Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics
arXiv:1003.1156 · doi:10.1007/s11005-010-0424-2
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 γconnecting points q_0 and q_1 in time t, we define a formal power series V_γ(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_γ) satisfies Schrödinger's equation, and explain in what sense the t\to 0 limit approaches the δdistribution. As such, our construction gives explicitly the full \hbar\to 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.
21 pages. Many diagrams drawn in TikZ. To appear in Letters in Mathematical Physics