Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(14)

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

TheFadeev-Popovdeterminant G=detδG/δ ～detd2/dt2isinthiscaseindependentoftheintegrationvariables.Thedeltafunctionalin(3.4)thenpermitsonlytheN=constantpathstocontributetotheintegraloverN(t),leaving

1 ′′q q′′ Ch q′ =dN≥0δpδqqidt[pAq˙A NH](3.14)′expch0

afterdroppingtheconstantfactorsthatcancelinthedecoherencefunctional(3.9).Changingvariablesto

ds=Ndt,(3.15)

theGaussianfunctionalintegralsoverpmaybeperformed,leavingsimply

∞

q′′ Ch q′ =dN q′′N Ch q′0

0(3.16)

solongasweassumethecoarsegrainingdoesnotrestrictthevalueofN;otherwisetherangeoftheNintegralmustberestrictedappropriatelyaswell.Herewehavede ned

1 1q′′ q′′N Ch q′0 =δqqidt′exp

ch0 AdqGAB V,(3.17b)4ds

wherethepathintegralmeasureδqhasbeenrenormalizedintheusualmannerinducedbythemomentumintegrations.(Seetheappendixfordetails.)Thenotationonthelefthandsideof(3.17a)isinspiredbytheobservationthatthepathintegralin(3.17b)sharestheformofthatforthepropagatoroveratimeNofarelativisticparticleinapotentialV.(3.16)maybethoughtofasa“restrictedpropagator”fortheclassofpathsch,asshouldbeevidentfromtherestrictedfunctionalintegral(3.4).

Ingeneral,thecomplexityoftheminisuperspacepotentialVprecludesmuchfurtherex-plicitprogressinthenon-perturbativeevaluationof q′′ Ch q′ .Itisstillpossible,however,toshowthattheclassoperatorssatisfytheconstraint(3.5)forcoarsegrainingswhichdonotrestrictthevaluesoftheendpointsq′andq′′orthevalueofN.11Thisisdoneintheappendix.(3.8)isthusasclaimedindependentofthesurfacesonwhichwechoosetoimposetheboundaryconditionsΨandΦ.

IV.DECOHERENCEFUNCTIONALFORRECOLLAPSINGBIANCHICOS-MOLOGIES

ThegeneralformofthedecoherencefunctionalfortypeAhomogeneouscosmologiesisgivenin(3.9).Inthissectionweemploysomespeci cchoicesoftheinitialand nalconditionsapproriatetoΛ=0closed(BianchiIX)cosmologiestocastthedecoherencefunctionalintoasimplerandmorepracticallyusefulform.