Deformation quantization of noncommutative quantum mechanics and dissipation

TL;DR

This paper reviews the Weyl-Wigner formulation of noncommutative quantum mechanics, introduces suitable mathematical tools, and explores the transition from noncommutative to commutative regimes via a particle coupled to an oscillator bath.

Contribution

It develops a framework for noncommutative quantum mechanics using Weyl-Wigner formalism and proposes criteria for the noncommutative-commutative transition.

Findings

Defined a star-product and Moyal bracket for noncommutative quantum mechanics

Analyzed properties of noncommutative Wigner functions and their relation to classical distributions

Derived the master equation for a Brownian particle in this context

Abstract

We review the main features of the Weyl-Wigner formulation of noncommutative quantum mechanics. In particular, we present a $\star$-product and a Moyal bracket suitable for this theory as well as the concept of noncommutative Wigner function. The properties of these quasi-distributions are discussed as well as their relation to the sets of ordinary Wigner functions and positive Liouville probability densities. Based on these notions we propose criteria for assessing whether a commutative regime has emerged in the realm of noncommutative quantum mechanics. To induce this noncommutative-commutative transition, we couple a particle to an external bath of oscillators. The master equation for the Brownian particle is deduced.