Differential Galois Theory and Quantum Physics (Recent Developments in Dynamical Systems and their Applications)

Abstract

We survey my 2020 paper [18], on the relevance of the differential Galois theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself, formulated in the framework of differential Galois theory. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in closed form. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems. Moreover, I will point out several of the many open problems motivated from the above simple result: problems from quantum mechanics to quantum field theory.