Modelling the neutrino in terms of Cosserat elasticity

TL;DR

This paper presents a geometric interpretation of the Weyl equation using Cosserat elasticity, modeling space as an elastic continuum with rotational degrees of freedom, and proves its equivalence to the Weyl equation in stationary conditions.

Contribution

It introduces a novel geometric framework for the Weyl equation based on Cosserat elasticity, avoiding traditional spinor formalism and establishing an equivalence in stationary scenarios.

Findings

Modeling space as an elastic continuum with rotations reproduces Weyl equation.

The proposed Lagrangian admits a factorization crucial for the proof.

The framework provides a new geometric perspective on massless fermions.

Abstract

The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli matrices or covariant differentiation. We think of our 3-dimensional space as an elastic continuum and assume that material points of this continuum can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the space continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We choose a particular potential energy which is conformally invariant and then incorporate time into our action in the standard Newtonian way, by subtracting kinetic energy. The main result of our paper is the theorem stating that in the stationary setting our model is equivalent to a pair of Weyl equations. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.