Holomorphic Quantization on the Torus and Finite Quantum Mechanics

Abstract

We construct explicitly the quantization of classical linear maps of $SL(2,R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2,Z)$ to the subgroup $SL(2,Z)/{\Gamma }_l$, ${\Gamma }_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_l(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.

12 pages LaTeX (needs amssymb.sty). Version as will appear in J. Phys. A