Current Research Programme
My two main areas of research are:
(1) The application of differential geometric techniques to the renormalisation group equation in relativistic quantum field theory and statistical mechanics
(2) an investigation of the phase space structure of Einstein's General theory of Relativity.
Under a rescaling of lengths (or energy) physical quantities change according to their dimensions. Sometimes they change in unexpected ways, giving rise to anomalous dimensions. If we represent the parameters of a physical theory (e.g. temperature and external magnetic field in a ferromagnet, or electric charge and top quark mass in a quantum field theory) by points in a multi-dimensional space, then the change in the parameters under a change in scale can be viewed as tracing out paths in this space with tangent vectors which constitute a vector field - an inherently geometrical object. Under certain circumstances, this vector flow can exhibit chaotic behaviour. It is possible to define the concept of distances on the space of parameters, giving rise to a metric on the space (the metric is essentially the matrix of expectation values of the two-point functions of the theory). In general the space is curved, and this opens up the exciting possibility of applying all the mathematical machinery of differential geometry to the renormalisation group. For example, it is possible to write the renormalisation group equation in a manifestly general co-ordinate covariant way (two papers [1] and [2]) - somewhat in the spirit of general relativity. It is also possible to describe the vector flow of the renormalisation group as a Hamiltonian flow on a symplectic space . This is analogous to the Hamiltonian flow of classical mechanics, albeit irreversible flow in this case.
The current research programme aims at a fully geometrical description of the way in which physical parameters change under scale. Geometry has always been of central importance to the development of Theoretical Physics, in classical mechanics, thermodynamics, electromagnetism, general relativity and quantum field theory to name but a few examples. It is hoped that it will also be relevant to the physics of scale changes.
In addition I have an interest in recent developments in General Relativity, in particular the phase space structure and Hamiltonian formalism of the theory as well as the effects of curved space on quantum field theory .
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