On obtaining classical mechanics from quantum mech

arXiv:gr-qc/0606078v24Dec2006IMSc/2006/6/17OnobtainingclassicalmechanicsfromquantummechanicsGhanashyamDate1,∗1TheInstituteofMathematicalSciences,CITCampus,Chennai-600113,INDIAAbstractConstructingaclassicalmechanicalsystemassociatedwithagivenquantummechanicalone,entailsconstructionofaclassicalphasespaceandacorrespondingHamiltonianfunctionfromtheavailablequantumstructuresandanotionofcoarserobservations.TheHilbertspaceofanyquan-tummechanicalsystemnaturallyhasthestructureofaninfinitedimensionalsymplecticmanifold(‘quantumphasespace’).Thereisalsoasystematic,quotientingprocedurewhichimpartsabundlestructuretothequantumphasespaceandextractsaclassicalphasespaceasthebasespace.ThisworksstraightforwardlywhentheHilbertspacecarriesweaklycontinuousrepresentationoftheHeisenberggroupandonerecoversthelinearclassicalphasespaceR2N.Wereportonhowthepro-cedurealsoallowsextractionofnon-linearclassicalphasespacesandillustrateitforHilbertspacesbeingfinitedimensional(spin-jsystems),infinitedimensionalbutseparable(particleonacircle)andinfinitedimensionalbutnon-separable(Polymerquantization).Toconstructacorrespondingclassicaldynamics,oneneedstochooseasuitablesectionandidentifyaneffectiveHamiltonian.Theeffectivedynamicsmirrorsthequantumdynamicsprovidedthesectionsatisfiesconditionsofsemiclassicalityandtangentiality.PACSnumbers:04.60.Pp,98.80.Jk,98.80.Bp∗Electronicaddress:[e-mail]1Developingasemi-classicalapproximationtoquantumdynamicsisingeneralanon-trivialtask.Intuitively,suchanapproximationentailsanadequateclassofobservablequantities(eg.expectationvaluesofself-adjointoperators)whosetimeevolution,dictatedbyquantumdynamics,iswellapproximatedbyaclassicalHamiltonianevolution.Roughly,theadequateclassrefersto(say)basicfunctionsonaclassicalphasespace(symplecticmanifold)withaHamiltonianwhichisafunctionofthesebasicfunctions.Theaccuracyofanapproximationiscontrolledbyhowwelltheclassicallyevolvedobservablesstayclosetothequantumevolvedoneswithinagivenprecisionspecifiedintermsofboundsonquantumuncertainties.Havingadescriptionofthequantumframeworkassimilaraspossibletoaclassicalframeworkisobviouslyanaidindevelopingsemi-classicalapproximations.Suchadescriptionisindeedavailableandisreferredtoasgeometricalformulationofquantummechanics[1].Thequantummechanicalstatespace,aprojectiveHilbertspace,isnaturallyasymplecticmanifold,usuallyinfinitedimensional(finitedimensionalforspinsystems).Furthermore,dynamicsspecifiedbyaSchrodingerequationisaHamiltonianevolution.Thisistrueforallquantummechanicalsystems.Inaddition,thereisalsoasystematicquotientingproceduretoconstructanassociatedHamiltoniansystem(usuallyoflowerandmostlyfinitedimensions)whichviewsthequantumstatespaceasabundlewiththeclassicalphasespaceasitsbasespace.ThisworkselegantlywhenthequantumHilbertspaceisobtainedastheweaklycontinuousrepresentationofaHeisenberggroup.GenericallytheseareseparableHilbertspacesandtheextractedclassicalphasespacesarelinear,R2n.QuantummechanicalHilbertspaceshoweverariseinmanydifferentways.Forexample,the(kinematical)Hilbertspaceofloopquantumcosmologycarriesanon-weaklycontinuousrepresentationoftheHeisenberggroupandisnon-separable.Forexamplessuchasparticleonacircleandspinsystems,onedoesnotevenhavetheHeisenberggroup.Asemi-classicalapproximationisstillneededforsuchsystems.Likewise,inclassicalmechanics(evenforfinitelymanydegreesoffreedom),theclassicalphasespaceisnotnecessarilylinear(egthecylinderforparticleonacircle,reducedphasespacesofconstrainedsystemsetc).Itisimportanttodevelopaquotientingproceduretoconstructsuch,possiblynon-linear,classicalphasespacesfrommoregeneralquantumstatespaces.Inthisworkwedevelopsuchaprocedureandillustrateitforthreeexamples:arbitraryspin-Jsystem,particleonacircleandBohrorpolymerquantizationappearinginloopquantumcosmology(LQC).This2takescareofthekinematicalaspects.ToconstructanassociatedclassicaldynamicsonehasalsotoobtainaHamiltonianfunc-tion(aneffectiveHamiltonian)ontheclassicalphasespace.ThisisdonebychoosingasectionofthebundleandobtainingtheeffectiveHamiltonianonthebasespaceasapullbackofthequantummechanicallydefinedone.AneffectiveHamiltoniansodefined,dependsonthesectionchosen.Onecannowconstructtwotrajectoriesontheclassicalphasespace:(a)projectionofaquantumtrajectory(i.e.trajectoryinthequantumstatespace)ontothebasespaceand(b)atrajectoryinthebasespace,generatedbytheeffectiveHamiltonianfunction.Ingeneral,i.e.forarbitrarysections,thesetwotrajectoriesdonotcoincide.Theydosowhenthesectionistangentialtothequantumtrajectories(equivalentlywhenthesectionispreservedbyquantumdynamics).Sincetheclassicalstatesareobtainedfromex-pectationvalues(viathequotientingprocedure),fortheclassicaltrajectoriestoreflectthequantumone,withinacertainapproximation,itisnecessarythatthequantumuncertaintiesalsoremainboundedwithinprescribedtolerances.Inotherwords,thestatesinthesectionshouldalsosatisfyconditionsofsemiclassicality.InsectionI,werecallthebasicdetailsofthegeometricformulationfrom[1]anddescribethequotientingprocedureinageneralsetting.Ingeneraltheclassicalphaseisobtainedasasub-manifoldofthebasespaceof