Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics

TL;DR

This paper develops a Feynman diagram approach to nonrelativistic quantum mechanics, providing a rigorous link between classical paths, quantum evolution, and asymptotic expansions in ar, thereby justifying the heuristic Feynman path integral expansion.

Contribution

It introduces a formal diagrammatic series for the quantum propagator along classical paths, proving its satisfaction of Schrb6dinger's equation and its asymptotic relation to classical mechanics.

Findings

The diagrammatic series converges and satisfies Schrb6dinger's equation.

The approach explicitly describes the arb0b0b0b0b0b0b0b0b0b0b0asymptotics of the quantum propagator.

The construction justifies the heuristic Feynman path integral expansion.

Abstract

We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path \gamma connecting points q_0 and q_1 in time t, we define a formal power series V_\gamma(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_\gamma) satisfies Schr\"odinger's equation, and explain in what sense the t\to 0 limit approaches the \delta distribution. As such, our construction gives explicitly the full \hbar\to 0 asymptotics of the fundamental solution to Schr\"odinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams.