On Poincaré-Invariant Reduction and Exact Solutions of the Yang-Mills Equations

Abstract

e ~ Aν × ( ~ Aν × ~ Aμ) = ~0. Here ∂ν = ∂ ∂xν , μ = 0, 3, e = const, ~ Aμ = ~ Aμ(x) = ~ Aμ(x0, x1, x2, x3) is a three-component vector-potential of the Yang-Mills field. Hereafter, the summation over the repeated indices μ, ν from 0 to 3 is supposed. Raising and lowering the vector indices are peformed with the aid of the metric tensor gμν , i.e., ∂μ = gμν∂ν (gμν = 1 if μ = ν = 0, gμν = −1 if μ = ν = 1, 2, 3 and gμν = 0 if μ 6= ν). It should be said that there were several reviews devoted to classical solutions of YME. The solutions were obtained with the help of ad hoc substitutions suggested by Wu and Yang, Rosen, ’t Hooft, Corrigan and Fairlie, Wilczek, Witten (for more detail see review [5] and references cited therein). But, in fact, symmetry properties of YME were not used. It was known [6] that YME are invariant under the group C(1, 3) ⊗ SU(2), where C(1, 3) is the 15-parameter conformal group and SU(2) is the infinite-parameter special unitary group. Symmetry properties of YME were used and some new exact solutions of equation(1) were obtained by W. Fushchych and W.Shtelen in [7] (see also [3]). The present report is a continuation of the investigations which were realized by the author together with R. Zhdanov and W. Fushchych. With the aid of P (1, 3)-inequivalent ansatzes for the Yang-Mills field, which are invariant under three-dimensional subgroups of the Poincare group, reduction of YME to systems of ordinary differential equations is carried out and wide families of their exact solutions are constructed in [8, 9]. Here we carry out symmetry reduction of YME on four-dimensional subgroups of the Poincare group to functional equations. The symmetry group of YME contains as a subgroup the Poincare group P (1, 3) having the following generators: