Real-Time Thermal Propagators and the QED Effectiv

arXiv:hep-ph/9312226v27Dec1993NORDITA-93/78PG¨oteborgITP93-11hep-ph/9312226December1993Real-TimeThermalPropagatorsandtheQEDEffectiveActionforanExternalMagneticFieldPerElmfors,1,aDavidPersson2,bandBo-StureSkagerstam3,b,caNORDITA,Blegdamsvej17,DK-2100CopenhagenØ,DenmarkbInstituteofTheoreticalPhysics,ChalmersUniversityofTechnologyandUniversityofG¨oteborg,S-41296G¨oteborg,SwedencUniversityofKalmar,Box905,S-39129Kalmar,SwedenAbstractThethermalaveragedreal-timepropagatorofaDiracfermioninastaticuniformmagneticfieldBisderived.Atnon-zerochemicalpotentialandtem-peraturewefindexplicitlytheeffectiveactionforthemagneticfield,whichisshowntobecloselyrelatedtotheHelmholzfreeenergyofarelativisticfermiongas,anditexhibitstheexpecteddeHaas–vanAlphenoscillations.Aneffec-tiveQEDcouplingconstantatfinitetemperatureanddensityisderived,andcomparedwithrenormalizationgroupresults.Wediscusssomeastrophysicalimplicationsofourresults.1Emailaddress:[e-mail].2Emailaddress:[e-mail].3Emailaddress:[e-mail]edishNationalResearchCouncilundercontractno.8244-311.1IntroductionLargemagneticfieldsBcanbeassociatedwithcertaincompactastrophysicalobjectslikesupernovae[1,2]whereB=O(1010)T,neutronstars[3,4]whereB=O(108)T,orwhitemagneticdwarfs[4,5]inwhichcaseB=O(104)T.(AsareferencetheelectronmassinunitsofTeslaism2e/e=4.414·109T.)Ithasrecentlybeenarguedthataplasmaatthermalequilibriumcansustainfluctuationsoftheelectromagneticfields.Inparticular,fortheprimordialBig-BangplasmatheamplitudeofmagneticfieldfluctuationsatthetimeoftheprimordialnucleosynthesiscanbeaslargeasB=O(1010)T[6].Furthermore,amodelforextragalacticgammaburstsintermsofmergersofmassivebinarystarssuggestsmagneticfieldsuptotheorderB=O(1013)T[7].Amorespeculativesystemwhereevenlargermacroscopicmagneticfieldscanbecontemplatedaresuperconductingstrings[8].HereonemayconceivefieldsaslargeasB∼O(1014)T.IthasalsorecentlybeensuggestedthatduetogradientsintheHiggsfieldduringtheelectroweakphasetransitionintheearlyuniverseverylargemagneticfields,B=O(1019)T,maybegenerated[9].Ifoneencountersmagneticfieldsofthisorderofmagnitudethecompleteelectroweakmodelhastobeconsideredandtheconceptofelectroweakmagnetismbecomesimportant(forarecentaccountseee.g.Ref.[10]).Inthepresentpaperweconsider,however,magneticfieldssuchthatcalculationswithinQEDaresufficient.Ashorterversionofthisreporthasbeenpublishedelsewhere[11].Inmanyofthesesystemsonehastoconsidertheeffectsofathermalenvironmentandafinitechemicalpotential.InthispaperwederivetheappropriateeffectivefermionpropagatorandtheeffectiveactioninQEDforathermalenvironmenttreatedexactlyintheexternalconstantmagneticfieldbutwithnovirtualphotonspresent,i.e.weconsidertheweakcouplinglimit.CalculationsoftheQEDeffectiveLagrangiandensityinanexternalfieldhavebeenattemptedbeforeeitheratfinitetemperature[12,13]oratfinitechemicalpotential[14].Inthelattercase[14]theeffectiveactionisnotcompletebutthecorrectformispresentedhere.Atfinitechemicalpotentialandforsufficientlysmalltemperatures,theQEDef-fectiveactionshouldexhibitacertainperiodicdependenceoftheexternalfield,i.e.thewell-knowndeHaas–vanAlphenoscillationsincondensedmatterphysics.ThiswasnotobtainedinRef.[14].Elsewhere,theradiativecorrectionstotheanomalousmagneticmomenthasbeenestimatedinthepresenceoflargemagneticfieldsanditwasarguedthattheyareextremelysmall[15,16].1BymakinguseoftheeffectiveactionwederivetheeffectiveQEDcouplingasafunctionoftheexternalfield,thechemicalpotentialandtemperature.Inafuturepublicationwewilldiscussthefermionself-energyandradiativecorrectionstotheelectronsanomalousmagneticmoment,intermsoftheformalismderivedhere[17].2ThermalpropagatorsintheFurrypictureWeconsiderDiracfermionsinthepresenceofanexternalstaticfieldasdescribedbythevectorpotentialAμ.UsingstaticenergysolutionswemayrepresentthesecondquantizedfermionfieldintheFurrypicture[18].ItisgivenbyΨ(x,t)=Xλ,κbλκψ(+)λκ(x,t)+d†λκψ(−)λκ(x,t),(2.1)whereλisapolarizationindex,κdenotestheenergyandmomentum(orother)quantumnumbers(discreteand/orcontinuous)neededinordertocompletelycharacterizethesolutions,and(±)denotespositiveandnegativeenergysolutionsofthecorrespondingDiracequation,(i6D−m)ψ(±)λκ(x,t)=0,(2.2)whereDμ=∂μ+ieAμisthecovariantderivative.Thecreationandannihilationoperatorssatisfythecanonicalanti-commutationrelations{dλ′κ′,d†λκ}=δλ′λδκ′κ={bλ′κ′,b†λκ},(2.3)whileotheranti-commutatorsarezero.ThecompletenessrelationXλ,κψ(+)†λκ,a(x′,t)ψ(+)λκ,b(x,t)+ψ(−)†λκ,a(x′,t)ψ(−)λκ,b(x,t)=δabδ3(x′−x),(2.4)whereψ(±)λκ,adenotesthea-componentoftheDiracspinorψ(±)λκ,leadstothecanonicalanti-commutationrelationsforthefields{Ψa(x′,t),Ψ†b(x,t)}=δabδ3(x′−x).(2.5)Invacuum,thefermionpropagatoriSF(x′;x|m)isdefinedbyiSF(x′;x|m)=h0|TΨ(x′,t′)Ψ(x,t)|0i=θ(t′−t)Xλκψ(+)λκ(x′,t′)ψ(+)λκ(x,t)−θ(t−t′)Xλκψ(−)λκ(x′,t′)ψ(−)λκ(x,t),(2.6)2wheretheconjugatedspinorψ(±)λκisgivenbyψ(±)λκ=(ψ(±)λκ)†γ0.Sinceψ(±)λκ(x,t)satisfiestheDiracequation,onlythetimederivativeactingonthestepfunctionsgivesanon-zerocontribution,soonefindsthat(i6D−m)SF(x′;x|m)=11·δ4(x′−x).(2.7)Thereal-timepropagatoratfinitetemperatureTandchemicalpotent