Sedenion algebra for three lepton/quark generations and its relations to SU(5)

TL;DR

This paper explores sedenion algebra as a mathematical framework to naturally explain the existence of three fermion generations in particle physics, linking algebraic structures to the Standard Model extensions.

Contribution

It introduces a novel application of sedenion algebra to model three fermion generations, overcoming limitations of previous quaternion-based models.

Findings

Sedenion algebra contains multiple quaternion and octonion sub-algebras.

Each octonion sub-algebra corresponds to one fermion generation.

The model naturally predicts exactly three generations.

Abstract

In this work, we analyze two models beyond the Standard Models descriptions that make ad hoc hypotheses of three point-like lepton and quark generations without explanations of their physical origins. Instead of using the same Dirac equation involving four anti-commutative matrices for all such structure-less elementary particles, we consider in the first model the use of sixteen direct-product matrices of quaternions that are related to Diracs gamma matrices. This associative direct-product matrix model could not generate three fermion generations satisfying Einsteins mass-energy relation. We show that sedenion algebra contains five distinct quaternion sub-algebras and three octonion sub-algebras but with a common intersecting quaternion algebra. This model naturally leads to precisely three generations as each of the non-associative octonion sub-algebra leads to one fermion generation. Moreover, we demonstrate the use of basic sedenion.