Noncommutative 3D harmonic oscillator

TL;DR

This paper develops transformation matrices to relate noncommutative 3D harmonic oscillators to isotropic and anisotropic commutative oscillators, providing physical interpretations and methods to derive complex solutions from simpler ones.

Contribution

It introduces a systematic approach to express noncommutative oscillators in terms of commutative ones using transformation matrices, with physical insights and solution strategies.

Findings

Transformation matrices relate noncommutative and commutative oscillators.

Noncommutative parameters correspond to external magnetic fields.

Methods to derive complex solutions from simple cases.

Abstract

We find transformation matrices allowing to express non-commutative three dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following ``philosophy of simplicity'' approach. Non-commutative parameters have physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones.