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1、arXiv:math-ph/0703052v229Nov2007TheSquareoftheDiracandspin-DiracOperatorsonaRiemann-CartanSpace(time)∗E.A.Notte-CuelloDepartamentodeMatem´aticas,UniversidaddeLaSerena,Av.Cisterna1200,LaSerena-Chile.e-mail:enotte@userena.clW.A.RodriguesJr.InstituteofMathematics,StatisticsandScientificComputation,IMECC-UNICAMPCP6065,13083-859Campinas,SP,Brazil.e-mail:walrod@ime.unicamp.brQ.A.G.SouzaInstituteofMathematics,StatisticsandScientificComputation,IMECC-UNICAMPCP6065,13083-859Campinas,SP,Brazil.quin@ime.uni。
2、camp.brFebruary7,2008AbstractInthispaperweintroducetheDiracandspin-Diracoperatorsasso-ciatedtoaconnectiononRiemann-Cartanspace(time)andstandardDiracandspin-DiracoperatorsassociatedwithaLevi-CivitaconnectiononaRiemannian(Lorentzian)space(time)andcalculatethesquareoftheseoperators,whichplayanimportantroleinseveraltopicsofmodernMathematics,inparticularinthestudyofthegeometryofmodulispacesofaclassofblackholes,thegeometryofNS-5branesolutionsoftypeIIsupergravitytheoriesandBPSsolitonsinsomestringtheori。
3、es.Weob-tainageneralizedLichnerowiczformula,decompositionsoftheDiracandspin-DiracoperatorsandtheirsquaresintermsofthestandardDiracandspin-Diracoperatorsandusingthefactthatspinorfields(sectionsofaspin-Cliffordbundle)haverepresentativesintheCliffordbundlewepresentalsoanoticeablere-lationinvolvingthespin-DiracandtheDiracoperators.Keywords:Spin-Cliffordbundles,DiracOperator,LichnerowiczFormula∗toappear:ReportsonMathematicalPhysics60(1),135-157(2007).11IntroductionRecently,inseveralapplicationsoftheoreti。
4、calphysicsanddifferentialgeometry,inawayoranothertheDiracoperatoranditssquareonaRiemann-Cartanspace(time)hasbeenused.In,e.g.,[12]RapoportproposedtogiveaCliffordbundleapproachtohistheoryofgeneralizedBrownianmotion;in[1]AgricolaandFriedrichinvestigatetheholonomygroupofalinearmetricconnectionwithskew-symmetrictorsionandin[2]theyintroducedalsoanelliptic,second-orderoperatoractingonaspinorfield,andinthecaseofanaturallyreductivespacetheycalculatedtheCasimiroperatoroftheisometrygroup.Thesquareofthespin-Di。
5、racoperatoralsoappearsnaturallyinthestudyofthegeometryofmodulispacesofaclassofblackholes,thegeometryofNS-5branesolutionsoftypeIIsupergravitytheoriesandBPSsolitonsinsomestringtheories([5])andmanyotherimportanttopicsofmodernmathematics(see[3,6]).Someoftheworksjustquotedpresentextremelysophisticatedandreallycomplicatedcalculationsandsometimesevenerroneousones.ThisbringstomindthatasimpletheoryofDiracoperatorsandtheirsquaresactingonsectionsoftheCliffordandSpin-CliffordbundlesonRiemann-Cartanspace(times。
6、)hasbeenpresentedin[15],andfurtherdevelopedin[14].Usingthattheory,inSection2wefirstintroducethestandardDiracoperator∂|(as-sociatedwithaLevi-CivitaconnectionDofametricfieldg)actingonsectionsoftheCliffordbundleofdifferentialformsCℓ(M,g)andnext,insection2.1weintroducetheDiracoperator∂(associatedwithanarbitraryconnection∇)andalsoactingonsectionsofCℓ(M,g).Next,wecalculateinSection2.2inasimpleanddirectwaythesquareoftheDiracoperatoronRiemann-Cartanspace,andthenspecializetheresultforthesimplestcaseofascalar。
7、functionf∈sec^0T∗M֒→secCℓ(M,g)inordertocompareourresultswiththeonespresentedin[12].Wegivetwocalculations,oneusingthedecompositionoftheDiracoperatorintothestandardDiracoperatorplusatermdependingonthetorsiontensor(seeEq.(16)below)andanotherone,whichfollowsdirectlyfromthedefinitionoftheDiracoperatorwithoutusingthestandardDiracop-erator.NextwepresentarelationbetweenthesquareoftheDiracandthestandardDiracoperators(actingonascalarfunction)intermsofthetorsiontensorandinvestigatealsoinSection2.3therelatio。
8、nbetweenthoseoperatorsinthecaseofanullstraintensor.InSection3,wepresentabriefsummaryofthetheoryoftheSpin-Cliffordbundles(CℓℓSpine1,3(M,g)andCℓrSpine1,3(M,g))andtheirsections(spinorfields)andtheirrepresentativesinaCliffordbundle(Cℓ(M,g))following[10,14].WerecallinSection3.1someimportantformulasfromthegeneraltheoryofthecovariantderivativesofCliffordandspin-CliffordfieldsandinSection3.2werecallthedefinitionofthespin-Diracoperator∂s(associatedwithaRiemann-Cartanconnection∇)actingonsectionsofaspin-Cliffordbu。
9、ndle.Insection3.3weintroducetherepresentativesofspinorfieldsintheCliffordbundleandtheimportantconceptoftherepresentativeof∂s(denoted∂(s))thatactsontherepresentativesofspinorfields(see[10,14]fordetails).TomakeclearthesimilaritiesanddifferencesbetweenCℓ(M,g)and2CℓℓSpine1,3(M,g),wewrite,inSection4,Maxwellequationinbothformalisms.InSection5.1wefirstfindthecommutatorofthecovariantderivativeofspinorfieldsonaRiemann-Cartanspace(time)andcompareourresultwithonethatcanbefoundin[11],whichseemstoneglectaterm.Nexti。
10、nSection5.2wecalculatethesquareofthespin-DiracoperatoronaRiemann-CartanspacetimeandfindageneralizedLichnerowiczformula.InSection6,takingadvantagethatanyψ∈secCℓℓSpine1,3(M,g)canbewrittenasψ=A1ℓΞwithA∈secCℓ(M,g)and1ℓΞ∈secCℓℓSpine1,3(M,g)wefindtwonoticeableformulas:thefirstrelatesthesquareofthespin-Diracoperator(θa∇sea)actingonψwiththesquareoftheDiracoperator(θa∇ea)actingonA;thesecondformularelatesthesquareofthespin-Diracoperator(θa∇sea)actingonψwiththesquareofthestandardDiracoperator(θaDea).InSection7。
本文标题:The Square of the Dirac and spin-Dirac Operators o
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