The Dirac equation in (split-)octonions: origins, variants, and modern context

Abstract

Octonions and split-octonions have been used to express the Dirac equation in physics in several conceptually distinct ways. This review organizes them into four groups: the 2-factor, 3-factor, and projection representations, which use (split-)octonion basis elements natively to model a spacetime basis, and the conventional use of octonions to carry Dirac algebra acting on spinors. For each group we identify the originating construction, relate later variants to it, and point to modern applications and to recent methods for native split-octonionic analysis. Because the multiplication table of the (split-)octonions is not unique, forms that look different in print can coincide after a structure-preserving rotation and a relabeling of basis elements; we make such relations explicit and tabulate the basis conventions used across the sources. The 2-factor representation is treated in most detail: we document its origin and show that a recently proposed split-octonionic Dirac equation reduces to it up to such a rotation and relabeling, while crediting the independent contributions of that work.