Chaos and Quantum Chaos in Cosmological Models

TL;DR

This paper reviews the chaotic dynamics of certain cosmological models near singularities, explores their quantization, and discusses semi-classical solutions related to boundary conditions like no-boundary and wormhole states.

Contribution

It provides a comprehensive review of classical chaos, quantum properties, and semi-classical solutions of Bianchi type VIII and IX cosmological models using billiard dynamics.

Findings

Chaotic behavior of Bianchi models near singularities.

Quantum energy level statistics of the billiard models.

Explicit semi-classical solutions with boundary conditions.

Abstract

Spatially homogeneous cosmological models reduce to Hamiltonian systems in a low dimensional Minkowskian space moving on the total energy shell $H=0$. Close to the initial singularity some models (those of Bianchi type VIII and IX) can be reduced further, in a certain approximation, to a non-compact triangular billiard on a 2-dimensional space of constant negative curvature with a separately conserved positive kinetic energy. This type of billiard has long been known as a prototype chaotic dynamical system. These facts are reviewed here together with some recent results on the energy level statistics of the quantized billiard and with direct explicit semi-classical solutions of the Hamiltonian cosmological model to which the billiard is an approximation. In the case of Bianchi type IX models the latter solutions correspond to the special boundary conditions of a `no-boundary state' as proposed by Hartle and Hawking and of a `wormhole' state.