Mathematika

Research Article

MODELLING THE ELECTRON WITH COSSERAT ELASTICITY

James Burnetta1 and Dmitri Vassilieva2

a1 Department of Mathematics and Institute of Origins, University College London, Gower Street, London WC1E 6BT, U.K. (email: J.Burnett@ucl.ac.uk)

a2 Department of Mathematics and Institute of Origins, University College London, Gower Street, London WC1E 6BT, U.K. (email: D.Vassiliev@ucl.ac.uk)

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 analysing 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.

(Received November 29 2011)

(Online publication April 12 2012)

MSC (2010)

  • 35Q41;
  • 35Q74 (primary)