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

TL;DR

This work embeds the Standard Model's local degrees of freedom into the Euclidean Jordan algebra $\mathcal{H}_{16}(\mathbb{C})$ and reveals a division-algebra–driven Peirce decomposition that yields a $\mathbb{Z}_2^5$ grading. Internal octonionic structure produces the SM gauge factors $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y$ plus extra $\mathfrak{u}(1)$ components, while quaternionic blocks furnish independent $\mathfrak{so}(3)$ spatial rotations; gauge and Higgs content reside in diagonal blocks, two generations occupy outer off-diagonal blocks, and the third generation sits in inner off-diagonal blocks, with top-quark representations omitted in the base embedding. The construction yields a $256$-real-dimensional, pre-geometric setting with a $64$-dimensional momentum-space covariant derivative and a three-generation structure, suggesting links to quantum computing via Bott periodic particle physics and hinting at emergent gravity through the full multiplication algebra $M_{\mathbb{A}}$. The work also discusses projection-valued Peirce measurements, an observable algebra that precludes color, and the emergence of Lorentz-type structure from algebraic inner automorphisms, outlining future directions for anomaly cancellation and gravity integration.

Abstract

It is demonstrated how a set of particle representations, familiar from the Standard Model, collectively form a superalgebra. Those representations mirroring the 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 how this article fits into the larger picture of Bott Periodic Particle Physics, first introduced in [1], [2], [3].

Figures (6)

• Figure 1: Decomposition of $L_{\mathbb{C}\otimes\mathbb{O}}\simeq \mathbb{C}l(6)$ into $\mathfrak{su}(3)_C$ representations familiar from gluons, and three generations of quarks and leptons.
• Figure 2: Two generations of off-shell fermion representations in the outer off-diagonal blocks, including sterile neutrinos.
• Figure 3: Momentum-space covariant derivative in the diagonal blocks
• Figure 4: Third generation in the inner off-diagonal blocks
• Figure 5: Peirce decomposition of $\mathcal{H}_{16}(\mathbb{C})$ into familiar covariant derivative and fermionic vector spaces. Note: (1) Fermions are identified as off-diagonal objects within the Peirce decomposition. Physically, we say that they extend between two distinct sites, $s_i$ and $s_j$, each subject to an independent $\mathfrak{su}(2)$ transformation. This is a fundamentally different description than that of the Standard Model. Please see Section \ref{['disc']} for details. (2) Representations $(\nu_{\tau}, \tau)_L$ and $h\cdot \nu_{\tau R}^{\dagger}$ are identical under $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y$. For visual simplicity we identify each with a particular Peirce block. Also mostly for visual simplicity, we have labeled Peirce blocks with explicit generation labels. (3) A certain sign change within the weak hypercharge operator allows for the interchange of RH neutrino and charged lepton so that neutrinos may be positioned in the extreme $\mathcal{H}_{15}$ corners of the diagram. This was depicted in the first arXiv version of the article.