Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(4)

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

“BianchiI”through“BianchiIX”models.The(simply-connectedcovering)groupmanifoldwithitsnaturalmetricisthemanifoldforspatialgeometry.IntheBianchiImodels,forexample,thegroupisgeneratedbythetranslationsofthree-dimensional atspaceandthemanifoldisR3.IntheBianchiIXmodelsthegroupisSU(2),ckij=εijk(inappropriatecoordinates),andthemanifoldisthethree-sphereS3.TheclassicalFriedmann-Robertson-Walker(FRW)modelsarethemostfamiliarexamplesofBianchiuniverses:theopenFRWuniverseisofBianchitypeI,the atuniversetypeV,andtheclosedFRWuniverseisoftypeIX.

The“typeA”Bianchimodelsarethoseforwhichciij=0;therestarecalled“typeB”.WerestrictattentiontotypeAmodelsbecausetheactionprincipleforthetypeBBianchimodelsdeducedfromthatforgeneralrelativitybysubstitutionofthehomogeneityansatz(2.1)doesnotleadtothecorrectequationsofmotion[24];inthesecasesthehomogeneityofthespatialmetricobstructstheeliminationofboundarytermsproportionaltothetraceofthestructuretensor.TheBianchitypesI,II,VI0,VII0,VIII,andIXarealltypeA.Avarietyofmattercontentsareconsistentwithhomogeneity.Asanillustrativeexampleweshallrestrictattentiontoasingle,minimallycoupled,homogeneousscalar eldφ(t)togetherwithapositivecosmologicalconstantΛ.Fortheactionofthescalar eldwetake

SM[g,φ]= 1 g( φ)+Vφ(φ) 2

forsomepotentialVφ(φ).[Wefollow,asfaraspossible,theconventionsof[20]withrespecttosignature( ,+,+,+),de nitionsofthecurvaturetensors,theextrinsiccurvatureofahypersurface,etc..Weemployunitswhere =c=G=1.]

AcanonicalactionforthetypeABianchicosmologieswithscalarmattermaybearrivedatinthefollowingmanner.FirstnotethatforallthetypeAmodelswith“diagonal”matter(T0i=0),asinourexample,itispossibleclassicallytochoosetheone-formsσiin(2.1)sothatthematrixβijisdiagonal,andtheckijtaketheircanonicalvalues[18,19];theEinsteinequationsguarantee thatβ√remainsdiagonalastime passes.Itisthentraditionaltoparametrizeβasβ=diagβ++3β , 2β+.

Assumingthatthemetricremainsdiagonalquantummechanicallyisequivalenttosolvingtheclassicalmomentumconstraintsbeforequantization[10](cf.[20,equationE.2.34]);theonlyremaininggaugefreedomthenliesinthetime-reparametrizationsofthesurfacesofhomogeneity.Weshallassumethatthemetricremainsdiagonalinthesequel.However,thequantumtheorythusobtainedisnotobviouslyequivalenttoatheoryinwhichthemomentumconstraintisinsteadimposedasanoperatorcondition.4 Insertingadiagonalhomogeneousmetric(2.1)intotheactionS=(1/16π)(R 2Λ)+SManddoingthespatialintegrationoverastandardcoordinatevolumeof(4π)2(thecoordinatevolumeofBianchiIX’sSU(2)closedspatialmanifold;cf.[25,Box30.1])yieldsthereducedminisuperspaceaction

3α ˙++p β˙ +pφφ˙ LeS=dtpαα˙+p+β

4 (2.3)Theissueisthatwhilethemomentumconstraintimpliesthatthespatialmetricmayalwaysbediagonal-izedatanyonemomentoftime,theclassicalequationsofmotionarerequiredtoshowthatitremainssothereafter.TheseissuesarediscussedwithcharacteristiclucidityinsectionIVof[10].