Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(17)

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

Forarbitrarycoarsegrainings,itisnotimmediatelyevidentthatΨh(q)forq∈σ0mustbeinthespaceF(σ0)offunctionswhichfallo rapidlyatin nityonσ0,evenwhentheinitialwavefunctionsΨ(q)arerequiredtobeinF(σ0),aswehavedone.However,wewilltaketheargumentsofHiguchiandWald[10]asstronglysuggestivethatinitialstatesinF(σ0),propagatedbythefullpropagator q′′ Cu q′ o ofσ0andthenbacktoα→ ∞,indeedremaininF(σ0)whenthereisascalar eldpresent.13Forthepresent,wewillassumewhennecessarythatattentionisrestrictedtocoarsegrainingsforwhichallΨh(q)∈F(σ0)whenq∈σ0i.e.forwhichB|σ0 F(σ0).

C.FinalIndi erence

Boundaryconditionsof“ nalindi erence”shouldamount,inessence,tono nalbound-aryconditionatall,inthesensethattheendpointsofallpathsareweightedequally.The nalboundaryconditionthene ectivelydisappearsfromthedecoherencefunctional,justasitdoesinordinaryquantummechanics.

Letusbeslightlymorepreciseabouthowtodothis.

WhatweseekareacompletesetofsolutionstotheWheeler-DeWittequation(3.5)whichmayservetode neapositive“resolutionoftheidentity”inthespaceofbranchwavefunctionsB.Bythiswemeanthefollowing.First,notethatifwede ne

′′′′′′ ′′I(q2,q1)=p′′(4.2)iΦi(q2)Φi(q1)

i

′′′′forq1,q2∈σ0,thedecoherencefunctional(3.9)forthecaseofapureinitialstateΨmaybewrittenverysimplyintermsofthebranchwavefunctions(4.1)as

′D(h,h)=Np′′i(Φi Ψh)(Φi Ψh′)

i

=NΨh′ I Ψh.(4.3)

Inordertocapturethenotionof“ nalindi erence”,wenextsplitBintoorthogonalsectorsB±onwhichtheWheeler-DeWittproduct ispositiveornegativede nite.14Werequirefor nalindi erencethattheΦibechosensothat

I ψ±=±ψ±,(4.4)