A note on the short-time quantum propagator

TL;DR

This paper demonstrates that the Feynman propagator postulate in quantum mechanics can be derived from the Heisenberg picture, showing that neglecting second-order terms yields classical dynamics.

Contribution

It provides a proof that the Feynman postulate naturally arises within the Heisenberg picture of quantum mechanics.

Findings

Heisenberg picture includes the Feynman propagator postulate

Neglecting second-order terms leads to classical dynamics

Supports the classical-quantum correspondence in short-time evolution

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.