Научная статья на тему 'Квантовые бильярды с бранами'

Квантовые бильярды с бранами Текст научной статьи по специальности «Математика»

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Ключевые слова
КОСМОЛОГИЧЕСКИЕ БИЛЬЯРДЫ / БРАНЫ / УРАВНЕНИЕ УИЛЕРА-ДЕВИТТА / COSMOLOGICAL BILLIARDS / BRANES / WHEELER-DEWITT EQUATION

Аннотация научной статьи по математике, автор научной работы — Иващук В. Д., Мельников В. Н.

Рассмотрена космологическая модель типа Бианки-I в (n + 1)-мерной гравитационной теории с несколькими полями форм. В случае, когда принят анзатц с электрическими некомпозитными бранами, проанализировано уравнение Уилера-ДеВитта (УДВ), записанное в конформно-ковариантном виде. При определенных ограничениях найдены асимптотические решения уравнения УДВ вблизи сингулярности, которые сводят проблему к так называемому квантовому бильярду на (n 1)-мерном пространстве Лобачевского H^n-1.

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QUANTUM BILLIARDS WITH BRANES

Cosmological Bianchi-I type model in the (n + 1)-dimensional gravitational theory with several forms is considered. When electric non-composite brane ansatz is adopted the Wheeler-DeWitt (WDW) equation for the model, written in the conformally-covariant form, is analyzed. Under certain restrictions asymptotic solutions to WDW equation near the singularity are found which reduce the problem to the so-called quantum billiard on the (n 1)-dimensional Lobachevsky space H^n-1.

Текст научной работы на тему «Квантовые бильярды с бранами»

UDC 530.1; 539.1

QUANTUM BILLIARDS WITH BRANES

V. D. Ivashchuk a'b, V. N. Melnikov a'b

a Center for Gravitation and Fundamental Metrology, VNHMS, Ozyornaya St., 46, Moscow 119361, Russia. b institute of Gravitation and Cosmology, Peoples' Friendship University of Russia, Miklukho-Maklaya St., 6, Moscow

117198, Russia.

E-mail: ivashchuk@mail.ru

Cosmological Bianchi-I type model in the (n + 1)-dimensional gravitational theory with several forms is considered. When electric non-composite brane ansatz is adopted the Wheeler-DeWitt (WDW) equation for the model, written in the conformally-covariant form, is analyzed. Under certain restrictions asymptotic solutions to WDW equation near the singularity are found which reduce the problem to the so-called quantum billiard on the (n — 1)-dimensional Lobachevsky space Hn-1.

Keywords: cosmological billiards, branes, Wheeler-DeWitt equation.

1 Introduction

In this paper we deal with the quantum billiard approach for multidimensional cosmological-type models defined on the manifold (u_,u+) x Rn, where n > 3. In classical case the billiard approach was suggested by Chitre fl] for explanation the BLK-oscillations [2] in the Bianchi-IX model [3] by using a simple triangle billiard in the Lobachevsky space H2.

In multidimensional case the billiard representation for cosmological model with multicomponent "perfect" fluid was introduced in [4,5]. The billiard approach for multidimensional models with scalar fields and fields of forms was suggested in [6], see also [7] for examples of "chaotic" behavior in supergravitational models.

Recently the quantum billiard approach for a multidimensional gravitational model with several forms was considered in [8]. The asymptotic solutions to WDW equation presented in [8] are equivalent to those obtained earlier in [5].

Here we use another form of the WDW equation with enlarged minisuperspace which include the form potentials $s [9]. We get another version of the quantum billiard approach, which is different from that

of [8].

2 The model

Here we consider the multidimensional gravitational model governed by the action

Sa

2K2 Im dDzVMIfibi

-£ sesè (Fs)2} + S

YGH ;

(ns > 2) on a ^-dimensional manifold M, s G S. In (1) we denote |g| = |det(gMN)|, (Fs)2 =

Fs Fs gMiNi gMns Nns s (Z S where S

FMl...Mn cNl...Nna g ■■■g ' s G S wueie S

is some finite set of indices and Sygh is the standard York-Gibbons-Hawking boundary term.

Let us consider the manifold M = R* x Rn with the metric

g = we2Y(u)du <g> du + e2^^(u)e(i)dxi <g> dx

(2)

where R* = (u-,u+), w = ±1 mid e(i) = ±1 i = 1,..., n. The dimension of M is D = 1 + ^^or w = —1 and e(i) = 1 i = 1,... ,n, we deal with cosmological solutions while for w = 1, and e(1) = —1 e(j) = 1, j = 2,... ,n, we get static solutions (e.g. wormholes etc).

Let Q = Q(n) be a set of all non-empty subsets of {1,...,n}. For any I = {ii,... ,ik} € Q ii < ... < ik, we denote t(I) = dx11 A...Adxik, e(I) = e(i1).. .e(ik), d(I) = |I| = k .

For the fields of forms we consider the following non-composite electric ansatz

As = $st(Is), Fs = d$s A t(Is), (3)

where $s = $s(u) ^s ^^^^^h function on R* and Is € Q, s € S Due to (3) we have d(Is) = ns — 1, s € S.

The equations of motion for the model (1) with the fields from (2) and (3) are equivalent to equations of motion for the a-model governed by the action [9]

(1)

Sa = ^

where g = gMN dzM ® dzN is the metric on the manifold M, dim M = D, 6,s = 0 Fs = dAs = ^ FsM M dzM A ... A dzMns is a ns-form

ns! Mi...Mna s

J duN j Gab (X)XAXb}

(4)

where ^ = ^d N = exp(Y0 — 7) > 0 is modified lapse function with yo(^) = "=i X = (XA) =

(4i, € RN, N = n + m, m = |S| is the number of branes, X = dX/du and minisupermetric G = Gab (X )dXA ® dXB on minisupers pace M = RN is defined by the relation

where

a —

(N - 2) 8(N - 1) '

(14)

G = G + ^ese-2Us(0)d$s <g> d$s,

ses

where

G = Gij dfî <g> drfP, Gij = 5ij - 1,

(5)

(6)

N = n + m.

Here = (X) is the wave function

corresponding to the /-gauge (10) and satisfying the relation

yf = ebf yf=0, b = (2 - N)/2.

(15)

and

Us(4) = U^ = £ 4\

In (13) we denote by A[Gf^d R[Gf] the Laplace-Beltrami operator and the scalar curvature corresponding to the metric

Us

(U/) = Sus, (7) Gf = e2f G,

ie/s

(16)

s e S.

Here Si1 = 5ij is an indicator of i belonging to I: Si1 = 1 for i e I and Si1 = 0 otherwise; and es = e(Is)6s, s e S.

In what follows we will use the scalar product

respectively.

GG

(-, +,..., +) • We put

e2f = -(Gij 4 <j )-

(17)

(U,U ' ) = Gij UiU'j,

for U = (Ui),U' vers«

i,j = 1,...,n

(8)

where Gj 4i4j < 0-

In what follows we will use a diagonalization of 4 variables

(U/) € Rn, where (Gij) is the ij )

matrix inverse to the matrix (Gij ) Gij = Sij + ^ZD'

4i

sa ,

(18)

■qabzazb, where

3 Quantum billiard approach

First we outline two restrictions which will be used in derivation of the quantum billiard: (i) d(Is) < D - 2, (ii) es > 0, for all s.

Due to the first restriction we get

a = 0, ,.,n — 1, obeying Gij4>i4j (nab) = diag( — 1, +1,..., +1).

We restrict the WDW equation to the lower light cone V_ = {z = (z0, Z)|z0 < 0,nabzazb < 0} and introduce Misner-Chitre-like coordinates

z0 = -e-y° 1+ 7

2

(Us,Us) > 0, s € S.

Let us fix the temporal gauge as follows

Y0 - Y = 2/(X), N = e'

2f

(9)

(10)

—2e-y

1 - T2'

° y 1 - y2,

(19)

(20)

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where y0 < 0 and y2 < 1- We note that in these variables / = y°. We denote

where /: M ^ R is a smooth function. Then we obtain the Lagrange system with the Lagrangian

Gij = e-2f Gij.

Lf = ge2f Gab (X )X AX B

2

and the energy constraint Ef = 2e2f Gab (X )X AX B

0.

ij ij The following formula is valid

(H) G = -dy0 <g> dy0 + hL, where

4Srsdyr <g) dys

hL

(1 - yy

22

Using the standard prescriptions of covariant and Here the metric hL is defined on the unit ball Dn 1 =

(21) (22)

(23)

-l

conformally covariant quantization of the energy-constraint [10] we (WDW) equation [9

1 . a

M

{y e M"_1|y2 < 1}. The pair (D"_\hL) is one of

(n — 1) Lobachevsky space.

We use the following ansatz

Hf = (- 2MA [e2f G] + [e2f G]) = 0, (13) yf = e^)^.*' y0L,

(24)

l

where a = 0, ...,n — 1.

1 The inequalities (33) imply |vs| > 1 for al Is.

C= — m/). (25) The potential corresponds to the billiard B in the

ses multidimensional Lobachevsky space (Dn-1, hL). This

Here parameters Qs = 0 correspond to charge densities billiar(lis 311 open domain in which is defined by

of branes and e^-*' = exp(i £seS Qs$s). a set of Equalities:

Then the WDW is reduced to the following relation |y — v | < ^v2 — 1 = r (36)

(— -A[G] + 1 ^^ Q2ge_2f+2Us(^) + SV J s e S. The boundary dB is formed by parts of hyper-

2 2 seS J spheres with centers in vs and radii rs.

x^ol = 0 (26) ^e condition (34) is also obeyed for the

diagonalization (35) with where _

1 z0 = Ui^vVKUrU)!, (37)

SV = Ae_2f — -(n — 2)2 (27)

8 where U-vector is time-like (U, U) < 0 Mid (U, Us) < 0

seS

1

A = 8(N_1)^ (U s,U ) funCti0n ^0,L(y0,y)

( ) s,s'es

— (N — 2) ^(Us,Us)]. (28) (— 1A[<G] + SV^ =0 (38)

ses ^ '

It was shown in [6] that with y e B and the zero boundary condition ^0,L|dB =

0 imposed. Due to (22) we get A[G] = — (d0)2 + A[hL],

1 y^Q2e_2f+2Us(0) ^ v (29) where A[hL] = AL is the Laplace-Beltrami operator

2 ^S s ' corresponding to the (n — 1)-dimensional Lobachevsky

metric hL.

as y0 = / ^ —to. By splitting the variables

In this relation is the potential of infinite walls

which are produced by branes: ^0,l = ^0(y )^L(y) (39)

V = ^^ q (v2 — 1 — (y — v )2) ^q-j we are led to the asymptotic relation (for y0 ^ —to)

seS (()2 — Al + 2Ae_2«° + E — | (n — 2)2)

Here we use the notation (x) = +to for x > 0 Vv 7 /

and QTO(x) = 0 for x < 0. The vectors vs, s e S, x^0 = 0 (40)

3>n_1

belonging to Rn 1 are defined by the formulae

Vs = -Us/us0, (31)

equipped with the relations

where n-dimensional vectors us = (us0,us) = (usa) al^l = — E^L, ^l|sb = 0. (41)

Us

matrix (Si) from (18) e we assume that the 0Perator (—Al) with

a

usa = Sa Uts. (32) obeying

Due to condition (9) E > i(n — 2)2. (42)

(Us,Us) = -(us0)2 + (Us)2 > 0 (33)

4

This inequality was proved in [8] for billiards with finite

s

obeying Here we put

us0 > 0 (34) A < (43)

for all s € S. The inverse matrix (Sa) = (Sa)-1 defines Solving equation (40) we get for A < 0 the following

the the map inverse to ( 18) basis of solutions

za = Sa4i, (35) y0 = Biu (72|Aje-y°) , (44)

_ m _

4 Conclusion

Here we have done an overview of our approach from [11,12] by considering the quantum billiard for

n

factor-spaces in the theory with several forms. After adopting the electric non-composite brane ansatz with certain restrictions on parameters of the model we have deduced the Wheeler-DeWitt (WDW) equation for the model, written in the conformally-covariant form.

By imposing certain restrictions on parameters of the model we have obtained the asymptotic solutions to WDW equation which are of a quantum billiard form since they are governed by the spectrum of the Laplace-Beltrami operator on the billiard with the zero boundary condition imposed. The billiard is a part of the (n — 1)-dimensiond Lobachevsky space Hn_1.

Acknowledgement

This research has been supported in part by PFUR grant (No. 200312-1-174) in 2014.

References

[1] Chitre D. M. 1972 Ph. D. Thesis (University of Maryland).

[2] Belinskii V. A., Lifshitz E. M. and Khalatnikov I. M. 1970 Usp. Fiz. Nauk 102 463 [in Russian],

[3] Misner C. W. 1969 Phys. Rev. 186 1319.

[4] Ivashchuk V. D., Kirillov A. A. and Melnikov V. N. 1994 Russian Physics Journal 37 1102.

[5] Ivashchuk V. D. and Melnikov V. N. 1995 Class. Quantum, Grav. 12 809.

[6] Ivashchuk V. D. and Melnikov V. N. 2000 J. Math. Phys. 41 634.

[7] Damour T„ Henneaux M. and Nicolai H. 2003 20 R145.

[8] Kleinschmidt A., Koehn M. and Nicolai H. 2009 Phys. Rev. D 80 061701.

[9] Ivashchuk V. D. and Melnikov V. N. 1998 J. Math. Phys. 39 2866.

[10] Misner C. W. 1972 In Magic without Magic. John Archibald Wheeler, a collection of essays in honor of his sixtieth birthday ed. Klauder J. R. (Freeman, San Francisko).

[11] Ivashchuk V. D. and Melnikov V. N. 2013 Grav. Cosmol. 19 171.

[12] Ivashchuk V. D. and Melnikov V. N. 2014 Eur. Phys. J. C 74 2805.

Received 13.11.2014

where Biu(z) = (z),Kiu(z) are modified Bessel functions and

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E — 1(n — 2)2 > 0. (45)

It was shown in [11] that

^f ^ 0 (46)

as y0 ^ —to for fixed y G ^d $s G R, s G S, in the following two cases: i) B = K; ii) B = I, when 1 q> y/2\A\.

In [11] we have presented an example of quantum d = 9 billiard for D = 11 gravitational model with 120 "electric" 4-forms and have shown the asymptotic vanishing of the basis wave functions ^ 0, as y0 ^ —to, for any choice of the Bessel function B = K,I. The generalization of the model to electromagnetic composite case (when scalar fields were present) was done in [12].

В. Д. Иващук, В. H. Мельников КВАНТОВЫЕ БИЛЬЯРДЫ С ВРАНАМИ

Рассмотрена космологическая модель типа Биаики-1 в (n + 1)-мериой гравитационной теории с несколькими полями форм. В случае, когда принят анзатц с электрическими некомпозитными бранами, проанализировано уравнение Уилера-ДеВитта (УДВ), записанное в конформно-ковариантном виде. При определенных ограничениях найдены асимптотические решения уравнения УДВ вблизи сингулярности, которые сводят проблему к так называемому квантовому бильярду на (n — 1)-мерном пространстве Лобачевского Hn—1.

Ключевые слова: космологические бильярды, браны, уравнение Уилера-ДеВитта.

Иващук В. Д., доктор физико-математических наук, ведущий научный сотрудник. ВНИИ метрологической службы. Ул. Озёрная, 46, 119361 Москва, Россия. E-mail: ivashchuk@mail.ru

Мельников В. Н., доктор физико-математических наук, профессор, главный научный сотрудник. ВНИИ метрологической службы. Ул. Озёрная, 46, 119361 Москва, Россия. E-mail: melnikov@phys.msu.ru

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