Quantum Field Theory

423 Pages·2013·3.875 MB·English

by  Michael V. Sadovskii

DE GRUYTER
Michael V. Sadovskii
QUANTUM FIELD
THEORY
STUDIES IN MATHEMATICAL PHYSICS 17
De Gruyter Studies in Mathematical Physics 17
Editors
MichaelEfroimsky,Bethesda,USA
LeonardGamberg,Reading,USA
DmitryGitman,SãoPaulo,Brasil
AlexanderLazarian,Madison,USA
BorisSmirnov,Moscow,Russia

Michael V. Sadovskii
Quantum Field Theory
De Gruyter
PhysicsandAstronomyClassificationScheme2010:03.70.+k,03.65.Pm,11.10.-z,11.10.Gh,
11.10.Jj, 11.25.Db, 11.15.Bt, 11.15.Ha, 11.15.Ex, 11.30. -j, 12.20.-m, 12.38.Bx, 12.10.-g,
12.38.Cy.
ISBN978-3-11-027029-7
e-ISBN978-3-11-027035-8
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Preface
ThisbookistherevisedEnglishtranslationofthe2003Russianeditionof“Lectureson
QuantumFieldTheory”,whichwasbasedonmuchextendedlecturecoursetaughtby
theauthorsince1991attheUralStateUniversity,Ekaterinburg.Itisaddressedmainly
tograduateandPhDstudents,aswellastoyoungresearchers,whoareworkingmainly
incondensedmatterphysicsandseekingacompactandrelativelysimpleintroduction
to the major section of modern theoretical physics, devoted to particles and fields,
which remains relatively unknown to the condensed matter community, largely un-
awareofthemajorprogressrelatedtotheformulationtheso-called“standardmodel”
ofelementaryparticles,whichisatthemomentthemostfundamentaltheoryofmatter
confirmedbyexperiments.Infact,thisbookdiscussesthemainconceptsofthisfun-
damentaltheorywhicharebasicandnecessary(intheauthor’sopinion)foreveryone
startingprofessionalresearchworkinotherareasoftheoreticalphysics,notrelatedto
high-energyphysicsandthetheoryofelementaryparticles,suchascondensedmatter
theory. This is actually even more important, as many of the theoretical approaches
developedinquantumfieldtheoryarenowactivelyusedincondensedmattertheory,
andmanyoftheconceptsofcondensedmattertheoryarenowwidelyusedinthecon-
structionofthe“standardmodel”ofelementaryparticles.Oneofthemainaimsofthe
bookistoillustratethisunityofmoderntheoreticalphysics,widelyusingtheanalogies
betweenquantumfieldtheoryandmoderncondensedmattertheory.
Incontrasttomanybooksonquantumfieldtheory[2,6,8–10,13,25,28,53,56,59,
60],mostofwhichusuallyfollowratherdeductivepresentationofthematerial,here
we use a kind of inductive approach (similar to that used in [59,60]), when one and
thesameproblemisdiscussedseveraltimesusingdifferentapproaches.Intheauthor’s
opinionsuchrepetitionsareusefulforamoredeepunderstandingofthevariousideas
andmethodsusedforsolvingrealproblems.Ofcourse,amongthebooksmentioned
above,theauthorwasmuchinfluencedby[6,56,60],andthisinfluenceisobviousin
manypartsofthetext.However,thechoiceofmaterialandtheformofpresentationis
essentiallyhisown.ForthepresentEnglisheditionsomeofthematerialwasrewritten,
bringingthecontentmoreuptodateandaddingmorediscussiononsomeofthemore
difficultcases.
Thecentralideaofthisbookisthepresentationofthebasicsofthegaugefieldthe-
oryofinteractingelementaryparticles.Astothemethods,wepresentaratherdetailed
derivationoftheFeynmandiagramtechnique,whichlongagoalsobecamesoimpor-
tant for condensed matter theory. We also discuss in detail the method of functional
(path)integralsinquantumtheory,whichisnowalsowidelyusedinmanysectionsof
theoreticalphysics.
vi Preface
Welimitourselvestothisrelativelytraditionalmaterial,droppingsomeofthemore
modern (but more speculative) approaches, such as supersymmetry. Obviously, we
also drop the discussion of some new ideas which are in fact outside the domain of
thequantumfieldtheory,suchasstringsandsuperstrings.Alsowedonotdiscussin
anydetailtheexperimentalaspectsofmodernhigh-energyphysics(particlephysics),
usingonlyafewillustrativeexamples.
Ekaterinburg,2012 M.V.Sadovskii
Contents
Preface v
1 Basicsofelementaryparticles 1
1.1 Fundamentalparticles..................................... 1
1.1.1 Fermions ........................................ 2
1.1.2 Vectorbosons..................................... 3
1.2 Fundamentalinteractions .................................. 4
1.3 TheStandardModelandperspectives ........................ 5
2 Lagrangeformalism.Symmetriesandgaugefields 9
2.1 Lagrangemechanicsofaparticle ............................ 9
2.2 Realscalarfield.Lagrangeequations ......................... 11
2.3 TheNoethertheorem ..................................... 15
2.4 Complexscalarandelectromagneticfields .................... 18
2.5 Yang–Millsfields ........................................ 24
2.6 Thegeometryofgaugefields ............................... 30
2.7 Arealisticexample–chromodynamics ....................... 38
3 Canonicalquantization,symmetriesinquantumfieldtheory 40
3.1 Photons ................................................ 40
3.1.1 Quantizationoftheelectromagneticfield ............... 40
3.1.2 RemarksongaugeinvarianceandBosestatistics ......... 45
3.1.3 VacuumfluctuationsandCasimireffect ................ 48
3.2 Bosons ................................................ 50
3.2.1 Scalarparticles .................................... 50
3.2.2 Trulyneutralparticles .............................. 54
3.2.3 CPT-transformations .............................. 57
3.2.4 Vectorbosons..................................... 61
3.3 Fermions ............................................... 63
3.3.1 Three-dimensionalspinors ........................... 63
3.3.2 SpinorsoftheLorentzgroup ......................... 67
3.3.3 TheDiracequation ................................. 74
3.3.4 ThealgebraofDirac’smatrices ....................... 79
3.3.5 Planewaves ...................................... 81
viii Contents
3.3.6 Spinandstatistics .................................. 83
3.3.7 C,P,T transformationsforfermions .................. 85
3.3.8 Bilinearforms .................................... 86
3.3.9 Theneutrino ...................................... 87
4 TheFeynmantheoryofpositronandelementaryquantum
electrodynamics 93
4.1 Nonrelativistictheory.Green’sfunctions ...................... 93
4.2 Relativistictheory........................................ 96
4.3 Momentumrepresentation ................................. 100
4.4 Theelectroninanexternalelectromagneticfield ................ 103
4.5 Thetwo-particleproblem .................................. 110
5 Scatteringmatrix 115
5.1 Scatteringamplitude ...................................... 115
5.2 Kinematicinvariants...................................... 118
5.3 Unitarity ............................................... 121
6 Invariantperturbationtheory 124
6.1 SchroedingerandHeisenbergrepresentations .................. 124
6.2 Interactionrepresentation .................................. 125
6.3 S-matrixexpansion ...................................... 128
6.4 Feynmandiagramsforelectronscatteringinquantum
electrodynamics ......................................... 135
6.5 Feynmandiagramsforphotonscattering ...................... 140
6.6 Electronpropagator ...................................... 142
6.7 Thephotonpropagator .................................... 146
6.8 TheWicktheoremandgeneraldiagramrules .................. 149
7 Exactpropagatorsandvertices 156
7.1 FieldoperatorsintheHeisenbergrepresentationandinteraction
representation ........................................... 156
7.2 Theexactpropagatorofphotons ............................ 158
7.3 Theexactpropagatorofelectrons ........................... 164
7.4 Vertexparts ............................................ 168
7.5 Dysonequations ......................................... 172
7.6 Wardidentity ........................................... 173
Contents ix
8 Someapplicationsofquantumelectrodynamics 175
8.1 Electronscatteringbystaticcharge:higherordercorrections ...... 175
8.2 TheLambshiftandtheanomalousmagneticmoment ............ 180
8.3 Renormalization–howitworks............................. 185
8.4 “Running”thecouplingconstant ............................ 189
8.5 AnnihilationofeCe(cid:2) intohadrons.Proofoftheexistenceofquarks 191
8.6 Thephysicalconditionsforrenormalization ................... 192
8.7 Theclassificationandeliminationofdivergences ............... 196
8.8 Theasymptoticbehaviorofaphotonpropagatoratlargemomenta . 200
8.9 Relationbetweenthe“bare”and“true”charges ................ 203
8.10 TherenormalizationgroupinQED .......................... 207
8.11 Theasymptoticnatureofaperturbationseries .................. 209
9 Pathintegralsandquantummechanics 211
9.1 Quantummechanicsandpathintegrals ....................... 211
9.2 Perturbationtheory ....................................... 219
9.3 Functionalderivatives .................................... 225
9.4 Somepropertiesoffunctionalintegrals ....................... 226
10 Functionalintegrals:scalarsandspinors 232
10.1 Generatingthefunctionalforscalarfields ..................... 232
10.2 Functionalintegration..................................... 237
10.3 FreeparticleGreen’sfunctions ............................. 240
10.4 Generatingthefunctionalforinteractingfields ................. 247
10.5 '4 theory............................................... 250
10.6 Thegeneratingfunctionalforconnecteddiagrams .............. 257
10.7 Self-energyandvertexfunctions ............................ 260
10.8 Thetheoryofcriticalphenomena ............................ 264
10.9 Functionalmethodsforfermions ............................ 277
10.10 PropagatorsandgaugeconditionsinQED ..................... 285
11 Functionalintegrals:gaugefields 287
11.1 Non-AbeliangaugefieldsandFaddeev–Popovquantization ....... 287
11.2 Feynmandiagramsfornon-Abeliantheory .................... 293