Origin of fermion generations from extended noncommutative geometry

Abstract

We propose a way to understand the 3 fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e., the emergence of 3 fermion generations and the mass relationships among those generations. We define the coordinate fiber space of the bundle of the manifold as the space in which the classical noncommutative geometry is expressed, then the tensor product extension explicitly shows the contribution of structures in the non-coordinate base space of the bundle to the action. The quaternion extension plays an essential role to reveal the physical effect of the structure in the non-coordinate base space.

arXiv comment17 latex pages, no figure. Final version for publication

References in corpus (19)

• Gravity coupled with matter and foundation of non-commutative geometry
• The Spectral Action Principle
• Gravity and the standard model with neutrino mixing
• A Universal Action Formula
• Noncommutative Geometry and the standard model with neutrino mixing
• Why the Standard Model
• Fermion Hilbert Space and Fermion Doubling in the Noncommutative Geometry Approach to Gauge Theories
• Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
• Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification
• Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model
• Inner Fluctuations in Noncommutative Geometry without the first order condition
• Inner fluctuations of the spectral action
• Quanta of Geometry: Noncommutative Aspects
• Geometry and the Quantum: Basics
• Quantum field theory of gravity with spin and scaling gauge invariance and spacetime dynamics with quantum inflation
• Estimate of neutrino masses from Koide's relation
• Energy scale independence of Koide's relation for quark and lepton masses
• Theory of Quantum Gravity Beyond Einstein and Space-time Dynamics with Quantum Inflation
• Unification of gravity and quantum field theory from extended noncommutative geometry