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


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 


quantum mechanics Feynman diagrams formal integrals path integrals semiclassical asymptotics 



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.


  1. 1.
    Albeverio S., Höegh-Krohn R.: Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics, I. Invent. Math. 40(1), 59–106 (1977)zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    DeWitt-Morette C.: The semiclassical expansion. Ann. Phys. 97(2), 367–399 (1976)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Duru I.H., Kleinert H.: Quantum mechanics of H-atom from path integrals. Fortschr. Phys. 30(8), 401–435 (1982)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Evans, L.C., Zworski, M.: Lectures on semiclassical analysis. Available at (2007)
  5. 5.
    Feynman R.P.: Space–time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20(2), 367–387 (1948)CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. International Series in Pure and Applied Physics. McGraw-Hill, New York (1965)Google Scholar
  7. 7.
    Johnson, G.W., Lapidus, M.L.: The Feynman Integral and Feynman’s Operational Calculus. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York. Oxford Science Publications (2000)Google Scholar
  8. 8.
    Johnson-Freyd, T.: The formal path integral and quantum mechanics. J. Math. Phys. Available at or arXiv:1004.4305 [math-ph] (in press)
  9. 9.
    Johnson-Freyd, T., Schommer-Pries, C.: Critical points on a fiber bundle. Online forum discussion at (2009)
  10. 10.
    Klauder J.R., Daubechies I.: Quantum mechanical path integrals with Wiener measures for all polynomial Hamiltonians. Phys. Rev. Lett. 52(14), 1161–1164 (1984)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Kleinert H., Chervyakov A.: Integrals over products of distributions from perturbation expansions of path integrals in curved space. Int. J. Mod. Phys. A 17(15), 2019–2050 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Manuel C., Tarrach R.: Perturbative renormalization in quantum mechanics. Phys. Lett. B 328, 113–118 (1994)CrossRefADSGoogle Scholar
  13. 13.
    Milnor, J.: Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton (1963)Google Scholar
  14. 14.
    Polyak, M.: Feynman diagrams for pedestrians and mathematicians. In: Lyubich, M., Takhtajan, L.A. (eds.) Graphs and Patterns in Mathematics and Theoretical Physics. Proc. Sympos. Pure Math., vol. 73, pp. 15–42. American Mathematical Society, Providence (2005)Google Scholar
  15. 15.
    Reshetikhin, N.: Lectures on Quantization of Gauge Systems. ArXiv e-prints, arXiv: 1008.1411 [math-ph] (2010)Google Scholar
  16. 16.
    Takhtajan, L.A.: Quantum Mechanics for Mathematicians. American Mathematical Society, Providence (2008)Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA

Personalised recommendations