Gauge Fields with Positive-Definite Energy Density

The relationship between the couplings of the non‐Abelian gauge field and the internal holonomy group is investigated on the classical level, under the requirements of full local gauge invariance and positive definiteness of the gauge field's energy density. For the free gauge field it is found that each solution to the gauge field equation with self‐coupling is associated with a simple compact internal holonomy group. When the gauge field is coupled to other multiplet fields, an internal symmetry group is present in addition to the holonomy group $\mathcal{H}$ . In this case the internal holonomy group must also be simple and compact. In addition, homogeneity and isotropy of event space for closed systems of fields require that $\mathcal{H}$ be identical in group structure with either the symmetry group or a subgroup of the symmetry group and that all the field equations can be decoupled into sets of equations acting on multiplet fields transforming under irreducible representations of $\mathcal{H}$ . Thus, in local gauge theory, multiplet fields can be classified with respect to simple and compact internal holonomy groups. Some comments are offered on the relationship between internal holonomy and symmetry groups.