A Novel Approach To Particle Representations

This paper proposes a new approach to deriving a finite particle content, suitable for the construction of a gauge theory. Specifically, the outlined construction generates a finite set of irreducible gauge representations, which are interpreted as describing a full set of elementary particles. These representations are constructed from endofunctions between restricted representations of some symmetry group $G$ acting on some space $V$. As a proof of concept, we show how a set of irreducible representations arise as endofunctions on the vector space $V=\mathbb{C}^8$ equipped with the exceptional Lie group $G=G_2$ as its symmetry group. We discuss how the irreducible representations of our simple example compare to the various particle types of the Standard Model. The process through which the particle content is constructed yields adjoint, fundamental, and Higgs-like representations, thereby reproducing the essential types of particle transformations seen in the Standard Model. In particular we focus on the discrimination of gauge structures and the natural appearance of Higgs-like representations. Avenues to generalizing the construction are considered, and some inevitable consequences are discussed. We conclude by comparing our results to those of non-commutative geometry, commenting on key similarities and differences between the two approaches.