A teleparallel representation of the Weyl Lagrangian

TL;DR

This paper introduces a novel geometric formulation of the Weyl Lagrangian using coframes and torsion, avoiding spinors and matrices, and deriving the Weyl equation through a purely differential geometric approach.

Contribution

It provides a new teleparallel geometric representation of the Weyl Lagrangian using coframes and torsion, simplifying the mathematical framework.

Findings

Derives Weyl equation from a coframe-based Lagrangian

Avoids spinors, matrices, and covariant derivatives

Establishes a geometric variational principle for Weyl spinors

Abstract

The main result of the paper is a new representation of 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 variation of the resulting action with respect to the coframe produces the Weyl equation. 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.