The Multidimensional Berry-Hannay Model

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 $\mathcal{A}$ 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 $\mathcal{A}$, 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.

11 pages, preprint

References in corpus (1)

• The Two Dimensional Hannay-Berry Model

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• Particle on a Torus Knot: Anholonomy and Hannay Angle
• Bounds on certain Higher-Dimensional Exponential Sums via the Self-Reducibility of the Weil Representation
• Geometric Phases for Classical and Quantum Dynamics: Hannay angle and Berry Phase for Loops on a Torus
• The Higher-Dimensional Rudnick-Kurlberg Conjecture