Torus as phase space: Weyl quantization, dequantization and Wigner formalism

TL;DR

This paper develops a comprehensive framework for Weyl quantization, dequantization, and Wigner formalism on the torus phase space, providing explicit formulas and characterizations for quantum-classical correspondence without regularity constraints.

Contribution

It introduces explicit Weyl quantization on the torus, characterizes symbol equivalence classes, and presents a dequantization method applicable to matrices like Pauli matrices.

Findings

Explicit Weyl quantization formulas on the torus

Characterization of symbol equivalence classes

Dequantization procedure for quantum observables

Abstract

The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.