Arithmetical Chaos and Quantum Cosmology

TL;DR

This paper develops a quantum analysis framework for a cosmological billiard model, revealing automorphic properties of the wave function and spectral characteristics, with implications for quantum cosmology and arithmetical dynamical systems.

Contribution

It introduces a quantum formalism for the billiard representation in cosmology using Maass automorphic forms and mathematical results on arithmetical systems, providing new insights into spectral properties.

Findings

Wave function is a Maass-Hecke eigenform.

Quantum states follow Selberg asymptotics.

Level spacing distribution is Poissonian.

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.