The Unified Standard Model

Abstract

The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8,$\mathbb{C})$ of complex 8$\times$8 matrices. This algebra is well motivated by its close relation to the normed division algebra of octonions. (Anti-)particle states are identified with basis elements of the vector space M(8,$\mathbb{C})$. Gauge transformations are simply described by the algebra acting on itself. Our result shows that all particles and gauge structures of the Standard Model are contained in the tensor product of all four normed division algebras, with the quaternions providing the Lorentz representations. Interestingly, the space M(8,$\mathbb{C})$ contains two additional elements independent of the Standard Model particles, hinting at a minimal amount of new physics.