P000009
Group Geometric Algebras and the Standard Model
*Carl Brannen (Washington State University, USA)
We generalize the massless Dirac equation and Weyl equation to give particles with internal symmetry of the Standard Model fermions. The 2x2 or 4x4 complex matrices are matrix rings $R$ and can be generalized by a group $G$ to a group algebra $G[R]$. For a group of size N, this creates N times as many Pauli spin or gamma matrices and generalizes the wave equations. We consider point group symmetries for $G$ and show that using the full octahedral group and the 2x2 complex matrices gives a group algebra which generalizes the Weyl equation to the Standard Model fermions plus a dark matter particle. An alternative is to use the chiral octahedral group and the 4x4 complex matrices to generalize the massless Dirac equation. The method leaves a dark matter particle and its antiparticle. We describe the symmetry of dark matter.