Gauge Field of a Point Charge

Abstract

The problem and treatment of integration ambiguities in the conventionally defined Yang‐Mills charges is demonstrated explicitly, using a non‐Abelian solution of the Yang‐Mills equations for a point charge. The internal holonomy group $\mathcal{H}$ for this solution is noncompact and nonsemisimple, and the solution is not expected to have a direct physical meaning. However, it provides a convenient example showing important and quite unexpected features of gauge theories of the Yang‐Mills type, before quantization. It is found that the number of unambiguously definable and comparable charges is less than the dimension of $\mathcal{H}$ and less than the rank of $\mathcal{H}$ as well. If a gauge group $\mathcal{G}$ is present in the conventional manner, i.e., $\mathcal{H}\subseteq \mathcal{G}$ , this number of charges is less than the rank of the gauge group. Other interesting features of the solution found are: discreteness of certain components of the gauge field, as a result of regularity conditions together with the condition that the Yang‐Mills charge density vanishes outside a sphere of finite radius, and a harmonic oscillation of the other gauge components, while the observable charges are steady. Higher‐order charges are all found to be zero. No action principle is used and no a priori particle fields are introduced. Use is made of the differential‐geometric properties of gauge fields.