General solutions of one class of field equations

TL;DR

This paper derives general solutions for a class of invariant field equations in pseudo-Euclidean spaces, utilizing novel geometric objects to advance understanding in mathematical physics.

Contribution

It introduces new geometric tools, such as Clifford field vectors and an algebra of h-forms, to solve primitive field equations related to Dirac and Yang-Mills theories.

Findings

Derived general solutions for primitive field equations.

Introduced Clifford field vectors and h-form algebra.

Enhanced mathematical framework for gauge-invariant 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-K\"{a}hler algebra.