A Superalgebra Within: representations of lightest standard model particles form a $\mathbb{Z}_2^5$-graded algebra

TL;DR

This paper shows how standard model particles form a superalgebra linked to Jordan algebras, revealing internal and spacetime symmetries and suggesting connections to quantum computing.

Contribution

It introduces a superalgebra structure for standard model particles based on Euclidean Jordan algebra, with a natural division algebraic substructure and a $bZ_2^5$ grading.

Findings

The superalgebra includes gauge bosons and three fermion generations, excluding top quark irreps.

The algebra is isomorphic to $H_{16}(bC)$ and generated by division algebras.

Internal symmetries include $rak{su}(3)_C$, $rak{su}(2)_L$, $rak{u}(1)_Y$, and multiple $rak{u}(1)$s.

Abstract

It is demonstrated how a set of particle representations, familiar from the Standard Model, collectively form a superalgebra. Those representations mirroring the internal behaviour of the Standard Model's gauge bosons, and three generations of fermions, are each included in this algebra, with exception only to those irreps involving the top quark. This superalgebra is isomorphic to the Euclidean Jordan algebra of $16\times 16$ hermitian matrices, $H_{16}(\mathbb{C}),$ and is generated by division algebras. The division algebraic substructure enables a natural factorization between internal and spacetime symmetries. It also allows for the definition of a $\mathbb{Z}_2^5$ grading on the algebra. Those internal symmetries respecting this substructure are found to be $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y,$ in addition to four iterations of $\mathfrak{u}(1)$. For spatial symmetries, one finds multiple copies of $\mathfrak{so}(3)$. Given its Jordan algebraic foundation, and its apparent non-relativistic character, the model may supply a bridge between particle physics and quantum computing. We close by describing current research directions. This includes (1) detailing how this construction fits into the larger picture of Bott Periodic Particle Physics, first introduced in [1], [2], [3], and (2) detailing how the origin of this Peirce decomposition may be grounded in the unsung algebra $\mathbb{R} \oplus \mathbb{C} \oplus \mathbb{H} \oplus \mathbb{O}$.