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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Fang, Yan-Long and Vassiliev, Dmitri 2015. Analysis as a source of geometry: a non-geometric representation of the Dirac equation. Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue. 16, p. 165203.


    Chervova, O. Downes, R. J. and Vassiliev, D. 2014. Spectral theoretic characterization of the massless Dirac operator. Journal of the London Mathematical Society, Vol. 89, Issue. 1, p. 301.


    Deymier, P. A. Runge, K. Swinteck, N. and Muralidharan, K. 2014. Rotational modes in a phononic crystal with fermion-like behavior. Journal of Applied Physics, Vol. 115, Issue. 16, p. 163510.


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MODELLING THE ELECTRON WITH COSSERAT ELASTICITY

  • James Burnett (a1) and Dmitri Vassiliev (a2)
  • DOI: http://dx.doi.org/10.1112/S002557931200006X
  • Published online: 12 April 2012
Abstract
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.

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[1]J. M. Ball and A. Zarnescu , Orientable and non-orientable director fields for liquid crystals. Proc. Appl. Math. Mech. 7(1) (2007), 10507011050704.

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[4]J. Burnett and D. Vassiliev , Weyl’s Lagrangian in teleparallel form. J. Math. Phys. 50(10) (2009), 102501, 17pp.

[7]O. Chervova and D. Vassiliev , The stationary Weyl equation and Cosserat elasticity. J. Phys. A 43(33) (2010), 335203, 14pp.

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[10]J. B. Griffiths and R. A. Newing , Tetrad equations for the two-component neutrino field in general relativity. J. Phys. A 3 (1970), 269273.

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[14]T. Sauer , Field equations in teleparallel space-time: Einstein’s fernparallelismus approach toward unified field theory. Historia Math. 33(4) (2006), 399439.

[16]D. Vassiliev , Teleparallel model for the neutrino. Phys. Rev. D 75(2) (2007), 025006, 6pp.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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