Elementary Particle Theory. Volume 2, Quantum Electrodynamics

217 Pages·2019·2.92 MB·English

by  Eugene Stefanovich

EugeneStefanovich
ElementaryParticleTheory
De Gruyter Studies in
Mathematical Physics
|
Edited by
Michael Efroimsky, Bethesda, Maryland, USA
Leonard Gamberg, Reading, Pennsylvania, USA
Dmitry Gitman,São Paulo, Brazil
Alexander Lazarian, Madison, Wisconsin, USA
BorisSmirnov, Moscow, Russia
Volume 46
Eugene Stefanovich
Elementary
Particle Theory
|
Volume 2: Quantum Electrodynamics
MathematicsSubjectClassification2010
Primary:81-02,81V10,81T15;Secondary:47A40,81T18
Author
DrEugeneStefanovich
SanJose,California
USA
[email protected]
ISBN978-3-11-049089-3
e-ISBN(PDF)978-3-11-049320-7
e-ISBN(EPUB)978-3-11-049143-2
ISSN2194-3532
LibraryofCongressControlNumber:2018016481
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Contents
Listoffigures|IX
Listoftables|XI
Postulates,statements,theorems|XIII
Conventionalnotation|XV
Preface|XVII
1 Fockspace|1
1.1 Creationandannihilationoperators|1
1.1.1 Sectorswithfixednumbersofparticles|1
1.1.2 ParticleobservablesinFockspace|3
1.1.3 NoninteractingrepresentationofPoincarégroup|3
1.1.4 Creationandannihilationoperatorsforfermions|4
1.1.5 Anticommutatorsofparticleoperators|6
1.1.6 Creationandannihilationoperatorsforphotons|7
1.1.7 Particlenumberoperators|7
1.1.8 Continuousspectrumofmomentum|8
1.1.9 Normalordering|9
1.1.10 Noninteractingenergyandmomentum|11
1.1.11 Noninteractingangularmomentumandboost|12
1.1.12 Poincarétransformationsofparticleoperators|13
1.2 Interactionpotentials|15
1.2.1 Conservationlaws|15
1.2.2 Generalformofinteractionoperators|17
1.2.3 Fivetypesofregularpotentials|20
1.2.4 Productsandcommutatorsofregularpotentials|23
1.2.5 Moreaboutt-integrals|25
1.2.6 Solutionofonecommutatorequation|27
1.2.7 Two-particlepotentials|28
1.2.8 Momentum-dependentpotentials|31
2 ScatteringinFockspace|33
2.1 Toymodeltheory|33
2.1.1 FockspaceandHamiltonian|33
2.1.2 S-operatorinsecondorder|35
2.1.3 Drawingdiagramsintoymodel|36
VI | Contents
2.1.4 Readingdiagramsintoymodel|39
2.1.5 Scatteringinsecondorder|40
2.2 Renormalizationintoymodel|41
2.2.1 Renormalizationofelectronself-scatteringinsecondorder|41
2.2.2 Renormalizationofelectronself-scatteringinfourthorder|43
2.3 Diagramsingeneraltheory|46
2.3.1 Productsofdiagrams|46
2.3.2 Connectedanddisconnecteddiagrams|47
2.3.3 Divergenceofloopintegrals|50
2.4 Clusterseparability|52
2.4.1 Clusterseparabilityofinteraction|52
2.4.2 ClusterseparabilityofS-operator|54
3 Quantumelectrodynamics|57
3.1 InteractioninQED|57
3.1.1 Whydoweneedquantumfields?|58
3.1.2 Simplequantumfieldtheories|58
3.1.3 InteractionoperatorsinQED|60
3.2 S-operatorinQED|62
3.2.1 S-operatorinsecondorder|62
3.2.2 CovariantformofS-operator|66
3.2.3 Feynmangauge|68
3.2.4 Feynmandiagrams|70
3.2.5 Comptonscattering|72
3.2.6 Virtualparticles?|73
4 Renormalization|75
4.1 Tworenormalizationconditions|75
4.1.1 Noself-scatteringcondition|75
4.1.2 Chargerenormalization|78
4.1.3 Renormalizationbycounterterms|78
4.1.4 Diagramsofelectron–protonscattering|79
4.1.5 Regularization|80
4.2 Counterterms|81
4.2.1 Electron’sself-scattering|81
4.2.2 Electronself-scatteringcounterterm|83
4.2.3 Fittingcoefficient(δm)2|84
4.2.4 Fittingcoefficient(Z2−1)2|85
4.2.5 Photon’sself-scattering|86
4.2.6 Photonself-energycounterterm|87
4.2.7 Applyingchargerenormalizationcondition|89
4.2.8 Vertexrenormalization|90
Contents | VII
4.3 RenormalizedS-matrix|93
4.3.1 “Vacuumpolarization”diagrams|93
4.3.2 Vertexdiagram|93
4.3.3 Ladderdiagram|95
4.3.4 Crossladderdiagram|98
4.3.5 Renormalizability|101
A Usefulintegrals|103
B Quantumfieldsoffermions|107
B.1 Paulimatrices|107
B.2 Diracgammamatrices|108
B.3 DiracrepresentationofLorentzgroup|109
B.4 ConstructionofDiracfield|112
B.5 Propertiesoffunctionsuandv|114
B.6 Explicitformulasforuandv|115
B.7 Usefulnotation|118
B.8 Poincarétransformationsoffields|119
B.9 Approximation(v/c)2|120
B.10 Anticommutationrelations|122
B.11 Diracequation|123
B.12 Fermionpropagator|125
C Quantumfieldofphotons|129
C.1 Constructionofphotonquantumfield|129
C.2 Propertiesoffunctioneμ(p,τ)|130
C.3 Usefulcommutator|131
C.4 Commutatorofphotonfields|133
C.5 Photonpropagator|133
C.6 Poincarétransformationsofphotonfield|135
D QEDinteractionintermsofparticleoperators|139
D.1 Currentdensity|139
D.2 First-orderinteractioninQED|142
D.3 Second-orderinteractioninQED|142
E RelativisticinvarianceofQFT|155
E.1 RelativisticinvarianceofsimpleQFT|155
E.2 RelativisticinvarianceofQED|156
F LoopintegralsinQED|163
F.1 Schwinger–Feynmanintegrationtrick|163
VIII | Contents
F.2 Somebasicfour-dimensionalintegrals|164
F.3 Electronself-energyintegral|167
F.4 Vertexintegral|170
F.4.1 CalculationofM|172
F.4.2 CalculationofMσ |173
F.4.3 CalculationofMστ |174
F.4.4 Completeintegral|175
F.5 Integralforladderdiagram|178
F.5.1 CalculationofLI|179
F.5.2 CalculationofLII|181
F.5.3 CalculationofLIII|182
F.5.4 Completeintegral|184
G Scatteringmatrixin(v/c)2approximation |185
G.1 Secondperturbationorder|185
G.2 Vertexcontributioninfourthorder|187
H Checksofphysicaldimensions|191
Bibliography|193
Index|195
List of figures
Figure1.1 Operatorsin“indexspace”(page20)
Figure2.1 DiagramsforoperatorsV1andV1(t)(page37)
Figure2.2 Normalorderingoftheproductoftwodiagrams(page38)
Figure2.3 RenormdiagramsinVcVcVcVc(page44)
Figure2.4 DiagramofthecountertermQ (page44)
2
Figure2.5 RenormdiagramsinVcVc+VcVcVc(page45)
Figure2.6 TotheproofofLemma2.3(page49)
Figure2.7 Genericdiagraminahypotheticaltheory(page49)
− +
Figure3.1 Second-orderdiagramfore +p scattering(page72)
−
Figure3.2 e +γscatteringdiagrams(page73)
− +
Figure4.1 e +p scatteringdiagramsuptothefourthorder(page80)
Figure4.2 Electronself-scatteringdiagrams(page81)
Figure4.3 Photonself-scatteringdiagrams(page86)
Figure4.4 “Vacuumpolarization”diagrams(page89)
Figure4.5 Vertexdiagrams(page91)
Figure4.6 Ladderdiagram(page95)
Figure4.7 Cross-ladderdiagram(page98)
FigureA.1 Tothecalculationofintegral(A.10)(page104)
FigureF.1 Wickrotationintheintegral(F.6)(page164)
FigureF.2 Integrationareain(F.45)(page179)
https://doi.org/10.1515/9783110493207-201