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A doctoral thesis by Alejandro Jenkins, advised by Mark Wise, defended at Caltech in May 2006 (arXiv hep-th/0607239). It covers massless spin-1 and spin-2 particles, gauge and diffeomorphism invariance, the Weinberg-Witten theorem, Goldstone photons and gravitons, phenomenology of spontaneous Lorentz violation, dark energy and anthropic arguments, and the reverse sprinkler problem. It is someone else's work held in the archive, not by Phil.

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arXiv:hep-th/0607239 v1 29 Jul 2006Topics in Theoretical Particle Physics and Cosmology Beyond the Standard Model Thesis by Alejandro Jenkins In Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2006 (Defended 26 May, 2006) ii c∝ci∇cleco√y∇t2006 Alejandro Jenkins All Rights Reserved iii Para Viviana, quien, sin nada de esto, hubiera sido posible ( sic) iv If I have not seen as far as others, it is because giants were st anding on my shoulders. — Prof. Hal Abelson, MIT v Acknowledgments How small the cosmos (a kangaroo’s pouch would hold it), how p altry and puny in comparison to human consciousness, to a single indiv idual recollection, and its expression in words! — Vladimir V. Nabokov, Speak, Memory What men are poets who can speak of Jupiter if he were like a man , but if he is an immense spinning sphere of methane and ammonia must b e silent? — Richard P. Feynman, “The Relation of Physics to Other Scien ces” I thank Mark Wise, my advisor, for teaching me quantum ﬁeld th eory, as well as a great deal about physics in general and about the professional pra ctice of theoretical physics. I have been honored to have been Mark’s student and collaborat or, and I only regret that, on account of my own limitations, I don’t have more to show for it. I thank him also for many free dinners with the Monday seminar speakers, for his p atience, and for his sense of humor. I thank Steve Hsu, my collaborator, who, during his visit to C altech in 2004, took me under his wing and from whom I learned much cosmology (and wit h whom I had interesting conversations about both the physics and the business world s). I thank Michael Graesser, my other collaborator, with whom I have had many oppor- tunities to talk about physics (and, among other things, abo ut the intelligence of corvids) and whose extraordinary patience and gentlemanliness made it relatively painless to expose to him my confusion on many subjects. I accuse D´ onal O’Connell of innumerable discussions about physics and about such topics as teleological suspension, Japanese ritual suicid e, and the diﬀerence between white wine and red. Also, of reading and commenting on the draft of C hapter 2, and of quackery. I thank Kris Sigurdson for many similarly interesting discu ssions, both professional and unprofessional, for setting a ridiculously high standard o f success for the members of our vi class, and for his kind and immensely enjoyable invitation t o visit him at the IAS. I thank Disa El ´iasd´ ottir for many pleasant social occasions and for loudl y and colorfully supporting the Costa Rican national team during the 2002 Wor ld Cup. ¡Ticos, ticos! I also apologize to her again for the unfortunate beer spillin g incident when I visited her in Copenhagen last summer. I thank my oﬃcemate, Matt Dorsten, for patiently putting up w ith my outspoken fond- ness for animals in human roles, for clearing up my confusion about a point of physics on countless occasions, and for repeated assistance on comput er matters. I thank Ilya Mandel, my long-time roommate, for his forbeara nce regarding my poor housekeeping abilities and tendency to consume his supplie s, as well as for many interesting conversations and a memorable roadtrip from Pasadena to San Jos´ e, Costa Rica. I thank Jie Yang, with whom I worked as a teaching assistant fo r two years, for her superhuman eﬃciency, sunny disposition, and willingness t o take on more than her share of the work. I thank various Irishmen for arguments, and Anura Abeyesing he, Lotty Ackermann, Christian Bauer, Xavier Calmet, Chris Lee, Sonny Mantry, Mi chael Salem, Graeme Smith, Ben Toner, Lisa Tracy, and other members of my class and my res earch group whom I was privileged to know personally. I thank Jacob Bourjaily, Oleg Evnin, Jernej Kamenik, David M aybury, Brian Murray, Jon Pritchard, Ketan Vyas, and other students with whom I had occassion to discuss physics. I thank physicists Nima Arkani-Hamed, J. D. Bjorken, Roman B uniy, Andy Frey, Jaume Garriga, Holger Gies, Walter Goldberger, Jim Isenberg, Ted Jacobson, Marc Kamionkowski, Alan Kosteleck´ y, Anton Kapustin, Eric Linder, Juan Maldac ena, Eugene Lim, Ian Low, Guy Moore, Lubos Motl, Yoichiru Nambu, Hiroshi Ooguri, Kris hna Rajagopal, Michael Ramsey-Musolf, John Schwarz, Matthew Schwartz, Guy de T´ er amond, Kip Thorne, and Alex Vilenkin, for questions, comments, and discussions. I thank Richard Berg, David Berman, Ed Creutz, John Dlugosz, Lars Falk, Monwhea Jeng, Lewis Mammel, Carl Mungan, Frederick Ross, Wolf Rueck ner, Tom Snyder, and the other professional and amateur physicists who commented on the work in Chapter 6. I thank the professors with whom I worked as a teaching assist ant, David Goodstein, Marc Kamionkowski, Bob McKeown, and Mark Wise, for their pat ience and understanding. vii I thank my father, mother, and brother for their support and a dvice. I thank Caltech for sustaining me as a Robert A. Millikan grad uate fellow (2001–2004) and teaching assistant (2004–2006). I was also supported du ring the summer of 2005 as a graduate research associate under the Department of Energy contract DE-FG03-92ER40701. viii Abstract We begin by reviewing our current understanding of massless particles with spin 1 and spin 2 as mediators of long-range forces in relativistic quantum ﬁeld theory. We discuss how a description of such particles that is compatible with Loren tz covariance naturally leads to a redundancy in the mathematical description of the physics , which in the spin-1 case is local gauge invariance and in the spin-2 case is the diﬀeomor phism invariance of General Relativity. We then discuss the Weinberg-Witten theorem, w hich further underlines the need for local invariance in relativistic theories with mas sless interacting particles that have spin greater than 1/2. This discussion leads us to consider a possible class of mode ls in which long-range in- teractions are mediated by the Goldstone bosons of spontane ous Lorentz violation. Since the Lorentz symmetry is realized non-linearly in the Goldst ones, these models evade the Weinberg-Witten theorem and could potentially also evade t he need for local gauge invari- ance in our description of fundamental physics. In the case o f gravity, the broken symmetry would protect the theory from having non-zero cosmological constant, while the composite- ness of the graviton could provide a solution to the perturba tive non-renormalizability of linear gravity. This leads us to consider the phenomenology of spontaneous L orentz violation and the experimental limits thereon. We ﬁnd the general low-ene rgy eﬀective action of the Goldstones of this kind of symmetry breaking minimally coup led to the usual Einstein gravity and we consider observational limits resulting fro m modiﬁcations to Newton’s law and from gravitational ˇCerenkov radiation of the highest-energy cosmic rays. We co mpare this eﬀective theory with the “ghost condensate” mechanism , which has been proposed in the literature as a model for gravity in a Higgs phase. Next, we summarize the cosmological constant problem and co nsider some issues related to it. We show that models in which a scalar ﬁeld causes the sup er-acceleration of the ix universe generally exhibit instabilities that can be more b roadly connected to the violation of the null-energy condition. We also discuss how the equati on of state parameter w=p/ρ evolves in a universe where the dark energy is caused by a ghos t condensate. Furthermore, we comment on the anthropic argument for a small cosmologica l constant and how it is weakened by considering the possibility that the size of the primordial density perturbations created by inﬂation also varies over the landscape of possib le universes. Finally, we discuss a problem in elementary ﬂuid mechanics t hat had eluded a deﬁnitive treatment for several decades: the reverse sprinkler, comm only associated with Feynman. We provide an elementary theoretical description compatib le with its observed behavior. x Contents Acknowledgments v Abstract viii 1 Introduction 1 1.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Massless mediators 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Unbearable lightness . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Polarizations and the Lorentz group . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 The little group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Massive particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 2.2.3 Massless particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 The vector ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 2.3.1 Vector ﬁeld with j= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Vector ﬁeld with j= 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 Massless j= 1 particles . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Why local gauge invariance? . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.1 Expecting the Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.2 Further successes of gauge theories . . . . . . . . . . . . . . . . . . . 21 2.5 Massless j= 2 particles and diﬀeomorphism invariance . . . . . . . . . . . . 22 2.6 The Weinberg-Witten theorem . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.1 The j >1/2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.2 The j >1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 xi 2.6.3 Why are gluons and gravitons allowed? . . . . . . . . . . . . . . . . 25 2.6.4 Gravitons in string theory . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Emergent gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Goldstone photons and gravitons 32 3.1 Emergent mediators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Nambu and Jona-Lasinio model (review) . . . . . . . . . . . . . . . . . . . . 37 3.3 An NJL-style argument for breaking LI . . . . . . . . . . . . . . . . . . . . 41 3.4 Consequences for emergent photons . . . . . . . . . . . . . . . . . . . . . . . 50 4 Phenomenology of spontaneous Lorentz violation 52 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Phenomenology of Lorentz violation by a background sour ce . . . . . . . . . 54 4.3 Eﬀective action for the Goldstone bosons of spontaneous Lorentz violation . 56 4.4 The long-range gravitational preferred-frame eﬀect . . . . . . . . . . . . . . 58 4.5 A cosmic solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Some considerations on the cosmological constant problem 70 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Gradient instability for scalar models of the dark energ y withw<−1 . . . 73 5.3 Time evolution of wfor ghost models of the dark energy . . . . . . . . . . . 77 5.4 Anthropic distribution for Λ and primordial density per turbations . . . . . 78 6 The reverse sprinkler 88 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Pressure diﬀerence and momentum transfer . . . . . . . . . . . . . . . . . . 90 6.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . . . . . 93 6.4 History of the reverse sprinkler problem . . . . . . . . . . . . . . . . . . . . 97 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 01 Bibliography 103 xii List of Figures 2.1 Feynman diagram for scattering mediated by scalar ﬁeld . . . . . . . . . . . . 7 2.2 Schematic of Witten’s argument against an emergent theo ry of gravity . . . . 30 3.1 Diagrammatic Schwinger-Dyson equation . . . . . . . . . . . . . . . . . . . . 38 3.2 Primed self-energy in a theory with a four-fermion inter action . . . . . . . . 39 3.3 Fermion and antifermion energies at ﬁnite densities . . . . . . . . . . . . . . 42 3.4 Four-fermion vertex as two massive, zero momentum photo n exchanges . . . 44 3.5 Plots self-consistent equations for the fermion mass m. . . . . . . . . . . . . 46 3.6 Further plots of self-consistent equations for the ferm ion massm. . . . . . . 49 3.7 Radiative corrections for the eﬀective potential of the auxiliary ﬁeld Aµ. . . 50 3.8 Graphic representation of how radiative corrections gi ve a ﬁnite ∝an}b∇acketle{tAµ∝an}b∇acket∇i}ht. . . . 51 4.1 Test mass orbiting a source moving with respect to the pre ferred frame . . . 66 4.2 Modiﬁcation to gravity by perturbations in the CMB . . . . . . . . . . . . . 67 5.1 Tadpole diagram corresponding to the cosmological cons tant term . . . . . . 71 5.2 Piston ﬁlled with vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Eﬀective coupling of two gravitons to several quanta of t he scalar ghost ﬁeld 74 6.1 Closed sprinkler in a tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 Open sprinkler in a tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3 Fluid ﬂow in a pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.4 Force creating the ﬂow into the reverse sprinkler . . . . . . . . . . . . . . . . 93 6.5 Tank recoiling as water rushes out of it . . . . . . . . . . . . . . . . . . . . . 94 6.6 Machine gun in a ﬂoating ship . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.7 Water ﬂowing out of a shower head . . . . . . . . . . . . . . . . . . . . . . . 96 6.8 Illustrations from Ernst Mach’s Mechanik . . . . . . . . . . . . . . . . . . . . 98 1 Chapter 1 Introduction est aliquid, quocumque loco, quocumque recessu unius sese dominum fecisse lacertae. — Juvenal, Satire III He thought he saw a Argument That proved he was the Pope: He looked again, and found it was A Bar of Mottled Soap. “A fact so dread,” he faintly said, “Extinguishes all hope!” — Lewis Carroll, Sylvie and Bruno Concluded This dissertation is essentially a collection of the variou s theoretical investigations that I pursued as a graduate student and that progressed to a publi shable state. It is diﬃcult, a posteriori , to come up with a theme that will unify them all. Even the absu rdly broad title that I have given to this document fails to account at al l for Chapter 6, which concerns a long-standing problem in elementary ﬂuid mechanics. Ther efore I will not attempt any such artiﬁcial uniﬁcation here. I have made an eﬀort, however, to make this thesis more than co llation of previously published papers. To that end, I have added material that rev iews and clariﬁes the relevant physics for the reader. Also, as far as possible, I have compl emented the previously published research with discussions of recent advances in the literat ure and in my own understanding. Chapter 2 in particular was written from scratch and is inten ded as a review of the relationship between massless particles, Lorentz invaria nce (LI), and local gauge invariance. In writing it I attempted to answer the charge half-seriousl y given to me as a ﬁrst-year graduate student by Mark Wise of ﬁguring out why we religious ly follow the commandment of promoting the global gauge invariance of the Dirac Lagran gian to a local invariance in 2 order to obtain an interacting theory. Consideration of the role of local gauge invariance in quantum ﬁeld theories (QFT’s) with massless, interacting p articles also helps to motivate the research described in Chapter 3. Chapter 3 brings up spontaneous Lorentz violation, which is the idea that perhaps the quantum vacuum of the universe is not a Lorentz singlet (or, t o put it otherwise, that empty space is not empty). The idea that gravity might be mediated b y the Goldstone bosons of such a symmetry breaking is attractive because it oﬀers a p ossible solution to two of the greatest obstructions to a quantum description of gravi ty: the non-renormalizability of linear gravity, and the cosmological constant problem. The work described in Chapter 4 seeks to place experimental l imits on how large spon- taneous Lorentz violation can be when coupled to ordinary gr avity. This line of research is independent from the ideas of Chapter 3 and applies to a wide v ariety of models in which cosmological physics takes place in a background that is not a Lorentz singlet. Chapter 5 begins with a brief overview of the cosmological co nstant problem, one of the greatest puzzles in modern theoretical physics. The nex t three sections of that chapter concern original results that are connected to that problem . Section 5.2 in particular has applications beyond the cosmological constant problem, as it oﬀers a theorem that helps connect the energy conditions of General Relativity (GR) wi th considerations of stability. All of this work concerns both QFT and GR, our two most powerfu l (though mutually incompatible) tools for describing the universe at a fundam ental level. In Chapter 6 we consider an amusing problem about introductory college phy sics that, surprisingly, had evaded a completely satisfactory treatment for several dec ades. 1.1 Notation and conventions We work throughout in units in which /planckover2pi1=c= 1. Electrodynamical quantities are given in the Heaviside-Lorentz system of units in which the Coloumb p otential of a point charge q is Φ =q 4πr. We also work in the convention in which the Fourier transform and inverse Fourier 3 transform in ndimensions are f(x) =/integraldisplaydnk (2π)n/2˜f(k)e−ik·x;˜f(x) =/integraldisplaydnk (2π)n/2f(x)eik·x. Lorentz 4-vectors are written as x= (x0,x1,x2,x3), wherex0is the time component andx1,x2, andx3are the ˆx,ˆy, and ˆzspace components respectively. Spatial vectors are denoted by boldface, so that we also write x= (x0,x). Unit spatial vectors are denoted by superscript hats. Greek indices such as µ,ν,ρ , etc. are understood to run from 0 to 3, while Roman indices such as i,j,k, etc. are understood to run from 1 to 3. Repeated indices are always summed over, unless otherwise speciﬁed. We takegµνto represent the full metric in GR, while ηµν= diag(+1,−1,−1,−1) is the Minkowski metric of ﬂat space-time. Indices are raised and l owered with the appropriate metric. The square of a tensor denotes the product of the tens or with itself, with all the indices contracted pairwise with the metric. Thus, for inst ance, the d’Alembertian operator in ﬂat spacetime is /square=−∂2=−∂µ∂µ=−ηµν∂µ∂ν=−∂2 0+∇2. We deﬁne the Planck mass as MPl=/radicalbig 1/8πG, whereGis Newton’s constant. For linear gravity we expand the metric in the form gµν=ηµν+M−1 Plhµνand keep only terms linear inh. In Chapter 2 we will work in units in which MPl= 1. Elsewhere we will show the factors ofMPlexplicitly. We use the chiral basis for the Dirac matrices γµ= 0σµ ¯σµ0 , γ5= −1 0 0 1 , whereσµ= (1,σ), ¯σµ= (1,−σ), and theσi’s are the Pauli matrices σ1= 0 1 1 0 , σ2= 0−i i0 , σ3= 1 0 0−1 . All other conventions are the standard ones in the literatur e. In writing this thesis, I have used the ﬁrst person plural (“w e”) whenever discussing 4 scientiﬁc arguments, regardless of their authorship. I hav e used the ﬁrst person singular only when referring concretely to myself in introductory of parenthetical material. I feel that this inconsistency is justiﬁed by the avoidance of styl istic absurdities. 5 Chapter 2 Massless mediators Did he suspire, that light and weightless down perforce must move. — William Shakespeare, Henry IV, part ii , Act 4, Scene 3 You lay down metaphysic propositions which infer universal consequences, and then you attempt to limit logic by despotism. — Edmund Burke, Reﬂections on the Revolution in France 2.1 Introduction I have sometimes been asked by scientiﬁcally literate layme n (my father, for instance, who is a civil engineer, and my ophthalmologist) to explain to them how a particle like the photon can be said to have no mass. How would a particle with zero mass be distinguishable from no particle at all? My answer to that question has been that in modern physics a particle is not deﬁned as a small lump of stuﬀ (which is the mental image im mediately conveyed by the word, as well as the non-technical version of the classical d eﬁnition of the term) but rather as an excitation of a ﬁeld, somewhat akin to a wave in an ocean. In that sense, masslessness means something technical: that the excitation’s energy go es to zero when its wavelength is very long. I have then added that masslessness also means t hat those excitations must always propagate at the speed of light and can never appear to any observer to be at rest. Here I will attempt a fuller treatment of this problem. Much o f the professional life of a theoretical physicist consists of ignoring technical diﬃ culties and underlying conceptual confusion, in the hope that something publishable and perha ps even useful might emerge from his labor. If the theorist had to proceed in strictly log ical order, the ﬁeld would advance 6 very slowly. But, on the other hand, the only thing that can ul timately protect us from being seriously wrong is suﬃcient clarity about the basics. In modern physics, long-range forces (electromagnetism and gravity) are understood to be mediated by massless particles with spinj≥1. The description of such massless particles in quantum ﬁel d theory (QFT) is therefore absolutely central to our current understandi ng of nature. Therefore, I have decided to use the opportunity aﬀorded by t he writing of this thesis to review the subject. My goals are to elucidate why a relativis tic description of massless parti- cles with spin j≥1 naturally requires something like local gauge invariance (which is not a physical symmetry at all, but a mathematical redundancy in t he description of the physics) and to clarify under what circumstances one might expect to e vade this requirement. I shall conclude with a discussion of how these consideratio ns apply to whether some of the major outstanding problems of quantum gravity could b e addressed by considering gravity to be an emergent phenomenon in some theory without f undamental gravitons. Nothing in this chapter will be original in the least, but it w ill provide a motivation for some of the original work presented in Chapter 3. 2.1.1 Unbearable lightness In his undergraduate textbook on particle physics, David Gr iﬃths points out that massless particles are meaningless in Newtonian mechanics because t hey carry no energy or momen- tum, and cannot sustain any force. On the other hand, the rela tivistic expression for energy and momentum: pµ= (E,p) =γm(1,v) (2.1) allows for non-zero energy-momentum for a massless particl e ifγ≡/parenleftbig 1−v2/parenrightbig−1/2→ ∞, which requires |v| →1. Equation (2.1) doesn’t tell us what the energy-momentum i s, but we assume that the relation p2=m2is valid form= 0, so that a massless particle’s energy Eand momentum pare related by E=|p|. (2.2) Griﬃths adds that Personally I would regard this “argument” as a joke, were it n ot for the fact that [massless particles] are known to exist in nature. They do indeed travel at 7/A1 Figure 2.1: Feynman diagram for the scattering of two particles that int eract through the exchange of a mediator. the speed of light and their energy and momentum arerelated by [Eq. (2.2)] ([1]). The problem of what actually determines the energy of the mas sless particle is solved not by special relativity, but by quantum mechanics, via Planck ’s formulaE=ω, whereωis an angular frequency (which is an essentially wave-like prope rty). Thus massless particles are the creatures of QFT par excellence , because, at least in current understanding, they can only be deﬁned as relativistic, quantum-mechanical entiti es. Like other subjects in QFT, describing massless particles requires arguments that wou ld seem absurd were it not for the fact that they yield surprisingly useful results that have g iven us a handle on observable natural phenomena. We need massless particles because we regard interaction fo rces as resulting from the exchange of other particles, called “mediators.” Figure 2. 1 shows the Feynman diagram that represents the leading perturbative term in the amplitude f or the scattering of two particles (represented by the solid lines) that interact via the excha nge of a mediator (represented by the dashed line). We can calculate this Feynman diagram in QFT and match the result to what we would get in non-relativistic quantum mechanics f rom an interaction potential V(r) (see, e.g., Section 4.7 in [2]). The result is V(r) =−g2 4πe−µr r, (2.3) wheregis the coupling constant that measures the strength of the in teraction and µis the mediator’s mass. Therefore, a long-range force requires µ= 0. In order to accommodate the observed properties of the long-range electromagnetic and gravitational interactions, we also need to give the mediator a on-zero spin. We will see that this is non-trivial. 8 2.1.2 Overview In this chapter we shall ﬁrst brieﬂy review how one-particle states are deﬁned in QFT and how their polarizations correspond to basis states in ir reducible representations of the Lorentz group. We will emphasize the diﬀerence between the c ase when the mass mof the particle is positive and the case when it is zero. We shall pro ceed to use these tools to build a ﬁeldAµthat transforms as a Lorentz 4-vector, ﬁrst for m>0 and then for m= 0. We shall conclude that the relativistic description of a massless sp in-1 ﬁeld requires the introduction of local gauge invariance. Similarly, we will point out how t he relativistic description of a massless spin-2 particle that transforms like a two-index L orentz tensor requires something like diﬀeomorphism invariance (the fundamental symmetry o f GR). Our discussion of these matters will rely heavily on the treatment given in [3]. We will then seek to formulate a solid understanding of the me aning of local gauge invariance and diﬀeomorphism invariance as redundancies o f the mathematical description required to formulate a relativistic QFT with massless medi ators. To this end we will also review the Weinberg-Witten theorem ([4]) and conclude by co nsidering how it might be possible to do without gauge invariance and evade the Weinbe rg-Witten theorem in an attempt to write a QFT of gravity without UV divergences. 2.2 Polarizations and the Lorentz group We deﬁne one-particle states to be eigenstates of the 4-mome ntum operator Pµand label them by their eigenvalues, plus any other degrees of freedom that may characterize them: Pµ|p,r∝an}b∇acket∇i}ht=pµ|p,r∝an}b∇acket∇i}ht. (2.4) Under a Lorentz transformation Λ that takes pto Λp, the state transforms as |p,r∝an}b∇acket∇i}ht →U(Λ)|p,r∝an}b∇acket∇i}ht (2.5) whereU(Λ) is a unitary operator in some representation of the Loren tz group. The 4- momentum itself transforms in the fundamental representat ion, so that U†(Λ)PµU(Λ) = Λµ νPν. (2.6) 9 The 4-momentum of the transformed state is therefore given b y PµU(Λ)|p,r∝an}b∇acket∇i}ht=U(Λ)/bracketleftig U†(Λ)PµU(Λ)/bracketrightig |p,r∝an}b∇acket∇i}ht=U(Λ)Λµ νpν|p,r∝an}b∇acket∇i}ht= (Λp)µU(Λ)|p,r∝an}b∇acket∇i}ht,(2.7) which implies that U(Λ)|p,r∝an}b∇acket∇i}htmust be a linear combination of states with 4-momentum Λ p: U(Λ)|p,r∝an}b∇acket∇i}ht=/summationdisplay r′crr′(p,Λ)/vextendsingle/vextendsingleΛp,r′/angbracketrightbig . (2.8) If the matrix crr′(p,Λ) in Eq. (2.8), for some ﬁxed p, is written in block-diagonal form, then each block gives an irreducible representation of the Loren tz group. We will call particles in the same irreducible representation “polarizations.” T he number of polarizations is the dimension of the corresponding irreducible representatio n.1 2.2.1 The little group For a particle with mass given by m=/radicalbig p2≥0, let us choose an arbitrary reference 4- momentum ksuch thatk2=m2. Any 4-momentum with the same invariant norm can be written as pµ=K(p)µ νkν(2.9) for some appropriate Lorentz transformation K(p). Let us then deﬁne the “little group” as the group of Lorentz tr ansformations Ithat leaves the reference kµinvariant: Iµ νkν=kµ. (2.10) Then Eq. (2.8) can be approached by considering Drr′(I) =crr′(p=k,Λ =I) so that U(I)|k,r∝an}b∇acket∇i}ht=/summationdisplay r′Drr′(I)/vextendsingle/vextendsinglek,r′/angbracketrightbig (2.11) and deﬁning 1-particle states with other 4-momenta by: |p,r∝an}b∇acket∇i}ht=N(p)U(K(p))|k,r∝an}b∇acket∇i}ht, (2.12) 1Notice that in this choice of language a Dirac fermion has fou r polarizations: the spin-up and spin-down fermion, plus the spin-up and spin-down antifermion. 10 whereN(p) is a normalization factor. If we impose that ∝an}b∇acketle{tk′,r′|k,r∝an}b∇acket∇i}ht=δr′rδ3(k′−k) (2.13) for states with 4-momentum k, then ∝an}b∇acketle{tp′,r′|p,r∝an}b∇acket∇i}ht=N∗(p′)N(p)/angbracketleftbig k′,r′/vextendsingle/vextendsingleU†(K(p′))U(K(p))/vextendsingle/vextendsinglek,r/angbracketrightbig =N∗(p′)N(p)Dr′r/parenleftbig K−1(p′)K(p)/parenrightbig δ3(k′−k). (2.14) Since theδ-function in the second line vanishes unless k′=k, this implies that the overlap is zero unless p′=p, and theDmatrix in Eq. (2.14) is therefore trivial: ∝an}b∇acketle{tp′,r′|p,r∝an}b∇acket∇i}ht=|N(p)|2δr′rδ3(k′−k). (2.15) We wish to rewrite Eq. (2.15) in terms of δ3(p′−p), to which we have argued it must be proportional. It is not diﬃcult to show that d3p/p0is a Lorentz-invariant measure when in- tegrating on the mass shell p0=/radicalbig p2+m2. This implies that δ3(k′−k) =δ3(p′−p)p0/k0 and we therefore have that ∝an}b∇acketle{tp′,r′|p,r∝an}b∇acket∇i}ht=|N(p)|2δr′rδ3(p′−p)p0/k0. (2.16) Equation (2.16) naturally leads to the choice of normalizat ion N(p) =/radicalbig k0/p0. (2.17) 2.2.2 Massive particles A massive particle will always have a rest frame in which its 4 -momentum is kµ= (m,0,0,0). This is, therefore, the natural choice of reference 4-momen tum. It is easy to check that the little group is then SO(3), which is the subgroup of the Lorentz group that includes only rotations. The generators of SO(3) may be written as Ji=iǫijkxj∂k, (2.18) 11 which are the angular momentum operators and which obey the c ommutation relation /bracketleftbig Ji,Jj/bracketrightbig =iǫijkJk. (2.19) The Lie algebra of SO(3) is the same as that of SU(2), because both groups look identical in the neighborhood of the identity. In quantum mechanics, the intrinsic angular momentum of a particle (its spin) is a label of the dimensionality of th e representation of SU(2) that we assign to it. A particle of spin jlives in the 2 j+1 dimensional representation of SU(2). The generators of SO(1,3) may be written as Jµν=i(xµ∂ν−xν∂µ), (2.20) which are clearly anti-symmetric in the indices and which ob ey the commutation relation [Jµν,Jρσ] =i(ηνρJµσ−ηµρJνσ−ηνσJµρ+ηµσJνρ). (2.21) We may write the six independent components of Jµνas two three-component vectors: Ki=J0i;Li=1 2ǫijkJjk, (2.22) where Kis the generator of boosts and Lis the generator of rotations. Using Eqs. (2.21) and (2.22), one can immediately show that these satisfy the c ommutation relations: /bracketleftbig Li,Lj/bracketrightbig =iǫijkLk;/bracketleftbig Li,Kj/bracketrightbig =iǫijkKk;/bracketleftbig Ki,Kj/bracketrightbig =−iǫijkJk. (2.23) Let us deﬁne two new 3-vectors: J±=1 2(L±iK). (2.24) Using Eq. (2.23) we can write their commutators as /bracketleftig Ji ±,Jj ±/bracketrightig =iǫijkJk ±;/bracketleftig Ji ±,Jj ∓/bracketrightig = 0. (2.25) That is, both J+andJ−separately satisfy the commutation relation for angular mo - 12 mentum, and they also commute with each other. This means tha t we can identify all ﬁnite-dimensional representations of the Lorentz group SO(1,3) by pair of integer or half- integer spins ( j+,j−) that correspond to two uncoupled representations of SO(3). The Lorentz-transformation property of a left-handed Weyl fer mionψLcorresponds to (1 /2,0), while (0,1/2) corresponds to the right-handed Weyl fermion ψR. A massive Dirac fermion corresponds to the representation (1 /2,0)⊕(0,1/2). A Lorentz 4-vector (that is, a quantity that transforms unde r the fundamental repre- sentation of SO(3,1)), corresponds to (1 /2,1/2). This indicates that it can be decomposed into a spin-1 and a spin-0 part, since 1 /2⊗1/2 = 1⊕0. Or, to put it otherwise, a general Lorentz vector has four independent components, three of wh ich may be matched to the three polarizations of a j= 1 particle and one to the single polarization of a j= 0 particle. 2.2.3 Massless particles Since a massless particle has no rest frame, the simplest ref erence 4-momentum is k= (1,0,0,1). The corresponding little group clearly contains as a subgro up rotations about the z-axis. The little group can be parametrized as I(δ,η,φ)µ ν= Λ(δ,η)µ ρΛ(φ)ρ ν, (2.26) where Λ(φ)µ ν= 1 0 0 0 0 cosφsinφ0 0−sinφcosφ0 0 0 0 1 (2.27) and Λ(δ,η)µ ν= 1 +ζ δ η −ζ δ1 0 −δ η0 1 −η ζ δ η 1−ζ , (2.28) withζ=/parenleftbig δ2+η2/parenrightbig /2. 13 It can be readily checked that Λ(δ1,η1)µ ρΛ(δ2,η2)ρ ν= Λ(δ1+δ2,η1+η2)µ ν, (2.29) which implies that the little group is isomorphic to the grou p of rotations (by an angle φ) and translations (by a vector ( δ,η)) in two dimensions.2UnlikeSO(3), this group, ISO(2), is not semi-simple, i.e., it has invariant abelian subgroup s: the rotation subgroup deﬁned by Eq. (2.27) and the translation subgroup deﬁned by Eq. (2.2 8). This leads to the important consequence that massless one-p article states |p,r∝an}b∇acket∇i}htcan have only two polarization, called “helicities,” given by the co mponent of the angular momentum along its direction of motion. The physical reason for this i s that only the angular momen- tum component associated with the rotations in Eq. (2.27) ca n deﬁne discrete polarizations. Helicities are Lorentz-invariant, unlike the polarizatio ns of a massive particle. It is clear that massless particles in QFT are diﬀerent from m assive ones. It is possible to understand some of the properties of massless particles by c onsidering them as massive and then taking the m→0 limit carefully, but this discussion should make it appare nt that this limiting procedure is fraught with danger. We shall explore this issue in the construction of the vector ﬁeld. 2.3 The vector ﬁeld We seek a causal, free quantum ﬁeld Aµthat transforms like a Lorentz 4-vector. By analogy to the procedure used to obtain free quantum ﬁelds with spin 0 and 1/2 (see, e.g., Chapters 2 and 3 in [2], or Sections 5.2 to 5.5 in [3]), we start by writin g Aµ(x) =/integraldisplayd3p (2π)3/2/summationdisplay r/parenleftig ǫµ r(p)ar(p)eip·x+ǫµ∗ r(p)a† r(p)e−ip·x/parenrightig , (2.30) where the index rruns over the physical polarizations of the ﬁeld, while aanda†are the creation and destruction operators for particles of the corresponding momentum and polarization that obey bosonic commutation relations, and pµ=/parenleftig/radicalbig m2+p2,p/parenrightig . LetK(p) be the Lorentz transformation (boost) that takes a particl e of massmfrom 2The Lorentz transformation in Eq. (2.28) is, of course, not a physical translation. It just happens that the group of such matrices is isomorphic to the group of trans lations on the plane. 14 rest to a 4-momentum p. It can be shown that the measure d3p/p0is Lorentz-invariant when integrating on the mass-shell p2=m2. Since both pµandAµare Lorentz 4-vectors, we must have ǫµ r(p) =/radicalbiggm p0K(p)µ νǫν r(0). (2.31) Now consider the behavior of ǫµ r(0) under an inﬁnitesimal rotation. For our ﬁeld Aµ(x) in Eq. (2.30) to have a deﬁnite spin j, we must have that Lµ νǫν r(0) = S(j) rr′ǫµ r′(0), (2.32) where the three components of S(j)are the standard spin matrices for spin j. Equation (2.32) follows immediately from requiring ǫµ r(0) to transform under rotations as both a 4-vector and as a spin- jobject.3 For the rotation generators in the fundamental representat ion ofSO(1,3) we have: (Li)0 0= (Li)0 j= (Li)j 0= 0, (2.33) (Li)jk=iǫi jk. (2.34) Therefore, for ( L2)µ ν=/summationtext i(Li)µ ρ(Li)ρ ν, we have (L2)0 0= (L2)j 0= (L2)0 j= 0 ; ( L2)j k= 2ηj k. (2.35) Meanwhile, recall that, for the spin matrices, (S(j) 2)rr′=j(j+ 1)δrr′. (2.36) Using Eqs. (2.32), (2.35), and (2.36) we therefore obtain th at ǫi r(0) =j(j+ 1) 2ǫi r(0) ;j(j+ 1)ǫ0 r(0) = 0. (2.37) Equation (2.37), combined with Eq. (2.31), leaves us only tw o posibilities if the ﬁeld Aµ(x) in Eq. (2.30) is to transform as a 4-vector: 3It should perhaps also be pointed out that in Eq. (2.32) the in dices µ, νin the left-hand side indicate components of the three matrices Lideﬁned in Eq. (2.22). In Eq. (2.21) µ, νlabeled the matrices themselves. 15 •Eitherj= 0 andǫ0(0) is the only non-vanishing component, •orj= 1 and the three ǫi(0)’s are the only non-vanishing components This agrees with the claim made at the end of the previous sect ion, which we had based on 1/2⊗1/2 = 1⊕0. Let us explore both possibilities. 2.3.1 Vector ﬁeld with j= 0 For thej= 0 case we can chose the conventionally normalized ǫ0(0) =i/radicalbig m/2, which, by Eq. (2.31) gives ǫµ(p) =ipµ/radicalbigg 1 2p0. (2.38) One can then compare the resulting form for Aµ(x) in Eq. (2.30) to the form for a free scalar ﬁeld and conclude that this vector ﬁeld has the form Aµ(x) =∂µφ(x) (2.39) forφ(x) a free, Lorentz scalar ﬁeld. Notice that as the ﬁeld φhas a single physical polar- ization, so also does Aµ, and that even though our construction of the vector ﬁeld ass umed anm>0 in Eq. (2.31), the m→0 limit in this case is perfectly sensible.4 2.3.2 Vector ﬁeld with j= 1 Now consider the case where the vector ﬁeld has j= 1. Following the popular convention we write ǫµ r=±1(0) =∓1 2√m(ηµ 1±iηµ 2) (2.40) and ǫµ r=0(0) =/radicalbigg 1 2mηµ 3. (2.41) We may check that the raising and lowering operators S(1) ±=S(1) 1±iS(1) 2act appropriately on these polarization vectors. For a plane-wave propagatin g along the i= 3 spatial direction, r=±1 correspond to two transverse, circular polarizations of t he vector ﬁeld, while r= 0 corresponds to the longitudinal polarization. 4This kind of massless, spinless vector ﬁeld will appear agai n in the discussion of the “ghost condensate” mechanism in Chapter 4. 16 We may rewrite the ﬁeld Aµin terms of polarization vectors that are mass-independent by introducing ˜ǫµ r(0) =√ 2mǫµ r(0) (2.42) then we have that Eq. (2.30) becomes Aµ(x) =/integraldisplayd3p (2π)3/21/radicalbig 2p01/summationdisplay r=−1/parenleftig ˜ǫµ r(p)ar(p)eip·x+ ˜ǫµ r(p)a† r(p)e−ip·x/parenrightig , (2.43) where ˜ǫµ r(p) =K(p)µ ν˜ǫν r(0). The ﬁeld in Eq. (2.43) obeys the equation of motion /parenleftbig/square−m2/parenrightbig Aµ(x) = 0. (2.44) Notice also that pµ˜ǫµ r(p) =pµKµ ν(p)˜ǫν r(0) =/parenleftbig K−1(p)p/parenrightbig ν˜ǫν r(0) =m˜ǫ0 r(0) = 0 (2.45) implies that ∂µAµ= 0. (2.46) In the limit m→0 the boost K(p) becomes the identity and ˜ ǫµ r(p) = ˜ǫµ r(0) for all p. The ﬁeld then obeys both /squareAµ= 0 and∂µAµ= 0.5The fact that there are complications in this limit is revealed by using Eq. (2.31) and the form of ˜ ǫµ r(0)’s to obtain Πµν(p)≡1/summationdisplay r=−1˜ǫµ r(p)˜ǫν r(p) =ηµν+pµpν m2. (2.47) Notice that Πµνpν= 0, while Πµνkν=kµfork·p= 0, which means Πµνis a projection unto the space orthogonal to pµ. Equation (2.47) clearly is not ﬁnite as m→0. This will be a problem if we try to directly couple Aµto anything in a Lorentz-invariant way, Lint∝Aµjµ, (2.48) because then the rate at which Aµ’s would be emitted by the interaction would be propor- 5Therefore taking the m→0 limit of the spin-1 vector ﬁeld automatically gives us the m assless ﬁeld in the Lorenz gauge. 17 tional to /summationdisplay r|˜ǫµ r(p)∝an}b∇acketle{tjµ∝an}b∇acket∇i}ht|2= Πµν(p)∝an}b∇acketle{tjµ∝an}b∇acket∇i}ht∝an}b∇acketle{tjν∝an}b∇acket∇i}ht∗=∝an}b∇acketle{tjµ∝an}b∇acket∇i}ht∝an}b∇acketle{tjµ∝an}b∇acket∇i}ht∗+1 m2|p· ∝an}b∇acketle{tj∝an}b∇acket∇i}ht|2, (2.49) which clearly diverges as m→0 unless we impose that p· ∝an}b∇acketle{tj∝an}b∇acket∇i}ht= 0. That is, in the presence of an interaction of the form Eq. (2.48), we must require that the current to which the ﬁeld couples be conserved, ∂µ∝an}b∇acketle{tjµ∝an}b∇acket∇i}ht= 0, (2.50) in order to avoid an inﬁnite rate of emission. As emphasized earlier in this chapter, the spin of a massless particle must point either parallel or anti-parallel to its direction of propagation. These possibilities correspond to the longitudinal polarizations ˜ ǫµ ±1. A massless particle cannot have a longitudinal polarizati on ˜ǫµ 0. The requirement of current conservation in Eq. (2.50) ensu res that the longitudinal polarization decouples from the current jµin them→0 limit, so that it cannot be produced by the interaction in Eq. (2.48). 2.3.3 Massless j= 1particles Let us now try to construct a genuinely massless vector ﬁeld w ith non-zero spin j. To that eﬀect we adopt an arbitrary reference momentum k= (0,0,1) and a corresponding light-like reference 4-momentum k= (1,0,0,1). LetK(p) be now deﬁned as the Lorentz transformation that takes a massless particle with referen ce momentum kto a general momentum p. We can write this transformation as the composition of a rot ation (from the direction of kto the direction of p) followed by a boost along the direction of pthat scales the magnitude. Then ǫµ r(p) =K(p)µ νǫν r(k). (2.51) We now require that ǫµ r(k) transform as both a massless particle with helicity r=±j and as a 4-vector. For rotations by an angle φaround the axis of k, we must have eirφǫµ r(k) = Λ(φ)µ νǫν r(k), (2.52) where Λ(φ)µ νis the Lorentz transformation matrix corresponding to the r otation, given in 18 Eq. (2.27). For Eq. (2.52) to be true of a general φin thej= 1 case, we must have ǫµ ±1(k)∝(0,1,±i,0) (2.53) and we might as well normalize this solution to match the ˜ ǫµ r’s in Eq. (2.42), giving ǫµ ±1(k) =1√ 2(0,1,±i,0). (2.54) These are the same polarization vectors that we obtained pre viously in the m→0 limit of the massive vector ﬁeld. But the little group for massless particles is larger than th eO(2) =U(1) group rep- resented by Eq. (2.27), as was seen in Subsection 2.2.3. For o ur ﬁeld to transform as a 4-vector we would also require that ǫµ r(k) = Λ(δ,η)µ νǫν r(k), (2.55) where Λ(δ,η)µ νwas given in Eq. (2.28). Plugging in the polarization 4-vect ors in Eq. (2.54) we can see immediately that this is impossible because, unde r the transformation Λ( δ,η), ǫµ ±1(k)→ǫµ ±1(k) +kµ |k|δ±iη√ 2. (2.56) Thus we are forced to accept that the one-particle states of a massless spin-1 vector ﬁeld are not Lorentz-covariant under the action of their little g roup, but only covariant up to a term proportional to the reference kµ. If we then construct the general states using Eq. (2.51) and Aµ(x) =/integraldisplayd3p (2π)3/21/radicalbig 2p0/summationdisplay r=±1/parenleftig ǫµ r(p)ar(p)eip·x+ǫµ r(p)a† r(p)e−ip·x/parenrightig (2.57) we see that we are forced to accept that Aµ(x) transforms under a general Lorentz trans- formation Λ as: Aµ(x)→Λµ νAν(Λx) +∂µΩ(x,Λ) (2.58) where Ω is some function of the coordinates xand the parameters of the Lorentz transfor- mation Λ. 19 Equation (2.58) should, in my opinion, be regarded as a disas ter. Massless spin-1 quan- tum ﬁelds, which we need in order to explain the observed prop erties of the electromagnetic interaction, are incompatible with one of the most sacred pr inciples of modern physics: Lorentz covariance.6It is not, however, an irretrievable disaster, and in fact th ere will be a rich silver lining to it. We can “save” Lorentz covariance by announcing that two ﬁeld s related by the trans- formation Aµ→Aµ+∂µΩ (2.59) describe the same physics, so that the second term in Eq. (2.5 8) becomes irrelevant.7We can couple such an Aµif the interaction is of the form Lint∝Aµjµfor a conserved current jµ, because in that case the coupling is invariant under transf ormations of the form in Eq. (2.59). Notice that this requirement on the coupling of Aµagrees with what we imposed earlier, by Eq. (2.49), in order to avoid an inﬁnite rate of em ission for the vector ﬁeld in them→0 limit. It is easy to construct a genuinely Lorentz-covariant two-i ndex ﬁeld strength tensor that is invariant under Eq. (2.59): Fµν=∂µAν−∂νAµ. (2.60) Lorentz-invariant couplings to this ﬁeld strength would be gauge-invariant, but the presence of derivatives in Eq. (2.60) means that the resulting forces must fall oﬀ faster with distance than an inverse-square law (i.e., they cannot be long-range forces). 2.4 Why local gauge invariance? The Dirac Lagrangian for a free fermion, L=¯ψ(i∂ /−m)ψis invariant under the global U(1) gauge transformation ψ→eiαψ. This global symmetry, by Noether’s theorem, implies 6This statement may seem peculiar in light of the fact that the Lorentz group was ﬁrst discovered as the symmetry of the Maxwell equations of classical electrodyna mics. But those equations are written in terms of the ﬁelds EandB. The scalar and vector potentials ( A0andArespectively) enter classical electrodynamics only as computational aids. It is quantum mechanics which requires a formulation in terms of Aµ. 7This irresistibly brings to my mind a scene from the Woody All en movie comedy Bananas in which victorious rebel commander Esposito announces from the Pre sidential Palace that “from this day on, the oﬃcial language of San Marcos will be Swedish... Furthermor e, all children under 16 years old are now 16 years old.” 20 conservation of the current: jµ=¯ψγµψ . (2.61) In the established model of quantum electrodynamics, this L agrangian is transformed into an interacting theory by making the gauge invariance local: The phaseαis allowed to be a function of the space-time point x. This requires the introduction of a gauge ﬁeld Aµwith the the transformation property Aµ→Aµ+∂µα (2.62) and the use of a covariant derivative Dµ=∂µ−iAµinstead of the usual derivative ∂µ. This procedure automatically couples Aµto the conserved current in Eq. (2.61) so that the coupling is invariant under transformations of the form Eq ( 2.62). We than add a Lorentz- invariant kinetic term −F2 µν/4 for the ﬁeld Aµ. The generalization to non-abelian gauge groups is well known, as is the Higgs mechanism to break the ga uge invariance spontaneously and give the ﬁeld Aµa mass. This is what we are taught in elementary courses on QFT, but th e question remains: Why do we promote a global symmetry of the free fermion Lagran gian to a local symmetry? Equation (2.58) provides a deeper insight into the physical meaning of local gauge invariance: a massless particle, having no rest frame, cannot have its sp in point along any axis other than that of its motion. Therefore, it can have only two polar izations. By describing it as a 4-vector, spin-1 ﬁeld Aµ(which has three polarizations) a mathematical redundancy is introduced. This redundancy is local gauge invariance. A ﬁeld with local gauge symmetry is coupled to the conserved current of the corresponding global gauge s ymmetry in order to make the coupling locally gauge-invariant. The procedure describe d of promoting the global gauge symmetry to a local gauge invariance is therefore required i n order to couple fermions in a Lorentz-invariant way via a long-range, spin-1 force. 2.4.1 Expecting the Higgs Remarkably, local gauge invariance also comes to our aid in w riting sensible QFT’s for the short-range weak nuclear interaction. At low energies, thi s interaction is naturally described 21 as being mediated by massive, spin-1 vector ﬁelds. The Lagra ngian for such a mediator must look like L=−1 4F2 µν+1 2m2A2−AµJµ, (2.63) whereJµis the current to which it couples. But in the case of the weak n uclear interaction this current is not conserved. At energy scales much higher t han themin Eq. (2.63), we therefore expect the same problem we found in Subsection 2.3 .2 of a divergent emission rate for the longitudinal polarization, unless other higher-de rivative operators, which were not relevant at low energies, have come to our rescue. In the standard model of particle physics, the resolution of this problem is to make the mediators of the weak nuclear interaction gauge bosons, and then to break that gauge invariance spontaneously by introducing a scalar Higgs ﬁel d with a non-zero VEV, thus giving the bosons the mass that accounts for the short range o f the force they mediate. At high energies the gauge invariance is restored. The problem atic longitudinal polarization disappears and is transmuted into the Goldstone boson of the spontaneously broken sym- metry. Since the Goldstone boson has no spin, it does not have the problem of a divergent rate of emission. This is the reason why many billions of doll ars have been spent in the search for that yet-unseen Higgs boson, a search soon to come to a head with the turning on of the Large Hadron Collider (LHC) at CERN next year. 2.4.2 Further successes of gauge theories Gauge theories as descriptions of the fundamental particle interactions have other very attractive attributes. It was shown by ’t Hooft that these th eories are always renormalizable, i.e., that the inﬁnities that plague QFT’s can all be absorbe d into a redeﬁnition of the bare parameters of the theory, namely the masses and the coupling constants ([5]). Politzer ([6]) and, independently, Gross and Wilczek ([7]), showed t hat the renormalization ﬂow of the coupling constants in non-abelian gauge theories pro vides a natural explanation of the observed phenomenon of asymptotic freedom, whereby the nuclear interactions become more feeble at higher energies. It is also widely believed, though not strictly demonstrate d, that QCD, the theory in which the strong nuclear force is mediated by the bosons of anSU(3) gauge theory, accounts for conﬁnement, i.e., for the fact that the strongl y interacting fermions (quarks) 22 never occur alone and can appear only in bound states that are singlets ofSU(3). These successes illustrate what we meant when we said in Subsectio n 2.3.3 that having to accept local gauge symmetry was a disaster with a rich silver lining . For interesting accounts of the history of local gauge invariance in classical and quant um physics, see [8, 9]. 2.5 Massless j= 2particles and diﬀeomorphism invariance We could repeat the sort of procedure used in Subsection 2.3. 3 in order to try to construct Lorentz-covariant hµνout of the two helicities of a j= 2 massless ﬁeld. This procedure would similarly fail, requiring us to accept the transforma tion rule: hµν(x)→Λµ ρΛν σhρσ(Λx) +∂µξν(x,Λ) +∂νξµ(x,Λ). (2.64) Saving Lorentz covariance would then require announcing th at states related by a transfor- mation of the form hµν→hµν+∂µξν+∂νξµ(2.65) are physically equivalent. We can construct a four-index ﬁe ld strength tensor Rµνρσinvari- ant under Eq. (2.65) that is anti-symmetric in µ,ν, anti-symmetric in ρ,σ, and symmetric under exchange of the two pairs. But to accommodate a long-ra nge force we would need to couplehµνto a quantity Θµνsuch that ∂µ∝an}b∇acketle{tΘµν∝an}b∇acket∇i}ht= 0. (2.66) This Θµνis the stress-energy tensor obtained from translational in variance xµ→xµ−ξµ, (2.67) through Noether’s theorem.8Invariance under Eq. (2.65) corresponds to promoting the translational symmetry in Eq. (2.67) to a local invariance b y lettingξµbe a function of x. It turns out that the theory constructed in this way matches l inearized GR around a ﬂat background with hµνbeing the graviton ﬁeld. 8If there were another conserved Θ′µν, there would have to be another conserved 4-vector besides pµ, namely p′µ=/integraltext d3xΘ′0µ. Kinematics would then allow only forward collisions. 23 It is well known that one can reconstruct the full GR uniquely from linear gravity by a self-consistency procedure ([10, 11, 12]). Therefore a re lativistic QFT in ﬂat spacetime with a massless spin-2 particle mediating a long-range forc e essentially implies GR. In full GR the invariance under Eq. (2.65) is a consequence of the inv ariance of the theory under diﬀeomorphisms: xµ→x′µ(x). (2.68) Remarkably, we may therefore think of diﬀeomorphism invari ance as a redundancy required by the relativistic description of a massless spin-2 partic le. 2.6 The Weinberg-Witten theorem The Weinberg-Witten theorem9rules out the existence of massless particles with higher sp in in a very wide class of QFT’s ([4]). In their original paper, t he authors present their elegant proof very succinctly. This review is longer than the paper i tself, which may be justiﬁed by the importance of this result in further clarifying the ne ed for local gauge invariance in relativistic theories that accommodate long-range forces such as are observed in nature. Let|p,±j∝an}b∇acket∇i}htand|p′,±j∝an}b∇acket∇i}htbe two one-particle, massless states of spin j, labelled by their light-like 4-momenta pandp′, and by their helicity (which we take to be the same for the two particles). We will be considering the matrix elements /angbracketleftbig p′,±j/vextendsingle/vextendsinglejµ|p,±j∝an}b∇acket∇i}ht;/angbracketleftbig p′,±j/vextendsingle/vextendsingleTµν|p,±j∝an}b∇acket∇i}ht, (2.69) wherejµis a conserved current (i.e., ∂µ∝an}b∇acketle{tjµ∝an}b∇acket∇i}ht= 0) andTµνis a conserved stress-energy tensor (i.e. ∂µ∝an}b∇acketle{tTµν∝an}b∇acket∇i}ht= 0). 2.6.1 The j >1/2case If we assume that the massless particles in question carry a n on-zero conserved charge Q=/integraltext d3xJ0, so that (suppressing the helicity label for now) Q|p∝an}b∇acket∇i}ht=q|p∝an}b∇acket∇i}ht, (2.70) 9According to the authors, a less general version of their the orem was formulated earlier by Sidney Coleman, but was not published. 24 whereq∝ne}ationslash= 0, then evidently /angbracketleftbig p′/vextendsingle/vextendsingleQ/vextendsingle/vextendsinglep/angbracketrightbig =qδ3(p′−p). (2.71) Meanwhile, we also have that /angbracketleftbig p′/vextendsingle/vextendsingleQ/vextendsingle/vextendsinglep/angbracketrightbig =/integraldisplay d3x/angbracketleftbig p′/vextendsingle/vextendsinglej0(t,x)/vextendsingle/vextendsinglep/angbracketrightbig =/integraldisplay d3x/angbracketleftbig p′/vextendsingle/vextendsingleeiP·xj0(t,0)e−iP·x/vextendsingle/vextendsinglep/angbracketrightbig =/integraldisplay d3xei(p′−p)·x/angbracketleftbig p′/vextendsingle/vextendsinglej0(t,0)/vextendsingle/vextendsinglep/angbracketrightbig = (2π)3δ3(p′−p)/angbracketleftbig p′/vextendsingle/vextendsinglej0(t,0)/vextendsingle/vextendsinglep/angbracketrightbig ,(2.72) so that combining Eqs. (2.71) and (2.72) gives lim p′→p/angbracketleftbig p′/vextendsingle/vextendsinglej0(t,0)/vextendsingle/vextendsinglep/angbracketrightbig =q (2π)3, (2.73) which, by Lorentz covariance, implies that lim p′→p/angbracketleftbig p′/vextendsingle/vextendsinglejµ(t,0)/vextendsingle/vextendsinglep/angbracketrightbig =qpµ E(2π)3∝ne}ationslash= 0. (2.74) Notice that Eq. (2.74) implies current conservation, becau sep2= 0. For any light-like pandp′ (p′+p)2= 2(p′·p) = 2(|p′||p| −p′·p) = 2|p′||p|(1−cosθ)≥0, (2.75) whereθis the angle between the momenta. If θ∝ne}ationslash= 0, then ( p′+p) is time-like and we can therefore choose a frame in which it has no space component, s o that p= (|p|,p) ;p′= (|p|,−p) (2.76) (i.e., the two particles propagate in opposite directions w ith the same energy). In this frame, consider rotating the particles by an angle φaround the axis of p: |p,±j∝an}b∇acket∇i}ht →e±iφj|p,±j∝an}b∇acket∇i}ht;/vextendsingle/vextendsinglep′,±j/angbracketrightbig →e∓iφj|p,±j∝an}b∇acket∇i}ht. (2.77) 25 The Lorentz covariance of the matrix element of jµthen implies that e±2iφj/angbracketleftbig p′,±j/vextendsingle/vextendsinglejµ(t,0)/vextendsingle/vextendsinglep,±j/angbracketrightbig = Λ(φ)µ ν/angbracketleftbig p′,±j/vextendsingle/vextendsinglejν(t,0)/vextendsingle/vextendsinglep,±j/angbracketrightbig , (2.78) where Λ(φ) is the Lorentz transformation corresponding to a rotation by an angle φaround the direction of p. But Λ(φ) contains no Fourier components other than e±iφand 1, so Eq. (2.78) implies that the matrix elements vanish for j >1/2. In the limit p′→p, we then arrive at a contradiction with Eq. (2.74). Therefore no relativistic QFT with a conserved current can have massless spin-1 particles (eith er fundamental or composite) that have Lorentz-covariant spectra and are charged under the co nserved current. 2.6.2 The j >1case If the massless particles in question carry no conserved cha rge, we may still consider the matrix elements of the stress-energy tensor Tµν. By the same kind of argument as in Subsection 2.6.1 lim p′→p/angbracketleftbig p′/vextendsingle/vextendsingleTµν(t,0)/vextendsingle/vextendsinglep/angbracketrightbig =pµpν E(2π)3∝ne}ationslash= 0. (2.79) Notice again that this stress-energy is conserved because p2= 0. Then combining Eq. (2.77) with relativistic covariance imp lies that e±2iφj/angbracketleftbig p′,±j/vextendsingle/vextendsingleTµν(t,0)/vextendsingle/vextendsinglep,±j/angbracketrightbig = Λ(φ)µ ρΛ(φ)ν σ/angbracketleftbig p′,±j/vextendsingle/vextendsingleTρσ(t,0)/vextendsingle/vextendsinglep,±j/angbracketrightbig . (2.80) The fact that Λ( φ) contains only the Fourier components e±iφand 1 then implies that the matrix elements must vanish for j >1, contradicting Eq. (2.79) in the limit p′→p. Therefore no relativistic QFT with a conserved stress-ener gy tensor can have massless spin-2 particles (either fundamental or composite) that have Lore ntz-covariant spectra. 2.6.3 Why are gluons and gravitons allowed? Evidently, the Weinberg-Witten theorem does not forbid pho tons, because they carry no conserved charge. It also does not forbid the W±andZbosons because they are massive. But the Standard Model contains charged, massless spin-1 pa rticles (the gluons) as well as massless spin-2 particles (the gravitons). How is this po ssible? The resolution of this question helps to clarify the necessity for local gauge inva riance. 26 In a Yang-Mills theory, LYM=−1 4Fa µνFaµν+Lmatter(ψ,D µψ) (2.81) the gauge-invariant current jµ a=δSmatter δAaµ(2.82) is not conserved, because it obeys the equation Dµ∝an}b∇acketle{tjµ a∝an}b∇acket∇i}ht= 0, rather than ∂µ∝an}b∇acketle{tjµ a∝an}b∇acket∇i}ht= 0. Fur- thermore, ∝an}b∇acketle{tjµ a∝an}b∇acket∇i}htvanishes for one-particle gauge ﬁeld states. Therefore con sidering the matrix elements of this jµ abetween gauge boson states in Yang-Mills theory would avail us nothing because the limit in Eq. (2.74) would be zero. What we actually want is a current that measures the ﬂow of cha rge in the absence of matter (i.e., for the Yang-Mills bosons alone) and that is co nserved in the sense ∂µ∝an}b∇acketle{tJµ a∝an}b∇acket∇i}ht= 0: Jµ a=−Fµν cfcabAbν, (2.83) where thef’s are the structure constants of the gauge group. Conservat ion follows imme- diately from the equation of motion for Eq. (2.81). This is, i n fact, the conserved current obtained through Noether’s theorem from the global gauge in variance of Eq. (2.81) without matter. But the current in Eq. (2.83) is obviously not gauge- invariant. Therefore, under the action of a Lorentz transformation Λ, Jµ a→Λµ νJν a+∂µΩa (2.84) and it is not, consequently, Lorentz-covariant. If we tried making it Lorentz-covariant by introducing an unphysical extra polarization of the gauge b oson, then the theorem would fail because the helicities would not be Lorentz-invariant , invalidating the choice-of-frame procedure used to arrive at Eq. (2.77). To put this in another way, in a gauge theory the physical |p,±j∝an}b∇acket∇i}htstates are actually equivalence classes, because two states related by a gauge t ransformation represent the same physics. A technical way of thinking about this is that the ph ysical states are elements of the BRST cohomology ([13]). Therefore, matrix elements suc h as those in Eq. (2.69) are only well-deﬁned if the operator jµis BRST-closed, which requires the operator to be 27 gauge-invariant. It is well known that Yang-Mills theories do not allow the construction of gauge-invariant conserved currents. The case of the graviton is very closely analogous to that of t he Yang-Mills bosons. In Einstein-Hilbert gravity S=/integraldisplay d4x√−g[R+Lmatter(φ,∇µφ,gµν)], (2.85) where the ﬁeld φstands for all possible matter ﬁelds of any spin. The covaria nt stress-energy tensor Tµν=1√−gδSmatter δgµν(2.86) obeys ∇µ∝an}b∇acketle{tTµν∝an}b∇acket∇i}ht= 0 rather than ∂µ∝an}b∇acketle{tTµν∝an}b∇acket∇i}ht= 0, and ∝an}b∇acketle{tTµν∝an}b∇acket∇i}ht= 0 for any state with only gravi- tational ﬁelds. What we want is therefore not T, but rather Θµ ν=∂R ∂(∂µgαβ)(∂νgαβ)−gµ νR . (2.87) But recall that the Ricci scalar Rcontains not only the metric and its ﬁrst derivatives, but also terms linear in its second derivatives. In order to deﬁn e Θ we therefore need to do the usual trick of integrating by parts and setting the boundary terms to zero in order to get rid of the second derivatives in R. This means that Ris no longer a covariant scalar and therefore Θ is not a covariant tensor, but rather a pseudoten sor. It is well known that gravitational energy cannot be deﬁned i n a covariant way. For instance, the energy of gravity waves on a ﬂat background is l ocalizable only for waves trav- eling in a single direction, which is not a coordinate-invar iant condition (see, for instance, Chapter 33 in [14]). A general Lorentz transformation of the graviton ﬁeld hµνwill destroy this condition. This means that the stress-energy pseudote nsor Θµνfor gravitons involves a ﬁeldhµνthat does not transform like a Lorentz tensor. Its matrix ele ments are therefore not Lorentz-covariant. Once again, if we attempt to remedy t his by introducing unphysical extra polarizations of the gravitons, the Lorentz invarian ce of the helicity is lost. Otherwise stated, in a theory with diﬀeomorphism invarianc e like GR, the physical states are equivalence classes, because two states related by a coordinate transformation represent the same physics. The matrix elements in Eq. (2.69 ) are only well deﬁned if the operatorTµνis BRST-closed, but GR admits no local BRST-closed operator s, and thus 28 evades the Weinberg-Witten theorem. Notice that even in theories with a local symmetry, such as QC D or GR, the Weinberg- Witten theorem does rule massless particles of higher spin t hat carry a conserved charge associated with a symmetry that commutes with the local symm etry. For instance, the au- thors of [4] point out that their result forbids QCD from havi ng ﬂavor non-singlet massless bound states with j≥1, since ﬂavor symmetries commute with the SU(3) local gauge symmetry. Similarly, a j= 1 gauge theory cannot produce composite gravitons with Lorentz-covariant spectra, because translations in ﬂat Mi nkowski space-time commute with the gauge symmetry. Gauge theories admit the conserved, Lor entz-covariant Belinfante- Rosenfeld stress-energy tensor ([15]). 2.6.4 Gravitons in string theory String theories have a massless spin-2 particle in their spe ctrum. This discovery killed the original versions of string theory as possible description s of the strong nuclear interaction (which was the context in which they had been proposed) and ma de modern string theory a candidate for a quantum theory of gravity (see, for instance , Chapter 1 in [16]). The reason why this result does not violate the Weinberg-Witten theore m is that it is not possible to deﬁne a conserved stress-energy tensor in string theory. Consider a string propagating in a D-dimensional background space-time with metric gab, wherea,b= 0,1,...D −1. IfSis the action in the background, then Tab=1√−gδS δgab(2.88) is not well deﬁned because a consistent string theory requir es imposing superconformal sym- metry on the background, which in turn automatically requir esgabto obey an equation of motion (at low energies this equation of motion corresponds to the Einstein ﬁeld equation of GR). The functional derivative in Eq.(2.88) cannot be deﬁ ned because there is no con- sistent oﬀ-shell deﬁnition of the background action S: The exact equation of motion for gab in string theory does not come from extremizing the action wi th respect to the background metric, but rather from a constraint required for consisten cy.10 In general, we expect that a theory with emergent diﬀeomorphism invariance would 10I thank John Schwarz for clarifying this point for me. 29 not have a stress-energy tensor. The reason is that in the low -energy eﬀective action (i.e., in GR) the graviton couples to a stress-energy tensor which i s not observable because it is not diﬀeomorphism-covariant. If the fundamental theory itself has no diﬀeomorphism invariance, then it should not have a stress-energy tensor a t all (see [17]). 2.7 Emergent gravity The Weinberg-Witten theorem can be read as the proof that mas sless particles of higher spin cannot carry conserved Lorentz-covariant quantities . Local gauge invariance and dif- feomorphism invariance are natural ways of making those qua ntities mathematically non- Lorentz-covariant without spoiling physical Lorentz cova riance. It is possible and interesting nonetheless, to consider other ways of accommodating massl ess mediators with higher spin. Despite the successes of gauge theories, the fact remains th at there is no clearly compelling a priori reason to impose local gauge invariance as an axiom, and that such an axiom has the unattractive consequence that it makes our mathematica l description of physical reality inherently redundant (see, for instance, Chapter III.4 in [ 18]). Also, while local gauge invariance guarantees renormaliza bility for spin 1, it is well known that quantizing hµνin linear gravity does not produce a perturbatively renorma lizable ﬁeld theory. One attractive solution to this problem would b e to make the graviton a composite, low-energy degree of freedom, with a natural cut oﬀ scale Λ UV. The Weinberg- Witten theorem represents a signiﬁcant obstruction to this approach, because the result applies equally to fundamental and to composite particles. Indeed, ruling out emergent gravitons was the authors’ purpose for establishing that th eorem. In a recent public lecture ([19]), Witten has made the strong claim that “whatever we do, we are not going to start with a conventional theory of n on-gravitational ﬁelds in Minkowski spacetime and generate Einstein gravity as an eme rgent phenomenon.” His reasoning is that identifying emergent phenomena requires ﬁrst deﬁning a box in 3-space and then integrating out modes with wavelengths shorter tha n the length of the edges of the box (see Fig. 2.2). But Einstein gravity implies diﬀeomo rphism invariance, and a general coordinate transformation spoils the deﬁnition of our box. Witten’s conclusion is that gravity can be emergent only if the notion on the space-t ime on which diﬀeomorphism invariance operates is simultaneously emergent. This is a p lausible claim, but it goes beyond 30 b^, 89[m฀ [ Figure 2.2: Schematic representation of Witten’s argument that a gener al coordinate transformation spoils the box used to deﬁne the modes that are integrated out in order to identify the emergent low-energy physics for energy scales well below Λ UV. what the Weinberg-Witten theorem actually establishes. In 1983, Laughlin explained the observed fractional quantu m Hall eﬀect in two-dimensional electronic systems by showing how such a system could form an incompressible quantum ﬂuid whose excitations have charge e/3 ([20]). That is, the low-energy theory of the inter- acting electrons in two spatial dimensions has composite de grees of freedom whose charge is a fraction of that of the electrons themselves. In 2001, Zh ang and Hu used techniques similar to Laughlin’s to study the composite excitations of a higher-dimensional system ([21]). They imagined a four-dimensional sphere in space, ﬁ lled with fermions that interact via anSU(2) gauge ﬁeld. In the limit where the dimensionality of the r epresentation of SU(2) is taken to be very large, such a theory exhibits composit e massless excitations of integer spin 1, 2 and higher. Like other theories from solid state physics, Zhang and Hu’s proposal falls outside the scope of the Weinberg-Witten theorem because the proposed t heory is not Lorentz-invariant: The vacuum of the theory is not empty and has a preferred rest- frame (the rest frame of the fermions). However, the authors argued that in the three -dimensional boundary of the four-dimensional sphere, a relativistic dispersion re lation would hold. One might then imagine that the relativistic, three-dimensional world we inhabit might be the edge of a four-dimensional sphere ﬁlled with fermions. Photons and g ravitons would be composite low-energy degrees of freedom, and the problems currently a ssociated with gravity in the UV would be avoided. The authors also argue that massless bos ons with spin 3 and higher might naturally decouple from other matter, thus explainin g why they are not observed in nature. 31 In Chapter 3 we will discuss another proposal, dating back to the work of Dirac ([22]) and Bjorken ([23]) for obtaining massless mediators as the Gold stone bosons of the spontaneous breaking of Lorentz violation. Such an arrangement evades t he Weinberg-Witten theorem because the Lorentz invariance of the theory is realized non -linearly in the Goldstone bosons. Therefore the matrix elements in Eq. (2.69) will not be Loren tz-covariant. 32 Chapter 3 Goldstone photons and gravitons In this chapter we will address some issues connected with th e construction of models in which massless mediators are obtained as Goldstone bosons o f the spontaneous breaking of Lorentz invariance (LI). This presentation is based larg ely on previously published work [24, 25]. 3.1 Emergent mediators In 1963, Bjorken proposed a mechanism for what he called the “ dynamical generation of quantum electrodynamics” (QED) ([23]). His idea was to form ulate a theory that would reproduce the phenomenology of standard QED, without invok ing localU(1) gauge invari- ance as an axiom. Instead, Bjorken proposed working with a se lf-interacting fermion ﬁeld theory of the form L=¯ψ(i∂ /−m)ψ−λ(¯ψγµψ)2. (3.1) Bjorken then argued that in a theory such as that described by Eq. (3.1), composite “photons” could emerge as Goldstone bosons resulting from t he presence of a condensate that spontaneously broke LI. Conceptually, a useful way of understanding Bjorken’s prop osal is to think of it as as a resurrection of the “lumineferous æther” ([26, 27]): “empt y” space is no longer really empty. Instead, the theory has a non-vanishing vacuum expectation value (VEV) for the current jµ=¯ψγµψ. This VEV, in turn, leads to a massive background gauge ﬁeld Aµ∝jµ, as in the well-known London equations for the theory of superconduct ors ([28]). Such a background spontaneously breaks Lorentz invariance and produces thre e massless excitations of Aµ(the 33 Goldstone bosons) proportional to the changes δjµassociated with the three broken Lorentz transformations.1 Two of these Goldstone bosons can be interpreted as the usual transverse photons. The meaning of the third photon remains problematic. Bjorke n originally interpreted it as the longitudinal photon in the temporal-gauge QED, which becomes identiﬁed with the Coulomb force (see also [26]). More recently, Kraus and Tomb oulis have argued that the extra photon has an exotic dispersion relation and that its c oupling to matter should be suppressed ([30]). Bjorken’s idea might not seem attractive today, since a theo ry such as Eq. (3.1) is not renormalizable, while the work of ’t Hooft and others has demonstrated that a lo- cally gauge-invariant theory can always be renormalized ([ 5]). Furthermore, as detailed in Section 2.4, the gauge theories have had other very signiﬁca nt successes. Unless we take seriously the line of thought pursued in Chapter 2 that local gauge invariance is suspect because it is a redundancy of the mathematical description r ather than a genuine physical symmetry, there would not appear to be, at this stage in our un derstanding of fundamental physics, any compelling reason to abandon local gauge invar iance as an axiom for writing down interacting QFT’s.2Furthermore, the arguments for the existence of a LI-breaki ng condensate in theories such as Eq. (3.1) have never been soli d.3 In 2002 Kraus and Tomboulis resurrected Bjorken’s idea for a diﬀerent purpose of greater interest to contemporary theoretical physics: making a com posite graviton ([30]). They proposed what Bjorken might call “dynamical generation of g ravity.” In this scenario a composite graviton would emerge as a Goldstone boson from th e spontaneous breaking of Lorentz invariance in a theory of self-interacting fermion s. Being a Goldstone boson, such a graviton would be forbidden from developing a potential, t hus providing a solution to the “large cosmological constant problem:” the Λ hµ µtadpole term for the graviton would vanish without ﬁne-tuning (see Section 5.1). This scheme would als o seem to oﬀer an unorthodox avenue to a renormalizable quantum theory of gravity, becau se the fermion self-interactions 1In Bjorken’s work, Aµis just an auxiliary or interpolating ﬁeld. Dirac had discus sed somewhat similar ideas in [22], but, amusingly, he was trying to write a theory of electromagnetism with only a gauge ﬁeld and no fundamental electrons. In both the work of Bjorken and the work of Dirac, the proportionality between Aµandjµis crucial. 2According to Mark Wise, though, in the 1980’s Feynman consid ered Bjorken’s proposal as an alternative to postulating local gauge invariance. 3For Bjorken’s most recent revisiting of his proposal, in the light of the theoretical developments since 1963, see [29]. 34 could be interpreted as coming from the integrating out, at l ow energies, of gauge bosons that have acquired large masses via the Higgs mechanism, so t hat Einstein gravity would be the low energy behavior of a renormalizable theory. This p roposal would, of course, radically alter the nature of gravitational physics at very high energies. Related ideas had been previously considered in, for instance, [31]. In [30], the authors consider fermions coupled to gauge boso ns that have acquired masses beyond the energy scale of interest. Then an eﬀective low-en ergy theory can be obtained by integrating out those gauge bosons. We expect to obtain an eﬀective Lagrangian of the form L=¯ψ(i∂ /−m)ψ+∞/summationdisplay n=1λn(¯ψγµψ)2n +∞/summationdisplay n=1µn/bracketleftbigg ¯ψi 2(γµ→ ∂ν−γµ← ∂ν)ψ/bracketrightbigg2n +... , (3.2) where we have explicitly written out only two of the power ser ies in fermion bilinears that we would in general expect to get from integrating out the gau ge bosons. One may then introduce an auxiliary ﬁeld for each of these fer mion bilinears. In this example we shall assign the label Aµto the auxiliary ﬁeld corresponding to ¯ψγµψ, and the labelhµνto the ﬁeld corresponding to ¯ψi 2(γµ→ ∂ν−γµ← ∂ν)ψ. It is possible to write a Lagrangian that involves the auxiliary ﬁelds but not their derivatives, so that the cor- responding algebraic equations of motion relating each aux iliary ﬁeld to its corresponding fermion bilinear make that Lagrangian classically equival ent to Eq. (3.2). In this case the new Lagrangian would be of the form L′= (ηµν+hµν)¯ψi 2(γµ→ ∂ν−γµ← ∂ν)ψ−¯ψ(A /+m)ψ+... −VA(A2)−Vh(h2) +... , (3.3) whereA2≡AµAµandh2≡hµνhµν. The ellipses in Eq. (3.3) correspond to terms with other auxiliary ﬁelds associated with more complicated fer mion bilinears that were also omitted in Eq. (3.2). We may then imagine that instead of having a single fermion sp ecies we have one very heavy fermion, ψ1, and one lighter one, ψ2. Since Eq. (3.3) has terms that couple both 35 fermion species to the auxiliary ﬁelds, integrating out ψ1will then produce kinetic terms forAµandhµν. In the case of Aµwe can readily see that since it is minimally coupled to ψ1, the kinetic terms obtained from integrating out the latter must be gauge -invariant (provided a gauge- invariant regulator is used). To lowest order in derivative s ofAµ, we must then get the standard photon Lagrangian −F2 µν/4. SinceAµwas also minimally coupled to ψ2, we then have, at low energies, something that has begun to look like Q ED. IfAµhas a non-zero VEV, LI is spontaneously broken, producing th ree massless Gold- stone bosons, two of which may be interpreted as photons (see [30] for a discussion of how the exotic physics of the other extraneous “photon” can be su ppressed). The integrating out ofψ1and the assumption that hµνhas a VEV, by similar arguments, yield a low-energy approximation to linearized gravity. Fermion bilinears other than those we have written out expli citly in Eq. (3.2) have their own auxiliary ﬁelds with their own potentials. If those pote ntials do not themselves produce VEV’s for the auxiliary ﬁelds, then there would be no further Goldstone bosons, and one would expect, on general grounds, that those extra auxiliar y ﬁelds would acquire masses of the order of the energy-momentum cutoﬀ scale for our eﬀectiv e ﬁeld theory, making them irrelevant at low energies. The breaking of LI would be crucial for this kind of mechanism , not only because we know experimentally that photons and gravitons are massles s or very nearly massless, but also because it allows us to evade the Weinberg-Witten theor em ([4]), as we discussed in Section 2.7. Let us concentrate on the simpler case of the auxiliary ﬁeld Aµ. For the theory described by Eq. (3.3), the equation of motion for Aµis ∂L′ ∂Aµ=−¯ψγµψ−V′(A2)·2Aµ= 0. (3.4) Solving for ¯ψγµψin Eq. (3.4) and substituting into both Eq. (3.2) and Eq. (3.3 ) we see that the condition for the Lagrangians LandL′to be classically equivalent is a diﬀerential equation for V(A2) in terms of the coeﬃcients λn: V(A2) = 2A2[V′(A2)]−∞/summationdisplay n=1λn22nA2n[V′(A2)]2n. (3.5) 36 It is suggested in [30] that for some values of λnthe resulting potential V(A2) might have a minimum away from A2= 0, and that this would give the LI-breaking VEV needed. It seems to us, however, that a minimum of V(A2) away from the origin is not the correct thing to look for in order to obtain LI breaking. The Lagrangi an in Eq. (3.3) contains Aµ’s not just in the potential but also in the “interaction” ter mAµ¯ψγµψ, which is not in any sense a small perturbation as it might be, say, in QED. In o ther words, the classical quantityV(A2) is not a useful approximation to the quantum eﬀective poten tial for the auxiliary ﬁeld. In fact, regardless of the values of the λn, Eq. (3.5) implies that V(A2= 0) = 0, and also that at any point where V′(A2) = 0 the potential must be zero. Therefore, the existence of a classical extremum at A2=C∝ne}ationslash= 0 would imply that V(C) =V(0), and unless the potential is discontinuous somewhere, this would require t hatV′(and therefore also V) vanish somewhere between 0 and C, and so on ad inﬁnitum . Thus the potential Vcannot have a classical minimum away from A2= 0, unless the potential has poles or some other discontinuity. A similar observation applies to any fermion bilinear for wh ich we might attempt this kind of procedure and therefore the issue arises as well when dealing with the proposal in [30] for generating the graviton. It is not possible to sides tep this diﬃculty by including other auxiliary ﬁelds or other fermion bilinears, or even by imagining that we could start, instead of from Eq. (3.2), from a theory with interactions gi ven by an arbitrary, possibly non-analytic function of the fermion bilinear F(bilinear). The problem can be traced to the fact that the equation of motion of any auxiliary ﬁeld of t his kind will always be of the form 0 =−(bilinear) −V′(ﬁeld2)·2ﬁeld. (3.6) The point is that the vanishing of the ﬁrst derivative of the p otential or the vanishing of the auxiliary ﬁeld itself will always, classically, impl y that the fermion bilinear is zero. Classically at least, it would seem that the extrema of the po tential would correspond to the same physical state as the zeroes of the auxiliary ﬁeld. 37 3.2 Nambu and Jona-Lasinio model (review) The complications we have discussed that emerge when one tri es to implement LI breaking as proposed in [30] do not, in retrospect, seem entirely surp rising. A VEV for the auxiliary ﬁeld would classically imply a VEV for the corresponding fer mion bilinear, and therefore a trick such as rewriting a theory in a form like Eq. (3.3) shoul d not, perhaps, be expected to uncover a physically signiﬁcant phenomenon such as the sp ontaneous breaking of LI for a theory where it was not otherwise apparent that the fermion bilinear in question had a VEV. Let us therefore turn our attention to considering what would be required so that one might reasonably expect a fermion ﬁeld theory to exhibit the kind of condensation that would give a VEV to a certain fermion bilinear. If we allowed ourselves to be guided by purely classical intu ition, it would seem likely that a VEV for a bilinear with derivatives (such as ¯ψi 2(γµ→ ∂ν−γµ← ∂ν)ψ) might require non- standard kinetic terms in the action. Whether or not this int uition is correct, we abandon consideration of such bilinears here as too complicated. The simplest fermion bilinear is, of course, ¯ψψ. Being a Lorentz scalar, ∝an}b∇acketle{t¯ψψ∝an}b∇acket∇i}ht ∝ne}ationslash= 0 will not break LI. This kind of VEV was treated back in 1961 by Nambu and Jona-Lasinio, who used it to spontaneously break chiral symmetry in one of t he early eﬀorts to develop a theory of the strong nuclear interactions, before the adve nt of quantum chromodynamics (QCD) ([32]). It might be useful to review the original work o f Nambu and Jona-Lasinio, as it may shed some light on the study of the possibility of giv ing VEV’s to other fermion bilinears that are not Lorentz scalars. In their original paper, Nambu and Jona-Lasinio start from a self-interacting massless fermion ﬁeld theory and propose that the strong interaction s be mediated by pions, which appear as Goldstone bosons produced by the spontaneous brea king of the chiral symmetry associated with the transformation ψ∝ma√sto→exp (iαγ5)ψ. This symmetry breaking is produced by a VEV for the fermion bilinear ¯ψψ. In other words, Nambu and Jona-Lasinio originally proposed what, by close analogy to Bjorken’s idea, would be t he “dynamical generation of the strong interactions.”4 Nambu and Jona-Lasinio start from a non-renormalizable qua ntum ﬁeld theory with a 4Historically, though, Bjorken was motivated by the earlier work of Nambu and Jona-Lasinio. 38/A1 /BP/A2 /B7/A3 /BD/C8/C1 /BC/BP/A4 /B7/A5 /BD/C8/C1 /BC/B7/A6 /BD/C8/C1 /BC/BD/C8/C1 /BC/B7 /BM /BM /BM Figure 3.1: Diagrammatic Schwinger-Dyson equation. The double line re presents the primed prop- agator, which incorporates the self-energy term. The singl e line represents the unprimed propagator. 1PI′stands for the sum of one-particle irreducible graphs with t he primed propagator. four-fermion interaction that respects chiral symmetry: L=i¯ψ∂ /ψ−g 2[(¯ψγµψ)2−(¯ψγµγ5ψ)2]. (3.7) In order to argue for the presence of a chiral symmetry-break ing condensate in the theory described by Eq. (3.7), Nambu and Jona-Lasinio borro wed the technique of self- consistent ﬁeld theory from solid state physics (see, for in stance, [33]). If one writes down a Lagrangian with a free and an interaction part, L=L0+Li, ordinarily one would then proceed to diagonalize L0and treat Lias a perturbation. In self-consistent ﬁeld theory one instead rewrites the Lagrangian as L= (L0+Ls) + (Li− Ls) =L′ 0+L′ i, where Lsis a self-interaction term, either bilinear or quadratic in the ﬁelds, such that L′ 0yields a linear equation of motion. Now L′ 0is diagonalized and L′ iis treated as a perturbation. In order to determine what the form of Lsis, one requires that the perturbation L′ inot produce any additional self-energy eﬀects. The name “self- consistent ﬁeld theory” reﬂects the fact that in this technique Liis found by computing a self-energy via a perturbative expansion in ﬁelds that already are subject to that self-ene rgy, and then requiring that such a perturbative expansion not yield any additional self-ene rgy eﬀects. Nambu and Jona-Lasinio proceed to make the ansatz that for Eq . (3.7) the self- interaction term will be of the form Ls=−m¯ψψ. Then, to ﬁrst order in the coupling constantg, they proceed to compute the fermion self-energy Σ′(p), using the propagator S′(p) =i(p /−m)−1, which corresponds to the Lagrangian L′ 0=¯ψ(i∂ /−m)ψthat incorporates the proposed self-energy term. The next step is to apply the self-consistency condition usi ng the Schwinger-Dyson 39/A0 /CX /A6 /BC/BP/A1 /BD/C8/C1 /BC/BP/A2 /B7 /C7 /B4 /CV /BE/B5 Figure 3.2: Diagrammatic equation for the primed self-energy. We will w ork to ﬁrst order in the fermion self-coupling constant g. equation for the propagator S′(x−y) =S(x−y) +/integraldisplay d4zS(x−z)Σ′(0)S′(z−y), (3.8) which is represented diagrammatically in Fig. 3.1. The prim es indicate quantities that correspond to a free Lagrangian L′ 0that incorporates the self-energy term, whereas the unprimed quantities correspond to the ordinary free Lagran gianL0. For Σ′we will use the approximation shown in Fig. 3.2, valid to ﬁrst order in the co upling constant g. After Fourier transforming Eq. (3.8) and summing the left si de as a geometric series, we ﬁnd that the self-consistency condition may be written, i n our approximation, as m= Σ′(0) =gmi 2π4/integraldisplayd4p p2−m2+iǫ. (3.9) If we evaluate the momentum integral by Wick rotation and reg ularize its divergence by introducing a Lorentz-invariant energy-momentum cutoﬀ p2<Λ2we ﬁnd 2π2m gΛ2=m/bracketleftbigg 1−m2 Λ2log/parenleftbiggΛ2 m2+ 1/parenrightbigg/bracketrightbigg . (3.10) This equation will always have the trivial solution m= 0, which corresponds to the vanishing of the proposed self-interaction term Li. But if 0<2π2 gΛ2<1 (3.11) then there may also be a non-trivial solution to Eq. (3.10), i .e., a non-zero mfor which the condition of self-consistency is met. For a rigorous treatm ent of the relation between non- trivial solutions of this self-consistent equation and loc al extrema in the Wilsonian eﬀective potential for the corresponding fermion bilinears, see [39 ] and the references therein. 40 In this model (which from now on we shall refer to as NJL), we se e that if the interaction between fermions and antifermions is attractive ( g >0) and strong enough (2π2 gΛ2<1) it might be energetically favorable to form a fermion-antifer mion condensate. This is reason- able to expect in this case because the particles have no bare mass and thus the energy cost of producing them is small. The resulting condensate wo uld have zero net charge, as well as zero total momentum and spin. Therefore it must pai r a left-handed fermion ψL=1 2(1−γ5)ψwith the antiparticle of a right-handed fermion ψR=1 2(1+γ5)ψ, and vice versa. This is the mass-term self-interaction Li=−m¯ψψ=−m(¯ψLψR+¯ψRψL) that NJL studies. After QCD became the accepted theory of the strong interacti ons, the ideas behind the NJL mechanism remained useful. The uanddquarks are not massless (nor is u-dﬂavor isospin an exact symmetry) but their bare masses are believe d to be quite small compared to their eﬀective masses in baryons and mesons, so that the form ation of ¯uuand¯ddcondensates represents the spontaneous breaking of an approximate chir al symmetry. Interpreting the pions (which are fairly light) as the pseudo-Goldstone boso ns generated by the spontaneous breaking of the approximate SU(2)R×SU(2)Lchiral isospin symmetry down to just SU(2), proved a fruitful line of thought from the point of view of the phenomenology of the strong interaction.5 Condition Eq. (3.11) has a natural interpretation if we thin k of the interaction in Eq. (3.7) as mediated by massive gauge bosons with zero momentum and coupling e. For it to be reasonable to neglect boson momentum in the eﬀective th eory, the mass µof the bosons should be µ>Λ. Ife2<2π2theng=e2/µ2<2π2/Λ2, which violates Eq. (3.11). Therefore for chiral symmetry breaking to happen, the coupl ingeshould be quite large, making the renormalizable theory nonperturbative. This is acceptable because the factor of 1/µ2allows the perturbative calculations we have carried out in the eﬀective theory Eq. (3.7). This is why the NJL mechanism is modernly thought of as a model for a phenomenon of non-perturbative QCD. 5For a treatment of this subject, including a historical note on the inﬂuence of the NJL model in the development of QCD, see Chap. 19, Sec. 4 in [38]. 41 3.3 An NJL-style argument for breaking LI We have reviewed how NJL formulated a model that exhibited a n on-zero VEV for the fermion bilinear ¯ψψ. The next simplest fermion bilinear that we might consider i s¯ψγµψ, which was the one that Bjorken, Kraus, and Tomboulis conside red when they discussed the “dynamical generation of QED.” This particular fermion bil inear is especially interesting because it corresponds to the U(1) conserved current, and also because it is the simplest bilinear with an odd number of Lorentz tensor indices, so tha t a non-zero VEV for it would break not only LI but also charge (C), charge-parity (CP), an d charge-parity-time (CPT) reversal invariance. C and CP may not be symmetries of the Lag rangian, as indeed they are not in the standard model, but by a celebrated result CPT m ust be an invariance of any reasonable theory (see [41] and references therein). This i nvariance, however, may well be spontaneously broken, as it would be by any VEV with an odd num ber of Lorentz indices. Before proceeding, however, it may be advisable to try to dev elop some physical intuition about what would be required for a fermion bilinear like ¯ψγµψto exhibit a VEV. If we choose a representation of the gamma matrix algebra and use i t to write out ( ¯ψγµψ)2for an arbitrary Dirac bispinor ψ, we may check that ( ¯ψγµψ)2≥0 for the choice of mostly negative metric gµν= diag(1,−1,−1,−1). That is, ¯ψγµψis time-like. This has an intuitive explanation, based on the observation that ¯ψγµψis a conserved fermion-number current density. Classically a charge density ρmoving with a velocity vwill produce a current jµ= (ρ,ρv) (in units of c= 1). Therefore the relativistic requirement that the charg e density not move faster than the speed of light in any frame of reference implies that j2≥0. Considerations of causality make it natural to expect tha t something similar would be true of ¯ψγµψ. For any time-like Lorentz vector nµit is possible to ﬁnd a Lorentz transformation that maps it to a vector n′µwith only one non-vanishing component: n′0. For a constant current densityjµ, this means that for jµto be non-zero there must be a charge density j0, which has a rest frame. Therefore we only expect to see a VEV for ¯ψγµψif our theory somehow has a vacuum with a non-zero fermion number density. The consequ ent spontaneous breaking of LI may be seen as the introduction of a preferred reference frame: the rest frame of the vacuum charge. In the literature of ﬁnite density quantum ﬁeld theory and of color superconductivity 42 zero density finite density particlehole (antiparticle) E=0 Figure 3.3: Fermion and antifermion energies in QFT, at zero density (le ft) and at ﬁnite density (right). Finite density introduces a chemical potential te rm−f·¯ψγ0ψinto the fermion Lagrangian. (see, for instance, [34] and [35]), the Lagrangians discuss ed are explicitly non-Lorentz- invariant because they contain chemical potential terms of the formf·¯ψγ0ψ. This term appears in theories whose ground state has a non-zero fermio n number because, by the Pauli exclusion principle, new fermions must be added just a bove the Fermi surface, i.e., at energies higher than those already occupied by the pre-ex isting fermions, while holes (which can be thought of as antifermions) should be made by re moving fermions at that Fermi surface. The result is an energy shift that depends on t he number of fermions already present and which has opposite signs for fermions and antife rmions, as illustrated in Fig. 3.3. The physical picture that emerges is now, hopefully, cleare r: A theory with a VEV for ¯ψγµψis one with a condensate that has non-zero fermion number. Th is means that only theories with some form of attractive interaction between p articles with the same sign in fermion number may be expected to produce such a VEV. The situ ation is closely analogous to BCS superconductivity ([40]), in which a phonon-mediate d attractive interaction between electrons allows the presence of a condensate with non-zero electric charge. Note that in the NJL model, the condensate was composed of fermion-antif ermion pairs, and therefore clearly ∝an}b∇acketle{t¯ψγ0ψ∝an}b∇acket∇i}ht= 0, which implies ∝an}b∇acketle{t¯ψγµψ∝an}b∇acket∇i}ht= 0. It should now be clear why a VEV for ¯ψγµψ would break not only LI but also C, CP, and CPT. This picture al so helps to clarify the nature of the Goldstone bosons that we will be invoking as med iators of the electromagnetic interaction: They are density waves in the background “Dira c sea,” whose energy at inﬁnite wavelengths vanishes because they are then proportional to the broken boosts. 43 There is an easy way to write a theory that will have a VEV for a U(1) conserved current: to couple a massive photon to such a current via a pur ely imaginary charge. To see this, let us write a Proca Lagrangian for a massive photon ﬁeld with an external source: L=−1 4F2 µν+µ2 2A2−jµAµ. (3.12) The equation of motion for the photon ﬁeld is ∂µFµν=jν−µ2Aν. (3.13) At energy scales well below the photon mass µ, the kinetic term −F2 µν/4 may be ne- glected with respect to the mass term µ2A2/2. We may then integrate out the photon at zero momentum by solving the equation of motion Eq. (3.13) fo r the photon ﬁeld Aµwith its conjugate momenta Fµνset to zero, and substituting the result back into the Lagran gian in Eq. (3.12). The resulting low-energy eﬀective ﬁeld theor y has the Hamiltonian Heﬀective =j2 2µ2. (3.14) Nothing interesting happens if the source is a timelike curr ent density, since in that case Eq. (3.14) has its minimum at jµ= 0. But if we were to make the charge coupling to the photon imaginary (e.g., jµ=ie¯ψγµψforereal), then j2is actually always negative (recall that ( ¯ψγµψ)2is always positive) and we get a “potential” with the wrong si gn, so that the energy can be made arbitrarily low by decreasing j2. If we make jµdynamical by adding to the Lagrangian terms corresponding to the ﬁeld that sets up t he current, we might expect, for certain parameters in the theory, that the energy be mini mized for a ﬁnite value of jµ. By making the charge purely imaginary, our eﬀective theory a t energy scales much lower than the photon mass µwill look similar to Eq. (3.7), except that the four-fermion interaction in the eﬀective Lagrangian will be e2(¯ψγµψ)2/2µ2(with an overall positive, rather than a negative, sign). What this means is that fermio ns are attracting fermions and antifermions are attracting antifermions, rather than wha t we had in NJL (and in QED): attraction between a fermion and an antifermion. Condensat ion, if it occurs, will here produce a net fermion number, spontaneously breaking C, CP, and CPT.6 6Dyson argued that a theory with a long-range attraction betw een particles of the same fermion number 44/A1=/A2+/A3 Figure 3.4: The four-fermion vertex in the self-interacting theory may be seen as the sum of two photon-mediated interactions with a massive photon that ca rries zero momentum and is coupled to the fermion via a purely imaginary charge. Let us analyze this situation again more rigorously using se lf-consistent ﬁeld theory methods, following Nambu and Jona-Lasinio. For this we cons ider a fermion ﬁeld with the usual free Lagrangian L0=¯ψ(i∂ /−m0)ψand pose as our self-consistent ansatz: Ls=−(m−m0)¯ψψ−f¯ψγ0ψ. (3.15) The corresponding momentum-space propagator for L′ 0=L0+Lsis, therefore, S′(k) =i(k /−fγ0−m)−1. (3.16) Now let us suppose that the interaction term looks like Li=g 2(¯ψγµψ)2. (3.17) To obtain the Feynman rules corresponding to Eq. (3.17) we no te that this is what we would obtain in massive QED if we replaced the charge ebyieand the usual photon propagator by igµν/µ2, withg=e2/µ2. Therefore to compute the self-energy we will rely on the identity represented in Fig. 3.4. (In QED the second di agram on the right-hand side of Fig. 3.4 would vanish by Furry’s theorem, but in our case th e propagator in the loop will have a chemical potential term that breaks the C invaria nce on which Furry’s theorem depends.) would be unstable and used this to suggest that perturbative series in QED would diverge after renormaliza- tion of the charge and mass [42]. As we will see at the end of thi s section, the “photon” mass µwill prevent the instability in our case. 45 To leading order in g, the self-energy is Σ(0) = 2ig/integraldisplayd4k (2π)43(k0−f)γ0+ 3kiγi−2m k2 0−k2−m2+f2−2fk0+iǫσ(3.18) whereσ(a function of |k|,f, andm) takes values ±1 so as to enforce the standard Feynman prescription for shifting the k0poles: positive k0poles are shifted down from the real line, while negative poles are shifted up. At ﬁrst sight it might appear as if the self-energy in Eq. (3.1 8) could not be used to argue for the breaking of LI, because the shift in the integra tion variable k∝ma√sto→k′= (k0−f,k) would wipe out fdependence. This, however, is not the case, as we will see. We may carry out thedk0integration, for which we must ﬁnd the corresponding poles. These are located at k0=f±/radicalbig k2+m2. (3.19) From now on, without loss of generality, we will take fto be positive. The contour integral that results from closing the d0kintegral of Eq. (3.18) in the complex plane will vanish unless f <√ k2+m2, because otherwise both poles in Eq. (3.19) will lie on the same side of the imaginary axis. In light of the Feynman presc ription used for the shifting of the poles away from the real axis, it would then be possible to close the contour at inﬁnity so that there would be no poles in the interior. The pole-shif ting prescription, through its eﬀect on the dk0integral, is what introduces an actual fdependence into the expression for the self-energy. By the Cauchy integral formula, we have Σ(0) =−g 4π3/integraldisplay d3k/bracketleftigg 3√ k2+m2γ0+ 2m 2√ k2+m2 ×θ(/radicalbig k2+m2−f)−3 2γ0/bracketrightbigg , (3.20) where the second term in the right-hand side subtracts the co ntribution from closing the contour out at inﬁnity in the complex plane (note the branch c ut in the logarithm that results from computing that part of the contour integral exp licitly). We will introduce the cutoﬀ k2<Λ2to make the integral in Eq. (3.20) ﬁnite.7 7Carrying out the dk0integration separately from the spatial integral is legiti mate and useful in light of the form of Eq. (3.18), which does not lend itself naturally t o Wick rotation. But the use of a non-Lorentz- 46 -2000 -1000 0 1000 2000-1000-500050010001500 m (a)-2 -1 0 1 205101520 m (b)-2000 -1000 0 1000 2000-1000-500050010001500 m (c) -2000 -1000 0 1000 2000-1000-500050010001500 m (d)-2 -1 0 1 205101520 m (e)-2 -1 0 1 205101520 m (f) Figure 3.5: Plots of the left-hand side (in gray) and right-hand side (in black) of equation Eq. (3.25). Deﬁne α≡g 2π2. For each plot the parameters are: (a) Λ = 100, m0= 0,α= 0.001. (b) Λ = 100,m0= 15,α= 0.001. (c) Λ = 100, m0= 1200,α= 0.001. (d) Λ = 100, m0= 0,α= 0.002. (e) Λ = 100, m0= 15,α= 0.002. (f) Λ = 200, m0= 15,α= 0.001. Note that the Heaviside step function θ(√ k2+m2−f) in Eq. (3.20) is always unity if m>f , so that there will be no fdependence at all in Eq. (3.20) unless m≤f. Assuming thatm≤fwe have Σ(0) =−g 2π2/bracketleftig −(f2−m2)3/2γ0+m3log (f+/radicalbig f2−m2) −m3log (Λ +/radicalbig Λ2+m2) +mΛ/radicalbig Λ2+m2−mf/radicalbig f2−m2/bracketrightbigg . (3.21) As before, we use the Schwinger-Dyson equation Eq. (3.8), an d after summing up the invariant regulator may cause concern that any breaking of L I we might arrive at could be an artifact of our choice of regulator. An alternative is to regulate Eq. (3 .20) dimensionally by replacing d3kwithdd−1k. The resulting equations are more complicated and the depend ence on the range of energies where our non- renormalizable theory is valid is obscured, but the overall argument does not change. It is also possible to multiply the integrand in Eq. (3.18) by a cutoﬀ in Minkowski s paceθ(Λ2+k2) =θ(Λ2+k2 0−k2). For k2<Λ2 we get the same result as in Eq. (3.20). For k2>Λ2we must impose the condition that k2 0>k2−Λ2. It should be pointed out that previous work on LI breaking has used 3-momentum cutoﬀs in computing self-energies [56], although in that case there seems to be a physical interpretation for such a cutoﬀ which does not apply to the present discussion. The original work o f Nambu and Jona-Lasinio [32] considers cutoﬀs in Euclidean 4-momentum and in 3-momentum, arriving in both cases at similar conclusions. 47 right-hand side as a geometric series, we arrive at the self- consistency condition for our ansatz Eq. (3.15): m0−m−fγ0=−Σ(0) =g 2π2/bracketleftig −(f2−m2)3/2γ0 +m3log/parenleftigg f+/radicalbig f2−m2 Λ +√ Λ2+m2/parenrightigg +mΛ/radicalbig Λ2+m2 −mf/radicalbig f2−m2/bracketrightbigg . (3.22) Clearly Eq. (3.22) will not admit a non-trivial solution f∝ne}ationslash= 0 unlessgis positive, which agrees with our intuition that the theory must exhibit attra ction between particles of the same fermion number. The self-consistent condition Eq. (3. 22) may be separated into two simultaneous equations: f=g 2π2(f2−m2)3/2(3.23) and m0−m=gm 2π2/bracketleftigg m2log/parenleftigg f+/radicalbig f2−m2 Λ +√ Λ2+m2/parenrightigg + Λ/radicalbig Λ2+m2−f/radicalbig f2−m2/bracketrightbigg . (3.24) It is important to bear in mind that Eqs. (3.23) and (3.24) wer e written under the assump- tion thatf≥m. Forf <m thefdependence of the self-energy in Eq. (3.18) disappears. The trivial, Lorentz-invariant solution f= 0 to the self-consistent equations will always be present for any m, as should be the case when spontaneous breaking of a symmetr y is observed. Equation (3.23) can be readily solved for fas a function of m(imposing the condition thatfbe real and positive), and the resulting f(m) can be substituted into Eq. (3.24) to 48 yield m0−m=gm 2π2/bracketleftigg m2log/parenleftigg f(m) +/radicalbig f2(m)−m2 Λ +√ Λ2+m2/parenrightigg + Λ/radicalbig Λ2+m2−f(m)/radicalbig f2(m)−m2/bracketrightbigg . (3.25) Equation (3.25) cannot be solved algebraically, but we may s tudy some of its properties graphically. In Fig. 3.5 we have plotted the left-hand side a nd the right-hand side of Eq. (3.25) for various values of the parameters g,m0, and Λ. As plot (a) illustrates, m0= 0 impliesm= 0, i.e., we cannot dynamically generate both a chemical pot ential and a mass term. Form=m0= 0 we have f=π/radicalbig 2/g. (3.26) Plot (b) in Fig. 3.5 shows a 0 < m 0≪Λ for which the corresponding mwill be signiﬁcantly less than m0. Plot (c) in the same ﬁgure illustrates that a very large m0is needed before m>m 0, but such solutions are not physically meaningful because m0itself is already well beyond the energy scale for which our eﬀectiv e theory is supposed to hold. By comparing plot (b) to plot (e) we may see the eﬀect of increa singgfor a given m0and Λ. A comparison of plots (b) and (f) should illustrate the eﬀe ct of increasing Λ with the other parameters ﬁxed. The plots in Fig. 3.6 illustrate the progression, as the para meter Λ is increased for ﬁxedα, from an unstable theory in which bare masses m0on the order of Λ are mapped to m>Λ, to a theory that maps such bare masses to m<Λ. Such an analysis of Eq. (3.25) reveals that the condition for this mass stability is 0<2π2 gΛ2<1, (3.27) which is reminiscent of the condition Eq. (3.11) for chiral s ymmetry breaking in the NJL model (except that now the interaction has the opposite sign ). Combining Eq. (3.27) with Eq. (3.26) (which was exact for m0but may serve approximately for m0small) we arrive at the requirement 0<f2<Λ2, (3.28) which would surely have to hold if our theory were stable. Ind eed, we may interpret Eq. 49 -10 -5 0 5 10-7.5-5-2.502.557.510 m (a)-20 -10 0 10 20-15-10-505101520 m (b) -20 -10 0 10 20-15-10-505101520 m (c)-20 -10 0 10 20-15-10-505101520 m (d) Figure 3.6: Plots of the left-hand side (in gray) and right-hand side (in black) of equation Eq. (3.25). For all of them α≡g 2π2= 0.01. (a) Λ = m0= 2. (b) Λ = m0= 8. (c) Λ = m0= 12. (d) Λ =m0= 16. (3.28) as saying that if we pick physically good parameters g,m0, and Λ we will have a stable theory with ﬁnite chemical potential f. The parameters for plots (a), (b), (d), (e), and (f) in Fig. 3.5 all give examples of such stable theories. As in NJL, the good parameters involveg−1/2large with respect to Λ, suggesting that Eq. (3.17) should be a low-energy approximation to a non-perturbative interaction of a full r enormalizable theory that allows attraction between particles of the same fermion number sig n. The issue of how the form of the self-consistent equations wi ll depend on the choice of regulator for the integral in Eq. (3.18) is not an entirely st raightforward matter. But it seems to be a solid conclusion that, for positive fermion sel f-couplingg, the solutions to such self-consistent equations show the presence of LI-bre aking vacua. In the next section of this paper we oﬀer an alternative approach that strengthe ns this conclusion and that sheds further light on the issue of stability. 50 Γ[A] =V(A) +/A1+/A2+/A3+... Figure 3.7: Correction of the eﬀective potential of the auxiliary ﬁeld Aµfrom integrating out the fermion. The ﬁrst graph does not contribute by the Ward ident ity, while the second vanishes by Furry’s theorem. 3.4 Consequences for emergent photons The theory L=¯ψ(i∂ /−m0)ψ+g 2(¯ψγµψ)2(3.29) is equivalent to L′=¯ψ(i∂ /−A /−m0)ψ−A2 2g. (3.30) Since we argued that Eq. (3.29) may spontaneously break LI by giving a ﬁnite ∝an}b∇acketle{t¯ψγµψ∝an}b∇acket∇i}ht, we conclude that Aµin Eq. (3.30) would also have a ﬁnite VEV, since, by the algebr aic equation of motion, Aµ=−g¯ψγµψ. (3.31) This interpretation agrees with the observation that Eq. (3 .30) has a vector boson ﬁeld whose mass term carries the wrong sign if g >0, indicating that the zero-ﬁeld state is not a good vacuum. To ﬁnd the correct vacuum for the theory we must carry out the path integral over the fermion ﬁeld to obtain the eﬀective action Γ[A], and then minimize that quantity. Figure 3.7 shows the radiative corrections to Γ[ A] as a perturbative series, in terms of Feynman diagrams. The ﬁeld Aµis minimally coupled to ψ, so that the computation should proceed as in QED. By the Ward identity we do not expect a correction to the mass term forAµ, as long as an adequate regulator is used. But we do expect to g et terms in the eﬀective action that go as A4and higher even powers of the auxiliary ﬁeld. Since we have reason to believe that QED is stable for any valu e of the charge e, it therefore seems logical to expect that the eﬀective action f orAµin Eq. (3.30) gives it a ﬁnite time-like VEV, which would imply a ﬁnite VEV for ¯ψγµψin the theory of Eq. (3.29). We argued in the previous section that gmust be large for the theory described by Eq. (3.29) to be stable. This too seems natural in light of Eq. (3. 30), because a large gmakes 51 9→ ' Figure 3.8: Radiative corrections make the eﬀective potential Γ[ A] stable and give Aµa non-zero VEV. theA2term small, so that the instability created by it may be easil y controlled by the interaction with the fermions, yielding a VEV for Aµthat lies within the energy range of the eﬀective theory. Figure 3.8 schematically represents h ow the radiative corrections to the eﬀective action give a ﬁnite VEV for Aµ. Armed with Eq. (3.30) it would seem possible to carry out the p rogram proposed by Bjorken, and by Kraus and Tomboulis, in order to arrive at an a pproximation of QED in which the photons are composite Goldstone bosons. It is conc eivable that a complicated theory of self-interacting fermions, perhaps one with non- standard kinetic terms, might sim- ilarly yield a VEV for ¯ψi 2(γµ→ ∂ν−γµ← ∂ν)ψ, allowing the project of dynamically generating linearized gravity to go forward. It would have been more encouraging if we had been able to obta in a non-zero/angbracketleftbig¯ψγµψ/angbracketrightbig through a more natural mechanism than invoking an imaginary charge. Non-abelian gauge theories (such as QCD) exhibit attraction between particle s of the same fermion number (and, like abelian theories with imaginary charge, they exh ibit anti-screening). So far, however, attempts to ﬁnd a non-abelian gauge theory with non -zero/angbracketleftbig¯ψγµψ/angbracketrightbig have failed, possibly because in such theories the attraction between fe rmion and antifermion is stronger than the attraction between fermions (see, for instance, [4 3]). 52 Chapter 4 Phenomenology of spontaneous Lorentz violation What lies behind the Principle of Relativity? This is a philo sophical question, not a scientiﬁc one. You will have your own opinion ; here is ours. We think the Principle of Relativity as used in special relat ivity rests on one word: emptiness. Space is empty. — Edwin F. Taylor and John A. Wheeler, Spacetime Physics , Chap. 3 4.1 Introduction Lorentz invariance (LI), the fundamental symmetry of Einst ein’s special relativity, states that physical results should not change after an experiment has been boosted or rotated. In recent years, and particularly since the publication of w ork on the possibility of sponta- neously breaking LI in bosonic string ﬁeld theory ([44]), th ere has been considerable interest in the prospect of violating LI. More recent motivations for work on Lorentz non-invariance have ranged from the explicit breaking of LI in the non-commu tative geometries that some have proposed as descriptions of physical space-time (see [ 45] and references therein), and in certain supersymmetric theories considered by the strin g community ([46, 47]), to the possibility of explaining puzzling cosmic ray measurement s by invoking small departures from LI ([48]) or modiﬁcations to special relativity itself ([49, 50, 51]). It has also been suggested that anomalies in certain chiral gauge theories m ay be traded for violations of LI and CPT ([52]). Extensions of the standard model have been proposed that are meant to capture the low-energy eﬀects of whatever new high-energ y physics (string theory, non- commutative geometry, loop quantum gravity, etc.) might be introducing violations of LI 53 ([53]). Our own investigation of composite massless mediators in Ch apters 2 and 3 led us to consider the question of how a reasonable QFT might spontane ously break LI through a timelike Lorentz vector VEV ∝an}b∇acketle{t¯ψγµψ∝an}b∇acket∇i}ht ∝ne}ationslash= 0. This breaking of LI can be thought of conceptu- ally as the introduction of a preferred frame: the rest frame of the fermion number density. If some kind of gauge coupling were added to the theory withou t destroying this LI breaking, the fermion number density would also be a charge density, an d the preferred frame would be the rest frame of a charged background in which all process es are taking place. This allows us to make some very general remarks in Section 4.2 on t he resulting LI-violating phe- nomenology for electrodynamics and on experimental limits to our non-Lorentz-invariant VEV. This discussion will be based on work previously publis hed in [24]. Experimental data put very tight constraints on Lorentz vio lating operators that involve Standard Model particles [66], but the bounds are more model -independent on Lorentz vio- lation that appears only in couplings to gravity [67, 68]. On e broad class of Lorentz-breaking gravitational theories are the so-called vector-tensor th eories in which the space-time met- ricgµνis coupled to a vector ﬁeld Sµthat does not vanish in the vacuum. Consideration of such theories dates back to [69] and their potentially obs ervable consequences are ex- tensively discussed in [70]. These theories have an unconst rained vector ﬁeld coupled to gravity. Theories with a unit constraint on the vector ﬁeld w ere proposed as a means of alleviating the diﬃculties that plagued the original uncon strained theories ([71]). The phenomenology of these theories with the unit constrain t has been recently explored. It has been proposed as a toy model for modifying dispersion r elations at high energy ([72]). The spectrum of long-wavelength excitations is discussed i n [73], where it was found that all polarizations have a relativistic dispersion relation , but travel with diﬀerent velocities. Applications of these theories to cosmology have been consi dered in [74, 75]. Constraints on these theories are weak, as for instance, there are no corr ections to the Post-Newtonian parameters γandβ([76]). The status of this class of theories, also known as “æ ther- theories,” is reviewed in [77]. In Section 4.4 we will show that the general low-energy eﬀect ive action at the two- derivative level of the Goldstones of spontaneous Lorentz v iolation by a timelike vector VEV minimally coupled to gravity corresponds to the vector- tensor theory of gravity with the unit constraint. This will allow us to place observation al constraints of very general 54 validity on this kind of Lorentz violation, from solar syste m tests of gravity. This discussion will be based on work previously published in [54]. Finally, in Section 4.5 we shall discuss the physical meaning of this kind of Lorentz violation and it s relation to some other models that have appeared recently in the literature. 4.2 Phenomenology of Lorentz violation by a background source Following up on the idea presented in Chapter 3, imagine that the fermions of the universe have some interaction that plays the role of Eq. (3.17) in giv ing a VEV to ¯ψγµψ, and that in addition they have a U(1) gauge coupling (at this stage we have abandoned the proje ct of producing composite photons). Then the U(1) gauge ﬁeld may interact with a charged background and we would be breaking LI in electrodynamics by introducing a preferred frame: the rest frame of the background source. The possibility of a vacuum that breaks LI and has non-trivia l optical properties has already been investigated in [55, 56]. This work, however, d eals with signiﬁcantly more complicated models, both in terms of the interactions that s pontaneously break LI and of the optical properties of the resulting vacuum. To obtain a p henomenology for our own simpler proposal, we consider a free photon Lagrangian of th e form Lphoton 0 =−1 4F2 µν−jµAµ, (4.1) wherejµ=e∝an}b∇acketle{t¯ψγµψ∝an}b∇acket∇i}ht, thought of as an external source. The corresponding propag ator for the free photon is ∝an}b∇acketle{tT{Aµ(x)Aν(y)}∝an}b∇acket∇i}ht=Dµν F(x−y) +∝an}b∇acketle{tAµ(x)∝an}b∇acket∇i}htj∝an}b∇acketle{tAν(y)∝an}b∇acket∇i}htj, (4.2) whereDµν(x−y) is the connected photon propagator and ∝an}b∇acketle{tAµ(x)∝an}b∇acket∇i}htjis the expectation value ofAµin the presence of the external source. If we takejµconstant and naively attempt to calculate the classical exp ectation value of Aµin the presence of a constant source by integrating the Green function for electrodynam- ics, we will get a volume divergence. We may attempt to regula te this volume divergence 55 by introducing a photon mass µ, which gives the result ∝an}b∇acketle{tAµ(x)∝an}b∇acket∇i}htj=jµ µ2. (4.3) (It is trivial to check that this is a solution to /squareAµ−µ2Aµ=−jµ, the wave equation for the massive photon ﬁeld with a source.) This is not satisfact ory because the disconnected term in Eq. (4.2) will be proportional to µ−4and Feynman diagrams computed with our modiﬁed photon propagator would produce results that depen d strongly on what we took for a regulator. In fact the mass is physical and analogous to the eﬀective photon mass ﬁrst described by the London brothers in their theory of the e lectromagnetic behavior of superconductors [28]. (Using the language of particle phys ics we may say that, in the presence of a U(1) gauge ﬁeld, the VEV ∝an}b∇acketle{t¯ψγµψ∝an}b∇acket∇i}htspontaneously breaks the gauge invariance and gives a mass to the boson, as in the Higgs mechanism.) Photons in a superconductor propagate through a constant el ectromagnetic source. In a simpliﬁed picture, we may think of it as a current density se t up by the motion of charge carriers of mass mand charge e, moving with a velocity u. The proper charge density is ρ0. The proper velocity of the charge carriers is ηµ= (1,u)/√ 1−u2. The source is then jµ=ρ0ηµ=ρ0pµ/m, wherepµis the classical energy momentum of the charge carriers. We may think of mandρ0as deriving from the solutions to the parameters in a self-co nsistent equation such as we had in Eq. (3.25). The canonical energy momentum Pµof the system is Pµ=mηµ+eAµ=mjµ/ρ0+eAµ. As is discussed in the superconductivity literature (see, f or instance, Chap. 8 in [57]), the superconducting state has zero canonical energy momentum, which leads to the London equation jµ=−eρ0 mAµ. (4.4) With thisjµinserted into the right-hand side of /squareAµ=−jµ(the wave equation for the photon ﬁeld in the Lorenz gauge), we ﬁnd that we have a solutio n to the wave equation of a massiveAµwith no source and a mass µ2=eρ0/m: /squareAµ−eρ0 mAµ= 0. (4.5) 56 If we solve for Aµin Eq. (4.4) and substitute this back into Eq. (4.2), we get th at ∝an}b∇acketle{tT{Aµ(x)Aν(y)}∝an}b∇acket∇i}ht=Dµν F(x−y) +m2 e2j2jµjν. (4.6) Notice that if jµ(x) is not constant, then Fourier transformation of the second term in Eq. (4.6) will not yield, in Feynman diagram vertices, the usual energy-momentum conserving delta function. Therefore, presumed small violations of en ergy or momentum conservation in electromagnetic processes could conceivably be paramet rized by the space-time variation of the background source.1 With Eq. (4.6) and a rule for external massive photon legs, on e may then go ahead and calculate the amplitude for various electromagnetic proce sses with this modiﬁed photon propagator, and parametrize supposed observed violations of LI (see [59, 60, 61]) by jµ. If we can make an estimate of the size of the mass mof the background charges, experimental limits on the photon mass ( <2×10−16eV according to [62]) will provide a limit on the VEV of ¯ψγµψ, in light of Eq. (4.4). There are other consequences of a VEV ∝an}b∇acketle{t¯ψγµψ∝an}b∇acket∇i}ht ∝ne}ationslash= 0 on which we may speculate. Such a background may have cosmological eﬀects, a line of thought t hat might connect, for instance, with [63]. Also, it is conceivable that such a VEV might have s ome relation to the problem of baryogenesis, since it gives the background ﬁnite fermion n umber and spontaneously breaks CPT, a violation that can ease the Sakharov condition of ther modynamical non-equilibrium [64, 65]. 4.3 Eﬀective action for the Goldstone bosons of spontaneous Lorentz violation Here we begin by considering the general low-energy eﬀectiv e action for a theory in which Lorentz invariance is spontaneously broken by the VEV of a Lo rentz four-vector Sµ. With an appropriate rescaling, the VEV satisﬁes ∝an}b∇acketle{tSµSµ∝an}b∇acket∇i}ht= 1, (4.7) 1This line of thought could connect to work on LI violation fro m variable couplings as discussed in [58]. 57 since we assume the VEV of Sµis time-like. The existence of this VEV implies that there exists a universal rest frame (which we sometimes refer to as the preferred frame) in which Sµ=δµ 0. When the resulting low-energy eﬀective action is minimall y coupled to gravity, we shall see that it simply becomes the vector-tensor theory with the unit constraint. Objects of mass M1andM2in a system moving relative to the preferred-frame can experience a modiﬁcation to Newton’s law of gravity of the fo rm ([70, 78]) UNewton =−GNM1M2 r/parenleftbigg 1−α2 2(w·r)2 r2/parenrightbigg , (4.8) where wis the velocity of the system under consideration, such as th e solar-system or Milky Way galaxy, relative to the universal rest frame. The m ain purpose of this note is to computeα2in theories where Lorentz invariance is spontaneously brok en by the VEV of a four-vector. The VEV of Sµspontaneously breaks Lorentz invariance. But as rotationa l invariance is preserved in the preferred frame, only the three boost gener ators of the Lorentz symmetry are spontaneously broken. The low-energy ﬂuctuations Sµ(x) which preserve Eq. (4.7) are the Goldstone bosons of this breaking, i.e., those that sati sfy Sµ(x)Sµ(x) = 1. (4.9) In the preferred-frame the ﬂuctuations can be parameterize d as a local Lorentz transforma- tion Sµ(x) = Λµ 0(x) =1/radicalbig 1−φ2 1 φ , (4.10) where φis as vector with components φ1,φ2, andφ3. Under Lorentz transformations Sµ(x)→Λµ νSν(x) and the symmetry is realized non- linearly on the ﬁelds φi. Using this ﬁeld Sµ(x) we may then couple the Goldstone bosons to Standard Model ﬁelds. Since however, the constraints on L orentz-violating operators2 involving Standard Model ﬁelds are considerable [66], we in stead focus on their couplings to gravity, which are more model-independent because they a re always present once the Goldstone bosons are made dynamical. 2More correctly, operators that appear to be Lorentz violati ng when the Goldstone bosons φiare set to zero. 58 The Goldstone bosons are made dynamical by adding in kinetic terms for them. Since Lorentz invariance is only broken spontaneously, the actio n for the kinetic terms should still be invariant under Lorentz transformations. The only interactions relevant at the two-derivative level and not eliminated by the constraint E q. (4.9) are3 L=c1∂αSβ∂αSβ+ (c2+c3)∂µSµ∂νSν+c4Sµ∂µSαSν∂νSα. (4.11) Expanding this action to quadratic order in φi, one ﬁnds that the four parameters cican be chosen to avoid the appearance of any ghosts. In particula r, we require c1+c4<0.4 To leading order, the eﬀective action for the Goldstone boso ns is: L=1 2/summationdisplay i=1,2,3/bracketleftig/parenleftbig ∂µφi/parenrightbig2−α/parenleftbig ∂iφi/parenrightbig2/bracketrightig (4.12) whereα≡(c2+c3)/c1. By inserting a plane wave ansatz, φi(xµ)∝exp/parenleftbig iωx0−ikx3/parenrightbig , we see that we have 2 transverse waves, φ1andφ2, with speed v=ω/k= 1, and one longitu- dinal wave, φ3, withv=√1 +α. Since we’ve broken LI, massless particles no longer need to travel at light speed. For α>0, the longitudinal Goldstone boson is superluminal. We shall return to the issue of superluminality in Section 4.5. This agrees with the result, discussed in [30] and in Chapter 3, that spontaneous Lorentz violation gives us not only two transverse Goldstone bosons (which we could identify as emergent photons) but also an extra polarization with an unu sual dispersion relation. In [30], where the Lorentz-breaking VEV was imagined to be spac elike, that extra polarization was timelike. In our case it is a longitudinal polarization b ecause the VEV in Eq. (4.7) was chosen to be timelike. 4.4 The long-range gravitational preferred-frame eﬀect With gravity present the situation is more subtle. One expec ts the gravitons to “eat” the Goldstone bosons, producing a more complicated spectru m [79, 80]. The covariant 3The other possible term, ǫµνρσ∂µSν∂ρSσ, is a total derivative. 4Notice that in our convention Sµis dimensionless and the ci’s have mass dimension two. 59 generalization of the constraint equation becomes gµν(x)Sµ(x)Sν(x) = 1 (4.13) and in the action for Sµwe replace∂µ→ ∇ µ. Note that there is no Higgs mechanism to give the graviton a ma ss. For a gauge theory we have the covariant derivative Dµ=∂µ−ieAµ, so that (Dµφ)2gives a term proportional φ2A2, i.e., a gauge boson mass, when ∝an}b∇acketle{tφ∝an}b∇acket∇i}ht ∝ne}ationslash= 0. For in the case of gravity coupled to a vector ﬁeld we have ∇µSν=∂µSν+ Γν ρµSρ, (4.14) with Γν ρµ=1 2/parenleftbig ∂ρhν µ+∂µhν ρ−∂νhρµ/parenrightbig (4.15) so that there is no way to get a term proportional to S2h2. Compare this the ghost condensate mechanism described in [8 1], where L=P(X) for X≡gµν∂µφ∂νφ. If we assume that P′(X=c2 ∗∝ne}ationslash= 0) = 0, (4.16) then, in the preferred frame, this implies that ∝an}b∇acketle{tX∝an}b∇acket∇i}ht=c2 ∗=/angbracketleftig ˙φ2/angbracketrightig ∝ne}ationslash= 0 (4.17) and theX2term inP(X) gives a graviton mass ˙φ4h2 00. This is diﬀerent from our case, where we get ﬁve massless graviton polarizations with diﬀer ent propagation velocities. Going back to our model, we see that local diﬀeomorphisms can be used to gauge away the three Goldstone bosons. For under a local diﬀeomorp hism (which preserves the constraint Eq. (4.13)), S′µ(x′) =∂x′µ ∂xνSν(x) (4.18) and withx′µ=xµ+ǫµ,Sµ≡vµ+φµ, φ′µ(x′) =φµ(x) +vρ∂ρǫµ(4.19) 60 from which we can determine ǫµto completely remove φµ. Note that in the preferred frame, ǫican be used to remove φi. In this gauge, the constraint Eq. (4.13) reduces to S0(x) = (1 −h00(x)/2). (4.20) The residual gauge invariance left in ǫ0can be used to remove h00. This is an inconvenient choice when the sources are static. In a more general frame wi th∝an}b∇acketle{tSµ∝an}b∇acket∇i}ht=vµ, obtained by a uniform Lorentz boost from the preferred frame, the constra int Eq. (4.13) is solved by Sµ(x) =vµ(1−vρvσhρσ(x)/2). (4.21) Next we discuss a toy model that provides an example of a more c omplete theory, that at low energies reduces to the theory described above with th e vector ﬁeld satisfying a unit covariant constraint (4.13).5Consider the following non-gauge-invariant theory for a ve ctor bosonAµ, L=−1 2gµνgρσ∇ρAµ∇σAν+λ/parenleftbig gµνAµAν−v2/parenrightbig2. (4.22) Fluctuations about the minimum are given by gµν=ηµν+hµν, Aµ=vµ+ψµ. (4.23) This theory has one massive state Φ with mass MΦ∝λ1/2v, which is Φ =vµψµ+hµνvµvν/2. (4.24) In the limit that λ→ ∞ this state decouples from the remaining massless states. In the preferred frame the only massless states are hµν, andψi. Since we have decoupled the heavy state, we should expand A0=v+/bracketleftbig ψ0+vh00/2/bracketrightbig −vh00/2→v−vh00/2, (4.25) where in the last limit we have decoupled the heavy state. Not e that this parameterization ofA0is precisely the same parameterization that we had above for S0. In other words, 5For a related example, see [80]. 61 in the limit that we decouple the only heavy state in this mode l, the ﬁeld Aµsatisﬁes gµνAµAν=v2, which is the same as the constraint (4.13) with Aµ→vSµ. In the unitary gauge with φi= 0, the only massless degrees of freedom are the gravitons. There are the two helicity modes, which in the Lorentz-invar iant limit correspond to the two spin-2 gravitons, along with three more helicities that are the Goldstone bosons, for a total of ﬁve. The sixth would-be helicity mode is gauged away by the remaining residual gauge invariance. But the model that we started from does have a ghost, since we w rote a kinetic term forAµthat does not correspond to the conventional Maxwell kineti c action. The ghost in the theory is A0, which in our case is massive. The presence of this ghost mean s that this ﬁeld theory model is not a good high-energy completion for th e low-energy theory involving onlySµand gravity that we are considering in this section. We assum e that a sensible high energy completion exists for generic values of the ci’s. Now we proceed to compute the preferred-frame coeﬃcient α2appearing in the modiﬁ- cation to Newton’s law. The action we consider is S=/integraldisplay d4x√g(LEH+LV+Lgf), (4.26) with6 LEH=−1 16πGR (4.27) and LV=c1∇αSβ∇αSβ+c2∇µSµ∇νSν+c3∇µSν∇νSµ+c4Sµ∇µSαSν∇νSα.(4.28) This is the most general action involving two derivatives ac ting onSµthat contributes to the two-point function. Note that a coeﬃcient c3appears, since in curved space-time covariant derivatives do not commute. Other terms involving two deriv atives acting on Sµmay be added to the action, but they are either equivalent to a combi nation of the operators already present (such as adding RµνSµSν), or they vanish because of the constraint Eq. (4.13). We 6The coeﬃcients ciappearing here are related to those appearing in, for exampl e [73], by chere i= −cthere i/16πG. 62 assume generic values for the coeﬃcients cithat in the low energy eﬀective theory give no ghosts or gradient instabilities. As previously discussed, Sµsatisﬁes the constraint (4.13). We also assume that it does not directly couple to Standard Model ﬁelds. In the literatu re, Eq. (4.13) is enforced by introducing a Lagrange multiplier into the action. Here we e nforce the constraint by directly solving forSµ, as given by Eq. (4.21), and then insert that solution back in to the action to obtain an eﬀective action for the metric. In our approach there is a residual gauge invariance that in t he preferred-frame corre- sponds to reparameterizations involving ǫ0only. To completely ﬁx the gauge we add the gauge-ﬁxing term Lgf=−α 2(SρSσSµ∂µhρσ)2. (4.29) Neglecting interaction terms, in the preferred frame the ga uge-ﬁxing term reduces to Lgf=−α 2(∂0h00)2. (4.30) Physically, this corresponds in the α→ ∞ limit to removing all time dependence in h00 without removing the static part, which is the gravitationa l potential. This is a convenient gauge in which to compute when the sources are static. At the two-derivative level, the only eﬀect in this gauge of t he new operators is to modify the kinetic terms for the graviton. The dispersion relation for the ﬁve helicities will be of the formE=β|k|, where the velocities βare not the same for all helicities and depend on the parameters ci([73]). This spectrum is diﬀerent than that which is found in the “ghost condensate” theory, where in addition to the two massless gr aviton helicities, there exists a massless scalar degree of freedom with a non-relativistic d ispersion relation E∝ |k|2([81]). There exists a range for the ci’s in which the theory has no ghosts and no gradient instabilities ([73]). In particular, for small ci’s, no gradient instabilities appear if c1+c2+c3 c1+c4>0 andc1 c1+c4>0. (4.31) The condition for having no ghosts is simply c1+c4<0. The correction to Newton’s law in Eq. (4.8) is linear order in the source. Thus to determine its size we only need to ﬁnd the graviton propagato r, since the non-linearity of 63 gravity contributes at higher order in the source. In order t o compute that term we have to specify a coordinate system, of which there are two natura l choices. In the universal rest frame, the sources, such as the solar system or Milky Way galaxy, will be moving and the computation is diﬃcult. We instead choose to compute in t he rest frame of the source, which is moving at a speed |w| ≪1 relative to the universal rest frame. Observers in that frame will observe the Lorentz breaking VEV vµ≃(1,−w). In the rest frame of the source, a modiﬁed gravitational potential will be generate d. Technically this is because terms in the graviton propagator v·k≃w·kare non-vanishing. It is natural to assume that dynamical eﬀects align the universal rest frame where vµ=δµ 0with the rest frame of the cosmic microwave background. In a general coordinate system moving at a constant speed wit h respect to the universal frame the Lorentz-breaking VEV will be a general time-like v ectorvµ. Thus we need to determine the graviton propagator for a general time-lik e constant vµ. Since Lorentz invariance is spontaneously broken, the numerator of the gr aviton propagator is the most general tensor constructed out of the vectors vµ,kνand the tensor ηρσ. There are 14 such tensors. Writing the action for the gravitons as S=1 2/integraldisplay d4k˜hαβ(−k)Kαβ|σρ(k)˜hσρ(k) (4.32) it is a straightforward exercise to determine the graviton p ropagator Pby solving Kαβ|µν(k)Pµν|ρσ(k) =1 2/parenleftig ηρ αησ β+ησ αηρ β/parenrightig . (4.33) The above set of conditions leads to 21 linear equations that determine the 14 coeﬃcients of the graviton propagator in terms of the coeﬃcients ciand the VEV vµ. Seven equations are redundant and provide a non-trivial consistency check o n our calculation. Although it is necessary to compute all 14 coeﬃcients in orde r to invert the propagator, here we present only those that modify Newton’s law as descri bed previously (assuming stress-tensors are conserved for sources). These are Pαβ|ρσ Newton=/braceleftig Aηαβηρσ+ B(ηαρηβσ+ηασηβρ) + C(vαvβηρσ+vρvσηαβ) +Dvαvβvρvσ+ E(vαvρηβσ+vαvσηβρ+vβvρηασ+vβvσηαρ)/bracerightig .(4.34) 64 We ﬁnd that each of these coeﬃcients is independent of the gau ge parameter α. We also numerically checked that without the presence of the gauge- ﬁxing term the propagator could not be inverted. To compute the preferred-frame eﬀect coeﬃcient α2, we only need to focus on terms in the momentum-space propagator proportional to ( v·k)2. To leading non-trivial order in G(v·k)2and in theci’s we obtain, from the linear combination A+ 2B+ 2C+D+ 4E, g00= 1 + 8πGN/integraldisplayd4k (2π)41 k2/braceleftbigg 1−8πGN(v·k)2 k21 c1(c1+c2+c3)/bracketleftbig 2c3 1+ 4c2 3(c2+c3)+ +c2 1(3c2+ 5c3+ 3c4) +c1((6c3−c4)(c3+c4) +c2(6c3+c4))/bracketrightbig/bracerightbigg ˜T00(k),(4.35) where in the ﬁrst line kis a four-vector. Next we use vµ= (1,−w), place the source at the origin, substitute T00=Mδ(3)(x) or˜T00(k) = 2πMδ(k0) and use /integraldisplayd3k (2π)3kikj k4eik·x=1 8πr/bracketleftig δij−xixj r2/bracketrightig (4.36) to obtain g00= 1−2GNM r/parenleftbigg 1−(w·r)2 r28πGN 2c1(c1+c2+c3)/bracketleftbig 2c3 1+ 4c2 3(c2+c3)+ +c2 1(3c2+ 5c3+ 3c4) +c1((6c3−c4)(c3+c4) +c2(6c3+c4))/bracketrightbig/parenrightbigg ,(4.37) where we have only written those terms that give a correction to Newton’s law proportional to [w·r/r]2. We have also assumed that |w| ≪1 so that higher powers in w·r/rcan be neglected. The factor of 1 /c1in the preferred-frame correction to the metric arises because when c1→0 the “transverse” components of φihave no spatial gradient kinetic term. Similarly, the factor of 1 /(c1+c2+c3) arises because when c1+c2+c3→0 the “longitudinal” component of φihas no spatial gradient kinetic term. Either of these cases causes a divergence in the static limit.7 The coeﬃcients ciredeﬁne Newton’s constant measured in solar system experim ents and we ﬁnd that GN=G[1−8πG(c1+c4)]≃G 1 + 8πG(c1+c4), (4.38) 7This divergence can of course be avoided by considering high er-derivative terms in the action for the Goldstone bosons. This would then give non-relativistic di spersion relations for these modes, E∝ |k|nfor n >1, as was the case in [81]. 65 which agrees with previous computations to linear order in t heci’s after correcting for the diﬀerences in notation [74, 77]. The experimental bounds on deviations from Einstein gravit y in the presence of a source are usually expressed as constraints on the metric perturba tion. Since the metric is not gauge-invariant, these bounds are meaningful only once a ga uge is speciﬁed. In the litera- ture, the bounds are typically quoted in harmonic gauge. Her e, the preferred-frame eﬀect is a particular term appearing in the solution for h00. For static sources, the gauge transfor- mation needed to translate the solution in our gauge to the ha rmonic gauge is itself static. But since a static gauge transformation cannot change h00, we may read oﬀ the coeﬃcient of the preferred-frame eﬀect in the gauge that we used. By inspection α2=8πGN c1(c1+c2+c3)/bracketleftbig 2c3 1+ 4c2 3(c2+c3) +c2 1(3c2+ 5c3+ 3c4) +c1((6c3−c4)(c3+c4) +c2(6c3+c4))/bracketrightbig , (4.39) which can be compared with the experimental bound |α2|<4×10−7given in [78]. After [54] was published, Foster and Jacobson ([82]) carried out the fu ll computation of α2in terms of theciparameters in the vector-tensor theory with the unit constr aint and conﬁrmed that Eq. (4.39) is correct to leading non-trivial order. The experimental bound on α2is obtained by considering the torque that the eﬀect in Eq. (4.8) would exert on the plane of the orbit of a planet. For simplicity, let us consider a circular planetary orbit of radius r, moving around the sun, whose velocity wwith respect to the preferred frame we take to be aligned with the z-axis, as shown in Fig. 4.1. The average torque over one orbit is τ=−ˆxα2GNM1M2w2 4rsin 2θ0, (4.40) whereθ0is the inclination between the plane of planet’s orbit and th e axis of w. This torque would cause the planes or the planets in the solar system to precess at diﬀerent rates, unless all the orbital planes were perfectl y aligned or anti-aligned with the axis of w. If we consider, for instance, the orbits of Earth and Mercur y, whose planes are aligned to within a few degrees, and then consider Eq. (4.40) with 66 [\] Z qq e eUAq AASODQHW Figure 4.1: Diagram of a planet moving in a circular orbit of radius raround the sun (located at the origin), whose velocity with respect to the preferred fr ame is w. The inclination between the plane of the orbit and the axis of wisθ0(the minimum value of the polar angle θduring the planet’s orbit). •M1= solar mass • |w| ≃10−3(the sun’s speed with respect to the CMB rest frame) •sin 2θ0∼O(1) then the fact that Mercury and the Earth have maintained thei r approximate alignment over the age of the solar system ( ∼4.5×109years) gives us, roughly, the bound in the literature of |α2|∼<10−7. A considerably stronger constraint on the size of the ci’s can be derived from the fact that a particle moving faster than one of the graviton polari zations would lose energy through gravitational ˇCerenkov radiation. In particular, this gravitational ˇCerenkov radi- ation would limit the ﬂux of the highest-energy cosmic rays ( which are protons moving at nearly the speed of light). Depending on the exact assumptio ns regarding the abundance and distribution of cosmic ray sources, the resulting bound can range from G|ci|∼<10−15to G|ci|∼<10−31([67]). These limits, however, apply only if the extra gravi ton polarizations propagate subluminally. We will have more to say on this issu e in Section 4.5. 67/A1 Figure 4.2: Feynman diagram for the (negligible) modiﬁcation to gravit y by the coupling of the graviton to acoustic perturbations in the CMB. 4.5 A cosmic solid We know that the eﬀect considered in Section 4.4, the modiﬁca tion of gravity by the presence of a background Sµwith a rest frame, is present in nature, because the electrom agnetic radiation in the CMB has a conserved Poynting 4-vector: Pµ=1 8π/parenleftbig E2+B2,2E×B/parenrightbig . (4.41) This background Pµmodiﬁes gravity because gravitons can couple to acoustic pe rtur- bations in it, as shown in Fig. 4.2. This eﬀect is, however, co mpletely negligible, since the characteristic energy scale of the CMB is TCMB∼2.7 K, which means that this eﬀect is suppressed by a factor of/parenleftbiggTCMB MPl/parenrightbigg2 ∼10−64. (4.42) The question remains, however, whether there might be some o ther background that, unlike the CMB, couples strongly to gravity (and only to gravity, so as to explain why it has not been otherwise detected). The Goldstone bosons of spontane ous Lorentz violation would correspond to the sound waves in this background, and the mod iﬁcation to gravity comes, as it did in Fig. 4.2 from the mixing of the gravitons with thes e acoustic modes. In [73], the authors ﬁnd the propagation velocities of the ﬁv e graviton polarizations in vector-tensor theories with the unit constraint. In our lan guage, these are the velocities of 68 the two usual gravitons plus the three acoustic modes in the L orentz-violating background: 2 transverse traceless metric vtt= 1/(1−c13)→1, 2 transverse Goldstones vtrv= (c1−c2 1/2 +c2 3/2)/(c14)(1−c13)→c1/(c14), 1 longitudinal Goldstone vlgt=c123(2−c14)/c14(1−c13)(2 +c13+ 3c2) →c123/c14,(4.43) whereci...k≡G(ci+...+ck) and where the limits correspond to vanishing ci’s. Since, for generalci’s, there are two distinct sound speeds, one for the longitud inal and one for the transverse modes, our Lorentz-violating background fulﬁl ls the canonical deﬁnition of a solid.8The transverse sound speed is associated with a shear mode (a deformation which alters the shape but not the volume of a body). Linear shear wa ves are absent in a ﬂuid (see, for instance, Chapters III and VI in [83]). In Section 4.4 we emphasized the diﬀerence between our model , which we may now refer to as the “cosmic solid” model, and the “ghost condensa te” of [81]. In [81], Lorentz invariance is broken by a VEV for a spin-0 vector ﬁeld Aµ=∂µφwith a single degree of freedom, whereas in the cosmic solid model the Lorentz invar iance is broken by a spin- 1 vector ﬁeld Aµwith three degrees of freedom. Therefore the ghost condensa te has a single Goldstone boson, with non-relativistic dispersion relationsE∝ |k|2, and it gives the graviton a mass when minimally coupled to it, whereas the cos mic solid has three Goldstone bosons, with dispersion relations E∝ |k|, and it does not give the graviton a mass. It turns out that if the ghost condensate is gauged (i.e., if the ghost condensate ﬁeld φis minimally coupled to a U(1) gauge ﬁeld Aµ), then the two polarizations of the gauge ﬁeld provide the two extra degrees of freedom, and the resulting model is e quivalent to the cosmic solid ([84]). Whether the ghost condensate itself admits a high-e nergy completion is unresolved (see [85, 86]). It can be seen from Eq. (4.43) that the speeds of the Goldstone bosons can be made su- perluminal without introducing ghosts or other obvious pro blems in the low-energy eﬀective action. As pointed out in Section 4.4, if the Goldstones are r equired to be subluminal, then 8This was brought to my attention by Juan Maldacena and Ian Low . 69 α2no longer gives the strongest constraint on the size of the ci’s because a far more strin- gent limit applies, from the gravitational ˇCerenkov radiation of the highest-energy cosmic rays. Superluminal Goldstone bosons would evade that const raint. Whether superluminal- ity could result from a reasonable high-energy completion, and whether the initial value problem in the low-energy eﬀective action is well-posed in t he presence of superluminal modes, remain open questions. 70 Chapter 5 Some considerations on the cosmological constant problem 5.1 Introduction Consider Einstein-Hilbert gravity as an eﬀective theory, c ontaining all the terms compatible with its symmetries: S=/integraldisplay d4x√−g/bracketleftbig Lmatter(φ,gµν)−2Λ +M2 PlR+···/bracketrightbig , (5.1) whereMPl≡/radicalbig 1/8πGand Tµν≡1√−gδSmatter δgµν. (5.2) The equation of motion for the metric gµνis Rµν−1 2gµνR−Λgµν= 8πGT µν. (5.3) We would naturally expect that Λ∼M4 Pl∼/parenleftbig 1028eV/parenrightbig4. (5.4) If we let gµν=ηµν+1 MPlhµν, (5.5) 71/A1Figure 5.1: Feynman diagram of the coupling of a single graviton to the co smological constant Λ in Eq. (5.1). The blob may also be thought of as a collection of va cuum-to-vacuum quantum processes. wherehµνis the graviton ﬁeld, then the Λ term in Eq. (5.1) gives −√−g(2Λ) = −2Λ−Λ MPlhµ µ− O(h2). (5.6) The ﬁrst term in the right-hand side of Eq. (5.6) is an irrelev ant constant, but the second gives a tadpole diagram for the graviton, as shown in Fig. 5.1 . By Eq. (5.4) we would therefore expect this tadpole interaction to be of order M3 Pl. Alternatively, one can think of this tadpole diagram, shown in Fig. 5.1, as the coupling of a single graviton to the quantum-mechanical vacuum energ y. This corresponds to moving the Λgµνin Eq. (5.3) from the left-hand to the right-hand side and thi nking of it as the contribution to the matter Tµνfrom the quantum-mechanical vacuum energy. In quantum ﬁeld theory, each frequency mode of the free ﬁeld is a simple h armonic oscillator. Therefore, each mode has a zero-point energy E=ω/2. We clearly have to cut oﬀ the sum at some scale, but the successes of quantum ﬁeld theory so far sugges t the cut oﬀ scale can’t be much smaller than ∼1 TeV. In any case, we get a positive value of Λ (the “cosmological co nstant”) far, far in excess of what observation allows. To see qualitatively the eﬀect of l arge positive Λ, imagine vacuum energy inside a piston. Its energy density, ρ, is constant. If the piston is pulled out, as shown in Fig. 5.2, the total energy must increase: dE=ρdV. By energy conservation, we must have supplied that energy when we pulled on the piston: dW=Fdℓ=pdV=−dE. Therefore the piston would resist being pulled out: Pressure is negative, p=−ρ. The Newtonian limit of GR for a test mass on the edge of a unifor m sphere of radius r0 gives an acceleration g=4π 3G(ρ+ 3p)r0. (5.7) Therefore, the quantum vacuum energy would anti-gravitate . A value of Λ as large as 72 ) GOYDFXXP HQHUJ\ Figure 5.2: Consider a piston ﬁlled with vacuum energy, whose density is constant. By energy con- servation, the piston must resist being pulled out, and ther efore the vacuum energy exerts negative pressure. what we would expect on eﬀective theory or quantum mechanica l grounds would rip apart the universe, preventing it from developing any structure. It was long presumed that some unknown symmetry of quantum gravity would forbid the Λ term i n Eq. (5.1), thus naturally making the cosmological constant zero. In Chapter 3 we discu ssed one such idea: that the graviton was a Goldstone boson of spontaneous Lorentz viola tion, so that the broken Lorentz invariance protected it from getting any potential at all. Data on the accelerated expansion of the universe, however, has recently shown that there is a small but non-zero anti-gravitating term.[87, 88 ] Two possible approaches to this cosmological constant problem that will be of interest to us here are: •to imagine that the true Λ is zero, but that the universe conta ins some other ﬁeld, coupled only to gravity, which accounts for the accelerated expansion. •to imagine that the value of Λ varies over some landscape of po ssible universes, and that we naturally happen to live in one where Λ is small enough that structure (and therefore intelligent life) may form. The ﬁrst line of thought will lead us in Section 5.2 to conside r whether a cosmological scalar ﬁeld can have a pressure more negative than −ρ. In Section 5.3 we will consider how the ghost condensate of [81] would behave if it were responsible for the accelerated expansion of the universe. In Section 5.4 we will re-examine the second line of thought in light of the proposal that other parameters besides Λ vary over the lands cape of possible universes. 73 5.2 Gradient instability for scalar models of the dark energ y with w <−1 Matter whose equation of state satisﬁes w≡p/ρ < −1 violates a number of conditions, including the weak energy condition, generally assumed to a pply to any reasonable model of physics [89]. However, the observational data do not excl ude the possibility that the dark energy has w<−1 ([90, 91]). The results reported in [92] indicate that −1.26<w< −0.83 at 95% conﬁdence level. The possibility of w<−1 has been explored by numerous authors (see, for example, [93]–[103]). These models often contain a ﬁeld with an unusual kinetic term, which is referred to as a phantom or ghost ﬁeld. In this l etter we show that for w<−1, single scalar ﬁeld models of the dark energy generally have a wrong sign gradient kinetic term for ﬂuctuations about the homogeneous background. Thi s result is not dependent on general relativistic eﬀects and survives in the ﬂat-space l imit. Spatial inhomogeneities of the dark energy are tightly constrained by observations of the c osmic microwave background. In our analysis we will assume a time-dependent but spatiall y homogenous scalar back- ground, and show that for w <−1 spatial instabilities inevitably arise. Consider the low-energy eﬀective theory of a scalar ﬁeld coupled to gravi ty: S=/integraldisplay d4x√−g/bracketleftbig M2 PlR+P+UR+V Rµν(∂µφ)(∂νφ) +···/bracketrightbig , (5.8) whereP,U, andVare functions of the scalar ﬁeld φand its derivatives. (Because of the anti-symmetry of Rµνρσin its ﬁrst two and also in its last two indices, no non-vanish ing invariant can be formed from it using ﬁrst derivatives of φ.) Naively we might expect that the higher-dimensional couplings of φto the Ricci tensor would be suppressed by powers of the Planck mass MPl, making them irrelevant for cosmology after the Planck epoc h. However, such terms are generated by graphs such as that in Fi gure 5.3. Writing the metric asgµν=ηµν+hµν/MPl, we see that scalar-graviton interactions in Feynman diagr ams are suppressed by the Planck mass, but when these interactions a re reassembled into the Ricci tensor that suppression is absent. That is, the higher-dime nsional terms in Eq. (5.8) will appear suppressed only by powers of the characteristic ener gy scale of the scalar ﬁeld, M, which may be much smaller than MPl. We neglect terms in the action (5.8) that involve higher powe rs of the Ricci tensor. The 74/A1 Figure 5.3: The eﬀective couplings of two gravitons to several quanta of the scalar ﬁeld. The shaded region represents interactions involving only scalars. terms we consider are ones that can generate contributions t o the stress-energy tensor Tµν that are not suppressed by powers of MPl. SinceTµνis obtained by varying the action with respect to the metric, terms with more than one power of Rµνρσyield contributions that are themselves proportional to the Ricci tensor and therefo re vanish in the ﬂat-space limit. Assuming a spatially homogeneous background, only the time derivatives of φwill be non-vanishing in Eq. (5.8). It may be shown that in the limit MPl→ ∞ , the term Rµν(∂µφ)(∂νφ)Vcontributes a term to the stress-energy tensor, which can be reproduced by an appropriate change in the function U. Therefore we may restrict ourselves to V= 0 and consider the most general Uin order to analyze the ﬂat-space behavior of Eq. (5.8). It is always possible to perform a rescaling of the metric in E q. (5.8)gµν→e2wgµν, withw= log[1+U/M2 Pl], so that the Uterm in Eq. (5.8) disappears, being absorbed into a redeﬁnition of the Paction for the ghost scalar ﬁeld. (See, for example, [104].) The action resulting from this rescaling, up to terms suppressed by pow ers of 1/MPl, is then S=/integraldisplay d4x√−g/bracketleftbig M2 PlR+P/bracketrightbig . (5.9) The most general Lorentz-invariant scalar Lagrangian with out higher-derivative terms (which we will consider later) is L=P(X,φ), (5.10) whereX=gµν∂µφ∂νφ. (A potential term Vwould be the component of P(X,φ) that is independent of X.) Henceforth, P′(X,φ) will denote diﬀerentiation with respect to X. Since the scalar ﬁeld φis minimally coupled to gravity in Eq. (5.9), the stress-ene rgy tensor 75 is Tµν=−Lgµν+ 2P′(X,φ)∂µφ∂νφ , (5.11) and w=P(X,φ) T00=P(X,φ) −P(X,φ) + 2˙φ2P′(X,φ)=−1 +2˙φ2P′(X,φ) T00. (5.12) Forφto account for the dark energy, we must have T00>0. Then,w <−1 requires thatP′(X,φ)<0. Letφ0=φ0(t) be a solution to the equations of motion, and consider the ﬂuctuations about this solution: φ=φ0+π(x,t). When expanded in π, the eﬀective Lagrangian will contain a term L=−P′(X,φ)|∇π|2+···, (5.13) which implies that for P′(X,φ)<0 there will exist ﬁeld conﬁgurations with non-zero spatial gradients that have lower energy than the homogeneous conﬁg uration.1There is no direct connection between the sign of w+ 1 and that of the ˙ π2term in the eﬀective Lagrangian. IfP′(X,φ) is negative, a ﬁnite expectation value for the gradients ma y be obtained by adding higher powers of ( ∇π)2to theπLagrangian, but this is problematic because it gives rise to a spatially inhomogeneous ground state for the dark energy and would lead to inhomogeneities far larger than the limit of 10−5imposed by observations of the cosmic microwave background.2While a potential term such as m2φ2tends to conﬁne the gradients to regions of size 1 /m, in most models of the dark energy V′′(φ) must be small enough that these regions are of cosmological size. In thew<−1 case, it is possible, by adding higher-derivative terms to the Lagrangian, to avoid having ﬁnite spatial gradients lower the energy of t he ﬁeld. Consider, for example, L=P(X,φ) +S(X,φ)(/squareφ)2(5.14) in which case T00=−L+ 2[P′(X,φ)˙φ2+S′(X,φ)˙φ2(∂2φ)2+ 2S(X,φ)¨φ(∂2φ)−∂0(˙φS∂2φ)].(5.15) 1Here we mean energy constructed from the Hamiltonian for ﬂuc tations about the background ﬁeld conﬁguration. 2A condensate of gradients with a preferred magnitude, deter mined by the higher-order terms that stabi- lize Eq. (5.13), will spontaneously break the O(3) rotational symmetry down to O(2). The homotopy group π2[O(3)/O(2)] is non-trivial, which leads to the formation of global m onopole (hedgehog) conﬁgurations. 76 Setting the spatial gradients of φto zero, we have that ˙φ2Cgrad−2∂0(S¨φ)˙φ=(w+ 1) 2T00, (5.16) whereCgradis the coeﬃcient of −(∇π)2in theπLagrangian. If ∂0(S¨φ)˙φ>0, then a model may have both Cgrad>0 andw <−1. But for wsigniﬁcantly less than −1, this also requires ¨φ2to be at least of order M2˙φ2, unlessS(X,φ) is made unnaturally large. It is not clear how to treat these higher-derivative terms self-c onsistently beyond perturbation theory, so the dynamics of such models cannot be analyzed in a straightforward manner. The models we consider below have higher powers of ﬁrst deriv atives, but they satisfy the condition that ¨φ2≪(˙φM)2. Our analysis shows that w<−1 scalar models typically require a wrong sign ( ∇π)2term in the eﬀective Lagrangian. Previous analyses of ghost mode ls ([89, 105]) have focused on the problems associated with negative energy, particularl y with a kinetic term L=−(∂µφ)2 that has the wrong sign for boththe time and space derivatives. The classical equations of motion for such models do not exhibit growing modes of non- zero spatial gradients, although the energy of the ﬁeld is unbounded from below. Mode ls withw<−1 that do not have a wrong sign time-derivative kinetic term in the eﬀecti ve Lagrangian can result from a Lorentz-invariant action, as we demonstrate below. Howeve r, both Lorentz invariance and time translation invariance are spontaneously broken by a t ime-dependent condensate. In [81] a model with L=P(X) was proposed in which a ghost ﬁeld has a time-dependent condensate (from now on we take the Lagrangian to be a functio n of X only, and therefore invariant under the shift φ→φ+c). We use units in which the dimensional scale Mof the model is unity ( M∼10−3eV if the ghost comprises the dark energy). The ﬂat-space equation motion is ∂µ/bracketleftbig P′(X)∂µφ/bracketrightbig = 0. (5.17) Homogeneous solutions of the equations of motion with ˙φ2=c2were considered in [81]. In general, the existence of a ˙φcondensate allows for exotic equations of state, including w<−1. In what follows we let P(X) =−1 + 2(X−1)2+ (X−1)3, (5.18) 77 which leads to w<−1 withT00>0 whenX <1. The energy density is given by T00=H=∂L ∂˙φ˙φ− L= 2˙φ2P′(X)− L, (5.19) which is not necessarily minimized by a particular ghost con densateφ=ct, although it is a solution to the ﬂat-space equations of motion for any value ofc. This is possible because there is a conserved charge associated with the shift symmet ry, Q=/integraldisplay d3x P′(X)˙φ , (5.20) so conﬁgurations that do not extremize T00can still be stable. In fact, the Lagrangian describing small ﬂuctuations has the correct sign of ˙ π2ifP′(X) + 2XP′′(X)>0. This condition is satisﬁed in the region X < 1 by (5.18) given above. There is then a local instability to the formation of gradients, as required by ou r earlier results. 5.3 Time evolution of wfor ghost models of the dark energy Ghost models of the dark energy that approach w=−1 asymptotically make potentially interesting predictions for the time evolution of the equat ion of state for the dark energy. In a FRW universe, the equation of motion for the ghost ﬁeld is ∂µ/bracketleftbig a3(t)P′(X)∂µφ/bracketrightbig = 0, (5.21) wherea(t) is the FRW scale factor. If there is a value c2 ∗=˙φ2=Xsuch thatP′(c2 ∗) = 0, then Eq. (5.12) implies that w=−1 whenX=c2 ∗. The model described by Eq. (5.18) has c2 ∗= 1, and if we apply Eq. (5.21) to it, we see that if we start from X=c2 iwithciclose toc∗, then we are driven asymptotically towards X=c2 ∗andw=−1. In the model described by Eq. (5.18), we may be driven towards w=−1 either from above or from below, depending on whether we chose to start fr omc2 i>1 or fromc2 i<1. We have argued that w<−1 is problematic because of spatial gradient instabilities , so that the case in which we are driven to w=−1 from above is more interesting. 78 Near the asymptotic value c∗= 1 we have ˙π=P′(c2 i)ci 2P′′(c2∗)c2∗/parenleftigai a/parenrightig3 . (5.22) Thus, in this regime, w=−1−4P′′(c2 ∗)c3 ∗˙π P(c2∗)=−1−2P′(c2 i)c∗ci P(c2∗)/parenleftbigg1 +zi 1 +z/parenrightbigg3 . (5.23) Equation (5.23) oﬀers a prediction for the wparameter of the dark energy as a function of the redshift z, which could be tested by cosmological data. In summary, from Eqs. (5.12) and (5.13) we ﬁnd that in single s calar ﬁeld models of the dark energy with w <−1, the kinetic term for ﬂuctuations about the homogeneous background has a wrong sign gradient term. On the other hand, there is no direct connection between the sign of the ˙ π2kinetic term in the eﬀective Hamiltonian and the sign of w+ 1. 5.4 Anthropic distribution for cosmological constant and p ri- mordial density perturbations The anthropic principle has been proposed as a possible solu tion to the two cosmological constant problems: why the cosmological constant Λ is order s of magnitude smaller than any theoretical expectation, and why it is non-zero and comp arable today to the energy density in other forms of matter ([106, 107, 108]). This anth ropic argument, which predates direct cosmological evidence of the dark energy, is the only theoretical prediction for a small, non-zero Λ ([108, 109]). It is based on the observation that t he existence of life capable of measuring Λ requires a universe with cosmological structur es such as galaxies or clusters of stars. A universe with too large a cosmological constant eit her doesn’t develop any structure, since perturbations that could lead to clustering have not g one non-linear before the universe becomes dominated by Λ, or else has a very low probability of e xhibiting structure-forming perturbations, because such perturbations would have to be so large that they would lie in the far tail-end of the cosmic variance. The existence of t he string theory landscape, in which causally disconnected regions can have diﬀerent co smological and particle physics properties, adds support to the notion of an anthropic rule f or selecting a vacuum. 79 How well does this principle explain the observed value of Λ i n our universe? Careful analysis by [109] ﬁnds that 5% to 12% of universes would have a cosmological constant smaller than our own. In everyday experience we encounter ev ents at this level of conﬁ- dence,3so as an explanation this is not unreasonable. If the value of Λ is not ﬁxed a priori , then one might expect other fundamental pa- rameters to vary between universes as well. This is the case i f one sums over wormhole conﬁgurations in the path integral for quantum gravity ([11 0]), as well as in the string theory landscape ([111, 112, 113, 114]). In [114] it was emph asized that all the parameters of the low energy theory would vary over the space of vacua (“t he landscape”). Douglas ([112]) has initiated a program to quantify the statistical properties of these vacua, with additional contributions by others ([113]). In [115], Aguirre stressed that life might be possible in uni verses for which some of the cosmological parameters are orders of magnitude diﬀerent f rom those of our own universe. The point is that large changes in one parameter can be compen sated by changes in another in such a way that life remains possible. Anthropic predicti ons for a particular parameter value will therefore be weakened if other parameters are all owed to vary between universes. One cosmological parameter that may signiﬁcantly aﬀect the anthropic argument is Q, the standard deviation of the amplitude of primordial cosmolog ical density perturbations. Rees in [116] and Tegmark and Rees in [117] have pointed out that if the anthropic argument is applied to universes where Qis not ﬁxed but randomly distributed, then our own universe becomes less likely because universes with both Λ and Qlarger than our own are anthropi- cally allowed. The purpose of the work in this section is to qu antify this expectation within a broad class of inﬂationary models. Restrictions on the a prori probability distribution forQnecessary for obtaining a successful anthropic prediction for Λ, were considered in [118, 119]. In our analysis we let both Λ and Qvary between universes and then quantify the anthropic likelihood of a positive cosmological constant l ess than or equal to that observed in our own universe. We oﬀer a class of toy inﬂationary models that allow us to restrict the a priori probability distribution for Q, making only modest assumptions about the behavior of the a priori distribution for the parameter of the inﬂaton potential in t he anthropically allowed range. Cosmological and particle physics paramete rs other than Λ and Qare held 3For instance, drawing two pairs in a poker hand. 80 ﬁxed as initial conditions at recombination. We provisiona lly adopt Tegmark and Rees’s anthropic bound on Q: a factor of 10 above and below the value measured in our unive rse. Even though this interval is small, we ﬁnd that the likelihoo d that our universe has a typical cosmological constant is drastically reduced. The likelih ood tends to decrease further if larger intervals are considered. Weinberg determined in [108] that, in order for an overdense region to go non-linear be- fore the energy density of the universe becomes dominated by Λ, the value of the overdensity δ≡δρ/ρmust satisfy δ >/parenleftbigg729Λ 500¯ρ/parenrightbigg1/3 . (5.24) In a matter-dominated universe this relation has no explici t time dependence. Here ¯ ρis the energy density in non-relativistic matter. Perturbati ons not satisfying the bound cease to grow once the universe becomes dominated by the cosmologi cal constant. For a ﬁxed amplitude of perturbations, this observation provides an u pper bound on the cosmological constant compatible with the formation of structure. Throu ghout our analysis we assume that at recombination Λ ≪¯ρ. To quantify whether our universe is a typical, anthropicall y allowed universe, additional assumptions about the distribution of cosmological parame ters and the spectrum of density perturbations across the ensemble of universes are needed. A given slow-roll inﬂationary model with reheating leads to a Friedman-Robertson- Walker universe with a (late-time) cosmological constant Λ and a spectrum of perturbations that is approximately scale-invariant and Gaussian with a v ariance Q2≡ ∝an}b∇acketle{t˜δ2∝an}b∇acket∇i}htHC. (5.25) The expectation value is computed using the ground state in t he inﬂationary era and per- turbations are evaluated at horizon-crossing. The varianc e is ﬁxed by the parameters of the inﬂationary model together with some initial condition s. Typically, for single-ﬁeld φ slow-roll inﬂationary models, Q2∼H4 ˙φ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle HC. (5.26) This leads to spatially separated over- or underdense regio ns with an amplitude δthat for 81 a scale-invariant spectrum are distributed (at recombinat ion) according to N(σ,δ) =/radicalbigg 2 π1 σe−δ2/2σ2. (5.27) (The linear relation between Qand the ﬁltered σin Eq. (5.27) is discussed below.) By Bayes’s theorem, the probability for an anthropically al lowed universe (i.e., the probability that the cosmological parameters should take c ertain values, given that life has evolved to measure them) is proportional to the product o f thea priori probability distribution Pfor the cosmological parameters, times the probability tha t intelligent life would evolve given that choice of parameter values. Followi ng [109], we estimate that second factor as being proportional to the mean fraction F(σ,Λ) of matter that collapses into galaxies. The latter is obtained in a universe with cosm ological parameters Λ and σ by spatially averaging over all over- or underdense regions , so that ([109]) F(σ,Λ) =/integraldisplay∞ δmindδN(σ,δ)F(δ,Λ). (5.28) The lower limit of integration is provided by the anthropic b ound of Eq. (5.24), which gives δmin≡(729Λ/500¯ρ)1/3. The anthropic probability distribution is P(σ,Λ) =P(Λ,σ)F(σ,Λ)dΛdσ . (5.29) Computing the mean fraction of matter collapsed into struct ures requires a model for the growth and collapse of inhomogeneities. The Gunn-Gott m odel ([120, 121]) describes the growth and collapse of an overdense spherical region sur rounded by a compensating underdense shell. The weighting function F(δ,Λ) gives the fraction of mass in the inhomo- geneous region of density contrast δthat eventually collapses (and then forms galaxies). To a good approximation it is given by ([109]) F(δ,Λ) =δ1 δ+δmin. (5.30) Additional model dependence occurs in the introduction of t he parameter sgiven by the ratio of the volume of overdense sphere to the volume of the un derdense shell surrounding the sphere. We will set s= 1 throughout. 82 Since the anthropically allowed values for Λ are so much smal ler than any other mass scale in particle physics, and since we assume that Λ = 0 is not a special point in the landscape, we follow [122, 109] in using the approximation P(Λ)≃P(Λ = 0) for Λ within the anthropically allowed window.4The requirement that the universe not recollapse before intelligent life has had time to evolve anthropically rules out large negative Λ ([107, 124]). We will assume that the anthropic cutoﬀ for negative Λ is clos e enough to Λ = 0 that all Λ<0 may be ignored in our calculations. As an example of a concrete model for the variation in Qbetween universes, we consider inﬂaton potentials of the form (see, for example, [125]) V= Λ +λφ2p, (5.31) wherepis a positive integer.5We assume there are additional couplings that provide an eﬃcient reheating mechanism, but are unimportant for the evolution of φduring the inﬂationary epoch. The standard deviation of the amplitude of perturbations gives Q=A√ λφp+1 HC M3 Pl, (5.32) whereAis a constant, and φHCis the value of the ﬁeld when the mode of wave number kleaves the horizon. This φHChas logarithmic dependence on λandk, which we neglect. Randomness in the initial value for φaﬀects only those modes that are (exponentially) well outside our horizon. Throughout this section, we will set th e spectral index to 1 and ignore its running. Equation (5.32) then gives λ∝Q2. Next, suppose that the fundamental parameters of the Lagran grian are not ﬁxed, but vary between universes, as might be expected if one sums over wormhole conﬁgurations in the path integral for quantum gravity ([110]) or in the strin g theory landscape ([111, 112, 114, 113]). To obtain the correct normalization for the dens ity perturbations observed in our universe, the self-coupling must be extremely small. As the standard deviation Qwill be allowed to vary by an order of magnitude around 10−5, for this model the self-coupling in alternate universes will be very small as well. 4Garriga and Vilenkin point to examples of quintessence mode ls in which the approximation P(Λ)≃P(Λ = 0) in the anthropically allowed range is not valid [123]. 5Recent analysis of astronomical data disfavors the λφ4inﬂationary model ([126]), but for generality we will consider an arbitrary pin Eq. (5.31). 83 We may then perform an expansion about λ= 0 for the a priori probability distribu- tion ofλ. The smallness of λsuggests that we may keep only the leading term in that expansion. If the a priori probability distribution extends to negative values of λ(which are anthropically excluded due to the instability of the res ulting action for φ), we expect it to be smooth near λ= 0, and the leading term in the power series expansion to be ze roth order inλ(i.e., a constant). Therefore we expect a ﬂat a priori probability distribution for λ. The a priori probability distribution for Qis then P(Q)∝dλ dQ∼Q , (5.33) where the normalization constant is determined by the range of integration in Q. Note that this distribution favors large Q. On the other hand, if the a priori probability distribution for the coupling λonly has support for λ>0 thenλ= 0 is a special point and we cannot argue that P(Q)∝Q. However, since the anthropically allowed values of λare very small, thea priori distribution for λshould be dominated, in the anthropically allowed window, by a leading term such as P(λ)∼λq. Normalizability requires q>−1. Usingλ∝Q2, this givesP(Q)∼Q2q+1. Before proceeding, it is convenient to transform to the new v ariables: y≡Λ ρ∗; ˆσ≡σ/parenleftbigg¯ρ ρ∗/parenrightbigg1 3 . (5.34) Here ¯ρis the energy density in non-relativistic matter at recombi nation, which we take to be ﬁxed in all universes, and ρ∗is the value for the present-day energy density of non-relativistic matter in our own universe. For a matter-d ominated universe ˆ σis time- independent, whereas yis constant for any era. Here and throughout this section, a s ubscript ∗denotes the value that is observed in our universe for the cor responding quantity. The only quantities whose variation from universe to universe w e will consider are yand ˆσ. In terms of these variables and following [109], the probabi lity distribution of Eq. (5.29) is found to be P=NdˆσdyP (ˆσ)/integraldisplay∞ βdxe−x β1/2+x1/2, (5.35) where β≡1 2ˆσ2/parenleftbigg729y 500/parenrightbigg2/3 , (5.36) 84 andNis the normalization constant. Notice that, since x≥β, largeβimplies that P ∼e−β≪1. For a ﬁxed ˆ σ, largey implies large β. Thus, for ﬁxed ˆ σ, large cosmological constants are anthropically disfavor ed. But if ˆσis allowed to increase, then β∼ O(1) may be maintained at larger y. Garriga and Vilenkin have pointed out that the distribution in Eq. (5.35 ) may be rewritten using the change of variables (ˆ σ,y)∝ma√sto→(ˆσ,β) ([119]). The Jacobian for that transformation is a function only of ˆ σ. Equation (5.35) then factorizes into two parts: one depend ing only on ˆσ, the other only on β. Integration over ˆ σproduces an overall multiplicative factor that cancels out after normalization, so that any choice of P(ˆσ) will give the same distribution for the dimensionless parameter β. In that sense, even in a scenario where ˆ σis randomly distributed, the computation in [109] may be seen as an anthr opic prediction for β.6The measured value of βis, indeed, typical of anthropically allowed universes, bu t an anthropic explanation for βalone does not address the problem of why both Λ and Qshould be so small in our universe. Implementing the anthropic principle requires making an as sumption about the mini- mum mass of “stuﬀ” collapsed into stars, galaxies, or cluste rs of galaxies that is needed for the formation of life. It is more convenient to express the mi nimum mass Mminin terms of a comoving scale R:Mmin= 4π¯ρa3 eqR3/3 (by convention a= 1 today, so Ris a physical scale). We do not know the precise value of R. A better understanding of biology would in principle determine its value, which should only depend o n chemistry, the fraction of matter in the form of baryons, and Newton’s constant. In our a nalysis these are all ﬁxed initial conditions at recombination. In particular, we wou ld not expect Mminto depend on Λ orQ.7Therefore, even though the relation between MminandRdepends on present-day cosmological parameters, the value of this threshold will b e constant between universes be- cause it depends only on parameters that we are treating as ﬁx ed initial conditions. Thus, in computing the probability distribution over universes, we will ﬁxR. Since we don’t know what is the correct anthropic value for R, we will present our results for both R=1 and 2 Mpc. (Ron the order of a few Mpc corresponds to requiring that struct ures as large as our galaxy be necessary for life.) We then proceed to ﬁlter out perturbations with wavelength s maller than R, leading to 6We thank Garriga and Vilenkin for explaining this point to us . 7Note, however, that requiring life to last for billions of ye ars (long enough for it to develop intelligence and the ability to do astronomy) might place bounds on Q. See [117]. 85 a varianceσ2that depends on the ﬁltering scale. Expressed in terms of the power spectrum evaluated at recombination, σ2=1 2π2/integraldisplay∞ 0dkk2P(k)W2(kR) (5.37) whereWis the ﬁlter function, which we take to be a Gaussian W(x) =e−x2/2.P(k) is the power spectrum, which we assume to be scale-invariant. ( ForP(k) we use Eq. (39) of [109], setting n= 1). Evaluating (5.37) at recombination gives, for our universe , ˆσ∗=C∗Q∗. (5.38) The number C∗contains the growth factor and transfer function evaluated from horizon crossing to recombination and only depends on physics from t hat era. We assume Λ is small enough so that at recombination it can be ignored and th us we take the variation in ˆσbetween universes to come solely from its explicit dependen ce onQ. We may then use observations of Q∗andσ∗to determine ˆ σ=C∗Q, valid for all universes. We use the explicit expression for C∗that is obtained from Eqs. (39)-(43) and (48)-(51) in [109]. This takes as inputs the Hubble parameter H0≡100h∗km/s, the energy density in non-relativistic matter Ω ∗, the cosmological constant λ∗= 1−Ω∗, the baryon fraction Ωb= 0.023h−2 ∗, the smoothing scale R, and the COBE-normalized amplitude of ﬂuctuations at horizon crossing, Q∗= 1.94×10−5Ω−.785−0.05∗ln Ω∗∗ . As we have argued, the dependence of C∗on the cosmological constant is not relevant for our purposes. For our calculations we use Ω ∗= 0.134h−2 ∗, andh∗= 0.73, consistent with their observed best-ﬁt values ([127]). The smoothing s caleRwill be taken to be either 1 Mpc or 2 Mpc, and the corresponding values for C∗are 5.2·104and 3.8·104. The values chosen for the range of Qare motivated by the discussion in [117] about anthropic limits on the amplitude of the primordial density perturbations. The authors of [117] argue that Qbetween 10−3and 10−1leads to the formation of numerous supermassive blackholes, which might obstruct the emergence of life.8They then claim that universes withQless than 10−6are less likely to form stars, or if star clusters do form, tha t they would 8They also note that for Q >10−4formation of life is possible, but planetary disruptions ca used by ﬂybys may make it unlikely for planetary life to last billions of ye ars. 86 P(Q)∝1/Q0.9in the range P(Q)∝Qin the range Q∗/10<Q< 10Q∗Q∗/15<Q< 15Q∗Q∗/10<Q< 10Q∗Q∗/15<Q< 15Q∗ R= 1 Mpc R= 2 Mpc R= 1 Mpc R= 2 Mpc R= 1 Mpc R= 2 Mpc R= 1 Mpc R= 2 Mpc P(y<y ∗) 1·10−33·10−34·10−41·10−35·10−41·10−31·10−44·10−4 ∝an}b∇acketle{ty∝an}b∇acket∇i}ht/y∗ 1·1044·1034·1041·1041·1045·1034·1042·104 y5%/y∗ 9·10 4·10 3·1021·1022·1027·10 6·1022·102 Table 5.1: Anthropically determined properties of the cosmological c onstant. not be bound strongly enough to retain supernova remnants. S ince there is considerable uncertainty in these limits, we carry out calculations usin g both the range indicated by [117] as well as a range that is somewhat broader.9 Previous work on applying the anthropic principle to variab le Λ andQhas assumed a priori distributions P(Q) that fall oﬀ as 1 /Qkfor largeQ, withk≥3 [118, 119]. Such distributions were chosen in order to keep the anthropic pro bability P(y,Q) normalizable, and they usually yield anthropic predictions for the cosmol ogical constant similar to those that were obtained in [109] by ﬁxing Qto its observed value, because they naturally favor aQas small as its observed value in our universe. For instance, forP(Q)∝1/Q3in the rangeQ∗/10< Q < 10Q∗,P(y < y∗) = 5% for R= 1 Mpc, while P(y < y∗) = 7% for R= 2 Mpc.) However, if we accept the argument of Tegmark and Rees in [117 ] that there are natural anthropic cutoﬀs on Q, it follows that the behavior of P(Q) at largeQis irrelevant to the normalizability of P(y,Q). Furthermore, P(Q)∼1/Qkin the neighborhood of Q= 0 for k≥1 leads to an unnormalizable distribution, since the integr al/integraltext P(Q)dQblows up. In what follows we shall consider two a priori distributions: P(Q)∝Q, andP(Q)∝1/Q0.9 inside the anthropic window, motivated by the inﬂationary m odels we have discussed. The results are summarized in Table 5.1, where P(y < y∗) is the anthropic probabil- ity that the value ybe no greater than what is observed in our own universe, ∝an}b∇acketle{ty∝an}b∇acket∇i}htis the anthropically weighed mean value of y, andy5%is the value of ysuch that the anthropic probability of obtaining a value no greater than that is 5%. By comparison, for this choice of cosmological parameters, the authors of [109] ﬁnd that, forQﬁxed (or measured), the probability of a universe having a co smological constant no 9Notice that we are using the ranges indicated in [117] as abso lute anthropic cutoﬀs. Arguments like those made in [117] introduce some correction to the approxi mation made in [109] that the probability of life is proportional to the amount of matter that collapses i nto compact structures. Since we are largely ignorant of what the form of this correction is, we have appro ximated it as a simple window function. 87 P(Q)∝1/Q0.9in the range P(Q)∝Qin the range Q∗/10<Q< 10Q∗Q∗/15<Q< 15Q∗Q∗/10<Q< 10Q∗Q∗/15<Q< 15Q∗ R= 1 Mpc R= 2 Mpc R= 1 Mpc R= 2 Mpc R= 1 Mpc R= 2 Mpc R= 1 Mpc R= 2 Mpc P(Q<Q ∗) 8·10−48·10−42·10−42·10−41·10−51·10−51·10−61·10−6 ∝an}b∇acketle{tQ∝an}b∇acket∇i}ht/Q∗ 8 8 11 11 8 8 13 13 Q5%/Q∗ 4 4 6 6 5 5 8 8 Table 5.2: Anthropically determined properties of the amplitude for d ensity pertubations. greater than our own is much higher: P(y <0.7/0.3) =.05 and 0.1, forR= 1 Mpc and R= 2 Mpc, respectively.10 One can also ask what is the probability of observing a value f orQin the range Q∗/10< Q<Q∗, after averaging over all possible cosmological constants . Table 5.2 summarizes the resulting distribution in Q. In summary, inﬂation and a landscape of anthropically deter mined coupling constants provides (in some scenarios) a conceptually clean framewor k for variation between universes in the magnitude of Q. Since increasing Qallows the probability of structure to remain non-negligible for Λ considerably larger than in our own uni verse, anthropic solutions to the cosmological constant problem are weakened by allowing Qas well as Λ to vary from one universe to another. 10These numbers are taken from Table 1 in the published version of [109]. 88 Chapter 6 The reverse sprinkler Everything’s got a moral, if only you can ﬁnd it. — Lewis Carroll, Alice’s Adventures in Wonderland This chapter is based largely on [128]. Some followups that h ave appeared since the publication of that article include [129] and [130]. 6.1 Introduction In 1985, R. P. Feynman, one of most distinguished theoretica l physicists of his time, pub- lished a collection of autobiographical anecdotes that att racted much attention on account of their humor and outrageousness ([131]). While describin g his time at Princeton as a graduate student (1939–1942), Feynman tells the following story ([132]): There was a problem in a hydrodynamics book,1that was being discussed by all the physics students. The problem is this: You have an S-shap ed lawn sprinkler . ..and the water squirts out at right angles to the axis and ma kes it spin in a certain direction. Everybody knows which way it goes around ; it backs away from the outgoing water. Now the question is this: If you . .. p ut the sprinkler completely under water, and sucked the water in . ..which way would it turn? 1It has not been possible to identify the book to which Feynman was referring. As we shall discuss, the matter is treated in Ernst Mach’s Mechanik , ﬁrst published in 1883 ([137]). Yet this book is not a “hydrodynamics book” and the reverse sprinkler is presente d as an example, not a problem. In [147], John Wheeler suggests that the problem occurred to them whil e discussing a diﬀerent question in the undergraduate mechanics course that Wheeler was teaching a nd for which Feynman was the grader. 89 Feynman went on to say that many Princeton physicists, when p resented with the problem, judged the solution to be obvious, only to ﬁnd that o thers arrived with equal conﬁdence at the opposite answer, or that they had changed th eir minds by the following day. Feynman claims that after a while he ﬁnally decided what the answer should be and proceeded to test it experimentally by using a very large water bottle, a piece of copper tubing, a rubber hose, a cork, and the air pressure sup ply from the Princeton cyclotron laboratory. Instead of attaching a vacuum to suck the water, he applied high air pressure inside of the water bottle to push the water out thro ugh the sprinkler. According to Feynman’s account, the experiment initially went well, but after he cranked up the setting for the pressure supply, the bottle exploded, and “. .. the wh ole thing just blew glass and water in all directions throughout the laboratory .. .” ([13 3]). Feynman ([131]) did not inform the reader what his answer to t he reverse sprinkler problem was or what the experiment revealed before explodin g. Over the years, and partic- ularly after Feynman’s autobiographical recollections ap peared in print, many people have oﬀered their analyses, both theoretical and experimental, of this reverse sprinkler problem.2 The solutions presented often have been contradictory and t he theoretical treatments, even when they have been correct, have introduced unnecessary co nceptual complications that have obscured the basic physics involved. All physicists will probably know the frustration of being c onfronted by an elementary question to which they cannot give a ready answer in spite of a ll the time dedicated to the study of the subject, often at a much higher level of sophisti cation than what the problem at hand would seem to require. Our intention is to oﬀer an elem entary treatment of this problem, which should be accessible to a bright secondary sc hool student who has learned basic mechanics and ﬂuid dynamics. We believe that our answe r is about as simple as it can be made, and we discuss it in light of published theoretical a nd experimental treatments. 2In the literature it is more usual to see this problem identiﬁ ed as the “Feynman inverse sprinkler.” Because the problem did not originate with Feynman and Feynm an never published an answer to the problem, we have preferred not to attach his name to the sprin kler. Furthermore, even though it is a pedantic point, a query of the Oxford English Dictionary suggests that “reverse” (opposite or contrary in character, order, or succession) is a more appropriate desc ription than “inverse” (turned up-side down) for a sprinkler that sucks water. 90 Figure 6.1: A sprinkler submerged in a tank of water as seen from above. Th e L-shaped sprinkler is closed, and the forces and torques exerted by the water pre ssure balance each other. 6.2 Pressure diﬀerence and momentum transfer Feynman speaks in his memoirs of “an S-shaped lawn sprinkler .” It should not be diﬃcult, however, to convince yourself that the problem does not depe nd on the exact shape of the sprinkler, and for simplicity we shall refer in our argument to an L-shaped structure. In Fig. 6.1 the sprinkler is closed: Water cannot ﬂow into it or o ut of it. Because the water pressure is equal on opposite sides of the sprinkler, it will not turn: there is no net torque around the sprinkler pivot. Let us imagine that we then remove part of the wall on the right , as pictured in Fig. 6.2, opening the sprinkler to the ﬂow of water. If water is ﬂowing i n, then the pressure marked P2must be lower than the pressure P1, because water ﬂows from higher to lower pressure. In both Fig. 6.1 and Fig. 6.2, the pressure P1acts on the left. But because a piece of the sprinkler wall is missing in Fig. 6.2, the relevant pressure on the upper right part of the open sprinkler will be P2. It would seem then that the reverse sprinkler should turn to ward the water, because if P2is less than P1, there would be a net force to the right in the upper part of the sprinkler, and the resulting torque would make th e sprinkler turn clockwise. If Ais the cross section of the sprinkler intake pipe, this torqu e-inducing force is A(P1−P2). But we have not taken into account that even though the water h itting the inside wall of the sprinkler in Fig. 6.2 has lower pressure, it also has le ft-pointing momentum. The 91 Figure 6.2: The sprinkler is now open. If water is ﬂowing into it, then the pressures marked P1 andP2must satisfy P1>P2. incoming water transfers that momentum to the sprinkler as i t hits the inner wall. This momentum transfer would tend to make the sprinkler turn coun terclockwise. One of the reasons why the reverse sprinkler is a confusing problem is t hat there are two eﬀects in play, each of which, acting on its own, would make the sprinkler tur n in opposite directions. The problem is to ﬁgure out the net result of these two eﬀects. How much momentum is being transferred by the incoming water to the inner sprinkler wall in Fig. 6.2? If water is moving across a pressure gradien t, then over a diﬀerential time dt, a given “chunk” of water will pass from an area of pressure Pto an area of pressure P−dPas illustrated in Fig. 6.3. If the water travels down a pipe of cross-section A, its momentum gain per unit time is AdP. Therefore, over the entire length of the pipe, the water picks up momentum at a rate A(P1−P2), whereP1andP2are the values of the pressure at the endpoints of the pipe. (In the language of calculus,A(P1−P2) is the total force that the pressure gradient across the pipe exert s on the water. We obtain it by integrating over the diﬀerential force AdP.)3 3As some readers of [128] pointed out to us ([134, 135]), this s impliﬁed discussion ignores the fact that the cross-section of a ﬂuid ﬂow is not in general constant whe n a pressure gradient exists. For example, for an ideal, incompressible ﬂuid the velocity (and therefore, through Bernoulli’s equation, also the pressure) must be constant inside a pipe of ﬁxed cross-section A. In that case all of the acceleration of the ﬂuid would have to occur outside of the sprinkler tube, as the ﬂow narrow s down to a cross-section A. However, if P1 is the pressure of the ﬂuid at rest, then A(P1−P2) is still the correct expression for the rate at which the ﬂow is gaining momentum. In fact, the shape of the ﬂow into the reverse sprinkler will not be relevant to our discussion at all, as should become more clear from the di scussion of conservation of angular momentum 92 Figure 6.3: As water ﬂows down a tube with a pressure gradient, it picks up momentum. For steady ﬂow, the rate A(P1−P2) must be the same rate at which the water is transferring momentum to the sprinkler wall in Fig. 6.2, bec ause otherwise the total amount of momentum contained in the ﬂow of water would not be constan t. Therefore A(P1−P2) is the force that the incoming water exerts on the inner sprin kler wall in Fig. 6.2 by virtue of the momentum it has gained in traveling down the intake pip e. Because the pressure diﬀerence and the momentum transfer eﬀ ects cancel each other, it would seem that the reverse sprinkler would not move at all . Notice, however, that we considered the reverse sprinkler only after water was alrea dy ﬂowing continuously into it. In fact, the sprinkler willturn toward the water initially, because the forces will bal ance only after water has begun to hit the inner wall of the sprinkl er, and by then the sprinkler will have begun to turn toward the incoming water. That is, in itially only the pressure diﬀerence eﬀect and not the momentum transfer eﬀect is relev ant. (As the water ﬂow stops, there will be a brief period during which only the momentum tr ansfer and not the pressure diﬀerence will be acting on the sprinkler, thus producing a m omentary torque opposite to the one that acted when the water ﬂow was being established.) Why can’t we similarly “prove” the patently false statement that a non-sucking sprinkler submerged in water will not turn as water ﬂows steadily out of it? In that case the water is going out and hitting the upper inner wall, not the left inn er wall. It exerts a force, but that force produces no torque around the pivot. The pressure diﬀerence, on the other hand, conservation in Section 6.3. 93 (a) (b) Figure 6.4: The force that pushes the water must originally come from a so lid wall. The force that causes the water ﬂow is shown for both the regular and the reve rse sprinklers when submerged in a tank of water. does exert a torque. The pressure in this case has to be higher inside the sprinkler than outside it, so the sprinkler turns counterclockwise, as we e xpect from experience. 6.3 Conservation of angular momentum We have argued that, if we ignore the transient eﬀects from th e switching on and switching oﬀ of the ﬂuid ﬂow, we do not expect the reverse sprinkler to tu rn at all. A pertinent question is why, for the case of the regular sprinkler, the sp rinkler-water system clearly exhibits no net angular momentum around the pivot (with the a ngular momentum of the outgoing water cancelling the angular momentum of the rotat ing sprinkler), while for the reverse sprinkler the system would appear to have a net angul ar momentum given by the incoming water. The answer lies in the simple observation th at if the water in a tank is ﬂowing, then something must be pushing it. In the regular spr inkler, there is a high-pressure zone near the sprinkler wall next to the pivot, so it is this lo wer inner wall that is doing the original pushing, as shown in Fig. 6.4(a). For the reverse sprinkler, the highest pressure is outside t he sprinkler, so the pushing originally comes from the right wall of the tank in which the w hole system sits, as shown 94 Figure 6.5: A tank with an opening on its side will exhibit a ﬂow such that t he water will have an angular momentum with respect to the tank’s bottom, even tho ugh there is no external source of torque corresponding to the angular momentum. The apparent paradox is resolved by noting that the tank bottom oﬀers no inertial point of reference, becaus e the tank is recoiling due to the motion of the water. in Fig. 6.4(b). The force on the regular sprinkler clearly ca uses no torque around the pivot, while the force on the reverse sprinkler does. That the water should acquire a net angular momentum around the sprinkler pivot in the absence of an exte rnal torque might seem a violation of Newton’s laws, but only because we are neglecti ng the movement of the tank itself. Consider a water tank with a hole in its side, such as t he one pictured in Fig. 6.5. The water acquires a net angular momentum with respect to any poi nt on the tank’s bottom, but this angular momentum violates no physical laws because the tank is not inertial: It recoils as water ﬂows out of it. But there is one further complication: In the reverse sprink ler shown in Fig. 6.4, the water that has acquired left-pointing momentum from the pus hing of the tank wall will transfer that momentum back to the tank when it hits the inner sprinkler wall, so that once water is ﬂowing steadily into the reverse sprinkler, the tan k will stop experiencing a recoil force. The situation is analogous to that of a ship inside of w hich a machine gun is ﬁred, as shown in Fig. 6.6. As the bullet is ﬁred, the ship recoils, but when the bullet hits the ship wall and becomes embedded in it, the bullet’s momentum is tra nsferred to the ship. (We assume that the collision of the bullets with the wall is comp letely inelastic.) If the ﬁring rate is very low, the ship periodically acquires a velocity in a direction opposite to that of the ﬁred bullet, only to stop when that bul let hits the wall. Thus the ship moves by small steps in a direction opposite that of the b ullets’ ﬂight. As the ﬁring rate is increased, eventually one reaches a rate such that th e interval between successive 95 Figure 6.6: In this thought experiment, a ship ﬂoats in the ocean while a m achine gun with variable ﬁring rate is placed at one end. Bullets ﬁred from the gun will travel the length of the ship and hit the wall on the other side, where they stop. bullets being ﬁred is equal to the time it takes for a bullet to travel the length of the ship. If the machine gun is set for this exact rate from the beginning, then the ship will move back with a constant velocity from the moment that the ﬁrst bullet is ﬁred (when the ship picks up momentum from the recoil) to the moment the last bullet hit s the wall (when the ship comes to a stop). In between those two events the ship’s veloc ity will not change because every ﬁring is simultaneous to the previous bullet hitting t he ship wall. As the ﬁring rate is made still higher, the ship will again mov e in steps, because at the time that a bullet is being ﬁred, the previous bullet will not have quite made it to the ship wall. Eventually, when the rate of ﬁring is twice the inverse of the time it takes for a bullet to travel the length of the ship, the motion of the ship will be such that it picks up speed upon the ﬁrst two shots, then moves uniformly until the penul timate bullet hits the wall, whereupon the ship loses half its velocity. The ship will ﬁna lly come to a stop when the last bullet has hit the wall. At this point it should be clear how th e ship’s motion will change as we continue to increase the ﬁring rate of the gun.4 For the case of continuous ﬂow of water in a tank (rather than a discrete ﬂow of machine gun bullets in a ship), there clearly will be no intermediate steps, regardless of the rate of ﬂow. Figure 6.7 shows a water tank connected to a shower head. Water ﬂows (with a 4Two interesting problems for an introductory university-l evel physics course suggest themselves. One is to show that the center of mass of the bullets-and-ship sys tem will not move in the horizontal direction regardless of the ﬁring rate, as one expects from momentum co nservation. Another would be to analyze this problem in the light of Einstein’s relativity of simultanei ty. 96 Figure 6.7: A water tank is connected to a shower head, so that water ﬂows o ut. Water in the pipe that connects the points marked A and B has a right-pointing m omentum, but as long as that pipe is completely ﬁlled with water there is no net horizontal for ce on the tank. consequent linear and angular momentum) between the points marked A and B, before exiting via the shower head. When the faucet valve is opened, the tank will experience a recoil from the outgoing water, until the water reaches B and begins exiting through the shower head, at which point the forces on the tank will balanc e. By then the tank will have acquired a left-pointing momentum. It will lose that moment um as the valve is closed or the water tank becomes empty, when there is no longer water ﬂo wing away from A but a ﬂow is still impinging on B. A. K. Schultz ([136]) argues that, at each instant, the water ﬂowing into the reverse sprinkler’s intake carries a constant angular momentum aro und the sprinkler pivot, and if the sprinkler could turn without any resistance (either fro m the friction of the pivot or the viscosity of the ﬂuid) this angular momentum would be counte rbalanced by the angular momentum that the sprinkler picked up as the water ﬂow was bei ng switched on. As the ﬂuid ﬂow is switched oﬀ, such an ideal sprinkler would then lo se its angular momentum and come to a halt. At every instant, the angular momentum of the s prinkler plus the incoming water would be zero. Schultz’s discussion is correct: In the absence of any resis tance, the sprinkler arm itself moves so as to cancel the momentum of the incoming water, in th e same way that the ship in Fig. 6.6 moves to cancel the momentum of the ﬂying bullets. Resistance, on the other 97 hand, would imply that some of that momentum is picked up not j ust by the sprinkler, but by the tank as a whole. If we cement the pivot to prevent the spr inkler from turning at all, then the tank will pick up all of the momentum that cancels tha t of the incoming water. How does non-ideal ﬂuid behavior aﬀect this analysis? Visco sity, turbulence, and other such phenomena all dissipate mechanical energy. Therefore , a non-ideal ﬂuid rushing into the reverse sprinkler would acquire less momentum with resp ect to the pivot, for a given pressure diﬀerence, than predicted by the analysis we carri ed out in Section 6.2. Thus the pressure-diﬀerence eﬀect would outweigh the momentum-tra nsfer eﬀect even in the steady state, leading to a small torque on the sprinkler even after t he ﬂuid has begun to hit the inside wall of the sprinkler. Total angular momentum is cons erved because the “missing” momentum of the incoming ﬂuid is being transmitted to the sur rounding ﬂuid, and ﬁnally to the tank. 6.4 History of the reverse sprinkler problem The literature on the subject of the reverse sprinkler is abu ndant and confusing. Ernst Mach speaks of “reaction wheels” blowing or sucking air wher e we have spoken of regular or reverse sprinklers respectively ([137]): It might be supposed that sucking on the reaction wheels woul d produce the opposite motion to that resulting from blowing. Yet this doe s not usually take place, and the reason is obvious . .. Generally, no perceptib le rotation takes place on the sucking in of the air . .. If . ..an elastic ball, which ha s one escape-tube, be attached to the reaction-wheel, in the manner represented i n [Fig. 6.8(a)], and be alternately squeezed so that the same quantity of air is by turns blown out and sucked in, the wheel will continue to revolve rapidly in t he same direction as it did in the case in which we blew into it. This is partly due to the fact that the air sucked into the spokes must participate in the motion of the latter and therefore can produce no reactional rotation, but it also re sults partly from the diﬀerence of the motion which the air outside the tube assume s in the two cases. In blowing, the air ﬂows out in jets, and performs rotations. In sucking, the air comes in from all sides, and has no distinct rotation .. . 98 (a) (b) Figure 6.8: Illustrations from Ernst Mach’s Mechanik ([137]): (a) Figure 153 a in the original. (b) Figure 154 in the original. (Images in the public domain, cop ied from the English edition of 1893.) Mach appears to base his treatment on the observation that a “ reaction wheel” is not seen to turn when sucked on. He then sought a theoretical rati onale for this observation without arriving at one that satisﬁed him. Thus the bluster a bout the explanation being “obvious,” accompanied by the tentative language about how “generally, no perceptible rotation takes place” and by the equivocation about how the l ack of turning is “partly due” to the air “participating in the motion” of the wheel and part ly to the air sucked “coming in from all sides.” Mach goes on to say that if we perforate the bottom of a hollow cylinder . ..and place t he cylinder on [a pivot], after the side has been slit and bent in the manner i ndicated in [Fig. 6.8(b)], the [cylinder] will turn in the direction of t he long arrow when blown into and in the direction of the short arrow when sucked on. The air, here, on entering the cylinder can continue its rotation unimpeded , and this motion is accordingly compensated for by a rotation in the op posite direction ([138]). This observation is correct and interesting: It shows that i f the incoming water did not give up all its angular momentum upon hitting the inner wa ll of the reverse sprinkler, then the device would turn toward the incoming water, as we di scussed at the beginning of 99 Section 6.3.5 In his introduction to Mach’s Mechanik , mathematician Karl Menger describes it as “one of the great scientiﬁc achievements of the [nineteenth ] century” ([139]), but it seems that the passage we have quoted was not well known to the twent ieth-century scientists who commented publicly on the reverse sprinkler. Feynman ([ 131]) gave no answer to the problem and wrote as if he expected and observed rotation . Some have pointed out, however, that the fact that he cranked up the pressure until t he bottle exploded suggests another explanation: that he expected rotation and didn’t s ee it. This interpretation seems to be supported by a recent letter published by E. Creutz, who claims to have been the only other person at the Princeton cyclotron when Feynman carrie d out his experiment ([129]). Creutz, however, explicitly disclaims any knowledge of wha t Feynman’s own theoretical understanding of the problem was. In [140] and [141], the authors discuss the problem and claim that no rotation is observed, but they pursue the matter no further. In [142], it is suggest ed that students demonstrate as an exercise that “the direction of rotation is the same wheth er the ﬂow is supplied through the hub [of a submerged sprinkler] or withdrawn from the hub, ” a result that is discounted by almost all the rest of the literature. Shortly after Feynman’s memoirs appeared, A. T. Forrester p ublished a paper in which he concluded that if water is sucked out of a tank by a vacuum at tached to a sprinkler then the sprinkler will not rotate ([143]). But he also made the st range claim that Feynman’s original experiment at the Princeton cyclotron, in which he had high air pressure in the bottle push the water out, would actually cause the sprinkle r to rotate in the direction of the incoming water ([143]). An exchange on the issue of conse rvation of angular momentum between Shultz and Forrester appeared shortly thereafter ( [136, 144]). The following year L. Hsu, a high school student, published an experimental ana lysis that found no rotation of the reverse sprinkler and questioned (quite sensibly) Fo rrester’s claim that pushing the water out of the bottle was not equivalent to sucking it out ([ 145]). E. R. Lindgren also published an experimental result that supported the claim t hat the reverse sprinkler did not turn ([146]). 5In [149], P. Hewitt proposes a physical setup identical to th e one shown in Fig. 6.8(b), and observes that the device turns in opposite directions depending on wh ether the ﬂuid pours out of or into it. Hewitt’s discussion seems to ignore the important diﬀerence between such a setup and the reverse sprinkler. The issue has recently been investigated in [130]. 100 After Feynman’s death, his graduate research advisor, J. A. Wheeler, published some reminiscences of Feynman’s Princeton days from which it wou ld appear that Feynman observed no motion in the sprinkler before the bottle explod ed (“a little tremor as the pressure was ﬁrst applied . .. but as the ﬂow continued there w as no reaction”) ([147]). In 1992 the journalist James Gleick published a biography of Fe ynman in which he states that both Feynman and Wheeler “were scrupulous about never r evealing the answer to the original question” and then claims that Feynman’s answer al l along was that the sprinkler would not turn ([148]). The physical justiﬁcation that Glei ck oﬀers for this answer is misleading: Gleick echoes one of Mach’s comments in [137] th at the water entering the reverse sprinkler comes in from many directions, unlike the water leaving a regular sprinkler, which forms a narrow jet. Although this observation is corre ct, it is not very relevant to the question at hand. The most detailed and pertinent work on the subject, both the oretical and experimental, was published by Berg, Collier, and Ferrell, who claimed tha t the reverse sprinkler turns toward the incoming water ([150, 151]). Guided by Schultz’s arguments about conservation of angular momentum ([136]), the authors oﬀered a somewhat c omplicated statement of the correct observation that the sprinkler picks up a bit of angu lar momentum before reaching a steady state of zero torque once the water is ﬂowing steadil y into the sprinkler. When the water stops ﬂowing, the sprinkler comes to a halt.6 The air-sucking reverse sprinkler at the Edgerton Center at MIT shows no movement at all ([153]). As in the setups used by Feynman and others, this sprinkler arm is not mounted on a true pivot, but rather turns by twisting or bending a ﬂexi ble tube. Any transient torque will therefore cause, at most, a brief shaking of such a device. The University of Maryland’s Physics Lecture Demonstration Facility oﬀer s video evidence of a reverse sprinkler, mounted on a true pivot of very low friction, turn ing slowly toward the incoming water ([152]). According to R. E. Berg, in this particular se tup 6There are other references in the literature to the reverse s prinkler. For a rather humorous exchange, see [154] and [155]. Already in 1990 the American Journal of Physics had received so many conﬂicting analyses of the problem that the editor proposed “a moratorium on publ ications on Feynman’s sprinkler” ([156]). In one of her 1996 columns for Parade Magazine , Marilyn vos Savant, who bills herself as having the highest recorded IQ, oﬀered an account of Feynman’s experiment that , she claimed, settled that the reverse sprinkler does not move ([157]). Vos Savant’s column emphasized the co nfusion of Feynman and others when faced with the problem, leading a reader to respond with a letter to his local newspaper in which he questioned the credibility of physicists who address matters more comp licated than lawn sprinklers, such as the origin of the universe ([158]). 101 while the water is ﬂowing the nozzle rotates at a constant ang ular speed. This would be consistent with conservation of angular momentum e xcept for one thing: while the water is ﬂowing into the nozzle, if you reach and stop the nozzle rotation it should remain still after you release it. [But, in practice,] after [the nozzle] is released it starts to rotate again” ([162]). This behavior is consistent with non-zero dissipation of ki netic energy in the ﬂuid ﬂow, as we have discussed. Angular momentum is conserved, but onl y after the motion of the tank is taken into account.7An earlier, unpublished treatment of how dissipation cause s a steady-state torque on the reverse sprinkler is due to Titc omb, Rueckner, and Sokol ([163]). Rueckner also reports that the behavior of a sprink ler made to suck argon gas whose viscosity is adjusted by changing its temperature see ms to corroborate that higher viscosity leads to a larger steady-state torque. This exper iment, however, would need to be carried out more carefully to fully conﬁrm this eﬀect experi mentally ([164]). 6.5 Conclusions We have oﬀered an elementary theoretical treatment of the be havior of a reverse sprinkler, and concluded that, under idealized conditions, it should e xperience no torque while ﬂuid ﬂows steadily into it, but as the ﬂow commences, it will pick u p an angular momentum opposite to that of the incoming ﬂuid, which it will give up as the ﬂow ends. However, in the presence of viscosity or turbulence, the reverse sprink ler will experience a small torque even in steady state, which would cause it to accelerate towa rd the incoming water. This torque is balanced by an opposite torque acting on the surrou nding ﬂuid and ﬁnally on the tank itself. Throughout our discussion, our foremost concern was to emph asize physical intuition and to make our treatment as simple as it could be made (but not simpler). A question about what L. A. Delsasso called, according to Feynman’s recollec tion, “a freshman experiment” ([133]) deserves an answer presented in a language at the cor responding level of complication. More important is the principle, famously put forward by Fey nman himself when discussing 7In the late 1950’s and early 1960’s, there was some interest i n the related physics problem of the so-called putt-putt (or pop-pop) boat, a fascinating toy boat that pro pels itself by heating (usually with a candle) an inner tank connected to a submerged double exhaust. Steam bu bbles cause water to be alternately blown out of and sucked into the tank ([159, 160, 161]). The ship mov es forward, much like Mach described the “reaction wheel” turning vigorously in one direction as air was alternately blown out and sucked in. 102 the spin statistics theorem, that if we can’t “reduce it to th e freshman level,” we don’t really understand it ([165]). 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