Origin of a classical space in guantum cosmologies

4 Pages·0.13 MB·English

by  A. A. Kirillov

Gravitation & Cosmology, Vol. 7 (2001), No. 3 (27), pp. 211–214
(cid:13)c 2001 RussianGravitationalSociety
ORIGIN OF A CLASSICAL SPACE IN QUANTUM COSMOLOGIES1
A.A. Kirillov2 and G.V. Serebryakov3
Institute for Applied Mathematics and Cybernetics, 10 Ulyanova St., Nizhny Novgorod 603005, Russia
Received 31 March 2001
2
0 Theinfluenceofvectorfieldsontheoriginofclassicalspaceinquantumcosmologiesandonapossiblecompactification
0 process in multidimensional gravity is investigated. It is shown that all general features of the transition between
2 the classical and quantum evolution regimes can be obtained within the simplest Bianchi-I model for an arbitrary
n number of dimensions. It is shown that the classical space appears when the horizon size reaches the smallest
a of the characteristic scales (the characteristic scale of inhomogeneity or a scale associated with vector fields). In
J the multidimensional case the presence of vector fields completely removes the initial stage of the compactification

4 process which takes place in the case of vacuummodels [1].
2

One of the most important problems of quantum cos- whichisananalogueofJeans’wavelength(inthepresent
2
v mology is an adequate description of the origin of the paper we do not consider the inhomogeneous case, and
5 classical background space. Indeed, the transition from therefore we shall discuss the second possibility only).
4 apurequantumregimetoaquasiclassicaloneformsthe However, in multidimensional case the presence of vec-
2
initialpropertiesofthe earlyUniverseandthereforede- torfieldscompletelyremovestheinitialcompactification
2
termineswhetherthesubsequentstagescontainaninfla- stage,theprocessfoundinRefs.[1]. Thismaybeagood
1
0 tionary period and which kind of initial quantum state reason why the vector fields should not be included as
0 should be chosen for its realization [2]. In vacuum in- external fields, but should rather be composed from
/ homogeneousmodels, the originofclassicalbackground additional metric components in the compactification
h
t space was first investigated in Ref.[3]. It was found process.
-
p thatthebackgroundmetricbelongstotheclassofquasi- First, we note that the originof a backgroundspace
e isotropicspaces,andtheoriginoftheclassicalspacecor- is not a specific problem for quantum cosmology only.
h responds to the instant when the horizon size matches Such a problem does also exist in classical theory when
:
v the inhomogeneity scale of the space. However, a com- the gravitationalfield and matter sources are described
i plete investigationof the problem requires studying the by a probabilistic measure distribution with unstable
X
influence of different matter sources and also reserving statistical properties. Just this case was shown to be
r
a the possibility that our Universe has extra dimensions, realizedin approachingthe cosmologicalsingularity [6].
as predicted by a number of unified theories [4]. In the It turns out that in both cases (classical or quantum
latter case, the transition between quantum and classi- cosmology)the backgroundspaceformationmechanism
calregimesshouldincludetheso-calledcompactification is the same, while the nature of the statistical descrip-
stage [5]. And indeed, as was shown in Refs.[1], quan- tion is different. Moreover, estimates for the formation
tum evolution of inhomogeneous models in dimensions time of the backgrounddiffer by a factor which collects
smallerorequalthantenincludesaninitialstageofsuch all quantum corrections and depends on the choice of
a compactification. In this case the initial expansion of theinitialconditions(andalsoonthe specificschemeof
theUniverseproceedsinananisotropicwaywhen,along quantization). This ensures the correctness of our con-
theextradimensions,scalesdecrease,andtheexpansion sideration and provides a hope that whatever quantum
only occurs in three dimensions. gravitywillbeconstructedinthefuture,ourresultswill
In this paper we study the influence of vector fields survive, (save, probably, minor corrections).
on the origin of classical space and on the possible Let Aµ =(ϕ,Aα) beavectorfield(α=1,2,...,n),
compactification process in multidimensional gravity. and for the metric we shall use the standard decompo-
It turns out that, in general, classical space appears sition
when the horizon size reaches the minimal one among
ds2 =N2dt2 g (dxα+Nαdt)(dxβ +Nβdt). (1)
the scales: the characteristic inhomogeneity scale or − αβ
thecharacteristicscaleassociatedwiththevectorfields,
Then the action takes the form (in what follows we use
1Talk presented at the Second School-Seminar “Problems of the Planckian units)
Theoretical Cosmology”,Ulyanovsk,Russia,September 2000.
2e-mail: [email protected] I = dnxdt παβ ∂ g +πα ∂ A
3e-mail: [email protected] Z (cid:26) ∂t αβ ∂t α
212
+ϕ∂απα NH0 NαHα , (2) (12gαβπαπβ) induces precisely the same type of evolu-
− − (cid:27) tion(fromqualitativeandevenquantitativeviewpoints)
as that of the curvature terms.
where
Thus, in this approximation, the Einstein equations
1 1 1 formally coincide with the equations for homogeneous
H0 = παπβ (πα)2+ g παπβ +V ,(3)
√g (cid:26) β α− n 1 α 2 αβ (cid:27) Bianchi-I model, and it is a remarkable fact that, near
−
the singularity, a homogeneous Bianchi-I model with a
H = πβ +πβF , (4)
α −∇β α αβ vector field contains all qualitative features of the gen-
here F =∂ A ∂ A , V =g(1F Fαβ R), and eralinhomogeneousmodels. Inwhatfollowsweshalluse
R is thαeβscalaαr cβur−vatβureαwith the m4etαrβic g −. Varying the gauge Nα = 0 and, for the sake of simplicity, con-
αβ
the action with respect to ϕ, we find the constraint siderahomogeneousmodel,i.e.,allfunctionsdependon
∂ πα = 0, so it suffices to consider only the transverse time only. We also use the normalization of the space
α
partsfor A and πα. Thusinwhatfollowsweset ϕ=0 volume Vn = dnx = 1. Thus we find (within our
α S
and ∂ πα =0 to be satisfied. approximation)Rthe equations for the vector field
α
It is convenient to use the so-called generalized
∂ N
Kasner-like parametrization of the dynamical variables Eα = Aα = gαβπα, (9)
∂t √g
[6, 1]. The metric components and their conjugate mo-
∂
menta are represented as follows: πα =0. (10)
∂t
gαβ = exp{qa}ℓaαℓaβ, πβα = paLαaℓaβ, (5) Thelastequationgives πα =const,andinwhatfollows
Xa Xa we shall treat the quantities πα as external parameters
where Lαℓb = δb (a,b = 0,...,(n 1) ), and the vec- (which are actually eigenstates of the respective opera-
tors ℓa caonαtainoanly n(n 1) arbit−raryfunctions ofthe tors). The rest of the present paper repeats mainly the
spatiaαl coordinates. A fu−rther parametrization may be method suggested in Ref.[3]. Near the singularity, it is
taken in the form convenienttomakeuseofthefollowingparametrization
of the scale functions [6]:
ℓa =UaSb, Ua SO(n), Sa =δa +Ra (6)
α b α b ∈ α α α qa =lnR2+Q lng; Q =1, (11)
a a
where Rαa denotes the triangle matrix (Rαa =0 as a < X
α ). Substituting Eqs.(5), (6) into (2 ), we find the where we distinguish a slow function of time R, which
following expression for the action functional: characterizes the absolute value of the metric functions
[7, 8] and is specified by initial conditions (see below),
∂qa ∂Ra ∂A
I = (p +Tα α +πα α and the anisotropy parameters Q and lng = qa
Z a ∂t a ∂t ∂t q −
S 2nlnR canbeexpressedintermsofthenewsetPofvari-
NH0 NαHα)dnxdt, (7) ables τ, yi (i=1,2...(n 1)), as follows:
− − −
wsthraeirnetTaacαqu=ire2sPthbepsbtLruαbcUtabu,reand the Hamiltonian con- Qa(y)= n1 (cid:18)1+ 12y+iAyai2(cid:19), lng =−ne−τ11+yy22 (12)
−
2
H0 = 1 p2 1 p where Aai is a constant matrix, see, e.g., Refs.[6]. The
√g(cid:26) a− n 1(cid:18) a(cid:19) parametrization (12) has the range y2 < 1 and <
X − X −∞
+ 1 eqa(πa)2+V . (8) τfun<ct∞ion(0R≤allgow≤s1o)n,eatnodcaonvearpbpyrotphrisiaptearcahmoeicterizoafttiohne
2 (cid:27)
X
the whole classicallyallowedregionof the configuration
In the last equation the potential V collects all spa- space.
tial derivatives, and we have used the notation πa = The evolution (rotation) of the Kasner vectors re-
παℓa. In case n=3, the functions Ra areonly con- sultsinaslowtimedependenceofthefunctions πa,and
α α
Pnected by transformations of a coordinate system and it can be shownthat these functions are completely de-
mayberemovedbyresolvingthemomentumconstraints termined by the momentum constraints, while the evo-
Hα = 0 [6]. However, in the multidimensional case the lution of the scale functions is described by the action
functions Ra contain 1n(n 3) dynamicalfunctionsas
α 2 − ∂~y ∂τ
well. I = P~ +h
Itcanbeshownthatnearthesingularity,inthecase Z (cid:26)(cid:18) ∂t ∂t(cid:19)
n > 3, all spatial derivatives can be neglected in the N
e2τ[ε2 h2+U(τ,~y)] dt, (13)
leading order. Therefore, we neglect the potential V − n(n 1)Rn√g − (cid:27)
−
(the terms F and R) in the action (8). In the case
αβ
n = 3, the curvature term cannot be neglected. How- where ε2 = 1(1 y2)2P~2, and the potential term U
4 −
ever, in this case the kinetic term of the vector field (which comes from the kinetic term of the vector field)
213
has the following structure: as solutions to the eigenvalue problem for the Laplace–
Beltramioperator ε2 =∆+(n−2)2P (seeRefs. [10,1])
n − 4
U =n(n 1)R2e−2τ (πa)2gQa. (14) (n 2)2
− Xa=1 (cid:18)∆+kJ2 + −4 P(cid:19)ϕJ(y)=0, ϕJ(cid:12)∂K =0,
(cid:12)
Here the coefficients πa (projections of πα on the Kas- (cid:12) (18)
ner vectors) are slow functions of lng and characterize
where the Laplace operator ∆ is constructed via the
the initial intensity of the vector field. In the approxi-
metric δl2 = h δyiδyj = 4(δy)2(1 y2)−2. The eigen-
mationofdeeposcillations,when g 1 , this potential ij −
can be modelled by a set of potentia≪l walls: states ϕJ are classified by the integer number J and
obey the orthogonality and normalization relations
gQa →θ∞[Qa]=(cid:26) +0∞, , QQaa <>00,, (15) (ϕJ,ϕJ′)=ZKϕ∗J(y)ϕJ′(y)Dµ(y)=δJJ′, (19)
andisindependentoftheKasnervectors U∞ = θ∞(Qa). where Dµ(y) = a1n√hd2y(x) and an is the volume of
By solving the Hamiltonian constraint HP= 0 in K. Thus, an arbitrary solution Ψ to the Schr¨odinger
(13) we determine the ADM action [9], reduced to the equation i∂τΨ=HADMΨ takes the form
physical sector as follows:
Ψ= exp( ik τ)ϕ (y)C (20)
J J J
−
d~y XJ
I = (P~ H )dτ, (16)
y~ ADM
Z · dτ − where C arearbitraryconstants which are to be spec-
J
ified by initial conditions. Within our approximation,
where HADM h = √ε2+U is the ADM Hamil- these constantsalsorepresentarbitraryfunctions ofthe
≡ −
tonian and τ plays the role of time (τ˙ = 1), which vector field
corresponds to the gauge
C (A)= dn−1πC exp(iπαA ), (21)
J J,π α
N =n(n 1)Rn√g[2H ]−1e−2τ. Z
ADM ADM
− and the normalization condition reads ( dn−1A
J |
The applicability condition of the approximation CJ(A) 2= 1). The probabilistic distriRbution oPf the
|
(15) can be written as follows: variables y has the standard form P(y,τ)= Ψ(y,τ)2.
| |
The eigenstates ϕ determine stationary (in terms of
J
ε2 U (17) the anisotropy parameters Q(y)) quantum states and
≫
describe an expanding universe with a fixed energy of
as Q >δ >0 (δ 1). Thus, from the condition that
a the anisotropy.
≪
theapproximationofdeeposcillations(15)breaksatthe
For an arbitrary quantum state Ψ we can deter-
instant g 1, one finds that the function R should be mine the background metric ds2 . However, such a
chosen as∼follows: R2 = ε2[n(n 1)π2]−1e2τ (where h i
backgroundis stable and has sense only when quantum
−
π2 = (πa)2), and the inequality (17) reads g 1.
fluctuations around it are small. In case g 1, fluc-
≪
ThPesynchronouscosmologicaltimeisrelatedto τ by tuations well exceed the average metric, and≪the back-
means of the equation dt = N dτ, from which we
ADM ground is hidden [10]. Indeed, consider the moments of
fiRvenecdft.o[t3rh]e)fi,eelLsdt,i∼mε1ai/tseπt√hisegaA∼cDhtMa/rta0ec,ntewerrhigseytriecdestnc0as=ilteyrceL(lεantε=end−co1ton,s(ttch)fe., tsrciabluetfiuonnctinionhsaMihaiMi ciom=ehsRfrMomgM2tQhoisie. pTohinetlseaodfinthgecboinl--
liardatwhich Q takesaminimalvalue. Suchpointsare
i
and c is a slow (logarithmic) function of time (c 1
∼ attheboundaryofthebilliard,andtheminimalvalueof
as g → 1 ). Thus, in the synchronous time, the upper Q is Qmi in = 0. Since ϕJ(∂K) = 0, in the neighbour-
limit of the approximation (15) is t t . We note
∼ 0 hood of ∂K we have ϕJ ηJQ, and the probability
that from the physical viewpoint t corresponds to the ≈
0 density is
instant when the horizon size reaches the characteristic
scale related to the energy of the vector field, and both P (Q)= P(y,τ)δ(Q Q(y))√hdn−1y
τ
termsintheHamiltonianconstraint(thekineticenergies ZK −
of the anisotropy and of the vector field) acquire the B (τ)Qf(n) (22)
J
≈
same order.
as Q 0. Here f(n) = 2 for n > 3 and f(3) = 3/2
The physical sector of the configuration space (the →
(since in case n = 3 we get √h 1/√Q). Thus, in
variables ~y) is a realization of the Lobachevsky space, ∼
thelimit g 0,themomentsofthefunction aM are
and the potential U∞ bounds the part K = Qa → h i i
{ ≥ given by (M >0)
0 . Quantization of this system can be carried out as
}
follows. The ADM energydensity representsa constant 1
aM D (M,τ) , (23)
ofmotion, andthereforewe candefine stationarystates h i i≃ i (Mln1/g∗)f(n)+1
214
∗ References
where g∗ = g(τ,y ) and Di is a function slowly vary-
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J
gives the stationary probabilistic distribution P(y) =
[2] A.D.Linde,“ParticlePhysicsandInflationaryCos-
ϕ 2. Inthiscase D bkM(LM) isaconstant,where
| J| ≃ J J mology” (Harwood Academic, 1990).
b comes from the uncertainty in the operator ordering.
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→ → −∞
sion δ2 = ( a2 / a 2 1) (ln1/g∗)f(n)+1, which
h i h i h i − ∼ [4] M.B.Green,J.H.Schwarz,andE.Witten, “Super-
explicitly shows the instability of the average geome-
string Theory”, Cambridge Univ. Press, 1988.
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reaches th≪e order δ 1 at the instant t (Ln) kn−1
∼ ∼ J J [5] A.ChodosandS.Detweiler,Phys. Rev.D21,2176
(g 1)whentheanisotropyfunctions canbe described
(1980);
∼
by small perturbations a2 = R2(1+Q lng+...), and
i i M. Demian´ski, Z. Golda, M. Heller and M.
the Universe acquires a quasi-isotropic character. This
Szyd lowski, Clas. Qu. Grav. 3,, 1196 (1986); ibid.,
instantcanbeconsideredastheoriginofastableclassi-
5, 733 (1988);
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K. Maeda and P.Y. Pang, Phys. Lett. 180B, 29
of a stable classical geometry turns out to be forbid-
(1986).
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structing a self-consistent theory which could account
[7] C.W. Misner, Phys. Rev. Lett. 22, 1071 (1969).
for back reaction of particle creation in such models).
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scale functions can be obtained by setting P(y,τ) =1,
[10] A.A. Kirillov, Phys. Lett. 399B, 201 (1997).
which gives the replacement f(n) f(n) 2 in (23).
→ −
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well(withminorcorrectionsoftheestimates). Andsec-
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the structure of the configuration space in dimensions 920 (2000).
n > 3. The billiard turns out to be finite for an arbi-
trarynumberofdimensions[12],butthepaymentisthe
fact that the initial stage of the compactification pro-
cess, which was found in vacuum models [1], is absent.
Indeed, in the model considered above, the anisotropy
parameters have the range 1 Q 0 and remain al-
≥ ≥
ways positive (we recall that in the vacuum case these
functions couldhavenegativevalues, e.g.,they changed
in the range 1 Q (n 3)/(n+1)). This means
≥ ≥ − −
that the expansion always proceeds in such a way that
the lengths monotonicaly increase in all spatial direc-
tions. This probably represents a rather serious reason
why vector fields should not be included as external
fields in multidimensional gravity.
This research was supported by RFBR (Grant No.
98-02-16273)andDFG(GrantNo. 436Rus113/23618).