Massless Dirac equation as a special case of Cosserat elasticity

TL;DR

This paper introduces a novel elastic continuum model based on Cosserat elasticity that, under quasi-stationary conditions, is mathematically equivalent to the massless Dirac equation, offering a new perspective on neutrino modeling.

Contribution

It demonstrates that a conformally invariant Cosserat elastic model can be reformulated as a massless Dirac equation in the quasi-stationary regime, linking elasticity theory with quantum field equations.

Findings

Model is equivalent to massless Dirac equations under certain conditions

Lagrangian admits a factorization enabling the equivalence

Provides a new elastic interpretation of neutrino behavior

Abstract

We suggest an alternative mathematical model for the massless neutrino. Consider an elastic continuum in 3-dimensional Euclidean space and assume that points of this continuum can experience no displacements, only rotations. This framework is a special case of the so-called Cosserat theory of elasticity. Rotations of points of the continuum are described by attaching to each point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We write down a potential energy which is conformally invariant and then incorporate time in the standard Newtonian way, by subtracting kinetic energy. Finally, we rewrite the resulting nonlinear variational problem in terms of an unknown spinor field. We look for quasi-stationary solutions, i.e. solutions that harmonically oscillate in time. We prove that in the quasi-stationary setting our model is equivalent to a pair of massless Dirac equations. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.