Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics

Abstract

Connes' notion of non-commutative geometry (NCG) generalizes Riemannian geometry and yields a striking reinterepretation of the standard model of particle physics, coupled to Einstein gravity. We suggest a simple reformulation with two key mathematical advantages: (i) it unifies many of the traditional NCG axioms into a single one; and (ii) it immediately generalizes from non-commutative to non-associative geometry. Remarkably, it also resolves a long-standing problem plaguing the NCG construction of the standard model, by precisely eliminating from the action the collection of 7 unwanted terms that previously had to be removed by an extra, non-geometric, assumption. With this problem solved, the NCG algorithm for constructing the standard model action is tighter and more explanatory than the traditional one based on effective field theory.

5 pages, no figures, matches published version

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Cited by in corpus (22)

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• The geometry of physical observables
• Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model
• Noncommutative geometry, the Lorentzian Standard Model and its B-L extension
• A $U(1{)}_{B-L}$-extension of the Standard Model from Noncommutative Geometry
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• Twisted reality and the second-order condition
• Can Spectral Action be a Window to Very High Energies?
• Particle models from special Jordan backgrounds and spectral triples
• Real Spectral Triples on Crossed Products
• A Proposed $E_8\times E_8$ Kinematic Scaffolding for the Standard Model with Pre-Gravitation
• A critical survey of twisted spectral triples beyond the Standard Model
• Type-II two-Higgs-doublet model in noncommutative geometry