The methodology to be investigated here is the stochastic bridge: a stochastic process with defined end-points in the past and in the future. This allows time to be treated in a reversible way, thus creating a new conceptual approach to quantum dynamics, applicable to the most challenging problem in quantum theory, which is the treatment of reversible, unitary evolution, in large interacting systems. This also involves the theory of forward-backwards stochastic equations. These are a type of stochastic equation in which a diffusion process runs simultaneously forwards and backwards in time. As well as being found in financial modeling, and stochastic control theory, these equations occur in quantum systems in phase-space. The project will implement techniques for solving these equations, and test their accuracy.
Among the skills that will be learned are a mastery of curved space geometry, stochastic equations, Fokker-Planck equations, numerical modeling, and writing reliable and testable computer code on Github, as part of a larger project team. The project will lead to testable theoretical predictions in ultra-cold atom physics, photonics, and quantum circuits, with the cooperation of experimental groups in these related areas, and possible exchange visits to laboratories.