Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(12)

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

andwrite HΨi(q)= GAB 2

α

Havingintroducedtheproduct wenowde neΨ(α, q).(3.7)

Φi|Ch|Ψj =Φi(q′′) q′′ Ch q′ Ψj(q′).(3.8)

Thisde nitionappearsat rstsighttodependonthechoiceofsurfaceonwhich isde ned,butinfactitdoesnot,solongasthecoarsegrainingchdoesnotrestrictthepathsonthosesurfaces.Thisisbecausetheclassoperators(3.4)forsuchcoarsegrainingsgenerallysolve(3.5)aswellasΨandΦ,afactwedemonstrateintheappendix.First,however,wecompletethede nitionofthegeneralizedquantumtheoryofhomogeneousminisuperspacecosmologies.