Gauge Field of a Point Charge

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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 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 H and less than the rank of H as well. If a gauge group G is present in the conventional manner, i.e., H⊆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.

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