The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6

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, 1996, Trayling and Baylis, 2004, and Furey, 2016.

References in corpus (3)

• Exceptional quantum geometry and particle physics
• Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra
• Charge quantization from a number operator

Cited by in corpus (17)

• Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• Left-Right symmetric fermions and sterile neutrinos from complex split biquaternions and bioctonions
• Spin(11,3), particles and octonions
• The Unified Standard Model
• Subgroups of Clifford algebras
• Braids, 3-manifolds, elementary particles: number theory and symmetry in particle physics
• Quantum gravity effects in the infra-red: a theoretical derivation of the low energy fine structure constant and mass ratios of elementary particles
• Octonions, trace dynamics and non-commutative geometry: a case for unification in spontaneous quantum gravity
• CKM matrix parameters from the exceptional Jordan algebra
• Braided fermions from Hurwitz algebras
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• A Proposed $E_8\times E_8$ Kinematic Scaffolding for the Standard Model with Pre-Gravitation
• An Algebraic Roadmap of Particle Theories, Part I: General construction
• Hermitian Geometry of Complex Multivectors, Determinants and Orientations
• Quantum number conservation: a tool in the design and analysis of high energy experiments
• The Standard Model Symmetry and Qubit Entanglement