Noncommutative Conformally Coupled Scalar Field Cosmology an(2)

Pa=

+

a

=NH,(4)

2aa˙

N

.(5)

AsthePoissonBracketsfortheclassicalphasespacevariableswehave

{a,χ}=0,{a,Pa}=1,{χ,Pχ}=1,{Pa,Pχ}=0.

(6)

Theequationsofmotionforthemetricandmatter eldvariablesa,Pa,χandPχthatfollowfrom(4)and(6)

are

a˙={a,H}= NPa/2a, ˙a={Pa,H}=2N, P

FromnowonweshalladoptconformaltimegaugeN=a.Thegeneralsolutionof(7)foraandχinthisgauge

is

a(t)=(A+C)cos(t)+(B+D)sin(t),

(8)

χ˙={χ,H}=NPχ/2a, P˙χ={Pχ,H}= 2Nχ/a.

(7)

χ(t)=(A C)cos(t)+(B D)sin(t),

wherethesuper-HamiltonianconstraintH≈0imposestherelation

AC+BD=0.

(9)

(10)

Asitcanbeseen,theclassicalcommutativesolutionsarenecessarilysingularinthepastandinthefuture.

Figs.1(a),(b),(c)and(d)presentplotsofthesolutionfora(t)inthedashedcurvesforgivenvaluesofA,B,andC[Dis xedby(10)].

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

III.

NONCOMMUTATIVEDEFORMATIONOFTHECLASSICALMODEL

LetusintroduceanoncommutativeclassicalgeometryinouruniversemodelbykeepingtheHamiltonianwiththesamefunctionalformas(4),

but

now

valued

on

noncommutative

variables,

2

P

aχ2nc

anc+ncH

=

N

4anc

2

Pχc,χnc=χc+

θ

2

Thesituationhereis,inacertainsense,similartothatofgeneralizedtheoriesofgravitation.Therearetwodi erentframesforphysicalinterpretation,astheEinsteinframeandtheJordanframe[26].Thereisamappingbetweenthem,astheconformaltransformationthatmapsthetheJordanintotheEinsteinframe.However,asinthegeneralizedtheoriesofgravitation,thenoncommutativecosmologiesinthedi erentframesarenotphysicallyequivalent.Thereasonisthesameasthere:thetheoriesarenotthesamebecausethephysicalobjectstheyrefertoarenotidentical.

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

Theequationsofmotionforthevariablesac(t),χc(t),Pac(t)andPχc(t)are

a˙c={ac,H}= 1

2

˙χc={Pχc,H}= 2χc θPac.P

1 θ

2

(15)

Pχc+θac,

Accordingtothevaluesofθ,thesystem(15)allowsthreetypesofsolutions,whichwedescribebelow.

1.

Case|θ|<1

Thegeneralsolutionsforac(t),χc(t),Pac(t)andPχc(t)intheθ<1caseare

√ √

1 θ2t+B1 θ2tac(t)=eθtA

+e

θt

√

χc(t)=eθtA

√ e θtC

√C

1 θ2t+D

1 θ2t+D1

θ2t +2e

√

1 θ2t+B

√

Dcos Dcos

√

1 θ2t

(16)

,

1 θ2t1 θ2t

,

(17)

Pac=2e

θt

BcosBcos

θt

1 θ2t,(18)

Pχc=2e

θt

From(13),(16)-(19)wecancalculateanc(t)andχnc(t)as

√ √

1 θ2t+Banc(t)=eθtA

√

+e θtC

θt

1 θ2t

+2e

θt

1 θ2t

.(19)

1 θ2t1 θ2t

√

1 θ2t+D

1

θ2t

,

(20)

√

χnc(t)=eA √ e θtC

TheconstraintH≈0inthepresentcasecanbewrittenas

(AC+BD)

√

1 θ2t+D

√+B

1 1 θ2t

θ2t

.

(21)

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

a t

50403020100 10

4

a

a t

30

151050

5

2.5 2 1.5 1 0.5 c

0.5

d

t

2520151050 1

0.5

0.5

1

1.5

t

2

2

4

t

50403020100 10 b

a t

4

2

2

4

t

a t

FIG.1:ThetypicalbehaviorofthescalefactorofthenoncommutativeFRWuniverseintheNC-framerealization(thicklines)incontrastwithC-framerealization(thinlines).Thescalefactorofthecommutativecounterpartappearsplottedinthedashedlines.(a):θ=3/4,A=5,B=3andC=6.(b):θ=1,A=3,B=2andC=1.(c):θ=3/2,A=4,B=3andC=1.(d):θ=3/2,A= 1.2,B=2andC= 1.

2.Caseθ=±1

Whenθ=±1,thesolutionsforac(t),χc(t),Pac(t)andPχc(t)are

ac(t)=Acosht+Bsinht,

(23)

χc(t)=±Bcosht±Asinht,(24)

Pac=2(D+At)cosht+2(C+Bt)sinht,(25)

Pχc= 2(C+Bt)cosht 2(D+At)sinht.

Asthecorrespondinganc(t)andχnc(t),wehave

anc(t)=(A+C+Bt)cosht+(B+D+At)sinht,

(26)

(27)

χnc(t)=±(B+D+At)cosht±(A+C+Bt)sinht.(28)

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

TheconstraintH≈0inthepresentcasecanbewrittenas

A2 B2+2(AC BD)=0.

(29)

Asitcanbeseen,theuniversesolutionsinthecaseθ=±1arecharacterizedbyaqualitativebehaviorthatdi ersfromtheonecorrespondingtothecase|θ|<1.Thereexistnon-singularbouncingsolutionsforbothanc(t)andac(t),asdepictedinFig.1(b).However,thecorrespondencebetweentheNC-andC-framescanbebrokenforsomevaluesoftheintegrationconstants.Non-singularsolutions …… 此处隐藏：4185字，全部文档内容请下载后查看。喜欢就下载吧 ……