Group Geometric Algebra and the Standard Model

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.