Kerner equation for motion in a non-Abelian gauge field

TL;DR

This work surveys the history and structure of isospin-carrying particle dynamics in Yang-Mills and gravitational fields, tracing Kerner's non-Abelian geodesic framework and Wong's subsequent flat-space equations, where the isospin evolves covariantly via $D_s Q$ and the spacetime motion experiences a YM force. It presents a coherent, fiber-bundle and symplectic formulation of the Kerner-Wong system, derives conserved quantities through van Holten's covariant method, and analyzes explicit models in Wu-Yang monopole and diatomic-molecule fields, including a Runge-Lenz vector in the monopole case with a tuned potential $V(r)$. The results reveal an $O(4)/O(3,1)$ dynamical symmetry for Wu-Yang monopoles outside a monopole core, while non-Abelian diatom configurations generally break this symmetry by rendering the isospin projection $q$ non-conserved. Overall, the paper clarifies the physical meaning of isospin dynamics, the role of gauge versus internal symmetries, and the conditions under which classical conserved quantities and hidden symmetries persist in non-Abelian gauge backgrounds.

Abstract

The equations of motion of an isospin-carrying particle in a Yang-Mills and gravitational field were first proposed in 1968 by Kerner, who considered geodesics in a Kaluza-Klein-type framework. Two years later the flat space Kerner equations were completed by considering also the motion of the isospin by Wong, who used a field-theoretical approach. Their groundbreaking work was then followed by a long series of rediscoveries whose history is reviewed. The concept of isospin charge and the physical meaning of its motion are discussed. Conserved quantities are studied for Wu-Yang monopoles and for diatomic molecules by using van Holten's algorithm.

Figures (2)

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• Figure 2: Souriau's framework: the worldline in ${\bf M}$ is the projection of a characteristic sheet of the 2-form $\widetilde{\bf \Omega} = {\bf d\tilde{{\bm{\omega}}}}$ on the evolution space, $\mathcal{V}$. Factoring out the characteristic foliation tangent to $\bf{\ker}\,\tilde{\hbox{\boldmath$\omega$}}$, $\mathcal{V}$ projects to the space of motions, $\mathcal{U}$, to which the 2-form $d\tilde{\omega}$ projects as a symplectic form ${\bf \Omega}$ and correspond to the worldlines in $M$.