Abstract
We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on \({\mathbb{R}^n}\), with magnetic and potential terms. In particular, for each classical path γ connecting 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 δ distribution. 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.
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Acknowledgements
This project was suggested by N. Reshetikhin, who provided support and suggestions throughout all stages of it. K. Datchev, C. Schommer-Pries, G. Thompson, and I. Ventura provided valuable discussions. I would like to also thank the anonymous referee for alerting me to the work of H. Kleinert and collaborators. I am grateful to Aarhus University for the hospitality. This work is supported by NSF grant DMS-0901431.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Johnson-Freyd, T. Feynman-Diagrammatic Description of the Asymptotics of the Time Evolution Operator in Quantum Mechanics. Lett Math Phys 94, 123–149 (2010). https://doi.org/10.1007/s11005-010-0424-2
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DOI: https://doi.org/10.1007/s11005-010-0424-2