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Gravitation & Cosmology, Vol. 2 (1996), No. 1 (5), pp. 17–26

c 1996 Russian Gravitational Society

MULTIDIMENSIONAL QUANTUM COSMOLOGY:QUANTUM WORMHOLES, THIRD QUANTIZATION,

INFLATION FROM ”NOTHING”, ETC.1

A. Zhuk

Department of Theoretical Physics, University of Odessa, 2 Petra Velikogo Str., 270100 Odessa, Ukraine

Received 9 September 1995

A multidimensional cosmological model with a space-time consisting of n (n ≥ 2) Einstein spaces M i is investigated

in the presence of a cosmological constant Λ and m (m ≥ 1) homogeneous minimally coupled scalar fields as amatter source. Classes of models integrable at classical as well as quantum levels are found. These classes areequivalent to each other. Quantum wormhole solutions are obtained for them and the third quantization procedureis performed. An inflationary universe arising from a classically forbidden Euclidean region is investigated for amodel with a cosmological constant.

1. Introduction

We believe that the multidimensional approach is mostadequate for describing quantum gravitational pro-

cesses at high energies. Modern theories of unifiedphysical interactions use ideas of hidden (or extra) di-mensions. In order to study different phenomena atthe early stage of the universe one should use these the-ories or at any rate models keeping their main ideas.In any case, multidimensional models have to explainthe observed four-dimensionality of the space-time atpresent. This is realized in models where one or anumber of internal spaces are compact and contractedto Planckian scales during the evolution of the uni-verse (dynamical compactification) or the symmetrybetween the external and internal dimensions is bro-ken from the very beginning and the internal spaces

are static and compactified at Planck’s length (sponta-neous compactification). A further possibility is givenin quantum theory where only the external spaces canbe created by tunneling while the internal dimensionsmay be hidden because they stay behind a potentialbarrier.

Of special interest are exact solutions since theycan be used for a detailed study of the evolution of ourspace, the compactification of the internal spaces andthe behavior of matter fields.

One of the most natural mulidimensional cosmo-logical models (MCM) generalizing the Friedmann-

1Talk presented at the Internat. School-Seminar “Founda-

tions of Gravitation and Cosmology”, Odessa, 4–10 Sept. 1995.

Robertson-Walker (FRW) universe is given by a toymodel with the topology R × M 1 × . . . × M n whereM i (i = 1, . . . , n) denote Einstein spaces. One of thesespaces, say M 1 , describes the external space but all

the others are internal ones. A gauge-covariant formof the Wheeler-DeWitt (WDW) equation [1, 2] for amodel with this topology was proposed in [3] and someintegrable models were studied in [4]-[6].

In the present paper we consider a general modelwith a cosmological constant Λ and m (m ≥ 1) ho-mogeneous minimally coupled scalar fields ϕ(a) (a =1, . . . , m) with the potentials U (a)(ϕ(a)). We showthat some integrable models considered before [4]-[6] and some new ones are equivalent to each other.Among solutions of the WDW equations for thesemodels there are ones which describe tunneling uni-

verses, in particular, birth of universes from classicallyforbidden Euclidean region (birth from ”nothing” [7]).Quantum wormholes [8] representing a special classof solutions of the WDW equation are constructedfor these models as well. Full sets of orthonormalsolutions in these models provide a possibility of per-forming third quantization [9] and getting a spectrumof created universes. An inflationary universe arisingdue to quantum tunneling is investigated for a modelwith a cosmological constant. Parameters of the modelwhich ensure inflation of the external space and dy-namical compactification of internal ones are found. In

particular, the dimension of internal spaces should bed > 40. It is shown that the tunneling from “nothing”is in this case strongly suppressed because of a very



18 A. Zhuk

large spatial volume of the arising universe.

2. General description of the model

The metric of the model

g = − e2γ(τ )dτ ⊗ dτ +ni=1

e2βi(τ )g(i) (2.1)

is defined on the manifold

M = R × M 1 × . . . × M n ,

where the manifold M i with the metric g(i) is an Ein-stein space of dimension di , i.e.,

Rmini [g(i)] = λig(i)mini , i = 1, . . . , n; n ≥ 2.

(2.2)

The total dimension of M is D = 1 +ni=1 di . This

describes the case where the topology of a factorizedspace-time manifold is assumed from the very begin-ning and compactification of internal spaces is de-scribed as shrinking to or freezing on the Planck scale.

Here we study the general model with the cosmo-logical constant Λ and m (m ≥ 1) non-interactinghomogeneous minimally coupled scalar fields ϕ(a) (a =1, . . . , m) with potentials U (a)(ϕ(a)). The action of themodel is adopted in the following form:

S =1

2κ2 dDx

|g|(R[g] − 2Λ) + S ϕ + S GH (2.3)

where R[g] is the scalar curvature of the metric (2.1)and κ2 is a D -dimensional gravitational constant.S GH is the standard Gibbons-Hawking boundary term

[10]. S ϕ =ma=1 S

(a)ϕ is the action of m non-interacting

minimally coupled homogeneous scalar fields

S (a)ϕ =

dDx

|g|−1

2gMN ϕ(a)

,M ϕ(a),N − U (a)(ϕ(a))

.

(2.4)

For the metric (2.1) the action (2.3) reads

S = µ dτ L (2.5)

with the Lagrangian L

L = 12

e−γ+γ0

Gij β iβ j + κ2ma=1

ϕ(a)

2− V. (2.6)

Here γ 0 =ni=1 diβ i and the overdot denotes d/dτ .

The components of the minisuperspace metric read [3]

Gij = diδij − didj (2.7)

and the potential is given by

V = eγ+γ0−1

2

ni=1

θi e−2βi

+ κ2ma=1

U (a)

(ϕ(a)

) + Λ

(2.8)

where θi = λidi . If M i are spaces of constant curva-ture, then θi may be normalized in such a way thatθi = kidi(di − 1), ki = ±1, 0. The parameter µ =

ni=1 V i/κ2 , where V i is the volume of M i , and we

may put µ = 1 [11].

The constraint equation reads

−∂L

∂γ =

1

2e−γ+γ0

(Gij β iβ j + κ2

ma=1

ϕ(a)

2+V = 0.

(2.9)

The minisuperspace metric G = Gijdβ i⊗ dβ j maybe diagonalized in different coordinate frames. In thepresent paper we use two of them. In the first one theminisuperspace metric reads [4, 5, 12]

G = −dv0 ⊗ dv0 +n−1i=1

dvi ⊗ dvi, (2.10)

where

q1v0 = (d1 − 1)β 1 +

ni=2

diβ i,

q1v1 = [(D − 2)/(d1Σ2)]1/2

ni=2

diβ i,

q1vi = [(d1

−1)di/d1ΣiΣi+1)]

1/2n

j=i+1

dj(β j

−β i),

i = 2, . . . , n − 1. (2.11)

Here we used the notations q1 = [(d1 − 1)/d1]1/2

andΣi =

nj=i dj . In the second coordinate frame

G = −dz0 ⊗ dz0 +

n−1i=1

dzi ⊗ dzi , (2.12)

where [3, 6, 13]

z0 = q−12

ni=1

diβ i,

zi = [di/ΣiΣi+1)]1/2

nj=i+1

dj(β j − β i),

i = 1, . . . , n − 1. (2.13)

Here q2 = [(D − 1)/(D − 2)]1/2 . The spatial volume

of the universe is proportional to v =ni=1 adii where

scale factors ai = exp β i . In the coordinates (2.13) ittakes the following form:

v =ni=1

adii = exp (q2z0). (2.14)



Multidimensional Quantum Cosmology: Wormholes, Third Quantization, Inflation, etc. 19

3. Wheeler-DeWitt equations. Integrable

cosmologies

At the quantum level the constraint (2.9) is modi-fied into the WDW equation [1, 2]. Now, we considerclasses of cosmological models which are integrable atclassical as well as quantum levels and show the equiv-alence of these models.

3.1. Models with one non-Ricci-flat factor space

In this subsection we consider the integrable case of a MCM where only one of the factor spaces M i , sayM 1 , is non-Ricci-flat: θ1 = 0, θi = 0, i = 2, . . . , n.Using the coordinates (2.11), we obtain for free scalarfields (it is clear that in the case of free scalar fields itis sufficient to take m = 1) the following form of theconstraint (2.9):

− v02

+

n−1i=1

vi2

+ ϕ2 − θ1 e2q1v0

= 0 , (3.1)

where we have used the harmonic time gauge γ = γ 0and the scalar field ϕ(1) ≡ ϕ is redefined: κϕ → ϕ .The WDW equation in this case reads [4, 5, 12, 14]:

− ∂

∂v0∂

∂v0+

n−1i=1

∂

∂v i∂

∂v i+

∂ 2

∂ϕ2+ θ1 e2q1v

0

Ψ = 0.

(3.2)It is easy to solve this equation by separation of vari-ables

Ψ(v) = eipvΦ(v0) (3.3)

where p = ( p1, . . . , pn) is a constant vector, v =(v1, . . . , vn−1, vn = ϕ), pv =

ni=1 piv

i and pi = pi .The substitution of (3.3) into (3.2) gives

−1

2

d

dv0

2+

1

2θ1 e2q1v

0

Φ = εΦ , (3.4)

where

ε =1

2

ni=1

( pi)2 . (3.5)

3.2. Models with a cosmological constant

Here we consider the integrable case of a MCM withall Ricci-flat factor spaces: θi = 0 (i = 1, . . . , n), inthe presence of the cosmological constant Λ and freescalar field as a matter source. Using the coordinates(2.13) and the harmonic time gauge, we get for (2.9)

−(z0)2 +n−1i=1

(zi)2 + ϕ2 + 2Λ e2q2z0

= 0 , (3.6)

which is modified into the WDW equation [3, 6, 13]− ∂

∂z0∂

∂z0+n−1i=1

∂

∂z i∂

∂z i+

∂ 2

∂ϕ2− 2Λ e2q2z

0

Ψ = 0.

(3.7)

We are seeking a solution of (3.7) in the form

Ψ(z) = eipzΦ(z0) . (3.8)

where p = ( p1, . . . , pn) and z = (z0, . . . , zn−1, zn =ϕ). Φ(z0) satisfies the equation

−1

2

d

dz0

2− Λ e2q2z

0

Φ = εΦ (3.9)

with ε defined by (3.5).It is easy to see that the models of the subsections

3.1 and 3.2 are equivalent to each other up to the evi-dent substitutions:

vi ↔ zi, i = 0, . . . , n − 1,

θ1/2 ↔ −Λ, q1 ↔ q2 . (3.10)

Thus to study the quantum behavior of these modelsit is sufficient to consider only one of them, e.g., thatof Subsec. 3.2.

3.3. Exact scalar-field cosmologies

Here we consider a special class of integrable MCMwith m (m

≥1) scalar fields. The action of these

models is given by Eqs. (2.3), (2.4) where Λ = 0 . TheLagrangian (2.6) for these models reads

Ls =1

2e−γ+γ0

Gij β iβ j + κ2

ma=1

ϕ(a)

2

+ eγ+γ0

1

2

ni=1

λidi e−2βi − κ2

ma=1

U (a)(ϕ(a))

. (3.11)

The energy-momentum tensor of matter with theaction S =

dDx

|g|L is defined by

T ik = −2

∂L

∂g ik + gikL. (3.12)

Using this formula, we get the nonzero components of the scalar field ϕ(a) energy-momentum tensor:

T (a)0

0 = −12

e2γ

ϕ(a)2 − U (a)(ϕ(a)) ≡ −ρ(a), (3.13)

T (a)mi

mi= 1

2 e2γ

ϕ(a)2 − U (a)(ϕ(a)) ≡ P (a),

i = 1, . . . , n , (3.14)

where we have introduced the energy density ρ(a) andthe pressure P (a) corresponding to the scalar fieldϕ(a) . Now suppose that these quantities are connected

by the equation of state

P (a) =

α(a) − 1

ρ(a), (3.15)



20 A. Zhuk

where α(a) = const, a = 1, . . . , m.

It is not difficult to prove that these models areequivalent to cosmological models in the presence of anm-component perfect fluid with the energy-momentumtensor

T M N =ma=1

T (a)M

N , (3.16)

T (a)M

N = diag−ρ(a), P (a)δm1

k1, . . . , P (a)δmn

kn

(3.17)

where the τ -dependent pressure and energy density of each a-th perfect fluid component are connected by theequation of state (3.15) and the conservation equationsare imposed on each component separately:

T (a)M

N ;M = 0. (3.18)

The nontrivial conservation equations (3.18) are

ρ(a) + γ 0

ρ(a) + P (a)

= 0, a = 1, . . . , m , (3.19)

which results in

ρ(a) = A(a) e−α(a)γ0 = A(a)v−α

(a)

, a = 1, . . . , m (3.20)

where A(a) = const and the spatial volume v is givenby (2.14).

To prove the proposition on the equivalence be-tween these models, we first note that the Klein-

Gordon equations

∂

∂τ

e−γ+γ0

∂ϕ(a)

∂τ

+ eγ+γ0

∂U (a)

∂ϕ(a)= 0, (3.21)

following from the Lagrangian (3.11) are equivalent tothe conservation equations (3.19) if the relations (3.13)and (3.14) are valid. Second, using the relations (3.13)– (3.15) and (3.20), we obtain

∂Ls∂γ

=∂Lρ∂γ

(3.22)

andd

dτ

∂Ls

∂ β i− ∂Ls

∂β i=

d

dτ

∂Lρ

∂ β i− ∂Lρ

∂β i, i = 1, . . . , n , (3.23)

where the Lagrangian Ls is given by (3.11) and theeffective Lagrangian Lρ corresponding to the modelswith the energy-momentum tensor (3.16) – (3.18) is[15, 16, 17]

Lρ =1

2e−γ+γ0Gij β iβ j

+ eγ+γ01

2

n

i=1

λidi e−2βi

−κ2

m

a=1

ρ(a), (3.24)

with ρ(a) defined by (3.20).

To obtain integrable models, we consider the Ricci-flat case (λi = 0, i = 1, . . . , n). Then, in the z -coordinates (2.13) the Lagrangian and the constraintread, respectively:

Lρ = −12

z02 + 1

2

zi2

− κ2ma=1

A(a) exp

(2−α(a))q2z0

(3.25)

and

− 12

z02

+ 12

zi2

+ κ2ma=1

A(a) exp

(2−α(a))q2z0

= 0, (3.26)

where we use the harmonic time gauge γ = γ 0 . Theequations of motion

zi = 0, i = 1, . . . , n − 1 (3.27)

have the first integrals

zi = pi. (3.28)

The constraint (3.26) can be rewritten as follows:

v = ±√

2q2v

ε + κ2v2

ma=1

ρ(a)(v)

, (3.29)

where the parameter ε is defined by a relation similarto (3.5). Using Eqs. (3.29) and (3.13), it is easy to get

ϕ(a) as a function of the spatial volume [18]:

ϕ(a) = ±

α(a)/2

q2

ρ(a)(v)dv

ε + κ2v2ma=1 ρ(a)(v)

+ ϕ(a)0 .

(3.30)

Inverting this expression, we can find the spatial vol-ume as a function of the scalar field ϕ(a) : v = v(ϕ(a))and consequently a dependence of the energy densityρ(a) on the scalar field ϕ(a) : ρ(a) = ρ(a)(ϕ(a)). To re-construct the potentials U (a) we can write them withthe help of the relations (3.13) – (3.15) in the form

U (a)(ϕ(a)) = 12

(2 − α(a))ρ(a)(ϕ(a)) (3.31)

where a = 1, . . . , m and

ρ(a)(ϕ(a)) = A(a)

v(ϕ(a))−α(a)

. (3.32)

If α(a) = 0, then ϕ(a), U (a) and ρ(a) are con-stant. This scalar field is equivalent to the cosmolog-ical constant Λ ≡ κ2U (a) with the equation of stateP (a) = −ρ(a) . For α(a) = 2 we have U (a) ≡ 0. Thescalar field ϕ(a) in this case is equivalent to the perfectfluid with the ultrastiff equation of state P (a) = ρ(a) .

It is clear that it is hardly possible to integratemodels with an arbitrary set of exponents in the re-lations (3.25) and (3.26). Therefore, we consider now




the integrable case of a two-component (m = 2) scalarfield with arbitrary α(1) ≡ α = 0, 2 and α(2) = 2 ,which corresponds to the the two-component perfectfluid with the energy density

ρ = ρ(1) + ρ(2) = A(1)v−α + A(2)v−2 . (3.33)

The potentials U (a) are in this case [18] U (2) ≡ 0 and

U (1)(ϕ(1)) =

=(2 − α)Λ

2κ2exp

∓

√2ακq2(ϕ(1)−ϕ

(1)0 )

,

E = 0, Λ > 0;

= U (1)0

sinh

κq2(2 − α)√

2α(ϕ(1) − ϕ

(1)0 )

−2α/(2−α),

E > 0, Λ > 0;

= −U (1)0

sinκq2(2

−α)

√2α (ϕ(1)

− ϕ(1)0 )

−2α/(2−α),

E > 0, Λ < 0;

= U (1)0

cosh

κq2(2 − α)√

2α(ϕ(1) − ϕ

(1)0 )

−2α/(2−α),

E < 0, Λ < 0 (3.34)

where Λ = κ2A(1) and

E = ε + κ2A(2). (3.35)

As usual, at the quantum level the constraints∂Ls/∂γ = 0 and ∂Lρ/∂γ = 0 are modified into the

WDW equations. It is easier to investigate the quan-tum behavior of the geometry of these models study-ing the WDW equation obtained from the effectiveLagrangian (3.25). For the integrable two-componentmodel (3.33) we get

− ∂

∂z0∂

∂z0+

n−1i=1

∂

∂z i∂

∂z i

− 2κ2A(2) − 2κ2A(1) e(2−α)q2z0

Ψ = 0. (3.36)

Separating the variables in the form

Ψ(z) = eipzΦ(z0), (3.37)

where p = ( p1, . . . , pn−1) is a constant vector and z =(z1, . . . , zn−1), we get for Φ(z0) the equation

−1

2

d

dz0

2− κ2A(1) e(2−α)q2z

0

Φ = E Φ (3.38)

with E defined by (3.35).It is again easily seen that the models of Subsec-

tions 3.2 and 3.3 are mutually equivalent up to thesubstitutions

ε ↔ E, Λ ↔ κ2A(1),

2q2 ↔ (2 − α)q2. (3.39)

4. Quantum wormholes

Using the equivalence between the above three mod-els, we can consider only one of them to study thequantum behavior of the universe. Let us consider the

second one. A simple analysis of Eq. (3.9) shows [6]that the quantum behavior strongly depends on thesigns of Λ and ε . For example, if Λ > 0, then forε ≥ 0 only Lorentzian regions exist. For ε < 0 ex-ist both the Lorentzian and Euclidean regions. In thiscase quantum transitions with topology changes takeplace (tunneling universes or birth from “nothing”). If Λ < 0, then for ε ≤ 0 only a Euclidean region exists;for ε > 0 exist both Lorentzian and Euclidean regions.In this case quantum transitions with topology changestake place (quantum wormholes).

Let us consider the latter case in more detail. Solv-

ing (3.9), we get

Φ(z0) = Bi√2ε/q2

√−2Λq−12 exp(q2z0)

, (4.1)

where√

2ε/q2 = |p|/q2 and B = I, K are modifiedBessel functions. The general solution of Eq. (3.7) hasthe following form:

Ψ(z) =B=I,K

dn p C B(p)exp(ipz)Bi|p|/q2

×√−2Λq−12 exp(q2z0)

, (4.2)

where the functions C B (B = I, K ) belong to an ap-

propriate class.Quantum wormholes represent a special class of

solutions of the WDW equation with the followingboundary conditions [8]:

(i) the wave function is exponentially damped for largespatial geometry,

(ii) the wave function is regular when the spatial ge-ometry degenerates.

We restrict our consideration to real values of pi, (i = 1, . . . , n) . This corresponds to real geometriesin the Lorentzian region. In this case we have ε

≥0.

If Λ > 0, the wave function (3.8) Ψ = eipz)Φ(z0)where Φ(z0) defined by (4.1) is not exponentiallydamped when the spatial volume v → ∞ , i.e. thecondition (i) for quantum wormholes is not satisfied.Ψ oscillates and may be interpreted as correspondingto the classical Lorentzian solutions.

For Λ < 0 the wave function (3.8) is exponentiallydamped for large v only when B = K in (4.1). But inthis case the function Φ oscillates an infinite numberof times when v → 0. Thus the condition (ii) isnot satisfied.The wave function describes a transitionbetween the Lorentzian and Euclidean regions.

The function

Ψp(z) = eipzK i|p|/q2√

−2Λq−12 eq2z0

(4.3)



22 A. Zhuk

may be used for constructing quantum wormhole solu-tions. We consider the superposition of singular solu-tions

Ψλ,n(z) =1

π

+∞

−∞dkΨq2kn(z) e−ikλ, (4.4)

where λ ∈ R, n is a unit vector (n2 = 1) and thequantum number k is connected with the quantumnumber ε = 1

2|p|2 by the formula 2ε = q22k2 . The

calculation gives [6, 13]

Ψλ,n(z) = exp

−

√−2Λ

q2exp(q2z0)cosh(λ − q2zn)

.

(4.5)

It is not difficult to verify that Eq. (4.5) leads to so-lutions of the WDW equation (3.7), satisfying the

quantum-wormhole boundary conditions. Similar quan-tum wormholes for the model of Subsection 3.1 wereobtained in [5, 12].

These results are a straightforward generalizationof the discussion in [19] – [21] to the multidimensionalcase. Therefore, the set of wave fnctions Ψp and Ψλ,nare spanning the same space of physical states and areboth bases of the Hilbert space of the model in thecorresponding representation. A connection betweenthese bases Ψp and Ψλ,n is given by Eq. (4.4).

The function

Ψm,n = H m(x0)H m(x1)exp−12 (x0)2+(x1)2 ,(4.6)

where H m are the Hermite polynomials and

x0 = (2

q2)1/2(−2Λ)1/4 exp

q2z0

coshq2zn

2, (4.7)

x1 = (2

q2)1/2(−2Λ)1/4 exp

q2z0

sinhq2zn

2, (4.8)

m = 0, 1, . . . are also solutions of the WDW equationwith the quantum-wormhole boundary conditions. So-lutions of this type are called discrete spectrum quan-tum wormholes [5, 8, 12, 13], [19] – [21] and form adiscrete basis for the Hilbert space of the system.

5. Third quantization

The WDW equations (3.2), (3.7) and (3.36) are sim-ilar to the scalar field equation in curved space-time.These equations are gauge-covariant (conformally co-variant) [3]. It is not difficult to show [4, 22] that theminisuperspace metric G (2.7) is conformally equiva-lent to the Milne metric and for a special gauge theWDW equations (3.2), (3.7) and (3.36) coincide withthe field equation for a scalar field conformally coupledto a Milne space-time.

By analogy with quantum field theory, it might beworthwhile to perform the second quantization of theuniverse wave function Ψ expanding it on the creation

and annihilation operators. The WDW equation it-self is a result of quantization of geometry and matter.Thus, the quantization procedure for the wave functionΨ is called third quantization [9]. Similar to quantumfield theory in curved space-time, we can expect thatthe vacuum state in a third-quantized theory is unsta-ble and creation of particles (in our case, universes)from the initial vacuum state takes place. To per-form the procedure of scalar field quantization againstthe time-dependent gravitational field background weshould specify a vacuum state. This is quite a prob-lem. Since there is no global timelike Killing vectorand hence there is no global vacuum state, it is onlypossible to define different vacuum states which are ingeneral inequivalent to each other and have differentphysical nature. In the third quantization procedurewe have a similar situation. Different Fock spaces con-

structed from exact solutions of the WDW equationsare inequivalent to each other. It is natural to definean initial vacuum state with respect to the orthonor-mal set of mode solutions which are positive frequencymodes in the limit of vanishing spatial volume, v → 0[23]. As a result, the birth of particles (universes) from“nothing” may have place where “nothing” is the ini-tial vacuum state defined above.

Let us now consider the model of Subsection 3.2with Λ > 0, which corresponds to a scalar field with apositive squared mass. It is easy to find two completeset of modes [4, 22]:

Ψp = 1(2π)n/2

√π2q2 sinh (π

√2ε/q2)

1/2× eipzJ −i√2ε/q2

√2Λ

q2eqz

0

(5.1)

and

Ψp =1

(2π)n/2

√π

2√

q2exp(π

ε/2/q2)

eipzH (2)

i√2ε/q2

√2Λ

q2eqz

0

, (5.2)

which are orthonormal:Ψp, Ψp

= −i

z0=const

Ψp

↔∂ z0 Ψ∗

pdz = δ(p− p).

(5.3)

The modes (5.1) are excited states above the Hartle-Hawking vacuum state [24] with ε = 0 [6]. As bothsets (5.1) and (5.2) are complete, they are related bythe Bogoliubov transformation

Ψp = αεΨp + β εΨ∗p

, (5.4)

where the Bogoliubov coefficients are

αε =

exp(π√2ε/q2)

2sinh(π√

2ε/q2)

1/2

(5.5)




and

β ε =

exp(−π

√2ε/q2)

2sinh(π√

2ε/q2)

1/2. (5.6)

The coefficients β ε are nonzero. Thus two Fock spacesconstructed with the help of the modes Ψp and Ψp

are not equivalent and we have two different third-quantized vacuum states (voids): |0 > and |0 > . Themodes (5.1) have the asymptotes

Ψp ∼ exp

i(pz−√

2εz0)

, v → 0 , (5.7)

which are positive-frequency modes with respect to theconformal “time” z0 . Thus the vacuum |0 > , definedwith respect to these modes, is connected with the min-isuperspace conformal Killing vector ∂ z0 . The modes(5.1) are no longer positive-frequency ones as v

→ ∞.

In this limit the modes (5.2) have the asymptotes

Ψp ∼ exp

i(py−√

2Λy0)

, v → ∞ (5.8)

where

y0 = q−12 exp(q2z0); yi = zi, i = 1, . . . , n . (5.9)

The modes (5.2) in this limit are positive-frequencyones with respect to the “time” y0 .

Since the vacuum states |0 > and |0 > are notequivalent, the birth of universes from “nothing” mayhappen, where “nothing” is the vacuum state |0 > .If

|0 > is the initial state when v

→0, then an

observer defined with respect to the vacuum state |0 >will detect in the limit v → ∞

nε = |β ε|2 =

exp(2π√

2ε/q2) − 1−1

(5.10)

universes in mode p (recall that 2ε = p2 ). This is

precisely the Planck spectrum for radiation at the tem-perature T = q2/2π .

Now consider the model with Λ < 0. It is easilyseen that we can get the WDW equation (3.7) fromthe action

S =1

2 dn+1zΨH Ψ, (5.11)

which coincides with the action for a scalar field in theMinkowski space-time with the potential

V (Ψ) =M 2

2Ψ2, (5.12)

where

M 2(z) = 2Λexp(2q2z0). (5.13)

If Λ < 0,then M 2 < 0 and this model has an unsta-ble vacuum state. The energy spectrum is unboundedfrom below. The theory is well defined if we add a

self-interaction term. Then

V (Ψ) = −ν 2

2Ψ2 +

λ

4Ψ4 , (5.14)

where we define

M 2 ≡ −ν 2 = −2|Λ| exp (2q2z0) . (5.15)

A minimum of the potential (5.14) takes place at

Ψ0 = ± ν √λ

= ± 2|Λ|λ

exp(q2z0). (5.16)

It follows from this expression that a symmetry break-ing takes place dynamically, because

Ψ0 → 0 if v → 0. (5.17)

The depth of wells at the minima is V (Ψ=Ψ0) =−ν 4/(4λ) = −(Λ2/λ)exp(4q2z0). The square of massof Ψ-excitations after the symmetry breaking becomespositive:

m2(Ψ = Ψ0) =d2V

dΨ2

Ψ=Ψ0

= 2ν 2. (5.18)

Let us now consider the field Ψ = Ψ − Ψ0 which de-scribes oscillations near the minima of the potentialV (Ψ). This field satisfies the equation

− ∂

∂z0∂

∂z0+

ni=1

∂

∂z i∂

∂z i− 2ν 2

Ψ

= j(z0) + 3λΨ0Ψ2 + λΨ3 (5.19)

where the source j is

j(z0) =∂

∂z0∂

∂z0Ψ0 = ±q22

2|Λ|

λexp(q2z0). (5.20)

As follows from Eq. (5.18), the field¯Ψ has positivesquared mass which depends on the “time” z0 . Thus

in the linear approximation and without a source termbirth from “nothing” takes place, just as for the caseΛ > 0. We should make in Eqs. (5.1) and (5.2) theonly replacement: Λ → 2|Λ| . The presence of thesource term in Eq. (5.19) leads to production of addi-tional universes. The source term has its origin in thedependence of the classical minimum Ψ0 on “time”(see Eq. (5.20)).

The presence of the interaction terms ∼ Ψ2 and Ψ3

in (5.19) (respectively, ∼ Ψ3 and Ψ4 in the potentialV (Ψ)) provides the opportunity to consider processes

with topology alteration. For example, the cubic termin the potential is similar to the interaction term whicharises naturally in string theory. This term describesfission of a universe into two or fusion of two universesinto a single one.

It is important to note that the third quantiza-tion may affect the model topology choice. If werequire renormalizability of the third quantized the-ory, then, by analogy with the scalar field theory withself-interaction, it follows that its dimension should beequal or less than four [25]. In our case it means thatwe should take models with n ≤ 3, i.e. in the models

with no scalar field ϕ we can take at most four factorspaces M i and in the presence of a scalar field we canconsider at most three factor spaces.



24 A. Zhuk

6. Inflation from “nothing”

Inflationary models are now very popular in cosmologysince they explain why our universe is homogeneous,isotropic and almost spatially flat [26]. So, it might

be worthwhile to get inflationary models in multidi-mensional cosmology as well. However, unlike usual 4-dimensional cosmologies, in the multidimensional casewe should solve two problems simultaneously. Namely,it is necessary to get inflation of our external spaceand compactification of internal dimensions near thePlanck length LPl ∼ 10−33 cm to make them unob-servable at present.

Another interesting hypothesis consists in the pro-posal that an inflationary universe arose by quantumtunneling from a classically forbidden Euclidean re-gion. This process is called birth from “nothing” [7]

as in the previous Sec. 5, but its nature is quite differ-ent.

In the present section we study a multidimensionalinflationary universe which arose from “nothing” byquantum tunneling. It is clear that in the harmonictime gauge the solutions of the constraints (3.1), (3.6)and (3.26) and the equation of the form (3.27) areequivalent to each other, but inverting them we getquite a different behavior of the scale factors ai =expβ i in these models. Thus the classical behaviorof the universe should be analyzed separately for eachmodel.

Here we investigate the model of Subsection 3.2with the cosmological constant Λ > 0. As follows fromthe Sec. 4, in this case the quantum tunneling takesplace if ε < 0. As we require the reality of the metricin the Lorentzian region, the condition ε < 0 takesplace only for an imaginary scalar field ϕ [6].

In the harmonic time gauge the solution of the con-straint (3.6) reads

v = e(q2z0) =

|ε|/Λ

cos(q2

2|ε|τ ), |τ | ≤ π/2

q2

2|ε| , (6.1)

with a turning point at the spatial volume minimum

vmin =

|ε|/Λ ≡ vt . (6.2)

The analytic continuation τ L → −iτ E yields the solu-tion in the Euclidean region

v =

|ε|/Λ

cosh(q2

2|ε|τ ), −∞ < τ < +∞ (6.3)

with a turning point at the maximum: vmax = vt .

In the synchronous system (γ = 0) the scale factorsread [6]:

ai = Ai

cosh

t

T

σ f t

2T

σi, i = 1, . . . , n , (6.4)

where t is the proper time, σ = 1/(D − 1) and T =

[(D − 2)/2Λ(D − 1)]1/2 ;

f (x) = exp

−2 arctan e−2x

(6.5)

is a smooth monotonically increasing function with theasymptotes f (x) → e−π as x → −∞ , f (x) → 1 asx → +∞ , and at zero: f (0) = e−π/2 . The parametersσi satisfy the relations

ni=1

diσi = 0, (6.6)

ni=1

diσ2i + σ2

n+1 =D − 2

D − 1. (6.7)

The spatial volume is

v = ni=1

Adii

cosh

t

T (6.8)

and has a minimum at t=0. It is easily verified that

ni=1

Adii =

|ε|/Λ. (6.9)

The scale factors ai have their minima at

t(0)iT

= arsinhσiσ

= − ln

σiσ

+

σiσ

2+ 1

,

i = 1, . . . , n , (6.10)

from which it follows that sign t(0)i = − sign σi .Suppose now that the universe has arisen by tun-

neling from the Euclidean region and from the turningpoint t = 0 (see Eq. (6.8)) its behavior can be de-scribed by classical equations. For simplicity, considera model with two factor spaces (n = 2) where oneof them (say, M 1 ) is our external space. A general-ization to the case n > 2 is straightforward. Supposealso that, after the birth, the external space M 1 mono-tonically expands. Hence it follows from Eqs. (6.6) and(6.10) that t(0)1 < 0 (σ1 > 0) and t(0)2 > 0 (σ2 < 0).Let all dimensions at the moment of the universe cre-ation from “nothing” have equal rights:

a1(t = 0) = a2(t = 0) = 10x LPl, (6.11)

where 2 ≤ x ≤ 3. If we take x > 3, then the birthprobability becomes too small because of a too largespatial volume. If x < 2, then the scale factor a2 goesto LPl too fast and there is is no sufficient time for aninflation of the scale factor a1 . From (6.4) and (6.11)we get

Ai = 10x expπσi

2, i = 1, 2. (6.12)

Using these relations we find for the scale factor a1

that [27]

a1 ≈ 10x expπσ1

2(6.13)




if

4 <∼

t/T D − 1. (6.14)

To solve the flatness and horizon problems, the

scale factor a1 should expand during the inflation bya factor of 1030 [28]. Thus,

πσ1/2 >∼

70. (6.15)

It gives a lower boundary for the parameter σ1 . If the size of M 1 by the end of inflation is approximatelyequal to the presently observable size of the universe,i.e. ∼ 1028 cm, then

πσ1/2 ≈ 140. (6.16)

For the parameter σ2 we get

σ2 <∼ − d1d2

140π . (6.17)

It is easily seen that within the limits 140/π ≤ σ1 ≤280/π we have for the position of the minima of a2

t0(2)/T ≈ 6, (6.18)

if d2 d1 = 3. Here we consider the model wherethe space M 2 shrinks at the end of the inflation to itsminimum size near the Planck length, i.e.,

t0(2) = t∗ (6.19)

anda2(t0(2)) ≈ LPl. (6.20)

Thus, for the scale factor a2 we get

a2(t0(2)) ≈ LPl ≈ 10x expπσ2

2<∼

10x exp(−70d1d2

),

(6.21)

which leads to the estimate

70d1d2

<∼

x ln10. (6.22)

For example, if d1 = 3, πσ1/2 = 70 and 2 ≤ x ≤ 3 ,we get

45 ≥ d2 ≥ 30. (6.23)

In general, we find that to ensure inflation of the ex-ternal space the dimension of the internal space shouldbe d > 40, in accordance with the paper [29].

After inflation, the external space M 1 should expe-rience a power-law expansion and the internal one M 2should remain frozen near the Planck scale. A transi-tion to such a stage can be performed if the cosmolog-ical constant Λ goes very fast to zero. As a result, we

have a Kasner-like solution [30]:

ai = a(0)itαi , ϕ = ln tαn+1 + const,

whereni=1 diαi = 1 and

ni=1 diαi

2 = 1−αn+12 . In

particular, a solution with frozen internal spaces existswhen αi = 0 (i = 2, . . . n). In this case we get forthe external space α1 = 1/d1 . So the factor spaceM 1 expands as a FRW universe filled with ultra-stiff matter (for d1 = 3).

Now consider the probability of birth of the infla-tionary universe from “nothing”. The transition am-plitude between the states with zero spatial volumev = 0 at the moment τ i and some value of v at τ f isgiven by the path integral

< v, τ f |, τ i >=

[dg][dϕ] eiSL , (6.24)

where S L is a Lorentzian action and the path integralis taken over all trajectories between the points v =

0 and v . In our case the action S L is given by therelation (2.5) (for all Ricci-flat factor spaces and one-component free scalar field ( m = 1)) and we considerthe transition between “points” with v = 0 and theclassical turning point vt =

|ε|/Λ. To make the

oscillating integral (6.24) convergent it is necessary toperform a Wick rotation to the Euclidean time: τ L →−iτ E . The transition probability between the pointsv = 0 and vt is proportional to the squared modulusof the amplitude:

P ∼ | < vt, τ f |, τ i > |2. (6.25)

In the semiclassical limit

< vt, τ f |, τ i >∼ e−SE , (6.26)

where the Euclidean action for our model

S E =1

2κ2

τ f τ i

dτ −(z0)2 + 2|ε| + 2Λ e2q2z

0

− 1

2κ2v

v

τ i

= 2Λ

κ2

τ fτ i

dτ e2q2z0 − 1

2κ2v

v

τ i

(6.27)

is calculated on classical solutions of the Euclidean fieldequations (instantons) interpolating between the van-ishing geometry v = 0 and the turning point vt .

Classical solutions of the Euclidean field equationsin our model are given by the relation (6.3) whichshows that τ i = −∞ (v = 0) and τ f = 0 (v = vt).Substituting this relation into (6.27), we get [27]

S E = C

8π

ni=2

adi(c)i

−1 |ε|, (6.28)

where

C =

√2

q2− q2√

2

(6.29)



26 A. Zhuk

and we have taken into account that the D -dimensionalgravitational constant κ2 is connected with the New-ton constant GN by the relation

κ

2

= 8πGN

ni=2a

di

(c)i, (6.30)

(in (6.28) we put GN = 1). The parameter C > 0 forD > 3 and the presence of the boundary term in theaction (6.27) does not change the sign of S E . Let usestimate S E for the two-component (n=2) inflation-ary model. As follows from the relations (6.6), (6.9)and (6.12),

vt =

|ε|/Λ = Ad11 Ad22 = 10x(d1+d2). (6.31)

Then, with the help of the estimate (6.22) we get

|ε| = Λ

1/2

10

x(d1+d2) >∼

Λ

1/2

10

d1(x+70/ ln10)

. (6.32)Thus

S E = C

8πad2(c)2

−1 |ε| >

∼

C

8πΛ1/210d1(x+70/ ln10),

(6.33)

where a(c)2 ≈ LPl (see Eq. (6.20)). The quantum birthof the universe will not be suppressed if S E

<∼

1. Asfollows from (6.33), this is the case for Λ <

∼10−124cm−2 .

This quantity is much less than a possible value of the cosmological constant in the observable universe,Λ0

<∼

10−57 cm−2 . Thus, for realistic theories S E 1

and consequently a quantum birth of this system isstrongly suppressed. The reason is that there is alarge number of internal dimensions, as follows fromthe relation (6.31).

In some papers (see, e.g. [28]) it was suggested that,instead of the standard Euclidean rotation τ L → −iτ E ,the action (6.27) should be obtained by rotation in theopposite sense, τ L → +iτ E . However, in this casewe get the nonphysical result that the universe birthprobability is proportional to the volume of the arisinguniverse.

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Preprint of Astrophysical Institute at Potsdam, AIP95-03.

a. zhuk- multidimensional quantum cosmology: quantum wormholes, third quantization, inflation from...

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