Wigner Functions for Noncommutative Quantum Mechanics: a group representation based construction

TL;DR

This paper constructs and analyzes Wigner functions for noncommutative quantum mechanics using group representation theory, exploring different non-commutativity levels and their effects on quantum states.

Contribution

It introduces a group-theoretic method to construct Wigner functions for various noncommutative quantum systems, extending previous techniques to new non-commutativity scenarios.

Findings

Wigner functions reflect different non-commutativity levels.

The method unifies the treatment of standard and noncommutative quantum mechanics.

Star-products and marginal distributions are explicitly derived.

Abstract

This paper is devoted to the construction and analysis of the Wigner functions for noncommutative quantum mechanics, their marginal distributions and star-products, following a technique developed earlier, {\it viz\/,} using the unitary irreducible representations of the group $\g$, which is the three fold central extension of the abelian group of $\mathbb R^4$. These representations have been exhaustively studied in earlier papers. The group $\g$ is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom. The Wigner functions studied here reflect different levels of non-commutativity -- both the operators of position and those of momentum not commuting, the position operators not commuting and finally, the case of standard quantum mechanics, obeying the canonical commutation relations only.