Group Geometric Algebra and the Standard Model

Carl Brannen​

doi:10.4236/jmp.2020.118075

Abstract

We show how to generalize the Weyl equation to include the Standard Model fermions and a dark matter fermion. The 2 × 2 complex matrices are a matrix ring R. A finite group G can be used to define a group algebra G[R] which is a generalization of the ring. For a group of size N, this defines N Weyl equations coupled by the group operation. We use the group character table to uncouple the equations by diagonalizing the group algebra. Using the full octahedral point symmetry group for G, our uncoupled Weyl equations have the symmetry of the Standard Model fermions plus a dark matter particle. We describe the symmetry properties of dark matter.

Highlights

We will write the Weyl equation [1] for a single massless fermion as σ μ ∂μψ= I2 ∂ψ ∂t ±σx ∂ψ ∂x ±σy ∂ψ ∂y ±σz ∂ψ= ∂ z (1)where I2 is the 2 × 2 unit matrix, the σ j is the Pauli spin matrices
We show how to generalize the Weyl equation to include the Standard Model fermions and a dark matter fermion
Using the full octahedral point symmetry group for G, our uncoupled Weyl equations have the symmetry of the Standard Model fermions plus a dark matter particle

Summary

Introduction

Where I2 is the 2 × 2 unit matrix, the σ j is the Pauli spin matrices. The Pauli algebra has a basis of four. { } { } { } 2 × 2 complex m= atrices σ μ = σt ,σ x ,σ y ,σ z I2 ,σ x ,σ y ,σ z where each of the four σ μ has two nonzero complex elements. We will generalize the σ μ to σ μg so that they depend on a group element g. There will be 4N different σ μg where N is the number of elements in the finite group G.

Brannen DOI

Generations

Dark Matter Nomenclature