Leptons, Quarks, and Gauge Symmetries, from a Clifford Algebra

Abstract

Spinors having the discrete properties of the leptons and quarks in a family of the Standard Model, with the proper symmetries, are obtained using the left ideals of a Clifford algebra. This algebra is the complex Clifford algebra ${\mathbb {C}}{\ell }_6$ obtained from the exterior algebra of a complex three-dimensional vector space and its dual, this giving the ideal decomposition and representing the electric charges, the quark colors, and the proper ${\text {SU}}(3)$ symmetries. The Lorentz and Dirac algebras appear as subalgebras, their left actions on the ideals representing therefore the leptons and the quarks. Because the representation of the Dirac algebra on the minimal left ideals of ${\mathbb {C}}{\ell }_6$ is reducible, the weak symmetry emerges as well, with the isospins, hypercharges, and chirality. The electroweak symmetry is broken geometrically, without resulting in additional exchange bosons or other fermions. The bare Weinberg angle $\theta _W$ predicted by this model is given by $\sin ^2\theta _W=0.25$. The mass-related parameters and the three families of leptons and quarks are not yet obtained in this model.