Cosmological quantum billiards

TL;DR

This paper explores the quantization of D=11 supergravity via cosmological quantum billiards, revealing automorphic wavefunctions that tend to zero at singularities, offering new insights into quantum cosmology and gravity.

Contribution

It establishes a connection between supergravity quantization and automorphic forms in a cosmological billiard framework, providing novel perspectives on quantum singularities.

Findings

Wavefunctions involve automorphic (Maass wave) forms under W^+(E10)

Wavefunctions tend to zero approaching the initial singularity

Derived inequalities for Laplace eigenvalues of automorphic forms

Abstract

The mini-superspace quantization of D=11 supergravity is equivalent to the quantization of a E10/K(E10) coset space sigma model, when the latter is restricted to the E10 Cartan subalgebra. As a consequence, the wavefunctions solving the relevant mini-superspace Wheeler-DeWitt equation involve automorphic (Maass wave) forms under the modular group W^+(E10)=PSL(2,O). Using Dirichlet boundary conditions on the billiard domain a general inequality for the Laplace eigenvalues of these automorphic forms is derived, entailing a wave function of the universe that is generically complex and always tends to zero when approaching the initial singularity. The significance of these properties for the nature of singularities in quantum cosmology in comparison with other approaches is discussed. The present approach also offers interesting new perspectives on some long standing issues in canonical quantum gravity.