Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane

TL;DR

This paper derives gauge invariant energy spectra and degeneracies for a particle on a 2D noncommutative plane under magnetic field and harmonic potential, revealing how noncommutativity and magnetic field influence degeneracy and spectrum.

Contribution

It introduces a method to obtain gauge invariant energy eigenvalues and degeneracies using phase space transformations on the noncommutative plane, connecting to Landau levels and noncommutative harmonic oscillators.

Findings

Energy levels depend on magnetic field and noncommutativity parameter.

Degeneracies occur under specific proportionality conditions.

Spectrum reduces to Landau problem for certain parameter values.

Abstract

We obtain the gauge invariant energy eigenvalues and degeneracies together with rotationally symmetric wavefunctions of a particle moving on 2D noncommutative plane subjected to homogeneous magnetic field $B$ and harmonic potential. This has been done by using the phase space coordinates transformation based on 2-parameter family of unitarily equivalent irreducible representations of the nilpotent Lie group $G_{NC}$. We find that the energy levels and states of the system are unique and hence, same goes to the degeneracies as well since they are heavily reliant on the applied $B$ and the noncommutativity $\theta $ of coordinates. Without $B$, we essentially have a noncommutative planar harmonic oscillator under generalized Bopp shift or Seiberg-Witten map. The degenerate energy levels can always be found if $\theta$ is proportional to the ratio between $\hbar$ and $m\omega$. For the scale $B\theta = \hbar$, the spectrum of energy is isomorphic to Landau problem in symmetric gauge and hence, each energy level is infinitely degenerate regardless of any values of $\theta$. Finally, if $0 < B\theta < \hbar$, $\theta$ has to also be proportional to the ratio between $\hbar$ and $m\omega$ for the degeneracy to occur. These proportionality parameters are evaluated and if they are not satisfied then we will have non-degenerate energy levels. Finally, the probability densities and effects of $B$ and $\theta$ on the system are properly shown for all cases.