Unified theory of ideals

Unified field theories try to merge the internal symmetries of the standard model into a single group. Here we lay out something different. We give evidence that a theory may be ultraunified; that is, beyond aiming to unify the internal symmetries into a single group, those groups may be unified together with the quarks and leptons that they act on. Furthermore, the ($3+1$) Lorentz transformations may likewise be unified 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 division algebras, whose connection to the standard model was earlier advocated for in Dixon [Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics (Kluwer Academic Publishers, Dordrecht, 1994).]. Here we propose a principle of ideals, which states that particle sectors of the standard model are given by generalized ideals of the algebra. As we show, this method very quickly uncovers all of the Lorentz representations of the standard model and one full generation of quarks and leptons.

Figure 1

The unified theory of ideals in brief. Scalars, spinors, four-vectors, and field strength tensors are unified together with the Lorentz transformations that act on them by the algebra $\mathbb{C}\otimes \mathbb{H}$. In like manner, the internal degrees of freedom for a generation of quarks and leptons and the generators of $SU(3)\times SU(2)\times U(1)$ are all made from the algebra $\mathbb{C}\otimes \mathbb{O}$. (Note, though, that electroweak generators will require projectors in $\mathbb{C}\otimes \mathbb{H}$ so as to account for the difference between left- and right-handed spinors.) Finally, $\mathbb{C}\otimes \mathbb{H}$ and $\mathbb{C}\otimes \mathbb{O}$ combine, via a tensor product over $\mathbb{C}$, to form $\mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}=\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$, the tensor product of the only four normed division algebras over the real numbers.

Figure 2

Octonionic multiplication rules.

Figure 3

$\mathbb{C}\otimes \mathbb{H}$ decomposes into ideals several times, using three different multiplication rules. $\mathbb{C}\otimes \mathbb{H}$ breaks down into two ideals (left and right Weyl spinors here) under the multiplication $m(a,\underline{v})=a\underline{v}P+a^{*}\underline{v}{P}^{*}$. Under $m(a,\underline{v})=a\underline{v}{a}^{†}$, $\mathbb{C}\otimes \mathbb{H}$ reduces into Hermitian and anti-Hermitian parts. Under $m(a,\underline{v})=a\underline{v}\tilde{a}$, $\mathbb{C}\otimes \mathbb{H}$ breaks down into a scalar and a field strength tensor. In this formalism, all of the local spacetime degrees of freedom of the standard model come from an algebra of only four complex dimensions; nature recycles.

Figure 4

Eigenvectors of the weak isospin operator, $\frac{i{e}_7}{2}$, give a full generation of fermions as a basis for $\mathbb{C}\otimes \mathbb{O}$.