Conserved quantities in non-abelian monopole fields

TL;DR

This paper uses covariant Hamiltonian methods to identify conserved quantities for particles in non-Abelian monopole fields, extending known results and analyzing specific potentials and symmetries.

Contribution

It derives the most general scalar potential allowing a conserved Runge-Lenz vector in non-Abelian monopole fields, generalizing previous models and interpreting results in physical monopole contexts.

Findings

Identified scalar potentials with conserved Runge-Lenz vector for Wu-Yang monopole.

Constructed conserved angular momentum in non-Abelian fields of diatomic molecules.

Did not find a Runge-Lenz vector in the molecular field case.

Abstract

Van Holten's covariant Hamiltonian framework is used to find conserved quantities for an isospin-carrying particle in a non-Abelian monopole-like field. For a Wu-Yang monopole we find the most general scalar potential such that the combined system admits a conserved Runge-Lenz vector. It generalizes the fine-tuned inverse-square plus Coulomb potential, found before by McIntosh and Cisneros, and by Zwanziger, for a charged particle in the field of a Dirac monopole. Following Feh\'er, the result is interpreted as describing motion in the asymptotic field of a self-dual Prasad-Sommerfield monopole. In the effective non-Abelian field for nuclear motion in a diatomic molecule due to Moody, Shapere and Wilczek, a conserved angular momentum is constructed, despite the non-conservation of the electric charge. No Runge-Lenz vector has been found.