Towards a unified theory of ideals

Abstract

Unified field theories act to merge the internal symmetries of the standard model into a single group. Here we lay out something different. That is, instead of aiming to unify the internal symmetries, we demonstrate a sense in which the group transformations may be unified with the quarks and leptons that they act on. Similarly, the (3+1) Lorentz transformations may be united with the scalars, spinors, four-vectors and field strength tensors that they act on. These simplifications occur because the representations can be found in the form of an algebra acting on itself. The approach described in this paper is meant to tie everything into the Dixon algebra: $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the tensor product of the only four normed division algebras over $\mathbb{R}$. Here we demonstrate that the standard model's Lorentz representations may be cast as a special set of generalized ideals within the algebra $\mathbb{C}\otimes\mathbb{H}$. We then make an early attempt at extending this idea to one generation of quarks and leptons.