The Standard Model Algebra

arXiv link: https://arxiv.org/abs/1702.04336

A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model, and the electroweak and color gauge symmetries. The algebra is just the Clifford algebra of a complex six-dimensional vector space endowed with a preferred Witt decomposition, and it is already implicitly present in the mathematical structure of the Standard Model. The minimal left ideals determined by the Witt decomposition correspond naturally pairs of leptons or quarks whose left chiral components interact weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2(θW)=0.25.