Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(5)

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

whereHisthesuper-Hamiltonian

2224α6α6αH= p2α+p++p +pφ+eVβ(β+,β )+eVφ(φ)+eΛ

afterrescalingφ,Vφ,andΛbypositiveconstants((2.5)

2β4βtr1 2e+e,becauseVais3positivede nitewithaglobalminimumof0atβ=0,andistriangularlysymmetricabout

β=0.)ThehomogeneousisotropicFRWuniversemayberecoveredbysettingβ=0intheequationsforBianchiIX.Forthem(3)R=(3/2)e 2αandVβ= (6π)2.(NotetheusualFRWscalefactora=2eαafterchangingfromtheEuler-angleσibasistothestandardpolarcoordinatesonS3.)

Thecon gurationspacefortheseminisuperspacemodelsisspannedbythevariables(α,β+,β ,φ).Thatis,itisthesuperspaceofspatialgeometriesplusthespaceofscalar eldvalues.Theaction(2.4)maybeexpressedinamorecompactformbyintroducingthenotationqA,A=0,1,2,3forthefourvariables(α,β+,β ,φ),the atLorentzianDeWittmetricGAB=diag( 1,1,1,1),andarescaledlapsefunctionN(t)=e 3α(t)L(t)/24π.Then

1 AASpA,q,N=dtpAq˙ NH(2.8)

where

H=GABpApB+e4αVβ(β+,β )+e6α[Vφ(φ)+Λ]

=GABpApB+V(α,β+,β ,φ,Λ),(2.9)

andwehavetakenadvantageofthearbitrarinessintasacoordinatelabeltoassignthevalues0and1totheendsoftherangeofintegration,achoiceweshallmakewhereverconvenient.Inthisformtheanalogywitharelativisticparticlemovinginapotentialisclear[1,28]andweshallexploitthisinwhatfollows.

Theformofthemetric(2.1)isleftunchangedbyreparametrizationsofthetimet→f(t).Asalreadynoted,thisinvarianceistheremnantofthefour-dimensionaldi eomorphisminvarianceofthefulltheoryofgeneralrelativityoncethediagonalform(2.1)hasbeen xed.Correspondinglytheaction(2.8)isinvariantunderreparametrizationtransformationsoftheform

qA(t)→q A(t)=qA(f(t)),

pA(t)→p A(t)=pA(f(t)),

(t)=N(f(t))f˙(t),N(t)→N

(2.10a)(2.10b)(2.10c)