Standard model physics from an algebra?

TL;DR

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Contribution

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Abstract

This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Using nothing more than $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ acting on itself, we set out to find standard model particle representations. From the complex quaternionic portion of the algebra, we find generalized ideals, and show that they describe concisely all of the Lorentz representations of the standard model. From the complex octonionic portion of the algebra, we find minimal left ideals, and show that they mirror the behaviour of a generation of quarks and leptons under $su(3)_c$ and $u(1)_{em}$. We then demonstrate a rudimentary electroweak model which yields a straightforward explanation as to why $SU(2)_L$ acts only on left-handed states. This holds in the case of leptons. Finally, we demonstrate how $\mathbb{C}\otimes\mathbb{O}$ can generate a 64-$\mathbb{C}$-dimensional algebra, wherein we find the $SU(3)_c$ irreducible representations corresponding to three generations of quarks and leptons. We then conclude by showing how to arrive at all 48 electric charges.