Skip to main content
    • Aa
    • Aa


  • James Burnett (a1) and Dmitri Vassiliev (a2)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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

[3] C. G. Böhmer , R. J. Downes and D. Vassiliev , Rotational elasticity. Quart. J. Mech. Appl. Math. 64(4) (2011), 415439.

[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.

[9] D. Elton and D. Vassiliev , The Dirac equation without spinors. In The Maz’ya Anniversary Collection, Vol. 2 (Rostock, 1998) (Operator Theory: Advances and Applications 110), Birkhäuser (Basel, 1999), 133152.

[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.

[12] J. M. Nester , Special orthonormal frames. J. Math. Phys. 33(3) (1992), 910913.

[13] Y. N. Obukhov , Conformal invariance and space-time torsion. Phys. Lett. A 90(1,2) (1982), 1316.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 53 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2017. This data will be updated every 24 hours.