On the search for nonvacuum, nonself-dual solutions of Yang-Mills equations
The solutions to the Yang-Mills equations (YME), such as instantons and monopoles, are solutions of the famous self-duality equations [see \textit{I. A. Polyakov}, Pis'ma Zh. Eksp. Teor. Fiz. 20, 430 (1974; per bibl.), \textit{G. 't Hooft}, Nucl. Phys. B79, 276-284 (1974; MR 54\#1923), \textit{A. A. Belavin, A. M. Polyakov, A. S. Schwarz} and \textit{Yu. S. Tyupkin}, Phys. Lett. B 59, 85-87 (1975; MR 55\#7152)]. The author attempts to generalize these equations and to find new solutions to the YME. The generalization of the self-duality equations is as follows (for simplicity, the indices are omitted) \[ \ast F+ iF = (\ast\mathcal F+ i\mathcal F)\phi, \] where \(\phi\) is an unknown scalar function, \(\mathcal F\) is a solution to the YME and \(F\) is the required field. Both sides of this equation are operated with the operator \(D\) connected with the field \(F.\) If the condition \(D[(\ast\mathcal F+ i\mathcal F)\phi]=0\) is required to hold, the YME for the field \(F\) will be satisfied. The group \(SO(0,3)\) is used as a gauge group. The above tensor equation is transformed into two scalar equations for two scalar functions. The possibility of existence of solutions to the YME that are nonvacuum, non-self-dual, real-valued, possess a finite action, and are time-dependent is discussed.
scientific article; zbMATH DE number 1120311