Noncommutative Spacetime and Quantum Mechanics

arXiv:quant-ph/0404028v15Apr2004NONCOMMUTATIVESPACETIMEANDQUANTUMMECHANICSJaroslawWawrzyckii∗InstituteofNuclearPhysics,ul.Radzikowskiego152,31-342Krak´ow,PolandFebruary1,2008AbstractInthispaperwewillanalyzethestatusofgaugefreedominquantumme-chanics(QM)andquantumfieldtheory(QFT).Alongwiththisanalysiscom-parisonwithordinaryQFTwillbegiven.Wewillshowhowthegaugefreedomproblemisconnectedwiththespacetimecoordinatesstatus—theverypointatwhichthedifficultiesofQMbegin.AnaturalsolutionoftheabovementionedproblemwillbeproposedinwhichwegiveaslightlymoregeneralformofQMandQFT(incomparisontotheordinaryQFT)withnoncommutativestructureofspacetimeplayingfundamentalroleinit.WeachieveitbyreinterpretationoftheBohr’scomplementarityprincipleontheonehandandbyincorporationofourgaugefreedomanalysisontheother.WewillpresentageneralizationoftheBargmann’stheoryofexponentsofrayrepresentations.Itwillbegivenanexampleinvolvingtimedependentgaugefreedomdescribingnon-relativisticquantumparticleinnonrelarivisticgravitationalfield.Inthisexamplewein-ferthemostgeneralSchr¨odingerequationandproveequalityofthe(passive)inertialandthegravitationalmassesofquantumparticle.1GeneralIntroductionProbablyitwillbehelpfultogiveabriefoutlineofthispaperprovidingthegen-erallineofreasoning.Fordetails,however,thereadermustconsulttheforegoingsections.OflatetherehasbeenproposedareformulationofthestandardQM(J.Wawrzy-cki,math-ph/0301005;Comm.Math.Phys.,toappear),whichisslightlymoregeneralincomparisontotheordinaryformofthetheory.Thisreformulationhasemergedinanaturalwayindescriptionofaquantumparticleinthenon-relativisticgravitationalfieldwithtimedependentgaugefreedom.Remember,please,thatthe∗Electronicaddress:[e-mail]12JAROSlAWWAWRZYCKIstatesofaphysicalsystemdonotcorrespondbi-uniquelytounitvectorsφoftherespectiveHilbertspaceHbuttotherays,sayφ={eiξφ}intheQMandQFT,whereξisanarbitraryrealnumber.1Observenow,please,thatinordinary(non-relativistic)QM,whenusingSchr¨odingerpicture,onecangoconsiderablyfurtherwiththisobservation.Namely,twoSchr¨odingerwavesψandeiξ(t)ψareindistin-guishableevenwhenξdependsontime,butonehavetoassumesimultaneouslythatSchr¨odingerwaveequationpossessatimedependentgaugefreedom.Letusrecallthattheintegraldefiningthescalarproductisoverthespacecoordinatesandonecantakeatimedependentfactorovertheintegralsign.Afterthis,however,theSchr¨odingerwavefunctionsshouldconstitutetheappropriatecrosssectionsofaHilbertbundleR△HovertimeR.TherepresentationsTrofacovarianceaswellasasymmetrygroupsactinR△HandtheirexponentsξintheformulaTrTs=eiξ(r,s,t)Trs,(1)dodependontimet∈R.Thus,atfirstsightthenaturalassumptionthattwoSchr¨odingerwavesdifferingbyatimedependentphaseareequivalentleadstoaratherstrangeconstruction,namely,theHilbertbundle—anobjectmuchmoreinvolvedthentheHilbertspaceitself.Onecanprove,however,thatinthenon-relativisticGalileaninvarianttheory,thismoregeneralassumptionleadsexactlytotheordinaryQM.ThewholestructuredegeneratesduetothespecificstructureoftheGalileangroup.Moreover,inthelesstrivialcaseofaquantumparticleinnon-relativisticgravitationalfield,whenthetimedependentgaugefreedomisindis-pensable,theresultsarequiteinteresting.Namely,onecaninferthemostgeneralwaveequationforthatparticleandproveequalityoftheinertialandgravitationalmass,theresultsconfirmedbyexperiments!InthelastcasetheMilnegroupplaystheroleoftheGalileangroup.Thisnon-relativisticgeneralizationpossessesalsoaverynaturalrelativisticex-tensionwhichcanbeincorporatedwithinQFTratherthenQM.InQFTonecanstillgoastepfurtheralongwiththislineofgeneralizingthequantummechanicalprinciples.Remember,please,thatintheFockconstructiontheFouriercompo-nentsoftheclassicalfieldconstitutetheargumentsofthewavefunction.Anywaytheargumentshavenothingtodowithordinaryspacetimecoordinates.Inotherwordsthespacetimecoordinatesaremereparametersortheso-calledc-numbersinHeisenbergcanonicalfieldquantization—justlikethetimeinordinarynon-relativisticQM.Oneshould,thus,assumethetwowavefunctionstobeequivalentwhenevertheydifferbyaspacetimedependentwavefunction.But,whentreating1Letusrecallthatallrelevantinformationcarriedbyφiscontainedinthesetofnumbers|(φ,ϕ)|2(φ,φ)(ϕ,ϕ),where(φ,ϕ)isthescalarproductoftheHilbertspaceH.Assuchφandeiξφareequivalentcontainingexactlythesameinformation.NONCOMMUTATIVESPACETIMEANDQUANTUMMECHANICS3thisassumptionseriously,thewavefunctionsshouldconstitutetheappropriatecrosssectionsofaHilbertbundleM△HoverspacetimeM.Accordinglytherepresenta-tionsTrofcovarianceorsymmetrygroupspossessspacetimedependentexponentsξ=ξ(r,s,p):TrTs=eiξ(r,s,p)Trs,withp∈M(J.Wawrzycki,math-ph/0301005;Comm.Math.Phys.,toappear).Inparticularoneisforcedtoextendtheordinaryclassificationtheoryofexpo-nentsξofrepresentationsactinginordinaryHilbertspacesoastoembracetheabovecaseofrepresentationactinginaHilbertbundlewithspacetimedependentξ.ItcanbeviewedasageneralizationoftheBargmann’stheory(V.Bargmann,Ann.Math54,1,1954)ofexponentsofrayrepresentationsactinginordinaryHilbertspacewithspacetime-independentξ.Thefactthatthesimplerformofthetheorywithtimedependentgaugefreedomgivesthecor