Classical solutions for SU(4) gauge fields: Interacting monopoles

Abstract

The notion of local idempotents is introduced, and their relation to a class of solutions for $\mathrm{SU}(n)$ gauge fields is pointed out. This class includes the known monopole-type solutions for SU(2) and SU(3) gauge fields---coupled to scalars and spinors. Next, these ideas are used to study solutions for SU(4) gauge fields. The following classes of solutions are studied. Corresponding to two commuting SU(2) subgroups of SU(4) one has two monopole-type contributions from the space components, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{W}}(x)$, of the gauge field. They are directly coupled among themselves, the remaining SU(4) components providing a tensor-type interaction. They are also coupled to a scalar field $\ensuremath{\Phi}(x)$ and the time component ${W}_{0}(x)$. Two different possibilities for $\ensuremath{\Phi}(x)$ and ${W}_{0}(x)$ are considered in detail. An exact solution is given for a point monopole interacting with a particular system of finite mass. Simple variational calculations are used to obtain finite mass for the total system. Brief remarks are added concerning other possibilities; e.g., how pseudoparticles can be studied from our point of view.