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SeriesEditors
R.FriendUniversityofCambridge M.ReesUniversityofCambridge D.SherringtonUniversityofOxford G.VenezianoCERN,Geneva
172.J.K¨ubler: Theoryofitinerantelectronmagnetism,Secondedition 171.J.Zinn-Justin: Quantumfieldtheoryandcriticalphenomena,Fifthedition
170.V.Z.Kresin,S.G.Ovchinnikov,S.A.Wolf: Superconductingstate-mechanismsand materials
169.P.T.Chru´sciel: Geometryofblackholes
168.R.Wigmans: Calorimetry–Energymeasurementinparticlephysics,Secondedition 167.B.Mashhoon: Nonlocalgravity
166.N.Horing: Quantumstatisticalfieldtheory
165.T.C.Choy: Effectivemediumtheory,Secondedition
164.L.Pitaevskii,S.Stringari: Bose-Einsteincondensationandsuperfluidity
163.B.J.Dalton,J.Jeffers,S.M.Barnett: Phasespacemethodsfordegeneratequantum gases
162.W.D.McComb: Homogeneous,isotropicturbulence-phenomenology, renormalizationandstatisticalclosures
160.C.Barrab`es,P.A.Hogan: Advancedgeneralrelativity-gravitywaves,spinning particles,andblackholes
159.W.Barford: Electronicandopticalpropertiesofconjugatedpolymers,Secondedition 158.F.Strocchi: Anintroductiontonon-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 heterogeneitiesinglasses,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: Advancesinthe Casimireffect
144.T.R.Field: Electromagneticscatteringfromrandommedia 143.W.G¨otze: 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-quantum coherencephenomenainelectron-holeandcoupledmatter-lightsystems
138.I.M.Vardavas,F.W.Taylor: Radiationandclimate
137.A.F.Borghesani: Ionsandelectronsinliquidhelium
135.V.Fortov,I.Iakubov,A.Khrapak: Physicsofstronglycoupledplasma
134.G.Fredrickson: Theequilibriumtheoryofinhomogeneouspolymers
133.H.Suhl: Relaxationprocessesinmicromagnetics
132.J.Terning: Modernsupersymmetry
131.M.Mari˜no: 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
117.G.Volovik: TheUniverseinaheliumdroplet
116.L.Pitaevskii,S.Stringari: Bose-Einsteincondensation
115.G.Dissertori,I.G.Knowles,M.Schmelling: Quantumchromodynamics
114.B.DeWitt: Theglobalapproachtoquantumfieldtheory
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
105.Y.Kuramoto,Y.Kitaoka: Dynamicsofheavyelectrons
104.D.Bardin,G.Passarino: TheStandardModelinthemaking
103.G.C.Branco,L.Lavoura,J.P.Silva: CPViolation
101.H.Araki: Mathematicaltheoryofquantumfields
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: Themathematicaltheoryofblackholes
51.C.Møller: Thetheoryofrelativity
46.H.E.Stanley: Introductiontophasetransitionsandcriticalphenomena
32.A.Abragam: Principlesofnuclearmagnetism
27.P.A.M.Dirac: Principlesofquantummechanics
23.R.E.Peierls: Quantumtheoryofsolids
QUANTUMFIELDTHEORYANDCRITICALPHENOMENA FIFTHEDITION
JEANZINN-JUSTIN
IRFU/CEA,Paris-SaclayUniversity and
FrenchAcademyofSciences
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries
c JeanZinn-Justin2021
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ToNicole
Preface
IntroducedasaquantumextensionofMaxwell’sclassicaltheory,quantumelectrodynamics(QED)hasbeenthefirstexampleofalocalquantumfieldtheory(QFT).Eventually, QFThasbecometheframeworkforthediscussionofallfundamentalinteractionsat themicroscopicscaleexcept,possibly,gravity.Moresurprisingly,ithasalsoprovided aframeworkfortheunderstandingofsecondorderphasetransitionsinstatisticalmechanics.Infact,ashopefullythisworkwillillustrate,QFTisthenaturalframeworkfor thediscussionofmostsystemsinvolvinganinfinitenumberofdegreesoffreedomwith localcouplings.ThesesystemsrangefromcoldBosegasesat thecondensationtemperature(abouttennanokelvin)toconventionalphasetransitions(fromafewdegrees toseveralhundred)andhighenergyparticlephysicsuptoTeVs,altogethermorethan twentyordersofmagnitude intheenergyscale.
Therefore,althoughexcellenttextbooksaboutQFThadalreadybeenpublished,I thought,manyyearsago,thatitmightnotbecompletelyworthlesstopresentaworkin whichthestrongformalrelationsbetweenparticlephysics andthetheoryofcriticalphenomenaaresystematicallyemphasized.Thisoptionexplainssomeofthechoicesmade inthepresentation.Aformulationintermsoffieldintegralsisadoptedtostudythe propertiesofQFT.Lessimportant,perhaps,ingeneralthespace–timemetricischosen Euclidean,asisnaturalforstatisticalmechanics,andinparticlephysicsoftenconvenient forperturbativecalculations,andnecessaryfornumericalsimulations.Thelanguageof partitionandcorrelationfunctionsisusedthroughout,eveninapplicationsofQFTto particlephysics.Renormalizationandrenormalizationgroup(RG)propertiesaresystematicallydiscussed,whereaslimitedspaceisdevotedto scatteringtheory.Onlyformal aspectsofQEDareconsidered,sinceexcellenttextbookscoverthissubjectextensively.
Forwhatfollows,notethat,inadeepquantumrelativisticcontext,onecanset = c = 1,andenergiesarethenproportionaltomomentaandmasses, andinverseofdistances.
InQFT,thebasicanalytictooltocalculatephysicalquantitiesisanexpansioninpowersoftheinteractions.Theinitial(orbare)Lagrangianof QEDgeneratesanexpansion intermsofthe barefine-structureconstant α0 = e2 0/4π c.Inastraightforwardperturbativecalculation,onediscoversthatallphysicalquantitiesareinfinite,the locality ofQED generating short-distancesingularities (onespeaksaboutultraviolet(UV)divergences).
Thissituationhastobecontrastedwithwhathappensinclassicalornon-relativistic quantummechanics(QM);there,thereplacementofmacroscopicbypoint-likeobjects leads,ingeneral,tonomathematicalinconsistencies,and isoftenaverygoodapproximation:theabsenceofthispropertywouldindeedhavemadeprogressinphysicsquite difficult.Tosummarize:inthelattertheories, phenomenaatverydifferentscales,toa goodapproximation,decouple.Mostsurprisingly,thisisnolongerthecaseinQFT.
InQED,aremedytotheinfinityproblemwasfoundempirically:onefirst regularizes QED(i.e. onerenderstheperturbativeexpansionfinite)byartificiallymodifyingthe theoryatshortdistance,orequivalentlyatlargemomentum,atascalecharacterizedby
alargemomentumcut-offΛ(ingeneral,introducingnon-physicalshort-distanceproperties).Inspiredbymethodsofcondensedmatterphysics,one thenre-expressesallphysical quantitiesintermsofthemeasuredfinestructureconstantandthephysicalmassesof particles,inplaceoftheoriginal(bare)parametersofthe Lagrangian.Afterthischange ofparametrization,thecut-offisremoved,andsomewhatmiraculously,orderbyorder inperturbationtheory,allotherphysicalquantitieshave afinitelimit.Moreover,the limitisindependentofthepreciseformoftheregularization.Thisstrangemethod, called renormalization,didsoonfindanexperimentalconfirmation:itledtopredictions agreeingwithincreasinglyimpressiveprecisionwithexperiments.
Therefore,itbecamethennaturaltosearchforother renormalizable QFTs,todescribe allinteractions.Thisledtoanothermajorachievement:arenormalizableQFTforall three,strong,weak,andelectromagneticinteractions.Theso-calledStandardModel (SM),whoseformalstructurewasproposedmorethanfortyyearsago,completelydescribesphysicsatthemicroscopicscale,andhasbeencomfortedin2012,inaspectacular way,bythediscovery,attheLargeHadronCollideroftheEuropeanCenterforNuclear Research(CERN),ofthelastmissingparticleofthemodel,theHiggsscalarboson.
TheimpressivesuccessofastrategybasedonlookingforrenormalizableQFTs,that ledtotheSM,thenslowlypromoted renormalizability asakindofadditionallawof nature.Inparticular,oncetheSMofweak,electromagnetic andstronginteractionswas established,mucheffortwasdevotedtocastgravityinthesameframework.Despite ingeniousattempts,norenormalizableformofquantumgravityhasbeenfoundyet.
InamasslessrenormalizableQFT,itisnecessarytointroduceareferencephysical energyscaleatwhichthephysicalcouplingconstantsaredefined.Itwasrealizedearly on,firstasamathematicalcuriosity,thatanRGcouldbeassociatedwithachangeof thereferencescaleatconstantphysicalproperties.TheRG describeshowthephysical (oreffective)couplingconstantsvarywiththereferencescale.
Eventually,itwasrealizedthatthispropertycouldalsobe usedtodiscusstheshortdistancepropertiesofsomephysicalprocesses.In asymptoticallyfree QFTs(wherethe freeQFTisanultraviolet(UV)RGfixedpoint),theseeffectivecouplingsbecomesmall atlarge-Euclideanmomentaand,therefore,perturbationtheory,improvedbyRG,can beused.Inparticlephysics,thetheoryofstronginteractions,basedon SU (3)gauge symmetry,sharesthisproperty.Later,WeinbergarguedthattheexistenceofUVRG fixedpoints(likeinnon-Abeliangaugetheories,ornon-linear σ models),thatis,the existenceoflimitsfortheeffectiveshort-distancecouplings,wasanecessarycondition fortheconsistencyofaQFTonallscales.However,mostofthefieldtheoriesproposed todescribestrong,electromagneticandweakinteractions arenotasymptoticallyfree.
Ofcourse,theexistenceofothernon-trivialfixedpointscannotbeestablishedin theframeworkofperturbationtheory.However,manynumericalsimulationsoffield theoriesonthelattice,whichmakenon-perturbativeexplorationspossible,havefailedto discovernon-trivialfixedpoints.Therefore,presumably, thepresentSM,whichdescribes sopreciselyparticlephysicsatpresentscale,isnotconsistentonallscales,andhasto bemodifiedatshorterdistance.
Thisalsosuggeststhatthepropertyofrenormalizabilityhasadifferentorigin. Somewhatsurprisingly,instatisticalphysics,QFThasalsobecomeanessentialtool fortheunderstandingofthecriticalbehaviourofalargeclassofsecond-orderphase transitionswithshort-rangeinteractions.Nearthecriticaltemperature,cooperative phenomenageneratealargescale,associatedwiththeso-called correlationlength.Moreover,thelarge-scalepropertiesofthesystembecomeindependentofmostofthedetails ofthemicroscopicdynamics.Firstattemptstoexplainthesepropertieswerebasedon
usualideas:adescriptiononlyinvolvingmacroscopicdegreesoffreedomadaptedtothe scaleofthecorrelationlength.Suchadescriptionnaturallyemergesinsimpleapproximationslike mean-fieldtheory.Itisconsistentwiththegeneralprobabilisticideathat averagesoveralargenumberofindependentstochasticvariablesobeyaGaussiandistribution.Thecorrespondinggeneralideasweresummarizedin Landau’stheoryofcritical phenomena.However,itslowlybecameclearthatthepredictionsofsuchatheorywere toouniversal,conflictingwithnumericalcalculationsofcriticalexponents,experimentaldataandexactresultsintwodimensions.Theseresultssupportedtheconceptofa morerestricted universality:broadclassesofsystemshaveindeedthesamelargedistanceproperties,but,unlikeinmean-fieldtheory,thesepropertiesseemedtodependon asmallnumberofqualitativefeatures,likedimensionofspace,numberofcomponentsof theorderparameter,symmetries,andsoon.Actually,ananalysisoftheleadingcorrectionstomean-fieldorGaussianapproximationsindeedrevealsthat,atleastinlow-space dimensions,theshortdistancenevercompletelydecouple, amostunusualsituation.
Toexplainthisremarkablephenomenon,thatis,thatlargedistancepropertiesofsecondorderphasetransitionsare,toalargeextent,short-distanceinsensitive,although degreesoffreedomonallscalesremaincoupled,Wilson,partiallyinspiredbysomeprior attemptsofKadanoff,introducedtheRGidea:startingfroma microscopicHamiltonian, oneintegratesout,recursively,thedegreesoffreedomcorrespondingtoshort-distance fluctuations,andgeneratesascale-dependenteffectiveHamiltonian.Universalitythen reliesupontheexistenceofIRfixedpointsinHamiltonianspace.Oneofthespectacular implicationsisthattheuniversalpropertiesofalargeclassofcriticalphenomenacan beaccuratelypredictedbythesameQFTmethodsthathadbeen inventedforparticle physics.Atleadingorderinthecriticaldomain,thephysicsofthefixed-pointHamiltoniancanbereproducedbyrenormalizable(orsuper-renormalizable)QFTs.
PredictionsobtainedfromaRGanalysisofsimplefieldtheorieslikethe(φ2)2 QFT havebeensuccessfullycomparedtoexperimentsaswellasnumericaldatafromlattice models.ThesameQFTmethodshavebeenshowntodescribevastlydifferentphysicalsystemsatcriticality,likeferromagnets,liquid–vapour,binarymixtures,superfluid heliumand,evenmoresurprisingly,statisticalpropertiesofpolymers.
IfQFThasledtoanunderstandingoftheconceptofuniversalityandmadethe calculationofmanyuniversalphysicalquantitiespossible,conversely,criticalphenomena haveshedanewlightontheoriginofrenormalizableQFTsand oftherenormalization processinparticlephysics.
Letusdescribeagain,ingeneralterms,therenormalizationmethod.Startingfroman initialLagrangian,assumedtobe renormalizable,expressedintermsof bareparameters, onegeneratesanexpansioninpowersoftheinteractions.The perturbativeexpansion isthenplaguedbyUVdivergences.Torendertheperturbativeexpansionfinite,one introducesalarge-momentumcut-offΛ,whichgeneratesanartificial,andsomewhat arbitraryshort-distancestructure,withnon-physicalproperties.Onethencalculates quantitiesatthephysicalscaleand tunesthebareparameters asfunctionsofthecut-off, insuchwaythatobservablesatthephysicalscalehaveafiniteinfiniteΛlimit.Since thebareparametershavegenerallynofiniteinfiniteΛlimit, thisledsomephysiciststo theparadoxicalconclusionthattheLagrangianinrelativisticQFTisnon-physical.The insistencefortakingtheinfinitecut-offlimitwasmotivatedbythewishtoconstructa renormalizedQFTphysicallyconsistentonallscales.Formalismsweredeveloped(like theBogoliubov–Parasiuk–Hepp–Zimmermann(BPHZ)formalism)togeneratedirectly arenormalizedperturbationtheory,inwhichtheLagrangianwasreducedtoadevice forgeneratingFeynmanrules.However,anotherinterpretationofthebareQFTand
therenormalizationprocedure,directlyinspiredfromthe functionalRGasappliedto criticalphenomenabyWilson,hasgainedstrengthovertheyears.Atveryshortdistance (thePlanckscale?),thefamiliarnotionoflocalQFTlosesitsmeaning.However,the necessarynon-localeffects,whichrenderthemorefundamentaltheoryfinite,arelimited tothismicroscopicscale(theequivalentoftheconditionofshort-rangeforcesinstatistical systems).Dynamicaleffects,ofanaturewhichatpresentcan onlybeguessed,generate atdistanceslargecomparedtothemicroscopicscale(equivalently,atlowerenergies) physicsassociatedwiththeappearanceofverylowmassparticles(comparedtothe Planckmass,forexample,allknownparticlesarealmostmassless).Atlargedistance, physicscanbedescribedbyan effectivelocalQFT,whoseactionisalinearcombinationof alllocalmonomialsconsistentwithsymmetries,andotherpossiblerequirements.With increasingdistance,atleastinamean-fieldlikeapproximation,allinteractionsscale asdimensionalanalysis(orpowercounting)indicates.Eventually,non-renormalizable interactionsbecomeverysmall,dimensionlessrenormalizableinteractionssurvive,and unprotectedterms,withpositivedimension,likemassterms,diverge.
Actually,thispicturehastobecorrected,becausetheUVdivergencesandthenecessityofintroducingalarge-momentumcut-off(asubstitutefortheunknownmicroscopic physics)showthatmicroscopicscaleandthephysicalscale donotcompletelydecouple. TheflowofinteractionshastobedescribedbyaRGthatinterpolatesbetweenthemicroscopicscaleandthephysicalscale.Still,asweshallargue,theflowofrenormalizable interactionsisonlylogarithmic,whiletheothertermshaveapower-lawbehaviour.Thus, themean-fieldanalysisofthehierarchyofinteractionsremainsqualitativelycorrect.
Thisschemehasthefollowingconsequences:itexplainsthe emergenceofrenormalizableQFTs.Theroleofrenormalizationtheoryistoprove,orderbyorderinperturbation theory,thatphysicsatthephysicalscaleis,toalargeextent,butnotcompletely,shortdistanceinsensitive(andthusinsensitivetotheprecisecut-offimplementation),butina moresubtlewayasinclassicalphysics.However, theinitial(bare)parametershavetobe consideredasbeinggiven,andonehas,therefore,toprove,beyondperturbationtheory, thatthechangeofparametrization,assumedintherenormalizationtheory,frombareto physicaleffectiveparameters,ispossible.Inparticular, thisleadstothe trivialityissue forIR-freetheories(likeQED):thephysicalchargedecreaseslogarithmicallywiththe cut-offand,therefore,the cut-offcannotbesenttoinfinity.Thecorrespondingcoupling constantsareexpectedtobesmall(thisisconsistentwiththesmallvalueof α).Moreover,mostQFTscannotbemadeconsistentonallscales.Thepossibilityofverysmall non-renormalizableinteractions,sinceproportionaltotheinverseofthepowerofcut-off, hastobeconsidered.GeneralRelativitymaybeofthisnature.
TheoriginoftheHiggsboson,whichisinvolvedinmostofthe parametersofthe SM,andthenecessary finetuning ofitsmasstermintheLagrangian,becomeessential physicsissues.Incontrasttocriticalphenomenainwhicha controlparameter,likethe temperature,canbeadjustedtomakethecorrelationlength large,inparticlephysics theexistenceofsmallmassparticleshastobeexplainedbygeneralpropertiesoftheunknownfundamentaltheory.Thisisthefamous hierarchy problem.Spontaneousbreaking ofacontinuoussymmetry,gaugeprinciple,andchiralinvariance(butwhosenaturalimplementationseemstorequireadditionalspacedimensions)aretheknownmechanisms whichgeneratemasslessparticles.Supersymmetrycanbehelpfultodealwithscalar bosons.Atpresent,thesetofgeneralconditionstobeimposedonanyfundamental theory,thatis,inthelanguageofcriticalphenomenathecompletedescriptionofthe universalityclassofparticlephysicshasnotbeenformulated.Thisisalsoafundamental problemoftheSM.
Ontheotherhand,sincethelargedistancephysicsis,toalargeextent,short-distance insensitive,therealnatureofthefundamentaltheorymayremain,intheforeseeable future,elusive,inthesamewayasapreciseknowledgeofthe criticalexponentsofthe liquid–vapourphasetransitiongiveslimitedinformation aboutrealinteractionsinwater. Thiswork,whichdoesnotclaimtoshedanylightonthesedifficultproblems,simply triestodescribeparticlephysicsandcriticalphenomenainstatisticalmechanicsina unifiedframework.
Chapters1–7dealwithfunctionalintegrals,perturbation theory,functionalmethods anddiscussgeneralpropertiesofscalarbosonQFTs.Chapters8–13provideanintroductiontorenormalizationtheorywiththesimpleexampleofthe φ4 QFTinfourdimensions, andRGequationsarederived.Compositeoperatorsandtheshort-distanceexpansion arediscussed.Relativisticfermionsaredescribed.Renormalizationpropertiesoftheories withsymmetriesarestudied,andspecificapplicationstoparticlephysicsareemphasized.
Chapters14–19aredevotedtocriticalphenomenainmacroscopicphasetransitions: generalproperties,mean-fieldapproximation,andmainlyapplicationsofQFTmethods andRG,withthecalculationofuniversalquantities,inparticular,withWilson–Fisher ε expansion,large N techniquesandnon-linear σ-modelfor O(N )-symmetricmodels.
WithChapters20–26,wereturntoparticlephysics.Chapter 20dealswithspontaneous fermionmassgeneration.Wethendiscussgaugetheories,Abelianandnon-Abelian.In particular,Chapter22brieflydescribestheSMofparticlephysics.Chapter26introduces Becchi–Rouet–Stora–Tyutin(BRST)symmetry,theZinn-Justin(ZJ)equationandthe proofoftherenormalizablityofgaugetheories.Chapter27 containsashortintroduction tosupersymmetry.Chapter28gathersthefewelementsofclassicalandquantumgravity neededinthework.
InChapters29–31,wefocusontwo-dimensionalfieldtheories,ofrelevancebothfor particleandstatisticalphysics.Chapter29isdevotedtoQFTsdefinedonhomogeneous spaces,andChapters30and31describeexactly-solvabletwo-dimensionalQFTs.
Chapter32providesanintroductiontofinite-sizeeffects,andChapter33tofinite temperaturerelativisticQFT.Chapters34–36dealwithstochasticevolutionequations, andtheirapplicationtocriticaldynamicsinphasetransitions.Chapters37–42describe theroleofinstantonsinQMandQFT,theapplicationofinstantoncalculustothe analysisoflarge-orderbehaviourofperturbationtheory, andtheproblemofsummation oftheperturbativeexpansion.Inparticular,Chapter41appliesthisinformationtothe evaluationofcriticalexponentsandseveralotheruniversalquantities.
Iamfullyawarethatthisworkislargelyincomplete.Myignoranceorlackofunderstandingofmanyimportanttopicsisofcoursemostlyresponsibleforthisweakness.A lackofspacehasalsoforcedmetoremoveanintroductiontolargerandommatrices,and preventedmefromaddingsomeothertopics.Anyway,Ibelievethatacompletesurvey ofQFTanditsapplicationsisbeyondthescopeofasinglephysicist.
Thisworkincorporatesnotesforlecturesdeliveredinnumeroussummerschools,most notably,Carg`ese1973,Bonn1974,Karpacz1975,BaskoPolje1976,andLesHouches 1982,aswellasforgraduatelecturesinuniversitieslikePrinceton,Louvain-la-Neuve, Berlin,Lausanne,Cambridge(Harvard),EcoleNormaleSup´erieure,Paris7,andsoon.
Conversely,sincesomerelevantmaterialthatIhavegatheredovertheyearscanno longerfindaplaceinthiswork,someelementshavebeenpublishedinfourreviews (intheformofPhysicsReports)andinthreecompanionvolumes,Refs.[6,64]andJ. Zinn-Justin, Fromrandomwalkstorandommatrices (OxfordUniv.Press2019).
Finally,commentsorcorrectionsaremostwelcome,andcanbesenttotheemail address:jean.zinn-justin@cea.fr.
Acknowledgements
ItisimpossibletolistallthephysicistsfromwhomIhavebenefitedinmylongcareerand whoseinfluencecan,therefore,befeltinoneformoranother inthiswork.Mymasters, M.FroissartandD.Bessis,guidedmyfirststepsinphysics.E.Br´ezinandJ.C.LeGuillou havecollaboratedwithmeformorethanfifteenyears,andwithoutthem,obviously,this workwouldneverhavebeenproduced.IalsothinkwithadeepemotionofB.W.Lee: theyearIspentworkingwithhimatStony-Brookwasoneofthe mostexcitingofmy lifeasaphysicist.
Wilson’sRGideasareamajorsourceofinspirationforthiswork.S.Coleman,A.A. Slavnov,R.Stora,K.Symanzik,A.N.Vasilev,amongothers, playedanimportantrolein myunderstandingofseveralaspectsofphysicsthroughtheirarticlesandlecturenotes,as wellasthroughprivatediscussions.T.D.LeeandC.N.Yanghaveconsistentlyhonoured mewiththeirfriendshipandhospitalityintheirinstitutions.Theirdeepremarkshave beenprecioustome.
Ihavelearnedmuchfromlecturesinseveralsummerschools, mostnotably,C.G. Callan,L.D.Faddeev,andD.GrossinLesHouchesschool1975
Severalcolleaguesagreedtoreadpartofthevariouseditionsbeforepublication,andI havebenefitedfromtheircriticisms,remarksandwisdom,E. Br´ezin,R.Stora,C.Bervillier,R.Guida,M.Moshe,O.Napoly,A.N.Vasilev,P.Zinn-Justin,andJ.-B.Zuber.
Moregenerally,mycolleaguesoftheSaclaytheorygroup,withwhomIhavehadso manydiscussions,inparticularC.deDominicis,E.Iancu,andC.Itzykson,havedirectly influencedthiswork.Finally,IalsowishtothankallothercolleagueswithwhomIhave collaboratedovertheyearsand,morespecifically,P.Ginsparg,R.Guida,S.Hikami, U.D.Jentschura,M.Moshe,andG.Parisi.
ThemanylecturesIhaveattendedinLesHouchesduringninesummershaveprovided mewithadditionalinspiration,andastayattheMassachusettsInstituteofTechnology (MIT),wherelecturenotesconcerningfinitetemperaturefieldtheorywereprepared,is gratefullyacknowledged.
S.ZaffanellaandM.Porneufweremosthelpfulinthepreparationofthefirstedition. Alldeservemydeepestgratitude.
Fullyrevisedforthe5th edition,Paris-Saclay,6February2021
Somegeneralreferencesforthewholework
Inadditiontotheworksexplicitlyquotedinthetext,anumberoftextbooksorreviews havebeenadirectsourceofinspiration:
S.Weinberg, TheQuantumTheoryofFields,2volumes,(CambridgeUniv.Press1995, 1996);
A.N.Vasiliev, FunctionalMethodsinQuantumFieldTheoryandStatistical Physics, (StPetersburg1976),Englishtranslation(GordonandBreach,Amsterdam1998);
L.D.FaddeevandA.A.Slavnov, GaugeFields:IntroductiontoQuantumFieldTheory (Benjamin,Reading,MA1980);
T.D.Lee, ParticlePhysicsandIntroductiontoFieldTheory (HarwoodAcademic,New York1981);
A.M.Polyakov, GaugeFieldsandStrings (HarwoodAcademic,NewYork1988);
J.DrouffeandC.Itzykson, Th´eorieStatistiquedesChamps (InterEditions1989),Englishversion: StatisticalFieldTheory,(CambridgeUniv.Press1989); Constructionoffieldtheoriesfromamorerigorouspointofviewisdiscussedin
R.F.StreaterandA.S.Wightman, PCT,Spin&StatisticsandAllThat (Benjamin, NewYork1964);
G.GlimmandA.Jaffe, QuantumPhysics:AFunctionalIntegralPointofView (Springer-Verlag,Berlin1981).
CriticalPhenomena,RandomSystems,GaugeTheories,ProceedingsofLesHouches SummerSchool1984,K.OsterwalderandR.Storaeds.(Elsevier,Amsterdam1986).
1Gaussianintegrals.Algebraicpreliminaries ...............1
1.1Gaussianintegrals:Wick’stheorem................. .1
1.2Perturbativeexpansion.Connectedcontributions.... ........3
1.3Thesteepestdescentmethod.....................4
1.4ComplexstructuresandGaussianintegrals........... ...5
1.5Grassmannalgebras.Differentialforms..............
1.6DifferentiationandintegrationinGrassmannalgebras.
1.7GaussianintegralswithGrassmannvariables.........
1.8Legendretransformation......................16
2Euclideanpathintegralsandquantummechanics(QM) .........18
2.1Markovianevolutionandlocality...................
2.2Statisticaloperator:Pathintegralrepresentation.. ..........20
2.3Explicitevaluationofapathintegral:Theharmonicoscillator......24
2.4Partitionfunction:Classicalandquantumstatistical physics.......25
2.5Correlationfunctions.Generatingfunctional.......
2.6Harmonicoscillator.CorrelationfunctionsandWick’s theorem......30 2.7Perturbedharmonicoscillator....................33
2.8Semi-classicalexpansion.......................35
A2Additionalremarks
A2.1Ausefulrelationbetweendeterminantandtrace......
A2.2Thetwo-pointfunction:Anintegralrepresentation..
A2.3Time-orderedproductsofoperators................
3Quantummechanics(QM):Pathintegralsinphasespace
3.1GeneralHamiltonians:Phase-spacepathintegral.....
3.2Theharmonicoscillator.Perturbativeexpansion.....
3.3Hamiltoniansquadraticinmomentumvariables........
3.4Thespectrumofthe O(2)-symmetricrigidrotator...........51
3.5Thespectrumofthe O(N )-symmetricrigidrotator...........52
A3Quantization.Topologicalactions:Quantumspins,magneticmonopoles .56
A3.1Symplecticformandquantization:Generalremarks... .......56
A3.2Classicalequationsofmotionandquantization...... ......58
A3.3Topologicalactions........................60
4Quantumstatisticalphysics:Functionalintegrationformalism .......64
4.1One-dimensionalQM:Holomorphicrepresentation.....
4.2Holomorphicpathintegral......................67
4.3Severaldegreesoffreedom.Bosoninterpretation..... .......71
4.4TheBosegas.Fieldintegralrepresentation.......... ....72
4.5FermionrepresentationandcomplexGrassmannalgebras ........80
4.6Pathintegralswithfermions.....................83
4.7TheFermigas.Fieldintegralrepresentation......... .....87
5Quantumevolution:Fromparticlestonon-relativisticfields ........90
5.1Timeevolutionandscatteringmatrixinquantummechanics(QM)....90
5.2Pathintegraland S-matrix:Perturbationtheory............92
5.3Pathintegraland S-matrix:Semi-classicalexpansions..........95
5.4 S-matrixandholomorphicformalism.................99
5.5TheBosegas:Evolutionoperator.................102
5.6Fermigas:Evolutionoperator...................103
A5Perturbationtheoryintheoperatorformalism ............104
6Theneutralrelativisticscalarfield ..................105
6.1Therelativisticscalarfield.....................105
6.2Quantumevolutionandthe S-matrix................110
6.3 S-matrixandfieldasymptoticconditions..............112
6.4Thenon-relativisticlimit:The φ4 QFT...............116
6.5Quantumstatisticalphysics....................118
6.6K¨allen–Lehmannrepresentationandfieldrenormalization.......122
7Perturbativequantumfieldtheory(QFT):Algebraicmethods ......125
7.1Generatingfunctionalsofcorrelationfunctions..... .......126
7.2Perturbativeexpansion.Wick’stheoremandFeynmandiagrams....127
7.3Connectedcorrelationfunctions:Generatingfunctional........129
7.4Theexampleofthe φ4 QFT....................131
7.5Algebraicpropertiesoffieldintegrals.Quantumfieldequations.....133
7.6Connectedcorrelationfunctions.Clusterproperties. .........139
7.7Legendretransformation.Vertexfunctions.......... ....141
7.8Momentumrepresentation.....................144
7.9Looporsemi-classicalexpansion..................146
7.10Vertexfunctions:One-lineirreducibility......... .....151
7.11Statisticalandquantuminterpretationofthevertexfunctional....152 A7Additionalresultsandmethods ...................155
A7.1Generatingfunctionalattwoloops................155
A7.2Thebackgroundfieldmethod...................156
A7.3ConnectedFeynmandiagrams:Clusterproperties..... .....157
8Ultravioletdivergences:Effectivefieldtheory(EFT) ..........160
8.1Gaussianexpectationvaluesanddivergences:Thescalarfield.....161
8.2DivergencesofFeynmandiagrams:Powercounting...... ....162
8.3Classificationofinteractionsinscalarquamtumfieldtheories.....164
8.4Momentumregularization.....................166
8.5Example:The φ3 d=6 fieldtheoryatone-looporder..........169
8.6Operatorinsertions:Generatingfunctionals,powercounting......173
8.7Latticeregularization.Classicalstatisticalphysics..........175
8.8EffectiveQFT.Thefine-tuningproblem...............176
8.9Theemergenceofrenormalizablefieldtheories........ ....179 A8Technicaldetails ..........................181
A8.1Schwinger’sproper-timerepresentation........... ...181
A8.2Regularizationandone-loopdivergences........... ...181
A8.3Moregeneralmomentumregularizations............. .184
9Introductiontorenormalizationtheoryandrenormalizationgroup (RG) .185
9.1Powercounting.Dimensionalanalysis............... .186
9.2Regularization.BareandrenormalizedQFT........... ..187
9.3One-loopdivergences.......................191
9.4Divergencesbeyondone-loop:Skeletondiagrams...... .....194
9.5Callan–Symanzikequations....................196
9.6Inductiveproofofrenormalizability............... ..198
9.7The φ2φ2 vertexfunction....................203
9.8Therenormalizedaction:Generalconstruction....... .....204
9.9Themasslesstheory.......................204
9.10HomogeneousRGequations:MassiveQFT.............208
9.11EFTandRG..........................210
9.12SolutionofbareRGequations:Thetrivialityissue... ......212
A9FunctionalRGequations.Super-renormalizableQFTs.Normalorder ..214
A9.1Large-momentummodeintegrationandfunctionalRGequations...214
A9.2The φ4 QFTinthreedimensions:Divergences...........216
A9.3Super-renormalizablescalarQFTsintwodimensions:Normalorder..218 10Dimensionalcontinuation,regularization,minimalsubtraction(MS).
Renormalizationgroup(RG)functions ................220
10.1Dimensionalcontinuationanddimensionalregularization.......220 10.2RGfunctions..........................224
10.3Thestructureofrenormalizationconstants......... ....226
10.4MSscheme...........................227
10.5RGfunctionsattwo-looporder:The φ4 QFT............230
10.6Generalizationto N -componentfields...............235 A10Feynmanparametrization .....................239
11Renormalizationoflocalpolynomials.Short-distanceexpansion(SDE) .240
11.1Renormalizationofoperatorinsertions............ ...240
11.2Quantumfieldequations.....................245
11.3Short-distanceexpansionofoperatorproducts...... .....248
11.4Large-momentumexpansionoftheSDEcoefficients:CSequations...253
11.5SDEbeyondleadingorder.Generaloperatorproduct... .....255
11.6Light-coneexpansionofoperatorproducts.......... ...256
12Relativisticfermions:Introduction ..................258
12.1MassiveDiracfermions......................258
12.2Self-interactingmassivefermions:Non-relativisticlimit........263
12.3FreeEuclideanrelativisticfermions.............. ..265
12.4Partitionfunction.Correlations................. .269
12.5Generatingfunctionals......................270
12.6Connectionbetweenspinandstatistics............. ..272
12.7Divergencesandmomentumcut-off................274
12.8Dimensionalregularization....................276
12.9Latticefermionsandthedoublingproblem........... ..276 A12Euclideanfermions,spingroupand γ matrices ...........280
A12.1Spingroup.Dirac γ matrices..................280 A12.2Theexampleofdimension4...................288
A12.3TheFierztransformation....................289
A12.4Tracesofproductsof γ matrices.................290
13Symmetries,chiralsymmetrybreaking,andrenormalization ......292
13.1Liegroupsandalgebras:Preliminaries............. ..293
13.2LinearglobalsymmetriesandWTidentities.......... ..295
13.3Linearsymmetrybreaking....................298
13.4Spontaneoussymmetrybreaking.................301
13.5Chiralsymmetrybreakinginstronginteractions:Effectivetheory...304
13.6Thelinear σ-model.......................306
13.7WTidentities..........................310
13.8Quadraticsymmetrybreaking...................313
A13CurrentsandNoether’stheorem ..................317
A13.1Currentsinclassical-fieldtheory................ .317
A13.2Theenergy–momentumtensor..................318
A13.3Euclideantheory:Dilatationandconformalinvariance.......320 A13.4QFT:Currentsandcorrelationfunctions........... ..322
A13.5Energy-momentumtensorandQFT...............323
14Criticalphenomena:Generalconsiderations.Mean-fieldtheory(MFT) .324 14.1Thetransfermatrix.......................325
14.2Theinfinitetransversesizelimit:Ising-likesystems .........328
14.3Continuoussymmetries......................331 14.4Mean-fieldapproximation....................332 14.5Universalitywithinmean-fieldapproximation....... .....337 14.6Beyondthemean-fieldapproximation...............342 14.7Powercountingandtheroleofdimension4............ .345 14.8Tricriticalpoints........................346 A14Additionalconsiderations .....................347
A14.1High-temperatureexpansion...................347
A14.2Mean-fieldapproximation:Generalformalism....... ....348
A14.3Mean-fieldexpansion......................351
A14.4High-,low-temperature,andmean-fieldexpansions.. .......352
A14.5Quenchedaverages.......................354
15Therenormalizationgroup(RG)approach:Thecriticaltheorynear fourdimensions ...........................357 15.1RG:Thegeneralidea......................358 15.2TheGaussianfixedpoint.....................363 15.3Criticalbehaviour:Theeffective φ4 fieldtheory...........366 15.4RGequationsnearfourdimensions................368 15.5SolutionoftheRGequations:The ε-expansion...........370 15.6Criticalcorrelationfunctionswith φ2(x)insertions..........372 15.7The O(N )-symmetric(φ2)2 fieldtheory..............376 15.8Statisticalpropertiesoflongself-repellingchains ..........377 15.9Liquid–vapourphasetransitionand φ4 fieldtheory.........382 15.10Superfluidtransition......................387
16Criticaldomain:Universality, ε-expansion ..............391
16.1Strongscalingabove Tc:Therenormalizedtheory..........392 16.2Criticaldomain:HomogeneousRGequations.......... ..396
16.3Scalingpropertiesabove Tc ....................396
16.4Correlationfunctionswith φ2 insertions..............399 16.5Scalingpropertiesinamagneticfieldandbelow Tc .........400 16.6The N -vectormodel.......................403
16.7Thegeneral N -vectormodel...................405
16.8Asymptoticexpansionofthetwo-pointfunction...... .....410 16.9Someuniversalquantitiesas ε expansions.............412 16.10Conformalbootstrap......................420
17Criticalphenomena:Correctionstoscalingbehaviour .........421
17.1Correctionstoscaling:Genericdimensions......... ....421 17.2Logarithmiccorrectionsattheupper-criticaldimension.......423
17.3Irrelevantoperatorsandthequestionofuniversality .........426 17.4Correctionscomingfromirrelevantoperators.Improvedaction....428
17.5Application:Uniaxialsystemswithstrongdipolarforces.......431
18 O(N) -symmetricvectormodelsfor N large .............436
18.1Thelarge N action.......................436
18.2Large N limit:Saddlepointequations,criticaldomain........438
18.3Renormalizationgroup(RG)functionsandleadingcorrectionstoscaling445 18.4Small-couplingconstant,large-momentumexpansions for d< 4....447
18.5Dimension4:Trivialityissuefor N large..............448
18.6The(φ2)2 fieldtheoryandthenon-linear σ-modelfor N large....449 18.7The1/N -expansion:Analternativefieldtheory...........453
18.8Explicitcalculations.......................455
19Thenon-linear σ-modelneartwodimensions:Phasestructure .....458
19.1Thenon-linear σ-model:Definition................459
19.2Perturbationtheory.Powercounting............... .461
19.3IRdivergences.........................463
19.4UVregularization........................464
19.5WTidentitiesandmasterequation................466 19.6Renormalization.........................469
19.7Therenormalizedaction:Solutiontothemasterequation......471 19.8Renormalizationoflocalfunctionals.............. ..474
19.9Alinearrepresentation......................475
19.10(φ2)2 fieldtheoryintheorderedphaseandnon-linear σ-model....476 19.11Renormalization,RGequations.................479
19.12RGequations:Solutions(magneticterminology).... ......480 19.13Resultsbeyondone-looporder..................486
19.14Thedimension2:Asymptoticfreedom..............488 20Gross–Neveu–YukawaandGross–Neveumodels ............489
20.1TheGNYmodel:Spontaneousmassgeneration.......... .489 20.2RGequationsnearfourdimensions................494
20.3TheGNYmodelinthelarge N limit...............498
20.4Thelarge N expansion......................501
20.5TheGNmodel.........................504
21Abeliangaugetheories:Theframeworkofquantumelectrodynamics(QED) 507
21.1Thefreemassivevectorfield:Quantization.......... ...507 21.2TheEuclideanfreeaction.Thetwo-pointfunction.... ......509 21.3Couplingtomatter.......................512 21.4Themasslesslimit:Gaugeinvariance............... 514 21.5Masslessvectorfield,gaugeinvariance,andquantization.......516 21.6Equivalencewithcovariantquantization........... ...519 21.7Gaugesymmetryandparalleltransport.............. 521 21.8Perturbationtheory:Regularization.............. ..522 21.9WTidentitiesandrenormalization................. 526 21.10Gaugedependence:Thefermiontwo-pointfunction... ......528 21.11RenormalizationandRGequations................531 21.12One-loop β functionandthetrivialityissue............532 21.13TheAbelianLandau–Ginzburg–Higgsmodel.......... ..535 21.14TheLandau–Ginzburg–Higgsmodel:WTidentities.... .....537 21.15Spontaneoussymmetrybreaking:Decouplinggauge... ......538 21.16Physicalobservables.Unitarityofthe S-matrix..........539 21.17Stochasticquantization:Theexampleofgaugetheories.......540 A21Additionalremarks ........................542 A21.1VacuumenergyandCasimireffect................542 A21.2Gaugedependence.......................545 A21.3DivergencesatoneloopfromSchwinger’srepresentation......546
22Non-Abeliangaugetheories:Introduction ..............548
22.1Geometricconstruction:Paralleltransport........ .....548 22.2Gauge-invariantactions.....................551
22.3Hamiltonianformalism.Quantizationinthetemporalgauge.....551 22.4Covariantgauges........................554
22.5Perturbationtheory,regularization.............. ..557 22.6Thenon-AbelianHiggsmechanism................559 A22MassiveYang–Millsfields .....................565
23TheStandardModel(SM)offundamentalinteractions ........567
23.1Weakandelectromagneticinteractions:Gaugeandscalarfields....568
23.2Leptons:MinimalSMextensionwithDiracneutrinos... .....570
23.3Quarksandweak–electromagneticinteractions...... .....573 23.4QCD.RGequationsand β function................576
23.5GeneralRG β-functionatone-looporder:Asymptoticfreedom....578
23.6Axialcurrent,chiralgaugetheories,andanomalies.. .......582
23.7Anomalies:Applicationsinparticlephysics........ .....591
24Large-momentumbehaviourinquantumfieldtheory(QFT) ......593
24.1The(φ2)2 Euclideanfieldtheory:Large-momentumbehaviour....593
24.2General φ4-likefieldtheories:d=4.................598
24.3TheorieswithscalarbosonsandDiracfermions....... ....600
24.4Gaugetheories.........................602
24.5Applications:Thetheoryofstronginteractions..... ......604
25Latticegaugetheories:Introduction .................607
25.1Gaugeinvarianceonthelattice:Paralleltransport.. ........607
25.2Thematterlessgaugetheory...................609
25.3Wilson’sloopandconfinement..................611
25.4Mean-fieldapproximation....................617
A25Gaugetheoryandconfinementintwodimensions ..........621
26Becchi–Rouet–Stora–Tyutin(BRST)symmetry.Gaugetheories: Zinn-Justinequation(ZJ)andrenormalization .............623
26.1STidentities:Theorigin.....................624
26.2FromSTsymmetrytoBRSTsymmetry..............626
26.3BRSTsymmetry:Moregeneralcoordinates.Groupstructure.....628
26.4Stochasticequations.......................630
26.5BRSTsymmetry,Grassmanncoordinates,andgradientequations...632
26.6Gaugetheories:Notationandalgebraicstructure.... ......635
26.7Gaugetheories:Quantization...................636
26.8WTidentitiesandZJequation..................639
26.9Renormalization:Generalconsiderations.......... ....641
26.10Therenormalizedgaugeaction..................642
26.11Gaugeindependence:Physicalobservables......... ....647
A26BRSTsymmetryandZJequation:Additionalremarks ........649 A26.1BRSTsymmetryandZJequation................649 A26.2CanonicalinvarianceoftheZJequation............ ..650
A26.3ElementsofBRSTcohomology.................651 A26.4FromBRSTsymmetrytosupersymmetry.............654
27Supersymmetricquantumfieldtheory(QFT):Introduction ......656
27.1Scalarsuperfieldsinthreedimensions.............. .656
27.2The O(N )supersymmetricnon-linear σ model...........661
27.3Supersymmetryinfourdimensions................662
27.4Vectorsuperfieldsandgaugeinvariance............. .666
28Elementsofclassicalandquantumgravity ..............670
28.1Manifolds.Changeofcoordinates.Tensors.......... ...671
28.2Paralleltransport:Connection,covariantderivative.........673
28.3Riemannianmanifold.Themetrictensor............. .677
28.4Thecurvature(Riemann)tensor.................678
28.5Fermions,vielbein,spinconnection............... .682
28.6ClassicalGR.Equationsofmotion.................684
28.7Quantizationinthetemporalgauge:Puregravity..... .....687
28.8Observationalcosmology:Afewcomments............ .690
29Generalizednon-linear σ-modelsintwodimensions ..........692
29.1HomogeneousspacesandGoldstonemodes............. 692
29.2WTidentitiesandrenormalizationinlinearcoordinates.......695
29.3Renormalizationingeneralcoordinates:BRSTsymmetry......699
29.4Symmetricspaces:Definition...................703
29.5Classicalfieldequations.Conservationlaws........ .....704
29.6QFT:PerturbativeexpansionandRG...............706
29.7Generalizations.........................711
A29Homogeneousspaces:Afewalgebraicproperties ..........713
A29.1Puregauge.Maurer–Cartanequations.............. 713
A29.2Metricandcurvatureinhomogeneousspaces......... ..714
A29.3Explicitexpressionsforthemetric............... .715
A29.4Symmetricspaces:Classification................. 717
30Afewsolvabletwo-dimensionalquantumfieldtheories(QFT) .....721
30.1Thefreemasslessscalarfield...................721
30.2ThefreemasslessDiracfermion..................725
30.3Thegauge-invariantfermiondeterminantandtheanomaly......728
30.4ThesGmodel..........................731
30.5TheSchwingermodel......................732
30.6ThemassiveThirringmodel...................736
30.7AgeneralizedThirringmodelwithtwofermions....... ....739
30.8The SU (N )Thirringmodel....................742
A30Two-dimensionalmodels:Afewadditionalresults ..........745
A30.1Four-fermioncurrentinteractions:RG β-function.........745
A30.2TheSchwingermodel:Theanomaly...............745 A30.3SolitonsinthesGmodel....................746
31O(2)spinmodelandtheKosterlitz–Thouless’s(KT)phasetransition ..747
31.1Thespincorrelationfunctionsatlowtemperature.... ......748
31.2Correlationfunctionsinafield..................749
31.3TheCoulombgasintwodimensions................750
31.4 O(2)spinmodelandCoulombgas.................755
31.5Thecriticaltwo-pointfunctionintheO(2)model..... .....756
31.6ThegeneralizedThirringmodel..................758
32Finite-sizeeffectsinfieldtheory.Scalingbehaviour ..........760
32.1RGinfinitegeometries......................760
32.2Momentumquantization.....................764
32.3The φ4 fieldtheoryinaperiodichypercube.............766
32.4The φ4 fieldtheory:Cylindricalgeometry.............772
32.5Finitesizeeffectsinthenon-linear σ-model.............776 A32Additionalremarks ........................782
A32.1Perturbationtheoryinafinitevolume.............. 782
A32.2Discretesymmetriesandfinite-sizeeffects......... ....783
33Quantumfieldtheory(QFT)atfinitetemperature:Equilibriumproperties 786
33.1Finite-(andhigh-)temperaturefieldtheory......... ....786
33.2Theexampleofthe φ4 1,d 1 fieldtheory...............790
33.3Hightemperatureandcriticallimits............... .796
33.4Thenon-linear σ-modelinthelarge N limit............799
33.5Theperturbativenon-linear σ-modelatfinitetemperature......804
33.6TheGNmodelinthelarge N expansion..............810
33.7Abeliangaugetheories:TheQEDframework........... .817
33.8Non-Abeliangaugetheories....................824
A33Feynmandiagramsatfinitetemperature ..............828
A33.1One-loopcalculations......................828
A33.2Groupmeasure........................830
34Stochasticdifferentialequations:Langevin,Fokker–Planck(FP)equations 831
34.1TheLangevinequation......................831
34.2Time-dependentprobabilitydistributionandFPequation......833
34.3Equilibriumdistribution.Correlationfunctions... ........835
34.4Aspecialclass:DissipativeLangevinequations..... ......838
34.5ThelinearLangevinequation...................839
34.6Pathintegralrepresentation...................842
34.7BRSTandsupersymmetry....................843
34.8Gradienttime-dependentforceandJarzynski’srelation........846
34.9MoregeneralLangevinequations.MotioninRiemannian manifolds..848 A34Markov’sstochasticprocesses:Afewremarks ............852
A34.1Discretespaces:Markov’sprocesses,phasetransitions.......852
A34.2Stochasticprocesswithprescribedequilibriumdistribution.....855 A34.3Stochasticprocessesandphasetransitions........ .....856
35Langevinfieldequations:Propertiesandrenormalization .......857
35.1LangevinandFokker–Planck(FP)equations.......... ..857
35.2Time-dependentcorrelationfunctionsandequilibrium........858
35.3RenormalizationandBRSTsymmetry...............861
35.4DissipativeLangevinequationandsupersymmetry.... ......864
35.5Supersymmetryandequilibriumcorrelationfunctions ........867
35.6Stochasticquantizationoftwo-dimensionalchiralmodels.......868
35.7LangevinequationandRiemannianmanifolds......... ...871
A35TherandomfieldIsingmodel:Supersymmetry ...........874
36Criticaldynamicsandrenormalizationgroup(RG) ...........875
36.1Dissipativeequation:RGequationsindimension d =4 ε ......876
36.2Dissipativedynamics:RGequationsindimension d =2+ ε ......880
36.3Conservedorderparameter....................882
36.4Relaxationalmodelwithenergyconservation........ ....883
36.5Anon-relaxationalmodel.....................886
36.6Finitesizeeffectsanddynamics..................888 A36RGfunctionsattwoloops .....................894
A36.1Supersymmetricperturbativecalculationsattwoloops.......894
37Instantonsinquantummechanics(QM) ...............899
37.1Thequarticanharmonicoscillatorfornegativecoupling.......899
37.2Atoymodel:Asimpleintegral..................901
37.3QM:Instantons.........................902
37.4Instantoncontributionsatleadingorder........... ...904
37.5Generalanalyticpotentials:Instantoncontributions.........908
37.6Evaluationofthedeterminant:Theshiftingmethod... ......909
37.7Zerotemperaturelimit:Thegroundstate............ .915 A37ExactJacobian.WKBmethod. .................916 A37.1TheexactJacobian......................916 A37.2TheWKBmethod.......................917
38Metastablevacuainquantumfieldtheory(QFT) ...........919
38.1The φ4 QFTfornegativecoupling.................920
38.2Generalpotentials:Instantoncontributions....... ......924
38.3The φ4 QFTindimension4...................926
38.4Instantoncontributionsatleadingorder........... ...927
38.5Couplingconstantrenormalization................ .931
38.6Theimaginarypartofthe n-pointfunction.............932
38.7Themassivetheory.......................933
38.8Cosmology:Thedecayofthefalsevacuum............. 934 A38Instantons:Additionalremarks ..................936
A38.1Virialtheorem.........................936
A38.2Sobolevinequalities......................937
A38.3InstantonsandRGequations..................939 A38.4Conformalinvariance......................940
39Degenerateclassicalminimaandinstantons .............942
39.1Thequarticdouble-wellpotential................. 942
39.2Theperiodiccosinepotential...................945
39.3Instantonsandstochasticdynamics................ 948
39.4Instantonsinstablebosonfieldtheories:Generalremarks......951
39.5Instantonsin CP (N 1)models.................953
39.6Instantonsinthe SU (2)gaugetheory...............956 A39Traceformulaforperiodicpotentials ................959
40Largeorderbehaviourofperturbationtheory
40.2Scalarfieldtheories:Theexampleofthe φ4 fieldtheory.......963
40.3The(φ2)2 fieldtheoryindimension4and4 ε ...........964
40.4Fieldtheorieswithfermions...................968
A40large-orderbehaviour:Additionalremarks ..............974
41Criticalexponentsandequationofstatefromseriessummation ....975
41.1Divergentseries:Borelsummability,Borelsummation ........975 41.2Boreltransformation:Seriessummation............ ..978
41.3Summingtheperturbativeexpansionofthe(φ2)2 fieldtheory.....980
41.4Summationmethod:Practicalimplementation........ ...983
41.5Fieldtheoryestimatesofcriticalexponentsforthe O(N )model....985
41.6Otherthree-dimensionaltheoreticalestimates..... .......986
41.7Criticalexponentsfromexperiments............... .987
41.8Amplituderatios........................989
A41Someothersummationmethods ..................990
A41.1Order-dependentmappingmethod(ODM)............990
A41.2Lineardifferentialapproximants................. 991
42Multi-instantonsinquantummechanics(QM) ............992
42.1Thequarticdouble-wellpotential................. 993
42.2Theperiodiccosinepotential...................1000
42.3Generalpotentialswithdegenerateminima.......... ...1004
42.4The O(ν)-symmetricanharmonicoscillator.............1007
42.5GeneralizedBohr–Sommerfeldquantizationformula.. .......1009
A42Additionalremarks ........................1011
A42.1Multi-instantons:Thedeterminant............... .1011
A42.2Theinstantoninteraction....................1012
A42.3Asimpleexampleofnon-Borelsummability.......... ..1014
A42.4Multi-instantonsandWKBapproximation........... .1016 Bibliography .............................1019 Index .................................1041
