A note on the short-time quantum propagator

Abstract

In the Feynman formalism of quantum mechanics one encounters a postulate, namely, that the propagator in an infinitesimal time-interval is the classical wave function. This postulate, which was later studied thoroughly by Holland, was recently highlighted by using the improved Makri-Miller propagator. The present note, whose conclusion is in agreement with that recent achievement, demonstrates that the Heisenberg picture of quantum mechanics invariably includes the Feynman postulate and is able to yield a proof for it. In other words, by starting out from the Heisenberg picture, it is proved that when terms of second-order in time can be neglected, the dynamics of a system is classical.