Path Integrals and Complex Systems
Our preferred calculation technique is the path integral method. In this method reality is described as the weighted average of all possible histories. Doing such sums requires some technique, in particular because the weight is a phase factor with the action calculated for each history as its argument. All this was first developed by Richard Feynman. Nowadays path integrals are used in all branches of modern theoretical physics: solid state physics, elementary particle physics, field theory, quantum gravity, … Members of our research group have made contributions to the historical development of this theory in the field of physics, so it’s kind of a tradition in Antwerp.
Our current research on this is centered around the development of path integral propagators for Wigner distributions, with which we attempt to improve upon the “truncated Wigner approximation” and transform it to a new powerful variational-perturbative approach.
Recently it became clear that this technique can also be applied in other areas, more specifically in complex systems, in situations in which stochastic processes are added to describe physical systems. This is also the case in econophysics and financial option pricing: here one wants to calculate expectation values for underlying goods, averaged over all possible histories of the share price. The existing financial models use stochastic differential equations, which we can rewrite as path integral propagators with challenging action functionals. The methods we know from quantum theory seem particularly useful to calculate these financial propagators and to prize options correctly and evaluate the risks much better than so far possible.