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).

All of Dimitri's talks are available here

On non-Hermitian quantum mechanics

Our recent post "On non-Hermitian quantum mechanics" arXiv:0804.2051 we critique some recent work in the field by Carl Bender et al [see for example Phys. Rev. Lett 98, 040403 (2007) and Rep. Prog. Phys. 70, 947-1018 (2007)]. Their work is of interest but, in our opinion, there is some misuse of terminology with respect to the label "non-Hermitian". To us it is more like they are doing conventional quantum mechanics with novel inner products.

There already exists in the literature comments on this topic (e.g. Ali Mostafazadeh arXiv:quant-ph/0310164 and arXiv:quant-ph/0407070 as well as David B. Fairlie and Jean Nuyts arXiv:hep-th/0412148). While these papers present valid arguments they miss the central point of our comment. Namely, that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian (and the whole exercise is somewhat redundant). Our argument is based on an observation that there exists in the literature some confusion over the definition of Hermiticity and the imposition of Schrödinger evolution on the evolution of state vectors. [This argument is not quite the same as the one presented in L.D. Landau and E.M. Lifshitz, “Quantum Mechanics (Non-relativistic Theory)” which, as we point out, also contains a flaw].

Holiday Halo

Just back from a great holiday that started with a 7 night cruse from Luxor to Aswan with Abercrombie & Kent. Absolutely excellent, would recommend them to anyone. Then we to Mt Siani with some good friends of ours for new year. It was there that we saw this Halo above St Catherine's monastery. Makes a rather pleasing photo, feel free to use this image to support your physics teaching if you want to (please remember to cite this blog).