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The Conceptual Framework of Quantum Field Theory PDF

793 Pages·2012·3.75 MB·English
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The Conceptual Framework of Quantum Field Theory Anthony Duncan 1 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. IffurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©AnthonyDuncan2012 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2012 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable ISBN 978–0–19–957326–4 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Preface In the roughly six decades since modern quantum field theory came of age with the introduction in the late 1940s of covariant field theory, supplemented by renormaliza- tion ideas, there has been a steady stream of expository texts aimed at introducing eachnewgenerationofphysiciststotheconceptsandtechniquesofthiscentralareaof moderntheoreticalphysics.Eachdecadehasproducedoneormore“classics”,attuned to the background, needs, and interests of students wishing to acquire a proficiency in the subject adequate for the beginning researcher at the time. In the 1950s the seminal text of Jauch and Rohrlich, Theory of Photons and Electrons, provided the first systematic textbook treatment of the Feynman diagram technique for quantum electrodynamics, while more or less simultaneously the first field-theoretic attacks on thestronginteractionswerepresentedinthetwo-volumeMesons and FieldsofBethe, de Hoffmann, and Schweber. The 1960s saw the appearance of the massive treatise by Schweber, Introduction to Relativistic Quantum Field Theory, which addressed in much greater detail formal aspects of the theory, including the LSZ asymptotic formalismandtheWightmanaxiomaticapproach.Thedominanttextofthelate1960s was undoubtedly the two-volume text of Bjorken and Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields, which combined a thorough introduction to Feynman graph technology (in volume 1) with a more formal introduction to Lagrangian field theory (in volume 2). In the 1970s the emergence of non-abelian gaugetheoriesastheoverwhelminglyfavoredcandidatesforasuccessfulfield-theoretic descriptionofweakandstronginteractionscoincidedwiththeemergenceoffunctional (path-integral) methods as the appropriate technical tool for quantization of gauge theories. In due course, these methods received full treatment with the appearance in 1980 of Itzhykson and Zuber’s encyclopedic Quantum Field Theory. In a similar way, the surge to prominence of supersymmetric field theories throughout the 1980s necessitated a full account of supersymmetry, which is the sole subject of the third volume of Weinberg’s comprehensive three-volume The Quantum Theory of Fields, the first edition of which appeared in 1995. With such a selection of classic expository treatises (not to mention many other fine texts not listed above—with apologies to authors of same!) one may well doubt theneedforyetanotherintroductorytreatmentofquantumfieldtheory.Nevertheless, inthecourseofteachingthesubjecttograduatestudents(typically,secondyear)over the last 25 years, I have been struck by the number of occasions on which important conceptual issues are raised by questions in the classroom which require a careful explanation not to be found in any of the readily available textbooks on quantum field theory. To give just a small sample of the sort of questions one encounters in the classroomsetting:“OftheplethoraofquantumfieldsintroducedtodescribeNatureat subatomic scales, why do so few (basically, only electromagnetism and gravity) have classicalmacroscopiccorrelates?”;“Iftherearemanypossiblequantumfieldsavailable iv Preface to ‘represent’ a given particle, can, or in what sense does, quantum field theory prescribe a unique all-time dynamics?”; “If the interaction picture does not exist, as implied by Haag’s theorem, why (or in what sense) are the formulas derived in this picture for the S-matrix still valid?”; “Are there non-perturbative phenomena amenable to treatment using perturbative (i.e., graph-theoretical) methods?”; and so on. None of these questions require an answer if one’s attitude in learning quantum field theory amounts to a purely pragmatic desire to “start with a Lagrangian and compute a process to two loops”. However, if the aim is to arrive at a truly deep andsatisfyingcomprehensionofthemostpowerful,beautiful,andeffectivetheoretical edifice ever constructed in the physical sciences, the pedagogical approach taken by the instructor has to be quite a bit different from that adopted in the “classics” enumerated above. In the present work, an attempt is made to provide an introduction to quantum field theory emphasizing conceptual issues frequently neglected in more “utilitarian” treatments of the subject. The book is divided into four parts, entitled respec- tively,“Origins”,“Dynamics”,“Symmetries”,and“Scales”.Althoughtheemphasisis conceptual—theaimistobuildthetheoryupsystematicallyfromsomeclearlystated foundationalconcepts—andthereforetoalargeextentanti-historical,Ihaveincluded twohistoricalchaptersinthe“Origins”sectionwhichtracetheevolutionofthemodern theoryfromtheearliest“penumbra”ofquantum-field-theoreticalphenomenadetected by Planck and Einstein in the early years of the twentieth century to the emergence, inthelate1940s,oftherecognizablestructureofmodernquantumfieldtheory,inthe form of quantum electrodynamics. The reader anxious to proceed with the business of logically developing the framework of modern field theory is at liberty to skim, or even entirely omit, this historical introduction. The three remaining sections of the book follow a step-by-step reconstruction of this framework beginning with just a few basic assumptions: relativistic invariance, thebasicprinciplesofquantummechanics,andtheprohibitionofphysicalactionata distance embodied in the clustering principle. The way in which these physical ingre- dients combine to engender some of the most dramatic results of relativistic quantum field theory is outlined qualitatively in Chapter 3, which also contains a summary of the topics treated in later chapters. Subsequent chapters in the “Dynamics” section of the book lay out the basic structure of quantum field theory arising from the sequential insertion of quantum-mechanical, relativistic, and locality constraints. The rather extended treatment of free fields allows us to discuss important conceptual issues (e.g., the classical limit of field theory) in greater depth than usually found in the standard texts. Some applications of perturbation theory to some simple theories and processes are discussed in Chapter 7, after the construction of covariant fields for general spin has been explained. A deeper discussion of interacting field theories is initiated in Chapters 9, 10, and 11, where we treat first general features shared by all interacting theories (Chapter 9) and then aspects amenable to formal perturbation expansions (Chapter 10). The “Dynamics” section concludes with a discussion of “non-perturbative” aspects of field theory—a rather imprecise methodological term encompassing a wide variety of very different physical processes. In Chapter 11 we attempttoclarifytheextenttowhichcertainfeaturesoffieldtheoryare“intrinsically” Preface v non-perturbative,requiringmethodscomplementarytothegraphicalexpansionsmade famous by Feynman. In the “Symmetries” section we explore the many important ways in which symmetry principles influence both our understanding and our use of quantum field theory. Of course, at the heart of relativistic quantum field theory lies an inescapable symmetry of critical importance: Lorentz-invariance, which, together with translational symmetry in space and time, makes up the larger symmetry of the Poincar´egroup.ThecentralityofthissymmetryexplainsthedominanceofLagrangian methods in field theory, even though from a physical standpoint the Hamiltonian would appear (as is typically the case in non-relativistic quantum theory) to hold pride of place. The role played by Lorentz-invariance in restricting the dynamics of a fieldtheoryisthemaintopicofChapter12,whichalsoincludesanintroductiontothe extension of the Poincar´e algebra to the graded superalgebra of supersymmetric field theory. Discrete spacetime symmetries, and the famous twin theorems of axiomatic field theory—the Spin-Statistics and TCP theorems—are the subject of Chapter 13. The discussion of global symmetries, exact and approximate, in Chapter 14 leads naturally into the very important topics of spontaneous symmetry-breaking and the Goldstone theorem. The “Symmetries” section of the book closes with a treatment of local gauge symmetries in Chapter 15, which imply remarkable new features not present in theories where the only symmetries are global (i.e., involve a finite-dimensional algebra of spacetime-independent transformations). With the final section of the book, entitled “Scales”, we come to perhaps the most characteristic conceptual feature of quantum field theory: the scale separation property exhibited by theories defined by an effective local Lagrangian. Given that essentially all of the information obtained from scattering experiments at accelerators concerns asymptotic transitions (i.e., the infinite time evolution of an appropriately prepared quantum state, terminated by a detection measurement) it is critically importantfortheoreticalprogressthattheprobabilitiesofsuchtransitionsnotdepend inasensitivewayoninteractiondetailsatmuchsmallerdistancesthanthosepresently accessibleinacceleratorexperiments(roughly,theinverseofthecenter-of-massenergy ofthecollisionprocess).Theinsensitivityoffieldtheoryamplitudestoourinescapable ignorance of the nature of the interactions at very short distances (or equivalently, high momentum) is therefore of central importance if we are to infer reliably an underlying microdynamics from the limited phenomenology available at any given time.Remarkably,inthisrespectquantumfieldtheoriesarefarkindertousthantheir classical(particleorfield)counterparts,wherenon-linearitiesalmostalwaysintroduce chaotic behavior which effectively precludes the possibility of accurate predictions of state evolution over long time periods. The technical foundations needed for examining these issues are taken up in Chapter 16, which contains an account of regularization, power-counting, effective Lagrangians, and the renormalization group. Applications to the proof of perturbative renormalizability, and a discussion of the “triviality” phenomenon (the absence of a non-trivial continuum limit), follow in Chapter 17. Chapters 18 and 19 then explore important features of the behavior of quantum field theories at short distance (e.g., the operator product expansion and factorization) and long distance (in particular, the complications in defining the correctphysicalstatespaceinunbrokenabelianandnon-abeliangaugefieldtheories). vi Preface To the beginning student, quantum field theory all too often takes on the appear- anceofamulti-headedHydra,withmanyintertwinedparts,theunderstandingofany one of which seems to require a prior understanding of the rest of the frightening anatomy of the whole beast. The motivation for the present work was the author’s desire to provide an introduction to modern quantum field theory in which this rich andcomplexstructureisseentoarisenaturallyfromafewbasicconceptualinputs,in contrasttothemoretypicalapproachinwhichLagrangianfieldtheoryispresentedas a theoretical fait accompli and then subsequently shown to have the desired physical features. Much(perhapsmost)oftheattitudetowardsquantumfieldtheoryexpressedinthis bookistheresultofinnumerableconversations,overfourdecades,withcolleaguesand students. For laying the foundations of my knowledge of field theory I wish especially tothankmypredoctoralandpost-doctoralmentors(StevenWeinbergandAlMueller, respectively).InthecaseofthepresentworkIamextremelygratefultoEstiaEichten, MichelJanssen,AdamLeibovich,MaxNiedermaier,SergioPernice,andRalphRoskies for reading extensive parts of the manuscript, and for many useful comments and suggestions. Any remaining solecisms of style or content are, of course, entirely the responsibility of the author. Contents 1 Origins I: From the arrow of time to the first quantum field 1 1.1 Quantum prehistory: crises in classical physics 1 1.2 Early work on cavity radiation 3 1.3 Planck’s route to the quantization of energy 8 1.4 First inklings of field quantization: Einstein and energy fluctuations 14 1.5 The first true quantum field: Jordan and energy fluctuations 18 2 Origins II: Gestation and birth of interacting field theory: from Dirac to Shelter Island 30 2.1 Introducing interactions: Dirac and the beginnings of quantum electrodynamics 31 2.2 Completing the formalism for free fields: Jordan, Klein, Wigner, Pauli, and Heisenberg 40 2.3 Problems with interacting fields: infinite seas, divergent integrals, and renormalization 46 3 Dynamics I: The physical ingredients of quantum field theory: dynamics, symmetries, scales 57 4 Dynamics II: Quantum mechanical preliminaries 69 4.1 The canonical (operator) framework 70 4.2 The functional (path-integral) framework 86 4.3 Scattering theory 96 4.4 Problems 106 5 Dynamics III: Relativistic quantum mechanics 108 5.1 The Lorentz and Poincar´e groups 108 5.2 Relativistic multi-particle states (without spin) 111 5.3 Relativistic multi-particle states (general spin) 114 5.4 How not to construct a relativistic quantum theory 121 5.5 A simple condition for Lorentz-invariant scattering 125 5.6 Problems 130 6 Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity 132 6.1 Clustering and the smoothness of scattering amplitudes 133 6.2 Hamiltonians leading to clustering theories 138 6.3 Constructing clustering Hamiltonians: second quantization 144 6.4 Constructing a relativistic, clustering theory 149 6.5 Local fields, non-localizable particles! 159 viii Contents 6.6 From microcausality to analyticity 164 6.7 Problems 169 7 Dynamics V: Construction of local covariant fields 171 7.1 Constructing local, Lorentz-invariant Hamiltonians 171 7.2 Finite-dimensional representations of the homogeneous Lorentz group 173 7.3 Local covariant fields for massive particles of any spin: the Spin-Statistics theorem 177 7.4 Local covariant fields for spin-1 (spinor fields) 184 2 7.5 Local covariant fields for spin-1 (vector fields) 198 7.6 Some simple theories and processes 202 7.7 Problems 215 8 Dynamics VI: The classical limit of quantum fields 219 8.1 Complementarity issues for quantum fields 219 8.2 When is a quantum field “classical”? 223 8.3 Coherent states of a quantum field 228 8.4 Signs, stability, symmetry-breaking 234 8.5 Problems 238 9 Dynamics VII: Interacting fields: general aspects 240 9.1 Field theory in Heisenberg representation: heuristics 241 9.2 Field theory in Heisenberg representation: axiomatics 253 9.3 Asymptotic formalism I: the Haag–Ruelle scattering theory 268 9.4 Asymptotic formalism II: the Lehmann–Symanzik–Zimmermann (LSZ) theory 281 9.5 Spectral properties of field theory 289 9.6 General aspects of the particle–field connection 297 9.7 Problems 304 10 Dynamics VIII: Interacting fields: perturbative aspects 307 10.1 Perturbation theory in interaction picture and Wick’s theorem 309 10.2 Feynman graphs and Feynman rules 314 10.3 Path-integral formulation of field theory 325 10.4 Graphical concepts: N-particle irreducibility 341 10.5 How to stop worrying about Haag’s theorem 359 10.6 Problems 371 11 Dynamics IX: Interacting fields: non-perturbative aspects 374 11.1 On the (non-)convergence of perturbation theory 376 11.2 “Perturbatively non-perturbative” processes: threshhold bound states 386 11.3 “Essentially non-perturbative” processes: non-Borel-summability in field theory 400 11.4 Problems 411 Contents ix 12 Symmetries I: Continuous spacetime symmetry: why we need Lagrangians in field theory 414 12.1 The problem with derivatively coupled theories: seagulls, Schwinger terms, and T∗ products 414 12.2 Canonical formalism in quantum field theory 416 12.3 General condition for Lorentz-invariant field theory 421 12.4 Noether’s theorem, the stress-energy tensor, and all that stuff 426 12.5 Applications of Noether’s theorem 431 12.6 Beyond Poincar´e: supersymmetry and superfields 443 12.7 Problems 464 13 Symmetries II: Discrete spacetime symmetries 469 13.1 Parity properties of a general local covariant field 470 13.2 Charge-conjugation properties of a general local covariant field 474 13.3 Time-reversal properties of a general local covariant field 477 13.4 The TCP and Spin-Statistics theorems 478 13.5 Problems 485 14 Symmetries III: Global symmetries in field theory 487 14.1 Exact global symmetries are rare! 489 14.2 Spontaneous breaking of global symmetries: the Goldstone theorem 492 14.3 Spontaneous breaking of global symmetries: dynamical aspects 495 14.4 Problems 507 15 Symmetries IV: Local symmetries in field theory 509 15.1 Gauge symmetry: an example in particle mechanics 509 15.2 Constrained Hamiltonian systems 512 15.3 Abelian gauge theory as a constrained Hamiltonian system 519 15.4 Non-abelian gauge theory: construction and functional integral formulation 529 15.5 Explicit quantum-breaking of global symmetries: anomalies 544 15.6 Spontaneous symmetry-breaking in theories with a local gauge symmetry 552 15.7 Problems 565 16 Scales I: Scale sensitivity of field theory amplitudes and effective field theories 569 16.1 Scale separation as a precondition for theoretical science 570 16.2 General structure of local effective Lagrangians 571 16.3 Scaling properties of effective Lagrangians: relevant, marginal, and irrelevant operators 574 16.4 The renormalization group 581 16.5 Regularization methods in field theory 588 16.6 Effective field theories: a compendium 595 16.7 Problems 608

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