A Stroll through Quantum Fields (antecedent of Quantum Field Theory - From Basics to Modern Topics)

552 Pages·2019·8.107 MB·English

by  François Gelis

A Stroll Through
QUANTUM FIELDS
F ¸ G
RANCOIS ELIS
INSTITUT DE PHYSIQUE THE´ORIQUE
CEA-SACLAY

Contents
1 BasicsofQuantumFieldTheory 1
1.1 Specialrelativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Freescalarfields,Modedecomposition . . . . . . . . . . . . . . . 6
1.3 Interactingscalarfields . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 LSZreductionformulas . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Fromtransitionamplitudestoreactionrates . . . . . . . . . . . . . 17
1.6 Generatingfunctional . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 PerturbativeexpansionandFeynmanrules . . . . . . . . . . . . . . 27
1.8 Calculationofloopintegrals . . . . . . . . . . . . . . . . . . . . . 33
1.9 Ka¨llen-Lehmannspectralrepresentation . . . . . . . . . . . . . . . 36
1.10 Ultravioletdivergencesandrenormalization . . . . . . . . . . . . . 38
1.11 Spin1/2fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.12 Spin1fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.13 Abeliangaugeinvariance,QED . . . . . . . . . . . . . . . . . . . 57
1.14 Chargeconservation,Ward-Takahashiidentities . . . . . . . . . . . 60
1.15 Spontaneoussymmetrybreaking . . . . . . . . . . . . . . . . . . . 63
1.16 Perturbativeunitarity . . . . . . . . . . . . . . . . . . . . . . . . . 71
2 Functionalquantization 85
2.1 Pathintegralinquantummechanics . . . . . . . . . . . . . . . . . 85
2.2 Classicallimit,Leastactionprinciple. . . . . . . . . . . . . . . . . 89
2.3 Morefunctionalmachinery . . . . . . . . . . . . . . . . . . . . . . 89
2.4 Pathintegralinscalarfieldtheory . . . . . . . . . . . . . . . . . . 96
2.5 Functionaldeterminants. . . . . . . . . . . . . . . . . . . . . . . . 98
2.6 Quantumeffectiveaction . . . . . . . . . . . . . . . . . . . . . . . 101
2.7 Two-particleirreducibleeffectiveaction . . . . . . . . . . . . . . . 107
2.8 EuclideanpathintegralandStatisticalmechanics . . . . . . . . . . 114
i
ii F.GELIS–ASTROLLTHROUGHQUANTUMFIELDS
3 Pathintegralsforfermionsandphotons 119
3.1 Grassmannvariables . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.2 Pathintegralforfermions . . . . . . . . . . . . . . . . . . . . . . . 125
3.3 Pathintegralforphotons . . . . . . . . . . . . . . . . . . . . . . . 127
3.4 Schwinger-Dysonequations . . . . . . . . . . . . . . . . . . . . . 130
3.5 Quantumanomalies . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4 Non-Abeliangaugesymmetry 143
4.1 Non-abelianLiegroupsandalgebras . . . . . . . . . . . . . . . . . 144
4.2 Yang-MillsLagrangian . . . . . . . . . . . . . . . . . . . . . . . . 152
4.3 Non-Abeliangaugetheories . . . . . . . . . . . . . . . . . . . . . 157
4.4 Spontaneousgaugesymmetrybreaking . . . . . . . . . . . . . . . 162
4.5 θ-termandstrong-CPproblem . . . . . . . . . . . . . . . . . . . . 168
4.6 Non-localgaugeinvariantoperators . . . . . . . . . . . . . . . . . 176
5 QuantizationofYang-Millstheory 187
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.2 Gaugefixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.3 Fadeev-PopovquantizationandGhostfields . . . . . . . . . . . . . 191
5.4 Feynmanrulesfornon-abeliangaugetheories . . . . . . . . . . . . 193
5.5 On-shellnon-AbelianWardidentities . . . . . . . . . . . . . . . . 197
5.6 Ghostsandunitarity . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6 Renormalizationofgaugetheories 211
6.1 Ultravioletpowercounting . . . . . . . . . . . . . . . . . . . . . . 211
6.2 Symmetriesofthequantumeffectiveaction . . . . . . . . . . . . . 212
6.3 Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.4 Backgroundfieldmethod . . . . . . . . . . . . . . . . . . . . . . . 223
7 Renormalizationgroup 231
7.1 Callan-Symanzikequations . . . . . . . . . . . . . . . . . . . . . . 231
7.2 Correlatorscontainingcompositeoperators . . . . . . . . . . . . . 234
7.3 Operatorproductexpansion. . . . . . . . . . . . . . . . . . . . . . 237
7.4 Example: QCDcorrectionstoweakdecays . . . . . . . . . . . . . 241
7.5 Non-perturbativerenormalizationgroup . . . . . . . . . . . . . . . 248
CONTENTS iii
8 Effectivefieldtheories 259
8.1 Generalprinciplesofeffectivetheories . . . . . . . . . . . . . . . . 260
8.2 Example: Fermitheoryofweakdecays . . . . . . . . . . . . . . . 264
8.3 Standardmodelasaneffectivefieldtheory . . . . . . . . . . . . . . 267
8.4 EffectivetheoriesinQCD. . . . . . . . . . . . . . . . . . . . . . . 274
8.5 EFTofspontaneoussymmetrybreaking . . . . . . . . . . . . . . . 284
9 Quantumanomalies 295
9.1 Axialanomaliesinagaugebackground . . . . . . . . . . . . . . . 295
9.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
9.3 Wess-Zuminoconsistencyconditions. . . . . . . . . . . . . . . . . 314
9.4 ’tHooftanomalymatching . . . . . . . . . . . . . . . . . . . . . . 318
9.5 Scaleanomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
10 Localizedfieldconfigurations 327
10.1 Domainwalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
10.2 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
10.3 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
10.4 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
11 Moderntoolsfortreelevelamplitudes 357
11.1 Shortcomingsoftheusualapproach . . . . . . . . . . . . . . . . . 357
11.2 Colourorderingofgluonicamplitudes . . . . . . . . . . . . . . . . 358
11.3 Spinor-helicityformalism . . . . . . . . . . . . . . . . . . . . . . . 364
11.4 Britto-Cachazo-Feng-Wittenon-shellrecursion . . . . . . . . . . . 374
11.5 Tree-levelgravitationalamplitudes . . . . . . . . . . . . . . . . . . 385
11.6 Cachazo-Svrcek-Wittenrules . . . . . . . . . . . . . . . . . . . . . 395
12 Worldlineformalism 407
12.1 Worldlinerepresentation . . . . . . . . . . . . . . . . . . . . . . . 407
12.2 Quantumelectrodynamics . . . . . . . . . . . . . . . . . . . . . . 413
12.3 Schwingermechanism . . . . . . . . . . . . . . . . . . . . . . . . 417
12.4 Calculationofone-loopamplitudes . . . . . . . . . . . . . . . . . . 420
iv F.GELIS–ASTROLLTHROUGHQUANTUMFIELDS
13 Latticefieldtheory 431
13.1 Discretizationofbosonicactions . . . . . . . . . . . . . . . . . . . 432
13.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
13.3 Hadronmassdeterminationonthelattice . . . . . . . . . . . . . . 441
13.4 Wilsonloopsandconfinement . . . . . . . . . . . . . . . . . . . . 442
13.5 Gaugefixingonthelattice . . . . . . . . . . . . . . . . . . . . . . 446
13.6 Latticeworldlineformalism . . . . . . . . . . . . . . . . . . . . . 450
14 Quantumfieldtheoryatfinitetemperature 457
14.1 Canonicalthermalensemble . . . . . . . . . . . . . . . . . . . . . 457
14.2 Finite-T perturbationtheory . . . . . . . . . . . . . . . . . . . . . 458
14.3 Largedistanceeffectivetheories . . . . . . . . . . . . . . . . . . . 477
14.4 Out-of-equilibriumsystems . . . . . . . . . . . . . . . . . . . . . . 492
15 Strongfieldsandsemi-classicalmethods 501
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
15.2 Expectationvaluesinacoherentstate . . . . . . . . . . . . . . . . 503
15.3 Quantumfieldtheorywithexternalsources . . . . . . . . . . . . . 509
15.4 ObservablesatLOandNLO . . . . . . . . . . . . . . . . . . . . . 510
15.5 Green’sformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
15.6 Modefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
15.7 Multi-pointcorrelationfunctionsattreelevel . . . . . . . . . . . . 531
Chapter 1
Basics of Quantum
Field Theory
1.1 Special relativity
1.1.1 Lorentztransformations
Specialrelativityplaysacrucialroleinquantumfieldtheories1. Variousobserversin
framesthataremovingataconstantspeedrelativetoeachothershouldbeableto
describephysicalphenomenausingthesamelawsofPhysics. Thisdoesnotimply
that the equations governing these phenomena are independent of the observer’s
frame,butthattheseequationstransforminaconstrainedfashion–dependingonthe
natureoftheobjectstheycontain–underachangeofreferenceframe.
LetusconsidertwoframesFandF ,inwhichthecoordinatesofagivenevent
′
arerespectivelyxµandx′µ. ALorentztransformationisalineartransformationsuch
thattheintervalds2 dt2−dx2 isthesameinthetwoframes2. Ifwedenotethe
≡
coordinatetransformationby
xµ =Λµ xν , (1.1)
′ ν
1Anexceptiontothisassertionisforquantumfieldmodelsappliedtocondensedmatterphysics,where
thebasicdegreesoffreedomaretoaverygoodlevelofapproximationdescribedbyGalileankinematics.
2Thephysicalpremisesofspecialrelativityrequirethatthespeedoflightbethesameinallinertial
frames,whichimpliessolelythatds2=0bepreservedinallinertialframes.Thegroupoftransformations
thatachievesthisiscalledtheconformalgroup.Infourspace-timedimensions,theconformalgroupis15
dimensional,andinadditiontothe6orthochronousLorentztransformationsitcontainsdilatationsaswell
asnon-lineartransformationscalledspecialconformaltransformations.
1
2 F.GELIS–ASTROLLTHROUGHQUANTUMFIELDS
thematrixΛofthetransformationmustobey
g Λµ Λν =g (1.2)
µν ρ σ ρσ
whereg istheMinkowskimetrictensor
µν
+1
−1
g   . (1.3)
µν
≡ −1
 
 −1
 
 
Notethateq.(1.2)impliesthat
Λµ = Λ−1 µ . (1.4)
ν ν
(cid:0) (cid:1)
IfweconsideraninfinitesimalLorentztransformation,
Λµ =δµ +ωµ (1.5)
ν ν ν
(withallcomponentsofωmuchsmallerthanunity),thisimpliesthat
ω =−ω (1.6)
µν νµ
(with all indices down). Consequently, there are 6 independent Lorentz transfor-
mations, threeofwhichareordinaryrotationsandthreeareboosts. Notethatthe
infinitesimaltransformations(1.5)haveadeterminant3equalto+1(theyarecalled
proper transformations), and do not change the direction of the time axis since
Λ0 =1 0(theyarecalledorthochronous). Anycombinationofsuchinfinitesimal
0
≥
transformationssharesthesameproperties,andtheirsetformsasubgroupofthefull
groupoftransformationsthatpreservetheMinkowskimetric.
(cid:13)csileGsiocnarF
1.1.2 RepresentationsoftheLorentzgroup
Moregenerally,aLorentztransformationactsonaquantumsystemviaatransforma-
tionU(Λ),thatformsarepresentationoftheLorentzgroup,i.e.
U(ΛΛ )=U(Λ)U(Λ ). (1.7)
′ ′
ForaninfinitesimalLorentztransformation,wecanwrite
i
U(1+ω)=I+ ω Mµν . (1.8)
µν
2
3Fromeq.(1.2),thedeterminantmaybeequalto 1.
±
1.BASICSOFQUANTUMFIELDTHEORY 3
(Theprefactori/2inthesecondtermoftherighthandsideisconventional.) Since
theω areantisymmetric,thegeneratorsMµνcanalsobechosenantisymmetric.
µν
Byusingeq.(1.7)fortheLorentztransformationΛ−1Λ Λ,wearriveat
′
U−1(Λ)MµνU(Λ)=Λµ Λν Mρσ , (1.9)
ρ σ
indicatingthatMµνtransformsasarank-2tensor. Whenusedwithaninfinitesimal
transformationΛ=1+ω,thisidentityleadstothecommutationrelationthatdefines
theLiealgebraoftheLorentzgroup
Mµν,Mρσ =i(gµρMνσ−gνρMµσ)−i(gµσMνρ−gνσMµρ). (1.10)
When(cid:2)necessary,it(cid:3)ispossibletodividethesixgeneratorsMµνintothreegenerators
Jiforordinaryspatialrotations,andthreegeneratorsKifortheLorentzboostsalong
eachofthespatialdirections:
Rotations: Ji 1ǫijkMjk ,
≡ 2
Lorentzboosts: Ki Mi0 . (1.11)
≡
Inafashionsimilartoeq.(1.9),weobtainthetransformationofthe4-impulsionPµ,
U−1(Λ)PµU(Λ)=Λµ Pρ , (1.12)
ρ
whichleadstothefollowingcommutationrelationbetweenPµandMµν,
Pµ,Mρσ =i(gµσPρ−gµρPσ),
Pµ,Pν =0. (1.13)
(cid:2) (cid:3)
(cid:2) (cid:3)
1.1.3 One-particlestates
Letusdenote p,σ aone-particlestate,wherepisthe3-momentumofthatparticle,
andσdenotesitsotherquantumnumbers. Sincethisstatecontainsaparticlewitha
(cid:12) (cid:11)
definitemome(cid:12)ntum,itisaneigenstateofthemomentumoperatorPµ,namely
Pµ p,σ =pµ p,σ , withp0 p2+m2 . (1.14)
≡
Consider(cid:12)nowt(cid:11)hestate(cid:12)U(Λ(cid:11)) p,σ . Wehavpe
(cid:12) (cid:12)
PµU(Λ) p,σ =U(Λ) U(cid:12)−1(Λ(cid:11))PµU(Λ) p,σ =Λµ pνU(Λ) p,σ . (1.15)
(cid:12) ν
(cid:12) (cid:11) ΛµνPν (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12)
Therefore,U(Λ) p,σ isa|neigen{sztateofm}omentumwitheigenvalue(Λp)µ,and
wemaywriteitasalinearcombinationofallthestateswithmomentumΛp,
(cid:12) (cid:11)
(cid:12)
U(Λ) p,σ = C (Λ;p) Λp,σ . (1.16)
σσ′ ′
σ′
(cid:12) (cid:11) X (cid:12) (cid:11)
(cid:12) (cid:12)
4 F.GELIS–ASTROLLTHROUGHQUANTUMFIELDS
1.1.4 Littlegroup
Any positive energy on-shell momentum pµ can be obtained by applying an or-
thochronousLorentztransformationtosomereferencemomentumqµthatliveson
thesamemass-shell,
pµ Lµ (p)qν . (1.17)
ν
≡
Thechoiceofthereference4-vectorisnotimportant,butdependsonwhetherthe
particleunderconsiderationismassiveornot. Convenientchoicesarethefollowing:
m>0: qµ (m,0,0,0),the4-momentumofamassiveparticleatrest,
• ≡
m=0: qµ (ω,0,0,ω),the4-momentumofamasslessparticlemovingin
• ≡
thethirddirectionofspace.
Then, wemaydefineagenericone-particlestatefromthosecorrespondingtothe
referencemomentumasfollows
p,σ N U(L(p)) q,σ , (1.18)
p
≡
where(cid:12)N i(cid:11)sanumericalpre(cid:12)facto(cid:11)rthatmaybenecessarytoproperlynormalizethe
(cid:12) p (cid:12)
states. Thisdefinitionleadsto
U Λ p,σ =N U L(Λp) U L−1(Λp)ΛL(p) q,σ . (1.19)
p
(cid:0) (cid:1)(cid:12) (cid:11) (cid:0) (cid:1) (cid:0) Σ (cid:1)(cid:12) (cid:11)
(cid:12) (cid:12)
NotethattheLorentztransformationΣ |L−1(Λ{zp)ΛL(}p)mapsqµintoitself,and
≡
thereforebelongstothesubgroupoftheLorentzgroupthatleavesqµinvariant,called
thelittlegroupofqµ. Thus,whenU(Σ)actsonthereferencestate,themomentum
remainsunchangedandonlytheotherquantumnumbersmayvary
U(Σ) q,σ = C (Σ) q,σ . (1.20)
σσ′ ′
σ′
(cid:12) (cid:11) X (cid:12) (cid:11)
(cid:12) (cid:12)
Moreover,thecoefficientsC (Σ)intherighthandsideofthisformuladefinea
σσ′
representationofthelittlegroup,
C (Σ Σ )= C (Σ )C (Σ ). (1.21)
σσ′ 2 1 σσ′′ 2 σ′′σ′ 1
σ′′
X
Massiveparticles: Inthecaseofamassiveparticles,thelittlegroupismadeof
theLorentztransformationsthatleavethevectorqµ =(m,0,0,0)invariant,which
isthegroupofallrotationsin3-dimensionalspace. Theadditionalquantumnumber
σisthereforealabelthatenumeratesthepossiblestatesinagivenrepresentationof
SO(3). Theserepresentationscorrespondtotheangularmomentum,butsinceweare
intherestframeoftheparticle,thisisinfactthespinoftheparticle. Foraspins,the
dimensionoftherepresentationis2s+1,andσtakesthevalues−s,1−s, ,+s.
···