Quantum billiards in multidimensional models with fields of forms

TL;DR

This paper explores quantum cosmological models with multiple fields, analyzing the Wheeler-DeWitt equation in higher dimensions, and demonstrates how solutions near singularities can be described as quantum billiards in hyperbolic space.

Contribution

It introduces a conformally covariant form of the Wheeler-DeWitt equation for multidimensional models with forms and analyzes asymptotic solutions as quantum billiards.

Findings

Asymptotic solutions reduce to quantum billiards in hyperbolic space.

Vanishing wave function at singularities is established for certain cases.

Examples include 2D and 9D quantum billiards related to supergravity models.

Abstract

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 which mimics SM2-brane sector of D=11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved.