ar X iv : h ep - t h / 03 12 27 2 v 1 2 3 D ec 2 00 3 Patching up the monopole potential

It is well known that a vector potential cannot be defined over the whole surface of a sphere around a magnetic monopole. A recent claim to the contrary is shown to have problems. It is explained however that a potential of the proposed type works if two patches are used instead of one. A general derivation of the Dirac quantization condition attempted with a single patch is corrected by introducing two patches. Further, the case of more than two patches using the original Wu-Yang type of potential is discussed in brief. While magnetic monopoles have not been seen experimentally, they have continued to be of interest to theoreticians. In standard electrodynamics, magnetic monopoles are not admitted, and the magnetic field has zero divergence. If a magnetic monopole is present at a point, the divergence is a delta function with support at that point, and it is not possible to introduce a vector potential which is nonsingular everywhere. Dirac[1] found that there is at least a string of singularities from that point to infinity, though the location of the string is arbitrary, like a branch cut. Instead of working with a string, Wu and Yang [2] showed that it may be more convenient to work with two different nonsingular potentials on two overlapping patches, with a gauge transformation connecting them in the overlap. In both approaches the quantum theoretic description of an electrically charged particle requires the quantization of its charge, if a monopole is assumed to exist. Recently there has been a claim [3] about the construction of a nonsingular vector potential for the monopole field using a single patch. This could only be possible if the basis of the Wu-Yang formalism were incorrect, and indeed that is what has been asserted. One of our aims is to reexamine this question and to show that the single-patch potential does not work. The use of two patches,