The Conceptual Framework of Quantum Field Theory

793 Pages·2012·3.75 MB·English

by  Anthony Duncan

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