The Multidimensional Berry-Hannay Model

TL;DR

This paper develops a quantum mechanical model on a multidimensional symplectic torus, introducing a canonical quantization that respects symmetries and provides a compatible Hilbert space representation.

Contribution

It constructs a simultaneous quantization of functions and symplectic group actions on a multidimensional torus, using the quantum torus and fixed points for rational parameters.

Findings

Existence of a unique fixed point for rational er on the representation set

Construction of a canonical projective equivariant quantization

Development of a compatible Hilbert space with group and algebra actions

Abstract

The aim of this paper is to construct the Berry-Hannay model of quantum mechanics on a 2n-dimensional symplectic torus. We construct a simultaneous quantization of the algebra $\cal A$ of functions on the torus and the linear symplectic group $\G = \Sp(\mathrm{2n},\Z)$. In the construction we use the quantum torus $\A$, which is a deformation of $\cal A$, together with a $\G$-action on it. We obtain the quantization via the action of $\G$ on the set of equivalence classes of irreducible representations of $\A$. For $\h \in \Q$ this action has a unique fixed point. This gives a canonical projective equivariant quantization. There exists a Hilbert space on which both $\G$ and $\A$ act in a compatible way.