Matrices de commutation tensorielle: de l'\'equation de Dirac vers une application en physique des particules

TL;DR

This paper explores tensor commutation matrices and their applications in transforming Dirac equations, matrix equations, and expressing the electric charge operator in particle physics, introducing new matrix representations and generalizations.

Contribution

It introduces rectangle Gell-Mann matrices and applies tensor commutation matrices to particle physics operators, extending previous mathematical frameworks.

Findings

Representation of Dirac equations using TCMs

Expression of the electric charge operator with TCMs

Introduction of rectangle Gell-Mann matrices

Abstract

We construct two sets of representations of the Dirac equation. They transform themselves from one to another in multiplying by the tensor commutation matrix (TCM) $2\otimes 2$. The Gauss matrix can lead us to the Cholesky decomposition. The TCMs can be used in order to obtain different transforms of some matrix equations to linear matrix equations of the form $AX=B$. A TCM $n\otimes n$ is expressed in terms of the $n\otimes n$-Gell-Mann matrices. In order to generalize this expression to which of TCMs $n\otimes p$, we introduced what we call rectangle Gell-Mann matrices. The electric charge operator (ECO) for eight leptons and quarks of the Standard Model (SM) of a same generation proposed by Zenczykowski is expressed in terms of the TCM $2\otimes 2$. An ECO including all the fermions of the SM is constructed in terms of TCM $4\otimes 4$. Then the eigenvalues of a TCM take sense.