On Particles and Primes

TL;DR

This paper explores the algebraic structures of primes in complex and quaternionic algebras, revealing analogies with particles in the standard model, including leptons and hadrons, through equivalence relations and fractional charges.

Contribution

It introduces a novel algebraic framework linking primes in C and H to particle-like structures, providing a new perspective on particle classification.

Findings

Weak and strong equivalence classes relate to leptons and hadrons.

Prime-based particles can be assigned fractional charges.

The framework models particles as algebraic entities derived from primes.

Abstract

Primes in the two complete associative normed division algebras C and H have affinities with structures seen in the standard model of particle physics. On the integers in the two algebras, there are two equivalence relations: a strong one, related to a U(1) and SU(3) symmetry allowing to permute and switch signs of the coordinates of the integers, as well as a weak relation with origins from units U(1),SU(2) in the algebra. Weak equivalence classes within the strong equivalence classes of odd primes in C case relate to leptons, the inert ones being neutrino like, and the split ones resembling electron-positron pairs. In the H case, for odd primes, the equivalence classes come in groups of two or three, leading to a caricature of hadrons featuring either mesons built by a quark pair or then baryons obtained by quark triplets. We can now list for every rational prime p all these particles and attach fractional charges to its constituents.