One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$

Abstract

Peering in from the outside, $\mathbb{A}:=\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$ looks to be an ideal mathematical structure for particle physics. It is 32 $\mathbb{C}$-dimensional: exactly the size of one full generation of fermions. Furthermore, as alluded to earlier in arXiv:1806.00612, it supplies a richer algebraic structure, which can be used, for example, to replace SU(5) with the SU(3)$\times$SU(2)$\times$U(1) / ${\mathbb{Z}}_6$ symmetry of the standard model. However, this line of research has been weighted down by a difficulty known as the fermion doubling problem. That is, a satisfactory description of SL(2,$\mathbb{C}$) symmetries has so far only been achieved by taking two copies of the algebra, instead of one. Arguably, this doubling of states betrays much of $\mathbb{A}$'s original appeal. In this article, we solve the fermion doubling problem in the context of $\mathbb{A}$. Furthermore, we give an explicit description of the standard model symmetries, $g_{sm}$, its gauge bosons, Higgs, and a generation of fermions, each in the compact language of this 32 $\mathbb{C}$-dimensional algebra. Most importantly, we seek out the subalgebra of $g_{sm}$ that is invariant under the complex conjugate - and find that it is given by $su(3{)}_C\oplus u(1{)}_{EM}$. Could this new result provide a clue as to why the standard model symmetries break in the way that they do?

arXiv commentThis article was first presented for Rutgers University in 2020. It was subsequently circulated widely amongst colleagues on the 16th of February, 2021. It was presented at the Perimeter Institute Conference on Octonions and the Standard model on the 15th of March, 2021. https://pirsa.org/21030013

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• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Gauge Theory on Fiber Bundle of Hypercomplex Algebras
• Three Generations and a Trio of Trialities
• Physics with non-unital algebras? An invitation to the Okubo algebra
• A Superalgebra Within: representations of lightest standard model particles form a ${\mathbb{Z}}_2^5$-graded algebra
• A New Division Algebra Representation of $E_6$
• Preons, Braid Topology, and Representations of Fundamental Particles
• An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints