Homogeneous decoherence functionals in standard an

arXiv:math-ph/9807024v123Jul1998HomogeneousdecoherencefunctionalsinstandardandhistoryquantummechanicsOliverRudolphaTheoreticalPhysicsGroupBlackettLaboratoryImperialCollegeofScience,TechnologyandMedicineSouthKensingtonPrinceConsortRoadLondonSW72BZ,EnglandandJ.D.MaitlandWrightbAnalysisandCombinatoricsResearchCentreMathematicsDepartmentUniversityofReadingReadingRG66AX,EnglandABSTRACTGeneralhistoryquantumtheoriesarequantumtheorieswithoutagloballydefinednotionoftime.Decoherencefunctionalsrepresentthestatesinthehistoryapproachandaredefinedascertainbi-variatecomplex-valuedfunctionalsonthespaceofallhistories.However,inpracticalsituations–forinstanceinthehistoryformulationofstandardquantummechanics–thereoftenisaglobaltimedirectionandthehomogeneousdecoherencefunctionalsarespecifiedbytheirvaluesonthesubspaceofhomogeneoushistories.Inthisworkwestudytheanalyticpropertiesof(i)thestandarddecoherencefunctionalinthehis-toryversionofstandardquantummechanicsand(ii)homogeneousdecoherencefunctionalsingeneralhistorytheories.Werestrictourselvestothesituationwherethespaceofhistoriesisgivenbythelat-ticeofprojectionsonsomeHilbertspaceH.Amongotherthingsweprovethenon-existenceofafinitelyvaluedextensionforthestandarddecoherencefunctionaltothespaceofallhistories,derivearepresentationforthestandarddecoherencefunctionalasanunboundedquadraticformwithanaturalrepresentationonaHilbertspaceandprovetheexistenceofanIsham-Linden-Schreckenberg(ILS)typerepresentationforthestandarddecoherencefunctional.aemail:[e-mail]:[e-mail]isaninvestigationintocertainaspectsofthehistoryapproachtoquantummechanics.Thehistoriesapproachtoquantumtheoryisapromisingnewapproachtoquantummechanics[1]-[24]whichhasledtoseveralinterestingdevelopments.Originally,theconsistenthistoriesapproachtoquantummechanicswasintroducedbyGriffiths[1]asatoolforinterpretingstandardnonrelativisticHilbertspacequantummechanics.Thisso-called‘consistenthistoriesinterpretation’hasbeenfurtherdevelopedandbroughttoitspresentformbyOmn`es[2].However,inthepresentpaperweareexclusivelyinterestedinanotheraspectofthehistoriesapproach,namelythatitisapotentialframeworkforaquantumtheorywheretimeplaysasubsidiaryrole.InaseriesofinterestingpapersGell-MannandHartle[3]-[6]havestudiedquantumcosmologyandthepathintegralformulationofrelativisticquantumfieldtheoryintermsoftheconceptsofthehistoriesapproach.Theyputforwardforthefirsttimetheideaoftakingtheconceptsoftheconsistenthistoriesapproachtoquantummechanicsasindependentfundamentalentitiesintheirownrightinageneralizedquantumtheory.Basedonthisidea,Ishamhasformulatedin[7]anaturalalgebraicgeneralizationoftheconsistenthistoriesapproach.Withhisso-calledgeneralhistoryquantumtheorieshehasbroadenedboththescopeandthemathematicalframeworkoftheconsistenthistoriesapproachtoquantummechanics.Standardquantummechanicsisbasedontheidealizednotionsofobservableandstateatasingletime.Isham’sgeneralhistorytheoriesprovideaframeworkinwhichthesenotionsarereplacedbytemporalanalogues,historiesanddecoherencefunctionals,respectively.Accordingly,ingeneralthehistoriesaremoregeneralobjectsthansimplytime-sequencesofsingle-timeeventsbutareregardedaseventsintrinsicallyspreadoutintime.Moreover,Isham’sgeneralquantumhistoriesprovideapossibleframeworkforformulatingaquan-tumtheorywithoutanexternalgloballydefinednotionoftime.InIsham’sapproachageneralhistoryquantumtheoryisformallycharacterizedbythespaceofhis-toriesontheonehandandbythespaceofdecoherencefunctionalsontheotherhand.Thehistoriesareregardedasfundamentalentitiesintheirownrightandareidentifiedwiththegeneraltemporalpropertiesofthequantumsystem.Isham’sapproachhassubsequentlybecomethesubjectofintensestudy[8]-[23].Inthehistoryapproachprobabilitiesareassignedtocompletehistories.However,atthebasisofthehistoriesapproachistheideathatanyprobabilityassignmenttoahistoryhmakessenseonlywithrespecttoasocalledconsistentsetofhistoriescontainingh.Dualtothenotionofhistoryisthenotionofdecoherencefunctional.Thedecoherencefunctionaldeterminestheconsistentsetsofhistoriesinthetheoryandtheprobabilitiesassignedtohistoriesintheconsistentsets.Morespecifically,adecoherencefunctionaldmapseveryorderedpairofgeneralhistoriesh,ktoacomplexnumberdenotedbyd(h,k).Thenumberd(h,k)isinterpretedinphysicaltermsasameasureofthemutualinterferenceofthetwohistorieshandk.Aconsistentsetofhistoriesconsistsofhistorieswhosemutualinterferenceissufficientlysmall,suchthatthediagonalvalued(h,h)canbeinterpretedastheprobabilityofthehistoryhinthisconsistentset.InthepresentworkwestudyboththehistoryversionofstandardquantummechanicsandIsham’sgeneralhistoryquantumtheories.InthelattercasewerestrictourselvestothesituationwherethespaceofhistoriesisgivenbythespaceofprojectionsonsomeHilbertspace.In[11]Isham,Linden1andSchreckenbergstudiedoperatorrepresentationsfordecoherencefunctionalsinthefinitedimen-sionalcase.Forinfinitedimensions,Wright[18]obtainedthecanonicalrepresentationofboundeddecoherencefunctionalsbyquadraticformsonvonNeumannalgebras.AsaspecialcaseofWright’sresultsin[18]