Magnetic charge and the charge quantization condition

Abstract

Two viewpoints concerning magnetic charge are distinguished: that of Dirac, which is unsymmetrical, and the symmetrical one, which embodies invariance under charge rotation. It is pointed out that the latter is not in conflict with the empirical asymmetry between electric and magnetic charge. The discussion is based on an action principle that uses field strengths and the vector potential $A$ as independent variables; a second vector potential $B$ is defined nonlocally in terms of the field strengths. This nonlocality is described by an arbitrary vector function $f^{\nu }\mathrm{}\left(y\right)\mathrm{}$, subject only to the restriction ${\partial }_{\nu }{f}^{\nu }\mathrm{}\left(y\right)=\delta \mathrm{}\left(y\right)\mathrm{}$ and the additional requirement of oddness, in the symmetrical formulation. The charge quantization conditions for a pair of idealized charges, $a$ and $b$, are inferred by examining the dependence of the action $W$ on the choice of the arbitrary mathematical function $f$, and requiring the uniqueness of $\exp \mathrm{}\left[\mathrm{iW}\right]\mathrm{}$. For the unsymmetrical viewpoint the half-integer condition of Dirac is obtained, $\frac{e_a{g}_b}{4\pi }=\frac{1}{2}n$, while the symmetrical formulation requires the integer condition $\frac{(e_a{g}_b-e_b{g}_a)}{4\pi }=n$. The Dirac injunction, "a string must never pass through a charged particle," is criticized as unnecessarily restrictive, owing to its origin in a classical action context. As simplified by a restriction to small momentum transfers, permitting the neglect of form-factor and vacuum-polarization effects, the dynamics of a realistic system of two spin- ½ dyons is shown to involve the same interaction structure used in the idealized discussion.