Saturday, November 22, 2008

Massless Dirac equation as a special case of Cosserat elasticity

It seems that this blog is being updated nowhere near as frequently as as should be. I was recently invited to give a talk at Recent Trends in Mathematical Sciences in Bahrain 2008 - giving me an ideal opportunity to add something to this page.

There were a number of excellent talks, most notable were those by Dmitri Vassiliev, Barry Sanders and Alexander Balinsky.

It is the content of the talk by Dimitri Vassiliev that I want to highlight here. On a personal note it was great to see him again - as I last saw him when he taught me as an undergraduate.

In this work Cosserat theory of elasticity was applied to the telleparalism approach of Carten and Einstein in order to try to achieve a unified field theory.
Einsiten and Carten first proposed this theory there were initially great hopes of a unified field theory - however, as they failed to produce any results - it quickly fell into disfavour. In essence Dimitri presents a promising new approach to this old idea. Here, a conformally invariant axial torsion of telleparallsim connections is used to generate a Lagrangian density. Application of the principle of least action is then applied in the usual way. Interestingly, the Lagranginan factorizes so and as a corollary we the positive and negative massless Dirac equations fall out.
Furthermore, despite the governing equations being non-linear, plane wave solutions are found to exist.
Hence, a new model for fermions (with and without mass) was presented where:
  • space time is treated as a Cosserat continuum;
  • the Lagrangian density and axial torsion are conformally invariant
  • via an analogous approach to that of Kaluza & Klien; both mass and electromagnetism can be self consistently introduced to the model.
Although this topic might be considered a little off the wall - it also seems well worth revisiting (many papers are available off arxiv.org).


1 comment:

Blair S said...

It sounds like an interesting talk. I always enjoyed Professor Vasiliev's style especially when he taught us Real Variable.