non- Perturb ative Method s coupling constants of the composite particle . Unfartunately, it has not been possible to implement this procedure in quantum field theories, because as we have seen Z = 0 means that the particle couples as strongly as possible to its constituents, and this rules out the use of perturbation theory . The condition Z = 0does prove useful in non-relativistic quantum mechanics ; for instance, it fixes the coupling of the deuteron to the neutron and proton .12 Although the spectral representation has been derived here only for a spinless field, it is easy to generalize these results to other fields . Indeed, in the next chapter we shall show that to order e2, the Z-factor for the electromagnetic field (conventionally called Z3) is given b y e2112 Z3= i-122 In M2e (where A > rn, is an ultraviolet cutoff), in agreement with the bound (10.7.22). 10.8 Di spersion R elations* The failure of early attempts to apply perturbative quantum field theory to the strong and weak nuclear forces had ledtheorists bythe late 1950s to attempt the use of the analyticity and unitarily of scattering amplitudes as a way of deriving general non-perturbative results that would not depend on any particular field theory . This started with a revival of interest in dispersion relations . In its original form,3a dispersion relation was a formula giving the real part of the index of refraction in terms of an integral over its imaginary part . It was derived from an analyticity property of the index of refraction as a function of frequency, which followed from the condition that electromagnetic signals in a medium cannot travel faster than light in a vacuum . By expressing the index of refraction in terms of the forward photon scattering amplitude, the dispersion relation could be rewritten as a formula for the real part of the forward scattering amplitude as an integral of its imaginary part, and hence via unitarity in terms of the total cross-section . One of the exciting things about this relation was that it provided an alternative to conventional perturbation theory ; given the scattering amplitude to order e2, one could calculate the cross-section and the imaginary part of the scattering amplitude to order e4, and then use the dispersion relation t o ' This section lies somewhat out of the book's main line of development, and may be ornitt(d in a first reading . 10.8Dispersion Relations 463 calculate the real part of the forward scattering amplitude to this order, without having ever to calculate a loop graph . The modern approach to dispersion relations began in 1954 with the work of Dell-Mann, Goldberger, and Thirring .14 Instead of considering the propagation of light in a medium, they derived the analyticity of the scattering amplitude directly from the condition of microscopic causality, which states that commutators of field operators vanish when the points at which the operators are evaluated are separated by a space-like interval . This approach allowed Goldberger15 soon thereafter to derive a very useful dispersion relation for the forward pion-nucleon scattering amplitude . To see how to use the principle of microscopic causality, consider the forward scattering in the laboratory frame ❑f a massless boson of any spin on an arbitrary target x of mass rra, > 0 and p, = 0 . (This has important applications to the scattering not only of photons but also pions in the limit m, = 0, to be discussed in volume II .) By a repeated use of Eq . (10 .3.4) or the Lehmann-Symanzik-Zimmerman theorem,3 the S-matrix element here i s .S== (2-A) 4r~c~ 3 1 ~~ li~nkz,0]itn~~~~o X jdx f d4ye-W-ye ik-x(jE1y)(iE3_,) (ij T ~A~(y), A(x)~ Y) - (10 .8.1)IN1 Here k and k 'are the in itial and final boson four-momenta ,with r )=01 co` = k'° ; A(x) is and Heisenberg-picture operator with a non-vanishing matrix element (CiA(x)jk) ={27r}-3/Z(2co)-1/2e`k"" between the one- boson state jk} and the vacuum ; and Nis the constant in this matrix element . In photon scattering A(x) would be one of the transverse com- ponents of the electromagnetic field, while for massless pion scattering it would be a pseudoscalar function of hadron fields . The differential operators -i ❑_, and -i ❑y are inserted to supply factors of ik'2 and ik-' that are needed to cancel the external line boson propagators . Letting these operators act on At(Y)and A(x), we hav e _1S__(2~)-~ 4cc~w'jNj2 limk~~~, limkfz~~ x dux fd4v e-W-y eik' .x(xj T~ P(y),J{x}IJoc} + ETC , (10.8.2) where J(x) -oxA (x), and `ETC' denotes the Fourier transform of equal time commutator terms arising from the derivative acting on the step functions in the time-ordered product . The commutators of operators like A(x) and At(y) (or their derivatives) vanish for x° = y° unless x = Y, so the `ETA' term is the Fourier transform of a differential operator acting on 64(x - y ), and is hence a polynomial function of the boson 464 10 Non-Perturbative Method s four-momenta . We are concerned here with the analytic properties of the S-matrix element, so the details of this polynomial will be irrelevant . Using translation invariance, Eq.(10.8.2) gives the S-matrix element as S=--21rW(k' - k )M(w), wher e M(co)~ fa) N~F(~)II(10.8.3) F(w) =_ f dx d"","-`~~xjT~P(O), J(x)~~a) + ETC , (10.8.4) it now being understood that V= cnfP, where ~ is a fixed four-vector with 1116, =0and The time-ordered product can be rewritten in terms of commutators in two different ways : Correspondingly, we can writ e where ~~((O)_-d4Xo(X°)(uILJf(O) ,J(x)] Iu} ~~w"x+ETC, F+(0))ff[~JC(4Cl,~(.~),]~(~}I~)eir~t~x 1 F_~~ jd4x~~~~~~~~~~~~~~~ ekde,x(10.8.5) (10.8.6) (10.8.8) (10.8.9) (aa.s.ia) Microscopic causality tells us that the integrands in (1 Q .8.7)and (10.8.8) vanish unless x~` is within the light cone, and the step functions then require that x1' is in the backward light cone in (10.8.7),so that x ' e>0, and in the forward light cone in Eq .(10.8.8),so that x'~ <0. We conclude that FA(co) is analytic for Im w > 0 and F R(uj) is analytic for Imo) 0, because in both cases the factor ek"I"X provides a cutoff for the integral over xP . (Recall that the `ETA' term is a polynomial, and hence analytic at all finite points .) We may then define a functio n FA(OJ) Im ui>0 FR(W)IMO) < 0 10.8Dispersion Relations 46 5 which is analytic in the whole complex r .) plane, except for a gut on the real axis . We can now derive the dispersion relation . According to Eq . (10.8.6), the discontinuity of (cv} across the cut at any real E i s Ifmo(w) /~~ vanis hes as lc.~l --* ova i n the upper or lower half-plane ,then by div iding by any polynomial P(e))of order n we obtain a function that vanishes for J cr~J --~ oo and is an alytic except for the cut on the real axis and poles at the zeroes wuof P(co) .(Where flw }itself vanishe sas ~c~l--* co ,we can take P(w) = 1.)According to the met hodof residues, wethen hav e w+(z)dz(10.8.13)P(w) ~, (co, - (o}Pr(c)~,] = 1~ °+ 2 ;ri1C(z -rte ) P (z ) where r .) is any point off the real axis, and C is a contour consisting of tw o segments : one running just above the real axis from -oo + if to +ao + i f and then around a large semi-circle back to -oo +ie, and the other just below the real axis from +oc -ic to -oc - ie and then around a large semi-circle back to +oa - ic . Because the function flz } f P(z) vanishes forJaJ--+ca, we can neglect the contribution from the large semi-circles . Using Eq.(10.8.12), Eq.(10.8.13) become s 2ni ( )(E) where Q (ej) is the (n - I)th-OTder polynomia l y, (c),,-(0) P '(rv,,) A dispersion relation of this form, with P(w) and Q(co) of order n and n --1 respectively, is said to have n subtractions . If we can take P t-1 then Q= 0, and the dispersion relation is said to be unsubtracted . If we now let w approach the real axis from above, Eq . (1U .8.1 4) give s 21r i as E-w-tFP(E ~ ) ) Recalling Eqs .(10.8.6)and (3.1.25), this is (E-(_o)P(E) (10.8.16) with 1 . J(E -ca) now interpreted as the principal value function 9/(E - c, )). 466 10 Von-P erturhati veMethod s This result is useful because the functions F+(E )may be expressed in terms of measurable cross-sections, Summing over a complete set of multi-particle intermediate states Pin Eqs .(10.8,9)and (10.8.10)(including integrations over the momenta of the particles in /3 )and using translation invariance again, we hav e F-(-E) = (27E)'I(1 IJ(O)Ia)I' 64(p .)(+ Elf pp) . (10.5.18) But the matrix elements for the absorption of the massless scalar boson B in B + oc - -* or its antiparticle B" in B" + oc ~ Pare -2i7MBI +x,# -(2n)-1 / 22EBeN (27r) 1- 2E BN(10.8.19) (10.8.0) Comparing with Eq. (3.4.15), we see that F+(E) may beexpressed in terms of total cross-sections" at energies TE 2 ( 2 7 E ) 36X+B,.(IEr F_(E) 0 (E)2E IN12 (2)z) The scattering amplitude (10.8.3)isnow, for real co 0, 2o)IN122(27r)3 (9(2~4JO[E- r:jP(E } ~ } )Ua+Bf(E)(1D.8.21) E dE . (1(} .8.23) Itis more usual to express this dispersion relation interms of the amplitude f (to) for forward scattering in the laboratory frame, defined sothat the laboratory frame di fferential cross-section in the forward direction isIf (caq2.This amplitude is given in terms ofM(w) by In some cases where selection ru les a llow t he transition x ~ +H and x ~ + B`, the functions F+(E) a lso contain terms proportiona lto <5[F] arising from the contr ibution o fthe one-partic le state a in the sum over intermediate states fl . This does not occur for transversely po larized photons, or for pseudosca lar pions in the limit mn --+ Q . 10.8Dispersion Relation s f (w)=-47r2cn M{w}-27r2iF(co)/IN 12,soEq. (10 .8.23) now rea ds +P(`O) 47r2acs 4ar9x+B(o)) ,I~"~' [ Ca+B(E ) (E - oj)P(E +)Ca+BJE)E dE ,(E + oj)P(-E)]467 where R(w) -_2in2 Q(o))I 12. The optical theorem (3 .6.4) tells us that the second term on the right-hand side equals iIm f(ro), so this can just as well be written in the more conventional for m Ref(w) = R(c)} +P(w) 4TE211 ~ax+~~~~ (Tx+B'(E)-]E dE , (10.8.24) I n particular, we see that R(w) is real if we choose P(w) real . The forward scattering amplitude also satisfies an important symmetry condition . By changing the integration variable in Eqs . (10,8.7)and (10,8.8) from x to -x and then using the translation-invariance propert y we see that for Im cca ~ 0, FA{-co) is the same as F R(ca), except for an interchange of Jwith P .That is , FA(-( o} = F R(w) fo rIm cci ~ 0, where a superscript c indicates that the amplitude refers to the scattering of the antiparticle B1 on a . (We leave it to the reader to show that this relation is not upset by the equal-time commutator terms in Eds .(10.8.7) and (10.8.$).)In the same way, we fin d FR(-(o} - FA(w) for I m w ~0 and for real r o Using these relations in (10 .8.6), and recal]ing that f (c)} is proportional to F(w), we find the crossing symmetry relation, that fo r realw .f(-0j) = f `(co) (10.8.25) We are free to take Pico} as any polynomial of sufficiently high order, but R{oj} then depends not only on P(w) but also on the values of F(w) at the zeroes of P(rv) . For P (rj) real and of nth order, the only free parameters in Eq. (10 .8.16) are the n real coefficients in the real (n -1)th order polynomial R(w) . Hence Eq . (10 .8.16)contains just n unknown real independent constants, the coefficients in the polynomial R(cq) for a given P(w) . We therefore wish to take the order n of the otherwise arbitrary polyomial P(ca) to be as small as possible . 468 10 Non-Perturbative Method s We might try taking P(a)) = 1, but this doesn't work . The analysis of Section 3.7suggests that the forward scattering amplitude should grow like cc )or perhaps as fast as ro ln2 co.In this case for ,f (w) f P ( w) to vanish as co - -+0, it is sufficient to take P(w) as a second order polynomial, so that R(w) is linear in co . Choosing P(E) = E2for convenience, Eq . (10.8.24) then becomes (02a2+B,(E)] dE .10 (E + w) E10. 8,26) with a and bunknown real constants . The crossing symmetry condition {10.8,25} tells us that the corresponding constants in the dispersion relation for the antiparticle scattering amplitude f17(w) are If we assume for instance that the cross-sections CT,+B(E) and a,,+,3c{E} behave for E --+cc as different constants times (InE)'', then (10.8.26) would giv e Ref { w}--f[9%+B(W) - Qa+ B{'()]Inu3-w{1nc:o}"+1 (10.$.28) so the real part of the scattering amplitude would grow faster than the imaginary part by a factor In u ). This is implausible ; we saw in Section 3.7that the real part of the forward scattering amplitude is expected to become much smaller than the imaginary part for m - +oc, as confirmed by experiment . We conclude that if a,,+$(E) and 6«+B{(E)do behave for E--*cc as constants times (In E )lthen the constants must be the same . Because we are concerned here with the high-energy limit, this result does not depend on the assumption that B is a massless boson, so in the same seise, the ratio of the cross-sections of any particleand itsantiparticle on a fixed target s houldapproach unity at high energy .This result is a somewhat generalized version of what is known as Pomeranchuk's thearem .16 (Pomeranchuk considered only the case r -- 0, while section 3.7 and the observed behavior of cross-sections both suggest that r = 2 is more likely, ) Although Pomeranchuk took his estimates of the asymptotic behavior of scattering amplitudes from arguments like those ❑f section 3 .7, today high energy behavior is usually inferred from Legge pole theary .17 It would take us too far from our subject to go into details about this ; suffice it to say that for hadronic processes the asymptotic behavior of f (co) as oi goes to infinity is a sum over terms proportional to where Yn(t) are a set of `Regge trajectories", each representing the exchange of an infinite family ❑f different one-hadron states in the collision process . The leading trajectory (actually, a complex of many trajectories) in hadron- Problems 4 69 hadron scattering is the `Porneron,' for which ac(Q) is close to unity . It is this trajectory that gives cross-sections that are approximately constant for E - *oa. According to Pomeranchuk's theorem, the Pomeran couples equally to any hadron and its antiparticle . We can estimate cx,(O) for the lower Regge trajectories from the spectrum of hadronic states . A necessary though not sufficient condition" for a mesonic resonance of spin jto occur at a mass inis that m 2 equals the value of t where one of trajectories ccn(t) equals j . Apart from the Pomeron, the leading trajectory in pion-nucleon scattering is that on which we find the j=Ip meson at m = 77 0MeV, the j = 3g meson at in :;--1694 Mel, and a j = 5 meson atin= 23 50 MeV . Extrapolating these values of a(t) down to t=0, we can estimate that this trajectory has oz(O) , :e D.S. This trajectory couples with opposite sign to 7r+ and g-, so for pion-nucleon scattering we expect f (cry) - f '(w) to behave roughly lik e For photon scattering there is no distinction between B and B", so here Eq. (10 .8.27) gives b = 0, and Eq .(10.8.26)read s ~~(oo(7(E) .fM=a+2p)cy E2_OJ2dE. (1 .$.29) This is essentially the original Kramers-Kranigl3relation . As we shall see in Section 13.5, for a target of charge e and mass inthe constant a has the known value Ref(0) _ -e2/m, Problem s 1. Consider a neutral vector field v~{x} . What conditions must b e imposed on the sum II~ti,(k) of one-particle-irreducible graphs wit h two external vector field lines in order that the field should b e properly renormalized and describe a particle of renormalized mas s m? How do we split the free-field and interacting terms in th e Lagrangian to achieve this ? 2.Derive the genera lizedWard identity that governs the electromag- netic vertex function of a c harged scalar fie ld. 3. What is the most general form of the matrix element *U21J~WIP161) of the electromagnetic current J,"(x) between two spin ~ one-particle states of different masses ml and rra2 and equal parity? What if the parities were opposite? (Assume parity conservation throughout .) 4.Derive the spectral (Kallen -Lehmann) repre sentat ionforthe vacuum expectation value {TfJy{x}.I''(Y)t~~O, where JP(x) is a complex conserved current . 470 10 Non-Perturbative Method s 5. Drive the spectral (Ka11en-Lehmann) representation for the vacuum expectation value (T~Tn(x) q),(Y)j}a, where W(x) is a Dirac field. 6. Without using any assumptions about the asymptotic behavior of the scattering amplitude or cross-sections, show that it is impossible for forward photon scattering amplitudes to satisfy unsubtracted dispersion relations . 7. Derive the spectral (Kailen-Lehmann) representation for a complex scalar field by using the methods of dispersion theory . 8. Use dispersion theory and the results of Section 8 .7 to calculate the amplitude for forward photon-electron scattering in the electron rest frame to order e4 . Reference s 1. W. H. Furry, Phys .Rev. 51, 125 (1937)- 2. H. Yukawa, Proc . Phys .-Math . Soc .Japan 17, 48 (1935) . 3, H . Lehmann, K . Symanzik, and W .Timmerman, Nuovo 'irnento 1, 205 (1955) . 4. Y. Takahashi, Nuovo Ciento, Ser . 10, 6, 370 (1957) . 5. J. C. Ward, Pays . Rev .78, 182 (1950) . 6. J. Schwinger, Phy s.Rev. Lett .3, 296 (1959) . 7. P. A. M.Dirac, Prod .Roy.Soc. (London) A117, 610 (1928) . S. M. N. Rosenbluth, Pays . Rev. 79, 615 (1950) . 9. C. Kallen, Hed v.Pays . Acta 25, 417 (1952) - Quantum.Electrodynamics (springer-Veriag, Berlin, 1972) ; H. Lehmann, NuovoCimento ll, 342 (1954) . 10. J. C. Howard and B . Jouvet, Nuovo Cime nto18, 466 (1960) ; M. J. Vaughan, R . Aaron, and R . D. Amado, Whys .Rev. 1254,1258 {1961} ; S. Weinberg, in Proceedings of the 1962 High-Energy Conference at CERN(CERN, Geneva, 1962) : p, 683 . 11. R. L. Stratonov ich,Srrv.Phys. Dakt .2, 416 (1957) ; J. Hubbard, Whys . Rev.Lett . 3, 77 (1959) . 12. S. Weinberg, Phys .Rev. 137, B672 (1965) . R~ ferences 471 13. H. A. Krarners, Arta Congr . Intern . Fisici,Como Nicola Zaniche l- lli, Bologna, 1927) ; reprinted in H . A. Kramers, Collected Scientifi c Papers North-Ho lland, Amsterdam, 1956) ; R. de Kronig, Ned. Tyd . Nat.Kun de9, 402 (1942) ; Ph ysica 12, 543 (1946) ; J. S. Toll, Th e Dispersio nRelation for Light and its Application to Problems In- volving Electron Pairs (Princeton Univers ity Ph . D. Thesis 1952) . For historical reviews, see J . D. Jackson in Dispersion Relations, ed. by G . R. Screaton (Oliver and Boyd, Edinburgh, 1961) ; M. L. Goldberge r in Dispersion Relations and Elementary Particles, ed. by C . De Wit t and R . Omnes (Hermann, Pa ris, 1960) . 14. Ce ll-Mann, L. Goldberger, and Thirring, P hys.Rev.95, 1612 (1954) . The non-perturbative nature of this result was shown by M . L. Goldberger, P hyS.Rev.97, 508 ( 1955) . 15. M. L. Goldberger, Pays, Re v. 99, 979 (1955) . 16. 1. Ia. Pomeranchuk, J. Expo . Theor. Whys .(USSR .)34, 725 (1958) . English version : Soviet Physics - JETP 34(7), 499 (1958) . For a generalization, see S . Weinberg, Phys . Rev .124, 2049 ( 1961) . 17. See, e . g., F. D.B. Collins, An Introduction to Regge Theory and High Energy Physics (Cambridge University Press, Cambr idge, 1977) . The original references are T Regge, Nuovo Cimento 14, 951 (1959) :18, 947 (1960) . 18, The graph of spin versus squared mass is known as a Chew-Frautschi plot ; see G . F. Chew and S . C. Frautschi, Pays, Rev.Lett .8, 41 (1962) . 11 One-Loop Radiative Corrections in quantum Electrodynamic s In this chapter we shall proceed to carry out some of the classic one-loop calculations in the theory of charged leptons massive spin12 particles that interact only with the electromagnetic field . There are three known species or 'flavors" of leptons : the electron and moon, and the heavier, more recently discovered tauon . For definiteness we shall refer to the charged particles in our calculations here as `electrons,' though most of our calculations will apply equally to muons and tauflns . After some generalities in Section 1 1.1,we will move on to the calculation of the vacuum polarization in Section 11 .2, the anomalous magnetic moment of the electron in Section 11 .3, and the electron self-energy in section 11 .4. Along the wad, we will introduce a number of the mathematical techniques that prove useful in such calculations, including the use of Feynman parameters, Wick rotation, and both the dimensional regularization of `t Hooft and Veltman and the older regularization method of Pauli and Villars . Although we shall encounter infinities, it will be seen that the final results are finite if expressed in terms of the renormalized charge and mass . In the next chapter we shall extend what we have learned here about renormalization to general theories in arbitrary orders of perturbation theory . 11.1Counterterm s The Lagrangian density for electrons and photons is taken in the fornn * = - ~~'~''FB FV- ~sI11m[~'~ + ieBARI+ MaI Wa where FB'' =_CPA" -- 0~'A$ ; A$ and Wa are the bare (i .e., unrenormalized) fields of the photon and electron, and -eB and mB are the bare charge an d " in this chapter we will not be making transformations between Heisenberg- and ]Interactian- picture operators, so we shall return to a conventional notation, in which the an upper case A and a lower case tp are used to denote the photon and charged particle fields, respectively . 472 11.2 Vacuum Polarization 473 mass of the electron . As described in the previous chapter, we introduce renormalized fields and charge and mass : W=Z_1/2 Y~ B A~:Z~ 1/2 A ~ e=Z+1/2eB3(11.1.2) (11.1.3) (11.1.4) (11.1.5) with the constants Z2, Z3, and bm adjusted so that the propagators of the renormalized fields have poles in the same position and with the same residues as the propagators of the free fields in the absence of interactions . The Lagrangian may then b ewritten in terms of renormalized quantities, as = p+1+2 , where Yo = - 'Flu" Fu ,,-fpLys OJU +mIT, and 2 is a sum of `caunterterms'(11.1.6) (11.1.7) (11.1.8) 4[Yu IV +Z26mFP-ie(Z2 - 1 )~~~ li"T. (11.1.9) It will turn out that all of the terms in 2 are of second order and higher order in e, and that these terms just suffice to cancel the ultraviolet divergences that arise from loop graphs . 11.2Vacuum Polarizatio n We now begin our first calculation of a radiative correction involving loop graphs, the so-called vacuum polarization effect, consisting of the corrections to the propagator associated with an internal photon line . Vacuum polarization produces measurable shifts in the energy levels of hydrogen, and makes an important correction to the energies of moons bound in atomic orbits around heavy nuclei . Also, as we shall see in Volume II, the calculation of the vacuum polarization provides a key 474 I] One-Loop Radiative Correction s u-q Figure l1 .2. The one-loop diagram for the vacuum polarization in quantum elec- trodynamics . Here wavy lines represent photons ; lines carrying arrows represent electrons . element in the calculation of the high energy behavior of electrodynarnics and other gauge theories . As i nSection 10 .5,we define 4271)4IIW"'(q)as the sum ofallconnected graphs with two external photon lines with polarization indices p and v and carrying four-momentum q into and out of the diagram, not including photon prop agato rs for the two external lines, and with the asteris kindicating that we exclude diagrams that can be disconnected by cutting through some internal photon line . The complete photon propagator d 'P4'(q)is given by Eq .(10.5.13): where d P'(q)isthe p hotonpropagator without radiative corrections . Our task here is to calculate the leading contributions to fl*P'(q) . Inlowest order there isaone-loop contribut ion to 11 .*, corresponding to the diagram in Figure 11 .1: p [(,4P2+M2- ie x[(21r)4ey1]~~~] [(2~)4e(11 .2.2)[...4)(P-q)2+~ie with the first minus sign on the right required by the presence of a fermion loop . More simply, this i s loop(21r)4 f(p2 +tn'-iE)((p -q)2 +M2 ie) - (1123 ) The first step in doing this integral is to use a trick introduced by Feynman .1 We use the elementary formul a 1 ,1 dx AB - [(1-)A+xB]2 I1.2 Vacuum Polarization 475 to w ritethe product of scalar p ropagators in Eq . (11 .2.3) a s 1 [(P2 + M2 _ je)(1 _ X) ++Jo 12 +((p -q)' + 1Y32-dE)XJdX f[p2p .qx+q2x1~ 1 [(p - qX)2 + m' - ic +q2X(I _ X)l ~dx. I (This is a spec ial case of a class of integrals given in the Appendix to this chapter .) We can now shift the variable of integration in momentum space p-'P+ qx, so that Eq .(11.2.3)become s 21 *POF(q) = 'e dx -i-F+ q2X(I _X)] ILoop (27z)4/ 'J d4p[p' + m' Using the results of the Appendix to Chapter 8, the trace here can easily be calculated a s =4[- (p + qx)P (p - q(t -x)f+ (p + qx)- (p - q(1 - x))q" Our next step is called a wick rotation .2As long as -q 2 < 4m2, the quantity m2 + q2x(I - x) is positive for all x between 0and 1, so the poles in the integrand of Eq.(11.2.5) are at p o = + p + m~ + q x{1 - x)-- ic, i.e., just above the negative real axis and just below the positive real axis. (See Figure 11 .2.) We can rotate the contour of integrations of pa counterclockwise without crossing either of these poles, so that instead of integrating1Pon the real axis from -co to +cc, we integrate it on the imaginary axis from --ioa to +ix . That is, we can write p '= ip', with p4 integrated over real values from -cc to +ao .(If an ie instead of -i ehad appeared in the denominator of the propagator, then we would have been setting p° = ---ip', with p~ again integrated over real values from -oo t o Slriclly speaking, this step is only valid in convergent integrals . In principle, in order to justify the shift of variables, we should introduce some regulator scheme to make all integrals converge, such as the dimensional regularization scheme discussed below . 476 11One-Loop Radiative Correction s Figure 11 .2. Wick rotation of the p ocontour of integration . Small x's mark the poles in the jcomplex plane ; the arrow indicates the direction of rotation of the contour of integration, from the real to the imaginary p°-axes . +. Thy effect would be a change of sign of ni i~p( q).) Eq .(11.2.5)now becomes 2r f ]too(2-n)4 f x[- (p + qx)P(p - q(1 - x))' + (p + qx)- (p - q(1 - x))qP" -(p + qx )'(P-q(1- x))"+m2qP¢], (11.2.7) where (cl'P)E=dP1dP'dP'dp" and all scalar products are evaluated using the Euclidean nor m a- b=alb' +a2b2+a3b3 +a40 with the understanding that q4 - -fq° .Also, jqP' can be taken as either the Kronecker delta, with the indices running over 1, 2, 3, 4, or as the usual Minkowski tensor, with the indices running over 1, 2, 3, 0 . The integral ( 11.2.7) is badly divergent . Eventually all infinities will cancel, but to see this it is necessary at intermediate stages of the calcula- tion to use some sort of regularization technique that makes the integrals finite . It would not do simply to cut off the integrals at some maximum momentum A,integrating only over pY with p 2 < A' , because this would amount to introducing a step function O(A2 - pl)into the electron prop- agator, and the Ward identity (10 .4.25) shows that in order to maintain 11.EVacuum Polarization 477 gauge invariance, and modification of the electron propagator must be accompanied with a modification of the electron-photon vertex . If fact, with an ordinary cutoff A,radiativcorrections would induce a photon mass, a clear violation of the requirements of gauge invariance . Experience has shown that the most convenient method for regulating divergent integrals without impairing gauge invariance is the dimensional regularization technique introduced by 't Hoot and Veltman3 in 1972, based on a continuation from four to an arbitrary number d of space#ime dimensions . This amounts to carrying out angular averages in integrals like (11.2.7) by dropping all terms that are odd in p, and replacing the terms that have even numbers of p-factors with* * eP, --*F2q'Ur Id, ep,plp, --*(P 2)2 rnp,11p, + q,q,cF + ~ifypjld(d + 2)(11.2.8) (11.2.9) Also, after writing the integrand in this way as a function only of p 2,the volume element d 4FEis to bereplaced with ~ dKa-ldre, where rc =_ , and Q, is the area of a unit sphere in d d imension s Q,j= 2.7rd/2 r(d/2) . (11 .2.10) The integral (11 .2.7 now converges for complex spacetitne dimension- ality d . We can continue the integral through complex d-values to d = 4, the infinities then reappearing as factors (d - 4)- 1 For the integral (11.2.7),dimensional regularization give s 1100 (-2K2 dqP"+2qOq'x(l-x)+ (K 2_ q2X(1-X))j7PC + M 217 P17 1 - Thy integrals over can be carried out for any complex d (or for any real d, aside from the even integers) . We use the well-known formulas (given in greater generality in the Appendix to this chapter) : a Jor~-'[K' +u']-'d~c = z (v2) ~-' F (d/2) F (2 - d/2 ) Ka+l [K 2 + v']-'dK= 2 {v2}'-1F( 1+df2)T(1-df2),(11.x.11) These expressions may most easily be derived by noting that their form i5 dictated by Lorentz invariance and the symmetry a mong the indices ju, v, p, etc ., while the factors may be found by requiring that both sides give the same result when contracted with is . 478 andfind rl *"n (q) ~ _loopJIOne-Loop Radiative Correction s 2e2S~~ (21z)a x['dx[(l ~/2) +(2qfq'x(I_X)_ q2qp,_X(I - X)+M2qp,7 ) (M2+ q2X(l - X))d-2 x r (d 12) r (z - d 12) The two terms in the integrand can be combined, usin g find *p¢ 4e2S~d ~i l aap~ ~~ - (27E) 4 iIf(d12)F (2-d12)(qPq'-q2qf',T) x dx x(1 - x)(2+ q2x(t -- x)}~-2. ( 11.2.130 Wenote the very important resultthat this contribution toTI*{'1 sa tisfies the re lation ~~)1_11i~0P(9) = 0 (11.2.14) that was derived in Section 14.5 on the basis of the conservation and neutrality of the electric current . It was precisely to achieve this result that we adopted the dimensional regularization scheme . The reason that dimensional regularization gives this result is that the conservation of current does not depend on the dimensionality of spacetime . The gamma function F(2 - d/2) in Eq . (11 .2.13 blows up for d - +4. Fortunately, as we saw in Section 11 .1, there is another term that must be added to CiW ff(q), arising from the term -'(Z3 - 1 )F},VFP"in the interaction Lagrangian . This term has a structure like Eq . (1 .2.13) so to order e2, the complete r1* has the formqPqq), (11 .2.15) jJ*Pa {q}~(q2qp,_ ~~~~)7r(42),{11.2.16} with 2)4e2nd (21r)4 2 1 0 (ZI- 1). {11.2.17} 11.2Vacuum Polarization 479 As we saw in Section 10.5,the definit ionof the renor malizedelectromag- netic field requ ires that ir(0) =0(inorder that the residue ofthe pole in the co mpletephoton propagator at q2 = 0 should be the same as for the bare propagator ,aside from gauge-dependent term s).Therefore ,to order e2 a (21r) SOthat, to order e2 1) 7E(q2) 4e2QdF (2,g)a(M2) 2 0 1 F (2 - a } dx x(1- x)2 fo 2X(I_X)) x [(m' +q_(rn2)"-2] (11.2.19) Now we can remove the regularization, allowing d to approach its physical value d= 4. As mentioned before, there is an infinity in the one-loop contribution, arising from the limiting behavior of the Gamma function F (2-i~-'I (2 _d12) where yis the Euler constant, y = 0.5772157 .The infinite part of Z3- 1 is given by using 1/(2 -r d/2) for F(2 - d1 2), and replacing deverywhere else by4: 4e'- 2n21 V3 - 1)e21 We shall see in Volume II that this result may be used to derive the leading term in the renormalization group equation for the electric charge . The poles atd= 4 obviously cancel in 7r{q2}, because for d = 4 both (m2+q2x(l-X))'-2and(M2)d-2have the same limit, unity . For the same reason, the term -7 jinF(2-d/2) cancels in the total 7r(q2), though it does make a finite contribution to Z3-- L There are other finite contributions to Z3- 1, that arise from the product of the pole in I"(2-d /2)with the linear terms in the expansion of S 2dr(d/2)around d= 4, but these also cancel in the total n(q2) . Indeed, in carrying out our dimensional regularization, we might have replaced (2n)-' with (2 -g)-d, and the factor Tr 1 =4 might have been replaced with the dimensionality 2-d/2of gamma matrices in arbitrary even spacetime dimensionalities d, and these too would have contributed to the finite part of Z2- 1, but not of n(q2). Moreover, e' cannot be supposed to be d-independent, because as shown by inspection ofEq.(11.2.13), it has the d-dependent dimensionality [mass]4-d . If we take e2CCtj4-d, where µ is some quantity with the units of mass, then 480 1 1One-Loop Radiative Correction s there are additional finite terms in Z2 - 1, arising from the product ofthe pole in F(2- d12) with the term (4 - d) In p in the expansion of µ4-d i n powers of 4 - d , but again, these cancel between Z3 - 1 and the one-loop contributions to iz(q2y The only terms that do contribute to7r(q2)in the limit d - +4 are those arising from the product of the pole inr(2 - d/2) with the linear terms in the expansion of (m2 + qlx(l `-` x))~-~ and(M)"-2inpowers of d - 4 : I 2 (M2+q2X(1_X))' -2_(M2)'-l - y~ 2 - 2}In I+q X(m2X) This gives at las t ir(q2)e~~721 fq2.x(1 - x )dx. x{1--~ x} In 1 +02m(11.2.22) The physical sign ificance of the vacuum polarization can be explored byconsidering its effect on the scattering of two charged particles of spin 11.The Feynman diagrams o f Figure 11 .3 make cont ributions to the scattering S-matrix element of t he for m Sa {1} 2 --i~lr52f) = (27r)-12/264(Pi++F2' - P1 - P 2)[ei2ir~ityui] q Sb(1}2 --* 1r,2'}= (27r)-12/264( Pi'+P21-P1 - P2 )L q2[i(2,g (q + q,,q,),n(q')] [e2(27r Y'U2 ] where el and e2 are the charges of the two particles being scattered ; 7z{q2} is calculated using for e in Eq . (11 .2.22) the magnitude of the charge of the particle circulating in the loop in Figure 11.3; and q Yis the momentum transfer q= F1-p11 = PT -p2 . Using the conservation property q,jijryAu1 = 0the two diagrams together yield an S-matrix element : x C[~2yu2] . (11.2.23) In the non-relativistic limit, ulFy°u1 while uj+y!uj ^~ 0, and likewise for particle 2 . Also, in this limit q0is negligible compared with jqj. Eq .(11.2.23)in this limit become s Q+b(lr2'4 1i2) 4712q~ Ll+ ~(7)134( Y11+ PT C1 aP2)6Ci616C262 11.2 Vacuum Polarization 481 This may be compared with the 5-matrix in the Born approximation due to a local spin-independent central potential V(r) : SBarn(i, 2 -> 1',2') - -27rib(Elf + E2, -E1- E2)TBorn(1, 2 -' 1'? 2) . (11.2.25) TBorn(1r 2 -+ 1% 2') = 6a ~~160~~2fc~'~Cl C~ 'JC2V(IX 1 f 7C(27E)- 12/2e-ip1, xie-02''"201'x1 e iP2'X2 Setting x jTX2+r, this give s ;:54(pv 5Earn +PT -PI-P2)6,r irrl6da') x.#X21) (1L2.26) (11.2.27) Comparing this wi th Eq . (11.2.23) shows t hat in the non-relativis tic limit the diagrams of Figure 11 .3 yield the same S-matr ix element as a potential V(r) such that 1+z(qJd3 rY(r) e !!q r=el e.2 or,invertingthe Fourier transfo rm, V(r) = e1e2d3q eiy-r .2(2,R)3 9(11.x.28) Eq.(11.2.28) is to first order in the radiative correction the same potential energy that would be produced by the electrostatic interaction of two extended charge distributions elq{x} and e2q(Y) at a distance r : V(Ir1) = elezd Ix~~~ 1(x)~1(Y) J 4rlx - + rl where ?I(r) =P(r) +d3 q7r(q2)elq.r ~(27T~(11.2.30) Note t hat Jderrl(r)=1 + 17t(O) = 1 , (11.2.31) so the total charges of particles 1 and 2, as determined from the long- range part of the Coulomb potential, are the same constants el and e2 that govern the interactions of the renormalized electromagnetic field . For 1rl * 0 the integral (11.2.30)can be carried out bya straightforward 482 11 One-Loop Radiative Correction s 2` 1'x` ]' (a) ( b) Figure 11 .3. Two diagrams for the scattering of charged particles . Here lines carrying arrows are charged particles ; wavy lines are photons . Diagram (b) repre- sents the lowest-order vacuum polarization correction to the tree approximation graph (a) . contour integrat ion: ~z ~Mr -mr 8, arfo I ~ TF-_ X_~ This expression is negative everywhere . However, we have seen that the integral of q(r) over all r equals + 1. Therefore, q (r)must contain a term (1+L)b3(r) that is singular at r = 0, with L chosen to satisfy Eq .(11.2.31): 2 3 1 r L = 873 ~ ~X(1 -X)dx 1 + ~~ ~xp-mr Jo xl -x x1 The complete expression for the charge distribution function is the n q(r) = (1 + L)r53( r) -e 33 10x(1- x} cox S7r xl+~rexp - mr (11.2.33)xxX The physical interpretation of this result is that a bare point charge attracts particles of charge of opposite sign out of the vacuum, repelling their antiparticles to infinity, so that the bare charge is partially shielded, yielding a renormalized charge smaller by a factor I/(1 + L) . As a check, we may note that if we cut off the divergent integral {11 .x.32} bytaking the integral to extend only over r a, we find that the part that is divergent for a ---),0is e2 Lo, = 12n2In c~-i (11 .2.34) Hence if we identify the momentum space cutoff Awith u ', the divergent 11.2 Vacuum Polarizatio n Bart of L is related to the divergent part of Z3- 1 by483 (11.2.35) because to order e' the renormalized charge (10 .4.18) is given b y 3 2 (11.2.36) Eq. (1 .2.35) is confirmed be low. Vacuum polarization has a measurable effect on muonic atom ic ene rgy levels . As we shall see in Chapter 14, the effect of Feynman graph (b) in Figure 11 .3 is to shift the energy of an atomic state with wave function yy(r) by ❑E= d3r❑v(r)IW(r)l 2 where AV(r) is the perturbation in the potential (11.2.28)- AV~9(11.2.37) (1x.2.38) This perturbation falls off exponentially for r >m-1. On the other hand, the wave function of electrons in ordinary atoms will generally be confined within a much larger radius a >m-1 ; for instance, for hydrogenic orbits of electrons around a nucleus of charge Ze we have a = 137/Zm (where here rra = m,),The energy shift will then depend only ❑n the behavior of the wave function for r <a. For orbital angular momentum ~, the wave function behaves like r 4for r a, soEq.(11.2.37) gives ❑Eproportional to a factor (Ma)2,'13 . The effect of vacuum polarization is therefore very much larger for e_0than for higher orbital angular momenta . For /,=0 the wave function is approximately equal to the constant y )(0) for r less than or of the order of m -1, so Eq.( 1.2.37) become s AE=IT(0)I2fd3r❑V(r) . (11.2.39) Using Eqs .(11.2.38) and (11.2.22), the integral of the shift in the potential (for e le.,= -Ze2)is Jder❑V(r)__Ze27T'(0) _ 15trt 2.(11.2.40) Also, in states of hydrogenic atoms w ith 0 and princ ipal quantum number n the wave function at the origin i s 2(ZaM)3~2 ~~)=4n n 484 11 One-Loop R adiative Correction s so the energy shift (11 .2.39) is 4Z4cclm❑E_15nn3(11.2.42) For instance ,in the 2 s state of hydrogen this energy shift is -x ,.122 x 10-7 eV,corresponding to a frequency shift d EJ27rhof - 37 .13 MHz . This is som etimes called the Uehling effe ct.4As discussed in chapter 1, such tiny energy shifts became measurable because in the absence of various radiative corrections the pure Dirac theory would pred ict exact degeneracy of the 2 s and 2p states of hydrogen .Asweshall see in chapter 14 ,most of the +1058 MHz `Lamb shift'between the 2s and the 2p states comes from other radiative correction s,but the agree ment between theory and experiment is good enough to verify the presence of the -37 .13MHz shift due tovacuum polarization . Although vacuum polarization contributes only a small part of the radiative corrections in ordinary atoms, it dominates the radiative correc- tions in muonic atoms ,inwhich a moon take sthe place of the orbiting electron .This is because most radiative corrections give energy shifts in muonic atoms that on dimensional grounds are proportional to m ., while the integrated vacuum polarization energy f der AV due to an electro n loop is st ill proportional torn. 2asin Eq .(11.2.40), giving anenergy shift proportional to m ~m~ ' z-- ( 210)2m~.However, in this case the muonic atomic radius is not much larger than the electron Compton wavelength , sothe approximate result (1 1.2.39)only gives the order of magn itude of the energy shift due to vacuum polarization . For the purposes of comparison with later calculations, note that if we had cut off the integral ( 11.2.7) at x = A, then in place of Eq.(1 1.2.20) we would have encountered an integral of the for m e2 n d-5dK&7-r2fi~KezPd-4_Aa-a bar2 d-4 where yis an infrared effective cutoff of the order of the mass of the charged particle circulating in the loop of Figure 11.1.(The easiest way to find the constant factor here is to require that the limit of this expression for d <3and A --* as matches Eq .(11.2.20).) With such an ultraviolet cutoff in place, we can pass to the limit d --+4, and obtai n (Z3-16,67r2ln(~/,u)  (1 1 .2.43) 11.3Anomalous Magnetic Moments andChargeRadii 485 Figure 11 .4. One-loop diagrams for the photon-lepton vertex function P . Here wavy lines represent photons ; other lines represent electrons or muons . Diagrams (a) and (b)are cancelled by lepton field renormalization terms ; diagram (c) arises from the vacuum polarization calculated in Section 11 .2; and (d)is the term calculated in Section 11 .3. 11.3 Anomalous Magnetic Moments and Charge Radi i For our next example, we shall calculate the shift in the magnetic moment and the charge radius of an electron or moan due to lowest-order radiative corrections . The one-loop graphs and renormalization corrections for the photon-lepton vertex are shown in Figure 11 .4.Of these graphs, those involving insertions in incoming or outgoing lepton lines vanish because the lepton is on the mass shell, as discussed in Section 10.3. The graph involving an insertion in the external photon line is the vacuum polarization effect, discussed in the previous section . This leaves one one-loop graph (the last in Figure 11A) that needs to be calculated here : JFP )41 [ -i _i4?_ 9) +m II00P(P" P)d4k[eY P(2 7r [(p .4(p )+ ~(27r)4(p-k)2 + MI (2n)4 V-ic] (11.3.1) where p' and p are the final and initial lepton four-momenta, respectively . (The contribution of the vertex connecting the external photon line and the internal lepton line is taken as y P, because a factor e(2 7r)4 was extracted in defining FP .) This integral has an obvious ultraviolet divergence, roughly like J'd4k/(k 2)2. Unlike the case of the vacuum polarization, here we do 486 11One-Loop Radiati ve Cor rectio ns not need a fancy regularization procedure like dimensional regulariza- tion to maintain the structure required bygauge invariance, because the photon is a neutral particle and so the integral may be rendered finite by suitable modifications of the photon propagator (for instance by including a factor M2 1(k2+ M2)with a large cutoff mass M), without having to introduce modifications elsewhere to maintain gauge invariance . In any case, as we shall see the anomalous magnetic moment and charge radii can be calculated without encountering any ultraviolet divergences at all . In what follows we shall leave the integrals for the vertex function in their infinite form, with it being understood that if necessary any divergent integrals can be expressed in terms of a cutoff mass M . We start by combining denominators, using a repeated version of the Feynman trick described in the Appendix to this chapte r ABC Jo /dv 2 Applied to the denominators in Eq . (11 .3.1),this give s 1 1 I 1 x 2 dx dy [ ((pl - k)2+M2y + ((p - k)'+M2_le)(X _Y) 2.F) X)]-3+ (k 1 x -32 dx dy[k-p'y_p(x_y))2+Frt2x2+ q2Y(x - y) - iEj a4 (11.3.3) where q = p - p' is the mornenturn transferred to the photon . Shifting the variable of integratio n theintegral ( 11.3.1) become s 2ie2 1 dx dy I dak 3 1111 loop(P!, P) = (27E)4 f()[k2+M2X2+ q2y(X -Y)-iel XTP[-i( io-Y)- v *X - Y)) +MITP X[-i( *1 -X+Y)- ~-fy)+MITP. (113A ) Our next step is a Wick rotation . As explained in the previous section, 11.3Anomalous Magnetic M oments and Charge radii 48 7 the -ie in the denominator dictates that when we rotate the k° contour of integration to the imaginary axis we must rotate counterclockwise, so that the integral over k° from -w to +oo is replaced with an integral over imaginary values from -ice to fiao, or equivalently over real values of k'= -iJc° from -aa to +ac . We also exploit the rotational symmetry of the denominator in Eq. (11 .3.4); we drop terms in the denominator of odd order in k, replace k 4d with q:'k2 /4, and replace the volume element A = idkldk3 dkldk4with 2i~' K3 dre, where K is the Euclidean length of the four-vector k . Putting this all together, Eq . (11 .3.4) now becomes ,47t2e2 1 x "'aT'A (27r)4 +'/P i (io - Y)- xx - Y)) + X X + Y)_ Y) + m Ij 14rpl X ~K 2+M2X2+q'y(x - y)] -3 (113.5) We are interested here only in the matrix element R'Tlu of the vertex function between Dirac spinors that satisfy the relation s u{[iMfr +m]= 0 , [i~+M]u =0. We can therefore simplify this expression by using the anticommutation relations of the Dirac matrices to move all factors fto the right and all factors fito the left, replacing them when they arrive on the right ❑r left with im . After a straightforward but tedious calculation, Eq . {11.3.5} then becomes ~47r2e2Li .x f r nelanplPf~~t~ - ~ L~J~ d ) o F, I I Y p [ _ IC2+2M2(X2- 4x + 2) +2q2(Y(X y) X) j +4im p' ~'(Y - x +xy°)+ 4im{J`(x2- xY- y)Iu X[2+n2x2 .+q2v(x_v)]_3 K(11.3.6) We next exploit the symmetry of the final factor under the reflection y --r x---Y.Under this reflection, the functions y - x +xy and x2-xy - y that multiply p "'and eare interchanged, so both may be replaced with their average : 2(y-x+xy)+2(x2_xy -- y} -- zxG-x). 4$8 11One-Loop Radi ativeCorrection s This gives fina lly -4g2e2i X 00 ' dx dy TC dreu tone laop W, P)"=2g)4 fo f o x - K' + 2rrt2(x2-- 4x +2)+2q2(Y(X- Y)X)] -2im (pl~' + p,)x(I - x) )U X [K 2 +mY +q2y(X _Y)] -3 .(11.3.7) Note that p~ and p "'new enter only in the combination p t'+ p'l`, as required by current conservation . There are other diagrams that need to be taken into account . Of course, there is the zeroth-order term y "inF.The term proportional to Z2 - 1 in the correction term (11.1.9)yields a term in I' F Also, the effect of insertions of corrections to the external photon propa- gator is a term : rvac pol~~~~- -I F)2- ~fev(Pr-~P' P) Yv (11 .3.9) The form of each of these terms is in agreement with the general result {10.6.10} (with H(qz) 0 ) To order e2, the form facto rs are 2 24~2e2 ~ ~ f'dy rc d~ F(q)=Z2+ ~(q) + 2~)4 ,lopdX ~ XIK2-2M2(XI- 4)c + 2) -2q2(y(X - y) + X)] 3 IK2+Fn2XI +q2Y(X- Y)] {i 1.x.11) G{q'}=-471 22 ~1dX fXdY~°`'4m2x(1_x) ~c3d~c 3(11.3.12e Jo J o (27c)4 IrcI +era2x2+q2Y(X -Y) ] where ar(q2) is the vacuum polarization function (11 .2.22). The integral for the form factor G(q2) is finite as it stands : 2 2 1 x 47T Jo Jo rya x + q2Y(X - Y) 11.3Anomalous Magnetic Moments and Charge Radii 48 9 Figure 11 .5. A two-loop diagram for the moon magnetic moment . Here the heavy straight line represents a rnuon ; the light wavy lines are photons ; and the other light lines are electrons . This diagram makes a relatively large contribution to the fourth-order muon gyromagnetic ratio, proportional to tn(m,,/m,) . This makes it easy to calculate the anomalous magnetic moment . As noted in Section 10 .6, it is only the yA term that contributes to the magnetic moment, so the effect of radiative corrections is to multiply the Dirac value e fZm of the magnetic moment by a factor F(O).But the definition of e as the true lepton charge requires tha t F(O) +G(O) = 1 , (11 .3.14) so the magnetic moment may be expressed a s Y = '(I G(O)) (11.3.15) FromEq. (11.3.13), w e find G(O)S~~ = 0.001161. ( 11.3.16) This is the famous a /2ncorrect ion first calculated by Schwinger.5 Ofcourse ,this is only the fir stterm in the radiative corrections to the magnetic moment .Even in just the next ❑rder, fourth order in e,there are so many terms that the calculation sbecome qu ite complicated .However , because of the large moan -electron mass ratio ,there is one fourth-order term in the magnetic moment of the muon that is somewhat larger than any of the others . It arises from the insertion of an electron loop in the virtual photon line of the second-order diagram ,as shown in Figure 11 .5. Theeffect of this electron loop is to change the photon propagator Ilk' in Eq . (11 .3. 1) to (1+ 7re (k2))/kI,where 7z,(k2) is giv enbyEq.(11,2.22), 490 1IOne-Loop Radiative Correction s but with the mass m taken as the electron mass : 't z 2X(I - X)In(I+ 2nJo Inspection of Eq . (11 .3.12) shows that in calculat ing the moan magnetic moment the effective cutoff on the virtual photon momentum k is m..The ratio m./m,is so large that forVof order rra~ wemay approximat e ln(m.~/ml){11.3.17} 7re(k 2).-.~3~~ x(1 -x )lr~(rr~~/rn~)=127E" with the neglected terms having coefficients of order unity in place of ]n(rn~/rr~~) . Since this is a constant, the change in TG(0) produced b y adding an electron loop in the virtual photon line is simply given by multiplying our previous result (11..16)for-G(O) by Eq .(11.3.17), so that now 2 4 1TaPy_2ms~2_ + ~_6 ~~In~+0(i) u 1)(11.3.18) (As we shall see in volume IZ, this argument is a primitive version of the method of the renormalization group .) The result (11 .3.18) may be compared with the full fourth-order result-6 e e2e 4 2rn~ $r r(2+97[lnE I~ 25 197 7t29~(3)2 6 + 24 + 23n In 2 + 0 ~~ ] ( 11.3.19) It turns out that the '0(1)' terms multiplying e4/96lr2 add up to -6.137, which is not very much smaller than ln(m~fn~e) =10.663, so the approx- imation (11 .3.18) gives the fourth-order terms only to a factor of order 2. The correct fourth-order result (11 .3.19) gives p,,= 1.00116546e/2m ., in comparison with the second-order result p,, = 1 .001161 e/2m,, and the current experimental value, 7 jU'U = 1.001 165923{S}e f 2rrzp . Nom let us turn to the other form factor . The integral in Eq . (f 1 .3.11) for F (q2) has an ultraviolet divergence . However, in order to satisfy the charge-non-renormalization condition (11 .3.14), it is necessary that Z2 take the value e247rze21 x o0 Z2_I + ~ dx dy rc3d~c 8712 (21r)4, K2-2m2(x2-4x+ 2)X ~K2 + M2X21(11.3-20) 11.3Anomalous Magnetic Mo mentsand ChargeRadii 491 (Recall that 7r(0) = 0 .) This is itself ultraviolet divergent, with an infinite part ~e d K Inserting Eq . (11 .3.20) back into Eq .(11.3.11)gives F(q2 Z47r2e2 (1 x 00 3 T7 JO dX fo X(11.3.21) [K2 + 2X2 +qly(X - y) ] [K2_7m2(x2 _4x+2)] [K2 + rn2X2]3 The in tegral over K is now conve rgent : F(q2) =i+ Xe221r2e21 (2n)dx Jxdy -m2[x2- 4x+Z] - q2Ly(x - y)+ i - x ] - InM2 X 2 +q2 Ax -Y) m2x2 11(11.3.23) However, we see that the integral over x and ynow diverges logarith- mically at x = 0and y = 0,because there are two powers of x and/or yin the denominators, and just two differentials dx d y in the numera- tor. This divergence can be traced to the vanishing of the denominator [K2+ M2 xz+ q2 Y{x -- Y}13inEq. (11.3.i 1) at x = 0, y = 0, and K = 0 . Because this infinity comes from the region of small rather than large rc, it is termed an infrared divergence rather than an ultraviolet divergence . We shall give a comprehensive treatment of the infrared divergences in Chapter 13 . It will be shown there that infrared divergences in the cross-section for processes like electron-electron scattering, such as those that are introduced by the infrared divergence in the electron form factor F(q2), are cancelled when we include the emission of low-energy photons as well as elastic scattering . Also, as we shall see in Chapter 14, when we calculate radiative corrections to atomic energy levels the infrared divergence in F (q2) is cut off because the bound electron is not exactly on the free-particle mass shell . For the present we shall continue our calculation by simply introducing a fictitious photon mass yto cut off th e+x'--4x+2 X2 492 11One-Loop Radiative Correction s infrared divergence in F(q2), leaving it for Chapter 14 to see how to use this result . With a photon mass p, the denominator k2 - iein Eq .(11.3.1)would be replaced with k2+ p2 -le. The effect would than be to add a term P2(1 - x) to the cubed quantity in the denominators of Eqs .(11.3.3)- (11.3.7), (11 .3.20), and (11 .3.22). Eq .(11.3.23)then is replaced with (27r)4 o -rn2[x2-4x+2]--q2[y(x - y) + 1- x] in 2[x2-4x+21 - In ~~~~+q2y(x - y) +p2{1 - x} m2x~+ JU2(1 - x)(11.3.24) This integral is now completely convergent . It can be expressed in terms of Spence functions, but the result is not particularly illuminating . For our purposes in Chapter 14, it will be sufficient to calculate the behavior of F(q ')for small q '. We already know from the Ward identity that F(O) = 1 - G(O) = 1 + e '/87r', so let us consider the first derivative F'(q2) at q2 = 0 . According to Eq . (11 .3.24), this i s Y(0) =n'{4} +2 (21r2 ~ J01cox foX dy ~~ The vacuum polarization contribution is given by Eq . (11 .2.22) a s e2 (0)= f Dac2m2(11.3.2F) Dropp ing all terms proportional to powe rs of ju/m inEq. (11 .3.25), we then have ' 2(y2)F'(0)~Z4~2m2[ln2+5 +41](11.3.27) with the term ~ the contribution of vacuum polarization . On the other hand, Eq . (11 .3.13) shows that G(q ') has a finite derivative at q2 = 0 , G'(0) z-e 48ir2m2(11.3.28) The y-integ ralis trivia l. The x-in tegral is most easily calc ulatedin the limi t'U m by dividing the ra nge of integration into two parts, one from 0 to s, w here p/m < s < 1, a ndthe seco nd from s to 1 . 11.4Electron Self-Energy 493 These results are most conveniently expressed in terms of the charge form factor Fi(q2 ), defined by the alter-native representation (10.6.15) of the vertex functio n ~(P " U' )Fl (P {7 Y)U(P Sa) -ii(p r7(T)[yFj (q2) +~i[I~} ~~y'7(rf Y)4i `L(qL)]u {p}Q). ( 11.3.29) According toEqs. (10.6.1.7)and (10.6.18), (11.3.30) For jq~~ < M2,this form factor is approximatel y F1(~2)~_I +e2 24~~1] 4~In~1+2+ M2 M2 S4(11.3.31) This may be expressed in terms of a charge radius a, defined bythe limiting behavior of the charge form factor for q2 --~- 0 : Fi(q2)-_~.1 --q2Q2/6. (11 .3.32) (This definition is motivated by the fact that the average of exp(iq ' x) over a spherical shell of radius a goes as f - q2a2f b for q2a2<1.) We see that the charge radius of the electron is given b y 2(1~) 2 1]a2~~~~~to ~# ~ -54(11.3.33) We will see in Chapter 14 that for electrons in atoms the role of the photon mass is played by an effective infrared cutoff that is much less than m, so the logarithm here is large and negative, yielding a positive value for a2 11,4Electro nSelf-Energ y We conclude this chapter with a calculation of the electron self-energy function . This b yitself does not have any direct experimental implications, but some of the results here will be useful in chapter 14 and volume II . As in Section 10 .3, we define i(2g)2[£`(P)]fl,"as the sum of all graphs with one incoming and one outgoing electron line carrying momenta p and Dirac indices oc and (3 respectively, with the asterisk indicating that we exclude diagrams that can be disconnected by cutting through some internal electron line, and with propagators omitted for the two external lines . The complete electron propagator is then given bythe sum 494 11 One-Loop Radiative Correction s Figure 11 .6. The one-loop diagram for the electron self-energy function . As usual, the straight line represents an electron ; the wavy line is a photon . where S(P) The sum is trivial, and gives-r~+ M,(1L4 .2) (11.4.3) In lowest or derthere is a one-loop contr ibution to 1', given by Figure 11.6: i(2,gI J.P(P)f d k [ (21T)4k2- le ~(27r)4 (p- k)l +m,'-icI or more simply 2 1i loop(P) =~.7~ 4 ~~~ [k2- xt +i 9 + m e)rP(1 t.4.4) (This is in Feynman gauge ; amplitudes with charged particles off the mass shell are not gauge-invariant .) For use in our calculation ❑f the Lamb shift, it will be convenient to use a method of regularization introduced byPauli and Villars .8 We replace the photon propagator (k '-i,-)-' wit h 1 1 k2+M2 so that the electron self-energy function i s 1ie2 A(27C)4 )12 - 14F x[YPH i + i 9 + MJ~P (p-k)2+m~ -to(11.4.5) Later we can drop the regulator by letting the regulator mass ygo to infinity . In chapter 14 we will also be interested in the case where ,u < M, . We again use the Feynman trick to combine denominators, and recall 1 1.4 Electron Self-Energy 495 thatyPy'y,,=--21'and y p7p= 4. This gives 2 P X dx fo ( {k - px}l+ pl(l - x) + M~x - ie}2 - - 1 11 .4.6) C Shifting the variable of integration k --*k + px and rotating the contour ❑f integration give s loop(P) =2 (27r)4f dd x [2i (1x) + 4r~] K Ka 1 1x - - {~c2+ p2x(1 - x) + ~n~ X}~(K2+ p2x(1 - x)+M2x+µi(1 -x))2 (11.x.7) The x-integral is trivial : _7r'e 2 l'i loop(P) fo x In ~2~(1 ~-~x)+r~~x+#~~(1 -~~ 11 .4.8 P2X(I- JiC) + FIZ eX The interaction {1 .119} also contributes a renormalization counterterm -(Z2-1)0j+Me)+Z26rra, in V{p}, with Z2 and 6m, determined by the condition that the complete propagator S'(p) regarded as a function of ij should have a pole at i j --rn, with residue unity . (As we shall see in the next chapter, this makes Vfinite as µ --.j,oo to all orders in e .) In lowest order, this give s bme 11 loo p 2rr~,7r2e21 (rn2x2 +Pz(1 - X) (7T)4JOdx [1 + act In M2 X2(11.4.9) Z2 - i ~ _~ G~~ loo p 21z2e['dx (1-x) In(27E)4'~X2 + ~I ~I -X) mx ---2p2( 1---X)2( 1 +X) (11.4.10) 496 11 One-Loop R adiativeCorrection s (To this order, we do not distinguish between Mme and Z25rn, .) Dropping terms that vanish for p2--*oo, Eqs . (11 .4.8)-(11 .4.10) yiel d 2e2f1 P 2(X) -7 1 1 iloop~P) = ( 2~)4dxt2i(1-x) j+4m'] In(P2x(1 X-) + M2 x (11.4.11) 61~.e_ Z2-1=2m,7z2e2 (27z)4 -27r 2e2 (27r)ldx11+x]InYI( 2 2X) IneX)I jodx (i -.x) InP2(1 -X) 2x2e(11.4.12) 2(1X2) Inspection then shows that in the complete self energy function the In p2 terms cancel, leaving us wit h order eAF)-Vilnap (P)-V2 - Ni i + me) + Z26M , z 2 ~( } 2rr~ax [i(1 - x)~+2m,j In1 ( Jo (p2X(1x 1 -m, [l+x]Inx 2X -(i j + me) [(I~ -- x) In x2 xX There is still a divergence from the behavior of the last term as x -- + 0, which can be traced to the singular behavior of the integral over the photon momentum k in Eq. (1t.4.5) at k2 = 0, when we takep2at thepoint p 2 =-m2 where we evaluated Z2 - I. Such infrared divergences will be discussed in detail in Chapter 13 . For the present, the point that concerns us is that the ultraviolet divergence has cancelled . The result (11 .4.9) for bm, is of some interest in itself . Note that bm,/m,>0,as we would expect for the electromagnetic self energy due to the interaction of a charge with its own field . But unlike the classical estimates of electromagnetic self-energy by Poirtcare, Abraham, and others, Eq. (1 1.4.9) is only logarithmically divergent in the limit p--+ao where the cutoff is removed . In this limi t e (21r)4In~ (11 .4.15) In our calculation of the Lamb shift in section 14 .3 we will beinterested Appendix Assorted In tegrals inthe opposite limit, y C rra, Here E q. (11.4.9) give s e2P1- P$,g 2,gm , Appendi x Assorted Int egrals497 (11.4.16) In order to combine the denominators of Npropagators, we need to replace a product likeDIrD2r...D,yi with an integral of a function that involves a linear combination of D1, D 2,... DN. For this purpose it is often convenient to make use of the formul a DID2..l7N-(N- 1)? dx1 dx2... dxnr_1. 10 fo . fo In this chapter we have used special cases of this formula for =2 and ITT = 3 . After combining denominators, sbiffing the four-momentum variable of integration, wick rotating, and using four-dimensional rotational invari- ance, we commonly encounter integrals of the for m A(k2)n (k2+v2) with (k2 + v2),ncoming from the combined propagator denominators, and (k2), coming from the propagator numerators and vertex momentum factors . This is divergent for 2n + 4 >2m, but the integral can be given a finite value by analytically continuing the spacetime dimensionality from 4 to a complex value d . To evaluate the resulting integral, we use the well-known formul a ( ~ }+V 2 r(~. where e=d + 2 n.We used this formula in the special cases n= 0, m= 2 and n = I'm = 2 in Section 11 .2. Ultraviolet divergences man ifest themselves in Eq.(11.A.2)as poles in the factor I`{m - (/ )= r (rrz-n -d/2) asd --- ~. 4with fixed integer n.For 2+ n =m,this factor goes a s 4-d 2 2. ~d-4+Y (11.A.3) where y = 0.5772157  - -is the Euler constant . The limiting behavior for 2 + n m can be obtained from (I1.A,3} and the recursion relation for mamma functions . 498 1 1 One-Loop Radiative Correction s Problems 1. Calculate the contributions to the vacuum polarization function 7r(q2) and to Z3of one-loop graphs involving a charged spinless particle of mass m, . What effect does this have on the energy shift of the 2s state of hydrogen, if m5 >> Zac .e? 2. Suppose that a neutral scalar field 0of mass mohas an interaction goWipwith the electron field . To one-loop order, what effect does this have on the magnetic moment of the electron? On Z2 ? 3. Consider a neutral scalar field 0with mass mo and self-interaction g03f 6. To one-loop order, calculate the S-matrix element for scalar- scalar scattering . 4. To one-loop order, calculate the effect of the neutral scalar field of Problem 2 on the mass shift 6m, of the electron . References 1. R. P. Feyntnan, Phys .Rev. 76,769 (1949) . 2. G.C.Wick, Phys . Rev .80, 268 (1950) . 3. G. 't Hooft and M . Veltman, Nucl.Fhys .B44, 189 (1972). 4. E. A. Uehling, Phys . Rev. 48, 55 (1935 . The one-loop function 7c(q2) was first given for q 2*0by J . Schwinger, Phys, Rev. 75,651 (1949) . 5.1Schwinger, Whys . Rev . 73, 416 (1948) . 6. This is calculated (including terms that vanish for m, { my) by H . Suura and E . Wichmann, Phys . Rev.105, 1930 (1967) ; A. Petermann , Pays . Rev .105, 1931 (1957) ; H. H. Elend, Phys . Lett .20, 682 (1966) ; 21, 720 (1966) ; G. W. Erickson and H . H. T. Liu, UCD-CN L-$ 1 report (1968) . 7.J. Bailey e tal. (CERN-Mainz--Daresbury Collaboration), Nucl.Phys. B150, I (1979) . These exper iments are done by observing the p reces- sion of the moon spin in a storage ring . 8, Pau li and F . Villars, Rev. Mod. Pays .21, 434 (1949) . Also see J . Rayski, Phys . Rev. 75, 1961 (1949) . 9. See, e .g., A.I. Miller, Theory of Relativity Emergence (1905) and Early Interpretation (1905-191 1) (Addison-Wesley, Reading, MA, 1981) : Chapter 1 . 12 General l enormaliz tion Theor y We saw in the previous chapter that calculations in quantum electrody- namics involving one-loop graphs yield divergent integrals over momen- tum space, but that these infinities cancel when we express all parameters of the theory in terms of `renormalized' quantities, such as the masses and charges that are actually measured . In 1949 Dyson] sketched a proof that this cancellation would take place to all orders in quantum electrody- namics . It was immediately apparent (and will be shown here in Sections 12.1 and 12 .2) that Dyson's arguments apply to a larger class of theories with finite numbers of relatively simple interactions, the so-called renar- malizabte theories, of which quantum electrodynamics is just one simple example . For some years it was widely thought that any sensible physical theory would have to take the form of a renormalizable quantum field theory . The requirement of renormalizability played a crucial role in the devel- opment of the modern `standard model' of weak, electromagnetic, and strong interactions . But as we shall see here, the cancellation of ultravi- olet divergences does not really depend on renormalizability ; as long as we include every one of the infinite number of interactions allowed by symmetries, the so-called non-renormalizable theories are actually just as renormalixable as renormalizahle theories . It is generally believed today that the realistic theories that we use to describe physics at accessible energies are what are known as `effec- tive field theories .' As discussed in Section 12 .3, these are low-energy approximations to a more fundamental theory that may not be a field theory at all . Any effective field theory necessarily includes an infinite number of non-renormalizable interactians . Ncvertheless, as discussed in Sections 12 .3 and 12 .x, we expect that at sufficiently low energy all the non-renormalizable interactions in such effective field theories are highly suppressed . Renormalixable theories like quantum electrodynamics and the standard model thus retain their special status in physics, though for reasons that are somewhat different from those that originally motivated the assumption of renormalizability in these theories . 499 500 1 2 General Renoralizatian Theor y 12.1 Degrees of Divergenc e Let us consider a very general sort of theory , containing interact ionsof varying types labelled i . Each interaction may be charac terized by the number n ;f offields of each type f,and b ythe number diof derivat ives acting onthese fields. We will start by calculating the `superficial degree of divergence' Dof an arbitrary connected one-particle irreducible Feynman d iagram in such a theory . This is the number of factors of momentum in the numerator minus the nu mber in the denominator of the integrand ,plus four forevery independent four-momentum over which we integrate .The superficial divergence is the actual degree of divergence of the integration over the region of momentum space in which the momenta of all inte rnal lines gotoinfinity together . That is, if D 0, then the part of the amplitude where all interna lmomenta go to infinity with a common factor K--*ova will d iverge like x rcD-'dre. In the same sense, an integral with degree of divergence D=0 is loga- rithmically divergent, and an integral with D <0 is convergent, at least as far as this region of momentum space is concerned . We will come back later to the problem posed b ysubintegrations that behave worse than the integral over this region . To calculate D, we will need to know the following about the diagram : If number of internal lines of field type f, Ef number of external lines of field type f, Ni number of vertices of interaction type i . We will write the asymptotic behavior of the propagator d f(k) of a field of type f in the form Af (k) - k-'+-"f (12.1.2) Looking back at Chapter 6,we see that sf = 0for scalar fields, s f= for Dirac fields, and sf= 1 for massive vector fields . More generally, it can be shown that for massive fields of Lorentz transformation type (A, B), we have s f = A + B .Speaking loosely, we may call s fthe `spin .' However, dropping terms that because of gauge invariance have no effect, the effective photon propagator r~,,,/k2has s f=0. A similar result applies to a massive vector field coupled to a conserved current, provided the current does not depend on the vector field .It can also be shown that, in the same sense, the graviton field g ,,,, has propagator also with s f= D. 12.1 Degrees of Divergence 501 According to (12 .1.2), the propagators make a total contribution to D equal to Y:If (2 sf -2) f(12.1.3 Also, the derivatives in each interaction of type i introduce dimomentum factors into the integrand, yielding a total contribution to D equal t o Nj di . (12.1.4) Finally, we need the total number of independent momentum variables of integration . Each internal line can be labelled with afour-momentum, but these are not all independent ; the delta function associated with each vertex imposes a linear relation among these internal momenta, except that one delta function only serves to enforce conservation of the external momenta . Thus, the momentum space integration volume elements contribute to D a ter m 4 I f- Nj- 1 (12 .1.5) f which, of course, is just four times the number of independent loops in the diagram . Adding the contributions (1 2.1.3), (12 .1.4), and (12 .1.5), we find f Eq. (111.6)is not very convenient as it stands, because it gives a value for D that seems to depend on the internal details of the Feynman diagram . Fortunately, it can be simplified b yusing the topological identitie s 2If +Ef Nj nif , (12.1.7) (Each internal line con tributes two of the lines attached to vertices, while each external line contributes only one .) Using Eq . (12 .1.7) to el iminate If, we see that Eq . (12.1.6) become s f i where d ; is a parameter characterizing interactions of type i(12.1.8) (12.1.9) This resu lt could have been obtained by simple dime nsional ana lysis, without considering the structure of Feynman diagrams . The propagator 502 12 General Renormalization Theor y of a field isafour-dimensional Fouriertransform of the vacuum ex- pectation value of a time-ordered product of a pair of free fields, so a conventionally normalized field f w hose dimens ionalit y*in powers of mo- mentum is 'fwill have a propagator of d imensionality -4-}- 2-9f, Hence if the propagator behaves like k -2+2,f when k is m uchlarger than the mass, then the field must have a dimensionality with -4+2 -qj = -2+2sf, or9f=1+ s f .An interaction iwith nif s uch fields a nd diderivatives will then have dimensionality dd +l:fraif (I +s f).But the action m ust be dimen- sionless ,soeach term in the Lagrangian density must have dimensionality +4 to cancel the dimensional ity -4 of d4x. Hence the interaction must have a coupling constant ❑f dimens ionality 4 -d,- Ef nif (1+sf),which is just the pa rameter❑j. The momentum space amplitude corresponding to a connec tedFeynman graph with Efexter nallines of t ype f i sthe Fourier transform over 4X:fEfcoordinates of a vacuum expe ctation value of the time-ordered product offields with a total dimensionality J:fEf (1+sf), soit has d imens ionality fEf (-3+sf ). Of this dimens ionality, -4comes from a momentum space delta function, and EfEf(-2 + 2sf) is the dimensionality of the propagators for the external lines ,so the momen- tum space i ntegral itself together with allcoupling constant factors has dimensionalit y I:Ef(-3 +sf)-(-4) - 1:-Ef(-2 + 2,5f) = 4-EEf(sf +1) f f J The coupling constants for a given Feynman graph have total dimension- ality i Ni Aia leaving the momentum space integral with dimensionality 4 - EfEf(sf+ 1) - j NA .As long as we are interested in the region of integration where all momenta go to infinity together, the degree of divergence of the momentum space integral is its dimensionality, thus justifying Eq.(12.1.8). If all interactions have d j ~!: 0, then Eq . (12 .1.8) provides an upper bound on Dthat depends only on the numbers of external lines of each type, i .e., on the physical process whose amplitude is being calculate d D:<4 - E f(sf+ 1). (12 .1.10) f For example, in the simple version of quantum electrodynamics studied in the previous chapter, the Lagrangian included terms of the types shown in Table 12 .1.Allinteractions here have d ; 0, and hence a Feynman diagram with E,external photon lines and E,external Dirac lines wil l In this chapter, `dimensionality' will always refer In the dimensionality in powers of mass or momentum, in units with h = c = 1 . We are using fields that are conventionally normalized, in the sense that the term in the free-field [,agrangian with the largest number of derivatives (which determines the asymptotic behavior of the propagator) has a dimensionless coefficient . 12.1Degrees of Divergence 503 Table 12 .1. Terms in the Lagrangian density for quantum electrodynamics . Here di, nr_,, and nj,, are the numbers of derivatives, photon fields, and electron fields in the interaction, and d iis the dimensionality of the corresponding coefficient . (Recall that sr -=4,S'° ~.~ Interactio n ieip-,Ay_a V3 I)FpvFPv -(Z2 -- 1)iPkdini~ni e 0 1 2 2 2 0 1 0 2 0 0 2❑i 4-1 - 3= U 4-2-2 0 4-1-3=0 4-3= I have superficial degree of divergence bounded by Eq . (12 .1.10): D::~4-T ~ Ee-EY(12.1.11) Only a finite number of sets of external lines can yield superficially divergent integrals, these will be enumerated in Section 12 .2. We are going to show that the limited number of divergences that appear in theories with❑i0for all interactions are automatically removed by a redefinition of a finite number of physical constants and a renormalization of fields . For this reason, such theories are called renormah zable .In Section 12 .3 we will catalog all the renormalizable theories, and discuss the significance of renormalizability as a criterion for physical theories . The term `renormaliza6le' is also applied to individual interactions . Renormali7able interactions are those with dz 0, whose coupling con- stants have positive or zero dimensionality . Sometimes one distinguishes between interactions with d j = 0 , called simply renormalizable, and those with Al > 0,called superrenar malizahle. Since adding additional fields or derivatives always lowers Az, there can only be a finite number of renormalizable interactions involving fields of any given types . We have seen that all the interactions in the simplest version of quantum electro- dynamics are renormalizable, with the tip terms superrenormalizable . On the other hand, if any interaction has Aj 0, the degree of di- vergence (12 .1.8) becomes larger and larger the more such vertices we include . No matter how large we take the various E f, eventually with enough vertices of type i for which d j < 0, E q.(12.1.8) will become positive (or zero), and the integral will diverge . Such interactions, whose couplings have negative-definite dimensionality, are called non-retzormaliZable ;"the- [inperturbative statistical mechanics, non-renol`mali7ahlc interactions are called irrelevant, because they become less important in the limit of low energies . Renormalirahle and super-renormalixable interactions are cared marginal and relevant, respectively . 504 12 General Renarmalxzation Theor y Figure 12 .1. Some two-loop graphs for Compton scattering . Here straight lines are electrons ; wavy lines are photons . The momentum space integra] for diagram (a) is convergent, while for (b)and (c) it is divergent, due to the subintegration associated with the subgraphs surrounded by dotted lines . cries w ith any non-renorm alizable interactions are also known as non- renormali 7able. But this does not mean that such theories are hopeless ; we shall see that these divergences may also be absorbed into aredefinition of the parameters of the theory ,but here weneed an infinite number of couplings . It should be kept i nmind that wehave here calculated the degree of divergence of Feynman diagrams aris ingonly from regions of momentum , space inwhich allinternal four-momenta go to infinity together . Diver- gences can also arise from regions inwhich only the four-m omenta o f lines belonging to some subgraph go to infinity . For instance, in quantum electrodynamics Eq.(12. 1AZ) gives D~ - 1 for C ompton scatte ring(where Ee- 2, ET = 2), and indeed graphs li ke Figure 12.1(a) are convergent , but a graph like Figure 12.1(b) or12.1(c) is logarithmically divergent, because these graphs contain subgr aphs (indicated by dotted boxes) with D ~ 0. We can think of the divergence of these graphs as being due to an anoma- lously bad asymptotic behav ior that occurs when the eight components of thetwo independent internal four-momenta of these graphs go to infinity on a p articular four-dimensional subspace ,namely ,that s ubspa ce in wh ich the only four-momentum actually going to infin ity is the one circulating intheloops that are in serted in the internal lines or at an electron-photon vertex . ,It has been shown that the requirement for the actual convergence of 12.2Cancellation of Divergences 505 the amplitude corresponding to any graph is that power-counting should give D <0 not only for the complete multiple integral for the whole amplitude, but also for any subintegration defined by holding any one or more linear combinations of the loop momenta fixed . (The graphs shown in Figures 12.1(b)and 12 .1(c) fail this test because D 0for the subintegrations in which only the momenta for the loops within the dotted squares are integrated .) We will not repeat the rather long proof here, because it is well treated in earlier books,3 and in any case the method of proof has little to do with how we actually do calculations . The next section will describe how this requirement is fulfilled . 12.2 Cancellation of Divergence s Consider a Feynman diagram, or part of a Feynman diagram, with positive superficial degree of divergence, D ~ 0 . The part of the momentum space integral where all internal momenta go to infinity together will then diverge, like F'°V-1dk.If we differentiate D+ 1 times with respect to any external momentum, we lower the net number of momentum factors in the integrand b yD + 1,' and hence render this part ❑f the momentum space integral convergent . There may still be divergences arising from subgraphs, like those in Figures 12.1(b)and 12 .1(c) ; for the moment we will ignore this possibility, returning to it later in this section . Since differentiation D +1 times renders the integral finite, it follows that the contribution of such a graph or subgraph can be written as a polynomial of order D in external momenta, with divergent coefficients, plus a finite remainder . To see how this works without irrelevant complications, consider the logarithmically divergent one-dirnensional integra l jdk withD = I - I = 0 .Differen tiating once give s dk 1 (k+q)2 q so _0~(q)= -Ing+c . ' For instance, if an internal scalar field line carries a momentum k + p, where p is a linear combination of external four-momenta and kis a fou r-momentum variable of integration . then the derivative of the propagator [ (k+p)3 +m 2]-1with respect to p+` givc1 -2 (kj,+p")L(k+p) '+M, which goes as k-3 rather than k-2 for k --- >oo. 506 12 General Renormalization Theor y The constant c is obviously divergent, but the rest of the integral is perfectly finite . In exactly the same way, we can evaluate the D = 1 integral k dk-a+bq+q Inqa k+ q with divergent constants a and b. Now, a polynomial term in external momenta is just what would be produced byadding suitable terms to the Lagrangian : if a graph with Ef external lines of type fhas degree of divergence D~0,then the ultraviolet divergent polynomial is the same as would be produced by adding various interactions i with nif= Ef fields of type fand d ;:!9Dderivatives . If there already are such interactions in the Lagrangian, then the ultraviolet divergences simply add corrections to the coupling constants of these interactions . Hence these infinities can becancelled byincluding suitable infinite terms in these coupling constants . All that we ever measure is the sum of the bare coupling constant and the corresponding coefficient from one of the divergent polynomials, so if we demand that the sum equals the (presumably finite) measured value, then the bare coupling must automatically contain an infinity that cancels the infinity from the divergent integral over internal momenta, (One qualification : where the divergence occurs in a graph or subgraph with just two external lines, which appears as a radiative correction to a particle propagator, we must demand not that some effective coupling constant equals its measured value, but rather that the complete propagator has a pole at the same position and with the same residue as for free particles .) In this way, all infinities are absorbed into a redefinition of couplings constants, masses, and fields . For this renormalization program to work, it is essential that the La- grangian include allinteractions that correspond to the ultraviolet diver- gent parts of Feynman amplitudes . (There are exceptions to this rule in supersymmetric theories .4) The interactions in the Lagrangian are, of course, limited byvarious symmetry principles, such as Lorentz invari- ance, gauge invariance, etc ., but these constrain the ultraviolet divergences in the same way . (It takes some work to prove that non-Abelian gauge symmetries constrain infinities in the same way that they constrain inter- actions . This will be shown in Volume II .)In the general case, there are no other limitations on the ultraviolet divergences, so the Lagrangian must include everypossible term consistent with symmetry principles. But there is an important class of theories with only a finite number of interactions, where the renormalization program also works . These are the so-called renormalizable theories, whose interactions all have ❑jL:0. 12.2 Cancellation of Divergences 507 Eq. (l x..1.8) then gives ~~~ ~ Ef (5f +1), so divergent polynomials arise in only a limited number of Feynman graphs or subgraphs : those with few enough external lines so that D ~ 0. The contribution of such divergent polynomials is just the same as would be produced by replacing the divergent graph or subgraph with a single vertex arising from a term in the Lagrangian with Effields of type f and 0,1,... D derivatives . But, comparing with Eq .(12.1.9),we see that these are precisely the same as the interactions that satisfy the rertor malizuba lity. requirement ~j~0, or in other words , f In order for all infinities to cancel in a renormalizable theory, it is usually necessary that all renormalizable interactions that are allowed by symmetries must actually appear in the Lagrangian ."For instance, if there is a scalar (or pseudoscalar) field 0and fermion field yi with interactions ip-WO (or V,'Ys~k* then we cannot exclude an interaction 04 ; otherwise there would be no counterterm to cancel the logarithmic divergence arising from fermion loops with four attached scalar or pseudoscalar lines . Let's see in more detail how the cancellation of infinities works in the simplest version of quantum electrodynamics . Eq . (12 .1.11 ) shows that the only graphs or subgraphs that could possibly yield divergent integrals are the followin g Ee = 2,Ey= 1 This is the electron-photon vertex T'~~ )(p',P)- {The superscript (indicates that this includes only contributions from graphs with loops .)Ithas D = 0, so its divergent part is momentum-independent . Lorentz invariance then only allows this divergent constant to be proportional to ~ ., so VA() + I'~`[f 1 (12 .2.1) with L a logarithmically divergent constant, and r(f ~ finite . This does not uniquely define the constant L, since we can always move a finite ter m In xddi#ian, interactions and mass terms that are not allowcd by global symmetries may appear in the Lagrangian, as long as they are super renormaliaable, that is, with d ;>0. This is because the presence of a superrenormalizable coupling lowers the degree of divergence, so that the symmetry breaking does not affect those divergences that are cancelled by the strictly rennrrnalizahle couplings with d~ = D . Note that it is the bare strictly rennrmalixabJc couplings that would exhibit the symmetry ; renarmaliaed couplings that are defined in terms of mass-shell matrix elements generally show the effect of symmetry breaking . 508 12General Renormali zation Theor y 6L y,,from T~f) to Lys . To complete the definition, we may note that as shown in Section 9 .7, the mass-shell matrix element of I`A(p, p)and hence ofF(J)(p,P)between mass-shell Dirac spinors is proportional to the same matrix element of ,41, so we may define L by the prescription tha t for p2+ m~ = 0. This is the electron self-energy insertion V(p). It has D= 1, so its divergent part is linear in the momentum carried by the incoming and outgoing fermion . Lorentz invariance (including parity conservation) will only allow it to be a function of ~, so we may write the loop contribution as 11"( P) = A- {i j+ rra}B + E(.f)( J)s (12.23 where A and B are divergent constants ,and F-W is finite .Again, this does not uniquely define the constants A and B,because wecan always shift E[f}by a finite first-order polynomial in J.Wewill define A and B by the prescript ion that Z(f] =0 for i ~ _ -m. (12.2.4) Actually, B is not a new divergent constant . As long as we use a regularization procedure that respects current conservation, F.and Ewill be related by the Ward identity (10,4 .27) pj, and therefore Op Taking the matrix element of this equation between i~(p, or') and u(p, a) and using Eqs . (12 .2.2) and (12 .2.4), we fin d L= B, Ey -2, E~=0(12.2.6) This is the photon self-energy insertion II~ ,(q). It has D --2, so its divergent part is a second-order polynomial in q . Lorentz invariance only allows I ];,to take the form of a linear combination of q,,,and q ,,qy with 12.2 Cancellation of Divergences 509 coefficients depending only on q2, so the loop contributions take the for m =C11p+ C2 1pvq2+C3quqv + finite terms , where C 1,C2, and C 3are divergent constants . As long as we use a regularization technique that respects current conservation, we must hav e q111141(q) ==0. The same must then be true for the divergent terms, so CIgV+(C2+C 3)ql q, must be finite for all q . It follows that Ci and C2+C3must be finite, and can therefore be lumped into the finite part of II~ ;)(q). Thu s where g(q2) is finite and C is the sole remaining divergence in To pin down the definition of C, we may move any finite constant g(O) into C, so that 7r(D) =0. Ey =4,E, =0(12.2.8) This is the amplitude for scattering of lightby light .It has D = 0, so using Lorentz invariance and Bose statistics, it may be written (there is no non-loop contribution ) M~vpfT-K(qjuuqpy+qjupqjg +npTq wp)+finite term s with K a potentia lly divergent constant .However, current conservation gives and so K (qvqpd+ qpqwa + q ,tjUp) is finite . In order for this to be true for q * 0, K must itself be finite . This is a nice example of the role o f symmetry principles in the renorrnalixatian program ; if K had turned out to be infinite it could not be removed byrenormalization of the coupling constant for an interaction (AyAg)3, because no such interaction is allowed by gauge invariance, but Kis finite because of current conservation conditions that are imposed by gauge invariance . E7 =l.,E, =0 and E,= 1,Fy= D,1,2 These have D= 3 and D=5,3and z, respectively, but Lorentz invariance makes all such graphs vanish , E? = 3,Ee _0 This has D = 1, but vanishes because of charge-conjugation invariance . 51 0 12Genera l Rsrtarrnalizatinn Theor y The reader will perhaps have noticed that the independent divergent constants A, B, C are in one-to-one correspondence with the independent parameters Z2, Z1, and 6min the counterterm part (11 .1.9) of the La- grangian for quantum electrodynamics . These counterterms make a direct contribution Z;brn-(Z2 - i )(i#+m) to E" (P). The requirement that the position and residue of the one-particle pole be the same as in the free-field propagator means we must choose Z2 and bm so that the total V(P) satisfies Eq . (12 .2.4), i .e., ~~6M= -A, {12 .2.9} so that the complete electron self-energy insertion is just the finite function W(P) I(P)=If.fl(F).(12.2.11) Also, Y2 makes a direct contribution to I",, equal to V2 - 1)Y,, . Using Eq. (12 .2.6), we see that the full vertex i s This is not only finite, but satisfies the conditio n a(P, a {)Fi{(P?P)U(R, CT)=F1 (P, C') 7p U (As 6f ) a(12.2.12) as can also b eseen from Eqs .(10.6. Z 3)and (10 .6.14). Finally, Y7makes a contribution -(Z3 -1)(R2r~~jv -R ,,gti,)to II~,, (q)- In order that the photon propagator should have a pole with the same residue as for free fields we need the coefficient of q2 ~pu - q ,,vq,in the total II ,..,(q)to vanish, s o Z3=I+C and the photon propagator is then finite :(12.2.14) (12.2.15) So far, we have only checked that the divergences, arising from the re- gion of momentum space in which all internal momenta are large (and with generic ratios), are polynomials in external momenta that are cancelled by suitable counterterms . Such graphs are called superficially convergent . Before we conclude that all ultraviolet divergences actually are removed by renormalization, we need to consider the ultraviolet divergences aris- ing in higher-order graphs when some subset of the momentum space integration variables rather than all of them go to infinity . For instance, in quantum electrodynamics the superficial divergences in subintegrations come from subgraphs that are either photon self-energy parts II', or elec- tron self-energy parts 1', or electron-electron-photon vertices I'4 . The 12.2Cance llation ofDivergence s (ZZ -1)2(~~-~) a (Z3-1)overlap511 Figure 12 .2. Some fourth-order graphs for the photon self-energy in quantum electrodynamics that involve overlapping divergences . Lines carrying arrows are electrons ; wavy lines photons . The crosses mark the contribution of counterterms . problem with such divergences is that they cannot be removed by dif- ferentiati on with respect to external momenta ; we a re left with terms where the derivat ives act only on internal lines in the parts of the graph which are not in the divergent subgraphs, and therefore donot reduce the degree of diver genceofthese subgraphs .As mentioned in the previous section, a graph or sum of graphs is actually convergent only if it and all its sub integrations are super ficially convergent ,in the sense of counting powers of momentum . But wherever such a divergent subgraph appears , it always comes accompanied with an infinite count erterm .Inclectrody- namics, these are the terms inEq.X11.1.9: a term -(Z3-1) (q2rj~"-q{jqu) ,V(q); a term Z26 m,-V2 - 00 j +rte) for each V(p);andfor each I-I* a term (Z2---1)7,' for each I' P.Just as for the graph asa whole, these counterterms cancel the infinit ies from the divergent s ubgraph s.1 Unfortunately ,there is a flaw in this simple argument -the possibilit y of overlapping divergences .That is ,it is p ossiblethat t wo divergent subgraphs may share an internal l ine, so that w ecannot regard them as independent d ivergent integrals . Inquantum electrodynamics this happens only when two electron-electron-photon vertices overlap inside a photon or an electron self-energy insert ion, as sho wn in Figures 12 .2 and 12 .3. A comp letetreatment of renormalization that takes account of overlap- t The sharing of a line in two self-energy insertions or in a self-energy insertion and a vertex part would not leave enough external lines to attach such a subgraph to the rest of the diagram . Nistoricaliy, the Ward identity (10 .4.26) was used to by-pass the problem of overlapping divergences in the electron self-energy, by expressing the electron self-energy in terms of the vertex function, where overlapping divergences do not occur . This approach will not be followed here, as it is unnecessary, and in any case does not solve the problem for the self-energy of the photon or other neutral particles . 512 12 General Rertarmalizattorz t'heor' y {2z -02---~ (22-])Z Z3-1 ) overlap Figure 12 .3. Some fourth-order graphs for the electron self-energy in quantum electrodynamics that involve overlapping divergences . Wavy lines are photons, other lines are electrons . The crosses mark the contribution of counterterms . ping divergences should include a prescription for eliminating superficial ultraviolet divergences, not only in the overall integration but in all subin- tegrations as well, together with a proof that this prescription is (at least formally) implemented by renormalization of masses, fields, and coupling constants . The theorem of Ref . 2 then ensures that all Green's functions of renormalized fields are finite when expressed in terms of renormalized masses and couplings . The first proof that the renormalization of fields, couplings, and masses renders the whole integration and all its subinte- grations superficially divergent was offered by 5alatn .' A more specific prescription for eliminating ultra-violet divergences was given byBogali- ubov and Farasiuk,6and corrected by Hepp,7and shown by them to be equivalent to a renormalization of fields, masses, and coupling constants . Finally, Zimmerman8 proved that this prescription does eliminate all su- perficial divergences in the whole integration and all its subintegrations, and used the theorem of Ref . 2 to conclude from this that the renormalized Feynman momentum-space integrals are convergent . Briefly, the `HPHZ' prescription for eliminating superficial divergences requires that we consider all possible ways (called 'forests') of surrounding a whole graph and/or its subgraphs with boxes that may be nested within each other but do not overlap . (An example is given below .) For each forest we define a subtraction term by replacing the integrand for any subgraph of superficial divergence D within a box (starting with the innermost boxes and working outwards) with the first D + 1 terms of its Taylor series expansion in the momenta flowing into or out of that bvx, T As described here, this prescription applies to unrenormalixable as well as renorrnalixable theories . In renormalizable theories it implies that there is no subtraction unless the box contains one of the limited number of graphs corresponding to renorrnalicabZe terms in the Lagrangian . 12.2 Cancellation of Divergences 513 The subtracted Feynman amplitude is given by the original graph, minus all these subtraction terms, including the subtraction term for a forest consisting of a single box surrounding the whole graph . It is fairly easy to see that the subtracted Feynman amplitude that is calculated in this way is the same as would be obtained by replacing all fields, coupling constants, and masses in the original Lagrangian by their renormalized caunterparts . The difference between this procedure and the sort of renormalization we carried out in Chapter 11 is that the renormalized fields, coupling constants, and masses are defined in terms of amplitudes at a unconventional renormalization point, where all four- momenta vanish . (In this respect, the any-dimensional divergent integrals discussed at the start of this section provide an elementary example of the BPHZ method of separating divergent terms .) But there is nothing special about this renormalization point ;❑nce a Feynman amplitude is made convergent by expressing it in terms of these unconventional renormalized quantities, it can be rewritten in terms of conventionally renormalized fields, couplings, and masses without introducing new infinities . It is not necessary to use the BPHZ subtraction prescription in practice . Replacing fields, masses, and couplings with their renormalized counter- parts (defined using any convenient renormalization points) automatically provides counterterms that cancel all infinities . Instead of proving that the BPHZ subtraction prescription really does make all integrals converge, we shall just look at one example that shows how renormalization works, even in the presence of overlapping divergences . Consider the fourth-order contribution to the photon self-energy inser- ',M} shown in Figure 12 .2. (The forests here consist of the whole tion W integral over p and p', the subintegration over p alone, and the subinte- gration over p' alone .) Including the corresponding counterterms for the vertex parts and photon field renormalization, this has the valu e 2~r ( P-T p') , x T r{S (pr) Yv S (Pr + q) x'P S (p +q) Yp S (F} Y p e' d4pTr ~y, S(p + q),ip S(p) -2 V2 - 02 (27C) -(Z3-I)overiap (q 2qfu}'- qtjq v), (12.2.16) where S(p)-j-ij+M] 1 Ip2 +M2 - a,_]; ~ZZ- 02 is the term in Z2 - 1 of second order in e ; and V3 - 1)Qveflap is a logarithmically divergent constant of fourth order in e that cancels the terms in [II *jiu(q)]overlap of second order in qf .. The factor 2 in the second term arises because there is a renormalization counterterm Z2 - 1 for each of the two vertices in the second-order photon self-energy . Note, however, that the first term here 514 12 General Renormalization Theor y can be thought of either as the insertion of a vertex correction given by the p'-integral into a photon self-energy given by the p-integral, or as the insertion of a vertex correction given by the p-integral into a photon self- energy given by the p'-integral, but not as the insertion of two independent vertex corrections, because there is only one photon propagator . To see how to handle the infinities in Eq. (12,216), note tha t ie2 RZ2 -02 + ,R21 '~"P- ~2_g)4Pr2ic"r'~S W) ,'ft S(p')71, (12.2.17) where R2 is a finite remainder . (Lorentz invariance tells us that the integral on the right is proportional to y~,. The difference between this integral and (Z2. - 021 . equals the complete renormalized electron-electron-photon vertex to second order in e at zero electron and photon momenta, and is therefore finite .) This allows us to rewrite Eq. (12.2.16)in the for m e~ [ri*Pv(q)]overlap -- (2Ttgd4p j d'p l x _ '~ i ~~Tr {S (P'} "~~ S(Pf + q) yP S(p + q)f'uS(p) ~p} (P P T 12iieTrfS(P')YvS(Pf)f'P S (P+q) 7'jj S (P)YpIP P2 ie (S(P)Y'US(P)TP~ ] ie-2R21 (27C) 4 -V3- i}averlap(R2qkcv-quql,). (12 .2.8) First consider integration over p' alone . Each of the first two terms is logarithmically divergent, but their difference is finite . The third term is also logarithmically divergent (with a gauge-invariant regulator), but the divergence in this term (unlike the first two terms) takes the form of a second-order polynomial in q, with the remainder finite . This remaining divergence is cancelled by the term -(Z3 - 1)(q2-ql`q") that cancels all second-order terms in the expansion of I 1*u(q} So the p'-subintegration gives a finite result . The symmetry of Eq .(12.2.1+5)shows that in exactly the same way the p subintegration also gives a finite result . Generic subintegrations over p and p' with ap + bp' held fixed (where a and h are arbitrary non-zero constants) are manifestly convergent, and the integration over p and p' together is made finite by the counterterm -(Z3 - t) (q'-- ql`q').Thus Eq .(12.2.18) and any of its subintegrations satisfy the power-counting requirements for convergence, and therefore 12.2 Cancellation of Divergences 5 15 according to the theorem quoted in the previous section, the whole expression actually converges . **~ In electrodynamics there is a natural definition of renormalized cou- plings as well as renormalized masses and fields . This is not always the case . For example, consider the theory of a single reai scalar field 4(x) with Lagrangian density 1im W_ g(12.2.19)22 To one-loop order, the S-matrix for scalar-scalar scattering is given by the Feynman rules as (12.2.20) where 2 2 _42,E)4 F(q I q2 --3, q'q') = -i(27c)'g + i(27r)4gj (27r)4 xjA [(qi +k)2 + M2_ie][(q, - k)2+M2- je ] -ql and qt, q2 and q ', q~ are the incoming and outgoing four-momenta . Com- bining denominators and rotating the k°-integration contour as usual, this is F=g-- 1 672 ,/k3dk die Ck2 + rra2 - x( 1 x) } J O + [k2 + M 2 _tX(I -X)] -2 + [k2 + M2 _ UX(IX)] -2 (12 .2.22) where s,t,anduare the Mandelstam variable s } 2 ')2'(12.2.23) related by s + t + u = 4m2 ; also, x is the Feynman parameter introduced in combining denominators . With an ultraviolet cutoff at k T A, this gives 516 12 General Renormalization Theor y the result (for A > m) 2 1A2~ g ~ 32~Z ~ dx Inm2 - sx(I - x ) ~ A2 . (12 .2.24+ n m2 - tx(j- x)+ In m2 - ux(1 - x) ~ ~ Wecan de fine the renormalized coupling gx as the value of Fat any point s, t,uwe like, provided we stay in the region where F is real . For instance , suppose that in order t omaintain the symmetry among the scalars ,we choose to renorm alize at the off-mass-shell pointlpi = p ~=pig = p~2 ~2, s-t=u = -4~.2J3. Defin ing the renorrrralized coupling g Ras the value of F at this point, we hav e 3g22)j In ~ ~ 1 -- i ~~x In ~~ (13-.X) +M 2+ A i t The cutoff dependence then cancels in Eq . (12 .224) to order g~, leaving a finite formula for F in terms of g R F = gA_9~~ ~~ d. In nay+4x(1 ` x) ~2 /33 2 K J o n~ sx I X ) + In ('"2+4x(I- x)'U2/3+ In(in2+4x(1 - x ),u'13 + m2- t(l - x) m2- ux(1 - x ) (12.2.26) Here y~ may b etaken to be and real quantity greater than -3m2, in which range gR is real . The explicit {u-dependence in Eq . (12 .2,25) is, of course, cancelled b ythe p-dependence of the renormalized coupling . This freedom to change the renormalization prescription (which of course exists also in electrodynamics and other realistic theories will be of great importance to us when we come to the renormalization group method in Volume 11. 12.3 Is Renormalizability Necessary ? In the previous section we found a special class of theories having only a finite number ❑f terms in the Lagrangian, to which the renormaliza- tian program is nevertheless applicable . These are theories in which al l ~ Going back over the derivation of Eq . (12 .2,25), one may check that in this derivation we have not used the conditionsp~= p~ = p'i2= pz2 = -m2, so F-q, (12 .2.24) is valid whatever we take for the external line masses . 1 2.3IsReraorrnalazabilixy Necessary? 517 interactions satisfy the renormalizability conditio n f where di and nif are the numbers of derivatives and fields of type f in interactions of type i, and s fis (with some qualifications) the spin of fields of type f . For renormalization to work in such theories, it is also usually necessary that all renormalizable interactions that are allowed by symmetry principles should actually appear in the Lagrangian . Itis important that there are only a limited number of such interaction types .❑ibecomes negative if we have too many fields or derivatives, or fields of too high spin . Barring special cancellations, there are no renorinalizable interactions at all involving fields with s f~ 1, because the only possible term in the Lagrangian with ❑j~0that involves such a field along with two or more other fields would involve a single s f=1 field along with two scalars and no derivatives, which would not be Lorentz- invariant . We shall see in volume II that general, massless, spin one gauge fields in a suitable gauge effectively have s f=0, like the photon . Also, in Volume II we shall see that even massive gauge fields may effectively have s f=0,depending on where their mass comes from . Leaving aside these special cases, Table 12 .2 gives a list of all renormalizable terms in the Lagrangian density that are allowed by Lorentz invariance and gauge invariance involving scalars (s = 0),photons (s = 0), and spin 21 fermions {5_ 21}  We see that the requirement of renormalizability puts severe restrictions on the variety of physical theories that we may consider . Such restrictions provide a valuable key to the structure of physical theories . For instance, Lorentz and gauge invariance by themselves would allow the introduc- tion of a `Pauli' term proportional to in the Lagrangian of quantum electrodynamics, which would make the magnetic moment of the electron an adjustable parameter, but we exclude such terms be- cause they are not renarmalizable . The successful predictions of quantum electrodynamics, such as the calculation of the magnetic moment of the electron outlined in Section 11 .3, may be regarded as validations of the principle of renormalizabilit .Thy same applies to the standard model of weak, electromagnetic, and strong interactions, to be discussed in Volume II; there are and number of terms that might be added to this theory, such as four-fermion interactions among quarks and leptons, that would invalidate all the predictions of the standard model, and are excluded only because they are non-renormalizahle . Must we believe that the Lagrangian is restricted to contain only renormalizable interactions? As we saw in the previous section, if we include in the Lagrangian all of the infinite number of interactions allowed 5 18 1 2General Renormalization Theor y Table 12 .2. Allowed renormalizable terms in a Lagrangian density involving scalars 0, Dirac fields p,and photon fields AP. Here n r fand di are the number of fields of type f and the number of derivatives in an interaction of type i, and djis the dimensionality of the associated coefficient , Scalar s 1 3 4 1 0 0 0 0 by symmetries, then there will be a counterterm available to cancel every ultraviolet divergence . In this sense, as said earlier, non-renormalizable theories are just as renormalizable as renormalzzable theories, as long as we include all possible terms in the Lagrangian . In recent years it has become increasingly apparent that renormaliz- ability is not a fundamental physical requirement, and that in fact any realistic quantum field theory will contain non-renormalizable as well as renormalizable terms . This change in point of view can be traced in part to the continued failure to find a renormalizable theory of gravitation . In the general class of metric theories of gravitation governed by Einstein's principle of equivalence there are no renormalizable interactions at all - generally covariant interactions must be constructed from the curvature tensor and its generally covariant derivatives, and hence, even in a `gauge 'nif Photons 0 0 0 0 0 2 0 2 0 0 iSpin 2 0 0 0 0 0 0 0 2 0 L 2di❑i 0 0 1 0 0 0 1 03 0 1 0 0 0 0 0 0 0o2 01100-11 0 03 04 00POAlt o2Au~~~ OVY Fj! VFla 'f' Y ` ollo,MW q)r"Amv 1 2.3Is Re narmalazQhi lity Necessary? 519 where the graviton propagator goes as k-2, these interactions involve too many derivatives of the metric for renorrnaliz bility . In particular, we can easily see that genera] relativity is non-renormalizable from the fact that its coupling constant 87~Gv = (2 .43 x 1Q1 8GeV)-2 has negative dimen- sionality . Even if nothing else did, the cancellation of divergences due to virtual gravitons would require that the Lagrangian contain all interac- tions allowed bysymmetries -not only interactions involving gravitons, but involving any particles . But if renormalizability is not a fundamental physical principle, then how do we explain the success of renormalizable theories like quantum electrodynamics and the standard model? The answer can be seen by simple dimensional analysis . We have already noted that the coupling constant of an interaction of type i has dimensionalit y when ❑d is the index (12.1.9).Non-renormalizable interactions are just those whose coupling constants have the dimensionality of negative powers of mass . Now, it is not unreasonable to guess from (12.3.1)that the coupling constants not only have dimensionalities governed by ❑;, but are roughly of orde r where M is some common mass . (This is found to be actually the case in the effective field theories discussed below and in more detail in Volume II .)In calculating physical processes at a characteristic momentum scale k M ,the inclusion of a non-renormalizable interaction of type i with❑i < 0 will introduce a factor gj NIA,, wh ich on dimensional grounds must be accompan ied by a factor k-A!, and so the e ffect of such an interaction is suppressed* for k < Nf by a factor (k/!I{I}-°j < 1 . (This argument will be made more carefully using the method of the renormalization group in Volume II.) The success of the renormalizable theories of electroweak and strong energies shows only that is very much larger than the ene rgy scale at which these theories have been tested . 'ft is essential at this point to assume that the ultraviolet divergences have been removed by rcnnr maliaalion, so that there are no factors of an ultraviolet cutoff A to miss up our dimensional analysis . Otherwise, dimensional analysis tells us that for A -+ oc, each additional non-renormalizable coupling constant factor g, with Ai < 0 would be accompanied with a gxowing factor A- '~ . This dimensional argument led Heisenberg, very early to c lassify interactions according to the dimensionality of their coupling canstants, and to suggest ... that new effects might arise at energies of order gas for instance at the energy G f 1 `2.. 300 GeV, where GF . is the four-fermion coupling constant of the Fermi beta decay lheary . After the development of renormaliration theory it was noted by Sokoto et al .13 that the non-renarmaliza6le theories are those whose coupling constants have negative dimensionality . 520 1 2 General R enormaliaatiorz Theor y For instance, the leading non-renormalizable corrections to the conven- tional electrodynamics of electrons or moons would bethose interactions of dimension 5, which are suppressed by only one factor of 11M . There is just one such interaction allowed by Lorentz, gauge, and CP invariance, a Pauli term of ❑rder (ie/ 2M) i~ [yp,MW F .According to Eqs . (10,6 .24), (10.6.17), and (1 0.6.19), such a term would contribute an amount of order 4elM to the magnetic moment of the electron or muon . The calculated value of the magnetic moment of the electron agrees with experiment to within terms of order 10-"4eJ2rra,, so M must be greater than about 8x101°m ,=4x107GeV . This limit may be weakened if other symmetries restrict the form of the non-renormalizable interaction . For instance, the conventional Lagrangian of quantum electrodynamics is invariant under a chiral trans- formation V -+ y5ya, except for a change ❑f sign of the fermion mass term -rrt~ ;ip.If we assume that the full Lagrangian is invariant under a formal symmetry W --+ 7s1P, M"-rra, then a Pauli term in the Lagranglan would have to appear with an extra factor m f , so that its contribution to the magnetic moment would be only of order 4em/ 2 . Because of the extra factor of in, here it is the moan rather than the electron that provides the most useful limit on M . The calculated value of the magnetic moment of the moon agrees with experiment to within terms of order 10-$e/ 2m, so M must be greater than about $ x I 01nai,= 3 x 103GeV . In any case, if Mis anywhere near as large as 10 "GeV, then we are certainly justified in neglecting any non-renormalizable interactions that might appear in quantum electrodynamics . These considerations help us to cope with some of the puzzles associated with higher-derivative terms in the Lagrangian . For instance, in the general theory of a real scalar field 0, we would expect to find terms in the Lagrangian density of the form 0O110. Any one such term would make a direct contribution to the scalar self-energy function II*(q') proportional to(q')n. If we were to include this contribution to all orders, but ignore all other effects of non-renormalizabie interactions, then the propagator ❑'(q')= 11 (q'+m'- II*(q2)) would not have the simple pole in q2 at negative q2 expected from the general arguments of Section 10.7, but n such poles (some of which may coincide), generally at complex values of q2.But if the non-renormalizable term 0❑n0has a coefficient of orde r -2(n-l), where M m, then the extra poles are at q2 of orderM2, where it is illegitimate to ignore the infinite number of other non-renormalizable interactions that must also appear in the Lagrangian . Thus the appearance of higher-derivative terms in a general non-renormalizable Lagrangian is not in conflict with the general principles underlying quantum field theory that were used in Section 1 0.7. But by the same token, we also cannot use higher-derivative terms to avoid ultraviolet divergences altogether, as 12.3Is Reraormaliaabitity Necessary? 521 has been repeatedly proposed . A term f-2(',-1)00n0 in the Lagrangian density provides a cutoff at momenta q2 M2, but at these momenta we cannot ignore all the other non-renarmalizable interactions that must be present . Although highly suppressed, non-renormalizable interactions may be detectable if they have effects that would otherwise be forbidden . For instance, we will see in section 12 .5 that the symmetries of charge-con- jugation and space-inve rsion invariance are an automatic consequence of the structure of the electromagnetic interactions that is imposed by gauge invariance, Lorentz invariance, and renormalizabi lity, but we can easily imagine non-renormalizable terms that would violate these symmetries, such as an electron electric d ipole moment term FpTst7p, Yw] tpF~'', or the Fermi xnteractivn Ys~~ VTpY,uW. It is widely believed today that the conservation of baryon and lepton number is violated by very small effects of highly su ppressed non-renormalizable interactions . Another example of a detecta ble non-renormalizable interac tion is p rovided by gravitation . As mentioned before, gravitons have no renarmalizable interactions at all . But, of course, we detect gravi tation, because it has the special property that the gravitational fields of all the particles in a macroscopic body add up coherently . Although non-renormalizable theories involve an infinite number of free parameters, they retain considerable p redictive power : 12 they allow us to calculate the non-analytic parts of Feynman amplitudes, like the In q and q In q terms in the one-dimensional examples at the beginn ing of the previous section . Such calculations just reproduce the results required by the axiom of S-matrix theory, that the S-matrix has ❑nly those singu larities requiredby unitarity . Paradoxically, it is just in the case where symmetry principles forbid renormalizable interactions that non- renarrnalizable quantum field theo- ries prove the most useful . In such cases we can derive a useful perturba- tion theory by expanding in powers of k1M .This has been worked out in detail for the theory of low-energy pions,12,13 to be discussed in detail in volume II, and t he theory of low-energy gravitons .14 For a s impler example, consider the theory of a real scalar field, satisfying the principle of invariance under the field translatio n OW -f(x}+e with iF an arbitrary constant . This symmetry forbids and renormalizable interactions or scalar mass, but it allows an infinite number of non- renQt-amalizable derivative interactions 522 12 G eneral Renormalization Theory M-4, and ` ...' denotes terms with more derivatives or fields . where g ~ (For simplicity, it is assumed here that the theory also has a symmetry under the reflection According to the above dimensional anal- ysis, the graph for a general reaction in which all energies and momenta are of order k <M is suppressed b ya factor (k/ 1) ',wher e i i with rt;and dithe numbers of scalar fields and derivatives in an interaction of type i, and Vj the number of vertices for these interactions in our graph . For k 1, the dominant contributions to any process are those with the smallest value of v . The formula forv can be put in a more useful form by using the familiar topological identities for a connected graph : Yt=I - L.+1, Vjnj = 21 + E, where 1, E, and L are the numbers of in ternal lines, external lines, and loops in our graph . Comb ining these relat ions give s Now, the field translation symmetry requires that every field must be accompanied with at least one derivative, so the quantity d i-ray as well as L is non-negative for all interactions . Thus for a given process (that is, a fixed ❑umber E of external lines) the dominant terms will bethose constructed solely from tree graphs (i .e., L = 0), and interactions with the minimum number di =nj of derivatives . That is, in leading, order we can take the Lagrangian density to depend only on . rst derivatives of the field . Higher-order corrections may involve loops and/or interactions with more derivatives on some fields . But to any given order v in k f FYI, we need only consider a finite number of graphs, those with L <(4 - 2E + v)/4, and only a finite number of interaction types . For instance, scalar scalar scattering is given in leading order bythe one-vertex tree graph calculated using the interaction --g( O-yOa'10 )2in first order . According to our formula for v, the leading correction, suppressed at low energy by a factor (kIM)2, arises from another single-vertex tree graph, produced by an interaction with two additional derivatives of the farm** Opc?,00P6 406 ~O.The next corrections, suppressed at low en- ergy by two further factors of k f 11rI, arise both from the one-loop diagram of Figure 12 .4 (including permutations of external lines), calculated us- ing only the interaction - g(Op Ocpo)2, and also from tree graphs wit h In accordance with the remarks of Section 7 .7, we are excluding interactions involving 00, because the field equation for 0can be used to express such interactions in terms of the oihers . 12.3IsR enorrraa lizabi li tyNecessary " 523 \ { F \ J1 ! t i f~` xf r ~ r ti { t i Figure 12 .4. One-l oop diagram for scalar -scalar sca ttering in the theory wit h derivative quad rili-near interactions . a single vertex arising from a quartic interaction with eight derivatives, whose couplings contain infinite parts that cancel the ultraviolet diver- gence from the loop graph .t The loop graph also yields finite terms in the scattering amplitude proportional to terms like S4 In s+ t4 In t+U4In u, s2t2It,u + t2u2 In s+ U'S2In t, etc ., with calculable coefficients proportional to g2. These finite terms simply represent the correction to the lowest-order scattering amplitude needed to ensure the unitarity of the S-matrix, but perturbative quantum field theory is by far the easiest way of calculating them . Although non-renorrnalizable theories can provide useful expansions in powers of energy, they inevitably lose all predictive power at energies of the order of the common mass scale M that characterizes the various couplings . If eve were to take these expansions literally, the results for S-matrix elements would violate unitarity bounds for E There seem to be just two possibilities about what happens at such energies . One is that the growing strength of the effects of the non-renormalizable couplings somehow saturates, avoiding any conflict with unitarity .15 The other is that new physics of some sort enters at the scale M . In this case, the non-renormalizable theories that describe nature at energies E <Al are just e ffective ,field theories rather than truly fundamental theories . Probably the earliest example of an effective field theory was derived in the 1930s b yEuler etal.,16 as a theory of low-energy photon-photon interactions . {fee Section ] .3.)In effect, they calculated the contribution to photon-photon scattering of Feynman diagrams such as Figure 12 .5, and found that at energies much less than in, the scattering of light by light was the same as would be calculated with an effective Lagrangia n 2.2 2 4[(E2-B2)2+ 7(E ~B)2 e eE e B+ higher orders in 2 &meme These are the only ultraviolet divergences encountered in one-loop graphs if we ase dimensional regularization . For other methods of regularization there are also q UiICtYC and quadratic di- vergences, which are cancelled by counterterms in the four-scalar interac tions with four or six derivatives . 524 1 2 General R ennrma lazation Theor y Figure 12 .5. Diagram for photon-photon scattering, whose effect at low energy can be calculated from the effective Lagrangian of Euler et a1."Straight lines are electrons ; wavy lines are photons . Euler et u! . used this effective La rangian only in the tree approximation, to calculate the leading terms in photon interaction matrix elements . It was not until much liter that such Lagrangians, though non-renormalizable, were used beyond the tree appraximation.12,17 In modern jargon, we say that in deriving this Lagrangian the electron is `integrated out', because in the one-loop approximation we hav e exp i ,.,(E,B)d4X)11dy,(x) exp(iYQE n(We,A)dx A more general procedure is simply to write down the most general non- renormalizable effective Lagrangian, use it to calculate various amplitudes as an expansion in energies and momenta, and then choose the constants in the effective Lagrangian bymatching the results it gives for these amplitudes to those derived from the underlying theory . We will encounter effective field theories again, especially in considering broken symmetries in Volume II . As we shall see, effective field theories are useful even where they cannot be derived from an underlying theory, either because the theory is unknown, or because its interactions are too strong to allow the use of perturbation theory . Indeed, even if we knew nothing about the properties of charged particles, the scattering of photons at sufficiently low energy would have to be described by an effective Lagrangian consisting of the terms (E2 -B 2)2 and (E  B)2, because these are the unique quartic Lorentz- and gauge-invariant terms with no derivatives acting on E and B . Terms with such derivatives would be suppressed at low photon energies Eby additional factors of E/M, wher e f is some typical mass of the charged particles that are being integrated out. We can go further : we shall see that effective field theories are useful even where the light particles they describe are not present in the underlying theory at all, but composites of the heavy particles that are 1 2.4 The Floating Cutoff 525 integrated out .The und erlyin gtheory might not even be a field theory at all-the problem of incorporating gravitation has led many theorists to believe that, infact, it is a string theory . But wherever an effective field theory comes f rom,it is i nevitably a non-renormal izable theory . 12.4 The Floating Cutoff ` Before closing this chapter, it is worth commenting on the relation be- tween conventional renarmalization theory and an approach pioneered by Wilson? In Wilson's method one imposes a 'floating' finite ultraviolet cut-off (either sharp or smooth) at momenta with components of order A, and instead of taking A-y ao, one requires that the bare constants of the theory (those appearing in the Lagrangian) depend on Ain such a way that all observable quantities are 11-independent . It is convenient to work with dimensionless parameters . If a bare coupling or mass parameter gt(11 )has dimensionality [mays]°j, we define the corresponding dimensionless parameter j b y Ordinary dimensional analysis tells us that the value of Sd at one value At of the cutoff can be expressed as a function of the values of the jat another value Aof the cutoff, and the dimensionless ratio A' /11: ,~#j(A') = Fj(§(A), A/A) . (12.4.2) Na dimensional parameters other than A' and A can appear in F, because no ultraviolet or infrared divergences can enter here ; the difference be- tween the constants at Aand at A'arises from diagrams whose internal lines are restricted to have momenta between Aand A'.Differentiating Eq. (12 .4.2) with respect to A'and then setting A'equal to Ayields a differential equation for ~~ i(A)=fliff(A) ) 7 (12 .4.3) where flr(g)= [a/az F&,z)],=j .The functions fl;()may be calculated for small couplings in perturbation theory . This is Wilson's version of the `renormalization group' equation, which will be discussed in somewhat different terms in Volume H . The Lagrangian for and finite value of the cutoff defines an effective field theory, in which instead of (or in addition to) integrating out `heavy ' This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 526 12 General Renormalization Theor y particles like the electron in the work of Euler et a l., one integrates out a ll particles with momenta greater than A. Even if one starts with a theory with a finite number of coupling parameters lr~' at some cutoff A 0, at any other value of the cutoff the differential equation (12 .4,3) will generally yield non-zero values for all couplings allowed by symmetry principle .*' We now distinguish between the renormalizable and unrenormalizable couplings, labelled ,,and respectively, with a running over the finite number Nof couplings (including masses) for which ❑a>_ 0, and n running over the infinite number of couplings with dimensionalities A, < 0. We want to show that if the couplings 1a(Ao) and ,(Ao)at some initial cutoff'value A0lie on a generic N-dimensional initial surface Yo, then (with some qualifications) for A < A0 they will approach a fixed surface 51"that is independent of both A0and the initial surface .' This fixed surface is stable, in the sense that from and point on the surface, the trajectory generated by Eq . (12 .4.3) stays on the surface . Such a stable surface defines a finite-parameter set of theories whose physical content is cutoff-independent, which as argued in the previous section, is the essential property of renormalizable theories . Furthermore, this construction shows that a generic theory defined with cutoff A0will look for A A0 like a renormalizable theary .4 To prove these results, consider and small perturbation &'0i(A) in the values of the 1 j{ }satisfying Eq . (12 .4.3). It will satisfy the differentia l equation where a fliff) Mjj0v) =_awj(12.4.4) (12.4.5 This equation couples the re nortnalizable and unreno rmaliz able couplings , making it difficult to see the difference in their behavior . To decouple them , weintroduce the linear comb ination s (12.4.6) "The only known exceptions lcti this rule are in theories based au supeFSymmetry ¢ t This theorem is due to Palchinski .19 What follows here is a shortened and less rigorous version . In Polchinski's proof, the initial surface is taken to he that with all non-renormalicablc couplings vanishing . As we shall we here, the couplings approach the same fixed surface for generic initial surfaces .) Ofcourse some theories have symmetries and a field content that do not allow any renormalizable interactions . This is the case for theories containing only fermion fields, or only the gravitational field. Such theories lor A~ R 0look like free-field theories . 12.4 The Floating Cutqff 527 where S° are the values of the renormalizable couplings at cutoff 11 O, which we shall use as coordinates for the initial surface, and IV, are the values of the non-renormalizable couplings at cutoff Aderived from the differential equation (12 .4.3), with initial value for Ao at the point on the initial surface with coordinates rya . To calculate the derivative of ~„ with respect to A, we note that the derivatives 04 ~10-S satisfy the same differential equation (12 .4.4) as the It is an elementary exercise then to show tha t whered~,: NMI ~ m Wn J G-i 0170 ab01§a (aoo MhM -(12.4.7) (12.4.8) Now we must estimate the elements of ,,,. For a free-field theory no cutoff is needed, so for very small coupling all bare parameters gi(A) become A-independent . Hence for small couplings the dimensionless parameters ~#z simply scale asA-A~, and the matrix Mj .j is given by It follows that the matrix 1V ., is given approximately by --d ,6,.... The defining characteristic of the non-renormalizable couplings is that ❑, <0, so Eq . (12 .4.7) tells us that, at least for couplings in some finite range, where Nnm is positive-definite, the ,decay for A < AO like positive powers of A /11o.In this limit, then, the perturbations are related by ab ub~'`~~ ~~(12.4.10) I n particular, if we make a small change in the initial surface Yo and/or the starting point on that surface and/or the initial cutoff AO,such that the perturbations M%,, in the renormalizable couplings vanish at some cutoff A < A0, then the perturbations in all the other couplings at cutoff Aalso vanish . Thus the non-renormalizable couplings n (11)forA < 11o can depend only on the renormalizable couplings Wa{A}, not separately on the initial surface or the starting point on that surface or the initial cutoff A0.At cutoff A AOall the couplings therefore approach an N-dimensional surface Y, with coordinates a( A), which is independent of both the initial surface and of A0.Note that the non-renormalizable couplings S„ are not generally small on Y; the important point is that they become functions of the renormalizable couplings . Changes in A with Aremaining much less than A Owill change the couplings, but the couplings will remain close to Y(at least as long as the couplings do not 528 1 2 General Renormalizat tan Theor y become so large that 21 T,,, i s no longer apositive-definite m atrix).Hence Yis a stable surface, as was to be proved . Wehave seen that all physical quantities may be expre ssed in terms of A and the ' %,,(A), and are A-independent .This is true in part icular of theNconvent ional re-normalized couplings and masses , likeeand m , in quantum electrodynamics . But we can then invert t his relation ,and express the §n(A) in terms o f the c onventional parameters and A.In this way wecan justify the usual renormaliza tion program :all physical quantities are expressed inacutoff- independent way in terms of the convent ional renormalized couplings and masses . The Wilson approach has some advantages in practice .Onedoes not have to worry about subintegrations and overlapping divergences ;the momentum cutoff applies to all internal lines .Also, some of the non- renormalization theorems of super symmetry theor y,which tell us that certain couplings are not affected by rad iative corrections, work only for the cutoff-dependent bare couplings . 20 On the other hand ,there are disadvantages tothe Wilson approach . One must give up the spec ial simplicities of work ing with renormalizab le theories like guantum,electrodynamics ;once one starts integrating out particle swith momenta above some scale A, the resulting effective field theory will contain all Lorentz- and gauge-invariant interactions ,with A-dependent couplings .(Nevertheless , in physical processes atenergies E< A, the dominant couplings will still be the renormalizable ones .) Also ,the cutoff gener ally de stroys man ifest gauge invar iance,and either manifest Lorentz invariance or unitarity .None of this is a problem in condensed matter physics, the original conte xt of Wilson's approach, because noone would expect a realistic condensed matter theory to be strictly xenormali zabie,and there are no fundamental physical principles that are necessar ily violated bya cutoff . In fact, in crystals there is a cutoff onphonon momenta, pro vided b ythe inverse lattice spacing. Atbottom, the difference between the convent ional and t heWilson approach i sone of m athematical conv enience rather than ❑f physical interpretation . Indeed, conventional renormalization already provides a sort of adjustable cutoff ;when we express our answer in terms of coupling constants that are defined as the values of phy sical amplitudes at some momenta of order M (as for the scalar field theory discussed in the previous section), the cancellations that make integrals converge begin to operate at virtual momenta of order ,u. Conversely, the A -dependent coupling constants of the Wilson approach must ultimately b eexpressed in terms of observable masses and charges, and when this is done the results are, of course ,the same as those obtained b yconventional m eans. 12.5 Accidental Symmetries 529 12.5Accidental Symrnetries* In Section 12 .3 we saw that there are good reasons to adopt renormaliz- a61e field theories as approximate descriptions of nature at sufficiently low energy . It often happens that condition of renormalizability is so stringent that the effective Lagrangian automatically obeys one or more symmetries, which are not symmetries of the underlying theory, and may therefore violated by the suppressed non-renarmalizable terms in the effective La- grangian . Indeed, most of the experimentally discovered symmetries of elementary particle physics are `accidental symmetries' of this sort . A classic example is provided by the inversions and flavor conservation in the electrodynamics of charged leptons . The most general renormal- izable and gauge- and Lorentz-invariant Lagrangian density for photons and fields tpiof spin ~ and charge -e takes the for m R f ~ ~ E 1 ~ti~.f (12 .5.1) -~ ij~,~IWx,~ - where i, j are summed over the three lepton flavors (e,p, and T), ~iL and ~i,R are the left- and right-handed parts of the field Wj, defined by V!-a= 2G+YSh)i, 1P Ri= 120 -Y5)Vi n (12.5.2) and ZL, ZR, and M are numerical matrices . We are not assuming anything about lepton flavor conservation, so the matrices ZLij, ZRijand Mij need not be diagonal . Also we are not assuming anything about invariance under P, C, or T invariance, so there is no necessary relation between ZL and ZR, or between M and Mt . The only constraints on these matrices come from the reality of the Lagrangian density, which requires that ZLjjand ZRj} are Hermitian, and from the canonical anticommutation relations, which require that ZL;jand ZRj are positive-definite . Now suppose we replace the lepton fields yap,, ~Rwith new fields V L', ~~ defined by TL=SLWij tPR = SRT xa (12 .5.3) where S L,Rare n on-singul ar matricesthat can be chosen as we like. The Lagrangian den sity when expre ssed in t erms of the se new fields then t akes This sectio n lies somewhat o utof the boo k's main line of development, and may be om ittedin a first readi ng. 530 12Genera l Renormalization wear y the same form as in Eq .(12.5.1),but with new matrice s We can choose S Land SR so that Z~, =Z~ = 1 . (Take ~'~,,~ _ ~TL, RD~,,R> where UL,Rare the unitary matrices that diagonalize the positive-definite Hermitian matrices ZL R, and the DLY are the diagonal matrices whose elements are the inverse square roots of the eigenvalues of ZL,R .) Now make another transformation, to lepton fields w' defined b y (12.5.5) The Lagrangian density again takes the same form when expressed in terms of these new fields, with new matrice s zL = S~fS~ ,Z~ = ~ s~ , f" = S~ ~ . {12 .5.6} This time we take S{ . R unitary, so that again ZL'=Z~-1. We choose these unitary matrices so that M" is real and diagonal .(By the polar decomposition principle, M' like any square matrix may be put in the form ' =VH, where Vis unitary and H is Hermitian . Take S~, = S~ Vt and choose S R'as the unitary matrix that diagonalizes H .) Dropping primes, the Lagrangian density now takes the for m iz3Fvj1" -I : iV~LiVRf N-RiWLr i i(I2.J.7) where rtai are real numbers, the eigenvalues of the Hermitian matrix H . Finally, this can be put in the more famil iar for m r i With the Lagrangian taking this form, it is now apparent that any renor- maliza6le Lagrangian for lepton electrodynamics automatically conserves P, C, and T, as well as the numbers of leptons (minus the numbers of antileptons) of each flavor : electron, muon, and tauon .*` In particular, despite the appearance of E q.(12.5.1), this theory does not allow such processes as it -} e+y. The reader may perhaps worry whether it is correct to identify the lepton fields as the ipr (previously called W") appearing in Eq, (12 .5,8), which obviously conserves lepton flavor, rather than the V)j appearing in Eq .(12.5. 1), which seems to allow processes like p --* e + y . This was first shown by Feinberg, Kabir, and myself 21 F'einberg2= had earlier noted that weak interaction effects in a theory with only one neutrino species would give rise to an observable rate for the pr[rcecs p -e+ y, a difficulty that was only resolved by the discovery of a second neulrint) species . Problems 531 Such worries may be put aside ; as stressed in Section 10 .3, there is no one field that can be identified as the field of the electron or muon . In fact, although Eq .(12.5.1)yields a nnn-vanishing matrix element for radiative decay of lepton 1 into lepton 2 off the lepton mass shell, bytaking the lepton momenta on the mass shell we find a vanishing S-matrix for all such processes even when calculated using Eq .(12.5.1). It was essential in deriving these results that the same electric charge appeared in Eq .(12.5.1)for both the left- and right-handed parts of the lepton fields or, in other words, that both left- and right-handed parts of the lepton fields transform in the same way under electromagnetic gauge transformations . As we shall see in Volume II, for similar reasons the modern renormalizable theory of strong interactions known as quantum chromodynamics automatically conserves C, and (aside from certain non- perturbative effects) P and T, as well as the numbers of quarks (minus the numbers of antiquarks) of each quark flavor . We shall also see in Volume 11that the simplest version of the renormalizable standard model of weak and electromagnetic interactions automatically conserves lepton flavor (though not C and P) for reasons similar to those described here for electrodynamics . It remains an open possibility that non-renormalizable interactions arising from higher mass scales may violate any of these conservation iaws . Proble ms 1. List all the renorrnalizable (or superrenorrnalizable) Lorentz- invariant terms in the Lagrangian of a single scalar field for space- time dimensionalities 2, 3, and 6. 2. Show how t he ove rlapping divergence in the e lectron self-energy is cance lled in qua ntum e lectrodynamics . 3.Consider the theory of a scalar field 0 and spinor field tp,with interaction Ham iltonian g V )yr. Write the one-loop part of the scalar self-energy function II'(q) as a divergent polynomial in p +, plus an expl icitconvergent integral . 4. Suppose that the quantum electrodynamics of electrons and photon s is actually an effective field theory, derived by integrating out un - known particles of mass M >m, Assume gauge invariance an d Lorentz invariance, but not invariance under C, P, or T . What ar e the non-renormalizable terms in the Lagrangian of leading order i n 11M ?Ofnext to leading order'' 532 1 2 General Renormalizarion Theor y Reference s 1. F. J. Dyson, Phys . Rev . 75, 4$6, 1736 (1949) . A historical perspective is provided by Renormalizarion, ed. by L . M. Brow(Springer-Verlag , New York, 1993) . For a comprehensive modern treatment, see J. Collins, Renormalization (Cambridge University Press, Cambridge , 1984) . 2. S. Weinberg Pays . Rep .118, 838 (1959) . This proof relied only o n general asymptot ic properties of the integrands of Feynman graph s in Euclidean momentum space, obtained by dick rotation of al l integration contours . The proof was s implified through the use o f more detailed properties of the integrand, by Y . Hahn and W . Zim - merman, Commun .Math .Pays .10, 330 (1968), and then extended t o Minkowskjan momentum space by W . Zimmerman, Commun . Math. Phys .11, 1 (068) . 3. J. D. Bjorken and S . D. Drell, Relativistic Quantum Fields (McGraw- Hill, New York, 1965): Sections 10and 19 .11. 4. See, e .g,, J . Ness and J . Bagger, S upersym metry a nd Supergravity (Princeton University Press, Princeton, 1953), and original references quoted therein . 5. A. Salam, Phys . Rev. 82, 217 (X951 ); Phys . Rev . 84, 426 (1951 ).; R T . Matthews and A . Salam, Phys . Rev. 94, 185 (1954) . 6. N.N.Bogoliubov and D . Parasluk, Arta Math .97, 227 (1957 ) 7.K. Hepp, Comm .Math. Ph i . 2, 301 (1966), Hepp remarks that `it is difficult to find two theorists whose understanding of the essential steps of the proof [of Bogoliubov and Parasiuk] is isomorphic,' but Hepp's paper is itself not easy to read . S. W. Zimmerman, Comm .Math .Phys . 15, 2 08(1969). Also see Timmerman, in lectures on Elementary Particles and Quantum Field Theory - Brandeis University Summer Institute in Theoretica lPhysics (.I.'F. Press, Cambridge, 1970). 9. W. Heisenberg, Z.Physik 110,251 (1938) . 10. W. HeisenbergRef .6and Z.Physik 113, 61 (1939 ). 11. S. Takata, H . Umezawa, and S. Kamefuchi, Prog, Theor .Pays, 7, 327 (1952) . 12.S. Weinberg, Physica 96A, 327 (1979) R~f8re1ZCBS 533 13. J. Gasser and H . I,eutwyler, Ann . Whys . (NY) 158, 142 (1984) ;Nucl. Pays .B250, 465 (1985) . 14. J.F.Donoghu e,Pays.Rev.D 50 , 3874 (1994) . 15. One possible way that this can happen is through the phenomenon of 'asymptotic safety' ; see S.Weinberg, in General Relativity - An Einstein Centenary Survey, ed . by S . Hawking and W . Israel (Cambridge University Press, Cambridge, 1979) :Section 16.3. 16. H. Euler and B . Kockel, Naturwiss .23, 246 (1935) ;W. Heisenberg and H . Euler, Z.Physik 98, 714 (1936) . 17. The Euier et al.effective Lagrangian has been used in one-loop calculations by J. Halter, Phys .Lett.B 31 6,155(1993) . 18. K. G. Wilson, Pays .Rev. B4, 3174, 3184 (1971) ;Rev.Mod .Phys.47, 773(1975) . 19.J. Polchinski, Nucl. Whys .B231, 269 (1984) ; lecture inRecent Direc- tions in Particle Theory - Proceedings of the 1 992 TASI Conference, ed. byJ.Harvey and J . Polchinski (World Scientif ic, Singapore, 1993) : p. 235 . 20. Novikov, A . Shifman, A . I. Vainshtein, and V I . Zakharov, Nuc2 . Pays .B229, 381 (1983) ; Y. A. Shifman and A . I. Vainshtein, Nucl . Phys .B277,456 (1986) ; and references quoted therein . See also M. A. Shifman and A .I. Vainshtein, Nucl .Fhys.B359, 571 (1991) . 21.G. Feinberg, P . Kabir,andS. Weinberg,Phys.Rev.Lett.3,527 ( 1959) . 22. G. Feinberg ,Whys.Rev.110,1482 (1958) . 13 Infrared Effect s In the study of radiative corrections a special role is played by those corrections due to "soft' photons : photons whose energy and momentum are much less than the masses and energies characteristic of the process in question . Not only are these corrections often so large that they must be summed to all orders of perturbation theory ; they are so simple that this summation is not difficult . The contribution of photons of infinitely long wavelength takes the form of divergent integrals, but as we shall see these `infrared divergences' all cancel .' In most of this chapter we will deal with photons interacting with charged particles of arbitrary type and spin, including particles like atomic nuclei that have strong as well as electromagnetic interactions . But it is not difficult to adapt the calculations presented here to the infrared effects of other massless particles, such as the gluons of quantum chromodynamics . In Section 13 .4 we shall explicitly consider very general theories of massless particles, and will show the cancellation of infrared divergences on general grounds . After these generalities, we shall return to photons, and take up two topics of practical impor#ance : the scattering of soft photons by charged particles with arbitrary non-electromagnetic interactions and arbitrary spin, and the treatment of heavy charged particles like atomic nuclei as a source of an external electromagnetic field . 13.1Soft Photon Am plitudes In this section we shall derive a universal formula that gives the amplitude for emission of any number of very-low-energy photons in a process -i involving any number of higher-energy charged particles of any types . Let us start with the amplitude for emission of just one soft photon . If we attach the sift photon line with outgoing momentum q and polarization index itto an outgoing charged-particle line that leaves some connected Feynman diagram for the process a -~P, as in Figure 13 .1(a), then w e 534 13.1Soft Photon Amplitudes 535 must multiply the S-matrix element for ac with an additional charged- particle propagator carrying the momentum p +q that the charged particle had before emitting the photon, together with the contribution of the new charged-particle-photon vertex . For charged particles of spin zero, mass m,and charge +e, these factors ar e [i(27t )4 e(2pu + qu)] [-i i which in the limit q --i~ 0 becomes e pp pq-ie(13.1.1) We are freely redefining the scale of the positive infinitesimal e, being careful only to keep track of its sign .) This resul tis actually true for charged particles of any sp in. For instance, for a par ticle of spin ~ and charge +e, we must replace the coefficient function u( p,u) for the outgoing charged particle wit h ii(p,a)[-Qn)4e.~jl~[_i -iq +4) +M In the lim itq--*0 the numerator of the propagator is given by a sum of dyads : aF so we have a sum of equal-momentum matrix elements of YP, given b y and again the effect again is to multiply the matrix element for the process g -+ P by the factor (13.1.1).More generally, for any spin in the limit q --* 0 the four-momentum p +q of the new internal charged particle line approaches the mass shell, so the numerator of the propagator approaches a sum of dyads of coefficient functions which convert the new vertex matrix into a factor proportional to p 'and a unit matrix in helicity indices, leading again to the factor (13.1.1).Furthermore, as we saw in Chapter 10, higher-order corrections do not affect either the residue of the mass-shell poles in the propagators or the matrix element of the electric current between sates of the same particle at equal momentum, so (B .1.1) gives the correct factor associated with the emission of a soft photon from an outgoing charged-particle line to all orders of perturbation theory . The same reasoning applies to a photon emitted from an incoming charged-particle line of the process x except that after the incoming particle emits a photon of four-momentum q the charged-particle line has 536 13 I nfrared Effects R ~ q U (a) (b) Figure 13 .1. Dominant graphs for the emission of soft photons in an arbitrary process oc -- -~ P. Straight lines are particles in the states cc and #(including possible hard photons) ; wavy lines are soft photons . four-momentum p -q, so in place of (13. 1. 1) we find afacto r epp pq- ie(~312) The photon ca nalso, of course, be emitted from an internal line of the process a c-+ fl, but in this case there is no factor that goes as (p ' q) -1 for q --*Q. Th e amplitude MIA) (the S-matrix without the energy- momentum conservation delta function) for em itting a single soft photon with four-momentum q and polarization index u in the process a --+ P is therefore given in the lim it q-+0bymultiplying the matrix element ifla for x --* # with a s um of terms ti ke (13 .1.1)and (13.1.2),one foreach outgo ing or incoming charged particle : where p ,and +e, are the four-momentum and charge of the nth particle in the initial and final states, and q,is a sign factor with the value +1for particles in the final state fland -1 for particles in the initial state a . Before going on to consider the emission of more than one soft photon, it is worth mentioning an important feature2 of the formula (13 .1.3). To 13.1 So ft Photon Amplitudes 537 calculate the amplitude fo remission of a photon of definite helicity, we must contract this expression with the corresponding photon polarization vector e,,(q,±). But as we saw in Section 5 .9, e,'( 9, ±) is not a four-vector ; under a Lorentz transformation A ;',„ the polarization vector is transformed into 11 Ve° (q,±) plus a term propo rtional to q~` . In order for this last te rm not to spoil Lorentz invariance, it is therefore necessary that ~a (q) should vanish when contracted with q .. But for q --*0, (13 .1.3) give s qMI(q) --+ M~.1:qnen (13.1.4) M The coefficient of M#a on the right-hand side is just the total charge in the final state minus the total charge in the initial state, so the condition that it vanishes is just the condition that charge is conserved . Thus without any independent assumptions about gauge invariance, we see that for particles ofspin one and m asszero, Lorentz invariancerequires theconservation of whatever couplingconstant like electric charge governs the interaction Qf these particles at low energies . Incidentally, the amplitude for emitting a soft graviton of four-momen- turn gand tensor indices u, v in a process a is given by a formula3 analogous to (13.1.3): fl~xPn* q lln where f ,is the coupling constant of the soft graviton to particles of type n. Lorentz invariance here requires that this vanish when contracted with q,,. But q.~,~(q)~M#a qn.fq p~ , (13 .1.6) so the sum f p vis conserved . However, the only linear combinat ion of the four-momenta that can be conserved without forbidding all n on- trivial scattering processes is the total four-momentum, so in order for (13.1.6)to vani sh,the f nmust all be equal . (The common value of all f,, may be identified a s8nGN,where G _NisNewton's constant of gravitat ion.) Thus Lorentz invariance requires the result that low-energy masslessparticle sofspin two couple in the same way to all forms of energy and momentum .This goes a long way toward show ing that Einstein's principle of equ ivalence i sa necessary con sequence of Lorentz invariance as applied to massles sparticles of spin two .Likewise ,the amplitude for emitting a soft massle ss particle of four-momentum q and spin j 3 in a process a -># is of the form 538 13 Inf~ared Efjectti Lorentz invariance here requires that the sum ):g,p; pNO,. .must be con- served . But no such quantity can be conserved without prohibiting all non-trivial scattering processes, so the g ,,must all vanish . Massless parti- cles of spin j ~ 3 may exist, but they cannot have couplings that survive in the limit of low energy, and in particular they cannot mediate inverse square law forces . Now let us consider the emission of two soft photons . The contribution to the matrix element from a graph in which the two photons are emitted from different external lines of the process a --+ ft is given bymultiplying the matrix element for ~ -~ fl by a product of factors like (13.1.1)or (13.1.2).Perhaps surprisingly, the same is true even if the two photons are emitted from the same external line . For example, if photon Iis emitted from an external line of charge +eand energy-momentum four-vector p after photon 2 we get a facto r )7 e po,q epl'n p - qj - i~el ~P - W2+ ql) - iqE l while if photon 2 is emitted after photon Ithe factor i s p - q2 - iqel ~p - (q, + q2) - iqel (See Figure 13 .2. Again, n is +1 or -1 according to whether the charged- particle line is outgoing or incoming .) These two factors add up t o qeP"' ~ eP11` P- qi-iqe pg 2- aqE which is dust a product of the same factors encountered in the emission of a single photnn . More generally, in emitting an arbitrary number of photons from a single external line we encounter a sum of the for m This identity may be proved by mathematical induction . We have already seen that it is true for two photons . Suppose that it is true for N - Iphotons . For N photons we may then write the Qum over permutations as a sum over the choice of the first photon to be emitted together with a sum over permutations of the remaining photons : [F'qI-ixj,-]-1 [p ' (RL + q~) - [p'(Rj+q2+... + R ,,w)- ire]-1 + permutations N N I -iqe] 11[p, q, - ine]-'I: I : ~ Sir h' ~4' -1 Syr r=1 ct l) - I a-1N a _ r- [p  qy - i1e ]q-,- ir e] f j 5=1 as was to be proved . 13.EVirtual Soft Photons 539 n q, Figure 13 .2. Graphs for the emission of two soft photons from the same outgoing charged particle . Straight lines are ha rd particles ; wavy lines are soft photons . +permutat ions [p.q1- iqeJ-i[p.q2 - 1~1_-]-'tp.q3 -~~-EF1... ( 13.1.7) It follows then that the amplitude 11f1{~~ -~'~`` (q1 ... q~) for emitting N very soft photons with polarization indices µt, . µ,?v and four-momenta qi,... qNin the process oc is given in the limit q -i 0 by multiplying the matrix element Mfl, for a by a product of factors like that in (13.1.3), one for each photon : J%r ~r}7M e,P~ii' r=1 npn q, - iqn6) 13,2 Vi rtualSoft P hotons We shall now use the results of the previous section to calculate the effect to all orders of radiative corrections involving virtual soft photons exchanged among the charged particle lines of a process x -+ fl , as in Figure 13.3. By a `soft' photon we mean one that carries momentum less than A, where A is some convenient dividing point chosen low enough to justify the approximations made in the previous section . We shall find that these soft photons introduce infrared divergences, so as a stop-gap 540 13Infrared Effect s a Figure 13.3. A typical dominant graph for the radiative corrections due to virtual soft photons to the S-matrix for the process a -i~ P.Straight lines are particles in the states x and #(including possible hard photons) ; wavy lines are soft photons . we will have to introduce also a lower bound ~on the photon momenta . It is important to recognize the difference between these two limits on the soft photon momenta . The upper cutoff Ajust serves to define what we mean by `soft' photons ; the A-dependence of the soft-photon radiative corrections is cancelled by the A-dependence of the rest of the amplitude, which is calculated including only virtual photons with momenta greater than A.On the other hand, the lower cutoff ).must eventually be removed by taking A, -40; as we shall see, the infrared divergences in this limit will be cancelled by the effects of real soft photon emission . For each virtual soft photon we must supply a propagator facto r ~~ ~,,,g{13.2.1}(2n)4 q2- iE then multiply the ampl itude (13 .1,8) with the product of t hese propagators, contract photon polarization indices, and in tegrate over p hoton four- momenta . In addition for N virtual photons we must divide by a factor 2NN!, because the sum over a llplaces to which we may attach the two ends of the soft photon lines includes spurious sums over the N! permutations of the photon lines and over interchanges of the two ends of these lines . The ef fect of radiative corrections involving N soft photons 13.EVirtual Soft Photons 541 is then to multiply the matrix element 1#,, for the process without such radiative corrections by a facto r 1 1 E eyL,~~nqmJnm (13.2.2) wher e ,I,,,,,=-On  PM) 1!~ lql snW-i6][pM- q-jqn-,][-pm ,q - iqm 61. (13 .2.3) Note that we have changed the sign of p,,-q in the denominator in (13.2.3), because if we define q as the momentum emitted by line n then -qIs the momentum emitted by line m . Summing over N,we conclude that the matrix element for a process including radiative corrections due to any number of soft photons with momenta 19~ ~ Ais given by 2(27r)4 Ee. em qn?lm Jnm (13.14) nm where Ma is the amplitude including virtual photons only with momenta greater In A. The integral over q° in (13 .2.3) may be done by the method of residues . The integrand is analytic in q0except for four poles, a t q°=191- q ° _ -1 91+ ie, where v,, P ,/ptand likewise for vim . If particle n is outgoing and particle m is incoming, then q, = +1, q. = -1, so by closing the qo contour in the upper half-plane we avoid the contributions from the pokes atq0-- v,' q-xqn6 or q° = v,  q +in,,e. Similarly, if n is incoming and m is outgoing we can avoid these two poles byclosing the contour in the tower half-plane . In these two cases it is only one of the poles at q " = ±(j91- ;E") that contributes, and we find a purely real integral : d3q brim= - 7rPry 'Pm ~ ~ . A L1 41sA 1 91 (En - q-pn)(E m q ' pm) (farqn --- i?in= ± 1) (13.2.5) On the other hand ,if particles n and gyn .are both outgoing or both incoming , then the poles at v n- q- itj,cand v, ,' q+ iqnc lie on opposite sides of the real qo-axis, and we cannot avoid a contribution from one o f 542 13Inftared Effect s them whichever way we close the contour : 1~~ ~n En - L ' IqlICI~q~~)(m 4 - Pm ) ~iiPIn(A)(for q,,= q,, _ ± Z), (13.2,b) where fln,, is the relative velocity of particles n and m in the rest frame of either : m2m2 p, PM2(] 3..7) The imaginary term in Eq . (13.2.6)leads to an infrared-divergent phase factor in Eq. (13.2.4),which drops outwhen wetake the absolute value of thematrix element to calculate the rate forthe process a --->fl.(This infinite phase factor is the relativistic counterpart of the well-known feature of non-relativistic Coulomb scattering ,that the outgoing wave part of the Schrodinger wave function ha sadependence on the radial coordinate r of theform exp(ipr- iv In r )/r instead of exp(ipr)/r, w here v is the product of charges divided by the relative velocity . 5)The reaction rate is affected by the real part of J ,,,,which for all q,and q,,takes the valu e Re )d3q f~q~ iII CI ( qPn)(?nqPm) An elementary calculation give s ReJmn-2n?InI+finmIn(A) &m (I-Nnm(13.2.8) (13.2.9) Using this in the absolute value squared of Eq . (13 .2.4) gives the effect of soft virtual photons on the rates for the process 1711, a s ~r~~= ~ T `~Lflex , ( 13.2.10) where i,~~ and I~~~ are the rates for the process a including radiative corrections of soft photons only with momenta greater than Aor l1, respectively, and A is the exponen t 8~r ,~P'O flotol -I-&M Note that this makes sense only because the correction factor ( ~fA)' has turned out to be the ratio of a function of 1 . to the same function 13.2Virtual Sqft Photons 543 ofA, since the two rates in Eq .(13.2.10) can only depend on ;,and A, respectively . The exponent A is always positive . For instance, in the scattering of a single charged particle by a neutral particle or an external potential, we must add terms in Eq . (1 3 .2.1i } where both n and inare the initial or the final charged particle (in which case qnq,= +1 and #,,, = 0), orrt is the initial or final charged particle and m is the other one (in which case r107M = -1 and &„, =P, where 1 > fi > 0.) This give s 2eA 4 ~ I n which is positive for all I > ft > D. Because A is positive the effect of th e infrared divergences introduced by soft virtual photons when summed to all orders is to make the rate for any given charged particle process ac vanish in the limit 0 . Before we go on to consider how soft real photon emission cancels these infrared divergences, we should pause to note a technicality in the above calculation which as far as [know has always been ignored in the literature . In calculating these radiative corrections we have included diagrams in which the virtual photon is absorbed and emitted at the same external charged-particle line, as well as those in which it is emitted and absorbed at different lines . But as we learned in Chapter 10, in calculating the S-matrix we are not supposed to include radiative corrections arising from insertions of self-energy subgraphs in external lines . This might suggest that we should drop the terms n = m in Eq . (13 .2.11), but then the cancellation of infrared divergences we will find in the next section would not be complete . The resolution of this problem can be found in the observation that soft virtual photons produce infrared divergences not only directly, but also through their effect on the renormalization constants Z, of the charged particle fields . The renormalization constant Z,z is the one usually called Z-Yin theories like quantum electrodynamics with a single charged field of spin 1.)It is the counterterms proportional to Z, - 1 that cancel the effect of radiative corrections in external lines . To be specific, the renormalized field of a charged particle of type n is a factor Z, 1/2 times the unrenormalized field, so when we calculate the S-matrix using renormalized fields (corresponding to the omission of radiative corrections in external lines) we are introducing an extra factor fl,,Z,1 ''~~, the product running over all charged particles in the initial and final states, ( Of course there are also factors Z ._1 /2 for neutral particles, but these are not 544 1 3 ItZfrared Effect s infrared -divergent .)In a slight lydifferentnota tion, this facto r is ~~k/2 f where Zfis the field renormalization constant for fields of type f,Efis the number of external lines of type f, and the product now runs over all charged field types . However, these field renormalization constants also appear in the interiors of diagrams ; expressing an interaction of type i that involves lV ifcharged-particle fields of type f in terms of renormalized fields introduces an infrared-divergent facto r rj(Zf )N Ff /2. (For instance, the caunterterm -ie(Z2 -1 )A,tp°111inEq.(11.1.9) together with the ordinary electromagnetic interaction -fet~~Pzp yields a total in- teraction -iZ2eFpyPy), It was the infrared divergence in this Z2 factor that was responsible for the infrared divergence arising from the second term in the brackets in Eq .(11.3.23), and the last term in Eq . {11 .4,14} .) There is also an infrared divergence in the propagators of the renormalized fields ; the propagator of a renormalized charged field of type fwhen expressed in terms of the propagator of the unrenormalized field introduces a factor Zf11Putting this all together, the total number of factors of Zffor each charged field type f, introduced by the counterterms to interactions and to radiative corrections on internal as well as external lines, i s where I fand Efare the numbers of internal and external lines of type f, and Vi is the number of vertices for interactions of type i . We have already noted in Section 6 .3 that this quantity vanishes for each f . Thus the counterterms which cancelled the radiative corrections on external lines are themselves cancelled by the Zffactors arising from internal lines and vertices . Eq .(13.2-11) is therefore correct as it stands, including the terms with n =m. 13.3 RealSoft Phot ons; Cancellation o f Diverge nces The resolution of the infrared divergence problem encountered in the previous section is found in the observation that it is not really possible to measure the rate r#, for a reaction ot -- +Pinvolvang definite numbers of photons and charged particles, because photons of very low energy can always escape undetected . What can be measured is the rate I'pa(E, E?) 13.3Real Soft Photons;Cancellation of Divergences 545 for such a reaction to take place with no unobserved photon having an energy greater than some small quantity E, and with not more than some small total energy E Tgoing into any number of unobserved photons . (Of course, ES E T. In an experiment without soft photon detectors, one can rely on measurements of the energies of the `hard' particles in oc and Pto put a limit E Ton the total energy going into soft photons, and in this case we just set E = E T,) We now turn to a calculation of this rate. The S-matrix for emitting Nreal soft photons in a process oc --* fi is obtained by contracting each of the Nphoton polarization indices 91, Y2, on the amplitude (13.1.8)with the appropriate coefficient func- tion e~(9ah) where q is the photon momentum, h = ±1 is its helicity, and c"is the corresponding photon polarization 'vector'! This gives a photon emission matrix element (the S-matrix element with delta function omitted) a s PIZ 11Pn q r,r=t n (The superscript ~ is to remind us tha tthese amplitudes are to be calcu lated with an infrared cutoff A" on the momenta of virtual photons . Eventually we shall take ~-+ 0. The presence of soft virtual photons does not interfere with the result (13.3.1)because of the factorization discussed in Section 13.1.)The differential rate for emitting Nsoft photons into a volumefl,d3qr of momentum space is given by squaring this matrix element, summing over helicities, and multiplying with j lF d3q ,,. We recall from Eq. (8.5.7)that for q '=0,the helicity sums take the for m 61`(q, ~)_,V*(9, h)= q,~v +q,,c,+ q.c., ( 13.3.2 h=tl where c = -g l2lql2and cO= 1I2I QIThe charge conservation c ondition E,,qne, = 0allows us to drop the terms in Eq . (13 .3.2)invol ving q .or q,,, We are using e" instea dof e Pfor p hotonpolarization vectors, to avoid confusion wit houruse here of e„ for e lectric cha rges. 546 1310rared Effect s yielding a differential rate'* 3qe_ E 770menem( P 'PM)dJF~,(qi, V)=FIXNd r=l To calcu late the differential rate for the emission of Nsoft with defin ite energ ies co, = J q,J we must integrate Eq . (1 .x.3) directions of the photon momenta qr, These integra ls are the those we encountered in the in tegrals (13 .2.8), ~~27r 2In 1 ~~~ -77(Pn'per )f (En - q",q~,-=+1 - fl(13-3.3) photons over the same a s (13.3.x) Integrating Eq . (13 .3.3) over photon directions thus gives the differential rate for photons of energy c01,... cf)N a N d" NdF, (a) I ...O)N) A(a --* P) ( 13 .3.5) fl0, (ON whe re A(a ~ P) is the same constant encountered in the p revious section : ~(!xOnomqOm to 1 + fln'n 287rrIrn XIm ~ - Nrrx We see from Eq .(13.3.5)that an unrestricted integral over the energies of the emitted photons would introduce another infrared divergence . However, unitarity demands that if we use an infrared cutoff for the momenta of the virtual photons (as implied b ythe superscript A)then we must use the same infrared cutoff for the real photons . To calculate the rate r~ ,(E, E T)for the reaction oc with not more than energy Egoing into and one unobserved photon and not more than energy E Twing into any number of unobserved photons (with E and E Tchosen small enough to justify the approximations used in deriving Eq. (13 .3.1)), we must integrate Eq . (13.3.5)over all photon energies, subject to the limits E~ cc~~ ~ Aand E . r.~r ~ E r, then divide by N!because this integral includes configurations that differ only by permutations of the Nsoft photons, and finally sum over N. This gives wrN=ON! E'7o3r7~ . r[t]rCF{r=1(13-3-6) This integral would factor into the product of Nintegrals over the indi- vidual oi,were it not for the restriction ~:,r~),~ E .This restriction ma y » The result for qJdl p{ q}lI'p in the case N = 1 cnrre5ponds to the distribution of energy emitted classically by a discontinuously changing current density Fctiur-nectar P(x) = Ell (°) cS3{v- V, t}p;;e„fEn, with the sum here ru nning only over particles in tie in itial state for r <0 and in the final state for t > 0 . 13.3Real Hof i Photons ; Cancellation qfDivergences 547 be implemented byincluding as a factor in the integrand a step function f)(~T -- c~r) = 1 dusin ~- Uexp i u rr~)r (13.3.7) a~.:x u r Eq. (13.3.6)then become s 170(E, Er)= 1 ~ d uyin Eauexp A(ajEdcjer{ou n -~ (13.3.8) The integral in the exponent can b edone in the limit A<ETby writing it as the sum of the integral of (ehou- w o), in which we can set ~ = 0 , and the integral of i/o), which is trivia] . Resealing the u and w variables, this gives for A < E : F)., (E, ET)__*,~W EIETAO~--*fl))i-A(13.3-9) sin u -xdroW- 1))where f- .I)r-. U 0 0 ).flx;A) du A2O(x - ~) rd x....(13.3.10) = 1 - In -~+ 2.. . x m 1 -- r .~ For E and E Tof the same order and A<1the factor .F,(Ef ET, A)in Eq.(13.3.9)is close to unity ; for instance , Because A(ac ---~ ~} 0, the factor in Eq. (13 .3.9)becomes infinite in the limit D . However, Eq .(13.2.10) shows that the rate 171 vanishes in this limit : )A(x~~1~r~IX ~ rn#~X. Using this in Eq .(13.3.9)shows that the infrared cutoff ~drops out in the limit ;. E: FA,,(E,E F(EIET ;A(uA(13.3,11)flT)--* A) Fflg e We remind the reader that the energy A is just a convenient dividing point between 'soft* photons which are taken into account explicitly in Eq. (13 .3.11) and `hard' photons whose effects are buried in T-'A The right-hand side ❑f Eq . { ] 3 .3. l 1)is independent of Abecause I'~x ~ AA. However, in theories with a small coupling constant like quantum electrodynamics it is frequently a good strategy to take Ato be sufficiently 548 13Infrared Effect s small compared with the typical energies involved in the collision so that the approximations made here apply for photon energies less than A, but large enough so that A(ac ln(W/A) <1. Then it may be a good approximation to calculate T'~~ in lowest-order perturbation theory, with the dominant radiative corrections for E Agiven by the factor (EfA)l inEq. (13.3.11). *~* The same cancellation of infrared divergences occurs for soft gravitons .3 The rate for any process ac --* # , with not more than an energy Egoing into soft gravitons, turns ❑ut to be proportional to EB, wher e G 1 +fl~B= r~nr 7,~m.,~rra,~ In (13.3.12) 27Cnrtt#2 i ~ f~ nm 13.4General Infr aredDivergences The infrared divergence due to soft photons that we have been considering up to now in this chapter is just one example of a variety of infrared divergences that are encountered in various physical theories . Another example is provided by quantum electrodynamics with massless charged particles . Here even after the cancellation of infrared divergences due to soft photons, we find a logarithmic divergence in the exponent A in Eq. (13 .3.11). According to Eqs .(13.2.11) and (13.2.7), for a process in which all charged particles are electrons, in the limit m, -- + 0the exponen t goes as 12_ 1 1 :12CFn'Pml Inm,2A enemqn)lm In 4 7,2 Ee. 4 7~ M 2 nn n#M e2~ (In the last step we have used the charge conservation condition , ,e„r~,~ _ 0.) The infrared divergence in this formula arises from soft photons that are emitted in a direction parallel to the momentum of one of the ` lard' electrons in the in itial or final state, but it occurs a lso even if the photon like the electron is not soft, because the propagator denominator (p„ ± q)' vanishes for pn q2= 0 if p, is parallel to q . To be a little more spec ific, for pn = q2= 0 the integral of t his fac tor'over photon directions take s This factor is not squared, because the divergence occurs only in the interference between this term in the S-matrix element and terms in which the photon is emitted from some other charged particle line m :An. For m =n the integral (13.2.$) is proportional to m2 n. 13.4 General Infrared Divergences 549 the form 271 sink d 0 qP2q where 0is the angle between the momenta of the photon and the charged particle .This integral diverges logarithmi cally at 0 = 0. Of course, in the real world the re are no massless ele ctrically charged particles, but in reactions in which the typical value E 2of the scalar products gyp„-p,,lis much larger than rn,2, it isOf interest to identify the places where large ln(m,f E ) factors appear . The dominant radiativ e 2correction in this case isoften given by theterm - ln( me/E) E,, e2,,/27r in A . More importantly, in quantum chromodynamics there are mas sless particles ,the gl uons,that carry a conserved quantum number known as color that is an alogo us to electric charge, so that infrared divergences arise from the emission of parallel hard gluons from hard gluons or other hard c oloredpart icles in the init ial or final states . These infrared divergences are not, in general ,elim inated by summing over su itable sets of final st ates. However, Lee and Nauenberg6have pointed out that the infrared divergences can be made to cancel if we not only sum over su itable final states, but also assume a certain prob- abilistic distribution of initial states.What follows is a modified version of their argument, wh ich will immed iately make clear why in the case of electrodynamics with massive electrons it was sufficient t osum over final states . For these purposes itis convenient to return to `old-fashioned' pertur- bation theory, in which the S-matrix is given by Eq . (3.2.7)and Eq.(15.3) as Shy - 6{b-a)-2a7r6{E,, - Eb}Tb,,, ( 13.4.1) where 00 E,- E ci,fiE) (Ea -E ft + iE). ( (13.4.2) (The integrals over c l--c, should be understood to include sums over the spins and types of particles in these states as well as integrals over the three-momenta of these particles .) Infrared divergences arise from (and only from) the vanishing of one or more of the energy denominators in this expression . However, not all vanishing energy denominators give rise to infrared divergences . A general intermediate state c may have EG = E ,,,but usually this is just one point in the interior of the range of integration, and the integral over this range is rendered convergent by the prescription implied 550 13Infrared Eftet s by the k in the denominator .In order for an intermediate state cto produce an infrared divergence, it is necessary that the energy E .= E a be reached at the e ndpoint of the range of integration . This happens for instance if the first intermediate state c ,in Eq . (13 .4.2) consists of the particles in the initial state a, with any of the massless particles in this state replaced with jets, consisting of any number of nearly parallel mastless particles with a total momentum equal to that of the particle the het replaces . In this case, the endpoint at which Eel= EQ is the point in momentum space at which all of the massless particles in each jet are parallel . More generally, we can have any number of the massless particles in a replaced with jets of nearly parallel massless particles, plus any number of additional soft massless particles . The set of all such states will becalled D(u) . (To be precise, we need to introduce a small angle 0 as well as a small energy Ato define what we mean by `nearly parallel' and `soft' . We will not bother to show the dependence of the set D (a)on0 and A.)The states in D(a) are `dangerous', in the sense that the vanishing ❑f the energy denominator E . - E .,at the endpoint can introduce an infrared divergence ; the endpoint at which Et ;,=Ea is the point at which all massless particles in each jet are parallel, and all soft massless particles have zero energy , Furthermore, if cr, c,, are each in the set D(u),then an intermediate state c,+i in D(a)is also dangerous in the same sense . On the other hand, if some intermediate state c .isnotin D{a}, then a later state c kwith k >m would not be dangerous even if it belonged to the set D(a), because the configuration of hard particles or jets with three-momenta equal to those of particles in the state a would be just an ordinary point inside the range of integration . In exactly the same way, we may define a set of states D(b) in which one or more of the massless particles in the state b are replaced with jets of nearly parallel massless particles, each having the same total three-momentum as the particle it replaces, and we add any number of soft massless particles . An intermediate state c„z is dangerous if it belongs to the set D(h) and if the later states c kwith k >m all belong to D(b) . To isolate the effects of these dangerous states we rewrite Eq.(13.4.2 in the for m Tba= VbU +E ( V [1~90 + Yb +Y'~a,bV](13.4.3) where ~Pb, and are the projection operators respectively on D (a), D(b), and on all other states . (It is assumed here that none of the charged particles in h have momenta close to that of some charged particle in a, so that the sets D(a) and D(b) do not overlap .) Now, for A --3, 0 and D --+ 0, the dangerous intermediate states occupy so little phase space that they may be neglected wherever they do not lead to infrared divergences . The 13.4General Infrared D ivergences power series (13 .4.3) therefore become s ~a x o f, Thu- r=0s~0►~=0 XEa - HO iel Ya- VI') [Ea --H4+icbava,b551 (13.4.4) This mould be exact if all of the projection operators 4,,,b between the leftmost and the rightmost were replaced with +b+ and if h and Y Q on the left and right were replaced with b+,~41Q, but as remarked above this would have a negligible effect on the final result when Aand 0are sufficiently small . Eq. (13 .4.4) may be written in a more compact form : Tha= (Q ITS'20ha , where, for future use, we define S2x and S2flfor general states v and #as, (016 r=0 ~~ and Ts is the `safe' operator " (T5)dc"~ ,qa Ed --- H O-{- i~ ~.7a r V [ "'J(t 3.4,b) Eh-H-0--ic V db ').-(13.4.7) (13.4.8) Allof the infrared divergences have now been isolated in the two operator factors flh and S2~ . To eliminate these infrared divergences it is now only necessary to note that if it were not for the projection operators on the dangerous states, the operators f2band SSA would be just the unitary operators that according to Eq.(3.1.16 convert free-particle states into `cut' or `in' states, respectively . These operators are therefore unitary if confined to the subspaces D(fl) and D(a) of states that would be dangerous for some given final state fl and some given initial state or . That is, for general a and f3 (13.4.9) (13.4.10) ' In ( 92h)dbwe are using the fact that Tr,a is calculated with F b= Eu, and in (Tv)d,we are using the fact that the projection operators make ( QA),.,, vanish unless E ,is very' close to E .. Also, the factors SZb+and i~~ in Lq. (13 .4,5) make ~ V,.d=:YO,,,y. 552 13 Infrared Effect s The transition rate is theref~re free of i~frareddivergences of summed over the subspaces of states that would be dangerous for any given final and initialstates Pand ac: Sar=D(a) bED(fl ) Tr f9p Ts Y~, Tf I(Ts)bai 2 (13 .4.11) aED(a)bED({3) In order to be satisfied that this really does solve the general problem of infrared divergences, it is necessary to argue that it is only sums like that in Eq . (13 .4.11) that are experimentally measurable . It is plausible that we should have to sum over dangerous final states in order to have a measurable transition rate, since it is not possible experimentally to distinguish an outgoing charged (or colored) massless particle from a jet of massless particles with nearly parallel momenta and the same total energy,7together with an arbitrary number of very soft quanta, all with the same total charge (or color) . The sum over initial states is more problematic . Presumably one may argue that truly massless particles are always produced as jets accompanied by an ensemble of soft quanta that is uniform within some volume of momentum space . However, to the best of my knowledge no one has given a complete demonstration that the sums of transition rates that are free of infrared divergences are the only ones that are experimentally measurable . This problem does not arise in quantum electrodynamics (with massive charged particles), where as we have seen it is only necessary to sum over final states in order to eliminate infrared divergences . The reason for this difference can be traced to the fact that in electrodynamics the states a, b, c,.. - are direct products of states (labelled with Greek letters) with fixed numbers of charged particles and hard photons, times states containing only soft photons having energy less than some small quantity A. Then for a reaction in which some set of soft photons f is produced in a reaction or among charged particles and hard photons, Eq . (13 .4.5) simplifies to Tflf,.x =(n-(Val oo)fo(TS)PU (13.4.12) where 0denotes the soft photon vacuum, and 92± are calculated as before, but in the reduced Hilbert space consisting only of soft photons, and with the interactions of these photons taken as the interaction Hamiltonian with all charged particles in the fixed states indicated by the arguments P or cc . Just as before, these operators are unitary in the `dangerous' Hilbert 13.5 Soft Photon Scattering 553 space 9 ofsoft photon s, sod ~C_Q = RTs)flyl,(c+(tc+())00 =I(Ts)~,Xl2(13.4.13) without having to sum over initial states . 13.5 Soft Photon Scattering " In our treatment in this chapter of soft photon interactions, we have up to now considered only processes in which the soft photons are emitted or absorbed ina process x --* fl which was going on a nyway . It is also possible to make useful general statements about processes in which the process a -} fl is trivial, and the soft photons play an essential part in producing an interesting reaction .We will consider here the simplest and most important example of this sort, the scattering of a soft photon from a massive particle of arbitrary type and spin ,where at and #are single-part icle states .The complication here i sthat the leading term inthe softphoton scatte ring amplitude doe snot come from the pole term, but from non-pole term sthat are related to the pole terms by the cond ition ofcurrent conservation . The S-mat rix for photon scatter ingmay be put in the form (2,g )I '~q where qand q' are the in itial and final photon four-momenta, p and p' are the initial and final target four-momenta, i and ~'are the initial and final photon h elicities, cj9',Ar) and F,,(q,~)are the corresponding photon polarization vectors, and aand a' are the initial and final target spin z-components .According to the theorem of Section 6.4,the amplitude M'','may be expres sed a s CF ).~ [Z~.~.~} ~ The reason that we are now not encountering any factor ~EJA)Alike that in Eq .(13.3,11) is that we are here identifying the maximum energy E of the real soft photon states over which we are summing with the maximum energy h of the `dangerous' soft photon slates over which we sum in calculatin g This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 554 13 Inftared Effect s where P(x) is the electromagnetic current, and the dots indicate possible seagull' terms such as those in the theory of charged scalar fields in which the two photons interact at a single vertex instead of with separate currents . We now repeat the standard polology arguments described in Chapter 10and already used in Section 13.1.Inserting a complete set of intermediate states between the current operators in Eq. (13.5.2), integrating over x and isolating the one-particle intermediate states give s GP(P'ap'-q)G}'(P'- q,p) E(g'-- q) -E(p') +qQ -ie+Nv" ( q;p',p) , (13 .5.3) where G~ is the one-particle matrix element of the curren t lT and N" represents the contribution of states other than the one-particle state itself, plus any seagull terms . (Eq . (13 .5.3)is to be understood in the sense of matrix multiplication, with spin indices not shown explicitly .) About IV''~ we know very little, except that it does not have the singularity at qP--+0 exhibited by the first two terms, and therefore may be expanded in powers of q Y. We now use the current conservation (or gauge invariance) conditions : (13.5.5) (115.6) (13.5.7) Applied to Eq . (3 .5.3), these conditions yield the condition we need on Nv;t {i3.s.} We also note that M'M satisfies the `crossing sytnme try° conditio n `" (q; p',P) = M "►'(p' -p-q;p', p) (13 .5,9) and since the pole terms in Eq, (13.5.3)evidently satisfy this condition, so also does NVy: N"(q ~P', p) = N1v (P{ - P -q;P{a p) . (13.5.10) We will use these conditions to determine the first terms in the expansion of N'P in powers of momenta . First we need to say something about the expansion of the one-particle current matr ix elements GP( P',P) in powers of the momenta p' and p. 13.5 Soft Phot onScattering 55 5 Space inversion invariance (to the extent that it is applicable) tells us that the expansion of G° and GI (with i =1,2,3) contain terms respectively of only even and odd order in the momenta . According to Eq . (10 .6.3), the term in 47~+,ff of zeroth order in momenta is where e is the particle charge . The current conservation cond ition then tel ls us that to second order in momenta, fpf2 2 2m 2 m The terms in Gof first order in momenta are thus given by e(p'+p)JQ!,,/2rrz, plus a possible first-order term orthogonal to p' -p, which rotat ional invariance tells us m ust be p roportional to (p' - p) xJ,f,,, where J is the familiar s pin matrix of the charged particle . Summarizing these resu lts, we have the expans ions G4{p',p)= el +quadratic , r e1 ~~ G( PaR) = 2m (p'+ R) + J x (p' - p) + cubic ,(13.5.11) (13.5.12) where `1' is the unit spin matrix ,and `quadratic' and`cubic' refer to the order of the neglected terms in powers of the small momenta p and p '. The coefficient y1j in Eq . (13 .5.12) is real b ecause the current is Hermitian . With the coefficient written in this way (with j the spin of the charged particle ),µ isthe quantity known a sthe magnetic moment of the particle . Now let us return toN"p,and consider the expansion of Eq . (13 .5.8)in power sof the small momenta q ~I,p and p '.Taking v = 0 inEq. (13.5.5) shows that q ,N'P is at least quadratic in these sma llquantities . There is noconstant vector orthogonal to q I`,soNOP must be at least of first order insmall momenta . The crossing symmetry c onditio n(13.5. 1D)then tells us that 10must also beatleast of first order in small momenta . Taking y = i i n Eq .(13.5.8)and using Eq . (13 .3.12) then telly us that and hence2i qkN`k = _e+quadratic z IV's= -eb;k+linear .in(13.5.13) Since G Iis at least of first order in the small momenta, so are the pole terms in Eq .(13.5.3)for0k, leaving us in zero order with only the non-pole term 0 2 6 (13.5.14) 556 13Infrared Effect s From this we can calculate the soft photon scattering cross-section . But there is no need for this calculation ; now that we know that the photon scattering amplitude in the limit of zero momentum depends only on the target particle mass and charge, and is of second order in the charge, we can immediately use the results of any second-order calculation of the photon scattering cross-section for target particles of any given spin, such as our result (8.7.42)for the differential photon scattering cross-section in quantum electrodynamic s da4 dQ~ 32a r1raa2(1 + case 0). (13.5,15) We now see that this is a universal formula, valid in the low-energy limit for target particles of mass m and charge e and of arbitrary type and spin, even if these particles are composite and strongly interacting, li ke atomic nuclei . Drell-Mann and Goldberger and Low8 have shown that these results may be extended to give the next-to-leading term in the soft photon scattering amplitude in terms of the target particle's mass, charge, and magnetic moment . 13.6 The External Field Approximation * It is intuitively obvious that a heavy charged particle like the nucleus of an atom acts approximately like the source of a classical external field . In this section we will see how to justify this approximation, and will gain some idea of its limitations . Consider a Feynman diagram or a part of a Feynman diagram in which a heavy charged particle passing through the diagram from the initial to the final state emits Noff-shell photons with four-momenta ql> q2, ... qrr and polarization indices [11,P2,',UN.The sum of all such graphs or subgraphs (not including the Nphoton propagators) yields an amplitud e fd4xid 4 X2- d4XNe~qj,xje`q2`2 ---e-qN `jv x (p', cT'I T~J" (X 1)x J" (X2)}...JP~~(XN) IIP,(,T) ~§u1jux... i1,v(q1,q25...q1v;P) ,C(13.G.1) with the matrix element calculated including all interactions in which the heavy particle may participate, including strong nuclear forces . This amplitude has a multiple pole at q j, q2,... qN --* 0, arising from terms i n This section lies somewhat out ofthe book's main line of development, and may be omitted in a first reading . 13.6The Exter nal Field Approximation 557 the matrix elements of the product of currents in which the intermediate states consist of just the same heavy particle as in the initial and final states . This multiple pole dominates (13 .6.1)when all components of qi, q2,, - - q1v are small compared with all energies and momenta associated with the dynamics of the (perhaps composite) heavy particle . In this case the methods of Section 10 .2 give .' N- 1 X 64(1 +q1+q2+...+qN-F) ff~ Q~...Qh~-l lff2~~~..~~, ff b-l a~(P )X Cr dI Q +permutations (13 .6.2) whe re f,"(F) 2p°(2~}~ ~ P'ff'IJ~{ O}JP,C) (13.5.3) and `+ permutations' indicates that we are to sum over all permutations of the Nphotons . For applications to atomic systems it is important to recognize that (13.6.1)applies for particles of arbitrary spin that have strong as well as electromagnetic interactions, like atomic nuclei . We also note that for particles of arbitrary spin and charge fie, the matrix elements of the electric current between states of equal four- momenta aret PO(27z)3 so that ,1,1Y! L Z8pl~~fff (13.6.5) The important th ing about Eq . (13 .6.5)is that these matrices all commute , " IIn perturbation theory, the de nominators come from the denomi nators of t he propa gators: {P+i`Rl+...Rr}2+m2-tE-r 2p' '(41+-- 9t)-! e -~ 2A'{41 +...qr)-ie, while the numerators of the propagators provide factors uuT that together with the photon emission vertex matrices yield the matrix elements (13.6.3). The matrix 10differs from the matrix G"of the previous section by a facto r This is most easily proved by first noting that in the Lorentz frame in which the particle is a t rest, rotational invariance requires that the matrix elements of the current have vanishing space components and a time component proportional to with no other dependence on aor0r. The constant of proportionality is supplied by Eq .(10.6.3), and a Lorentz transformation then gives Eq, (13 .6.4). 558 13Infrared Eft~ct s so their product can be factored out of the sum over permutations : P~(2,g)l(27E)4 6 4(P,+q,+q2+... +Irv- F )6rY',rT x + permutations (13.6.6) To leading order in the q s the delta function here may be written (13.6.7) Fortunately, it turns out that the result of summing over permutations here is much simpler than the individual terms . For p` {qj+... + qrv} = 0, we have 1 +permutations = (2irz)nr- ',J (p ' R1) 6 (p ' q2)...6 (P 'qN-i) (13.6. 8) For instance, for N= 2 this reads : The genera] result (13 .6.8)can b eobtained most easily as the Fourier transform of the identit y B(x, -12) O(T2 - -13)...O(IN -1-TN) + permutations = 1. Inserting Eq.(13.6.$)in Eq . (13 .6.6) gives our final result for the amplitude (13.G.1): x63(PI+qi+q2+...+9N-P)J(F'4i)6(P.q2)...6 (P'qN).(13.6.9) This result applies to relat ivistic as we ll as slowly moving heavy particles, and can be used to derive the `Weizsacker- ilfiams' approximation9 for charged-particle scattering . In the special case of a non -relativistic heavy charged part icle, with C AS<p°, Eq . (13 .6.9) further simplifies t o S~f'}2.. -I~i,ti 1~1a q21... [~N }P1 ~(Zf')N(2?Z)~r18P1nF12...hF~~ti x63(P{ +q,+q2 +... +qN -P}6( Ri) 6(q°)...J (q~,)6,+,~ (13 .6. 10) 13.6The E xter nal Field Approximatio n where n is a unit t ime-like vecto r Now suppose that a single heavy non-relativistic particle of charge Ze with normalized momentum space wave Function Q(P) appears in both the initial and final states . Using the Fourier representation of the delta function in Eq . (13 .6.9), the matrix element of W in this state i s d3Fd'P{ ~~+ (P')X, (P)~; ~'.._~w(q I, q2y... qtr;P) N jr=1 where y)( X) is the coordinate space wave function . Because of the factorization in Eq .(13.6.11), the effect of including a heavy charged particle in this state is then the same as that of adding any numbers of a new kind of vertex in the momentum space Feynman rules, in which light Dirac particles of charge -e such as electrons interact with an external field, with each such vertex contributing to the overall amplitude a fa tarp (now including the photon propagator and electron- photon vertex } -iIl[(27T) .ei i'l d4q ~(21r)4 q2 (13..13) where k and k' are the initial and final electron four-momenta . The complete scattering amplitude must then be averaged over the heavy particle position X, with weight function J:, I W, (X) I2. The factor (13 .6.13) is the same as would be produced bya new term in the interaction Lagrangian cxt(x) ='4"{-~~ ~el"W, (13.6.14)559 (13.6.11) (13.6.12) where Je= -ae'I'},P"Y isthe electric current of the electron s,and VPis an external vector potentia l .~ (~) I Id4 e iq,27rZe nYb(q')e-'T ' (2?1)4q, q- 8 eI -(13.6-15) Thefirst factor here is the usua lfactor of iaccom panyi ng the cons tantsin the interaction Lagrangianof theheavy c harged particle i n the Fe ynman r ules. 560 13Infrared Effect s This, of course, is just the usual Coulomb potential : (13.6. 16) If there is more than one heavy charged particle (as in a molecule) we must express P'(x) as a sum of terms like (13.6.16), each with its own charge Ze and position X , It is useful to keep in mind what diagrams we are summing in using the external field approximation . Consider the interaction of a singe electron (relativistic or non-relativistic) with a single heavy charged particle such as a proton or deuteron .If we ignore all other interactions, then the Feynman diagrams for the scattering of the electron due to its interaction with the external field are just those with any number of insertions of the electron-external field vertex (13 .6.14) in the electron line . (See Figure 13.4,) But as shown bythe sum over permutations in Eq . (13.6.2), these diagrams in the external field approximation come from diagrams in the underlying theory in which the photons attached to the electron line are attached to the heavy charged particle line in all possible orders . (See Figure 13.5.)The `uncrossed ladder' diagrams (labelled L) of Figure 13.5do not dominate this sum unless the electron as well as the heavy charged particle is non-relativistic . (These diagrams include contributions from terms in old-fashioned perturbation theory whose intermediate states contain the same particles as the initial and final states, leading to small energy denominators when the electron and heavy charged particle are both non-relativistic, while all other diagrams of Figure 13.5correspond to intermediate states with either extra photons, electron-positron pairs, or heavy particle-antiparticle pairs, which are suppressed bylarge energy denominators .) The uncrossed ladders can be summed by solving an integral equation, known as the Bethe-Salpeter equation,"D but there is no rationale for selecting out this subset of diagrams unless both particles are non-relativistic, in which case the Bethe-Salpe#er equation just reduces to the ordinary non-relativistic Schrodinger equation, plus relativistic corrections associated with the spin-orbit coupling that can be treated as small perturbations . It must be said that the theory of relativistic effects and radiative corrections in bound states is not yet in entirely satisfactory shape . In the derivation of the external field (13.6.16)we evaluated the inter- action ❑f the heavy charged particle with the electromagnetic field only to leading order in the photon momentum . There are corrections of higher order in the photon momentum arising from the heavy particle's magnetic dipole moment, electric quadrupole moment, etc . Also, of course, there are radiative corrections arising from Feynman diagrams beyond those of Figure 13 .4, such as diagrams in which photons are emitted and absorbed 13.6The External Field Approximation +-~561 Figure 13 .4. Diagrams for the scattering of an electron by an external electro- magnetic field . Here straight lines represent the electron ; wavy lines ending in crosses represent its interaction with an external field . L LL Figure 13 .5. Diagrams for the scatteriDg of an electron by a heavy, charged target particle, which in the limit of large target mass yield the same result as the diagrams of Figure 13 .4. Here the single straight line is the electron ; the double straight line is the heavy target particle ; and wavy lines are virtual photons . Diagrams marked `L' are called uncrossed ladder graphs ; they dominate the sum when the electron as well as the target particle is non-relativistic . 562 13Infrared E L{'ts from the electron line or electron loops are inserted into photon lines . We shall see in the next chapter that in bound states the diagrams of Figure 13.4 must be included to all orders, but all other corrections of higher order in photon momenta or e may be included as perturbations to these diagrams . Problems 1. Consider the process e+ + e- --} art + 7r R in the center-of-mass fram e at an energy of k GeV and scattering angle of 90° . Suppose tha t by measuring the energies of the final pions we determine that a n energy of not more than E T !Gel is emitted in soft photons . How does the reaction rate depend on ET? 2. Consider a massless spinless particle described by a scalar field , whose interaction Lagrangian density is of the form O( x)J(x),wher e J(x) involves only massive particle fields . Derive a formula for th e rate of emission of arbitrary numbers of soft scalar particles in a process oc --* fl , with these soft scalars having a total energy less tha n some small quantity Er . Include radiative corrections due to sof t scalars with energy less than some small quantity A. 3. Derive a formula for the next term beyond Eq.(13.5.14) for low- energy photon scattering on an arbitrary target . 4. Suppose that a spin one particle of very small mass m is described b y a vector field Ve(x), which couples only to a much heavier fermio n described by a Dirac field fi(x), with an interaction Lagrangia n density of the form g V g~p7,t~p .Suppose that the heavy fermio n normally decays into other particles that have no interaction wit h Yu, releasing an energy W much greater than m . Consider such a decay process, but where an additional VP particle is emitted alon g with the other decay products, with the vector particle energy les s than an upper limit E, in the range E m . How does the rat e of this process depend on Eand m? Ignore radiative corrections . 5. Prove that Eq . (13 .6.8) holds when p -{qt + ...gar} = 0 . Referen ces 1.F. Bloch and A . Nordsieck, P hys. Re v. 37, 54 (1937) ; D. R. Bennie, S.C. Frautschi, and H . Suura, An n.Phys . (NY) 13, 379 (1955) . Re f erertces 563 2. S. Feinberg, Phyrs .Lett .9, 357 (1964) ;Phys . Rev. 135, B1049 (1964) . 3. S. Weinberg, F hys.Rev. 140, B515 (1965) . 4. This phase was encountered in the perturbation series for non- relativistic coulomb scattering, by R . H. Dalitz, Proc . Roy . Soc . London 206, 509(1951) . 5.See, e.g., L.1. Schiff, QuantumMechanics (McGraw-Hill, New York, 1949) : Section 20 . 6. T. D. Lee and Nauenberg, Pays . Rev .133, B1549 (1964) . Also see T . Kinoshita, J.Math .Whys . 3, 650 ( 1962 );G.Sterman and S. Weinberg, Whys .Rev. Lett . 39, 1416 (1977) . 7. Cr. Sterman and S.Weinberg, Ref G . S. F. E. Low, Pays .Rev. 96, 1428 (1954) ; dell-Mann and M . L. Goldberger, Phys . Rev .9G, 1433 (1954) . Also see S . Weinberg, in Lectures on Elementary P artic lesand Quantum Field Theory - 1970 Brandeis Summer Institutein Theoretical Physics, ed.by S. Weser, M . Grisaru, and H, Pendleton (MIT Press, Cambridge, MA, 197 0). 9. E. J. Williams, Kgl.Dan.Vi d.Sel,Mat.-fys.Medd. XIII, No . 4(1935) . 10. H. A. Bethe and E . E. Salpeter, Phys . Rev. 82, 309 (1951); 84, 1232 (1951) . 14 Bound States in External Field s In our calculations of radiative corrections i n Chapter 11 we went just one step beyond the lowest order in pertur bation theory .However, there is a very important class of problems where even the simplest calculation requires that from the beginning we consider classes of Feyn nian diagrams ofarbitrarily high order in coupl ingconstants like e . These problem s are those involving bound states -inelectrodynamics, either ordinary atoms and mo lecule s,orsuch exot icatoms as positronium or mu onium . Itis easy to see that such problems necessaril yinvolve a breakdown of ordinary pertur bation theory . Consider for in stance the amplitude for electron-proton scattering as a function of the center-of-mass energy E . As shown in Section 10 .3,theexisten ce of a bound state like the ground state of hydrogen implies the ex istence of a pole in this amplitude at E= m.~+rn,-13 .6 eV .However ,no single term in the perturbation ser ies forelectron-proton scattering has such a pole . The pole therefore can only arise from a di vergence ofthe sum over all d iagram sat center-of-mass energies near m .p+ tee. The reason for this divergence of t he perturbation series is also ea syto see,especially if for the moment weconsider the time-ordered diagrams of old-fa shioned perturbation theor yinstead of F eynman diagrams .Suppose that in the center-of-mass s ystem the electron and proton both have momenta of magnitude q < m e,and consider an intermediate state in which the electron and proton moment a are different but also of order q . The energy denominator factor contributed by th is state will be of order [q~1mJ -'.Each such state will also contribute amatrix element of the Coulomb interaction of order e2lq2 the Four ier transform ofe2/r), and the corresponding momentum space integration will contri bute a factor of order q3. Putting this together ,we see that each addit ional Coulo mb interaction cont ributes an overall factor ❑f orde r [q~~m, .]-' [e'lq2l tq-'] = e'mel q Thus the perturbat ion theory sh ould break down when q is less than or of the order of e2m,,or in other words when the kinetic and potentia l 564 1 4.1 e Dirac Equation 565 energies, which are of order q11M.,are no larger than about m,, which, of course, is of the order of the binding energy of hydrogen . Our problem here is to learn how to use perturbation theory to evaluate radiative corrections in bound-state problems, summing to all orders those diagrams that need to be summed to all orders, and keeping only a finite number of those that don't , 14.1 The Dirac Equatio n We shall limit ourselves in this chapter to problems in which bound states arise because of the Coulomb interaction of electrons (or muons) with heavy charged particles such as atomic nuclei . As shown in Section 13 .6, this interaction may be taken into account by adding to the interaction Lagrangian a term' representing the effects of a c-number external vector potential P(x) : -iC(Z2 - ~ ~~~TYuXFi(14.1.1) which is obtained by replacing the quantum vector potential A` with At+&& in the interaction part of Eq.(I1,1.6). For instance, for a single heavy particle of charge Ze at the origin , II(f4,1 .2) It is the interaction (14.1. 1) that must be taken into account to all orders . In th is section we will cons ider the theory w ith only th is interaction, leaving radiative corrections to subsequent sections . Physicists learn in kindergarten t oapproach this sort ofproblem b y solving the wave equation of Dirac in the presence of the external field .It might seem unnecessary toderive this equation h ere,but as emphasized in Chapter 1, Dirac's or iginal moti vation forthis equation as a sort of relativist icSchr6dinger equat ion does not stand up to inspection .Also, in thecourse of our derivation wewill d iscover the normalization conditions that have to be imposed ❑n the solut ions of the Dirac equation ,which seemed somewhat ad hoc in Dira c'sapproach . The solutions of the D irac equati ondiscussed here will be important ingred ients in our treatment of radiative correct ions in t he next section .-- avtwyqllAv -- OwA O ' In this chapter we return to the use of an upper case T to denote the electron field in the Heisenberg picture, reserving a lower case W for the Dirac field with time-dependence governed solely by the c-num ber external field 566 14 Bound Statesin External F ields We will work here in a version of the Heisenberg picture, inwhich the time-dependence of operators is determined by a Hamiltonian including the external field interaction (14 .1.1), but no other interactions . The electron field W(x) in this picture satisfies the field equatio n -c? f'ax; +M+iery1~`~~1 ~~) W(x) = 0 . (14.1.3) This is not the Dirac equation in the original sense of Dirac, because ~)(.x) here is not a c-number wave function but a quantum operator . The c-number Dirac wave functions are defined b y UNW ((DO, Y)WON ) VN(X) (ON, T000)(14.1.4) {14.f.5} where ON are a complete orthonormal set of state-vectors ,withq)()the vacuum .Itfollows immed iately from Eq .(14.1.3)that these functions satisfy the homogeneous Dir ac equatio n ~~ +tn+iey -(X)j UN(X) =[Y ex~ IVNM -0. (14.1.6) We can also derive a normalization condition from the equal-tune anti- commutation relations for the Dirac field . These are unaffected by the interaction (14 .1.1), and therefore take the same form as for the free fields : MX# ~~,IN3~,0= iY° 61(X-Y). (14 .1.7) Taking the vacuum expectation value and inserting a sum over the states ON, we find UN(XyOUNl~~0+ VN(Xk t)V11j(y}0 = 6' (aCY) (14 .1.8) N !Y it being understood that the sum over Nincludes an integral over contin- uum states as well as a sum over any discrete bound states . We are chiefly interested in the case of a time-independent external field, like (14 .1.2). In such cases, the states ONmay be taken as eigenstates of the Hamiltonian (including the interaction (1 .1.1)} with energies EN. Time-translation invariance then tells us that the UN(x) and i N(x)have the time-dependence : UN X,~}= e-iE,y[UN 4lX)a V~1+(x,t) = e +iEj,~f~,1V~3~C). (14.1.9) E f The homogeneous Dirac equations (14 .1.6) then become 1 4.1 the Dirac Equatio n iyo[y -V+m+iej~,2/1(x)] UN(X)=EN UN W, i'l' [T -V+m+ielj~&/A(x)] VN(X)= -ENVN(X)567 (14.1.11) The minus sign on the right-hand side of Eq . (14 .1.1 1) shows that the uNare the famous `negative-energy' solutions of Dirac . As shown by Eq. (14 .1.8), these negative-energy solutions are needed to make up a complete set of wave functions .Of course, for moderate external fields there are no negative-energy states in the theory, so all E Nare positive, but there is still an important difference between the states with non- vanishing UNorVN: the definitions (l 4. 1.4) and (14.1.5) show that a state can have UN * 0 or VN 7~ 0 only if it has charge -e or +e, respectively . It is in this sense that negative-energy solutions of the Dirac equation have something to do with the existence of antiparticles . However, this argument has nothing to do with the details of the Dirac equation, or even with the spin of the electron . From the Dirac wave equations (14 .1.10) and (14 .1.11)we can easily see that wave functions of different energy are orthogonal . That is , ) = V -(Ul i'lN N N YUM) sa if Ix12(u~ iy°yUM)remains bounded as JxJ ~ 0 and xJ ~ oo, the n Xx(=0ifEN~E~. (14 .1.12) With similar boundary conditions for the U .N. we find in the same way tha t J d3X Jd'x(4(X)vM(X)) UN(X)V-?If(X))=D i f = 0ifEN* EL, EN:~ -E* M.(14.1.13) Taking N = M, Eqs . (14.1.12)and (14 .1.13)tell us that the energies are all real. Dropping the complex con jugation of EV in Eqs .(14.1.12)-(14 .1.14), we see that us of different energy are orthogonal , vs of different energy are orthogonal ,and (as l ong as the potential is not strong enough toproduce negative energy states) all us are orthogonal to all vs. By a suitable choice of the di screte quantum numbers that characterize the states along with theenergy, w e can then always arrange tha t fdux (i(Xi(X)) = 0ifN 7~- M, dux(4(x)VM(x))=0ifN =~ M, dux(4(x)i(x)) =0 .(14.1.15) (14,1.16) (14.1. 17) 568 14 Bou nd States in E xternal Field s Multiplying Eq . (14 .1.8) on the right with um(y) or vm(Y), we find then that these wave functions must satisfy the normalization condition s (14.1.18) fd3y (ui1(y)uY)) _jd3y (4(y)VM(y)) =6uM -, where 6NM is a product of Kronecker deltas and momentum space delta functions, with normalization adapted to that used in defining Ejv, in such a way that N 6NM =1.These normalization conditions have nothing directly to do with any probabilistic interpretation of the Dirac wave functions, but arise instead from the anticommutation relations (14 .1.7) for the fields . Let us now specialize to the case of a pure electrostatic external field with A = 0 . In our standard representation of the Dirac matrices, we hav e 0-6 ~ 0 11'=~or 010 where a is the usual three-vector of 2 x 2 Pauli matrices, and `1' and `D' here are the 2 x 2 unit and zero matrices . We introduce two-component wave functions f vand gNby settin g UN-1( fN + i9N The energy eigenvalue condition (1 .1.10) then takes the farm : {U'V)gN = -(EN + e ° -rn.)Irv(14.1.19) (14.1.20) (14.1.21) In the non-relativistic case where ed4r Zoc <1 the binding energy ENis of orderZ2ac2rn, while the gradient operator is of the orde r of Z aM,so g Nis smaller than f N by a factor of order Z.(To find the positron wave functions v Nwe replace ENeverywhere by -E N, sn in this case f Nis smaller than g Nby the same factor .) We shall return to this non-relativistic case at the end of this section . Physical states may be classified as even or odd under space inversion : P(DN-qNON, (14.1.22) where qNis a sign factor, +1. Recall that with the intrinsic parity of the electron defined to be +1, the Dirac field has the space-inversion propert y PW{x, t}P-l= 0V(-xa t) so Eqs . {14 .1.4} and (14 .1.22) show that the Dirac wave functions satisfy the parity condition 14.1TheDirac Equation or equivalentl y fN(X)=t1rrf N(-X)~ 9N(X)=-nn~gN(-X)569 Note that the parity of the state is the same as the parity of fi(x), not 9N(~), Where the potential ~4`) is rotationally invariant, the solutions of the wave equations here may be classified according to their total angular momentum jand parity q. For a given j, the components ff and g may b eexpanded in spherical harmonics with orbital angular momentum /= ,j+ land ~ -j-_', but for a definite parity I-- Eqs . (14 .1.24) show that we can only have e= j + ~ in f and j ± ~ in g . The usual rules of angular-momentum addition then show that for a state of total angular momentum j, total angular-momentum z-component P, and parity the `large' two-component wave function f has the for m {14.i.2} where Cand Yare the usual Clebsch-Gordan coefficients and spherical harmonics .2 Also, given any wave function of definite total angular mo- mentum and parity we can construct another wave function with the same jand u but opposite parity byapplying the operator or  x, so the `small' components may be put in the form f C17 (j + Y.-F It is conventional to define the orbital angular momentum e of the state as the orbital angular momentum of the `large' components f (x) , /'0=i:~ 12 , (14.1.27) so that the parity is always (-1)'. Inserting Eqs. (14.1.25)and (1 4.1.26) inEqs.(14.1.20) and (14 .1.21) yield sthecoupled differential equation s dF k-1 dr r where for parity q =~-1 3,(14.1.29) (14.1.30) 570 14 Bound States in E xternal Field s Let us now concentrate on the simple Coulomb field (14 .1.2), for which e.~v'l° = Zoc/r . The treatment of the Dirac equation in this case is familiar, so it will be summarized briefly here just for completeness . It is easy to see that the solutions near the origin go as r", with s2= k2 -- ~2a2 . (Note that k 2>1, so the exponent s is real for Za 1 .) We must reject the solutions with s 0as being inconsistent with the normalization condition (14 .1.15). The condition that the wave functions do not blow up for r -> ao then fixes the allowed values of the energy eigenvalues : ~ ~a~E~,J= gyn I + n_ j_ ~ + ~~+ 1 ~ ~ _ Z 2,, 2 where n is a `principal quantum number' wit h j+ 1<n. (14.1.32) It is noteworthy that these energies do not depend on the parity or e, but only on n and j . For each n and jthere are two solutions, corresponding to the two signs of k or the two possible parities, except that for n = j + ~ we only have k >0 and parity (- I)j-;, so that j - ~ . With Eq .(14.1.32), this is the same as the familiar non-relativistic restriction that n - 1. For light atoms with Z a<1,Eq.(X4.1.31) yields the power serie s z2a2 E=»x 12n2Z4x43~ n4(8-12j+ 1 + ...(14.1.33} The first two terms, of course, just represent the rest energy and the binding energy as given by the non-relativistic Schrodinger equation . The leading term that depends on j as well as n is the third term, the first relativistic correction . For n = 1 there is only one value of the total angular momentum, j and since here n _ j+1there is also only one parity, (-1)J- 7 1, corresponding to 0. It is therefore difficult to see the effects of the relativistic corrections in Eq . (14 .1.33) in the n = I states of hydrogenic atoms, though as we shall see in section 14 .3, this has recently become possible . On the other hand, for n = 2 we have a state with both parities (i .e., 2s112 and 2P ,/2} as well as a 2P3/2 state with j and negative parity . Eq . (14 .1.33) gives the splitting between the p states in hydrogen as aMe - 4.5283 x10-5eV (14 .1,34)32 Such relativistic line splitting is known as the .fine structure of the atomic state . From the beginning it was known that this prediction is in good agreement with the observed fine structure . On the other hand, the Dirac 14. 1 The Dirac Equation 571 equation does not yield and energy difference in the 2s112 and 2pl/2 states, so this is a good place to look for the effects of further corrections, to be considered in Section 14 .3. Before closing this section we shall consider the approximate forms for the wave functions and matrix elements in the no-n-relativistic case for a general electrostatic potential .o. (For a Coulomb potential, this is the limit Zoc < 1 .)Since here EN+m ^f 2m >~e.i~101, the `small' components of the electron wave function are given approximately in terms of the large components by 9N:,' (or ' V ).fN/2me. (14.1.35) Eq. (14 .1.21) then becomes just the non-relativistic Schrodinger equatio n v2 - - et d{'A~~ (EN-M)frrZm,(14.1.36) Since there is no longer any coupling between spin and orbital degrees of freedom in the equation for f N, we may find a complete set of solutions of this equation in the form fN= Z IV VWW where XNis a two-component constant spinor, and IPN(x) is an ordinary one-component solution of the Schrodinger equation . However, we often work with states that have definite values of the total angular momentum j, for which , f N is (for non-vanishing orbital angular momentum) a sum of such terms . In the non-relativistic approximation, the four-component Dirac wave function takes the form 1(1+ isV/2m)f N ~~2--(1 -isV/2m)fN and Eq . (14 .1.18) gives the normalization conditio n where d3, f e(14.1.37) In relating matrix elements in an external field to free-particle matrix elements, It is useful to note that the momentum space wave function in an energy eigenstate Nmay be writte n UN(P)=Qiz)-"2d 3 p e--~~~~~~(~) ~ U ( P~ 0VN(P)1d a (14 .1.39) 572 14 Bound States in Exterrta! Field s where u( p, a-)is the free-partic le Dirac sp inor 72 (1 + por/2m,) Za 1 0X+2 0 and NR) is the Fourier transform of the two-cornponen tSchrod inger wave function fN(p)--(27r)-"' f d3pe _iP.XAW, In closing, as an aid in calculating the effects of various perturbations, we note that the leading terms in the electron matrix elements of the sixteen independent 4 x 4 matrices ar e (UMUN) ( JM'fN)~4773(Vf ~ ff ff VfN ) e (um[;'0,y]Irv)^'~[(v,f ~T iffiv)+ (f~a a - vfnr)] 14.2 Radiative Corrections in External Fie lds(14.1.40) (14.1,41) (14.1.42) (14.1.43) (14.1.44) (14.1.45) (14.1.4?) We now consider radiative corrections to the results of the previous section, due to the interaction of electrons with the quantum electromagnetic field as well as the external field of the heavy charged particles . These radiative corrections can be calculated using Feynman diagrams of the usual sort, with the whole effect of the external field being to modify the propagator of the electron field in the presence of the external electromagnetic field (and to supply the external-field-dependent renormalization counterterms shown in Eq . (14 .1.1)). To he specific, the effect of inserting any number 1 4.2 Radiative Corrections inExternal Fields 573 of vertices corresponding to the first term of the interaction (14 . 1.1) in an internal electron line of any graph is to replace the bare coordinate space propagator -iS(x - Y) with a corrected propagato r -iS,,t (X,Y) =-= -is(X- Y) + (-i), f d4Z,S(x-zj)ey1' .4,,(z1)s(z1 -A +(-i), d42l d4z2 S (x - zl)eYPcl,{zi}S(zl- z2)e"°J~/V(a2)S(a2--T Y) +...t where as usual S(X - A~ ~ P+ m eiP'(xl k~) (Z~)+M2- iEP~(14.2.1) (We must write S, vas a function of x and y rather than of x - Y, because the external field invalidates translation invariance .) The theorem proved in Section 6 .4 tells us that Eq. (14 .2.1) is the same a s with the subscript s lon the right indicating that the vacuum state (Do and electron field W(x) are to be defined in a Heisenberg picture in which the only interaction taken into account is the interaction (14 .1.1) with the external field . Inserting a complete set of intermediate states ON in Eq. (14 .2.2) yields an expression for the propagator in terms of the Dirac wave functions UNand VNintroduced in the previous sectio n N M It is also possible to obtain the propagator (14 .2.2) as the solution of the inhomogeneous Dirac equation : which follows from the field equations (14 .1.3) and anticommutation re- lations (14 .1.7), or formally from the perturbation series (14 .2.1). Also Eq. (14 .2.3) tells us that the propagator satisfies boundary conditions-, its Fourier decomposition contains only `positive frequency terms' pro- portional to exp{-iE(x° -- y O)} with E>0for x° - y° -# oo, and only `negative frequency terms' proportional to exp{+iE(x° - y° )) with E >0 for x 'o-° --*-oo. The inhomogeneous Dirac equation with these bound- ary conditions may be used to obtain a numerical solution 4 for this propagator even in cases where the external field is too strong to allow the use of the perturbation series (14 .2.1). Once the propagator 5 .,,,(x,Y) has 574 14 Bound States inExter nal Fields been calculated, the amplitudes for scattering in an external field can be calculated using ordinary Feynman diagrams, but with S,,~(x, Y) in place ofS{x - y} (and with .sit-dependent renormalization counterterms inserted where appropriate) . Now let us see how to use the perturbation series with this corrected propagator to calculate the shifts of bound-state energy levels . Consider the full electron propagator S 'W(x, y}, involving interactions of the electron with the quantum electromagnetic field as well as the external field : with T(x) the electron field in a Heisenberg picture including all inter- actions, and Q()the vacuum eigenstate of the full Hamiltonian . For a time-independent external potential we can find a complete orthonormal setON of eigenstates of the full Hamiltonian with energies E.Inserting a sum over these states in the operator product in Eq.(14.2.5), we fin d N whereN (N,''(xst)Qo)=_e+~E'NryNM (14.x.$) (The sum includes an integral over continuum states as well as a sum over discrete bound states . As before, UVand VNare non-zero only if the state ONhas charge -e or +e, respectively .) We can redefine the propagator as a function of energy rather than tim e fco { (Time-translation invariance dictates that S I(x,y)is a function of x0 - yo but not of x°andy'separately .) From Eq . (14.2.6) we see tha t UjVW UN (Y) E~; -E-a~VN W 11'N(r) E'+E-le(14.2.10) In particular, S,~(x, Y ;E)has a pole at any electron bound-state energy, and also at the negative of any positron bound-state energy . Of course, positrons do not have bound states in the coulomb field of an ordinary positively charged nucleus .) 14.ERadiative Corrections in External Fields 57 5 Let us now consider the lowest-order radiative corrections to the corm- plete propagator . The Feynman rules here give the complete propagator to this order as S, = S ,~ + 6 S.,f, with a correction ter m ~5.d (X,Y) _ d42d4w 5~,v (x, z )Y.* w)St,j(w,y), {14.2.11} where i E* is the sum ❑f all one-loop diagrams with one incoming and one outgoing electron line (excluding final electron propagators) calculated using W(x,y)in place of S (x - Y )for internal electron lines, plus second- order renormalization counterterms . Using energy variables in place of time variables, this i s where Y~(z?~'a E) = J d {14.2.3} The effect of these radiative corrections is to change the wave functions to U N=UN + 6UN and VN = VN + 6 z:Nand the bound-state energies to E~ = EN+6EN, so that the complete propagator i s 6U lY ~~~ U 7A'(J)+i~i1~(r7)bL6N(Y ) + E N-E 1:6VAXPAY) +VN(X)JV'IV(Y) NEN+E (EN-~)~1:VAXPA Y}6Erv (14 .2.14) N(kN+ E) ~ (We are dropping the ic terms because we are now not taking Etoliein the continuum of scattering states .) We see that the shift JEj,r of an electron bound-state energy is given bythe coefficient of -u N(xYiN(Y)1(EN-E)2 in the complete propagator . To calculate this, we note that Eq.(14.2.3) gives 1:UN(XA 1'v(Y) N EN-E - i6 Inserting this in Eq. (14 .2.12) givesVN(XPN( Y) 1:'EN + E - ieN 6S.Ax,y-E)=1:UnF~~~ RM(Y)11 NM (EN - E)(Em - E) X d32 dew uN(z) 2:*W(z,w; E )u~w(w) + ..(14.2.15) (14.2.16) 576 14 Bound States in Exter nal Field s Figure 14 .1. Lowest-order Feynrnan diagrams for the electron self-energy func- tion 2 :~(x,y) in the presence of an external field. Here double straight lines represent electron propagators S ,,,that include effects of the external field ; single straight lines are the incoming and outgoing electron lines ; wavy lines are virtual photons ; the cross represents renormalization counterterms . where the dots denote additional terms involving at least one negative- energy pole . Comparing the coefficients of ( EN - E)-2here and in Eq. (14.2.14), we fin d 6EN = - dux dayjj,(x)1:~a(X,y;Irv) UN(Y) . (14 .2.17) The UNare solutions of the homogeneous Dirac equation satisfying the normalization condition (14 .1.8), so this is very much like ordinary first- order perturbation theory, but with - 2:* in place of a perturbation to the Hamiltonian . Generally 6EN turns out to be complex . This is simply a consequence of the instability of atomic energy levels to radiative decay to lower levels ; we saw in chapter 3that an unstable state of energy E and decay rate Fproduces poles in various amplitudes at the complex energy E- d'/2 . The imaginary part of Eq . (14 .2. Z7) therefore equals -17/ 2, while its real part gives the energy shift . The Feynman diagrams for Vare shown in Figure 14 .1. (Note that there are new photon tadpole diagrams here because the external field breaks the Lorentz invariance and charge-conjuga#ion invariance that forbids such diagrams in the ❑rdinary Feynman rules .) Application of the 14.2 Radiative Corrections inExternal Field s position space Feynman rules to these diagrams give s -ley,64(X _ Y)]jd4z [-iD(x -z)] Tr~[-iSAz,z)] [ey,]) -V 2 -1)(y,011+)64(X- Y) + ism64(X- J')577 with the renormalization constants V2 - 1 ),V3- 1}, and bm calculated to second order in e. The minus sign in the second term is theusual one that accompanies closed fermio nloops .) For strong external fields with Za of order unity it is nece ssary to calculate the configuration space electron propagator S aand the integrals inEqs. (14 .2.17) and (14.2.18) numerically.4However, for weak fields we can use the first few terms of the series (14 .2. 1) in E q. (14 .2.18) and calculate these integrals in closed form .For th ispurpose ,it is more convenient to work in momentum space, defining : S,~(xj) =(21t)-4 f d4p~ d4p dl~'xe-P'yS,,,((P,,, P) , UN(X)= (2-K)-112 f d3peip-XUN(P)(14.2.19) (f 4.2.2U) (14.2.21) (We are here committing the impropriety of using the same symbol for a function and its Fourier transform, leaving it to the displayed arguments of the functions to indicate which is which .) Then Eqs . (14 .2.1) and (14.2. 1S) become -tO+M 578 and1 4Bound States in External Field s ie2A(p' (27r)4jk2 - i6- + 21Ai (21r)4(p'p,)2 +ie(Z3 - 1)[(P f2 r r + ,- P)] (4.2.24) Because the external field is time-independent ,S,I(x, y) and Y,,'{x,y} can depend on x0and y°only t hrough the difference x°-Y 0, SOS ',,I (p', p) and E~{p', p} a swell as ,ss~P(p'-p) must be proportional to d (p'~) -p0) The ene rgy shift is then given by Eqs . (14 .2.17) and (14-2 .13) a s 6EN= -f d' pj X p (ptI(14.2.25) (14.2.26) (14.2.28) with 1~(P', p; E~r)given byEqs. (14 .2.23), (14 .2.24), and (14 .2.27'). This is the master formula we shall use in the next section to calculate energy shifts in weak external fields . 14.E T he La mb Shift in L ight Atoms Let us now consider radiative corrections to the energy levels of a non- relativistic electron in a general electrostatic field, such as an electron in the Coulomb field of a light nucleus with Zx <1. It is natural in this limit to treat the Coulomb field as a weak perturbation, but we will see that this would lead to an infrared divergence, re lated to that found in Section 11 .3. The infrared divergence is really fictitious, because the four-momenta p, EN and p', EN are not on the electron mass shell, but it does force us to proceed with some care . Usually this problem is dealt with by dividing the integral over virtual photon energies into aloes-energy range, within which we can treat the electrons non-relativistica .lly but must include effects to all orders in the 14.3 The Lamb Shift in LightAtoms 579 external field, and a high-energy range, within which we have to include relativistic effects but may include only effects of lowest order in the external field . Instead, we shall here introduce a fictitious photon mass U, chosen to be much larger than typical electron kinetic energies, but much less than typical electron momenta . For a Coulomb field, this amounts to the requirement that (hoc)'Mg P CZIm"(14.3.1) We write the photon propagator in the first two terms of Eq . (14 .2.24) (including the formulas for the counterterms Z2 - 1 and Z26 rra)in the form V i, [k2+,U2 - ie]~P - ic k2+4"2i6i - The energy shift is correspondingly a sum of two terms, a `high-energy' and a `low-energy' term : The high-energy term is calculated by using the first term in the photon propagator (14 .3.2 in the first three terms in Eq. (14 .2.24), and adding the result to the last two terms (the vacuum polarization terms) in Eq . (14 .2.24) which are not infrared divergent anyway ; the low-energy term is calculated using the second term in Eq.(14.3.2) in the first three terms of Eq . (14 .2.24). One advantage of this procedure is that we shall be able to use the results of the relativistic calculations of Sections 11 .3 and 11 .4 directly, without the rather tricky conversion from a photon mass to an infrared energy cutoff . Of course, in the end we shall have to check that the dependence on the photon mass p in the high-energy and low-energy contributions to the energy shift cancel, leaving the total energy shift u-independent . A Nigh-Energy Ter m Because y is taken much larger than the atomic binding energies we can here keep only terms of lowest order in the external field . The one-loop ra- diative correction to atomic energy levels in a general time -independent ex- ternal vector potential v,/P(x) is given in momentum space by Eq . (14 .2.28), with the self energy insertion Z ( F',p)given byEqs. (14 .2.24) and (14.2.23). The terms of zeroth order in the external field simply cancel : the 6m term cancels the first term with V=0; the Z3 - 1 term cancels the third term with W=0;and the Z2 - i terms vanish because u(p) satisfies the Dirac equation . The term in 1.w(P', p )that is of first order in t dAmay be put in the form 1, 1 (14.13) 580 with14Bound States i n External Field s ie' A (27r)4 k2 +y + me +tn. (p'-k)2 +m~---~ i~~ (P - k)2 + M~ -iF + (Z 2 ae21 X A Tri I+Me 'U[-i I - i P! + i fi + me] VI 12+rn ~ ~+ ' ~---p)2+ M2j ~ I [~ -Z3- I [(p_PI)2quv P)'u(Pf - V . (1~ - p ) Comparison of the first two terms with Eqs .(11.3.1)and (11.3.8)and the second two terms with Eqs . (11.3.9),(11.2.3), and (11 .2.15) reveals that Vi(p', p) is the complete one-loop vertex function, including vacuum polarization and all counterterms, whose mass-shell matrix elements we have already calculated in Section 11.3. Using Eqs . (14 .2.26) and (14 .2.25), this contribution to the energy shift (14 .2.28) is given b y [~ENlhrgr,eriergy= ie dap' dip (ux (P')ri (P',EN,P,EN) U rrW} x 1{ P' -p). (14 .3.5) (This coul dhave been guessed at, by simp ly re placing the ,~P in the interaction of the e lectron w iththe external fiel dwith I,,.) As discussed in Section 1 4.1,because Zoc I we can approx imate theDirac wave func tion uIv in Eq . (14 .3.5) a s [Urr(P}lay_ Ua(P,6)[fN(R)I, (14.3.6) where fiv is the non-relativistic two-component wave function of an electron in the external Coulomb field, and u(P, a) is the four-component normalized solution of the momentum space Dirac equation (14.3.7) for spin z-component ar . Since u N(P)approximately satisfies the free- particle Dirac equation, Eq .(10.6.15)gives the general form of the matrix element of1-111,as FIMW ) LYE` + ri(P%P)IUrv(R) 14.3 The Lamb Shiji in Light Atoms 58 1 where q - p' - p . The wave functions uN(p) fall off very rapidly for Zam,, so we only need F I(92 ) and F 2(q2) in the limit Jq21 C rra2 . In this limit, Eqs .(11.3.31), (10.6.18), and (11.3.16) giv e 2 e2q~(!~)2 3 ~l {4 ~ ~ 24a~2nay In2+ 5-f- 4 , e m e 2) e2 02(q16rr~,7r2 22 ~'I n m ..2e~+5+4 1]Lit us first consider the contribution of the F'1 term in Eq. (14,x .8), which makes by far the largest contribution to the energy shift, and raises the most interesting problems in its calculation . For a pure electrostatic field with V = 0, Eqs . (14 .3.5), (14 .3.8), and {1 .3.9} giv e 2 2 [6ENIFi=`l24Tr2M2In22 +5~4e e xI d'pl f d'p UN(P) -ildo ('t- P)) 7 11(1 _ p)2 UN(P) (14.3.11) To ca lculate this con tributtnn, we may use the leading term i nthe non- relativist ic ma trix element (1 4.1.41), and f ind r e2 L6ENJF~ ~"247~2Y TZe XJ~3P1 orin pos ition space(14.3.9) (14.3.10) Jd3p f ' (e) eQe"(p' - p) [p' _ pl2fN(p)(14.3.12) 16ENIFeIn(EI)+2+3d3XfNI(X) [eV2_Q111(X)1fN(X) . 2 2 2 24nme me 5 4 In particular, for the Coulomb potential (14 .1.2), we have e~2S/°(x)= -Ze263(x), and the label Nconsists of a principal quantum number n and angular-momentum quantum numbers j, era, while (11 .2.41) gives [f`nj,v(4)]6 = 2(ZO(Mofn}~~26e,46,,"J4~. The energy shift (14 .3.11) is the n 2Z4~X5me 3~n3In?n2e2 3 54(14.3.1.4) (The lack of dependence of 6E on the total angular momentum z- component m is guaranteed by rotational invariance .) The term ~ in the brackets in Eqs . (14 .3.12) and {14 .3. 3} arises from vacuum polariza- tion, and yields just the energy shift calculated somewhat heuristically in Section 11 .2. 5$2 1 4Bound States in External Field s Before going on to calculate the magnetic and low-energy contributions to the energy shift, it is worth noting that the result we have ❑6tained so far yields a fair order-of- magnitude estimate of the Lamb shift without further work . We can anticipate that the low-energy terms will contain a term proportional to ln (ulftwith a coefficient such as to cancel the ~- dependence in Eq . (14 .3.12). The constant B here is an energy which must be included to make the argument of the logarithm dimensionless ; since it is the binding of the electron in the atom that will eventually provide our infrared cutoff, we may guess that B is a typical atomic binding energy, of order B -(Za)'rri, The total energy shift in a state Nwith principal quantum number n and orbital angular momentum Z7 is therefore of the form 6EN2Z'!Xsmc'~In(Z414) 6,., + O(l) ] For the 2s state of hydrogen, the logarithmic term alone gives(14.3.15) s 1 me dE2, ^- 12~ In la cy= 5.5x 10-' eV = 1300 MHz x 2~~t . As we shall see, the ` 0(1)' terms in Eq . (14 .3.15) will lower the total energy shift by about 25 %. Now let us consider the contribution of the F2 term in the matrix element of ri, which as we saw in Section 10 .6 may be interpreted as a radiative correction to the magnetic moment of the electron . Using Eqs. (14 .3.10), (14 .3.8), and (14 .3.6) in Eq . (14 .3.5), we find that this term gives an energy shif t [j E~Ta~~ ~2.r~2~~ J dip' r~'p(~tv(Pf ~ [Y', ~'L~IU~r( P)) Xe..CV'U (A,-P) (P, - P)v U4 .3.16) or in posi tionspace ie2 Id3x ON (X) 71UN(X)) e,~Fp, (X ) where For a pure electrostatic field with V- Q, this is W ie23 L6El1If.2 ~~( lYrirOI#~{~r(JiL)~ ❑[L''~~(~C)]32a~ rn~(14.3.17) (14.3.18) (14.3.19) In the non-relativistic limit L i <1, we can use the approximate result 14.3TheLamb Shift in Light Atom s (14.1.43), which here reads583 M.f.~vfN)-i(V fN'X () f'N-if (CTx v f1)]-(14.3.20) Using this in Eq.(1.3.5) and integrating by parts, this part of the energy shift is 2 327[2 M~2 E +2if~(x)a -(V(eWO(x)) x VfN(x))] (14 .3.21) Combining Eqs. (14.3.12) and (143 .21) gives the total high-energy contri- bution to the energy shift in an arbitrary electrostatic potential sa~ ~ [MN]high energy = e2 (j~2 ) +213X f1t 22r~~e me 5 24;T ie2 (14.3.22) BLow-Energy Ter m The low-energy contribution to the energy shift is obtained from the first three terms in Eq.(14.2.24), making the replacement in the photon propagator 0--i e k2 +M This substitution will eventually serve to cut off the integral over com- ponents of the photon four-momentum kat values of order p, but it is not possible to see this until we carefully take mass renormalization into account, so we shall defer making any non-relativistic approximations until then . Also, we are now including photon momenta as small or smaller than the binding energies of the atomic states, so we must treat the electrostatic forces responsible for this binding to all orders . Instead of working with the momentum space formula (14 .2.24), it will be convenient to return to the configuration space formula (142 .1$). This gives the low- energy contribution to the electron self-energy function a s X.F11X, A low energy :---d8'T'S ,si(xfy)~ pDlx + 6lYl, (P) 64(X ` )f 584 14 Bound States in External Field s where Dix - Y; fir) is the modified photon propagato r 1f f](x - Y ;p)- 2~~ d 4ke"")k21 ie ~ 1 '- IF(14.3.25) () ~ and the counterterms Z2(Y)-I and Srn(u) are calculated using this modified propagator . Converting from time to energy variables, the low-energy contribution to the function (14 .2.13) is the n ie2 [V,st(x,y; E)j low energy 2TZ4fA 1'PSAx, y;E- ko) yj, 1 ~ 1e (Z2(A) - 1) (y - V+iyOE +iey',91, +m,) AX - Y) The low-energy contribution to the energy shift is then given by Eq. (14 .2.17) a s I~EN]iowetter d'xId3.YRxW[Y_~(x,y;EA lowenergy UN(Y) _ -ie2 d4k d3x dayuN(x),IP S .a(x,y; err -V)y~UN(Y) (27r)4I ~ik.(x-Y) -bm'(p) dux FIN(X)UN(X) (14.3.27) Note that the terms proportional to Z2({U)-1 have dropped out because the Dirac wave function u .NW satisfies the Dirac equation Eq . (14 .1.10). For the electron propagator in the presence of the coulomb field, we use Eq. (14 .2.15): UM~XAM( Y) VM~~~VM(Y)S~q y; E) with the sums in the first and second terms running over all one-electro n and one-positron states, respectively . The k° integrals can be done most easily by closing the contour of integration with a large semi-circle in the lower half plans in the first term and in the upper half-plane in the second term : fdko(k2 +1 ( EN1±koi- ) i7t 1 14.3The Lamb Shift in Light Atoms 585 and likewise if pis replaced with zero . The energy shift (14 .3.27) now becomes e2 3[6 E N]lowenergy2(2~~-3dk M IMN(k)*FpMN(k)( Jkl(EM - EN + iC ) Z kl+ ~(EM - EN+ k +Y2 -i6) 1-rN(k)``~)Mrv(k)(Ikl(Em +Ear + I ke -ie} -bm,(P)f d3X ONMUNM , where MN(k) f d y e- RM(Y)Y'UN(Y)(14.3.28) (14.3.29) rPm,v(k) =[dy ~~~k~y N(3')Y~UN(Y). (14 .3.30) (Of course, the `sum' over M in Eq.(14.3.28) receives contributions only from electron states in the first term and positron states in the second term .) Eq . (14 .3.28) could have been derived more directly from old- fashioned perturbation theory ; the energy denominators E M- EN + (0 and E M+EN+ co are the result of subtracting the energy E Nof the initial state from the energy of an intermediate state consisting of either an electron of energy EM and a photon of energy co, or else a positron of energy EM, a photon of energy c), and both the final and the initial electron . (See Figure 14 .2.) Before making any approximations to Eq . (14 .3.28), it will be convenient to express the time components of the matrix elements ~'~~y and NN in terms of the corresponding space components, using relations' derive d ToderiveEq.[14.3.31),note tha t ,vf.,v(k)j d3X eW(X),/ UN ),O=4 Eq. (I43,32) is derived in the same way . 586 14Bound States in E xternalFields .......'N--- --------- Figure 14 .2. Old-fashioned perturbation theory diagrams for the low-energy part of the electron energy shift . Here solid lines are electrons ; wavy lines are photons ; the dashed line cuts through particle lines corresponding to the immediate states appearing in the first two terms in (14 .3.28) from the conservation of the electric curren t ki fiMv(k) =(EN + Em)r.°~~v(k). (14.3.32) Furthermore, by using the completeness relation (14 .1.8) it is straightfor- ward to show that E[~r' (k)l'+ and E[11WN(k)~'(Em -EN) MroMN{k}12] =1 r°~rr(k)12(Em +EN)1 ~-Fo*,Y(k) k - rMN(k)-rolw*N(k) k - rMN(k) -ik dux RN(X) YUv(X)=0, (14.3.34) the last step following from the parity condition (14 .1.23). In this way, 14.3The La mb Shiftin Light Atom, 58 7 Eq. (14 .3.28) can be rewritten [6E1V1low energy = e2 (IrMN(k)13-I~-IFMN(k )R21k2 2() M !I ( m NII ) (FMN(k)1' - Ik -rApv(k) 12/ (k' + y')) ( ~ I I (M NI I) (IuNk2 --Jk'r.WN(k)Iz1(k2+p2))-- Vk-F+- y-7 (EM + EN + V"k ~+W ) e22( ' I ( 2 , z}M + 6me(P) dux uN(x)UN(x) , (14 .3.35) In the next-to-last term, we have used the elementary integra l d3k I I ) =2y712 . (if- So far, this has been an exact rewriting of Eq . (14 .3.28). We must now invoke several approximations . First, consider mass renormalization . We have already calculated 6m,(,u) to order a in Section 11 .4; it is 2 bme(y) =2m,7r2e2JdX [I + x] In2 +(1 -X) i 4.3.36rra2x2( ) F Although in Section 11 .4 we regarded yas a regulator mass, to be taken much larger than m, we can just as well use Eq . (14 .3.36) to provide a vale for bm,(M) in the case that interests us here, y < m, In this limit Eq. (t4 .3.36) gives 6mAU} 2ow[1 21rm + ... {14 .3.37} We also recall that for Z a 1, UN(X) is given by Eq . (14 .1.37) a s I [ (I -a. v/2 +...)f N(X)(14.3,38 ) where dots indicate terms of higher order in hoc ; v is the non-relativistic velocity operator -aV/rn, ; and fi(x) is a two-component spinor solution 5 88 14 Bound States in External Field s of the non- relativistic Schrodinger equation, normalized according to Eq. (14 .1.8) so that fI.frr( X)IZ ~ ~ --- ~(v ')NN +... (14 .3.39) This gives the coefficient of 6me(y)in Eq . (14 .3.35) a s We note immediately that the leading term in --6m,(p)f dux uNUN cancels the term ap/2 in Eq. (14,3,35), Indeed, we could have anticipated this cancellation, because the term ocpJ2 in Eq. (14 .3.35) survives in the limit hoc--+ 0 , and the definition of m,(M) as the renormalized electron mass implies that there must be no energy shift in this limit . By the same argument we can anticipate that the term of order ap 2/me in 5m~(y) (which is larger than of order a(Z!X]4M" and therefore cannot simply be neglected) cancels the second and third terms in Eq . (14 .3.35), which are also of this vrder,'« On the other hand, the product of the ocp 2Ime The cancellation can be shown a s follows . We anticipate that the second and th ird term s in Eq. (14.13 5) are small enough sothat we they can beevaluated using the extreme nonrelativistic app roximation fluN(x) = u N{x) for the Dirac equation satisfied by u N{x).On the other hand, although the Coulomb force may be neglected in the positron wave fun ctions z m(x), the sum over M in the third term receives important contributions from relativistic positrons ,so we use the approximation v p,ff{x} 2~ (2n)-3~2 a(P,cr)eW' where a (p,ff) isthe po sitron spinor introduced insect ion 5 .5, normal ised so that v (P, d+)v(P, ¢) Thus the sums over Min the second and third terms ofEq.(14.3.35) are approximately given b y Ef'~M*N(k)f" ,+me M2Vfk + me6ij( 1#!e ~ ifil,- (' :)M 2 k f+ e To leading order in p{mr, the second and third terms in Eq. (14335) are then, respectively , 4me(2n)3 kka +i 2 2 k2+ ml 4gm" and e2 -1k2+ mP - me 2 207P ki k2 + P1 (Eq. (14.3,32) n.sles out t he possibi lity that a relativ istic correction to the lat ter exp ression m ight not be su ppresse dby the factor k2/M1 , tha tappears i nthis expression for IkI 2 < Mv} T hese two terms are cancel led by the term +3g2f4nine i n-bmr(p) f d'xiiv(x)u1V( x). Finally, we note tha t relativistic co rrections tothe above estimates for t he sums over positron states wou ld invo lve additionalfactors of r,,fcl sz (Za)" yielding con tribu tions of order a(Zx)2p2fine < o!(Zx)¢m" whic hjustifies t he non-rela tivistic approximations use dhere . 1 4.3 The Lamb Shift in Light Atoms 589 term in ~me with the second term in the matrix element (14 .3.40) is of order (Za)25cP2f Frt' <x(ZY)4,, and may therefore be neglected . To order a(Zoc)4m" the only remaining effect of mass renormalization is to leave us with the product of the leading term in 6m .,(p) with the term of order(ZOc)2 in f dux u NUN 871 2 1G,7r (This is the effect of mass renormalization on the electron kinetic energy, mentioned in Section 1 .3.) It turns out that this is oust the negative of what the first term in Eq . (14 .3.35) would he if we neglected the difference between energy levels . To see this, note that the integral in this term is effectively cut off at J k~ -p < Z arn,so we can evaluate the matrix element I'MN(k) in the limit k-+ 0 . To lowest order in Za, Eq . (14 .1.42) gives r~~~~~ ~ (V) IV (14.3.41) and using the completeness of the solutions fN of the non-relat ivistic Schrad inger equation ,we then hav e so to this orde r e2 2(27r~~r~MN (k)F MN(k)r'~~i O NN M A(~][MN(k)12 - Ik - rMN(k)12 /k2) k2 I-- Ik-IFMN(k) I'l(k'+u2)) k2+ M2 e22 (1..-k2/3(k2+/12,) Y() f 3kz k2'+'02 16ar(14.3.42) Thus after mass renormalization we are left with just the first term in Eq.(14.3.35), less the same with energy differences E N-Emdropped : 590 PE Nhowenergy14 B oundStatesin External Field s e 2(2ar)3M x (jrMN(k)j' - Ik -FMN(k)ll/(k2+ [j1))_- Again using Eq . (14 .3-41), this i s -e2 [6EN 1lowencrgy =2(Z7c)~~(~~EN)lVMrvl2 X A ~ 3( ) (14.3.44) though typical values ofthe electron momentum are much larger Even than typ ical atomic energy differences, this isnot true of typical values ofjkjin this integral , because the integral would be infrared divergent if we did not keep the E M - EN terms i n the denominators .The integral in Eq. (14 .3. 4)may be evaluated in the limit u>> jEm-EN~"-'(Z oc)2rraeby dividing the range ofintegration ofJklinto two segments, from zero to Aand from ).toinfinity, w ith~ chosenso thatI Em - EN~ ~, <cbut otherwise arbitrary . In t his way we fin d 1C* 0k'dk2 [3k2EA1 - EN+ k) - iiF ~--~k213(k2 +1U2) 2t2[iny - ) +I+inB(EN-E)3 21EM-ENl 6 The imaginary term here reflects the possibility of decay of the atom in state Nto states M ❑f lower energy . This term contributes to the decay rate, given bythe imaginary part of the energy shift . We are interested here in the real part of the energy shift, and so will drop this imaginary term in what follows .Eq.(14.3.44) now give s eY 'I6ENllowenergy -~772 (Em-ENY►'MN 1' In21E~v~EMI + 6 14.3 The Lamb Sh ift in Light Atom s C Total Energy Shift591 We need to make a connection between the sum in Eq . (4 .3.45) and the matrix element in the high-energy term (14 .3.22). For this purpose, let's first see what value the sum in Eq.(14.3.45) would have if we could ignore the logarithm . We note that (EM - EN)VNM = [v,H]rr , s o j:(EM - EN)IVMN12=' E ([v', HI jvM VMiN+VNi M [H, v ~]my) Al M i 2[p', [p', HI2m, ( ) NN The only term in the non-relativistic Hamiltonian H that does not com- mute with the momentum operator p is the potential term, -ev/°(x), so this gives T(EM-EN)fVnrrrvl '=-e 2~(V2dx))~v~r (14.3.46) Inspection of Eqs . (14 .3. 5) and (14 .3.22 now shows that the term propor- tional to In gin the high-energy term is cancelled bythe term proportional to In g in the low-energy term : 6 EN =P E N]highenergy + [~EN]1aw e nergy 2 2J:(Em - EN) ~ VMN Inme - +5-ii M(21E_N-Em~) e2 ( (14.3.47) So far, this has been for a general electrostatic field .4°(x) . Let us now specialize to a pure Coulomb field, wit h .sad°(x) =Ze/fix (14.3.48) In this case, Eq . (14 .3.46) read s 2 2m2(61(x)))NN e Af_ ~~2 (f(O)fNw)) .(14.3.49) e This is non-vanishing only for ~, = 0 .Also, the matrix element in the last term in Eq . (f 4,3 .47) has the valu e V(e,w°(x)) X P)NN-Ze(~~~~)NN ' (14.3.50) which is non-vanishing only for ~ =# 0, It is therefore useful at this point to divide our consideration between the two cases, e=0and ~'* 0. 592 i.e=01 4 Bound States in External Field s It is convenient here to define a mean excitation energy ❑EN IVMrrl2(Em-EN) InI rv - Eml =1n dE N IVMNl2(Em-Irv ) M M ~~2 1n AEN(f(O)fO) . (14 .3.51) For s-wave hydrogenic states, the label Nconsists of a principal quantum number n and spin z-component rrt, and [f'nm (O)l,_ 2(ZocmQ/n)~~'&,m/ ~4a, S o (Zm) 3(fO)fz(O))= 1 ae 7r n(14.3.52) Using Eqs . (14.3.51) and (14 .3.52) in Eq . {14.3.47} gives the energy shift in these states a s [6Ejn,(=o= 4a( 3n3meIn 2dEme LL+30 M,~~.a ii. f*D For non-vanishing orbital angular momentum the sum (14 .3.49) van- ishes, so the definition (14 .3.51) is inappropriate . Instead, it is conventional here to define a mean excitation energy ❑ENby 'InIEN-EmI=_2(Z17)4M,2AEN n (Z20(2me) (Because Eq . (14 .3.49) vanishes, it makes no difference what units are used to measure EN-EMinEq. (14 .3.54).} Also, in a state of total angular momentum j and orbital angular momentum ?, the scalar product a -L has the familiar value j(j+ 1) - ~( e+1) - ~, and for principal quantum number n the operator 1 /r3 has the expectation valu e 2Z3a3m3 (]( ) Putting this all together in Eq . (14 .3-47), we have for 0 : 4OG~ZOL)41~i3e ~~~~~'~ ~ 3 7rrt3In z2 LX 2 me It only remains to use these results to give numerical energy shifts . The mean excitation energies here mus t(14.3.53) (14.3.55) value for the becalculated 1 4.3 The Lamb Shift in Light Atoms 59 3 numerically ; using non-relativistic hydrogen wave functions, they have the values ❑E2V= 0.9704293186(3 )Ry, where I Ry =m,acZ /2 =13.6057 eV . Eq. (14 .3.53)thengives PE]is tbEl2s=4a~ ~ In Me + 19 = 3.3612 x 10-5eV 37r(2 ) AE1$ 3 0 _ 27rh x 81Z7 .4114Hz , (14.3.57) asmInme+~9 - 4 .2982 x10-5eV 67r 2AE2s TO = 27,hx1039.31MHz, (14.3.58) while Eq .(14.3.56)gives -In - - _-5.32G1X lu eOc 5me CCU ~e 1 -.8 61c 2dE2p 8 2,ah x - 12.88MHz . (14.3.59) The classic Lamb shift is the energy difference between the 2s and 2pi states of the hydrogen atom, states that would be degenerate in the absence of radiative corrections . Our calculation has give n [bE]2s.-[6E]2p,{z=4.35152 x 10-5 eV = 2,gh x 1052.19 MHz . This is numerically close (though analytically not identical) to the old result of Kroll and Lamb and French and Weisskopf,7 which they ob- tained using the techniques of old-fashioned perturbation theory . Earlier in this section we made a crude estimate of 1300 MHz by considering only the high-energy contribution to the 2s energy shift, with an infrared cutoff guessed to be of order a2 me= 2 Ry . We can now see that this was an overestimate, arising mostly from the fact that the true value of the effective infrared cutoff ❑E25 = 16 .64 Ry is considerably larger than we had guessed . On the other hand, as described in section 1 3, in 1947 Hans Bethe8was able to make a rather good estimate of the Lamb shift, 1040 MHz, by considering only the low-energy contribution to the 2s energy shift, with an ultraviolet cutoff guessed to be m ,(Bethe made the first estimate of the excitation energy, AE2SC--17.8 Ry .) The calculation of the Lamb shift described here has been improved bythe inclusion of higher-order radiative corrections and nuclear size 594 14 Bound States inExterraa! Field s and recoil effects . At present the greatest uncertainty is due to a doubt about the correct value of the rms charge radius rp of the proton . For rp = 0 .862 x 10-13 cm or rp = 0.$05x 10-1 1cm, one calculation' gives a Lamb shift of either 1057 .87MHz or 1057.85MHz, while anotherla gives either 1057.83 MHz or 10 57.865 MHz . Given the uncertainty in proton radius, the agreement is excellent with the present experimental value," 1057.845(9 )MHz . The accuracy of this experimental value is limited chiefly by the -100MHz natural line width of the 2p state in hydrogen, so further improvements here will be very difficult . In the last few years there has been an important improvement in measurements of the energy shift of the Is ground state itself, by direct comparison of the frequency of the is-2s resonance with four times the frequencies of the 2s-4 .y and 2s-4d two-photon resonances . These s and dstates are much narrower than the 2p state, so these frequency differences can be measured more accurately than the classic Lamb shift . For a brief while it seemed that there was a discrepancy here between theory and experiment . Calculationsla,13showed that for a proton radius rp= 0.$62(1 l .) x10-13 cm or 0.805(l 1) x 10-13cm, the inclusion of proton size and other corrections increases the theoretical is energy shift from the above result of 8127 .4 MHz to 8173 .12(6) MHz or 8172 .94(9) MHz, respectively . This result for a proton radius r . = 0,862{i1} x 10-1-1cm, which is believed to be more reliable, was not quite in agreement with the measured value13of 8172, 86{5} MHz . But a later calculation14 that adopts this proton radius and includes terms of order Y2 (~.!X)5yields energy shifts for the Is,2s,and 4s states in agreement with experiment . So apparently quantum electrodynamics wins again . Probl ems I. Consider a charged scalar particle of mass m 0, described by a field O(x) whose only interaction is with an external time-independen t electromagnetic field _Q/A(x) . Let TNbea complete set of normalize d one-boson or one-antiboson states with energies Fes, and defin e UN(X)e- ;ENt =(00,0(X, tK N) and VN(x)e~ENr =ON, 0 (X, 00o), wher e (Da is the vacuum . Show that the UNand v .N together form a complete set, and give formulas for the coefficients of u Nand VNin the expansion of a general function f (x) . 2. Suppose we include radiative corrections in the theory of Problem 1. Let iCi' (x, y)be the sum of all diagrams with one incoming and one outgoing charged scalar line (excluding final scalar propagators) to order a . Derive a formula for the shift in the energies E Nof the References 595 one-boson states due to these radiat ive corrections, in terms o€' UNW and II *(x,y) . 3. Use the results of Section 14 .3 to calculate the radiative decay rate ❑f the 2p states of hydrogen . 4. Suppose that the electron has an interaction with a light scalar field 0, of the form gOVW, . Suppose that the scalar mass rrao is in the range (ZI)2M,<rn'O<Zarn'. Calculate the change in the energy of the 1 sstate of hydrogenic atoms due to this interaction . 5. Carry out the calculation of Problem 4 for rno- 0. Reference s 1. P. A. M. Dirac, Proc .Roy. Soc .(London) A117, 610 (1928) . 2. See, e .g.,A. R. Edmonds, Angular Momentum in Quanturn Mechanics, (Princeton University Press, Princeton, 1957) ; M, E . Rose, Elementary TheoryofAngularMomentum (John Wiley & Sons, New York, 1957 ). 3. See, e .g., L. I. Schiff, Quantum Mechanics, (McGraw-Hill, New York, 1949) : Section 43 . The original references are C . G. Darwin, Procy. Roy, Soc . (London) A118, 654 (19 28); ibid ., A124, 621 (1928) ;W. Gordon, Zeit.fPhys. 48, 11 (1928). 4. G. E. Brown, J . S. Langer, and G . W. Schaefer, Proc . Roy . Soc . (London) A 251, 92 (1959) ; E . Brown and D . F. Mayers, Prat . Roy.Soc. (London) A 25 1, 105 (1959) ; A. M. Desiderio and W . R. Johnson, Ph y5.Rev.A3, 1267 (1971) . S. R. W. Huff, Phys . Rev .186, 1367(1969). 6. N. M. Kroland W E . Lamb, Whys .Rev.75, 388 (1949 ) 7.J.B.FrenchandV. F. Weissk opf,Phys. Rev .75,1240(1949 ) S. H.A.Bete ,Phys. Rev .72,339(1947) . 9. J. R. Sapirstein and D . R. Bennie, in Q uanturnElectrodynamics, e d. by I Kinoshita (World Scientific, Singapore, 1990) :p. 575, and references quoted therein . 10. H. Grotc h,Foundations of Physics 24, 249 (1994) . 596 14 Bound States in External Field s 11.S. R. Lundeen and F . M. Pipkin, Whys .Rev.Lett . 46, 232(198 1); S. R. Lundeen and F . M. Pipkin, Metrologia 22,9 (1986). For a review, see F . M. Pipkin, in Quantum Electrodynamics, ed.byI Kinoshita (World scientific, Singapore, 1990) :p.697. 12. M. Weitz, A . Huber, F . Schmidt-Kaleir, D . Lei fried, and T . W. Hanschs Whys .Rev. Lett . 72,328 (1994) . 13.M. Weitz, F . Schmidt-Kaler, and T . W. Hansch, Phys.Rev. Lett . C$, 1120 (1992), and Ref 1 1 14. K. Pachucki, Pays .Rev. Lett . 72,3154 (1994) . Author Inde x Where page numbers are given in italics, they refer to publications cited in bibliographies and lists of references . Aaron, R.470 Aharony,A. 168 Aitken, A. C.106 Amado, R. D.470 Anderson, C .D. 12, 30, 43,45 Ararnaki ,S.39 Argyres,P. C.257 Artie, E.424 Bagge r, J..338, 532 Bailey,J.498 Bakami jian,B.189 Borgmann ,V.106 Belinfan te, F .J.44, 316, 338 Herein ,F. A. 403, 404,424 Bergman n,P.G. 338 Bethe, H . 29, 33, 36, 45,47,560,563, 593,59 6 Beyer ,R. T.39,43 Bhabha, H . J. 29, 45 Bloch, F. 33, 46, 562 Bjorken, J.D. 532 Blackett,P. M.13 Bagoliuhav,N. N. 512, 53 2 Bohr, N . 3, 11, 19, 32, 42, 44, 168, 198, 256 Born, M . 3, 15,16,17,18,19, 23, 25, 40,42,43,46,115,166, 292 Breit,G. 36,48,162,167,168 de B roglie, L .3,40 Brown, G.E.596 Brown, L . 39, 40,45,46, I67, 53 2 Burgo yne, N.257 Butler, C . C. 30,45,123,167 Coo, T . Y. 39Carlson, J . F. 29, 3 0,32, 45 Cartan, E . 256 Casimir, H . B. G. 338 Cassen, B .167 Chadwick, J . 45 Chew, G . 47, 471 Chinowsky, Vii 167 Christenson, J . H. 106, 161 Cohen, R . S_44, 256 Collins, P . D.B. 471, 532 Compton, A . H. 41, 362, 364, 504 Condon, E . U. 167 Conversi, M .30,45 Crichton, J . 14. 189 Cronin, J . W. 106,132,167 curie-Joliat, I .45 Dalitz, R . H. 141, 168,563 Dancoff, S . M. 33, 34, 46 Darwin, C . G. 10, 4 1,42,596 Davissan, C. J.3,40 De Witt, C . M. 377, 418, 42 4, 471 Deans, W . M.41 Dedijer, S . 257 Deser, S.563 Desiderio, A . M.596 DeWitt, B . S. 39, 4 24 Dirac, P . A. M. 4, 5, 6,7, S, 9, 10, 11, 12, 13, 14, 18, 19, 22, 23, 24, 27, 28, 29, 32, 33, 34, 39, 4 0,41, 42, 43, 44, 46> 47, 49, 105, 200, 213, 218, 256, 292, 325, 3 28, 329, 33 0, 338, 345, 376, 4 24, 457, 470, 489, 5 65,566, 567,596 Donoghue, J . F. 533 597 598 Author Inde x Drell, S .D. 532 Dresden, M . 167 Drinkwater,J. W.47 Drixhl ,K.424 Dyso n, F.J. 37, 4 8, 106, 144, 168, 258, 259, 287, 291, 376, 499, 53 2 Eckert, 153, 156, 162, 16 5 Edmonds,A.R. 106 .167,168, 257, 596 Ehrenfest, P.15, 29,43,45 Einstein,A.12,13, 18, 19, 43,55, 518 Elsasser, W 3, 40 Epstein,S.46 Erick son,G. W.498 Euler,H. 32, 46, 523, 524, 526, 53 3 Fabri, E .168 Faddeev, L . D. 190, 376, 378, 424 Feenberg, E . 167,16$ Feinberg, G .167, 53 0, 533 Feinberg, J . 338 Fermi, E . 11,19, 23, 39, 42, 4 3, 44, 2 92 Ferretti, B . 167 Feynman, R . P. 33, 36, 37, 38, 47, 48, 259, 276, 280, 28 6, 353, 354, 355, 364, 375, 37 6, 377, 440, 411, 413, 417, 423, 424, 426, 430, 459, 472, 474, 48 6, 495, 559, 57 2 Fierz, M . 20, 40, 44, 46, 47, 257 Fitch, V . L.106, 133, 167 Flanders, H . 106, 375 Fock, V, 22, 23, 41,44, 375 Foley, H . M. 48 Frautschi, S.C.471, 562 Fredenhagen, K . 424 French, J . S. 35, 3 9, 47, 593, 596 Friedman, J .1. 106,167 Frohlich, H . 48 Froissart, M .168 Fukuda, H . 47 Furry, W . H. 23, 27, 28,32,44, 46, 428, $ 7Q Gaberdiel, R. 424 Garnow, G . 7, 39, 42 Garwin, R.106,167 Gasse r, J. 533Ge11-Mann, M .123,132, 167, 291, 463, 471, 55 6,Sfi3 Georgi, H . 257 Gerrner, L . H. 3, 4 0 Gerstein, 1. S. 424 Glimm, J . 423 Goldberger, M . L. I66, 463, 471, 55 6, 563 Goldstone, J. 178, 18 9 Gordon, W . 4, 7, 1 0, 13, 25, 27, 41, 42, 200, 211, 23 9, 277, 59 6 Goudsmit . S. 5, 10, 41 Green, M . B.375 Grisaru, M . 563 Grotch, H . 596 Gudehus, T . 257 Gursey, F . 106 Haag, R . 424 Hafstad, L . D. 167 Hahn, Y . 53? Halter, J . 533 Hansch, T . W. 596 Harvey, J . 533 Hawing, S . W 53 3 Heisenberg, W 3, 4, 10, 15, 16, 17, 20, 21, 23, 24, 25, 29, 31, 32, 33, 4 0,41, 42, 43 . 44 }45, 46, 109, 166, 2 72p 519, 532, 53 3 Heitler, W . 29, 45, 48 HePp, K . 512, 53 2 Herzberg, G . 45 Heydenberg, N .167 Hibbs, A . R. 423 Harking, J . G. I06 Hoddeson, L . 39, 40, 46, 167 Howard, J . C. 470 Hubbard, J, 461, 470 Huber, A . 596 Huff, R . W 596 Hugenholtz, N . M. 178,19 0 Infield, L .46 Inonii, E . 62, 105 Inoue, T .45 Israel, W . 533 1to,D.48 Author inde x J acki,R.40,424 Jackson,J.D.471 Jaffe, A.423 Jarnni er, M. 39 Bauch,J.M.257 Johnson, W .R.596 J Q11O L iF45 Jordan ,P. 3,10,15, 16, 17,18, 19 , 23, 25,40,41,43,44,292 Joos,H.257 Jost, R. 258 Jouvet, B. 470 Julian ,R. S.48 Kabir,P.530, 532 Kahn,B.48 Kaiser,G.257 Ka11en, G.457, 470 I{amef uchi, S. 424, 532 Kanesawa,S.48 Kemmer ,N.31,46 Kinoshita ,T.563, 596 Klein, 0 .4, 7, 13, 25, 27,29,41,44, 200, 21 1, 239, 277, 368, 37 5 Kola, Z .46,48 Kockel,B.31,46, 533 Kollath,R.16$ Kramers, H.A. 35,47, 81, 106, 1 68, 469,471 Kroll, N. M. 38, 47, 593, 596 de ECr onig,R. 469, 471 Kubo, R . 190 Kusch,P.48 Laidlaw, M . G. G. 418, 424 Lamb, W . E. 35, 36, 38, 47, 48, 593, 596 Landau, L . U. 106, 168, 418, 452 Lande, A . 41 Lang, S . 424 Langer, J .S.596 Latter, C . 30 Lederman, L . 106, 16 7 Lee, T .D. 106 , 127, 132, 167, 178, 189, 228,424,549,563 Lehmann, H . 438, 457, 463, 470 Liebfried, D . 596Leinaas, J . M. 424 Leutwyler, H. 533 Lewis,H. W. 46 Lifshitz, E . M. 106,168 Lippmann,B. 111, 166 Liu,H.H. T. 498 London, E 37 5 Low, F.E.291, 556, 563 Liiders, G. 245, 257 Ludwig, G .40 Lundeen ,S.R. 596 Lyubarski, G. Ya. 257 Mackey, G . W. 105 Mandela#am, S . 424 Marshak, R . E. 38, 45 Maskawa, T . 329, 330, 338 Matthews, P . T. 532 Mayers, D . F. 596 Mehra, J . 39, 4 2 Michelson, A . A. 41 Miller, A . I.39, 44, 4 98 1lriiyamoto, Y . 47 Miller, C . 29, 45, 4 6, 166 1Vf yrheirn, J . 424 Nafe, J . E. 48 Nagel, D . E.48 Nakajima, H . 329, 330, 338 Nappi, C . R. 257 Nash, C .106 Nauenherg, M . 549, 563 Neddermeyer, S . H. 30, 45 Ne'eman, Y .123,167 Nelson, E. S. 48 Newton, R . G.166 Nishina, Y .29,44, 3 68, 375 Nardheim, L . 30, 45 Nardsieck, A . 33, 46, 562 Novikov, V . 533599 Occhialini, G . P, S . 13, 3 0,45 Dhnuki, Y . 424 Omnes, R . 471 Oppenheimer, J . R. 12, 23, 27, 28, 29, 30,31, 35, 3 7, 38, 43, 44, 45, 46 Osterwalder, K .424 600 Author Inde x Pachucki, K . 596 Pais, A . 39, 13 2, 167 Pancini, E . 30, 45 Parasiuk, 0. 512, 532 Paschen, F . 41 Pasternack, S.47 Pauli, W 3, 11, 14, 19, 20, 21, 22, 23, 24, 26, 28, 31,40, 42, 43 .46, 47, 245, 257, 292, 494, 498, 51 7 Peierls, R . E. 44, 46, 47, 168 Pendleton, H . 563 Perlmutter, A .. 257 Petermann, A .498 Piccioni, 0. 30, 4 5,167 Pipkin, F . M. 596 Placzek, G .168 Podolsky, B . 22, 44 Polchinski, J . 424, 526, 533 Pomeranchuk, I . Ia. 468, 469, 471 Popov, V . N. 377, 4 24 Power, C . F. 30, 45 Present, R . S. 167 Rani,I. 1. 36, 48 Racah, G.45 Ram sauer ,165 Rarita, W . 232, 257 Rechenbe rg,H. 45 Regge, T . 468, 469, 471 Retherford,R. C.47 Richardson,Q. 47 Roberts, J . E.424 Robert son,H.P.42 Rochester, G.D. 30, 45, 123,167 Rnhrlich,F.257 Rose,M.E.46,106, 167, 168, 257,596 Rosenbluth ,M. N.457,470 Ros enfeld ,L.19,44,198, 256,338 Riiger ,S. M.424 Sakata, S . 45, 53 2 Salam, A, 39, 42, 512, 532 Salpeter, E . E.560, 563 Sapirstein, J. R. 596 Schaefer, G . W. 596 Schearer, J . F. 41 Schiff, L . I. 41, 43,168, 563, 596Schmidt-Kaler, F . 596 Schrader, R . 424 Schradinger, E . 3, 4, 6, 13, 27, 41, 109 Schroer, B. 257 Schubert, K . R. 167 Schur, I . 257 Schwarz, J . H. 375 Schweber, S .S.39,40 Schwinger, J. 14, 33, 35, 3 6, 37, 38, 40, 43, 46, 47, 48, 111, 166, 232, 257, 259, 291, 3 75,376,424, 470, 489, 498 Screatan, R .471 Seiler, R . 257 den, S.106 Berber, R .46, 48 Shifman, M . A.533 Skater, J . C. 42 Sommerfield, A. 3, 5, 6, 11, 41 Spaarnay, M . J.338 Stachel, J .44, 256 Steinberger, J . 167 Sterman, G . 563 Stevenson, E . C. 30 Stoner, E . C. 42 Stara, R . 39 Stratonovich, R . L. 461, 470 Streater, R . F.257, 258 Street, J . C. 30 Suura, H . 498, 56 2 Swieca, J . A.257 Syrnanzlk, K . 438, 464, 47 0 't Hooft, G . 377, 47 7, 424,498 Takahashi, Y . 447, 4 7 0 Tamrn, I . 12, 43, 44, 375 Tate, J . E.424 Tatti, T .48 Telegdi, V . L. I06,167 Thirring, W . 463, 471 Thomas, L . H. 42, 189 Thomson, J . J. 369 Titchmarsh, E . C.190 Toll, J . S.471 Tomonga, S .-I. 36, 37, 38, 40, 46, 47, 48,259,376 AuthorIndex Townsend 165 Tung, VLF -K .106 Turlay, R .106, 167 Tumhull, H . 106 Tuve, M . A.167 Ueh ling,E.A.34,46,47,484,498 LThlenheck ,G.E.5,41 Urnezawa , H.532 Vainshtein, A . I.533 Vaughn, M . J. 4 70 Selo, G . 257 Veltman, M . 477, 498 Villars, F . 494, 498 van der Waerden, B . 1. 40,43, 257 Waller, I, 31, 4 5 Ward, J .C. 447, 47 0 Watson, K . M.130, 166, 167 Webb, N. 45 Weinberg, S . 40, 42, 47, 103, 16 7,168, 190, 25 6, 257, 338, 375, 424, 470, 471, 532, 533, 56 3 Weinrich, M .106, 16 7 Weisskopf, V . F. 24, 2 6, 28, 31, 32, 34, 35, 37, 4 0, 43, 44, 45, 46, 47, 593, 596 Weitz, M . 596 Wentzel, G . 40, 42, 46 VVess, J . 335, 532601 West, T . 375 Weyl, H . 12, 43, 37 5 Wheeler, J . A. 4, 33, 4 6, 47, 166 Whittaker, E . 40 Wichmann, E . H. 159, 256, 49 8 Wick, G . C. 105, 291, 475, 486, 497, 498 9Vightinan, A. S.105,257, 258 aligner, E . P.19,39, 4 2, 43, 44, 51, 62, 68, 91, 93, 10[}, 1 03, 104, 105, 106, 108, 15 3, 156, 162, 1 68, 236 Wilczek, F . 424 Williams, E . J. 563 Williams, R . C. 47 Williams, E . 47 Wilson, W . 41, 52 5,528, 533 Witten, E.40,257, 37 5 Wu, C . S.106, 127, 16 7 Yang, C . N.146, 127, 132, 167, 168, 178, 184, 228, 257, 37 5 Yang . C. P. 168 Yennie, D . R. 562, 596 Yukawa, H . 30, 45, 159, 436, 470 Zacharias, J . R. 48 Zakharov, V . I. 533 Zimmerman, 438, 463, 470,512, 532 Zurnino, B . 257 Zwanziger, D . 257 Subject Index abefian groups, defined, 55 absorption of photons, 18 accidental symmetries, 529-31 action, 299, 3 07 alpha decay, 16 0 anomalous Seeman effect, 5 annihilation and creation operators , 16, 19- 20, 23-4, 26-8, 169, 173 7 antiparticles, 13-14, 2 3-8, 104, 149-50, 199,56 7 antiunitary and antilinear operators, defined, 5 1 anyans, 4Z Q auxiliary fields, 302-3, 314 axial gauge, 34 6 baryon number, defined, 122 Belinfante tensor, 316 Berez-in integration, defined, 403 beta decay, 23, 29, 127, 146, 2 28, 519 Bhabha (electron-positron) scattering, 29 Boltzmann H-theorem, 151 boasts, defined, 61 Born approximation, 115, 156 Bose-Einstein statistics, 11, 172, 418- 20 BPHZ prescription, 512-13 braid group, 420 Bettie-Saipeter equation, 560 bound states, see composite particles Breit-Wigner formula, 1 62-3 bremstrahlung, 29 broken symmetry, 443, 451canonical commutation and anticflM- m utation relations, 16,19-22, 293- 8, 52 9 canonical transformations, 329 Casimir effect, 29 7 causality, 145, 1 98, 463 center-of-mass frame, 139 central charges, 8 3 charge, see electric charge charge conjugation (C) accidental symmetry, 521,530-1 defined, X21, 131- 2 for photons, 427-8 intrinsic charge conjugation phases, 131 non-conservation, 132 transformation of creation operators, 177 transformation of Dirac fields, 226-7 transformation of ferminn bilinears, 229 transformation of general irreducible fields, 241- 2 transformation of scalar fields, 2 06 transformation of vector fields, 213 also see specific particle type s Chew-Frautschi plot, 469, 471 chiral transformation, 520 circular polarization, 359 Clebsch-Gordan coefficients, 124, 152 , 154,156,233-4,242,569 Clifford algebra, 21 4 closed p-forms, defined, 369 cluster decomposition principle, 169, 177-89, 197, 25 9 602 Subject Inde x C06° decay, 127,130-1 coherent states, 199 color, 54 9 compact and non-compact groups, 231 composite particles, 110, 461-2 Compton (electron-photon) scattering, 29, 362-9 connected amplitudes, 17 8-82, 270, 282, 286,389,41 3 conservation law s for angular momentum, 118 for charge, 11 9, 199, 427 for current, 212, 307, 47$, 58 6 for energy and momentum, 117-18, 425- 7 limitations, 253, 537- 8 also see specific symmetries and con- served quantitie s constraints, 325-3 1 in electrodynamics, 344, 346-7 cosmic rays, 29, 123 Coulomb gauge, 25 1, 346r50, 365 Coulomb energy, 350, 353, 355-6, 560 C P-invarianc e for degenerate rnultiplets, 104 ion-conservation in K°-k° decay, 132- 3 counterterms in quantum electrody- namics, 472- 3 CPT-invariance, 104, 13 3, 244-6, 459 creation operators, see annihilation and creation operator s cross sections defined, 137- 9 high energy limit, 158-9 partial wave expansions, 155-6 crossing symmetry, 269, 467, 554 cumulants, 17 8 Dalitz plot, 141 dangerous states, 550- 2 decay rates, general formula, 13 6r7 Abaryon, 165 d+ function, 202 ❑Ffunction, 27 6 de Itham cohomology, 370603 density matrix, 36 0 differential forms, seepµforrns diffraction scattering, 148-9, 158 dimensional regularization, 449, 477- 80,49 7 dimensionality, of fields and couplings, 502,519,525- 7 Dirac brackets, 329-31, 332-7 in electrodynamics, 3 47-9 Dirac equation, 1, 6-14, 225,565- 72 Dirac matrices, 8-9, 214-1 9 slash notation, defined, 358 traces, 361, 372- 4 Dirac representation of homogeneous Lorentz group, 213-1 9 dispersion relations, 460, 462-9 distorted wave Born approximation, 146- 7 dotted and undotted indices, 230 duality, 232, 37 1 Dyson series, 144, 259-6 0 eclipsing binaries, 368 effective field theories, 499, 523-5 electric charge, 34 1 conservation, 122, 537 renormalization, 342, 442-8, 473, 480- 3 electric charge radius, defined, 493 electric dipole moments, 81, 521 electro n charge radius, 49 3 classical theory, 31, 36 9, 496 magnetic moment, 6, 14, 14, 3 6, 457, 468, 520 spin, 6-9 elliptic polarisation, 36 0 energy shifts of atomic states, 3 1-2, 57 8 1s energy shifts, 59 4 also see Lamb shift, Ueh]ing effect, Muonic atom s energy-momentum tensor, 31 12 entropy, 151 equivalence principle, 537 exact p-forms, 36 9 604 Subject Index excitation energies, in hydrogen, 592-3 exclusion principle, 1 1 exterior derivatives, 36 9 external fields, 266, 287-94, 412-13, 556-62,572- 8 Euclidean path integrals, 384 Euler constant, 479, 497 Euier- Lagrange equations, 30 0 Faddeev equations, 18 8 Fermi-Dirac statistics, 12, 1 71-2, 267- 70,418-2 0 Feynman diagrams, 3 6-7, 259-91 for electrodynamics, 355-8 Feynman gauge, 355, 417 Feynman parameters, 474, 486, 497 fields, seequantum field s field equations, 20{], 211-12, 239 ;also see quantum field s field renormalization, 331-2, 436-42, 452,461,473,479,484,543-4 field-translation-invariant scalar theory, 521- 3 fine structure, 4-6, 57 0 fine structure constant, 2, 5 first class constraints, see constraints flavors (of leptons), 529 floating cutoff, 52 5-8 forests, 5 12-13 form factors, 452-7, 485-93, 580-2 Froissart bound, 1 59 functionals, notation, 299 Furry's theorem, 428-9, 50 9 Galilean invariance, 62, 145 js, 217-1 8 Gaussian integrals, 420-3 Grossmann variables, defined, 401 gauge transformations, 251-2, 3393 , 345,370,448-52 also see Lorentz gauge, Coulomb gauge, temporal gauge, axial gauge, unitarity gauge, Feynman gaug e general relativity, 255, 31 2, 316, 369, 518- 19, 52 1 generators of symmetries, 307-14global symmetries, defined, 3 07 gluons, 54 9 graviton, 73-4, 253, 521, 537, 548 groups, defined, 52;also see abelian groups, hnrnotnpY groups, Lie groups, little groups, representa- tions, semi-simple group s Hamiltonian, for complex scalar field, 22 for Dirac equation, 8 for electradynarnics, 349-50 for free particles, 17 6 for interacting Dirac field, 3 23 for interacting scalar fields, 199, 3 02 for interacting vector field, 32 1 for one-dimensional scalar field, 15- l7 Heisenberg picture, 1 09, 288, 297, 425 helicit y defined, 7 2 limitations for massless particle fields, 253- 4 limited to integers and half-integers, 90 Hilbert space, defined, 49 hole theory, 12, 29, 3 1 homotopy groups and classes, 89-90, 96-100,419-2 0 defined, 1U 0 Hubbard-Stratonovich transformation, 461 hypercharge, 123 hyperons, f 23 discovered, 3 0 `in' and `out' states, 1 07-12 , 116 induced emission of photons, 18 induced representations, 6 5 infinities, see ultraviolet divergences, infrared divergence s infrared divergences, 32-3, 491, 496, 539-53 Inbnii-VVigner contraction, 62 interaction picture, 143, 28 7 for derivative coupling, 318-2 0 for Dirac field, 323-5 SubjectIndex for electrodynamics, 35 3 for interacting scalar field, 304-5 for vector field, 320- 3 internal symmetries, 121- 2 invariant subgroups and subalgebras, 70, 8 8 irreducible representations, defined, 64 irrelevant couplings, 503 isospin (isotopic spin), 12 3 J/V meson, 225 Jacobi identity, 83 jets, 550, 55 2 K mesons, 12 3,132-3 discovered, 3 0 Kallen-Lehmann representation, 457- 62 kinetic theory, 15 0-1 Klein-Gordon-Schr6dinger equation, 1, 13, 2 1 Kramers degeneracy, 81 Kramers-Kronig relation, 46 9 Lagrangians, 24, 297-30 6 for complex scalar field, 21 for interacting scalar field, 302 integration by parts, 305-6 irregular, 32 6 Lamb shift, 34-6, 484, 578-94 Lee-Nauenberg theorem, 54 9-53 Legendre transformations, 297, 3 01 lepton number, 122, 530 leptons, 47 2 Lie groups, 53-5 linear polarization, 359 Lippmann-Schwinger equation, 111, 142 little groups, 64-6 m 7~ 0 : 68- 9 = 0: 69-73 local symmetries, 342 defined , 64 for di fferent momenta, 65-6 formasslessparticle s, 24 8 also seegauge transformations loops ,186-7,270,282-3,358, 413605 Lorentz (or Landau) gauge, 212, 346, 418 Lorentz transformation s action on creation operators, 177 action on general states, 5 7 action on one-particle states, 62-73 defined, 55-7 homogeneous Lorentz group, defined, 57; general representations, 229-33 in canonical formalism, 314-1 8 in off-shell matrix elements, 42 7 in perturbation theory, 145, 277-9, 259 of cross-sections and decay rates, 138 of 5-matrix, 11 6-21 Poincare algebra, 58-61 Paincare group, defined, 5 7 proper orthochronous Lorentz group, 58 LSD theorem, see reduction formul a M-matrix, defined, 11 7 magnetic moments, of spin 1/2 par- ticles, 45 6-7, 485-9 0, 520, 555- 6 also seeelectron, moa n Majorana fermions, 226, 242 Mandelstam variables, 515 many time formalism, 22 marginal couplings, 50 3 mass renormalization, 4392, 473, 495, 587- 9 matrix mechanics, 3-4 , Maxwell equations, 1, 252, 339, 342, 370 Moller (electron-electron) scattering, 2 9 momentum operator, 61, 310-11 moon, 3 8 discovered, 3 0 magnetic moment, 489-90, 52 0 muonic atoms, 483- 4 N14 molecular spectrum, 29 negative energy `states', 10-14, 23-7, 567 neutron, 2 9 neutron-proton scattering, 158 606 Subject Inde x neutron scattering on complex nu- clei, 157, 16 0 Noether's theorem, 30 7 non-compact groups, seecompact and non-compact group s non-linear a-model, 337, 377, 3 92-3 normal ordering, 175, 200, 262 nuclear forces, 29-30, 434- 6 atd-fashioned perturbation theory, see perturbation theor y w meson, 2 27 one-particle-irreducible, defined, 439 optical theorem, 14 8 QSterwalder-Sc hrader axioms, 384, 424 overlapping divergences, 5 11-15 p-forms, 3 69-72 pair production, 29 paCaSti#t15tIC5, 420 parity and space inversion (P) accidental symmetry, 521, 5 30-1 defined, 5 8 intrinsic parities, 125- 7 non-conservation in weak inter- actions, 12 7 of bound electron states, 568-9,586 transformation of creation operators, 177 transformation of Dirac fields, 221, 224- 5 transformation of general irreducible fields, 239-4 0 transformation of J .,and P,,, 74-6 - transforrnation of one-particle states, 76r9, 103 transformation of scalar fields, 205 -6 transformation of vector fields, 213 also see specific particle type s partial wave expansions, 151-9 path integrals, 259, 37 0r7, 524 derivation, 378-8 4 for derivative coupled scalars, 392-2 for fermions, 399-41 3 for massive vector fields, 393-5 for non-linear (7-model, 392-3for quantum electrodynamics, 413 18 for S-matrix, 385-9 Lagrangian version, 389-9 5 used to derive Feynman rules, 395-8 Pauli matrices, defined, 21 7 Pauli term, 14, 517, 5 20 Pauli-pillars regulator, 486, 494 perturbation theor y old-fashioned, 142, 259, 549, 564, 585, 59 3 time-dependent, 142- 5 also seeDyson series, Feynman dia- grams, path integral s phase shifts, 129-30, 155 phase space factors, 139-41 photon, 3 charge conjugation phase, 229 helicity and polarization, 73-4, 25{}- 1, 359 -61, 368 masslessness, 343, 452, 477 photon-photon scattering, 32, 509, 523- 5 propagator, 3 53-.5 also see soft photons, quantum elec- trodynamics , pion-nucleon coupling constant, 435 pions, 13, 38, 435-6, 52 1 71t decay, 127 7C° decay, 13 1 intrinsic charge-conjugation phase of 7z",131, 229 intrinsic parity, 127 prediction and discovery, 30 scattering on nucleons, 463, 46 9 Paincare group and algebra, see Lorentz transformations Paincare's theorem, 99, 370 Poisson brackets, defined, 32 7 polar decomposition theorem, 88, 53 0 polarizatio n massive spin one particle, 2 09--14, 212 masslessness, 343 photon, 73- 4 Polchinski's theorem, 526-8 Subject Index poles in sc attering amplitude s, 428-36, 554, 56 4 Pomeranchuk's theorem ,468 posit ivity of energy ,75-6,302 positron ,12-13, 5 67-5,585 positronium, 227 power counting theorem ,505 primary constraint s,seeconstraints principal value function, 113 probab ilities in quantum mechanics, 7 , 27-8, 33, 5 0 projective representations ,53,$1-91, 9b-lOD propagat ors,262,274r$0, 397-8, 506, 574-8 proton charge, 44 5 charge radius, 38, 594 form factors, 457 identified as holes, 12 in nuclei, 2 9 PT-non- conservation, 130- 1 quantum chromodynamics, 123, 531, 549 quantum electrodynamics, 2 9,31-8, 339-62, 413-18, 472-97, 503,507- 15,529-31,564-9 4 also see quantum fields, electron, photon, gauge invariance, ultra- violet divergences, infrared diver- gence s quantum fields, Dirac, 23-4, 219-29 early theory, 1 5-31 free fields, 191-20 0 general quantum fields, 233-44 massless particles, 246-55 redefinitions, 331- 2 scalar, 21-2, 24--8, 201-6 uniqueness of irreducible fields, 238 vector, 207-1 2 quantum mechanics, 49-50 quarks, 217, 53 1 radiative corrections, seerenormaliza- tion, quantum electrodynamics,607 ultraviolet divergences, infrared divergences, self-energy functions, vertex function, vacuum polariza- tion, soft photon s Ramsauer-Townsend effect, 1 65 ltarita-- chwinger field, 232, 423 rates, general formula, 134-6 rays, defined, 49-5 0 reduction formula, 43 6--8 redundant couplings, 331-- 2, 522-3 Regge trajectories, 468-9 relativistic wave equations, 1-14 relevant couplings, 53 0 renormalizability, 37, 499, 502-3, 516- 25 renormalization, 34-8, 506r16 also seeelectric charge, field renor- rnalization, mass renormalization representation s of groups, defined, 5 3 of homogeneous Lorentz group, 229- 33 renormalization group, 490, 516, 525 resonances, 15 9-f5 p-mesons, 165, 225, 227, 469 Riemann-Lebesgue theorem, 181 Rosenbluth formula, 457 rotation matrices (D~rn(A)), 68 Rydberg unit (Ry), defined, 59 3 5-matrix, 4, 33 C. CP and CPT, 131-4 defined, 11 3 internal symmetries, 121-4 Lorentz invariance, 11 6r21 parity, 124- 7 PT, 13 0--1, 133 S-operator, 11 4 time reversal, 127-30 unitarity, 113-14, 147-51 also seeT-matrix, M-matri x scattering amplitude (f), 148, 153-4, 466- 7 scattering lengths, defined, 157 Schrodinger picture, 1 09 Schwinger action principle, 288 SuNect Inde x Schwinger terms, 44 9 second class constraints, see constraints secondary constraints, see constraints self-energy functions (II', E'), 439 , 473-80, 493-6, 508-9, 512- 15, 575-9, 58 3 semi-simple Lie groups and algebras, defined, 70, S b separable interactions, 1 65 Shelter Island Conference (1947), 34-8 simply connected spaces, defined, 84 SL(2, C) group, 87, 9 0 soft photons, 534--48, 55 3-6 space irsion, see parity and space inversio n spectral functions, see Kallen-Lehmann representatio n spherical hamianios ; 153, 569 spin matrices, defined, 23 0 spin-statistics connection, 23 8 spin sums, 210, 224, 2307, 252, 3 60-1, 365, 54 5 Spin(d) group, 90 spontaneous emission of photons, 17- 19, 59 0 standing wave states, 166 strangeness and strange particles, 123 see K mesons, hyperon s string theory, 1, 15, 244, 37 1, 525 strong interactions, 3 0 structure constants, defined, 54 SU(2) group, 8$, 123, 13 0 SU(3) symmetry, 12 3 subtractions in dispersion relations, 460, 465 superficial divergences, 5 00-2, 510 superrenormalizable interactions, 503, 507 super selection rules, 53, 9 0-1 supersymmetry, 325, 5 06, 52 6 symmetries, 5 0-5, 91-6, 306r14, 42 5 also see conservation laws, groups, Lorentz invariance, parity, charge conjugation, time reversal, internal symmetries, isaspirt, S U(3)T-matrix, 111, 11 6, 141-2, 152, 549 'r-0 problem, 12 7 temporal gauge, 346 Thomson scattering, 369, 55 6 threshold behavior, 157-$ time-ordered products, defined, 143, 280 time reversal (T ) consequences for S-matrix, 127-3 0 defined, 5 8 non-conservation, 134 transformation of creation operators, 177 transformation of degenerate multi- plets, 100- 4 transformation of Dirac fields, 227- 8 transformation of general irreducible fields, 242- 3 transformation of Jugand P., 74-6 transformation of one-particle states, 77-8 1 transformation of scalar fields, 2 06 transformation of vector fields, 213 topolog y of Lorentz and rotation groups, 86- 90 also see homotopy groups, simply connected spaces, de Rham co- homolog y traces, seeDirac matrices tree graphs, defined, 283 two-cocycles, defined, 8 2 Uehling effect, 34, 484, 58 1 ultraviolet divergences, 32-8, 47 6--7, 482-3, 485, 491, 497, 505 unitarity gauge, 34 6 unitarity of S'-matrix, 113-16,129, 147-- 51, 155, 161, 52 1 universal covering group, 9 0 vacuum energy, 2 4,26 vacuum polarization, 32,34,473-85, 581 vacuum state, 24, 27, 176 Subject inde x vertex function {I }, 446, 488 ,507- 8,579-80 wave funct ion,386r8 W± particles, 20 7 Ward and Ward-Takahashi identities, 445-8, 476, 50$, 51 1 Watson's theorem, 130 wave pickets, 109-10 Nick rotation, 475-6609 Wick's theorem, 261 aligner--Eckart theorem, 153 Wigner rotation, 68-73 VVigner three- j symbols, 23 7 Wilson's renortnalizatian method, 525- 8 Z° particle, 159-60, 207 Z2group, $8