Dirac equation as a special case of Cosserat elasticity

TL;DR

This paper presents a geometric model for the electron using coframes and torsion, showing it reduces to the Dirac equation under certain conditions without relying on spinors or matrices.

Contribution

It introduces a spinor-free geometric framework for the Dirac equation via a torsion-based Lagrangian, simplifying the mathematical tools needed.

Findings

Model is equivalent to the Dirac equation in a specific case

Lagrangian proportional to axial torsion squared

Avoids use of spinors, Pauli matrices, covariant derivatives

Abstract

We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (field of orthonormal bases) and a density. The electron mass and external electromagnetic field are incorporated into our model by means of a Kaluza-Klein extension. Our Lagrangian density is proportional to axial torsion squared. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. We prove that in the special case with no dependence on the third spatial coordinate our model is equivalent to the Dirac equation. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.