Coxeter element and particle masses

TL;DR

This paper constructs a family of commuting Hermitian operators related to simple Lie algebras, with eigenvalues corresponding to eigenvector coordinates of the Cartan matrix, linking algebraic structures to particle mass models.

Contribution

It introduces a novel set of commuting Hermitian operators for simple Lie algebras, connecting algebraic eigenvectors to physical particle mass representations.

Findings

Eigenvalues match eigenvector coordinates of the Cartan matrix.

Operators commute and are Hermitian, ensuring real eigenvalues.

Provides a new algebraic framework for understanding particle masses.

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$ over $\mathbb{C}$, $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $\mathfrak{h}$ whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of $\mathfrak{g}$.