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Peer-reviewedCC BYJournal of Mathematical Physics

The standard model, the exceptional Jordan algebra, and triality

  • Latham A. Boyle1,2
  1. 1Higgs Centre for Theoretical Physics, University of Edinburgh , Edinburgh EH9 3FD, and , Waterloo, Ontario N2L 2Y5,
  2. 2Perimeter Institute for Theoretical Physics , Edinburgh EH9 3FD, and , Waterloo, Ontario N2L 2Y5,
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Abstract

Jordan, Wigner, and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 “ordinary” families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing relationship between the complexification of this algebra and the standard model of particle physics, its minimal left-right-symmetric SU(3) × SU(2)L × SU(2)R × U(1) extension, and Spin(10) unification. This suggests a geometric interpretation, where a single generation of standard model fermions is described by the tangent space (C⊗O)2 of the complex octonionic projective plane, and the existence of three generations is related to SO(8) triality.

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Cite

BibTeXRIS
BibTeX
@article{boyle2026standard,
  title = {{The standard model, the exceptional Jordan algebra, and triality}},
  author = {Latham A. Boyle},
  journal = {Journal of Mathematical Physics},
  year = {2026},
  month = jul,
  doi = {10.1063/5.0336175},
  url = {https://doi.org/10.1063/5.0336175},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The standard model, the exceptional Jordan algebra, and triality
AU  - Latham A. Boyle
PY  - 2026
DA  - 2026/07/01
JO  - Journal of Mathematical Physics
DO  - 10.1063/5.0336175
UR  - https://doi.org/10.1063/5.0336175
AB  - Jordan, Wigner, and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 “ordinary” families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing relationship between the complexification of this algebra and the standard model of particle physics, its minimal left-right-symmetric SU(3) × SU(2)L × SU(2)R × U(1) extension, and Spin(10) unification. This suggests a geometric interpretation, where a single generation of standard model fermions is described by the tangent space (C⊗O)2 of the complex octonionic projective plane, and the existence of three generations is related to SO(8) triality.
SN  - 0022-2488
SN  - 1089-7658
SN  - 1527-2427
ER  -