Non-commutative oscillator with Kepler-type dynamical symmetry

TL;DR

This paper explores a 3D non-commutative oscillator with Kepler-like symmetry, revealing conserved quantities and spectrum determination through algebraic methods, connecting to classical Kepler problem features.

Contribution

It introduces a non-commutative oscillator model with a conserved Runge-Lenz vector and extends the dynamical symmetry to the conformal algebra o(4,2).

Findings

Trajectories are elliptical arcs reducing to Kepler hodographs in the commutative limit.

The model admits a conserved Runge-Lenz vector derived from a dual momentum space description.

The bound-state spectrum can be determined algebraically using the extended symmetry.

Abstract

A 3-dimensional non-commutative oscillator with no mass term but with a certain momentum-dependent potential admits a conserved Runge-Lenz vector, derived from the dual description in momentum space. The latter corresponds to a Dirac monopole with a fine-tuned inverse-square plus Newtonian potential, introduced by McIntosh, Cisneros, and by Zwanziger some time ago. The trajectories are (arcs of) ellipses, which, in the commutative limit, reduce to the circular hodographs of the Kepler problem. The dynamical symmetry allows for an algebraic determination of the bound-state spectrum and actually extends to the conformal algebra o(4,2).