Non-abelian gauge invariant classical lagrangian formalism for point electric and magnetic charge

Abstract

The classical electrodynamics of electrically charged point particles has been generalized to include non-Abelian gauge groups and to include magnetically charged point particles. In this paper these two distinct generalizations are unified into a non-Abelian gauge theory of electric and magnetic charge. Just as the electrically charged particles constitute the generalized source of the gauge fields, the magnetically charged particles constitute the generalized source of the dual fields. The resultant equations of motion are invariant to the original “electric” non-Abelian gauge group, but, because of the absence of a corresponding “magnetic” gauge group, there is no “duality” symmetry between electric and magnetic quantities. However, for a class of solutions to these equations, which includes all known point electric and magnetic monopole constructions, there is shown to exist an equivalent description based on a magnetic, rather than electric, gauge group. The gauge potentials in general are singular on strings extending from the particle position to infinity, but it is shown that the observables are without string singularities, and that the theory is Lorentz invariant, provided a charge quantization condition is satisfied. This condition, deduced from a stability analysis, is necessary for the consistency of the classical non-Abelian theory, in contrast to the Abelian case, where such a condition is necessary only for the consistency of the quantum theory. It is also shown that in the classical theory the strings cannot be removed by gauge transformations, as they sometimes can be in the quantum theory. An action principle is presented which leads to the particle and field equations of motion except for extra contributions arising from the possibility that the strings carry electric charge. Only solutions with electrically neutral strings are acceptable as monopoles. The action includes unphysical electrically charged fields defined on particle trajectories and on strings. The Lagrangian is gauge invariant and leads to electric current conservation via Noether's theorem.