Feynman-Diagrammatic Description of the Asymptotics of the Time Evolution Operator in Quantum Mechanics

Abstract

We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on ℝn, with magnetic and potential terms. In particular, for each classical path γ connecting points q0 and q1 in time t, we define a formal power series Vγ(t, q0, q1) in h{stroke}, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(Vγ) satisfies Schrödinger's equation, and explain in what sense the t → 0 limit approaches the δ distribution. As such, our construction gives explicitly the full h{stroke} → 0 asymptotics of the fundamental solution to Schrödinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams. © 2010 The Author(s).