Holomorphic Quantization on the Torus and Finite Quantum Mechanics

TL;DR

This paper develops a holomorphic quantization framework for classical linear maps on the torus, connecting it to finite quantum mechanics and explicitly characterizing the quantum generators and states for various discretizations.

Contribution

It explicitly constructs the holomorphic quantization of $SL(2, Z)$ on tori and relates it to finite quantum mechanics, including the determination of generators and eigenstates for arbitrary discretizations.

Findings

Explicit construction of holomorphic quantization for $SL(2, Z)$ on tori.

Connection established between holomorphic quantization and finite quantum mechanics.

Determination of quantum generators and eigenstates for all discretizations.

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.