Stochastic mechanics according to M. Nagasawa

Tuesday, August 1, 2017
Luca Curcuraci (Ph.D. student, University of Trieste)
Stochastic mechanics is a attempt to formulate quantum mechanics starting from the theory of stochastic process. The most famous formulation is due to E. Nelson [1]. However such formulation is not capable to recover quantum mechanics unless one postulates the so called Wallstrom condition [2] by hand. In this talk I will briefly present a different formulation of stochastic mechanics due to M. Nagasawa [3], [4]. Stating from the duality theory of Markov process, I will introduce the Schrodinger representation for a Markov process. The equation of motion for a quantum particle and their relation with the usual Schrodinger equation will be discussed. Time permitting, as example, I will consider the excited states of the Hydrogen atom and the issue relating the so called “ergodic decompositions” of the sample space. The solution of this issue suggest which is the real meaning of the Nagasawa trajectories.
[1] Nelson E. - Dynamical Theories of Brownian Motion, Princeton University Press, 1967; [2] Timothy C. Wallstrom - On the derivation of the Schrodinger equation from stochastic mechanics, Found. Physics. Lett., 1989; [3] M. Nagasawa - Schrodinger equation and diffusion theory, Brikhauser Verlag, 1993; [4] M. Nagasawa - Stochastic processes in quantum theory, Brikhauser Verlag, 2000.