Minkowski space Yang–Mills fields from solutions of equations in the three-dimensional Euclidean space

Abstract

Equations in the three-dimensional Euclidean space are derived by combining the Yang–Mills field equations with the conditions which are imposed on a Yang–Mills field in Bernreuther’s method of constructing Yang–Mills fields in Minkowski space from Yang–Mills fields in Euclidean space. By a proper ansatz for the Yang–Mills fields these equations are reduced to a single differential equation. The differential equation is identical with merons’ equation in Euclidean space if we consider solutions of the latter equation which are functions of the ratio t/ρ, where t is the Euclidean time and ρ is the three-dimensional radius. One such solution is the single meron solution in Euclidean space. Starting from this and applying the method we get the de Alfaro–Fubini–Furlan solution in Minkowski space. Then, a more general ansatz is considered, which leads to a system of three nonlinear differential equations.