Quantization of a torus phase space

TL;DR

This paper explicitly performs the quantization of torus phase space using geometric quantization techniques, demonstrating the construction of quantum states and operators, and establishing the Hilbert space dimension related to phase space area.

Contribution

It introduces a detailed method for quantizing torus phase space, combining left and right invariant vector fields, and applies Dirac, Wu, and Yang's arguments for monopoles.

Findings

Quantization of $R^2$ and $S^1 imes S^1$ phase spaces is explicitly achieved.

The physical Hilbert space dimension equals the phase space area divided by Planck's constant.

The techniques are applicable to any symplectic manifold.

Abstract

Quantization of $R^2$ and $S^1 \times S^1$ phase spaces are explicitly carried out tweaking the techniques of geometric quantization. Crucial is a combined use of left and right invariant vector fields. Canonical bases, operators and their actions are explicitly presented. Arguments of Dirac and also Wu and Yang for monopoles are applied for obtaining the quantization of the phase space area. Equivalence of states in the infinite dimensional prequantum Hilbert space resulting in a physical Hilbert space of dimension equal to the phase space area in units of the Planck constant is demonstrated. These techniques can be applied to any manifold with a symplectic structure.