Quantum billiards in multidimensional models with fields of forms
V.D. Ivashchuk, V.N. Melnikov^{1}, V.D. Ivashchuk, V.N. Melnikov^{2}
(1) Center for Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya 46, Moscow, 119361, Russia
(2) Institute of Gravitation and Cosmology of Peoples' Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia
Abstract
A Bianchi type I cosmological model in (n + 1)-dimensional gravity with several forms is considered. When the electric non-composite brane ansatz is adopted, the Wheeler-DeWitt (WDW) equation for the model, written in a conformally covariant form, is analyzed. Under certain restrictions, asymptotic solutions to the 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}. Two examples of quantum billiards are considered: a 2-dimensional quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard for an 11D model with 120 4-forms, whichmimics the SM2-brane sector of D = 11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved.
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