Quantum Logic and the Histories Approach to Quantu

arXiv:gr-qc/9308006v19Aug1993Imperial/TP/92-93/39QuantumLogicandtheHistoriesApproachtoQuantumTheoryC.J.IshamBlackettLaboratoryImperialCollegeSouthKensingtonLondonSW72BZUnitedKingdomAugust1993AbstractAnextendedanalysisismadeoftheGell-MannandHartleaxiomsforagen-eralised‘histories’approachtoquantumtheory.Emphasisisplacedonfindingequivalentsofthelatticestructurethatisemployedinstandardquantumlogic.Particularattentionisgivento‘quasi-temporal’theoriesinwhichthenotionoftime-evolutionislessrigidthaninconventionalHamiltonianphysics;theoriesofthistypeareexpectedtoarisenaturallyinthecontextofquantumgravityandquantumfieldtheoryinacurvedspace-time.Thequasi-temporalstructureiscodedinapartialsemi-groupof‘temporalsupports’thatunderpinsthelatticeofhistorypropositions.Non-trivialexamplesincludequantumfieldtheoryonanonglobally-hyperbolicspacetime,andasimplecobordismapproachtoatheoryofquantumtopology.Itisshownhowthesetofhistorypropositionsinstandardquantumtheorycanberealisedinsuchawaythateachhistorypropositionisrepresentedbyagenuineprojectionoperator.Thisprovidesvaluableinsightintothepossiblelatticestructureingeneralhistorytheories,andalsoprovidesanumberofpotentialmodelsfortheoriesofthistype.1Introduction1.1PreliminaryremarksAgoodillustrationofthegreatdifficultyencounteredwhenattemptingtoconstructafullycomprehensiveaccountofquantumgravityisthestrikinglackofanythingremotelyresemblingan‘axiomatic’schemeforsuchatheory.Thepointatstakeisnotoneofmathematicalrigourperse,butrathertheneedforaclearstatementaboutwhatoneisreallytryingtodo.Thisuncertaintyisreflectedinthewiderangeofdeclaredviewsonhowtoapproachthesubjectand,indeed,onwhatwouldconstituteasuccessfulresolutionoftheproblem.Thelackofanyquasi-axiomaticschemeisregretable,buthardlysurprising,notleastbecauseofthefrequentlyexpressedviewthatthewell-developedspace-timeconceptsofclassicalgeneralrelativitymaybeinappropriateatscalesofthePlancklengthandtime.Oneexampleisthewell-known‘problemoftime’thatarisesinattemptstoconstructacanonicaltheoryofquantumgravity1.Thisproblemfirstappearsinthecanonicalapproachtoclassicalgeneralrelativitywherethenotionoftimeasanexternalparameterdisappearsandhastoberecoveredinsomewayfromtheinternalcontentsofthesystem.Intheanalogousquantumtheoryopinionsdifferonwhether(i)theconceptof‘time’isfundamental;or(ii)ournormalideaoftimeisacoarse-grainedconceptthatemergesonlyabovethePlanckscaleinatheoryofquantumgravitythatisfundamentally‘timeless’.Intheformercase,amajorissueiswhethertheidentificationofaninternaltimeshouldbemadebefore,orafter,quantisation.Notunsurprisingly,theproblemoftimeisofmajorimportanceinquantumcosmology,particularlyinattemptstodescribetheoriginoftheuniverseassometypeofquantum-mechanicalevent.OneofthemoreadventuroussuggestionsforresolvingthisissuewasmaderecentlybyGell-MannandHartleinthecontextofthe‘consistenthistories’approachtoquantumtheory.Theconsistenthistoriesprogrammeforstandardquantumtheoryiscurrentlygeneratingsubstantialinterest,inspiredmainlybyamajorseriesofpapersbyGell-MannandHartle[7,8,6,14,15,9,16]whichpartlyfollowonfrom,andarepartlyindependentof,seminalworkbyGriffiths[11]andbyOmn`es[30,31,32,33,34,35].Thisnewperspectiveonquantumtheoryisofimportanceforavarietyofreasons,notleastofwhichisthepossibilityitaffordsofescapingthethrallsoftheCopenhageninterpretationwithitsinfamousmeasurementproblemandassociatedconceptofstate-vectorreductioninducedbyanexternalobserver.Whetherornotthishopeisjustifiedhasbeenthesubjectofmuch,sometimesacrimonious,debate,althoughmostworkersinthefieldagreethat,attheveryleast,theideaofconsistenthistoriesprovidesvaluableinsightintotherelationbetweenclassicalphysicsandquantumphysicsand,inparticular,thesenseinwhichtheformermaybesaidto‘emerge’fromthelatter.However,themotivationofthepresentpaperisnotadesiretoeludetheclutchesof1Forrecentreviewssee[20,18].1CopenhagenepistemologybutratherthesuggestionofGell-MannandHartletousethemethodologyofconsistenthistoriestotackletheproblemoftimeinquantumgravity.Atafirstglance,itseemsoddtoaddresstheproblemoftimeusingthenotionof‘history’,aconceptthatissurelyintrinsicallytemporal?ButGell-MannandHartlerefutethisbyassertingthata‘history’neednotnecessarilyberegardedasatime-orderedstringofeventsorpropositions:itcouldappearasafundamentaltheoreticalentityinitsownright.Asimpleexample[16]ofthisideaisgivenbyaglobally-hyperbolicLorentzianmetricγonamanifoldM.Anysuchspace-timecanbeexpressed(inavarietyofways)asaone-parametert(time)familyofRiemanniangeometriesgtonaspatialmanifoldΣwith,topologically,M≃Σ×IR.Inthissense,γisapossiblehistoryoftheuniverse:i.e.,apatht7→gtintheinfinite-dimensionalspaceof3-geometriesonΣ.Butsupposenowthatγisnotgloballyhyperbolic.Thenotionoftimeisstillwell-definedlocallyon(M,γ)but,sincetherearenoglobaltimefunctions,itisnolongerpossibletointerpretγasatemporalsequenceofthree-geometries;however,γmaystillbeperfectlyacceptableasaspace-timegeometry,andhenceasa‘generalisedhistory’oftheuniverse.NotethatametricofthistypemaystillpossesssomespacelikehypersurfacesthatdivideMintotwopiecesand,tothatextent,someglobaltimestr