Integrable Toda system as a quantum approximation to the anisotropy of Mixmaster

TL;DR

This paper introduces a regularisation method that approximates the quantum dynamics of the Mixmaster universe using an integrable Toda system, enabling explicit eigenfunction construction and a new perturbative analysis.

Contribution

It presents a covariant Weyl-Heisenberg integral quantization approach that simplifies the anisotropy potential to an integrable Toda system, facilitating quantum analysis.

Findings

Approximate quantum dynamics using Toda system

Explicit eigenfunctions for the approximated system

New perturbative approach to quantum Mixmaster dynamics

Abstract

We present a regularisation approach to the study of the quantum dynamics of the Mixmaster universe which allows to approximate the anisotropy potential with the explicitly integrable periodic 3-particle Toda system. This approach is based on a covariant Weyl-Heisenberg integral quantization. Such a procedure naturally amplifies the dynamical role of the underlying Toda system by smoothing out the three canyons of the anisotropy potential. Since the respective eigenfunctions can be explicitly constructed, our finding paves the way to a novel perturbative approach to the quantum Mixmaster dynamics.