Research
My research is split between the foundations of quantum statistical physics and studying discrete spacetime quantum systems as models of quantum field theories.
Foundations of Quantum Statistical Physics
Statistical physics is a remarkably useful tool. It allows us to understand and predict the behaviour of systems with far too many degrees of freedom to study from first principles. For this to work, some assumptions must be made, regardless of whether the framework is classical or quantum. One such assumption is the principle of equal a priori probability, which says roughly that an isolated system in equilibrium is equally likely to be in any accessible state. Recently, however, it was argued by Popescu et al. that in quantum systems, this assumption is unnecessary. This is encouraging: maybe the fact that nature is quantum makes these extra assumptions unnecessary.
Another assumption made in statistical physics is that physical systems generally reach equilibrium. Remarkably, it was shown by Reimann and Linden et al. that quantum systems generally reach equilibrium anyway, so naively it seems like we don't need to assume that things will equilibrate. Unfortunately, almost nothing is known about how long the process of equilibration actually takes. This is something my supervisor and I studied in this paper. (The published version is in the New Journal of Physics here.) We found an upper bound for the time it takes to reach equilibrium. Viewed another way, this also upper bounds the time a system can spend far away from its equilibrium state.
Discrete Spacetime Quantum Systems
Combining quantum mechanics and special relativity is not easy, and the result, quantum field theory, has yet to be put on a firm mathematical footing. One of the reasons for this is that working in continuous space leads to infinities. So it could be easier in some ways to study analogous systems in discrete space. Also, if we assume that there is a maximum speed of propagation of information, like in special relativity, then we are basically forced to make time discrete too.
So we would like to find quantum systems in discrete spacetime that become equivalent to quantum field theories in the limit as the discrete spacetime becomes continuous. These would be useful as simpler mathematical models and also might allow efficient simulation of quantum field theories. Plus, some people believe that spacetime may in fact be discrete at some small scale, so quantum systems in discrete spacetime could give an accurate description of fundamental physics. Some interesting work has already been done by Bisio et al. with this idea in mind.
Another question we could ask is whether you can say anything about causal discrete spacetime quantum systems without assuming a particular model. My supervisor and I studied this question for systems of fermions here. One of the things we found was that, analogous to this result for QCA, the evolution of these fermions can always be viewed as a product of local unitaries in a simple way. Furthermore, these systems can be represented by lattices of qubits evolving causally, which is particularly useful from the point of view of quantum simulation. Hopefully, these results could help with the goal of constructing discrete spacetime quantum field theories that are causal.
