The Three Faces of $U(3)$

TL;DR

This paper explores the structure of the $U(3)$ gauge group, revealing three distinct representations that can be interpreted as three different matter field species with identical quantum numbers, within a Yang-Mills framework.

Contribution

It introduces a novel interpretation of $U(3)$ gauge symmetry as three matter field species, linked via a permutation symmetry, in a Yang-Mills model with massive fermions.

Findings

Three non-trivial homomorphisms induce three representations of $U(3)$.

A permutation symmetry relates the three covariant derivatives.

The model can be viewed as three matter field species with the same quantum numbers.

Abstract

$U(n)$ is a semi-direct product group that is characterized by non-trivial homomorphisms mapping $U(1)$ into the automorphism group of $SU(n)$. For $U(3)$, there are three non-trivial homomorphisms that induce three separate defining representations. In a toy model of $U(3)$ Yang-Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. By employing a $\mathrm{mod}\,3$ permutation of the vector space carrying the defining representation induced by a ``large'' gauge transformation, 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 $U(3)$ quantum numbers. One can interpret this as three species of matter fields in the defining representation.