Leptons, Quarks, and Gauge from the Complex Clifford Algebra $\mathbb {C}\ell _6$

Abstract

A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra $\mathbb {C}\ell _6$, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras 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 $\theta _W$ given by $\sin ^2\theta _W=0.25$. The model shares common ideas with previously known models, particularly with Chisholm and Farwell (Clifford (Geometric) algebras: with applications to physics, mathematics, and engineering. Birkhäuser Boston, Boston, pp 365–388, 1996), Trayling and Baylis (Clifford Algebras: applications to mathematics, physics, and engineering. Birkhäuser Boston, Boston, pp 547–558, 1996), and Furey (Standard Model Physics from an Algebra? Preprint arXiv:1611.09182, 2016).