Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(15)

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

A.InitialandFinalConditions

Asalreadynoted,itistobeexpectedthattheinitialquantumconditionsoftheuniverseare xedbysometheoryofcosmologicalboundaryconditions.Knowledgeofthistheoryisnot,however,requiredintheconstructionofthedecoherencefunctionalforcosmology.Weshall,therefore,illustratetheprocessofpredictionbasedonthedecoherencefunctional(3.9)withsomesimplechoicesofinitialstate.Thepracticalsigni canceofsuchpredictionsdependsentirelyonwhetherthechoseninitialstatesarerepresentativeoftheboundaryconditionsontheactualphysicaluniverse.

Mostoftheextanttheoriesoftheinitialstateoftheuniverse[6,42]produceabound-arystateconsistinginasingleinitialwavefunction.WewillthereforeinourexamplesconcentrateentirelyonthecaseofapureinitialstateΨ.

RecallfromSectionIIIA1thattheBianchiIXcosmologicalhistorieshave“ends”atwhichweimposeboundaryconditions{Ψi,p′i}and{Φi,p′′i}.Inordertocorrespondtoaconventionalnotionofcosmologicalboundaryconditionsforclosed,Λ=0universes,weimposetheboundaryconditionsonasuitablesurfaceσ0oflarge,negativeα(=α0),i.e.whentheuniverseisverysmall.BecausethewavefunctionsΨiandΦisatisfytheconstraint(3.5),itdoesnotmatteronwhichsurfacetheyareimposedsolongasthecoarsegrainingsunderconsiderationdonotinvolveregionsofminisuperspaceintersectingthosesurfaces,asnotedabove.Inthissense,then,thecosmologicalhistories“begin”and“end”atsmallspatialvolume,justastheclassicalhistoriesdo.(SeeFigure1.)

Maintainingaclosecorrespondencewithordinaryquantummechanics,weshallchoose nalboundaryconditionswhichare“indi erent”tothepaths,inasensetobemadepreciseinSectionIVC.Therewillthene ectivelybeno nalconditionsatallonthecosmologicalhistories.

Asforthechoiceoftheinitialboundaryconditions,wewillexaminetwoinstructiveexamplesinthesequel.Inordertoillustratehowthepresentsum-over-historiesconstructioncanpredictsemiclassicalbehaviourinasuitablelimitasnaturallyassum-over-historiesformulationsalwaysdo,asa rstexampleweconsiderasingleinitiallocalizedWheeler-DeWittwavepacketΨWPwhichisdesignedtopreferaparticularclassicalpathoverallothersinthesemiclassicallimit.Inapartitionoftheminisuperspacehistoriesbyclassicalpaths,asteepestdescentsevaluationofthepathintegralfortheclassoperatorsthenrevealsthattheprimarycontributiontothedecoherencefunctionalcomesfromtheclassoperatorcorrespondingtothecoarse-grainedclassofpathscontainingtheclassicalpathpreferredbytheinitialconditionΨWP.

Next,weshallturnourattentiontothesemiclassicalpredictionsofthedecoherencefunctionalwithaninitialstateofthemoretraditionalWKBform.Asistobeexpected,undersemiclassicalcoarsegrainingssuchachoiceofinitialstateleadstoadistributionofclassicaltrajectorieswithinitialvalueweightedaccordingtotheWKBprefactor,andinitialmomentumgivenbythegradientofthephase.

Letusnowproceedwiththedetails.