Table Of ContentINTERNATIONAL SERIES OF MONOGRAPHS
ON PHYSICS
SERIES EDITORS
J. BIRMAN CITY UNIVERSITY OF NEW YORK
S. F. EDWARDS UNIVERSITY OF CAMBRIDGE
R. FRIEND UNIVERSITY OF CAMBRIDGE
M. REES UNIVERSITY OF CAMBRIDGE
D. SHERRINGTON UNIVERSITY OF OXFORD
G. VENEZIANO CERN, GENEVA
International Series of Monographs on Physics
158.F.Strocchi:Anintroduction tonon-perturbativefoundationsofquantumfieldtheory
157.K.H.Bennemann,J.B.Ketterson:Novelsuperfluids,Volume2
156.K.H.Bennemann,J.B.Ketterson:Novelsuperfluids,Volume1
155.C.Kiefer:Quantumgravity,Thirdedition
154.L.Mestel:Stellarmagnetism,Secondedition
153.R.A.Klemm:Layeredsuperconductors, Volume1
152.E.L.Wolf:Principlesofelectrontunnelingspectroscopy, Secondedition
151.R.Blinc:Advancedferroelectricity
150.L.Berthier,G.Biroli,J.-P.Bouchaud,W.vanSaarloos,L.Cipelletti:Dynamical heterogeneities in
glasses,colloids,andgranularmedia
149.J.Wesson:Tokamaks,Fourthedition
148.H.Asada,T.Futamase,P.Hogan:Equationsofmotioningeneralrelativity
147.A.Yaouanc,P.DalmasdeR´eotier:Muonspinrotation,relaxation,andresonance
146.B.McCoy:Advancedstatisticalmechanics
145.M.Bordag,G.L.Klimchitskaya,U.Mohideen,V.M.Mostepanenko:AdvancesintheCasimireffect
144.T.R.Field:Electromagneticscatteringfromrandommedia
143.W.Go¨tze:Complexdynamicsofglass-formingliquids—amode-couplingtheory
142.V.M.Agranovich:Excitationsinorganicsolids
141.W.T.Grandy:Entropyandthetimeevolutionofmacroscopicsystems
140.M.Alcubierre:Introductionto3+1numericalrelativity
139.A.L.Ivanov,S.G.Tikhodeev:Problemsofcondensedmatterphysics—quantumcoherence
phenomenainelectron-holeandcoupledmatter-lightsystems
138.I.M.Vardavas,F.W.Taylor:Radiation andclimate
137.A.F.Borghesani:Ionsandelectronsinliquidhelium
136.C.Kiefer:Quantumgravity,Secondedition
135.V.Fortov,I.Iakubov,A.Khrapak:Physicsofstronglycoupledplasma
134.G.Fredrickson:Theequilibriumtheoryofinhomogeneous polymers
133.H.Suhl:Relaxationprocessesinmicromagnetics
132.J.Terning:Modernsupersymmetry
131.M.Marin˜o:Chern–Simonstheory,matrixmodels,andtopologicalstrings
130.V.Gantmakher:Electronsanddisorderinsolids
129.W.Barford:Electronicandopticalpropertiesofconjugatedpolymers
128.R.E.Raab,O.L.deLange:Multipoletheoryinelectromagnetism
127.A.Larkin,A.Varlamov:Theoryoffluctuationsinsuperconductors
126.P.Goldbart,N.Goldenfeld,D.Sherrington:Stealingthegold
125.S.Atzeni,J.Meyer-ter-Vehn:Thephysicsofinertialfusion
123.T.Fujimoto:Plasmaspectroscopy
122.K.Fujikawa,H.Suzuki:Pathintegralsandquantumanomalies
121.T.Giamarchi:Quantumphysicsinonedimension
120.M.Warner,E.Terentjev:Liquidcrystalelastomers
119.L.Jacak,P.Sitko,K.Wieczorek,A.Wojs:QuantumHallsystems
118.J.Wesson:Tokamaks,Thirdedition
117.G.Volovik:TheUniverseinaheliumdroplet
116.L.Pitaevskii,S.Stringari:Bose–Einsteincondensation
115.G.Dissertori,I.G.Knowles,M.Schmelling:Quantumchromodynamics
114.B.DeWitt:Theglobalapproachtoquantumfieldtheory
113.J.Zinn-Justin:Quantumfieldtheoryandcriticalphenomena,Fourthedition
112.R.M.Mazo:Brownianmotion—fluctuations,dynamics,andapplications
111.H.Nishimori:Statisticalphysicsofspinglassesandinformationprocessing—an introduction
110.N.B.Kopnin:Theoryofnonequilibriumsuperconductivity
109.A.Aharoni:Introductiontothetheoryofferromagnetism, Secondedition
108.R.Dobbs:Heliumthree
107.R.Wigmans:Calorimetry
106.J.Ku¨bler:Theoryofitinerantelectronmagnetism
105.Y.Kuramoto,Y.Kitaoka:Dynamicsofheavyelectrons
104.D.Bardin,G.Passarino:Thestandardmodelinthemaking
103.G.C.Branco,L.Lavoura,J.P.Silva:CPViolation
102.T.C.Choy:Effectivemediumtheory
101.H.Araki:Mathematical theoryofquantumfields
100.L.M.Pismen:Vorticesinnonlinearfields
99.L.Mestel:Stellarmagnetism
98.K.H.Bennemann:Nonlinearopticsinmetals
94.S.Chikazumi:Physicsofferromagnetism
91.R.A.Bertlmann:Anomaliesinquantumfieldtheory
90.P.K.Gosh:Iontraps
87.P.S.Joshi:Globalaspectsingravitationandcosmology
86.E.R.Pike,S.Sarkar:Thequantumtheoryofradiation
83.P.G.deGennes,J.Prost:Thephysicsofliquidcrystals
73.M.Doi,S.F.Edwards:Thetheoryofpolymerdynamics
69.S.Chandrasekhar:Themathematical theoryofblackholes
51.C.Møller:Thetheoryofrelativity
46.H.E.Stanley:Introductiontophasetransitionsandcriticalphenomena
32.A.Abragam:Principlesofnuclearmagnetism
27.P.A.M.Dirac:Principlesofquantummechanics
23.R.E.Peierls:Quantumtheoryofsolids
An Introduction to Non-Perturbative
Foundations of Quantum Field
Theory
Franco Strocchi
1
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof
OxfordUniversityPressintheUKandincertainothercountries
(cid:2)c FrancoStrocchi2013
Themoralrightsoftheauthorhavebeenasserted
FirstEditionpublishedin2013
Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin
aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe
priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted
bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics
rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe
aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe
addressabove
Youmustnotcirculatethisworkinanyotherform
andyoumustimposethissameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
ISBN 978–0–19–967157–1
Printedandboundby
CPIGroup(UK)Ltd,Croydon,CR04YY
Preface
The subject of these notes was the content of courses given at Scuola Normale
Superiore (Pisa) during the last ten years and addressed to graduate students in
theoretical physics.
Quantum field theory (QFT) has proved to be the most useful strategy for the
description of elementary particle interactions, and as such is regarded as a funda-
mental part of modern theoretical physics. The textbooks on QFT are hundreds and
someofthemexcellent,suchastherecentbookbyS.Weinberg.Inmostpresentations
theemphasisisontheeffectivenessofthetheoryinproducingexperimentallytestable
predictions, which at present essentially means perturbative QFT.
Clearly,itwouldbesillytounderestimatetheextraordinarysuccessofperturbative
quantum electrodynamics (QED) and of the perturbative standard model of elec-
troweakinteractions,theagreementwithexperimentsbeingveryimpressive.However,
also in view of the fact that QFT cannot be regarded as the final theory, it would
be somewhat reductive to disdain questions of consistency and/or of mathematical
soundness. The reasons are many.
Theperturbativeseriesisknowntodiverge,andactually,fortheprototypicmodel
of self-interacting scalar field (ϕ4-theory), on which most of the textbooks are based,
theperturbativeexpansionismisleading,sincethetheoryhasbeenprovedtobetrivial
ins+1dimensions(withs≥3).Clearly,thismakesitdifficulttodefineaQFTthrough
its perturbative expansion and raises conceptual problems such as the mathematical
consistency of such an expansion.
As a matter of fact, after more than fifty years of QFT we are still in the
embarrassing situation of not knowing a single non-trivial (even non-realistic) model
of QFT in 3+1 dimensions, allowing a non-perturbative control.
As a reaction to these consistency problems one may take the position that they
arerelatedtoourignorance ofthephysicsofsmall distances andthat QFTisonly an
effectivetheory,sothatradicallynewideasareneededforaconsistentquantumtheory
ofrelativisticinteractions(in3+1dimensions).Actually,proposals(likestringtheory,
non-commutative spacetime) have recently been put forward for possible descriptions
of the ultraviolet (UV) regime, leading to a theory more fundamental than QFT.
Inthisperspective,itmaybeusefultocriticallyre-examinetheconceptual motiva-
tions for the birth of QFT. Even if the difficulties of a relativistic quantum mechanics
(RQM)basedonrelativisticwaveequationsarementionedinanytextbookonQFT,a
general discussion of the conflict between quantum stability (i.e., spectral condition),
relativisticcovariance,andlocality(i.e.,localizationandfinitepropagationspeed)may
be of help in understanding the transition from one-particle Schr¨odinger quantum
mechanics, briefly relativistic wave mechanics (RWM), to QFT, with the necessary
involvement of infinite degrees of freedom. No arbitrariness is involved in the free
vi Preface
field case with mass gap, since the Fock representation is uniquely required by the
existence of the Hamiltonian, but there is still no non-perturbative control of a non-
trivialinteractionin3+1dimensions,compatiblywiththeconstraintsofthepositivity
of the energy and locality.
In Chapter 2 the basic ideas and the mathematical problems of the perturbative
expansion are outlined. The lack of convergence of the perturbative series is argued
alsoonthebasisofsimplelow-dimensionalmodels(Dysonargument,non-convergence,
asymptotic series, Borel summability). The general mathematical problems are dis-
cussed with reference to the mathematical inconsistency of the basic ingredients:
namely, the interaction picture, the canonical quantization, and the choice of the
Fock representation.
The non-perturbative and mathematical foundations of QFT are discussed in
Chapter3,intermsofgeneralphysicalrequirements,accordingtotheprinciplesofQM
and relativistic invariance (Wightman formulation). The emerging theory provides a
substitute for canonical quantization, which has been proved to be mathematically
inconsistentinthepresenceofnon-trivialinteractionsin3+1dimensions,andreduces
to it in the free field case.
General non-perturbative results, with relevant experimental implications, are
discussed in Chapter 4. In particular, the spin–statistics theorem, the PCT theorem,
the occurrence of Schwinger terms in current commutators and the axial anomaly,
which governs the π →2γ decay.
0
The analysis of the analyticity properties implied by the physical requirements
of positivity of the energy, relativistic invariance and locality allows for a non-
perturbative definition of the functional integral approach to QFT, which has proved
usefulforpracticalpurposesandforthenon-perturbativeconstructiveapproach.Since
the Feynman path integral does not have the property (σ additivity), which allows
computation of the integral by approximating the integrand, one needs the imaginary
time version and the Wick rotation leading to it requires suitable analyticity proper-
ties. The imaginary times (Schwinger) functions define the so-called Euclidean QFT
(Chapter 5), which displays remarkable analogies with classical statistical mechanics;
the full equivalence with the real-time formulation is guaranteed by the Osterwalder–
Schrader(OS)orreflectionpositivity,whichplaysacrucialroleinthelatticeapproach
to QFT.
InChapter6thenon-perturbativedefinitionandexistenceoftheS-matrixisbriefly
outlined through the derivation of the LSZ asymptotic condition from the general
principles of quantum mechanics and relativity, both in the case of mass gap (Haag–
Ruelletheory)andforfieldsdescribingmasslessparticles(Buchholztheory).Needless
tosay,anon-perturbativeapproachtotheS-matrixisacrucialissueforthecontrolof
the infrared problem in QED and for the discussion of color confinement in quantum
chromodynamics (QCD). A brief discussion of the infrared problem in QED is given
in Section4.
The general problem of quantizing gauge theories is discussed in Chapter 7, with
emphasisonthegeneralstructuresandmechanisms;inparticular,itisarguedthatthe
crucial property is not gauge invariance (which must be broken by a gauge fixing for
thestandardquantizationprocedures),butratherthelocalGausslaw(andtherelated
Preface vii
Ward identities), which is implied by gauge invariance (second Noether theorem) and
actually survives the gauge-breaking by a gauge-fixing.
Mostofthepeculiarpropertiesofgaugetheories,likethenon-localityofthecharged
states, the non-local field algebra, the superselection of Gauss charges associated
to the gauge group, the evasion of the Goldstone theorem (Higgs mechanism), the
“linearly rising quark potential” requiring a violation of the cluster property, and the
infraparticlestructure(i.e.,thelackofadefinitemass)ofthechargedparticlesinQED,
areallstrictlyrelatedtothelocalGausslawandcanbeunderstoodingeneral,without
reference to a particular Lagrangian model. In view of the relevance of the Higgs
mechanism for the standard model of elementary particle interactions, its realization
in the local renormalizable gauges and in the physical Coulomb gauge is discussed
in detail. As a result one obtains a non-perturbative proof of the Higgs mechanism:
namely, that only the vector bosons corresponding to the broken generators acquire
a mass, and their two-point function gives the Goldstone spectrum, so that there are
no massless Goldstone bosons as well.
The last chapter is devoted to the non-perturbative derivation of the vacuum
structure and chiral symmetry breaking in QCD, which govern the mass generation.
Ratherthanrelyingonthesemiclassicalapproximationintermsoftheinstantondilute
gas picture and on the topological classification of the gauge-field configurations in
the functional integral, we revisit the Jackiw strategy of exploiting the topological
structure of the gauge group. This avoids the problem of the zero functional measure
of the regular gauge-field configurations and allows a localization of the topological
structure,whichisprecludedfortheinstantonconfigurationsbytheirpropertyofmin-
imizing the action integral. A careful formulation of the temporal gauge avoiding the
mathematical inconsistencies and paradoxes, which invalidate the existing treatment,
provides a solution of the so-called U(1) problem and derives the θ vacuum structure
solely from the non-trivial topology of the gauge group, which is shown to force the
breaking of chiral symmetry in each θ sector.
The aim of these notes is to provide material complementary to the standard
teaching of QFT, addressed to students interested in questions of principle and in
thosefoundationalaspectswhichhaveanon-trivialphysicalimpact.Forthesereasons,
an effort is made to correct widespread prejudices and misconceptions which partly
invalidate the standard textbook presentation of crucial issues such as the infrared
problem,theHiggsmechanism,theinterplaybetweentopologyandvacuumstructure
in QCD, etc.
Moreover, the choice has been made of emphasizing the main ideas and problems
withoutaimingtoprovideanexhaustivetreatmentofthesubjectsdiscussed,forwhich
other more extended treatments, quoted in the footnotes, are available. The idea is to
provide basic information for a possible wide audience, leaving to the mathematically
minded reader the task of further deepening the outlined problems, according to his
taste and his mathematical education, e.g., by referring to the very systematic book
by J. Glimm and A. M. Jaffe.
The discussion of gauge field theories is largely based on collaborations and
illuminating discussions with Gianni Morchio, to whom I am greatly indebted. For
the general look to the non-perturbative foundations of QFT, the mentorship by
A. S. Wightman is gratefully acknowledged.
This page intentionally left blank
Contents
1 Relativistic quantum mechanics 1
1 Quantum mechanics and relativity 1
2 Relativistic Schro¨dinger wave mechanics 2
2.1 Relativistic Schr¨odinger equation 3
2.2 Klein–Gordon equation 4
2.3 Dirac equation 6
2.4 The general conflict between locality and energy positivity 8
3 Relativistic particle interactions and quantum mechanics 11
3.1 Problems of relativistic particle interactions 11
3.2 Field interactions and quantum mechanics 15
4 Free field equations and quantum mechanics 16
5 Particles as field quanta 20
6 Appendix: The Dirac equation 25
7 Appendix: Canonical field theory 29
2 Mathematical problems of the perturbative expansion 31
1 Dyson’s perturbative expansion 31
2 Dyson argument against convergence 34
2.1 ϕ4 model in zero dimensions 35
2.2 ϕ4 model in 0+1 dimensions 36
2.3 ϕ4 model in 1+1 and 2+1 dimensions 38
3 Haag theorem; non-Fock representations 39
3.1 Quantum field interacting with a classical source 40
3.2 Bloch–Nordsieck model; the infrared problem 42
3.3 Yukawa model; non-perturbative renormalization 44
4 Ultraviolet singularities and canonical quantization 48
5 Problems of the interaction picture 52
6 Appendix: Locality and scattering 53
6.1 Locality and asymptotic states 53
6.2 Scattering by a long-range potential 57
6.3 Adiabatic switching 60
6.4 Asymptotic condition 61
7 Wick theorem and Feynman diagrams 63
7.1 Compton and electron–electron scattering;
electron–positron annihilation 67
3 Non-perturbative foundations of quantum field theory 69
1 Quantum mechanics and relativity 69
2 Properties of the vacuum correlation functions 72
