Constant Solutions of Yang-Mills Equations and Generalized Proca Equations

Abstract

ABSTRACT. In this paper we present some new equations which we call Yang-Mills-Proca equations (or generalized Proca equations). This system of equations is a generalization of Proca equation and Yang-Mills equations and it is not gauge invariant. We present a number of constant solutions of this system of equations in the case of arbitrary Lie algebra. In details we consider the case when this Lie algebra is Clifford algebra or Grassmann algebra. We consider solutions of Yang-Mills equations in the form of perturbation theory series near the constant solution.

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

“…The general solution of the system (15) and its symmetries are discussed in [23], where we obtain the same system in the case of arbitrary Euclidean space R n . We remind these statements (Lemmas 3 and 4) here without proof for the convenience of reader.…”

“…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 ).…”

“…If we change the sign of some j k , k = 1, 2, 3, then we must change the sign of the corresponding b k , k = 1, 2, 3. Without loss of generality, we can assume that all expressions j k , k = 1, 2, 3, in (15) and (16) are nonnegative. In (15), the expressions b k , k = 1, 2, 3 will be nonnegative too.…”

“…Without loss of generality, we can assume that all expressions j k , k = 1, 2, 3, in (15) and (16) are nonnegative. In (15), the expressions b k , k = 1, 2, 3 will be nonnegative too. In (16), the expression b 3 will be nonnegative, the expressions b 1 and b 2 will be arbitrary real numbers.…”

“…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.…”