Quantum billiards in multidimensional models with fields of forms
V.D. Ivashchuk, V.N. Melnikov1, V.D. Ivashchuk, V.N. Melnikov2
(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
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 Hn - 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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