CC BY
29
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: [email protected]
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)
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
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: [email protected]
Мельников В. Н., доктор физико-математических наук, профессор, главный научный сотрудник. ВНИИ метрологической службы. Ул. Озёрная, 46, 119361 Москва, Россия. E-mail: [email protected]
