Noncommutative Conformally Coupled Scalar Field Cosmology an(5)

4

FIG.4:TheminimumvalueofthescalefactorofthenoncommutativequantumFRWuniversewith|θ|<1intheNC-framey=anc0asafunctionofitsvalueintheC-framex=ac0.Thethicklinereferstothecasewhereαθ<0,andthethinlinetothatwhereαθ>0.Thevaluesadoptedforαθareαθ= 1andαθ=1.Thediagonallineisalsoplotted.

propertyexhibitedinthiscase.From(85)wecanseethattoeachuniverseintheNCframetherecorrespondsatleastoneuniverseinthe C-frame,asitisshowninFig.4.Forαθpositive,thecorrespondenceisonetooneandexistsonlyforac0>

αθ/2,

wherethecurveanc0×ac0achievesitsminimumandthecorrespondenceisone-to-one.Thisvalueofac0marksthedivisionbetweenthetworegimesthatgoverntheNC-C-framecorrespondence:largeanc0andsmallac0andoflargeanc0andac0.Asimilarbehaviorwaspreviouslyfoundin[28]innon-relativisticBohmianquantummechanicswhenstudyingtheharmonicoscillator.Thecapabilityofthenoncommutativitye ectstopromotetheinterplaybetweenlargeandsmallscaledistanceswasinterpretedinthatreferenceasmanifestationofasortof“UR-UVmixing”intheoscillatororbits.

2.Example2

Thewavefunctionfunction(80)isasumoftwofunctions,

Ψ(ξ,η)=A1ξ 1Wµθ/2√

√

β,iν/2

2ξ2β

eiνη.

(86)

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

Thecorrespondingphaseis

S(ξ,η)=arctan

A1Wµθ/2√

β,iµ/2

β,iν/2

sin(νη)

cos(µη)+A2Wνθ/2√

dt

= 2ξ

β,iµ/2

2√A21Wµθ/2

Wνθ/2√

′√

β,iµ/2Wνθ/2

β,iν/2cos[(µ ν)η]

+2A1A2Wµθ/2√

β,iν/2

+A1A2(µ+ν)Wµθ/2√

√

β,iµ/2Wνθ/2

,(88)

dη

2ξ2

2√µA21Wµθ/2

β,iµ/2

β,iν/2

2√+A22Wνθ/2

β,iν/2

cos[(µ ν)η]

2.Againitsispossibleto ndbouncingsolutions,asitisshownin

Fig5(a),wherethesplittingbetweenNC-andC-frameevolutionsisquantitativelyirrelevant.Thee ectofnoncommutativityinthatcaseismanifestthroughthesuppressionofthesequenceofbouncesthatappearsinthecommutativecounterpartandbyshiftingthetimewheretheuniverseachievesitsminimumsize[seeFig.3(b)].

Thecasewhereµ= ν,whichinthecommutativecounterpartcorrespondstoastaticuniverse,herehasanon-trivialdynamics.OneexampleispresentedinFig.5(b),whereacyclicsolutionsimilartothatofFig.3(a)isdepicted.ThesplittingbetweenNC-andC-frameevolutionsinthiscaseisagainquantitativelyirrelevant.

B.

Caseθ=±1

Asintheclassicalcounterpart,thecaseθ=±1ismarkedbyapeculiarbehavior.Equation

(72)

inthiscaseisreducedtothe rstorderpartialdi erentialequation

i

dt

=0,

dη

2

ξ0cosh(±t+η0)+ξ0(±t+η0)sinh(±t+η0),(95)

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

χnc(t)=

1

|β|,iα/2

2iξ

2

|β|

eiαη+Bαξ 1Miαθ/2√

|β|,iα/2

2iξ|β|

2

eiαη.

(98)

Forthiswavefunction,theequationsofmotion(73)forξ(t)andη(t)canbeshowntobe

22iξ/|β|+1,iα/2

dξ

|β|ξ

2ξ

+

β

Wiαθ/2

√

|β|

,

(99)

dη

2ξ2

+θ.(100)

Theequations(99)and(100)canbesolvedanalyticallyinthelimitoflargeξ,wherethecontributioncoming

from thetermcontainingtheWhittakerfunctionsintherighthandsideof(99)canbeapproximatedby 2

dt=

|β|t

,

√

(102)

α

η(t)=24ξ0

1 e

2