Arithmetical Chaos and Quantum Cosmology

Abstract

In this note, we present the formalism to start a quantum analysis for the recent billiard representation introduced by Damour, Henneaux and Nicolai in the study of the cosmological singularity. In particular we use the theory of Maass automorphic forms and recent mathematical results about arithmetical dynamical systems. The predictions of the billiard model give precise automorphic properties for the wave function (Maass-Hecke eigenform), the asymptotic number of quantum states (Selberg asymptotics for PSL(2,Z)), the distribution for the level spacing statistics (the Poissonian one) and the absence of scarred states. The most interesting implication of this model is perhaps that the discrete spectrum is fully embedded in the continuous one.

35 pages, 4 figures. to be published on Classical and Quantum Gravity (scheduled for January 2009)

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Cited by in corpus (6)

• Supersymmetric quantum cosmological billiards
• Sugawara-type constraints in hyperbolic coset models
• The E10 Wheeler-DeWitt operator at low levels
• BKL maps and Poincaré sections
• Reflections on the hyperbolic plane
• Modular realizations of hyperbolic Weyl groups