Modeling the electron with Cosserat elasticity

TL;DR

This paper proposes a novel (1+2)-dimensional elastic continuum model for the electron based on Cosserat elasticity, demonstrating its equivalence to the Dirac equation and introducing a new geometric framework.

Contribution

It introduces a new elastic continuum model for electrons using Cosserat theory, linking it to the Dirac equation in 1+2 dimensions and analyzing related nonlinear PDEs.

Findings

Model is equivalent to the Dirac equation in 1+2 dimensions

Establishes a class of nonlinear PDEs reducing to linear first order equations

Provides a geometric interpretation of electron dynamics

Abstract

We suggest an alternative mathematical model for the electron in dimension 1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points 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 a coframe and a density. We then add an extra (third) spatial dimension, extend our coframe and density into dimension 1+3, choose a conformally invariant Lagrangian proportional to axial torsion squared, roll up the extra dimension into a circle so as to incorporate mass and return to our original (1+2)-dimensional spacetime by separating out the extra coordinate. The main result of our paper is the theorem stating that our model is equivalent to the Dirac equation in dimension 1+2. In the process of analyzing our model we also establish an abstract result, identifying a class of nonlinear second order partial differential equations which reduce to pairs of linear first order equations.