Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(21)

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

F.QuantumInitialStates:Examples

Wechooseour rstexampleinitialstateΨWPtobeapositivefrequencysolutionto(3.5)thatislocalisedwellwithinmixmaster’spotentialwallsontheinitialsurface,andalsoawayfromregionsofverylargeVφ.Suchaninitialstatewillpreferaparticularclassicalpathandpredictapproximatelyclassicalbehaviouralongthecorrespondingclassicalsolution.ΨWPbyassumptionsolves(3.5)withV≈0andislocalizednearsome q0onσ0.Itmaythenberepresentedas

+ WP+(pΨWP(q)=d3pΨ )Φp (q q0)

d3p1p·(q q 0) WP+=e iωp(α α0)ei Ψ(p )(4.25)32ωp(2π)

1/2 WP+ WP+(pforsomeΨ ),whereofcourseωp=|p |.TakingωpΨ(p )tobeaGaussian

centeredaroundsomep 0,forinstance,yieldsaΨWP(q)localizedonσ0about( q0,p 0)tothegreatestextentconsistentwiththeuncertaintyprinciple.

Alternatively,wecouldconsideraninitialstateofWKBform,

ΨWKB(q)=A(q)eiW(q),(4.26)

whereA(q)isinanappropriatesenseslowlyvaryingrelativetoW(q).Approximatecalcula-tionsofthewavefunctionscorrespondingtothevariousproposalsfortheinitialconditionoftheuniversetendtohavecomponentsofthisform.WealsorequirethatA(q)beofcompactsupportonσ0soastoensurethatΨ0∈F(σ0),andthatW(q)be“positivefrequency”inthesensethat W/ α<0onσ0.(Thiswillturnouttocorrespondtoan“expanding”initialconditiononthepathsthiswavefunctionde nes.)

Wewillexaminebothofthesechoicesinthesequel.First,however,weshallmoveontodiscussthecategoryofcoarsegrainingsde ningapproximatelyclassicalbehaviourthatweshallconsider.

V.APPROXIMATECLASSICALITY

Inthissection,weapplythedecoherencefunctional(4.22)forΛ=0,BianchiIXcos-mologiestocoarsegrainingswhichdistinguishbetweenthosepathsinminisuperspacewhichbehave(semi-)classically,andthosewhichdonot.We ndthatforsuitablechoicesoftheinitialconditionΨ,andforasuitableclassofde nitionsofthesemiclassicalcoarsegrain-ings,theuniverseispredictedtobehaveclassicallywithprobabilitynearone.Inparticular,Λ=0BianchiIXcosmologiesarepredictedquantum-mechanicallytorecollapsejustastheydoclassically[29].Thisissatisfying,notleastbecauseofthesurprisingpredictionofWald’s

[10,26]rigorouscanonicalquantizationofBianchiIXthatclosedquantumuniversesdonotrecollapseinaclassicalfashion.18(Foranotherdiscussionofdi cultiesinterpretingwavefunctionsforclassicallyrecontractingcosmologies,see[49].)