Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(10)

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

OnecouldpartitionthehistoriesintotheclasswhichremainclosetoasolutionoftheclassicalEinsteinequationbysomestandard(cf.(5.3))andtheclasswhichexhibitasigni cantexcursionawayfromclassicalbehavior.Theprobabilityofthe rstclassistheprobabilitythattheuniversebehavesclassicallyaccordingtothegivenstandard.WeshallemploysuchcoarsegrainingsinSectionV.

Onecouldpartitionthe ne-grainedhistoriesbyrangesofvaluesofthevolumeand anisotropyβ±theyassumeagiven“propertime”7Ldtaftertheinitialcondition.Theresultingprobabilitieswouldbetheprobabilitiesforthevolumesandanisotropiestheuniversecouldhaveatagivenpropertimefromtheinitialsurface.Thesearenotunliketheprobabilitiesthatwouldbeofinterestincomparingthepredictionsofarealisticquantumcosmologywithobservation.

Allofthecoarsegrainingsmentionedaboveareintomanifestlyreparametrizationinvari-antclassesof ne-grainedhistories.Mostarepartitionsthatarenotde nedbyalternativesthatareinanysensealternatives“atonemomentoftime”,andhencearenotde nedsimplybyobservablesonsuperspace.Rathertheyarespacetimealternativesreferringtopropertiesofhistoriesextendedovertime.Forexample,theprobabilitiesforclassicalbehaviorrefertowhetherasuitablycoarse-grainedhistoryapproximatelyobeystheEinsteinequationoveracourseoftime.

Coarse-grainedhistoriescorrespondtothephysicalquestionsthatmaybeaskedofasys-teminthefollowingsense.Askfortheprobabilitythattheuniversehasanyreparametriza-tioninvariantpropertyexpressibleinspacetimeterms.8Toanswerthisquestiononecon-sidersthepartitionofthe ne-grainedhistoriesintotheclasswhichhavethepropertyandtheclasswhichdonot.Ifthissetofcoarse-grainedhistoriesdecoheres,thenthequantummechanicspredictstheprobabilitythattheuniversehasthepropertyinquestion.Ifonecannottellwhetheragiven ne-grainedhistoryhasthepropertyornottheneitheritdoesnotmakesenseoritisnotexpressibleintermsofmetricand/ormatter eldcon gurationsalone.

Whileitiseasytoexhibitphysicallyinterestingsetsofalternativereparametrizationin-variantcoarse-grainedhistoriesforthesehomogeneousminisuperspacecosmologicalmodelsitismuchharderto ndsetsofsuchhistoriesthatdecohere.Thatisbecauseofthesmallnumberofdegreesoffreedomofthemodel.Coarse-grainingisessentialfordecoherence.Realisticmechanismsofdecoherencethataree ectiveinavarietyofinitialconditionsqual-itativelyinvolvethedissipationofphasesfromvariablesfollowedbythecoarse-grainingintovariablesthatareignored.9However,thepresentminisuperspacemodelsdonotpresentmanydegreesoffreedomtobeignored!