# Wigner functions for gauge equivalence classes of unitary irreducible representations of noncommutative quantum mechanics

## TL;DR

This paper develops a general method to construct Wigner functions for noncommutative quantum mechanics using gauge equivalence classes of unitary irreducible representations, unifying different gauge choices.

## Contribution

It introduces a comprehensive construction of Wigner functions for NCQM based on gauge equivalence classes of UIRs, encompassing specific gauge cases like Landau and symmetric gauges.

## Key findings

- Constructed Wigner functions for 2-degrees of freedom NCQM
- Unified gauge-dependent Wigner functions within a single framework
- Applied to specific gauge choices such as Landau and symmetric gauges

## Abstract

While Wigner functions forming phase space representation of quantum states is a well-known fact, their construction for noncommutative quantum mechanics (NCQM) remains relatively lesser known, in particular with respect to gauge dependencies. This paper deals with the construction of Wigner functions of NCQM for a system of 2-degrees of freedom using 2-parameter families of gauge equivalence classes of unitary irreducible representations (UIRs) of the Lie group $\g$ which has been identified as the kinematical symmetry group of NCQM in an earlier paper. This general construction of Wigner functions for NCQM, in turn, yields the special cases of Landau and symmetric gauges of NCQM.