The $\star$-value Equation and Wigner Distributions in Noncommutative Heisenberg algebras

TL;DR

This paper develops a phase-space formalism for quantum mechanics on noncommutative Heisenberg algebras, applying it to solve Landau and harmonic oscillator problems and deriving their spectra and Wigner distributions.

Contribution

It introduces a formalism connecting the Seiberg-Witten map with phase-space quantum mechanics on noncommutative algebras, providing exact solutions for specific models.

Findings

Derived correct energy spectra for Landau and harmonic oscillator in noncommutative space

Obtained Wigner distributions consistent with noncommutative quantum mechanics

Compared results with existing literature, confirming validity

Abstract

We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.