On Berry-hannay Equivariant Quantization of the Torus 0 Introduction 0.1 Motivation

The main goal of this paper is to construct the Berry-Hannay's model of quantum mechanics on a two dimensional symplectic torus. We construct a simultaneous quantization of the algebra of functions and the linear symplectic group G = SL 2 (Z). We obtain the quan-tization via an action of G on the set of equivalence classes of irre-ducible representations of Rieffel's 2-dimensional quantum torus A. For ∈ Q this action has a unique fixed point. This gives a canoni-cal projective equivariant quantization. There exists a Hilbert space on which both G and A act equivariantly. Combined with the fact that every projective representations of G can be lifted to a linear representation, we also obtain linear equivariant quantization. In the paper " Quantization of linear maps on the torus-Fresnel diffraction by a periodic grating " , published in 1980 (see [BH]), the physicists M.V. Berry 1 and J.Hannay explore a model for quantum mechanics on the 2-dimensional torus. One of the motivations was to study the phenomenon of quantum chaos in this model (see [R] for a survey). Berry and Hannay suggested to quantize simultaneously the functions on the torus and the linear symplectic group G = SL 2 (Z). They mentioned (see [BH],[M]) that the theta subgroup G θ ⊂ G is the largest that one can quantize and conjectured (see [BH],[M]) that the quantization of G should satisfy a multiplicativity property (be a linear representation of the group). In this paper we want to construct the Berry-Hannay's model. The central question is whether there exists a Hilbert space on which a deformation of the algebra of functions and the linear symplectic group G, both act in a compatible way. In this paper we give an affirmative answer to the existence of the quantiza-tion procedure and give explicit formulas. We show a construction (Theorem 0.3, Corollary 0.4 and Theorem 0.6) of the canonical equivariant quantization procedure for rational Planck constants. It is unique as a projective quan-tization (see definitions below). We show that the projective representation of G can be lifted in exactly 12 different ways to a linear representation (to obey the multiplicativity property). These are the first examples of such equivariant quantization. Our construction gives a counter example to the first Berry-Hannay's conjecture and a proof of the second (see [BH], [M]). Previously it was shown by Mezzadri and Kurlberg-Rudnick (see [M], [KR]) that one can construct …