Phase Space Formulation of Quantum Mechanics. Insi

PHASESPACEFORMULATIONOFQUANTUMMECHANICS.INSIGHTINTOTHEMEASUREMENTPROBLEMD.Dragoman*–Univ.Bucharest,PhysicsDept.,P.O.BoxMG-11,76900Bucharest,RomaniaAbstract:Aphasespacemathematicalformulationofquantummechanicalprocessesaccompaniedbyandontologicalinterpretationispresentedinanaxiomaticform.Theproblemofquantummeasurement,includingthatofquantumstatefiltering,istreatedindetail.Unlikestandardquantumtheorybothquantumandclassicalmeasuringdevicecanbeaccommodatedbythepresentapproachtosolvethequantummeasurementproblem.*Correspondenceaddress:Prof.D.Dragoman,P.O.Box1-480,70700Bucharest,Romania,email:[e-mail]enturyafterthediscoveryofthequantumanddespitetheindubitablesuccessofquantumtheoryincalculatingtheenergylevels,transitionprobabilitiesandotherparametersofquantumsystems,theinterpretationofquantummechanicsisstillunderdebate.Unlikerelativisticphysics,whichhasbeenfoundedonanewphysicalprinciple,i.e.theconstancyoflightspeedinanyreferenceframe,quantummechanicsisratherasuccessfulmathematicalalgorithm.Quantummechanicsisnotfoundedonafundamentalprinciplewhosevaliditymaybequestionedormaybesubjectedtoexperimentaltesting;inquantummechanicswhatisquestionableisthemeaningoftheconceptsinvolved.ThequantumtheoryoffersarecipeofhowtoquantizethedynamicsofaphysicalsystemstartingfromtheclassicalHamiltonianandestablishesrulesthatdeterminetherelationbetweenelementsofthemathematicalformalismandmeasurablequantities.Thissetofinstructionsworksremarkablywell,butontheotherhand,thesignificanceofevenitslandmarkparameter,thePlanck’sconstant,isnotclearlystated.Symptomaticforthepoorunderstandingofthefundamentalsofquantummechanicsistheexistenceof(atleast)ninedifferentformulations[1](includingtheHeisenbergandSchrödingerformulations,thepathintegralformulation,thepilotwaveandvariationalformulations),nottomentionthedifferentinterpretations.(Alistofessentialbibliographyforeachformulationcanbealsofoundin[1].)Althoughclassicalmechanicscanbeaswellformulatedinmanydifferentways(thereisaHamiltonian,aLagrangianformulation,andsoon),thedifferentformulationsofquantummechanicsareoftenbasedondifferentviewsaboutthephysicalreality.Perhapsthemosttroublesomeproblemofquantummechanicsismeasurement.Thereisalargenumberoftheoriesthattrytoexplaintheapparentcontroversybetweenthesuperpositionprincipleofquantummechanicsandtheprobabilisticresultsofmeasurement,i.e.toexplainhowtoreconciletheoccurrencewithacertainprobabilityofeigenvaluesofacertainoperatorbutexcludeanyevidenceforthesuperpositionoftheoperator’seigenstatesinwhichthequantumstatecanmathematicallybeexpanded.Thisdiscrepancybetweenmathematicsandmeasurementresultsisstillunderscrutiny.TheearliestattempttosolveitwasmadebyBohr(seethereprintsin[2])whodrawaborderbetweenthequantumsystem,subjecttothesuperpositionprinciple,andtheclassicalmeasuringdevice(includingtheobserver),towhichthisprincipledoesnotapply.AnotherapproachtothequantummeasurementproblemwasofferedbyvonNeumann[3]whoincludedalsothemeasuringapparatusinaquantumdescriptionbutwasforcedtopostulatethereductionorcollapseofthequantummechanicalwavefunctioninordertoexplainthemeasurementresults.Amorerecentexplanationofthewavefunctioncollapseisprovidedbytheinteractionofthequantumsystemwiththeenvironment,whichinducesthelossofphasecoherence(decoherence)ofthesuperpositionbetweenasetofpreferredstatessingledoutbytheenvironment(seethereviewin[4]).Themany-worldsinterpretationofquantummechanics[5],whichassumesthatatsuitableinteractionsthewavefunctionoftheuniversesplitsintoseveralbranches,orthesuperselectionrules,whichimplythattheessenceofaquantummechanicalmeasurementresidesinthenon-observationofacertainpartofthesystem(seethereviewin[6]),areexamplesofotherquantummechanicaltheoriesthatattemptedtosolvethemeasurementproblem.Thelistofsuchattemptsismuchlongerandstillgrowingbutasformanyissuesinquantummechanics,includingtheproblemsofdefiningthetimeorphase,thereisnoagreementuponthetheoryofmeasurement.Thepresentpaperoffersanothersolutiontothisunsettledissue,whichhastheadvantageofstartingfromaphysicalprincipleandaconsistentinterpretationofthemeasurementresults,andnotfromanothermathematicalformalism.Aswillbecomeclearerinthepaperthephasespaceformulationofquantumtheoryisthemostsuitableforoutliningtheskeletonofthisnewinterpretationofquantummechanicsand,inparticular,ofthemeasurementproblem.Thetheorypresentedinthispaperislaiddownasasetofpostulatesthataddressnotonlythemathematicalformulationbutalsotheontologicalviewpointoftheauthoraboutphysicalreality.2.PostulatesofthequantumtheoryThequantumtheorypresentedinthispaperisbasedonanumberofeightpostulates.Postulate1:Quantumparticleshavephysicalreality.ThroughquantumparticlesIunderstandlocalizedandindivisibleconcentrationsofenergy(suchasphotons)and/ormass(electrons,etc.).Theexistence,evolutionandinteractionofquantumparticlesdonotdependonthepresenceofanobserver.Postulate2:ThestateofaquantumparticlecanonlybedescribedbyavectorontheHilbertspaceorbyaquasi-probabilitydistributioninthephasespaceformulationofquantummechanics.Thispostula