Origin of fermion generations from extended noncommutative geometry

TL;DR

This paper uses extended noncommutative geometry to explain the origin of three fermion generations and their mass relationships, integrating tensor and quaternion extensions to reveal physical effects.

Contribution

It introduces a novel combination of tensor and quaternion extensions in noncommutative geometry to account for fermion generations and their mass patterns.

Findings

Three fermion generations emerge from the combined extensions.

Mass relationships among generations are derived from geometric structures.

Tensor and quaternion extensions are essential for physical effects in the model.

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.