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WHATISAQUANTUMFIELDTHEORY? Quantum field theory (QFT) is one of the great achievements of physics, of profound interest to mathematicians. Most pedagogical texts on QFT are geared toward budding professional physicists, however, whereas mathematical accounts areabstractanddifficulttorelatetothephysics.Thisbookbridgesthegap.While the treatment is rigorous whenever possible, the accent is not on formality but on explaining what the physicists do and why, using precise mathematical language. In particular, it covers in detail the mysterious procedure of renormalization. Written for readers with a mathematical background but no previous knowledge of physics and largely self-contained, it presents both basic physical ideas from specialrelativityand quantummechanicsandadvancedmathematicalconceptsin completedetail.ItwillbeofinteresttomathematicianswantingtolearnaboutQFT and, with nearly 300 exercises, also to physics students seeking greater rigor than theytypicallyfindintheircourses. michel talagrand istherecipientoftheLoèvePrize(1995),theFermatPrize (1997), and the Shaw Prize (2019). He was a plenary speaker at the International Congress of Mathematicians and is currently a member of the Académie des sciences (Paris). He has written several books in probability theory and well over 200researchpapers. “Thisbookaccomplishestheimpossibletask:Itexplainstoamathematician,inalanguage thatamathematiciancanunderstand,whatismeantbyaquantumfieldtheoryfromaphysi- cist’spointofview.Theauthoriscompletelyandbrutallyhonestinhisgoaltotrulyexplain thephysicsratherthanfilteringoutonlythemathematics,butisatthesametimeasmath- ematicallylucidasonecanbewiththistopic.Itisagreatbookbyagreatmathematician.” -SouravChatterjee,StanfordUniversity “Talagrand has done an admirable job of making the difficult subject of quantum field theoryasconcreteandunderstandableaspossible.Thebookprogressesslowlyandcarefully but still covers an enormous amount of material, culminating in a detailed treatment of renormalization.Althoughnoonecanmakethesubjecttrulyeasy,Talagrandhasmadeevery efforttoassistthereaderonarewardingjourneythoughtheworldofquantumfields.” -BrianHall,UniversityofNotreDame “A presentation of the fundamental ideas of quantum field theory in a manner that is both accessible and mathematically accurate seems like an impossible dream. Well, not anymore!Thisbookgoesfrombasicnotionstoadvancedtopicswithpatienceandcare.It is an absolute delight to anyone looking for a friendly introduction to the beauty of QFT anditsmysteries.” -ShaharMendelson,AustralianNationalUniversity “I have been motivated to try and learn about quantum field theories for some time but struggledtofindapresentationinalanguagethatIasamathematiciancouldunderstand. Thisbookwasperfectforme:Iwasabletomakeprogresswithoutanyinitialpreparation andfeltverycomfortableandreassuredbythestyleofexposition.” -EllenPowell,DurhamUniversity “Inadditiontoitssuccessasaphysicaltheory,quantumfieldtheoryhasbeenacontinuous sourceofinspirationformathematics.However,mathematicianstryingtounderstandquan- tumfieldtheorymustcontendwiththefactthatsomeofthemostimportantcomputationsin thetheoryhavenorigorousjustification.Thishasbeenaconsiderableobstacletocommuni- cationbetweenmathematiciansandphysicists.Itiswhy,despitemanyfruitfulinteractions, onlyveryfewpeoplewouldclaimtobewellversedinbothdisciplinesatthehighestlevel. Therehavebeenmanyattemptstobridgethisgap,eachemphasizingdifferentaspectsof quantumfieldtheory.Treatmentsaimedatamathematicalaudienceoftendeploysophisti- catedmathematics.MichelTalagrandtakesadecidedlyelementaryapproachtoanswering the question in the title of his book, assuming little more than basic analysis. In addition to learning what quantum field theory is, the reader will encounter in this book beautiful mathematicsthatishardtofindanywhereelseinsuchclearpedagogicalform,notablythe discussionofrepresentations ofthePoincaré groupandtheBPHZTheorem. Thebook is especiallytimelygiventherecentresurgenceofideasfromquantumfieldtheoryinproba- bilityandpartialdifferentialequations.Itissuretoremainareferenceformanydecades.” -PhilippeSosoe,CornellUniversity WHAT IS A QUANTUM FIELD THEORY? A First Introduction for Mathematicians MICHEL TALAGRAND UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781316510278 DOI:10.1017/9781108225144 ©MichelTalagrand2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 PrintedintheUnitedKingdombyTJBooksLtd,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Names:Talagrand,Michel,1952–author. Title:Whatisaquantumfieldtheory?:afirstintroductionfor mathematicians/MichelTalagrand. Description:Firstedition.|NewYork:CambridgeUniversityPress,2021.| Includesbibliographicalreferencesandindex. Identifiers:LCCN2021020786(print)|LCCN2021020787(ebook)| ISBN9781316510278(hardback)|ISBN9781108225144(epub) Subjects:LCSH:Quantumfieldtheory.|BISAC:SCIENCE/ Physics/Mathematical&Computational Classification:LCCQC174.45.T352021(print)|LCCQC174.45(ebook)| DDC530.14/3–dc23 LCrecordavailableathttps://lccn.loc.gov/2021020786 LCebookrecordavailableathttps://lccn.loc.gov/2021020787 ISBN978-1-316-51027-8Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Ifallmathematicsweretodisappear,physicswouldbesetbackexactlyoneweek. RichardFeynman Physicsshouldbemadeassimpleaspossible,butnotsimpler. AlbertEinstein The career of a young theoretical physicist consists of treating the harmonic oscillatorinever-increasinglevelsofabstraction. SydneyColeman Contents Introduction page1 PartI Basics 7 1 Preliminaries 9 1.1 Dimension 9 1.2 Notation 10 1.3 Distributions 12 1.4 TheDeltaFunction 14 1.5 TheFourierTransform 17 2 BasicsofNon-relativisticQuantumMechanics 21 2.1 BasicSetting 22 2.2 MeasuringTwoDifferentObservablesontheSameSystem 27 2.3 Uncertainty 28 2.4 FiniteversusContinuousModels 30 2.5 PositionStateSpaceforaParticle 31 2.6 UnitaryOperators 38 2.7 MomentumStateSpaceforaParticle 39 2.8 Dirac’sFormalism 40 2.9 WhyAreUnitaryTransformationsUbiquitous? 46 2.10 UnitaryRepresentationsofGroups 47 2.11 ProjectiveversusTrueUnitaryRepresentations 49 2.12 MathematiciansLookatProjectiveRepresentations 50 2.13 ProjectiveRepresentationsofR 51 2.14 One-parameterUnitaryGroupsandStone’sTheorem 52 2.15 Time-evolution 59 vii viii Contents 2.16 SchrödingerandHeisenbergPictures 62 2.17 AFirstContactwithCreationandAnnihilationOperators 64 2.18 TheHarmonicOscillator 66 3 Non-relativisticQuantumFields 73 3.1 TensorProducts 73 3.2 SymmetricTensors 76 3.3 CreationandAnnihilationOperators 78 3.4 BosonFockSpace 82 3.5 UnitaryEvolutionintheBosonFockSpace 84 3.6 BosonFockSpaceandCollectionsofHarmonicOscillators 86 3.7 ExplicitFormulas:PositionSpace 88 3.8 ExplicitFormulas:MomentumSpace 92 3.9 UniverseinaBox 93 3.10 QuantumFields:QuantizingSpacesofFunctions 94 4 TheLorentzGroupandthePoincaréGroup 102 4.1 NotationandBasics 102 4.2 Rotations 107 4.3 PureBoosts 108 4.4 TheMassShellandItsInvariantMeasure 111 4.5 MoreaboutUnitaryRepresentations 115 4.6 GroupActionsandRepresentations 118 4.7 QuantumMechanics,SpecialRelativityandthe PoincaréGroup 120 4.8 AFundamentalRepresentationofthePoincaréGroup 122 4.9 ParticlesandRepresentations 125 4.10 TheStates|p(cid:2)and|p(cid:2) 128 4.11 ThePhysicists’Way 129 5 TheMassiveScalarFreeField 132 5.1 IntrinsicDefinition 132 5.2 ExplicitFormulas 140 5.3 Time-evolution 142 5.4 LorentzInvariantFormulas 143 6 Quantization 145 6.1 TheKlein-GordonEquation 146 6.2 NaiveQuantizationoftheKlein-GordonField 147 6.3 RoadMap 150 6.4 LagrangianMechanics 151 6.5 FromLagrangianMechanicstoHamiltonianMechanics 156 Contents ix 6.6 CanonicalQuantizationandQuadraticPotentials 161 6.7 QuantizationthroughtheHamiltonian 163 6.8 UltravioletDivergences 164 6.9 QuantizationthroughEqual-timeCommutationRelations 165 6.10 Caveat 172 6.11 Hamiltonian 173 7 TheCasimirEffect 176 7.1 VacuumEnergy 176 7.2 Regularization 177 PartII Spin 181 8 RepresentationsoftheOrthogonalandtheLorentzGroup 183 8.1 TheGroupsSU(2)andSL(2,C) 183 8.2 AFundamentalFamilyofRepresentationsofSU(2) 187 8.3 TensorProductsofRepresentations 190 8.4 SL(2,C)asaUniversalCoveroftheLorentzGroup 192 8.5 AnIntrinsicallyProjectiveRepresentation 195 8.6 Deprojectivization 199 8.7 ABriefIntroductiontoSpin 199 8.8 SpinasanObservable 200 8.9 ParityandtheDoubleCoverSL+(2,C)ofO +(1,3) 201 8.10 TheParityOperatorandtheDiracMatrices 204 9 RepresentationsofthePoincaréGroup 208 9.1 ThePhysicists’Way 209 9.2 TheGroupP∗ 211 9.3 RoadMap 212 9.3.1 HowtoConstructRepresentations? 213 9.3.2 SurvivingtheFormulas 213 9.3.3 ClassifyingtheRepresentations 214 9.3.4 MassiveParticles 214 9.3.5 MasslessParticles 214 9.3.6 MasslessParticlesandParity 215 9.4 ElementaryConstructionofInducedRepresentations 215 9.5 VariegatedFormulas 217 9.6 FundamentalRepresentations 223 9.6.1 MassiveParticles 223 9.6.2 MasslessParticles 223 9.7 Particles,Spin,Representations 228