Modeling the electron with Cosserat elasticity

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.

References in corpus (5)

• Field equations in teleparallel spacetime: Einstein's Fernparallelismus approach towards unified field theory
• Rotational elasticity
• The stationary Weyl equation and Cosserat elasticity
• A teleparallel model for the neutrino
• Weyl's Lagrangian in teleparallel form

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• Spectral theoretic characterization of the massless Dirac operator
• Analysis as a source of geometry: a non-geometric representation of the Dirac equation
• QED Treatment of Linear Elastic Waves in Asymmetric Environments
• A `Third' Quantization Constructed for Gauge Theory of Gravity