Noncommutative Conformally Coupled Scalar Field Cosmology an(4)

← → ← → aχχccc ac2

g(ac,χc).(61)

ThecommutativecoordinatesacandχcarecalledWeylsymbolsoftheoperators aandχ ,respectively.A

Wheeler-DeWittequationthecanbeadoptedforthenoncommutativescalar elduniverseis

22 Ψ(ac,χc)+4a2 χ2 Ψ(ac,χc)=0, PPccχcac

(62)

whichisobtainedbyMoyaldeforming8(37).ByusingthepropertiesoftheMoyalproduct,itispossibleto

We study the implications of a noncommutative geometry of the minisuperspace variables for the FRW universe with a conformally coupled scalar field. The investigation is carried out by means of a comparative study of the universe evolution in four differen

10

8

6

Η

Η

4

2

a

Ξ 2

b

ΞFIG.2:The eldofdirectionsandselectedorbitscorrespondingtotheBohmiandi erentialequationsforthecommutativeFRWuniverseintwocases:(a):µ=0.6,ν= 0.58.Orbits:ξ0=1.2,η0=2andξ0=1.2,η0=4.(b):µ=0.6,ν=1.78.Orbits:ξ0=0.2,η0=3andξ0=1.7,η0=4.5.

writethe(62)as

2

22 a χ 2Ψ(ac,χc)=0,Pa PχΨ(ac,χc)+4

θ

2 ac,P

χ. a,P χc=P ac=PP

(63)

where

Equation(64)isnothingbuttheoperatorialversionofequation(13).Thenotationacandχcisnowjusti ed.

Thesesymbolsmatchexactlywiththecanonicalvariablesde nedby(13).Hereweareinfacewiththesamesituationasintheclassicalcase.Twoconsistentcosmologiesarepossible.Oneconsideringa candχ castheoperatorsassociatedwiththephysicalmetric,andtheotherconsideringa andχ. Intheapplicationproblemsweshallconsiderthetwopossibilities.

A.

NoncommutativeBohmianformalism

a= ac

(64)

Inordertodrawaparallelbetweenthenoncommutativequantumuniverseandtheotherthreeuniversetypes,

itisnecessarytohaveaprescriptionofhowtocomputeBohmiantrajectoriesinnoncommutativequantum

We study the implications of a noncommutative geometry of the minisuperspace variables for the FRW universe with a conformally coupled scalar field. The investigation is carried out by means of a comparative study of the universe evolution in four differen

Log a t

6543210

3000 a

10000

1000

3000

b 121086420

80 60 40 20

20

40

Log a t

FIG.3:ThetypicalbehaviorofthescalefactorofthecommutativeFRWuniverse.(a):µ=0.6,ν= 0.58,ξ0=1.2,andη0=4.(b):µ=0.6,ν=1.78,ξ0=1.7,andη0=4.5.

cosmology.ThesimplestwaytodothisisbyextendingtheBohmianformulationdiscussedinsection4alongthesamelinesproposedin[11].TheprocedureconsistsindepartingfromtheC-frameandusingthe“beable”mapping(46)toascribeanevolutionlawtothecanonicalvariables.Inourtimegaugeforthenoncommutativecosmology,N=anc(seesection3),theHamiltonian(11)reducessimplyto

2P2

h= anc a2(65)nc+χnc.4WecanthereforeusehtogeneratetimedisplacementsandobtaintheBohmianequationsofmotionforac(t)

andχc(t)as

dac

2

1 =Bi[h,χ c]=

S

1 θ2

χc

+θac.

(67)

dt

TheconnectionbetweentheC-andNC-framevariablesisestablishedbyapplyingthe“beable”mappingtotheoperatorialequations(64),thatis,byde ninga≡B( a)andχ≡B(χ ).OncethetrajectoriesaredeterminedintheC-frame,onecan ndtheircounterpartsintheNC-framebyevaluatingthevariablesaandχalongtheC-frametrajectories,

a(t)=B( a)|

ac=ac(t)

χc=χc(t)

=ac(t)

θ

2

acS[ac(t),χc(t)].(69)

Inwhatfollowsweshallillustratetheapplicationoftheformalisminnoncommutativequantumcosmology.

VI.

APPLICATIONTONONCOMMUTATIVEQUANTUMCOSMOLOGY

ByusingtherepresentationsPac= i acandPχc= i χc,wecanwritethenoncommutativeWheeler-DeWittequation(63)as

2 2

+4ac χ2β Ψ(ac,χc)=0,(70)c+4iθχc2 χc χc

We study the implications of a noncommutative geometry of the minisuperspace variables for the FRW universe with a conformally coupled scalar field. The investigation is carried out by means of a comparative study of the universe evolution in four differen

whereβ=1 θ2.Theseparationofvariablescanbemadebychangingtothenewsetofcoordinates

ac=ξcoshη,

whichallowustorewrite(70)as

2

β

χc=ξsinhη,

(71)

ξ

ξ2

2

η

4ξ2Ψ(ξ,η)=0.

(72)

ThecomputationoftheBohmiantrajectoriesisrenderedeasybyexpressingtheequationsofmotion(66)and(67)inthesamehyperboliccoordinatesasthewavefunction.Afterthechangeofvariables,theBohmianequationsofmotioncanbewrittenas

dξ

2

2 S(ξ,η)1 θ

dt=

1

η

+θ.

(73)

Theequations(68)and(69),responsiblebytheNC-C-framecorrespondence,canbewritteninthenewsetof

coordinatesas

anc(t)=ac(t)+

θ

2

θ

2

ξ 1coshη ηS[ξ(t),η(t)],

(74)

χnc(t)=χc(t)+

ξ 1sinhη ηS[ξ(t),η(t)].

(75)

BeforestartingourcomparativestudybycomputingBohmiantrajectoriescorrespondingtospeci csolutionsof(72),letusdiscussthecaseofrealwavefunctions.9WhileinthecommutativeBohmianquantumcosmologyrealwavefunctionsalwaysrepresentstaticuniverses,inthenoncommutativeBohmianquantumcosmologytheycanrepresentdynamicaluniverses,apropertypointedoutin[11]fortheKantowski-Sachsmodel.IntheFRWwithaconformallycoupledscalar eldunderconsideration,equations(73)tellusthattorealwavefunctionscorrespondalwaysanontrivialandidenticaldynamics.Moreover,from(74)and(75)wecanseethatwhenS=0theNC-andC-framerealizationsareindistinguishable,representingthesameuniverse.Thisuniverseisdeterminedbysolvingequations(73)andsubstitutingthesolutionsin(71).Asaresult,we nd

anc(t)=ac(t)=ξ0cosh(θt+η0),

(76)

χnc(t)=χc(t)=ξ0sinh(θt+η0).

plexwavefunctions,ontheotherhand,cangiverisetoagreatvarietyofdynamics,wherethedistinctionbetweentheframesofphysicalreali …… 此处隐藏：4699字，全部文档内容请下载后查看。喜欢就下载吧 ……