Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(9)

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

asα→∞.)Finally,weaddtherestrictionthatallpathspossessα˙(0)>0andα˙(1)<0onσ0,i.e.“expanding”initialand“contracting” nalconditions.

2.Coarse-grainedHistories

Coarse-grainedhistoriescorrespondtothephysicalquestionsthatmaybeaskedofasys-tem.Wethereforeallowascoarse-grainedsetsofalternativehistoriesanypartitionofthe ne-grainedhistoriesintoreparametrizationinvariantclasses{ch},h=1,2,···,becausereparametrizationinvarianceiswhatremainsinminisuperspaceofthedi eomorphismin-varianceofgeneralrelativity.TheclassesgenerallymaybethoughtofaspartitionsbyvaluesofreparametrizationinvariantfunctionalsF[qA,N]ofthepathsinCext.Explicitly,foranexclusivesetofranges{ h},h=1,2,···oftherealline,andasinglefunctionalFwede ne A Ach=(q(t),N(t))|F[q,N]∈ h.(3.2)Anypartitionmaybethoughtofasofthisformbecausewemayalwaysconsiderthefunc-tionalwhichhasthevaluehforpathsinchandasetofrangesthatbrackettheintegers.Simpleexamplesofinterestingpartitionsintodi eomorphisminvariantclasses–herereducedtoreparameterizationinvariantclasses–arereadilygiven:

Onecouldpartitionthehistoriesbyrangesofvaluesofthevolumeofthelargestvolumethreesurfaceofhomogeneity.Theresultingprobabilitiesareforthevaluesofthevolumeofmaximumexpansionoftheuniversereachedinthecourseofitsexpansionandcontraction.

Onecouldpartitionthehistoriesofthesehomogeneousspacetimesintotheclasseswhichhaveasurfaceofhomogeneitywithavolumelessthan andtheclassofthosewithnosuchsurface.Thevalueoftheprobabilitythatthereisasurfacewithvolumelessthan consideredas →0wouldbeonewayofassigningaprobabilitytotheuniversebecomingsingular.6