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