What Is a Quantum Field Theory?: A First Introduction for Mathematicians

758 Pages·2022·6.619 MB·English

by  Michel Talagrand

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
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Informationonthistitle:www.cambridge.org/9781316510278
DOI:10.1017/9781108225144
©MichelTalagrand2022
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andtotheprovisionsofrelevantcollectivelicensingagreements,
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Firstpublished2022
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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
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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
x Contents
9.8 AbstractPresentationofInducedRepresentations 232
9.9 ParticlesandParity 235
9.10 DiracEquation 236
9.11 HistoryoftheDiracEquation 238
9.12 ParityandMasslessParticles 240
9.13 Photons 245
10 BasicFreeFields 250
10.1 ChargedParticlesandAnti-particles 251
10.2 LorentzCovariantFamiliesofFields 253
10.3 RoadMapI 255
10.4 FormoftheAnnihilationPartoftheFields 256
10.5 ExplicitFormulas 260
10.6 CreationPartoftheFields 262
10.7 Microcausality 264
10.8 RoadMapII 267
10.9 TheSimplestCase(N = 1) 268
10.10 AVerySimpleCase(N = 4) 268
10.11 TheMassiveVectorField(N = 4) 269
10.12 TheClassicalMassiveVectorField 271
10.13 MassiveWeylSpinors,FirstAttempt(N = 2) 273
10.14 FermionFockSpace 275
10.15 MassiveWeylSpinors,SecondAttempt 279
10.16 EquationofMotionfortheMassiveWeylSpinor 281
10.17 MasslessWeylSpinors 283
10.18 Parity 284
10.19 DiracField 285
10.20 DiracFieldandClassicalMechanics 288
10.21 MajoranaField 293
10.22 LackofaSuitableFieldforPhotons 293
PartIII Interactions 297
11 PerturbationTheory 299
11.1 Time-independentPerturbationTheory 299
11.2 Time-dependent Perturbation Theory and the Interaction
Picture 303
11.3 TransitionRates 307
11.4 ASideStory:OscillatingInteractions 310
11.5 InteractionofaParticlewithaField:AToyModel 312