The Two Dimensional Hannay-Berry Model

Details The Two Dimensional Hannay-Berry Model The Two Dimensional Hannay-Berry Model

2003-12-13

arxiv.org/abs/math-ph/0312039v2

Physics

Mathematical Physics

11 pages

Scientific paper

The main goal of this paper is to construct the Hannay-Berry 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 quantization via an action of $\G$ on the set of equivalence classes of irreducible representations of Rieffel`s quantum torus $\Ad$. 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 $\Ad$ act equivariantly. Combined with the fact that every projective representation of $\G$ can be lifted to a linear representation, we also obtain linear equivariant quantization.

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