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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbar}$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t \to 0}$$\end{document} limit approaches the δ distribution. As such, our construction gives explicitly the full \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbar\to 0}$$\end{document} 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.