Quantum Field Theory

444 Pages·2019·7.46 MB·English

by  Sadovskii, Michael V.

MichaelV.Sadovskii
QuantumFieldTheory
Texts and Monographs in
Theoretical Physics
|
Edited by
Michael Efroimsky, Bethesda, Maryland, USA
Leonard Gamberg, Reading, Pennsylvania, USA
Michael V. Sadovskii
Quantum Field
Theory
|
2nd edition
PhysicsandAstronomyClassification2010
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
Author
Prof.Dr.MichaelV.Sadovskii
RussianAcademyofSciences
InstituteforElectrophysics
Amundsenstreet106
Ekaterinburg620016
Russia
[email protected]
ISBN978-3-11-064515-6
e-ISBN(PDF)978-3-11-064852-2
e-ISBN(EPUB)978-3-11-064526-2
ISSN2627-3934
LibraryofCongressControlNumber:2019936026
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
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We have no better way of describing elementary particles than quantum field the-
ory.Aquantumfieldingeneralisanassemblyofaninfinitenumberofinteracting
harmonicoscillators.Excitationsofsuchoscillatorsareassociatedwithparticles...
Allthishastheflavorofthe19thcentury,whenpeopletriedtoconstructmechanical
modelsforallphenomena.Iseenothingwrongwithit,becauseanynontrivialidea
is in a certain sense correct. The garbage of the past often becomes the treasure of
thepresent(andviceversa).Forthisreasonweshallboldlyinvestigateallpossible
analogiestogetherwithourmainproblem.
A.M.Polyakov,“GaugeFieldsandStrings”,1987[51]

Preface
ThisbookistherevisedEnglishtranslationofthe2003Russianeditionof“Lectureson
QuantumFieldTheory”,whichwasbasedonamuchextendedlecturecoursetaught
by the author since 1991 at the Ural State University, Ekaterinburg. It is addressed
mainlytograduateandPhDstudents,aswellastoyoungresearchers,whoarework-
ingmainlyincondensedmatterphysicsandseekingacompactandrelativelysimple
introductiontothemajorsectionofmoderntheoreticalphysics,devotedtoparticles
andfields,whichremainsrelativelyunknowntothecondensedmattercommunity.
Thelatterislargelyunawareofthemajorprogressrelatedtotheformulationtheso-
called“standardmodel”ofelementaryparticles,whichis—atthemoment—themost
fundamentaltheoryofmatterconfirmedbyexperiments.Infact,thisbookdiscusses
themainconceptsofthisfundamentaltheory,whicharebasicandnecessary(inthe
author’sopinion)foreveryonestartingprofessionalresearchworkinotherareasof
theoreticalphysics,notrelatedtohigh-energyphysicsandthetheoryofelementary
particles,suchascondensedmattertheory.Actually,anadditionalpointofthisbook’s
importance is that many of the theoretical approaches developed in quantum field
theoryarenowactivelyusedincondensedmattertheory,andmanyoftheconcepts
ofcondensedmattertheoryarenowwidelyusedintheconstructionofthe“standard
model”ofelementaryparticles.Oneofthemainaimsofthebookistoillustratethis
unity of modern theoretical physics, widely using the analogies between quantum
fieldtheoryandmoderncondensedmattertheory.
Incontrasttomanybooksonquantumfieldtheory[2,6,8–10,13,25,28,53,56,59,
60],mostofwhichusuallyfollowratherdeductivepresentationofthematerial,here
weuseakindofinductiveapproach(similartothatusedin[59,60]);thesameprob-
lem is discussed several times using different approaches. In the author’s opinion,
suchrepetitionsareusefulforadeeperunderstandingofthevariousideasandmeth-
odsusedforsolvingrealproblems.Ofcourse,amongthebooksmentionedabove,the
authorwasmuchinfluencedby[6,56,60],andthisinfluenceisobviousinmanyparts
ofthetext.However,thechoiceofmaterialandtheformofpresentationisessentially
hisown.ForthepresentEnglisheditionsomeofthematerialwasrewritten,bringing
thecontentmoreup-to-dateandaddingmorediscussiononsomeofthemoredifficult
cases.
Thecentralideaofthisbookisthepresentationofthebasicsofthegaugefield
theoryofinteractingelementaryparticles.Astothemethods,wepresentaratherde-
tailedderivationoftheFeynmandiagramtechnique,whichlongagoalsobecamevery
importantforcondensedmattertheory.Wealsodiscussindetailthemethodoffunc-
tional (path) integrals in quantum theory, which is now also widely used in many
aspectsoftheoreticalphysics.
We limit ourselves to this relatively traditional material, dropping some of the
moremodern(butmorespeculative)approaches,suchassupersymmetry.Obviously,
https://doi.org/10.1515/9783110648522-201
VIII | Preface
wealsodropthediscussionofsomenewideas,whichareinfactoutsidethedomainof
thequantumfieldtheory,suchasstringsandsuperstrings.Alsowedonotdiscussin
anydetailtheexperimentalaspectsofmodernhigh-energyphysics(particlephysics),
usingonlyafewillustrativeexamples.
Thesecondeditionofthisbookhasbeenexpandedwithboxespresentingbrief
summariesofthelivesandachievementsofthemajorfoundersandcontributorstothe
fieldof“QuantumFieldTheory”.Thebiographicaldetailscomplementthescientific
contentofthebookandcontextualizethediscoverieswithintheframeworkofglobal
researchinTheoreticalPhysics.Inmypersonalopinion,thisinformationisusefulfor
readersandlecturersalike.
Ekaterinburg,2019 M.V.Sadovskii
Contents
Preface|VII
1 Basicsofelementaryparticles|1
1.1 Fundamentalparticles|1
1.1.1 Fermions|2
1.1.2 Vectorbosons|3
1.2 Fundamentalinteractions|5
1.3 TheStandardModelandperspectives|5
2 Lagrangeformalism.Symmetriesandgaugefields|9
2.1 Lagrangemechanicsofaparticle|9
2.2 Realscalarfield.Lagrangeequations|10
2.3 TheNoethertheorem|15
2.4 Complexscalarandelectromagneticfields|20
2.5 Yang–Millsfields|26
2.6 Thegeometryofgaugefields|33
2.7 Arealisticexample:chromodynamics|40
3 Canonicalquantization,symmetriesinquantumfieldtheory|43
3.1 Photons|43
3.1.1 Quantizationoftheelectromagneticfield|43
3.1.2 RemarksongaugeinvarianceandBosestatistics|48
3.1.3 VacuumfluctuationsandCasimireffect|51
3.2 Bosons|54
3.2.1 Scalarparticles|54
3.2.2 Trulyneutralparticles|59
3.2.3 CPT-transformations|62
3.2.4 Vectorbosons|66
3.3 Fermions|68
3.3.1 Three-dimensionalspinors|68
3.3.2 SpinorsoftheLorentzgroup|73
3.3.3 TheDiracequation|80
3.3.4 ThealgebraofDirac’smatrices|86
3.3.5 Planewaves|89
3.3.6 Spinandstatistics|91
3.3.7 C,P,T transformationsforfermions|93
3.3.8 Bilinearforms|95
3.3.9 Theneutrino|95
X | Contents
4 TheFeynmantheoryofpositronandelementaryquantum
electrodynamics|103
4.1 Nonrelativistictheory.Green’sfunctions|103
4.2 Relativistictheory|107
4.3 Momentumrepresentation|111
4.4 Theelectroninanexternalelectromagneticfield|113
4.5 Thetwo-particleproblem|119
5 Scatteringmatrix|125
5.1 Scatteringamplitude|125
5.2 Kinematicinvariants|129
5.3 Unitarity|132
6 Invariantperturbationtheory|135
6.1 SchroedingerandHeisenbergrepresentations|135
6.2 Interactionrepresentation|136
6.3 S-matrixexpansion|141
6.4 Feynmandiagramsforelectronscatteringinquantum
electrodynamics|148
6.5 Feynmandiagramsforphotonscattering|153
6.6 Electronpropagator|156
6.7 Thephotonpropagator|159
6.8 TheWicktheoremandgeneraldiagramrules|162
7 Exactpropagatorsandvertices|171
7.1 FieldoperatorsintheHeisenbergrepresentationandinteraction
representation|171
7.2 Theexactpropagatorofphotons|173
7.3 Theexactpropagatorofelectrons|179
7.4 Vertexparts|183
7.5 Dysonequations|186
7.6 Wardidentity|187
8 Someapplicationsofquantumelectrodynamics|191
8.1 Electronscatteringbystaticcharge:higher-ordercorrections|191
8.2 TheLambshiftandtheanomalousmagneticmoment|195
8.3 Renormalization–howitworks|200
8.4 “Running”thecouplingconstant|204
+ −
8.5 Annihilationofe e intohadrons.Proofoftheexistenceof
quarks|206
8.6 Thephysicalconditionsforrenormalization|207
8.7 Theclassificationandeliminationofdivergences|211