The standard model, the Pati-Salam model, and "Jordan geometry"

Abstract

We argue that the ordinary commutative-and-associative algebra of spacetime coordinates (familiar from general relativity) should perhaps be replaced, not by a noncommutative algebra (as in noncommutative geometry), but rather by a Jordan algebra (leading to a framework which we term "Jordan geometry"). We present the Jordan algebra (and representation) that most nearly describes the standard model of particle physics, and we explain that it actually describes a certain (phenomenologically viable) extension of the standard model: by three right-handed (sterile) neutrinos, a complex scalar field $\phi$, and a $U(1{)}_{B-L}$ gauge boson which is Higgsed by $\phi$. We then note a natural extension of this construction, which describes the $SU(4)\times SU(2{)}_L\times SU(2{)}_R$ Pati-Salam model. Finally, we discuss a simple and natural Jordan generalization of the exterior algebra of differential forms.

arXiv commentv1: 16 pages; v2: references added v3: minor revisions, references added

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Cited by in corpus (19)

• Octonion Internal Space Algebra for the Standard Model
• Conjugation Matters. Bioctonionic Veronese Vectors and Cayley-Rosenfeld Planes
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• The geometry of physical observables
• Twisted Standard Model in noncommutative geometry I: the field content
• Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions
• The Mass Gap of the Space-time and its Shape
• Superselection of the weak hypercharge and the algebra of the Standard Model
• Superconnection in the spin factor approach to particle physics
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Causal Fermion Systems and Octonions
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Particle models from special Jordan backgrounds and spectral triples
• A critical survey of twisted spectral triples beyond the Standard Model
• Division Algebras, Triality, and Exceptional Magic
• Quantum Spacetime Imprints: The 24-Cell, Standard Model Symmetry and Its Flavor Mixing
• Unified products for Jordan algebras. Applications
• Real Part of Twisted-by-Grading Spectral Triples
• Algebraic structure underlying spherical, parabolic and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problem