Electroweak Quantum Numbers in the $D_4$ Root System

TL;DR

This paper introduces a novel framework using the $D_4$ root system and quaternion algebra to unify quarks and leptons, define electric charge, and explain electroweak interactions and fermion generations.

Contribution

It presents a new model linking $D_4$ root system, quaternion multiplication, and fundamental quantum numbers to unify particle properties and interactions.

Findings

Defined four fundamental quantum numbers from $D_4$ root system

Proved charge conservation via binary tetrahedral symmetry

Identified three fermion generations with quaternion imaginary dimensions

Abstract

We introduce four fundamental quantum numbers based on the $D_4$ root system, giving a unified description of quarks and leptons. These numbers will make it possible to define electric charge in a simple way. By postulating a fundamental symmetry given by the binary tetrahedral group acting on this root system, we can easily prove conservation of electric charge. Also, possible electroweak interactions are given by a simple rule where the 24-cell acts on itself by quaternion multiplication. The three generations of fermions in the Standard Model are hereby identified with the three imaginary dimensions of the quaternions.