A Deformation Quantization Theory for Non-Commutative Quantum Mechanics

TL;DR

This paper develops a deformation quantization framework for non-commutative quantum mechanics, expressing it as a Weyl calculus on double phase space, and proves a spectral theorem for the star-genvalue equation.

Contribution

It introduces a Weyl calculus formulation for non-commutative quantum mechanics and extends spectral analysis methods to this setting.

Findings

Star-product properties are characterized within the Weyl calculus framework.

A spectral theorem for the star-genvalue equation is established.

The approach generalizes previous deformation quantization models.

Abstract

We show that the deformation quantization of non-commutative quantum mechanics previously considered by Dias and Prata can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined, and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef.