ElementaryParticlePhysics
TheStandardTheory
J.IliopoulosandT.N.Tomaras
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ToAlexander,AvraandJiarui
5.6TheLorentzandPoincaréGroups
5.6.1TheLorentzgroup
5.6.2ThePoincarégroup
5.7TheSpaceofPhysicalStates
5.7.1Introduction
5.7.2Particlestates
5.7.3TheFockspace
5.7.4Actionofinternalsymmetrytransformationson F
5.7.5ActionofPoincarétransformationson F
5.8Problems
6ParticlePhysicsPhenomenology
6.3 β-DecayandtheNeutrino
6.41932:TheFirstTableofElementaryParticles
6.4.1Everythingissimple
6.4.2Conservationlaws–baryonandleptonnumber
6.4.3Thefourfundamentalinteractions
6.5HeisenbergandtheSymmetriesofNuclearForces
6.6FermiandtheWeakInteractions
6.7TheMuonandthePion
6.8FromCosmicRaystoParticleAccelerators
6.8.1Introduction
6.8.2Electrostaticaccelerators
6.8.3Linearaccelerators
6.8.4Cyclotrons
6.8.5Colliders
6.9TheDetectors
6.9.1Introduction
6.9.2Bubblechambers
6.9.3Counters
6.10NewElementaryParticlesandNewQuantumNumbers
6.10.1Unstableparticles
6.10.2Resonances
6.10.3 SU (2) asaclassificationgroupforhadrons
6.10.4Strangeparticles
6.11TheEightfoldWayandtheQuarks
6.11.1From SU (2) to SU (3)
6.11.2Thearrivalofthequarks
6.11.3Thebreakingof SU (3)
6.11.4Colour
6.12ThePresentTableofElementaryParticles
6.13Problems
7RelativisticWaveEquations
7.1Introduction
7.2TheKlein–GordonEquation
7.2.1TheGreen’sfunctions
7.2.2Generalisations
7.3TheDiracEquation
7.3.1Introduction
7.3.2WeylandMajoranaequations
7.3.3TheDiracequation
7.3.4The γ matrices
7.3.5Theconjugateequation
7.3.6Thestandardrepresentation
7.3.7Lagrangian,HamiltonianandGreenfunctions
7.3.8Theplanewavesolutions
7.4RelativisticEquationsforVectorFields
7.4.1TheMaxwellfield
7.4.2Massivespin-1field
7.4.3Planewavesolutions
7.5Problems
8TowardsaRelativisticQuantumMechanics
8.1Introduction
8.2TheKlein–GordonEquation
8.3TheDiracEquation
8.3.1Introduction
8.3.2Theconservedcurrent
8.3.3Thecouplingwiththeelectromagneticfield
8.3.4Thenon-relativisticlimitoftheDiracequation
8.3.5Negativeenergysolutions
8.3.6Chargeconjugation
8.3.7 CPT symmetry
8.3.8Chirality
8.3.9Hydrogenoidsystems
8.4Problems
9FromClassicaltoQuantumMechanics
9.1Introduction
9.2.1TheFeynmanpostulate
9.2.2Recoveringquantummechanics
9.3Problems
10FromClassicaltoQuantum Fields.FreeFields
10.2TheKlein–GordonField
11InteractingFields
CPT theorem
CPT andtheSpin-Statistics
11.5.1TheconfigurationspaceFeynmanrules
12ScatteringinQuantumFieldTheory
12.1TheAsymptoticTheory
12.1.1Theasymptoticstates
12.1.2Theasymptoticfields
12.2TheReductionFormula
12.3TheFeynmanRulesfortheScatteringAmplitude
12.5.1TheelectronComptonscattering
12.5.2Theelectron–positronannihilationintoamuonpair
12.5.3Thechargedpiondecayrate
13GaugeInteractions
13.1TheAbelianCase
13.2Non-AbelianGaugeInvarianceandYang–MillsTheories
17.3.1Tree-level(O(α2))electronCoulombscattering
17.3.3Infrareddivergenceofthephotonemissionamplitude
17.3.4The O(α3) measuredcrosssectionisinfraredfinite
17.5.1TheFaddeev-Kulishdressedelectronstates
17.5.2Coulombscatteringofadressedelectron
V
18.6.1Muondecay
18.6.2Otherpurelyleptonicweakprocesses
18.7Semi-LeptonicInteractions
18.7.1Strangenessconservingsemi-leptonicweak interactions
18.7.2Strangenessviolatingsemi-leptonicweakinteractions
19AGaugeTheoryfortheWeakandElectromagnetic Interactions
20.1.2Gargamelleandtheneutralcurrents
20.1.3Veryhighenergyneutrinobeams
20.2.1Introduction
20.2.2Experimentalevidenceforneutrinooscillations
20.2.3Theneutrinomassmatrix
20.2.4Neutrinolessdouble β decay
21.4.1Quantumchromodynamicsinperturbationtheory
21.4.2Quantumchromodynamicsandhadronicphysics
22TheStandardModelandExperiment
22.2.1Experimentaldiscoveryofcharmedparticles
22.2.2Stochasticcoolingandthediscoveryof W ± and Z
22.2.3Thediscoveryoftheheavyquarksandofthe Brout–Englert–Higgsboson
22.3HadronSpectroscopy
22.4TheCabibbo–Kobayashi–MaskawaMatrixand
Introduction
Thequestionofthemicroscopiccompositionofmatterhaspreoccupiednatural philosopherssinceatleastasfarbackasthe5thcenturyBC.The atomichypothesis, i.e.theexistenceof“elementary”constituents,isattributedtoDemocritusofAbdera (c.460–c.370BC).Determiningthenatureoftheseconstituentsandunderstanding theinteractionsamongthemisthesubjectofabranchoffundamentalphysicscalled thephysicsofelementaryparticles.Itisthesubjectofthisbook.Inthepastfew decadesthisfieldhasgonethrougha“phasetransition”.Atfirstsightthecriticaltime wassomewherebetweenthelate1960sandtheearly1970swhenthetheory,which becameknownas theStandardModel,wasfullyformulated.However,thisisonly partofthestoryandtheterm“phasetransition”maybemisleading.Ittookseveral decadesofintenseeffortbyexperimentalistsandtheorists,bothbeforeandafterthe “criticalpoint”,forthevariousingredientsofthisModeltobediscoveredandfor itsmainpredictionstobecomputedtheoreticallyandverifiedexperimentally.The agreementhasbeenimpressivetotheextentthatweshouldnomoretalkabout the StandardModel,butrather theStandardTheory.
Thistransitionhasbroughtprofoundchangesinourwayofthinkingandunderstandingthenatureofthefundamentalforces.Thechangesaresubtleandacasual observermaymissthem.SuperficiallytheStandardModelisnotfundamentallydifferentfromothermodelsthatpeoplehaveconsideredformanyyears.Theyareall basedonrelativisticquantummechanicsor,asitisusuallycalled,quantumfieldtheory.Thishasbeenthelanguageofelementaryparticlephysicssincetheearly1930s, whenEnricoFermiintroducedthenotionofaquantumfieldassociatedwithevery elementaryparticle.ThemainnewelementbroughtbytheStandardModelconcerns thenatureoftheinteractionsbetweenthesevariousquantumfields.Intheolddays theinteractionswerechosenonpurelyphenomenologicalgrounds,likethevariouspotentialfunctions V (x) usedinnon-relativisticquantummechanics.Thenewvision broughtbytheStandardModelisbasedongeometry:theinteractionsarerequired tosatisfyacertaingeometricalprinciple.Inthephysicists’jargonthisprincipleis called gaugeinvariance;inmathematicsitisabranchofdifferentialgeometry.This “geometrisation”ofphysicsisthemainlegacyoftheStandardModel.1
Itisthepurposeofthisbooktopresentandexplainthismodernviewpointto areadershipofwell-motivatedundergraduatestudents.Itisourimpressionthat,althoughtheStandardModeliswellestablishedinelementaryparticlephysicsandis
1
widelyused,itsunderlyingprinciplesarenoteasilyfoundinbooksthatundergraduatestudentsusuallyread.Ourambitionistoshowthatthistheoryismorethan anefficientwaytocomputescatteringamplitudesatlowestorderintheperturbation expansion.Thesubjectswecoverandthewaywechoosetopresentthemaredictated bythisgoal.Webelievethatitistimetointroduceundergraduatestudentstothese newconceptsandmethods.Andwemeanphysicsconcepts.Mathematicswillbeintroducedonlywhenitisabsolutelyneeded.Theemphasiswillbeonthetheoretical aspects,andthischoiceispartlyduetoourowncompetenceandpartlytolimitations ofspace.Agoodexpositionofexperimentaltechniqueswouldtakeasecondvolume.
Theplanofthebookisasfollows:WestartwithapresentationofDirac’stheoryof spontaneousemissioninatomicphysics.Thismakesitpossibletointroduceconcepts suchasthequantisedradiationfield,canonicalcommutationrelations,creationand annihilationoperators,gaugeinvarianceandgaugefixing,butalsotransitionprobabilitiesandFermi’sgoldenrule,andalltheseinaconcreteandwell-definedphysical context.IndeedtheproblemwewerefacingwasthatpresentingtheStandardModel requiredlengthychaptersofpureformalismbeforerealphysicsquestionscouldbe addressed.
Inthefollowingtwochapterswerecallsomeresultsfromclassicalfieldtheoryand weintroducethescatteringformalisminnon-relativisticquantummechanics.Chapter 5 presentsanelementaryintroductiontothetheoryofLiegroupsandLiealgebras, includingLorentzandPoincaré.Thestudentwhohasattendedacourseongroup theorymaygoveryfastthroughit.
Chapter6constitutestheintroductiontothephysicsofelementaryparticles.It isphenomenologicalandfollows,toacertainextent,thehistoricalevolutionofthe fieldduringthetwentiethcentury.Historyisnotourprimarygoal,butwebelieve thatithelpsinunderstandingthebirthanddevelopmentofnewideas.Thefollowing twochapterspresent,inasystematicway,theclassicalrelativisticwaveequationsfor fieldsoflowspinandtheattemptstousetheminordertobuildarelativisticoneparticlequantummechanics.Wederivethewellknownresultthatalltheseattempts pointunambiguouslytoasystemwithaninfinitenumberofdegreesoffreedom,i.e. toquantumfieldtheory.
Goingfromaclassicalfieldtheorytoitsquantumdescendantistheobjectof Chapters 9–12.WedecidedtodoitusingFeynman’spathintegralmethod.Several reasonsmadeuschoosethisapproach,althoughitissomewhatunorthodoxforan undergraduatetextbook.First,webelievethatthesumoverhistories,withitsrelationstostochasticprocesses,offersamoreprofoundvisionofthequantumworld. Second,itisbyfarthemostpracticalwaytoobtainthequantumtheoryofanonlinearconstrainedsystem,suchasaYang–Millstheory.Third,andveryimportant, itofferstheonlyquantisationmethodthatisnotrestrictedtotheperturbationexpansion.Thepathintegralformulationdoesnotassumethatthecouplingisweak.In fact,appropriatelytruncatedonaspace-timelattice,itbecomessuitablefornumerical simulationsinthestrongcouplingregime.Non-perturbativeresultsfromlatticesimulationshavealreadyreachedaremarkableprecision,andtheiragreementwiththe observedhadronicspectrumisimpressive.Furthermore,inthecomingyears,withthe continuingriseincomputingpower,theimportanceofthesecalculationsispredicted toincreaseaccordingly.
Asystematicintroductiontogaugeinvarianttheoriesandthephenomenonof spontaneoussymmetrybreakingarethesubjectsofChapters 13 and 14.Wewant toemphasisetheconceptualstepinvolvedintheintroductionofgaugeinvarianttheories.Whatwephysicistscall“gaugefields”arenotlikeanyotherfield.Thecorrect mathematicaldescriptionisgivenbydifferentialgeometry,butwepresentasimplifiedversionbasedonaformulationonaspace-timelattice.Latticefieldtheoryispoor man’sdifferentialgeometry.
AbookonparticlephysicstodaycannotbelimitedtothecalculationoftreelevelFeynmandiagrams.Theprecisionoftheexperimentsissuchthatameaningful comparisonrequiresustotakeintoaccounttheeffectsofhigherorders.Thiscannot bedoneconsistentlywithoutsomenotionsfromthetheoryofrenormalisationandthe renormalisationgroup.TheycanbefoundinChapter 15.Asimpletreatmentofthe infrareddivergencesassociatedwithmasslessparticlesisgiveninChapter 17
Chapters 16 to 21 presenttheStandardModel.Theycontainaone-loopcalculation oftheelectrongyromagneticratioinquantumelectrodynamics,thephenomenology oftheweakinteractions,the SU (2) × U (1) electroweakgaugetheoryandquantumchromodynamics.Adiscussionofneutrinophysics,withthepoorlyunderstood phenomenonofneutrinooscillations,canbefoundinChapter 20
WeendwithChapter 22,whichoffersapanoramaofthecomparisonoftheStandardModeltheoreticalpredictionswithexperimentalmeasurements,includingthe mostrecentresults.WhentheLargeHadronCollider(LHC)startedoperatingin2008 wewereallexpectingnewphysicstobearoundthecorner.Today,morethanadecade later,wemustadmitthatnocornerhasbeenfound.Thereasonswhywestillbelieve thattheremustbephysicsbeyondtheStandardModelarebrieflyexposedinthelast chapter.Finally,inanappendixweexplainthenotationweareusingandpresenta collectionofsomeusefulformulae.
Oneofus(J.I.)hasrecentlyco-authoredabookonquantumfieldtheory.2 Although thescopeandlevelofthispresentbookaredifferent,thereissomeoverlapinchapters thatarecommontoboth.
2“FromClassicaltoQuantumFields”,byLaurentBaulieu,J.I.andRolandSénéor,Oxford UniversityPress2017.
QuantisationoftheElectromagnetic FieldandSpontaneous PhotonEmission
2.1 Introduction
Thefirstgreatsuccessofquantummechanicswastheaccuratedescriptionofatomic spectra.However,thisverysuccessalsoshoweditslimitations.Indeed,bysolvingthe Schrödingerequation,wefindtheeigenstatesoftheHamiltonianwhichcorrespondto thestationarystates.Itfollowsthatalllevelsshoulddescribestablestatesoftheatom. Ontheotherhand,weknowexperimentally,thatonlythegroundstateisstable.All excitedstatesdecaybytheemissionofoneormorequantaofradiation–photons. Hereweshallanalysethesimplestcase
A(n) → A(0) + γ (2.1) inwhichthetransitiontothegroundstateisasingle-stepprocessaccompaniedby theemissionofonephoton. A(n) representstheatominthe n-thexcitedstateand A(0) thesameatominthegroundstate.Thephenomenonisknownas spontaneous emissionofradiation,anditisnotdescribedbytheSchrödingerequation.Itwasthe needtocomputetherateofsuchdecaysthatpromptedDiracin1927todevelopand usethequantumdescriptionoftheelectromagneticfield.
Inquantummechanicsthetimeevolutionofaphysicalsystemisgivenbythe operator U (t,t0)=e iH(t t0),where H istheHamiltonian.So,wemustfirstfindthe Hamiltonianwhich,whenappliedtothestate A(n),canyieldtheatominitsground state and aphoton.Inotherwords,thisHamiltonianshouldhavetermswithnon-zero matrixelementsbetweenstatescontainingdifferentnumbersofparticles.Sincethis willturnouttobeacentralthemeinoureffortstodescribethephenomenaweobserve inparticlephysicsexperiments,itwillbeuseful,asawarm-upexercise,tostartwith thisproblemofatomicphysics.Itwillallowustointroduce,inawell-definedphysical context,severalconceptsthatwillbeessentiallater.
2.2 ThePrincipleofCanonicalQuantisation
InthefollowingweshallfollowDiracandbuildupaformalism,whichwillmakeit possibleforparticlestobecreated,orabsorbed,asaresultoftheinteraction.This formalismwillturnouttobethatofaquantisedfieldtheoryandwewillstudyfirst thequantumtheoryoftheelectromagneticfield.
Letusstartwithabriefreminderoftheprincipleofcanonicalquantisation.Itis basedontheknowledgeofthephysicalsystemattheclassicallevel.Letusconsider,as anexample,asystemwithonedegreeoffreedom.Inclassicalmechanicsitisdescribed byageneralisedcoordinate q(t) anditscanonicalconjugatemomentum p(t).Thesocalled“canonicalquantisation”ofthissystemisgiven,bydefinition,bytheprescription accordingtowhich q and p arepromotedtooperators,actinginacertainHilbertspace, andsatisfyingtheequaltimecommutationrule1
wherewehaveusedthenotation ˆ A todenotetheoperatorcorrespondingtotheclassicalquantity A 2 Werecovertheclassicaltheoryinthelimit h → 0,inwhichthe operatorsbecomecommutingvariables.
Thisprescriptiontopassfromaclassicaltoaquantumsystemgeneraliseseasily to n degreesoffreedom.Thecommutationrelation(2.2)becomes
Aspecialcaseofasystemgivenby(2.3)consistsof α =1, 2,...,s degreesoffreedom livingoneachsiteofaspacelatticewith N pointsandlatticespacing a.Inthatcase theindex I isnaturallydenotedas I = {i,α} with i =1, 2,...,N labellingthelattice pointsand n = sN .For s =1,inparticular,theindices i and j denoteboththelattice pointsandthevariableateachpoint.Wehavestudiedsuchsystemsinstatistical mechanicswhereweoftenconsideredthelimit N →∞.Wecanalsoconsideran appropriatedoublelimit N →∞ and a → 0,inwhich i becomesthelabel x ofpoints ofthespatialcontinuumand qI (t) → qx,α(t),whichweshallwriteconvenientlyas qα(t, x).Wethusobtainasystemwith s continuousinfinitiesofdegreesoffreedom.In classicalphysicssuchasystemiscalleda classicalfield andthebestknownexample istheelectromagneticfield.
Iftheclassicaltheoryiswelldefinedandthecanonicalvariables qα(t, x) and pβ (t, x) correctlyidentified,thequantisationofsuchasystemis,inprinciple,straightforward: Thecommutationrelation(2.3)isreplacedby
withtheremainingequaltimecommutatorsoftwo q’s,aswellasoftwo p’sequalto zero.Wenoticethattheright-handsideoftherelations(2.4)isproportionaltothe Dirac δ-function,whichisnota“function”intheusualsenseoftheword.Inparticular, thesquare,oranypowerofit,cannotbedefined.Inmathematics,suchgeneralised functionsarecalled distributions.Itfollowsthattheoperators ˆ q and ˆ p mustalsobe
1Intheusualformulationofnon-relativisticquantummechanics,theso-called Schrödingerrepresentation,states |Ψ(t)⟩ dependontime,andevolveaccordingto |Ψ(t)⟩ = U(t, 0) |Ψ(0)⟩ = exp( iHt) |Ψ(0)⟩,whileobservables A withoutexplicittimedependencearerepresentedbytimeindependentoperators AS .Inthecommutationrelation(2.2)weusedthe Heisenbergrepresentation,inwhichthestatesarefixed |Ψ(0)⟩ = U 1(t, 0) |Ψ(t)⟩ andtheobservables AH (t)= U 1(t, 0)AS U(t, 0) aretime-dependent.At t =0 thetwocoincide: AH (0)= AS
2Inordertosimplifytheformulaewewilloftendropthe“hat”ifthereisnodangerofconfusion.
6 2QuantisationoftheElectromagneticFieldandSpontaneousPhotonEmission
representedbydistributions,andweexpecttheirpowerstobeilldefined.Wesay that ˆ q and ˆ p arenot operatorvaluedfunctions ofthespacepoint,butbecomeinstead operatorvalueddistributions.Thisleadstosomemathematicaldifficulties,whichare commontoallquantumfieldtheoriesandwhichareonlypartiallymastered.Weshall comebacktothispointquiteofteninthisbook.
Aspromised,wenextapplythisprogrammetotheelectromagneticfieldandobtain thequantumdescendantofMaxwell’stheory.Itwillbeourfirstexampleofaquantum fieldtheoryandtheonlyonewhichhasawellknownclassicallimit.
2.3 TheQuantumTheoryofRadiation
2.3.1 Maxwell’stheoryasaclassicalfieldtheory
ThesimplestversionofMaxwell’sequationstakestheform3
where E(t, x) and B(t, x) aretheelectricandmagneticfields,respectively,while ρ(t, x) and j(t, x) aretheexternalelectricchargeandcurrentdensities.Consistencyof(2.5), (2.6),requiresthat ρ and j satisfythecontinuitycondition: ∂ρ/∂t+∇·j =0.According toourrecipe,ifwewanttodescribethissystemasaclassicalfieldtheory,wemustfirst identifyasetofindependentvariables qα(t, x).Thesecannotbethesixcomponents of E(t, x) and B(t, x) becausetheyareconstrainedbythefirsttwoequations(2.5). Weshouldsolvetheseconstraintsandeliminatetheredundantvariables.
ItseemsthatafirststepinthisdirectionwastakenbyGaussin1835,longbefore Maxwellwrotehisequations.Itconsistsofintroducingthevectorandscalarpotentials A(t, x) and ϕ(t, x).Itwillbeconvenienttouseacompactrelativisticnotationinwhich x =(t, x) andintroducethefour-vectors jµ(x)=(ρ, j) and Aµ(x)=(ϕ, A).Wethen constructthetwo-indexantisymmetrictensor
Sincethederivativeoperator ∂/∂xµ willappearveryoften,weintroduceashort-hand notationforit: ∂/∂xµ = ∂µ.Similarly, ∂/∂xµ = ∂µ.Theelectricandmagneticfields aregivenintermsof Fµν by
3Weusethesymbol ∧ todenotethevectorproductoftwothree-dimensionalvectors: (a ∧ b)i = ϵijkaj bk
2.3TheQuantumTheoryofRadiation 7
where ϵijk(= ϵijk) isthethree-indexcompletelyantisymmetrictensor,equalto +1 if {ijk} formanevenpermutationof{123}.4 Itisnoweasytocheckthatthetwo inhomogeneousequations(2.6)combineto
whilethetwohomogeneousones(2.5)areautomaticallysatisfied.Theequation(2.9) followsfromavariationalprincipleappliedtotheaction
Canwechoosethefourcomponentsof Aµ asindependentdynamicalvariables? Theanswerisno,becausetheLagrangian(2.10)doesnotcontainthetimederivative of A0;inotherwords,thecanonicalconjugatemomentumof A0 wouldbeidentically zero.WeknowthatthisproblemisrelatedtothefactthattheLagrangiandensity in(2.10)isinvariantunderthetransformation Aµ(x) → Aµ(x)+ ∂µθ(x) with θ(x) anarbitraryfunctionof x.Inclassicalelectrodynamicswecallthisinvariance“gauge invariance”andwecanusethefreedomofchoosingaparticularfunction θ toreducethe numberofindependentvariables.InChapter 13 wewillstudythisprobleminamore generalcontext,butherewejustrecallthat,experimentally,anelectromagneticwave inemptyspacehasonlytransversedegreesofpolarisation.Therefore,wecanimpose thetransversalitycondition ∇ A(x)=0 (theso-called“Coulombgaugecondition”) underwhichthezerocomponentofthevectorpotentialsatisfies5
where ∆ istheLaplacian.ThisimpliestheCoulomblaw(hencethenameofthis condition)
whichshowsthat A0 isentirelygivenbytheexternalsourceanditisnotanindependentdynamicaldegreeoffreedom.6 Weareleftwiththespatialcomponents,which areconstrainedbytheCoulombcondition.Thesimplestwaytosolveitandobtainan unconstrainedsystemistotakethethree-dimensionalFouriertransform
intermsofwhichtheconstraintbecomes
4Inthisformula,raisingandloweringtheindicesofthree-dimensionalvectorsisperformedusing theMinkowskimetric,asweexplaininAppendix A
5AnotherchoiceistheLorentzcovariantgaugeconditionoftheform ∂µAµ(x)=0,theso-called “Lorenzgaugecondition”,firstintroducedbytheDaneLudvigLorenzin1867.
6Unlessnotedotherwise,wewillassumethatboththesourcesandthedynamicalfieldsvanishat infinity.
whichsuggeststochooseanorthonormalsystemofunitvectors ϵ(3)(k)= k/|k| and ϵ(λ)(k), λ =1, 2,satisfying
i.e. ϵ(3)(k) isparalleltothewavevectorandtheothertwoaretransversetoit.Becauseofthegaugecondition,inthisframethevectorpotentialhasonlytransverse components
inagreementwiththeexperimentalfactthattheelectromagneticwavesinemptyspace aretransverse.
Letussummarise:Formulatingthetheoryintermsofthevectorpotential Aµ, andmakingfulluseofitsgaugeinvariance,allowedustoshowthat,outofthesix componentsof E and B,onlytwoarereallyindependent,asexpectedfromtheknown propertiesofelectromagneticradiation.Thisresult,which,asweshallprovelater, followsfromgeneralinvarianceprinciplesofthetheory,madeitpossibletoformulate electromagnetismasadynamicalsystem.Foreach {k,λ}, A(λ)(t, k) isanindependent variable.TheassociatedcanonicalmomentumcanbecomputedfromtheLagrangian (2.10).Wefind
wherewehaveusedthestandardnotationofmechanicsinwhich“dot”meansderivativewithrespecttotime.
Intheabsenceofexternalsources,thevectorpotential A(x) intheCoulombgauge satisfiesthewaveequation
If A(k) isthefour-dimensionalFouriertransformof A(x),thewaveequation(2.18) becomesanalgebraicequation: k2A(k)=0,whichimpliesthat A(k)= F(k2) δ(k2) with F(k2) arbitraryfunctionsof k2,providedtheyareregularat k2 =0
Thegeneralsolutionof(2.18)canbeexpandedinplanewaves,i.e.functions oftheform e ik x with k · x = k0t k · x,andwiththefourvector kµ satisfying k2 ≡ k2 0 k2 =0.Itwillbeconvenienttointroducethenotation
dΩm(k)= d3k (2π)32Ek = d4k (2π)4 (2π)δ(k2 m 2)θ(k0) ,Ek = k2 + m2 (2.19)
whichistheLorentzinvariantmeasureonthepositiveenergybranchofthemass hyperboloidgivenby k2 = m2.Wethuswrite
where dΩ0(k) isthevaluefor m =0 oftheexpression(2.19).Itfollowsthatthe integrationin(2.20)isovertheboundaryofthepositiveenergy k2 =0 cone.
Wecaninvert(2.20)toexpressthecoefficientfunctions a(λ)(k) as
InProblem 2.1 weaskthereadertoverifythattheenergy H andmomentum P of thefieldintermsofthecoefficientfunctionsare
2.3.2 Quantumtheoryofthefreeelectromagneticfield–photons
Wearenowreadytoapplyourgeneralquantisationprescriptionandobtainthecorrespondingquantumtheory.Thecanonicalvariables(2.17)arepromotedtooperators satisfyingthecanonicalcommutationrelations(2.4).7 [A(λ)(t, k), ˙ A (λ′)† (t, k’
A(λ)(t, k), A(λ′)(t, k’)]=0 , [ ˙ A (λ
where † denotestheHermitianadjointoftheoperator.Thepresenceofthefactor (2π)3 isduetoourconventionontheFouriertransform.Here,andthroughoutthisbook, wehaveadoptedthesystemofunitsweintroduceinAppendix A inwhich c =¯h =1. Physicalunitswillberestoredonlywhenitisnecessary. Therelations(2.23)implyfortheoperatorscorrespondingtothecoefficientsofthe planewaveexpansion(2.20)thecommutationrelations
a(λ)(k),a(λ′)†(k′)]= δλλ
Theserelationsarestillformalbecausewehavenotyetdefinedthespaceinwhich theseoperatorsact.Inordertodoitweremarkthat,foreveryfixedvalueof {k,λ}, theoperators a(λ)(k) and a(λ)†(k),appropriatelyrescaled,satisfythecommutation relationsofannihilationandcreationoperatorsforaharmonicoscillatorwithfrequency Ek = |k|.Inotherwords,wecaninterprettheelectromagneticfieldinthevacuum asadoubly(oneforeachvalueof λ)continuousinfinitesetofindependentharmonic
7Weomitthesymbol ∧ ontheoperators.
2QuantisationoftheElectromagneticFieldandSpontaneousPhotonEmission oscillators.Followingtheexampleoftheharmonicoscillator,wedefinethespaceof statesasfollows:
1.Firstweassumetheexistenceofastatewhichisannihilatedbyallannihilation operators a(λ)(k),forbothvaluesof λ andall k.Weassumethisstatetobeunique andnormalisedtoone.Wedenoteitby |0 andwecallit thevacuumstate.
2.Startingfromthisstatewebuildexcitedstatesbyapplyingthecreationoperators onit.Forexample,thefirstexcitedstatesaregivenby
Usingthecommutationrelations(2.24)andtheequation(2.25),whichdefinedthe vacuumstate,wefind
Ofcourse,weexpectthestate |k,λ ,sinceitisastatewithfixedmomentum,to correspondtoawavefunctiondescribedbyaplanewaveand,therefore,tobenonnormalisable.Thisisthemeaningofthe δ functiononther.h.s.ofequation(2.27).As wedidinquantummechanics,wecanbuildnormalisablestatesusingwavepackets.
Givenafunction Φ(k) satisfying
wedefinethewavepacket
whichisnormalisedtoone.
Intheterminologyoftheharmonicoscillator, |k,λ isthestateofoneexcitation oftype {k,λ}.Weseethat,foreachvalueof λ,thespaceoftheone-excitationstates isourfamiliarHilbertspaceofsquare-integrablecomplex-valuedfunctions,whichwe denoteby H1 8
Inasimilarwaywebuild“multi-excitation”statesbyactingonthevacuumwith productsofcreationoperators |k1,λ1; k2,λ2; ... ; kn,λn = a(λn)†
Again,foreachsetofvalues {λi,i =1,...,n},wedenoteby Hn,λ theHilbertspace ofstatesinthedirectproductof H1 withitself n times.Theentirespaceofstatesis
8Moreprecisely,thespaceweuseinquantummechanicsisaray-space.Let |ψ⟩ beavectorin H1 normalisedto1 ⟨ψ|ψ⟩ =1 and C ∈ C C =0.Aunitrayassociatedto |ψ⟩ isthesetofallvectorsof theform C |ψ⟩.Inquantummechanicswehavelearnedthatallvectorsinthissetrepresentthesame physicalstatebecausethewavefunctions Ψ(x) and CΨ(x) areidentified.
thedirectsumofall Hn’s,forall n andallsets {λ}.Wecallthisspace theFockspace ofstates
where,fornotationalsimplicity,wehavedefined H0 astheone-dimensionalspace spannedbythevacuumstate,i.e.theray-spaceofcomplexnumbers.
ThephysicalmeaningofthestatesintheFockspacebecomesmoretransparentif weexpresstheHamiltonianandthemomentumoperatorsofthesystemintermsof a and a†,toobtainthequantumversionsoftherelations(2.22).Itiseasytoseethat, sincecreationandannihilationoperatorsdonotcommute,weend-upwith
Wenextuse(2.32)and(2.33)tocomputetheenergyandmomentumofeachstate in F.Butherewefaceasubtleproblem:Letusstartwiththeenergyofthevacuum state.Asusual,itisgivenbytheexpectationvalueoftheHamiltonian 0| H |0 .Using thecommutationrelations(2.24),weobtain
Thetroublecomesfromthecommutatorwhich,formally,isproportionalto δ3(0) andis,therefore,meaningless.Itiseasytounderstandtheoriginofthisproblem,both physicallyandmathematically.Firstwiththephysics:Werecallthattheenergyof asingleharmonicoscillatorwithfrequency ω isexpressedas H = ω(a†a +1/2) and itsmeanvalueinthegroundstateisjust ω/2, thezero-pointenergy oftheharmonic oscillator.Itisaquantummechanicalphenomenonand,assuch,inphysicalunits itisproportionalto h.Wehavenotedalreadythatthequantumelectromagnetic fieldintheabsenceofsourcesisequivalenttoaninfinitesetofharmonicoscillators satisfyingtherelativisticdispersionlaw ω = |k|.So,itisnotsurprisingthattheground stateenergyisinfinite;itistheinfinitesumofthezero-pointenergies.Nowwiththe mathematics:Adivergentexpressionoftenresultsfromamathematicalmistake;some quantityisilldefined.Indeed,inwritingtheLagrangiandensity,ortheHamiltonian, oftheclassicalelectromagneticfield,weusedexpressionssuchasthoseinequation (2.10),whichcontainthesquareofthefield Fµν (x).Thisisfinefortheclassicaltheory. Inthequantumtheory,however,wehavenotedalreadythatthefieldvariablesare distributions,andtheirsquare,oranyhigherpower,isnotwelldefined.Atthisstage theproblemisanuisanceratherthanacatastrophe.Sinceweunderstanditsoriginwe
