Peering in from the outside, $\mathbb{ A} := \mathbb{ R}\otimes\mathbb{ C}\otimes\mathbb{ H}\otimes\mathbb{ O}$ looks to be an ideal mathematical structure for particle physics. It is 32 $\mathbb{ C}$-dimensional: exactly the size of one full generation of fermions. Furthermore, as alluded to earlier in arXiv:1806.00612, it supplies a richer algebraic structure, which can be used, for example, to replace SU(5) with the SU(3)$\times$SU(2)$\times$U(1) / $\mathbb{ Z}_6$ symmetry of the standard model. However, this line of research has been weighted down by a difficulty known as the fermion doubling problem. That is, a satisfactory description of SL(2,$\mathbb{ C}$) symmetries has so far only been achieved by taking two copies of the algebra, instead of one. Arguably, this doubling of states betrays much of $\mathbb{ A}$'s original appeal. In this article, we solve the fermion doubling problem in the context of $\mathbb{ A}$. Furthermore, we give an explicit description of the standard model symmetries, $g_{ sm}$, its gauge bosons, Higgs, and a generation of fermions, each in the compact language of this 32 $\mathbb{ C}$-dimensional algebra. Most importantly, we seek out the subalgebra of $g_{ sm}$ that is invariant under the complex conjugate - and find that it is given by $su(3)_C \oplus u(1)_{ EM}$. Could this new result provide a clue as to why the standard model symmetries break in the way that they do?
Categories: hep-ph hep-th