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 R(n), with magnetic and potential terms. In particular, for each classical path. connecting points q(0) and q(1) in time t, we define a formal power series V(gamma) ( t, q(0), q(1)) in h, given combinatorially by a sum of diagrams that each represent finite- dimensional convergent integrals. We prove that exp( V(gamma)) satisfies Schrodinger's equation, and explain in what sense the t -> 0 limit approaches the delta distribution. As such, our construction gives explicitly the full h -> 0 asymptotics of the fundamental solution to Schrodinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams.