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arXiv · 2006.16265

The Standard Model, The Exceptional Jordan Algebra, and Triality

Latham Boyle
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Abstract

Jordan, Wigner and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 "ordinary" families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing relationship between the complexification of this algebra and the standard model of particle physics, its minimal left-right-symmetric $SU(3)\times SU(2)_{L}\times SU(2)_{R}\times U(1)$ extension, and $Spin(10)$ unification. This suggests a geometric interpretation, where a single generation of standard model fermions is described by the tangent space $(\mathbb{C}\otimes\mathbb{O})^{2}$ of the complex octonionic projective plane, and the existence of three generations is related to $SO(8)$ triality.

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Latham Boyle. 2020-06-29. The Standard Model, The Exceptional Jordan Algebra, and Triality. https://doi.org/10.1063/5.0336175

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