Braided fermions from Hurwitz algebras

Abstract

Some curious structural similarities between a recent braid- and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups that can be represented using the four normed division algebras are $B_2$ and $B_3^c$, exactly those required to represent a single generation of fermions in terms of simple three strand ribbon braids. These braided fermion states can be identified with the basis states of the minimal left ideals of the Clifford algebra $C\ell (6)$, generated from the nested left actions of the complex octonions $\mathbb{C}\otimes \mathbb{O}$ on itself. That is, the ribbon spectrum can be related to octonion algebras. Some speculative ideas relating to ongoing research that attempts to construct a unified theory based on braid groups and Hurwitz algebras are discussed.

arXiv commentProceeding paper for the 32nd International Colloquium on Group Theoretical Methods in Physics (Group32) held at Czech Technical University in Prague, Czech Republic in July 2018. 10 pages, 2 figures

References in corpus (11)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Quantum gravity and the standard model
• Generations: Three Prints, in Colour
• Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra
• The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6
• Braids, normed division algebras, and Standard Model symmetries
• The Elementary Particles as Quantum Knots in Electroweak Theory
• Particle Topology, Braids, and Braided Belts
• A Knot Model Sugugested by the Standard Electroweak Theory
• The Preon Sector of the SLq(2) Model and the Binding Problem
• Knots and Preons

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