Dynamical Symmetry of the Zwanziger problem in Non-commutative Quantum Mechanics

TL;DR

This paper demonstrates that the dynamical symmetry of the Zwanziger problem persists in non-commutative quantum mechanics, extending known symmetries of the hydrogen atom to a non-commutative coordinate framework.

Contribution

It introduces a non-commutative Hilbert space framework and describes the symmetry group structure using redefined operators, generalizing the classical symmetries to non-commutative geometry.

Findings

Bound state algebra is SO(4)

Scattering state algebra is SO(3,1)

Symmetry structure is preserved in non-commutative space

Abstract

The non-relativistic hydrogen atom and the Zwanziger problem have the same dynamical symmetry for bound and scattering states.We show that this is also true for a Hilbert space which is non-commutative in co-ordinates. The group structure is described using the redefined velocity operator and Laplace Runge-Lenz operator in terms of left and right handed representations of the non-commutative Hilbert space $ R_{\lambda}^{3}$.The bound state algebra is SO(4) and the scattering state algebra is SO(3,1).