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The Octonions
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The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.
Forward citations
Cited by 12 Pith papers
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Optimal measures for p-frame energies on spheres
Tight designs minimize p-frame energies over all probability measures for p between consecutive even integers, and the 600-cell does so on S3 for p in [8,10].
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Jordan Pair Quantum Theory and the Standard Model
The bi-Cayley hermitian Jordan triple yields the Standard Model gauge group and fermion representation through colinear minimal tripotents and their Peirce spaces.
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Leptonic CP Conservation and the Quark CP Phase from Octonionic Flavor Structure
Octonionic Cl(6) flavor structure yields φ12 = -2χ for quarks and real lepton amplitudes unless identity-flavor plane mixing occurs.
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Massless Representations in Conformal Space and Their de Sitter Restrictions
Introduces a canonical Clifford-split-octonion framework to construct massless ladder representations of U(2,2) and restrict them to Sp(2,2) with explicit invariant forms and operators.
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The Unified Standard Model
All Standard Model particle types and the SU(3)xSU(2)xU(1) gauge structure are placed inside the matrix algebra M(8,C), built from the octonions, with two leftover states.
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Left-right symmetry breaking in $E_6^L\times E_6^R$ occurs only in spacetime -- with possible implications for strong $CP$
Left–right exchange in E6^L×E6^R is spacetime parity, dissolving the second colour and yielding tree-level strong-CP conservation conditional on gravi-weak SU(2)_R.
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The Dirac equation in (split-)octonions: origins, variants, and modern context
Four approaches to the Dirac equation in (split-)octonions are classified; a 2024 split-octonionic equation is shown identical to the 2006 2-factor form by structure-preserving rotation and relabeling.
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Exceptional Periodicity and Magic Star Algebras. I : Foundations
The paper rigorously defines 'Magic Star algebras', periodic finite dimensional generalizations of e6, e7, and e8, which are Lie algebras only at the base level n=1.
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Fermion Mixing Matrices and the Exceptional Jordan Algebra
Using Hermitian elements of J3(OC) and cubic ladders for mass ratios as inputs, the paper constructs an effective bridge ansatz for two-generation mixing, deriving the local phase law φ12=-2χ in the quark sector with ...
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Fermion Mass Hierarchies and the Exceptional Jordan Algebra
Phenomenological deformation of an exceptional-Jordan framework that fits hierarchy exponent and normalizations to six charged-fermion mass ratios at MZ, yielding power-law relations while accommodating neutrino orderings.
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Octonions in Particle Physics through Structures of Generalised Proper Time
Generalised proper time as a higher-order polynomial yields octonionic exceptional-group structures whose symmetry breaking resembles Standard Model matter, with full unification deferred to a predicted E8 form.
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The Residual $288$ of the $E_8\times\omega E_8$ Program as Adjoint-Lineage Scaffolding Labels: an Ontology, and the Status of the Bifermionic Lagrangian
The residual 288 in the E₈×ωE₈ program is scaffolding labels not particles, with the bifermionic Lagrangian yielding sterile neutrinos and a second composite scalar.
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