Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(19)

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

α=2β+.Wellinsidethesewalls,wemaytakeVtobeessentiallyzeroforreasonableVφ.16

±Neartheoriginofthespacelikesurfaceσ0,theΦp (q)canthenbetakentohavetheform

±Φp (q)=1p·q e iωpαei ,n(2π)2ωp(4.13)

2whereωp=p 2.IntheasymptoticregionwherethepotentialVisnotnegligiblewemaychoosetomaintaineither(3.5)or(4.11),sinceourbranchwavefunctionsarebydesignessen-tiallyzerothere.Shouldwechoosetoadherestrictlytotherequirement(3.5)everywhere±onσ0,theΦp willofcoursebemorecomplicatedintheasymptoticregion.Practically,however,itisusuallyeasiesttoretaintheform(4.13)everywhere.Solongaswerestrictattentiontoinitialstatesandcoarse-grainingsforwhichB|σ0 F(σ0)thedistinctionwillbepracticallyirrelevantandtheexplicitform(4.13)canbeusedincalculatingthedecoherencefunctional.

Thebasisoffunctions(4.13)areorthogonalintheappropriatesense,

±±Φp Φp ′=±δp p ′,(n)(4.14)

(4.15)and

(nisthenumberof“spatial”minisuperspacedegreesoffreedom–three,inthepresentexample,β±and φ.)TheprojectionsI±of(4.7)maythenbeexplicitlyconstructedas (takingip′′dnp)i→ ′′′′I±(q2,q1)=±′′± ′′dnpΦp (q2)Φp (q1).± Φp Φ p ′=0.(4.16)Finally,wenotethattheorthonormalbases{Φ±i}arealsousefultorepresenttheinitialconditionΨ.Indeed,solutionstotheWheeler-DeWittequationinF(σo)maybewrittenas

++ Ψ=ciΦi+c iΦi

=Ψ+Ψ i+i(4.17)

where

whereFortheparticularchoiceofbasis(4.13)forF(σo),thislookslike + +(p )Φ (q)Ψ(q)=dnpΨ )Φp (q)+Ψ(pp ± ±(pΨ )=±Φp Ψ±c±i=±Φi Ψ.(4.18)(4.19)onσ0.(4.20)