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Letters in Mathematical Physics

, Volume 94, Issue 2, pp 123–149 | Cite as

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

  • Theo Johnson-FreydEmail author
Open Access
Article

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.

Mathematics Subject Classification (2010)

81T18 81S40 81Q15 

Keywords

quantum mechanics Feynman diagrams formal integrals path integrals semiclassical asymptotics 

Notes

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.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA

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