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arXiv:math/0605709v1[math.DG]28May2006ANOTEONTHESTANDARDMODELINAGRAVITATIONFIELD.R.A.SharipovAbstract.TheStandardModelofelementaryparticlesisatheoryunifyingthreeofthefourbasicforcesoftheNature:electromagnetic,weak,andstronginteractions.InthispaperweconsidertheStandardModelinthepresenceofaclassical(non-quantized)gravitationfieldandapplyabundleapproachfordescribingit.1.FermionfieldsoftheStandardModel.FermionfieldsoftheStandardModelaresubdividedintotwoparts:leptonfieldsandquarkfields.Leptonfieldsaresubdividedintothreegenerations.Thefirstgenerationisrepresentedbyanelectroneandanelectronicneutrinoνe,thesecondgenerationisrepresentedbyamuonμanditsneutrinoνμ,andthethirdgenerationisrepresentedbyatauonτanditsneutrinoντ.1-stgeneration2-ndgeneration3-rdgeneratione-neutrinoνeμ-neutrinoνμτ-neutrinoντelectronemuonμtauonτ(1.1)Inasimilarway,quarksaresubdividedintothreegenerations.Theyarerepresentedinthefollowingtablesimilartotheabovetable(1.1):1-stgeneration2-ndgeneration3-rdgenerationup-quarkucharm-quarkctop-quarktdown-quarkdstrange-quarksbottom-quarkb(1.2)Leptonsparticipateinelectromagneticandweakinteractions.TheseinteractionsaredescribedbytheU(1)×SU(2)symmetrywhichisspontaneouslybrokenaccord-ingtotheHiggsmechanism.Moreover,theybreakthechiralsymmetryonthelevelofDiracspinors.Thissymmetryisoftencalledtheleft-to-rightsymmetry,butweprefertosaythechiral-to-antichiralsymmetryorsimplythechiralsymmetry(seesomedetailsin[1]).Wedistinguishbetweenleptonwavefunctionsbymensofthegenerationindexenclosedintosquarebrackets:ψ[e],ψ[μ],ψ[τ].(1.3)2000MathematicsSubjectClassification.81T20,81V05,81V10,81V15,81V17,53A45.TypesetbyAMS-TEX2R.A.SHARIPOVThewavefunctions(1.3)havechiralandantichiralconstituentparts.ChiralpartsaredoubletswithrespecttoSU(2)symmetry:•ψaα111[e],•ψaα111[μ],•ψaα111[τ].(1.4)Theindicesin(1.4)meansthatwetakethespace-timemanifoldMequippedtheappropriatecomplexvectorbundlesDM,UM,andSUM(see[2]).Theindexa=1,2,3,4in(1.4)isaspinorindexassociatedwiththeDiracbundleDMor,moreprecisely,withsomeframe(U,Ψ1,Ψ2,Ψ3,Ψ4)ofDM.Theindexα=1,2isassociatedwithsomeframe(U,Ψ1,Ψ2)ofthetwo-dimensionalcomplexbundleSUM.Duetothepresenceofthisindexthewavefunctions(1.4)aresaidtobeSU(2)-doublets.Threelowerindicesin(1.4)arealwaysequaltounitybecausetheyareassociatedwithsomeframe(U,Ψ1)oftheone-dimensionalbundleUM.Theantichiralpartsofthewavefunctions(1.3)areSU(2)-singlets.Theircom-ponentshaveonespinorindexaandsixU(1)indicesequalto1:◦ψa111111[e],◦ψa111111[μ],◦ψa111111[τ].(1.5)Byusualconvention(see[3])antichiral(right)neutrinosarenotconsidered.Inthispaperwefollowthisconvention,thoughinsomepapersrightneutrinosareintroduced,e.g.in[4].Thewavefunctions(1.4)and(1.5)arechiralandantichiralinthesenseofthefollowingequalitiesrelatingthemwiththecomponentsofthechiralityoperatorH:4Xa=1Hba•ψaα111[q]=•ψbα111[e],4Xa=1Hba◦ψa111111[q]=−◦ψb111111[q].(1.6)Hereq=e,μ,τisthegenerationindex.ThechiralityoperatorHisanattributeoftheDiracbundle(seedetailsin[1]).Inphysicalliterature(see[3]asanexample),whentheflatMinkowskispaceistakenforthespace-timemanifoldM,thechiralityoperatorHisrepresentedbytheDiracmatrixγ5Hba=1000010000−10000−1=γ5=iγ0γ1γ2γ3.(1.7)OtherDiracγ-matricesaregivenbythefollowingformulas(see(1.13)in[5]):γb0a=0010000110000100,γb1a=000−100−1001001000,(1.8)γb2a=000i00−i00−i00i000,γb3a=00−10000110000−100.ANOTEONTHESTANDARDMODEL...3BymeansofthechiralityoperatorHwedefinetwoprojectionoperators:•H=id+H2,◦H=id−H2.(1.9)Hereidistheidentityoperator.Therefore,thecomponentsoftheprojectionoperators(1.9)aregivenbytheformulas•Hba=δba+Hba2,◦Hba=δba−Hba2.(1.10)Bymeansof(1.10)theequalities(1.6)arewrittenasfollows:4Xa=1◦Hba•ψaα111[q]=0,4Xa=1•Hba◦ψa111111[q]=0.(1.11)Theindicesaandbin(1.6),(1.7),(1.8),(1.9),(1.10),and(1.11)arespinorindices.TheyareassociatedwiththeDiracbundleDM.ThethirdindexoftheDiracma-tricesrepresentedbythenumbers0,1,2,3in(1.8)isaspacialindex,itisassociatedwiththetangentbundleTM.Likeinthecaseofleptons,quarkwavefunctionsaresubdividedintothreegen-erationsaccordingtothegenerationtable(1.2):ψ[1],ψ[2],ψ[3].(1.12)However,nowweuseanumericindexforgenerations,sinceψ[1]describesbothanup-quarkandadown-quark.Similarly,ψ[2]describesacharm-quarktogetherwithastrange-quarkandψ[3]describesatop-quarktogetherwithabottom-quark.Chiralandantichiralpartsofthewavefunctions(1.12)behavedifferentlywithrespecttotheSU(2)symmetry.ChiralpartsareSU(2)-doublets:•ψa1αβ[1],•ψa1αβ[2],•ψa1αβ[3].(1.13)Notethatin(1.13),incontrastto(1.4),wehaveonemoreindex.Theadditionalindexβ=1,2,3isresponsibleforcolor,itdescribesstronginteractionsofquarks.Forantichiralpartsofthewavefunctions(1.12)theindexαisomitted:◦ψa1111β[u],◦ψa1111β[c],◦ψa1111β[t],(1.14)◦ψaβ11[d],◦ψaβ11[s],◦ψaβ11[b].TheyareSU(2)-singlets.Thewavefunctions(1.13)and(1.14)arechiralandan-tichiralinthesenseofthefollowingequalities:4Xa=1Hba•ψa1αβ[q]=•ψb1αβ[q],q=1,2,3;4Xa=1Hba◦ψa1111β[q]=−◦ψb1111β[q],q=u,c,t;(1.15)4Xa=1Hba◦ψaβ11[q]=−◦ψbβ11[q],q=d,s,b.4R.A.SHARIPOVTheequalities(1.15)areanalogousto(1.6).Intermsoftheprojectionoperatorsintroducedbytheformulas(1.9)theyarerewrittenas4Xa=1◦Hba•ψa1αβ
本文标题:A note on the Standard Model in a gravitation fiel
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