Rotational Symmetry and Gauge Invariant Degeneracies on 2D Noncommutative Plane

• Main Authors: Rusli, M. N. N. M, Nurisya, M. S, Zainuddin, H, Umar, M. F, Jellal, A
• Format: Journal Article
• Language: English
• Published: 16-12-2021
• Subjects: Mathematics - Mathematical Physics; Physics - High Energy Physics - Theory; Physics - Mathematical Physics; Physics - Quantum Physics
• DOI: 10.48550/arxiv.2112.08666

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.