A note on the principle of least action and Dirac matrices

TL;DR

This paper explores how many physical Lagrangians can be represented as eigenvalues of matrices involving Dirac gamma matrices, analyzing the implications of the least action principle in this matrix framework across various theories.

Contribution

It introduces a matrix-based perspective on Lagrangians involving Dirac matrices and discusses the implications of the least action principle within this approach.

Findings

Lagrangians can be expressed as eigenvalues of Dirac matrix matrices

Application of least action principle to matrix Lagrangians analyzed

Examples include point particles, electrodynamics, gauge theories, and gravity

Abstract

Many Lagrangians of physical theories can be expressed as eigenvalues of certain, relatively simple, matrices involving Dirac gamma matrices. We give concrete examples for Lagrangian corresponding to a point particle coupled to electromagnetic field, electrodynamics, nonabelian gauge theories, extended objects and gravity. We also discuss (in case of a point particle) what are the implications of the least action principle applied to matrix Lagrangians.