## 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 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 h{stroke}, 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 → 0 limit approaches the δ distribution. As such, our construction gives explicitly the full h{stroke} → 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.

Original language | English (US) |
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Pages (from-to) | 123-149 |

Number of pages | 27 |

Journal | Letters in Mathematical Physics |

Volume | 94 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2010 |

## Keywords

- Feynman diagrams
- formal integrals
- path integrals
- quantum mechanics
- semiclassical asymptotics

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics