The intrinsic structure of Yang-Mills fields associated with an arbitrary finite-dimensional matrix Lie group and an arbitrary Lagrangian function is studied. The results are then specialized to the case of semisimple groups and gauge invariant Lagrangian functions. For the specialization, the Lie group becomes a gauge group that maps solutions onto solutions and hence defines an equivalence relation on the solution set. The resulting fiber space of solutions is shown to admit a canonical cross section consisting of all antiexact solutions. Thus any solution of the Yang-Mills field equations can be mapped onto an antiexact solution by the action of an appropriate gauge transformation, and two solutions can be mapped one onto the other only if their corresponding antiexact images are the same. It is further shown that every antiexact solution satisfies a matrix system of linear Riemann-Graves integral equations, and that any solution of the linear Riemann-Graves integral equations that satisfies a system of first-order integro-differential conditions constitutes a solution of the Yang-Mills field equations. The general solution of the linear integral equations is shown to be a power series expansion in the structure constants of the gauge group and the interaction current is shown to enter into the solution only in terms of its exact part.
1. MATHEMATICAL PRELIMINARIES
Most of the results obtained in this paper come about in a simple and direct manner by use of the calculus of exterior differential forms together with certain extensions that are given in this section. We assume that the reader is familiar with the standard structure of the exterior calculus so that we may use the following symbols without further remark:
A for the exterior product of forms, J for the inner multiplication of a vector with a form, d for the exterior derivative of a form.
The arena for our discussion is a 4-dimensional, flat, space-time manifold, M4, that is the Cartesian product of 3-dimensional Euclidean space with the real line. In the interests of simplicity, we work with a fixed global coordinate cover (9 I i = l,..., 41 relative to which the metric tensor of M4 is given (( gij)) = diag(1, 1, 1, -1). The summation convention is assumed with respect to lower case Latin indices with the range 1 through 4.