https://doi.org/10.1140/epjc/s10052-014-2805-7
Regular Article - Theoretical Physics
Quantum billiards in multidimensional models with branes
1
Center for Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya St., 46, 119361 , Moscow, Russia
2
Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, Miklukho-Maklaya St., 6, 117198 , Moscow, Russia
* e-mail: ivashchuk@mail.ru
Received:
10
January
2014
Accepted:
27
February
2014
Published online:
15
March
2014
A gravitational 
-dimensional model with 
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 
-dimensional Lobachevsky space. Two examples of quantum billiards are considered. The first one deals with 
-dimensional quantum billiard for 
model with 
four-forms which mimic space-like 
- and 
-branes of 
supergravity. The second one deals with the -dimensional quantum billiard for 
gravitational model with one scalar field, 
four-forms and 
three-forms which mimic space-like 
-, 
-, 
- and 
-branes in 
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 
model) but magnetic branes could not be neglected in calculations of quantum asymptotic solutions while they are irrelevant for classical oscillating behavior when all electric branes are present.
© SIF and Springer-Verlag Berlin Heidelberg, 2014
