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 norm-preserving linear isometry and a multiplication-preserving change of basis; we make such relations explicit and tabulate the basis conventions used across select sources. The 2-factor representation is treated in most detail: we document its origin and show that a recently proposed split-octonionic Dirac equation coincides with it after such an isometry and change of basis, while crediting the independent contributions of that work.
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BibTeX
@misc{koplinger2026dirac,
title = {{The Dirac equation in (split-)octonions: origins, variants, and modern context}},
author = {J. Köplinger},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.09703},
eprint = {2607.09703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Dirac equation in (split-)octonions: origins, variants, and modern context
AU - J. Köplinger
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2607.09703
UR - https://arxiv.org/abs/2607.09703
AB - 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 norm-preserving linear isometry and a multiplication-preserving change of basis; we make such relations explicit and tabulate the basis conventions used across select sources. The 2-factor representation is treated in most detail: we document its origin and show that a recently proposed split-octonionic Dirac equation coincides with it after such an isometry and change of basis, while crediting the independent contributions of that work.
ER -