The Unified Standard Model

TL;DR

This paper proposes a unified mathematical framework for particle physics using the algebra of 8x8 complex matrices, capturing the Standard Model's structure and suggesting potential new physics.

Contribution

It introduces a novel algebraic model based on M(8,ℂ) that encapsulates the gauge structure, particles, and symmetries of the Standard Model, linking them to division algebras.

Findings

Standard Model particles are represented within M(8,ℂ)

Gauge transformations are described by algebra acting on itself

Additional elements suggest minimal new physics

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.