Teleparallelism is a special case of Cosserat Elasticity where you only allow the points of a medium to rotate and not translate. First investigated by the Cosserat brothers it was later developed by Cartan and Einstein as a candidate for a "Theory of Everything". They never succeded and today it is only worked on by a handful of specialists.
In particular we are only looking at its applications to describing particle physics in a more geometrically intuitive manner. Thus far Dima, Olga and I have shown that using a Langrangian which is quadratic in axial Torsion is equivalent to a factorisation of the Dirac equation in 2 + 1 dimensions. Here we use a Klein extension to introduce mass and all our variables, including the metric (which is fixed and prescribed), do not depend on x3. We have also shown that the field equations are equivalent. In the case where we have a general dependence on all co-ordinates the field equations and Langragians are not equiavlent, but they do both admit the same stationary and quasi-stationary plane wave solutions. It is hoped though that when we compare the energy levels of say a hydrogen atom we get from our theory to that of the Dirac theory we end up with corrections that are smaller than the accuracy of possible measurments, suggesting that it is still a viable theory.
Dark spinors are Elko spinors, which are the eigenspinors of the charge conjugation operator, applied to dark matter or dark energy. They admit a richer helicity structer then normal Dirac spinors. It is this reason that they produce interesting results in a cosmological setting.
Christian and I have shown that they allow a non-trivial spin angular momentum tensor in a universe with Torsion. They act as a source of Torsion, when we solved the Einstein equations for this model it showed that Torsion is non-zero at some late-time in the universe but decays quickly enough that we can not see it today, which fits with current observations. Our latest work which we hope to publish soon, shows that they are also viable candidates for dark energy in that they have an equation of state which is dynamic and converges to -1. They also are able to cross the phantom divide which is heavly supported by observation. In the future we will continue to look at the various applications of this new and exciting particles and possible even construct a quantum field theory for them.
Classically to describe any reaction of particles the Smoluchowski equation is used. However in the case where only a finite number of particles are being used it breaks down, since it relies on the thermodynamic limit where flucations of particle populations is negligable.
Ian and I are working on a method constructed by ?, which uses the method of second quantization found in quantum field theory, to extract a new equation which is both complex and contains an additional term to that from the Smoluchowski equation. This additional term contains a complex white noise, turning our orriginal PDE into an SDE. Solving this and taking the real part gives a solution which is compatible with the real model. We hope to develop a general scheme for solving these equations and attempt to understand why this method works.
For my MSc thesis I did a literature review of Renormalisation techniques used in quantum field theory, focusing mainly on its applications in QED and QCD. I wrote this under the supervision of Ray Rivers at Imperial College London.
During my undergraduate years I worked on a couple of summer projects. The first was in the are of Knot theory. Here I helped Mitchel Berger write some Mathematica code for illustrating the magnetic flux flares found on the surface of the sun. They interestingly follow a lot of the same rules as found in Knot theory. The year after I worked with Ian Ford on modelling crowd control, we attempted to explain the strange sticking found when people are forced to travel through narrow tunnels with many people around them, i.e the tunnels found on the underground in London. We found that they had an intersting dip in their pressure versus volume graph, unfortunately there was not enough time to follow this up.