Self-dual covariant lagrangian formulation of electromagnetism with magnetic charges

Abstract

There are some subtle features associated with the quantum theory of magnetic monopoles. In order to describe the wave function of an electron in the field of a magnetic monopole, a vector potential A is needed. However it is well known (1,2) that there does not exist any vector potential free from anomalous singularities. There have been a number of Lagrangian formulations (1.3) intended to overcome this basic problem. Dirac's formulation is built up in terms of a vector potential which exhibits au unphysical string singularity. The formulation of WL~ and YANG is based on the observation (4) that a nonsingular vector potential can only be given on each of a set of overlapping regions of space-time. In both formulations there are somc important disadvantages. In the formulations a la Dirac the Lagrangian is highly singular due to the string-string terms. In the Wu-Yang formulation the second Maxwell equation cannot be obtained from the action and must be considered as a kinematical constraint. In addition, in both formulations, a completely asymmetrical t reatment to the electric and magnetic charges is given. Due to the dual symmetry one is allowed to write these Lagrangians with the role of the electric and magnetic charges exchanged, but a direct self-dual Lagrangian formulation has not been given yet. I t is a nontrivial problem to produce a, fully covariant formulation of electromagnetism with magnetic charges. However, a reduced noncovariant Lagrangian in terms of the dynamical degrees of freedom, may easily be written. To do this let us start from ordinary electromagnetism with the usual Lagrangian