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

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 V_γ(t, q₀, q₁) in h{stroke}, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_γ) 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.