Research Interests
Here is a short overview of some of my research interests
Here is a short overview of some of my research interests
Classical chaos is characterised by a sensitivity to initial conditions, for example the 3-body gravitational problem is known to be chaotic and is very dependant on the starting position and momentum of the planets, as this applet shows. If we imagine this gravitational problem on atomic scales then the effects of quantum mechanics become manifest. The natural question then arises as to what influences the classical chaos has on the quantum properties?
For closed stationary quantum systems the energy levels are discrete and infinite in number, therefore one can investigate the way these energy levels are spaced. Remarkably if the underlying classical dynamics are chaotic then the way the energies are distributed high in the spectrum appears to be the same as the eigenvalues for random ensembles of large Hermitian matrices. This was first proposed by Wigner and Dyson in the 50s and 60s and lead to the development of a field of Mathematics called random matrix theory (RMT).
[Image is of an eigenfunction of the cardioid billiard, taken from Arnd Bäcker's website].
One way of explaining these RMT observations is using the Gutzwiller trace formula which connects the quantum energy levels to periodic orbits of the corresponding classical system. In the three-body problem mentioned above these are the paths where the planet gets back to its starting point with the same momentum, they too are infinite in number and were first investigated by Poincaré. Through this formula we can turn problems about correlations between energy levels into correlations between different periodic orbits.
Symmetries are a crucial part of quantum mechanics; appearing everywhere from the spherical symmetry of the Hydrogen atom to the gauge invariance principles inherent in the standard model. In order to describe their effects in the quantum world requires representation theory, an area of Mathematics developed at the turn of the last century by Frobenius, Shur and others. Representation theory describes how the energy level of a symmetric quantum system fall into various subspectra, this is crucial in quantum chaos as it is now the investigation of energy level distributions within each subspectra that are important.
Quantum graphs are in essence a union of 1-dimensional quantum systems joined together in an appropriate fashion. They serve as excellent models for describing natural phenomena such as electrons in organic molecules, wave propagation in wired networks, Anderson localisation and various effects in mesoscopic systems.
In the past decade quantum graphs have also been extensively studied as a way of observing many of the features of quantum chaos in relatively simple systems. A graph consists of a set of vertices connected by various bonds or edges, so by increasing the number of connections in our graph we introduce more disorder and hence find the spectral properties of the graph approach those predicted by RMT. Quantum graphs offer many advantages over other models, for instance they too posses a trace formula (analogous to the Gutzwiller formula) which in this case are exact. Moreover due to their 1-dimensional roots it is possible to deduce the spectrum exactly using only a finite number of periodic orbits - a true quantum to classical correspondence!
For further reading please investigate the following.