General Solutions of One Class of Field Equations

Abstract

We find general solutions of some field equations (systems of equations) in pseudo-Euclidian spaces (so-called primitive field equations). These equations are used in the study of the Dirac equation and Yang-Mills equations. These equations are invariant under orthogonal O(p, q) coordinate transformations and invariant under gauge transformations, which depend on some Lie groups. In this paper we use some new geometric objects -Clifford field vector and an algebra of h-forms which is a generalization of the algebra of differential forms and the Atiyah-Kähler algebra.

On constant solutions of SU(2) Yang-Mills equations with arbitrary current in Euclidean space ℝn

“…The systems (15), (16) have the following symmetry. Suppose that (b 1 , b 2 , b 3 ) is a solution of (15) or (16) for known (j 1 , j 2 , j 3 ).…”

“…We remind these statements (Lemmas 3 and 4) here without proof for the convenience of reader. We present general solution of the system (16) and its symmetries in Lemmas 5 and 6.…”

“…The system of equations (16) with nonnegative parameters j 1 ≥ 0, j 2 ≥ 0, j 3 ≥ 0 has the following general solution:…”

“…Covariantly constant solutions of the Yang-Mills equations are discussed in [14,16,25], constant solutions of the Yang-Mills-Proca equations are discussed in [15] using Clifford algebra formalism.…”

Lévy Laplacians in Hida Calculus and Malliavin Calculus

“…Remark 6. It would be interesting to investigate whether it is possible to use the Levy-Laplacian approach in some areas connected to the theory of gauge fields (see, e.g., [11,27,24,33,18]).…”