Quantum billiards with branes on product of Einstein spaces

TL;DR

This paper explores quantum billiards arising in a multidimensional gravitational model with branes and Einstein spaces, analyzing the Wheeler-DeWitt equation and its asymptotic solutions related to hyperbolic Kac-Moody algebras.

Contribution

It introduces a novel approach to quantum billiards in multidimensional gravity with branes, deriving asymptotic solutions to the Wheeler-DeWitt equation in this context.

Findings

Asymptotic solutions to the Wheeler-DeWitt equation are obtained.

Quantum billiards are characterized in hyperbolic space related to Kac-Moody algebras.

Classical metric solutions are derived from quantum asymptotics.

Abstract

We consider a gravitational model in dimension D with several forms, l scalar fields and a Lambda-term. We study cosmological-type block-diagonal metrics defined on a product of an 1-dimensional interval and n oriented Einstein spaces. As 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 the WDW equation are found in the limit of the formation of the billiard walls. These solutions reduce the problem to the so-called quantum billiard in (n + l - 1)-dimensional hyperbolic space. Several examples of quantum billiards in the model with electric and magnetic branes, e.g. corresponding to hyperbolic Kac-Moody algebras, are considered. In the case n=2 we find a set of basis asymptotic solutions to the WDW equation and derive asymptotic solutions for the metric in the classical case.