Quantum billiards in multidimensional models with branes

The European Physical Journal C, Mar 2014

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 \(M2\)- and \(M5\)-branes 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 \(120\) three-forms which mimic space-like \(D2\)-, \(D4\)-, \(FS1\)- and \(NS5\)-branes in \(D = 10\) \(II A\) 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.

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Quantum billiards in multidimensional models with branes

V. D. Ivashchuk 0 1 V. N. Melnikov 0 1 0 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 - 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)


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V. D. Ivashchuk, V. N. Melnikov. Quantum billiards in multidimensional models with branes, The European Physical Journal C, 2014, pp. 2805, Volume 74, Issue 3, DOI: 10.1140/epjc/s10052-014-2805-7