Three-dimensional superintegrable systems in a static electromagnetic field

TL;DR

This paper investigates three-dimensional superintegrable systems of charged particles in static electromagnetic fields, analyzing integrals of motion, gauge invariance, and quantum/classical properties to identify conditions for maximal superintegrability.

Contribution

It classifies superintegrable systems based on first order integrals, explores additional integrals of motion, and examines their quantum and classical characteristics, including nonpolynomial integrals.

Findings

Identification of conditions for superintegrability in static electromagnetic fields.

Discovery of nonpolynomial integrals ensuring maximal superintegrability.

Analysis of quantum spectra and classical dynamics for these systems.

Abstract

We consider a charged particle moving in a static electromagnetic field described by the vector potential $\vec A(\vec x)$ and the electrostatic potential $V(\vec x)$. We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely $(P_1,P_2)$, $(L_3,P_3)$ and $(L_1,L_2,L_3)$. For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.