Time reversal of Markov processes and relativistic quantum theory

As mathematical foundations of non-relativistic and relativistic quantum theory, time reversal and duality of Markov processes are discussed. Then a stochastic theory of non-relativistic quantum mechanics is briefly explained as an application. The main theme of this article is to apply stochastic methods to relativistic quantum theory, in particular, for the movement of relativistic spinless particles in an electromagnetic field. It is shown that the movement of relativistic quantum particles can be described with pure-jump Markov processes; moreover that the relativistic Schrödinger equation converges to the non-relativistic one as the speed of light tends to infinity and that the relativistic pure-jump Markov process converges to the Schrödinger diffusion process accordingly. Finally a statistical model for systems of infinitely many interacting relativistic (virtual) particles is discussed.