On the Geometry of Supersymmetric Quantum Mechanic

arXiv:0710.2881v1[hep-th]15Oct2007OntheGeometryofSupersymmetricQuantumMechanicalSystems∗D.Lundholm†DepartmentofMathematics,RoyalInstituteofTechnologySE-10044Stockholm,SwedenAbstractWeconsidersomesimpleexamplesofsupersymmetricquantumme-chanicalsystemsandexploretheirpossiblegeometricinterpretationwiththehelpofgeometricaspectsofrealCliffordalgebras.Thisleadstonat-uralextensionsoftheconsideredsystemstohigherdimensionsandmorecomplicatedpotentials.1IntroductionInthefollowing,asupersymmetricsystemwillmeanasupersymmetricquantummechanics(SUSYQM)accordingtothefollowingdefinition1:Onacomplexsep-arableHilbertspaceactsahamiltonianH,anumberofsuperchargesQj=1,...,N,andagradingoperatorKwhichsplitstheHilbertspaceH=Hb⊕Hfintoabosonicandafermionicsector.Theseoperatorsareself-adjointontheirre-spectivedomainsandsatisfytherelations{Qj,Qk}=2δjkH,K2=1,{Qj,K}=0,(1)where{A,B}:=AB+BAistheanticommutator.AclassicexampleofaSUSYQMwithageometricinterpretationisprovidedbytheDiracoperator[1,3,4].Thatisperhapsaratheruninterestingexampleintheflatspacecase,whereonlylocalgeometryisnon-trivial,butitsextensiontothesettingofcurvedmanifoldshasledtonewinsightsinglobalgeometryandindextheory.Here,wewillfocusonthelocalgeometryofsupersymmetricsystemswithSchr¨odinger-likehamiltonians.Inparticular,weareinterestedinhamiltoniansoftheformH=HB+HF,wheretheso-calledbosonicpartHBisanordinarySchr¨odingeroperatorandthefermionicpartHFisamatrix-oralgebra-valuedmultiplicationoperator.Sincesuchoperatorsinvolvealaplacian,theircorrespondingsuperchargeswillnecessarilyhavetoinvolvesomeformofDiracoperator.∗SupportedbytheSwedishResearchCouncil†[e-mail]1See[1,2]andreferencesthereinforadiscussionofpossibledefinitions.1deCrombruggheandRittenberg[5]havecarriedoutarathergeneralal-gebraicanalysisofSUSYQMhamiltonians,butwithfocusoncaseswhenthesuperchargesarelinearintheCliffordgenerators.ThisistrueforasingleDiracoperator,butwillapartfromthatgenerallynotbethecaseintheexamplesweareconsidering.Furthermore,whenstudyingthealgebraicpropertiesofsupersymmetricsystemsitiscommontoworkwithcreationandannihilationoperatorsaj,a†j,ck,c†kandconsideraFockrepresentationoftheseonaHilbertspacewithaparticleinterpretation.WewillontheotherhandsticktothealternativeSchr¨odingerrepresentationinvolvingcoordinatesandmomentaxj,pxjactingasmultiplicationanddiffentialoperatorsonanL2-space,andClif-fordgeneratorsek1,ek2actinginarepresentationofthecorrespondingCliffordalgebra.Wewillemphasizetherealgeometryinthesystemsweconsideranduseitto‘explain’theappearanceofcomplexstructures.Thisleadstotheiden-tificationofadditionalstructures,propertiesandpossibleextensionsofthesesystemswhichmightnothavebeenatallobviousfromtheconventionalcom-plexformulation.Wewillalsopointoutthatitispossibletofindanotionofsupersymmetricsystemeveninapurelyrealsettingwithnocanonicalcomplexstructure.Apartfromthepurelymathematicalinterestininvestigatingthestructureofthesetypesofsystems,onemotivationfromphysicsisthatitseemsworthwhiletoexplorethepossibilityofgivingmorecomplicated,butrelated,SUSYQMsystemssuchassupersymmetricmatrixmodelsamoregeometricinterpretation.2GeometricalgebraInordertoappreciatethegeometricinterpretationofthesystemswhichweconsideritishelpfultohaveavailablesometoolsandnotionsfromgeometricalgebra,i.e.Cliffordalgebrawithemphasisonthegeometryoftheunderlying,usuallyreal,vectorspace.ForamorecompleteintroductiontothegeometricaspectsofCliffordalgebra,seee.g.[6]or[7].GivenarealvectorspaceVofdimensiondwithanon-degeneratebilinearforma·b(e.g.aninnerproduct,Minkowskimetric,etc.)thereisanaturallyassociatedCliffordalgebraorgeometricalgebraG(V)inthefollowingway.ForanorthonormalbasisE={e1,...,ed}ofVweletG(V)(orjustG)denotethefreeassociativealgebrageneratedbyEwiththerelationse2i=ei·ei=±1andeiej=−ejei,i6=j.Hence,normalizedvectorssquaretounity,orthogonalvectorsanticommute,andG(V)=SpanR{1,ei,eiej,...,e1e2...ed}ij....(2)Asvectorspaces,G(V)isisomorphictothegradedexterioralgebraV∗V,andthereisacorrespondingexteriorproduct∧inG(V)withrespecttowhichthisisomorphismextendstothelevelofalgebras.Hencewecanidentifythesespaces.WeletG±denotethesubspacesofeven/oddgradesinG.ThehighestgradeelementI:=e1e2...ed=e1∧e2∧···∧eddeterminesanorientationforVandiscalledthepseudoscalar.AnarbitraryelementofGiscalledamultivector.TheorthogonalgroupshavespinrepresentationsembeddedinGascanbeseenbyactionofCliffordmultiplicationonV⊆G.Forexample,areflectionalong(i.e.inthehyperplaneorthogonalto)aunitvectorn∈Sd−1⊆Rdactingonavectorv∈Rdcanbewrittenv7→−nvn.Arotation(beingacomposition2ofanevennumberofreflections)hasthespinorrotorrepresentationv7→R†vR,whereR=n1n2...n2kisaproductofunitvectorsandthedaggerdenotesreversionoftheorderofanyCliffordproduct.TheDiracoperator∇:=dXj=1ej∂∂xj(3)inV(flatspace)isofcoursealsoanaturalobject,combiningthepropertiesofdifferentiationwiththepropertiesoftheCliffordproduct.Aspinorisanelementofa(oftenirreducible)representationofaspingroupandassuchcanalsobeviewedasanelementofthegeometricalgebraitself.Namely,anyirreduciblespinrepresentationcanbeconstructedbylettingthespingroupSpin⊆G