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MODELLING THE ELECTRON WITH COSSERAT ELASTICITY

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  12 April 2012

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

Abstract

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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.

MSC classification

Secondary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 35Q74: PDEs in connection with mechanics of deformable solids

Information

Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 349 - 370
Copyright
Copyright © University College London 2012

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