Covariant geometric quantization of non-relativist

arXiv:quant-ph/0012036v211Dec2000Covariantgeometricquantizationofnon-relativisticHamiltonianmechanicsGiovanniGiachetta†1,LuigiMangiarotti†2andGennadiSardanashvily‡3†DepartmentofMathematicsandPhysics,UniversityofCamerino,62032Camerino(MC),Italy‡DepartmentofTheoreticalPhysics,PhysicsFaculty,MoscowStateUniversity,117234Moscow,RussiaAbstract.WeprovidegeometricquantizationoftheverticalcotangentbundleV∗QequippedwiththecanonicalPoissonstructure.Thisisamomentumphasespaceofnon-relativisticme-chanicswiththeconfigurationbundleQ→R.ThegoalistheSchr¨odingerrepresentationofV∗Q.WeshowthatthisquantizationisequivalenttothefibrewisequantizationofsymplecticfibresofV∗Q→Rthatmakesthequantumalgebraofnon-relativisticmechanicsaninstantwisealgebra.Quantizationoftheclassicalevolutionequationdefinesaconnectiononthisinstantwisealgebra,whichprovidesquantumevolutioninnon-relativisticmechanicsasaparalleltransportalongtime.1IntroductionWestudycovariantgeometricquantizationofnon-relativisticHamiltonianmechanicssub-jecttotime-dependenttransformations.ItsconfigurationspaceisafibrebundleQ→Requippedwithbundlecoordinates(t,qk),k=1,...,m,wheretistheCartesiancoordinateonthetimeaxisRwithaffinetransitionfunctionst′=t+const.DifferenttrivializationsQ∼=R×MofQcorrespondtodifferentnon-relativisticreferenceframes.Incontrarytoalltheexistentquantizationsofnon-relativisticmechanics(e.g.,[14,19]),wedonotfixatrivializationofQ.Themomentumphasespaceofnon-relativisticmechanicsistheverticalcotangentbundleV∗QofQ→R,endowedwiththeholonomiccoordinates(t,qk,pk).ItisprovidedwiththecanonicalPoissonstructure{f,f′}V=∂kf∂kf′−∂kf∂kf′,f,f′∈C∞(V∗Q),(1)1E-mailaddress:[e-mail]2E-mailaddress:[e-mail]3E-mailaddress:[e-mail]1whosesymplecticfoliationcoincideswiththefibrationV∗Q→R[8,12].Givenatrivial-izationV∗Q∼=R×T∗M,(2)thePoissonmanifold(V∗Q,{,}V)isisomorphictothedirectproductofthePoissonmani-foldRwiththezeroPoissonstructureandthesymplecticmanifoldT∗M.AnimportantpeculiarityofthePoissonstructure(1)isthatthePoissonalgebraC∞(V∗Q)ofsmoothrealfunctionsonV∗QisaLiealgebraovertheringC∞(R)offunctionsoftimealone.OurgoalisthegeometricquantizationofthePoissonbundleV∗Q→R,butitisnotsufficientforquantizationofnon-relativisticmechanics.Theproblemisthatnon-relativisticmechanicscannotbedescribedasaPoissonHamiltoniansystemonthemomentumphasespaceV∗Q.Indeed,anon-relativisticHa-miltonianHisnotanelementofthePoissonalgebraC∞(V∗Q).ItsdefinitioninvolvesthecotangentbundleT∗QofQ.Coordinatedby(q0=t,qk,p0=p,pk),thecotangentbundleT∗Qplaystheroleofthehomogeneousmomentumphasespaceofnon-relativisticmechanics.ItisequippedwiththecanonicalLiouvilleformΞ=pλdqλ,thesymplecticformΩ=dΞ,andthecorrespondingPoissonbracket{f,g}T=∂λf∂λf−∂λf∂λf′,f,f′∈C∞(T∗Q).Duetotheone-dimensionalcanonicalfibrationζ:T∗Q→V∗Q,(3)thecotangentbundleT∗QprovidesthesymplecticrealizationofthePoissonmanifoldV∗Q,i.e.,ζ∗{f,f′}V={ζ∗f,ζ∗f′}Tforallf,f′∈C∞(V∗Q).AHamiltonianonV∗Qisdefinedasaglobalsectionh:V∗Q→T∗Q,p◦h=−H(t,qj,pj),(4)oftheone-dimensionalaffinebundle(3)[8,12].Asaconsequence(seeSection6),theevolutionequationofnon-relativisticmechanicsisexpressedintothePoissonbracket{,}TonT∗Q.ItreadsϑH∗(ζ∗f)={H∗,ζ∗f}T,(5)2whereϑH∗istheHamiltonianvectorfieldofthefunctionH∗=∂t⌋(Ξ−ζ∗h∗Ξ))=p+H(6)onT∗Q.Therefore,weneedthecompatiblegeometricquantizationsbothofthecotangentbundleT∗QandtheverticalcotangentbundleV∗Qsuchthatthemonomorphismζ∗:(C∞(V∗Q),{,}V)→(C∞(T∗Q),{,}T)(7)ofthePoissonalgebraonV∗QtothatonT∗QisprolongedtoamonomorphismofquantumalgebrasofV∗QandT∗Q.Recallthatthegeometricquantizationprocedurefallsintothreesteps:prequanti-zation,polarizationandmetaplecticcorrection(e.g.,[3,14,19]).Givenasymplecticmanifold(Z,Ω)andthecorrespondingPoissonbracket{,},prequantizationassociatestoeachelementfofthePoissonalgebraC∞(Z)onZafirstorderdifferentialoperatorbfinthespaceofsectionsofacomplexlinebundleCoverZsuchthattheDiraccondition[bf,bf′]=−id{f,f′}(8)holds.Polarizationofasymplecticmanifold(Z,Ω)isdefinedasamaximalinvolutivedistributionT⊂TZsuchthatOrthΩT=T,i.e.,Ω(ϑ,υ)=0,∀ϑ,υ∈Tz,z∈Z.(9)GiventheLiealgebraT(Z)ofglobalsectionsofT→Z,letAT⊂C∞(Z)denotethesubalgebraoffunctionsfwhoseHamiltonianvectorfieldsϑffulfillthecondition[ϑf,T(Z)]⊂T(Z).(10)Elementsofthissubalgebraareonlyquantized.Metaplecticcorrectionprovidesthepre-HilbertspaceETwherethequantumalgebraATactsbysymmetricoperators.ThisisacertainsubspaceofsectionsofthetensorproductC⊗D1/2oftheprequantizationlinebundleC→ZandabundleD1/2→Zofhalf-densitiesonZ.ThegeometricquantizationprocedurehasbeenextendedtoPoissonmanifolds[16,17]andtoJacobimanifolds[7].WeshowthatstandardprequantizationofthecotangentbundleT∗Q(e.g.,[3,14,19])providesthecompatibleprequantizationofthePoissonmanifoldV∗Qsuchthatthemonomorphismζ∗(7)isprolongedtoamonomorphismofprequantumalgebras.Incontrastwiththeprequantizationprocedure,polarizationofT∗QneednotimplyacompatiblepolarizationofV∗Q,unlessitincludestheverticalcotangentbundleVζT∗Q3ofthefibrebundleζ(3),i.e.,spansovervectors∂0.ThecanonicalrealpolarizationofT∗Q,satisfyingtheconditionVζT∗Q⊂T,(11)isthevertical