Particle Physics from Almost Commutative Spacetimes

Abstract

Our aim in this review article is to present the applications of Connes' noncommutative geometry to elementary particle physics. Whereas the existing literature is mostly focused on a mathematical audience, in this article we introduce the ideas and concepts from noncommutative geometry using physicists' terminology, gearing towards the predictions that can be derived from the noncommutative description. Focusing on a light package of noncommutative geometry (so-called 'almost commutative manifolds'), we shall introduce in steps: electrodynamics, the electroweak model, culminating in the full Standard Model. We hope that our approach helps in understanding the role noncommutative geometry could play in describing particle physics models, eventually unifying them with Einstein's (geometrical) theory of gravity.

104 pages, 5 figures, version 2 (minor changes and some additional references)

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• Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics
• Quantum Gravity signatures in the Unruh effect
• Krein spectral triples and the fermionic action
• Spectral dimensions from the spectral action
• Causality in noncommutative two-sheeted space-times
• Electron dynamics in noncommutative geometry with magnetic field and Zitterbewegung phenomenon
• The noncommutative geometry of Zitterbewegung
• On globally non-trivial almost-commutative manifolds
• The graded product of real spectral triples
• Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Noncommutative geometry, the Lorentzian Standard Model and its B-L extension
• Doubling of the Algebra and Neutrino Mixing within Noncommutative Spectral Geometry
• Going beyond the Standard Model with noncommutative geometry
• A new algebraic structure in the standard model of particle physics
• Spectral distances on doubled Moyal plane using Dirac eigen-spinors
• Revisiting Connes' Finite Spectral Distance on Non-commutative Spaces : Moyal Plane and Fuzzy Sphere
• Particle models from special Jordan backgrounds and spectral triples
• Spectral interactions between strings in the Higgs background
• Type-II two-Higgs-doublet model in noncommutative geometry
• Physical models from noncommutative causality
• Lifting Bratteli Diagrams between Krajewski Diagrams: Spectral Triples, Spectral Actions, and $AF$ algebras