Short-time quantum propagator and Bohmian trajectories

We begin by giving correct expressions for the short-time action following the work Makri-Miller. We use these estimates to derive an accurate expression modulo Δt2 for the quantum propagator and we show that the quantum potential is negligible modulo Δt2 for a point source, thus justifying an unfortunately largely ignored observation of Holland made twenty years ago. We finally prove that this implies that the quantum motion is classical for very short times.