The Three Faces of U(3)

Abstract

(Formula presented.) is a semi-direct product group characterized by nontrivial homomorphisms mapping (Formula presented.) into the automorphism group of (Formula presented.). For (Formula presented.), there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of (Formula presented.) Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a (Formula presented.) permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same (Formula presented.) quantum numbers. One can interpret this as three species of matter fields in the defining representation.