Quantum billiards in multidimensional models with branes
V. D. Ivashchuk
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V. N. Melnikov
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1
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Institute of Gravitation and Cosmology, Peoples' Friendship University of Russia
, Miklukho-Maklaya St., 6,
117198 Moscow, Russia
1
Center for Gravitation and Fundamental Metrology
, VNIIMS, Ozyornaya St., 46,
119361 Moscow, Russia
A gravitational D-dimensional model with l scalar fields and several forms is considered. When a cosmological-type diagonal metric is chosen, an electromagnetic composite brane ansatz is adopted and certain restrictions on the branes are imposed; the conformally covariant Wheeler-DeWitt (WDW) equation for the model is studied. Under certain restrictions asymptotic solutions to WDW equation are found in the limit of the formation of the billiard walls which reduce the problem to the so-called quantum billiard on the (D +l 2)-dimensional Lobachevsky space. Two examples of quantum billiards are considered. The first one deals with 9-dimensional quantum billiard for D = 11 model with 330 four-forms which mimic space-like M 2- and M 5branes of D = 11 supergravity. The second one deals with the 9-dimensional quantum billiard for D = 10 gravitational model with one scalar field, 210 four-forms and 120 threeforms which mimic space-like D2-, D4-, F S1- and N S5branes in D = 10 I I A supergravity. It is shown that in both examples wave functions vanish in the limit of the formation of the billiard walls (i.e. we get a quantum resolution of the singularity for 11D model) but magnetic branes could not be neglected in calculations of quantum asymptotic solutions while they are irrelevant for classical oscillating behavior when all 120 electric branes are present. This paper deals with the quantum billiard approach for D-dimensional cosmological-type models defined on the manifold (u, u+) RD1, where D 4. The billiard approach in classical gravity originally appeared in the dissertation of Chitr [1] for the explanation of the BLK-oscillations [2] in the Bianchi-IX model [3,4]. In
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this approach a simple triangle billiard in the Lobachevsky
space H 2 was used.
In [5] the billiard approach for D = 4 was extended to the
quantum case. Namely, the solutions to the WheelerDeWitt
(WDW) equation [6] were reduced to the problem of finding
the spectrum of the LaplaceBeltrami operator on a Chitrs
triangle billiard. Such an approach was also used in [7] in the
context of studying the large scale inhomogeneities of the
metric in the vicinity of the singularity.
A straightforward generalization of the Chitrs billiard to
the multidimensional case was performed in [810], where
a multidimensional cosmological model with a
multicomponent perfect fluid and n Einstein factor spaces was studied.
In [10] the search of oscillating behavior near the singularity
was reduced to the problem of proving the finiteness of the
billiard volume. This problem was reformulated in terms of
the problem of the illumination of the sphere Sn2 by
pointlike sources. In [10] the inequalities on the Kasner
parameters were found and the quantum billiard approach was
considered; see also [11,12]. The classical billiard approach
for multidimensional models with fields of forms and scalar
fields was suggested in [13], where the inequalities for the
Kasner parameters were also written. For certain examples
these inequalities have played a key role in the proof of the
never-ending oscillating behavior near the singularity which
takes place in effective gravitational models with forms and
scalar fields induced by superstrings [1416]. It was shown
in [17] that in these models the parts of billiards are related
to Weyl chambers of certain hyperbolic KacMoody (KM)
Lie algebras [1821]. This fact simplifies the proof of the
finiteness of the billiard volume. Using this approach the
well-known result from [22] on the critical dimension of pure
gravity was explained using hyperbolic algebras in [23]. For
reviews on the billiard approach see [16,24].
In recent publications [25,26] the quantum billiard
approach for the multidimensional gravitational model with
several forms was considered. The main motivation for this
approach is coming from the quantum gravity paradigm;
see [27,28] and references therein. It should be noted that
the asymptotic solutions to the WDW equation presented in
these papers are equivalent to the solutions obtained earlier
in [10]. The wave function ( KKN) from [25,26] corresponds
to the harmonic time gauge, while the wave function ( IM)
from [10] is related to the tortoise time gauge. (These
functions are connected by a certain conformal transformation
KKN = IM.) In [10,25,26] a semi-quantum approach
was used: the gravity (of a toy model) was quantized but
the matter sources (e.g. fluids, forms) were considered at the
classical level.1 Such a semi-quantum form of the WDW
equation for the model with fields of forms and a scalar field
was suggested earlier in [33].
In our previous publication [34] we have used another
form of the WDW equation with enlarged minisuperspace
which (...truncated)