Classical Theory of Magnetic Monopoles

Abstract

A classical system of $n$ electric and ${n}^{*}$ magnetic point charges is considered. The field equations (Maxwell-Lorentz equations, suitably generalized) and the particle equations are obtained by postulating duality invariance and coherence with the theory of only electric point charges. The particle equations together with the solutions of the field equations yield the (generalized) Lorentz-Dirac equations including radiation reaction. The question is then raised whether this system of equations can be derived from an action principle, as is the case for only electric or only magnetic charges. It is shown that the particle equations can be derived only from a nonlocal action integral. If an electric and a magnetic point charge are allowed to meet (crossing of world lines), they must do so with equal velocity (in magnitude and direction) at the instant of crossing. An action integral from which the field equations can be derived is not difficult to obtain, but it is proven that no action integral exists from which both the particle equations and the field equations can be derived. Nevertheless, there exist a local symmetric energy tensor and a corresponding angular momentum tensor which yield ten conservation laws when the field and particle equations hold.