Harmonic oscillator in twisted Moyal plane: eigenvalue problem and relevant properties

TL;DR

This paper investigates the quantum harmonic oscillator within a twisted Moyal space, analyzing its eigenvalues, degeneracy, and properties using a matrix basis and dynamical star product, revealing infinite degeneracy and coordinate-dependent energies.

Contribution

It introduces a detailed analysis of the harmonic oscillator in twisted Moyal space, including explicit computation of star actions and degeneracy structure, which is a novel extension of noncommutative quantum mechanics.

Findings

Harmonic oscillator states are infinitely degenerate.

Energies depend on coordinate functions.

Explicit star actions of creation and annihilation functions are derived.

Abstract

The paper reports on a study of a harmonic oscillator (ho) in the twisted Moyal space, in a well defined matrix basis, generated by the vector fields $X_{a}=e_{a}^{\mu}(x)\partial_{\mu}=(\delta_{a}^{\mu}+\omega_{ab}^{\mu}x^{b})\partial_{\mu}$, which induce a dynamical star product. The usual multiplication law can be hence reproduced in the $\omega_{ab}^{\mu}$ null limit. The star actions of creation and annihilation functions are explicitly computed. The ho states are infinitely degenerate with energies depending on the coordinate functions.