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6 - Cosmological quantum billiards

Published online by Cambridge University Press:  05 August 2012

By
Axel Kleinschmidt  and
Hermann Nicolai
Edited by
Jeff Murugan ,
Amanda Weltman  and
George F. R. Ellis
Axel Kleinschmidt
Affiliation:
Université Libre de Bruxelles
Hermann Nicolai
Affiliation:
Albert-Einstein-Institut
Jeff Murugan
Affiliation:
University of Cape Town
Amanda Weltman
Affiliation:
University of Cape Town
George F. R. Ellis
Affiliation:
University of Cape Town
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Summary

The minisuperspace quantization of D =11 supergravity is equivalent to the quantization of an 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 minisuperspace Wheeler-DeWitt equation involve automorphic (Maass wave) forms under the modular group W+(E10)≅ PSL2(0). 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 longstanding issues in canonical quantum gravity.

Introduction

The present chapter is based on [1], and elaborates on several issues and arguments that were not fully spelled out there. In that work, a first step was taken towards quantization of the one-dimensional “geodesic” E10/K(E10) coset model which had been proposed in [2] as a model of M-theory. Here, E10 denotes the hyperbolic Kac-Moody group E10 which is an infinite-dimensional extension of the exceptional Lie group E8, and plays a similarly distinguished role among the infinite-dimensional Lie algebras as E8 does among the finite-dimensional ones. The proposal of [2] had its roots both in the appearance of so-called “hidden symmetries” of exceptional type in the dimensional reduction of maximal supergravity to lower dimensions [3], as well as in the celebrated analysis of Belinskii, Khalatnikov and Lifshitz (BKL) [4] of the gravitational field equations in the vicinity of a generic space-like (cosmological) singularity.

Type
Chapter
Information
Foundations of Space and Time
Reflections on Quantum Gravity
 , pp. 106 - 124
Publisher: Cambridge University Press
Print publication year: 2012

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References

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