Weyl's Lagrangian in teleparallel form

TL;DR

This paper introduces a new geometric formulation of the Weyl (massless Dirac) Lagrangian using teleparallelism, avoiding spinors and matrices, and providing a variational interpretation of the Weyl equation.

Contribution

It presents a novel teleparallel representation of the Weyl Lagrangian using coframes and axial torsion, simplifying the geometric framework and removing the need for spinors.

Findings

Weyl Lagrangian expressed via axial torsion and coframes

Avoids spinors, Pauli matrices, covariant derivatives

Provides a variational geometric interpretation of Weyl equation

Abstract

The main result of the paper is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian - wedge product of axial torsion with a lightlike element of the coframe - and show that this gives the Weyl Lagrangian up to a nonlinear change of dynamical variable. 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. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J. B. Griffiths and R. A. Newing.