Wigner Measures in Noncommutative Quantum Mechanics

TL;DR

This paper investigates the properties of noncommutative Wigner measures in quantum mechanics, establishing conditions for their characterization, analyzing Gaussian cases, and exploring their relation to classical phase-space distributions.

Contribution

It provides necessary and sufficient conditions for noncommutative Wigner measures, analyzes Gaussian cases, and clarifies their relation to classical and quantum phase-space distributions.

Findings

Derived conditions for noncommutative Wigner measures

Analyzed properties of Gaussian noncommutative Wigner measures

Established relations between noncommutative, Liouville, and Wigner measures

Abstract

We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schr\"odinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures.