Weyl-Wigner Formulation of Noncommutative Quantum Mechanics

TL;DR

This paper develops a comprehensive phase space formulation of noncommutative quantum mechanics using Weyl-Wigner transform and Seiberg-Witten map, unifying previous approaches and extending to arbitrary dimensions.

Contribution

It introduces a systematic method to derive noncommutative quantum mechanics in phase space, generalizing existing models through a covariant formalism and isomorphism construction.

Findings

Constructed the extended starproduct and Moyal bracket for noncommutative phase space.

Proved the formalism's independence from the Seiberg-Witten map choice.

Unified and generalized previous phase space formulations of noncommutative quantum mechanics.

Abstract

We address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativity. By resorting to a covariant generalization of the Weyl-Wigner transform and to the Seiberg-Witten map we construct an isomorphism between the operator and the phase space representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended starproduct, Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of Seiberg-Witten map. Our approach unifies and generalizes all the previous proposals for the phase space formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some 2-dimensional spaces.