The formal path integral and quantum mechanics

Abstract

Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.

arXiv comment33 pages, many TikZ diagrams, submitted to _Journal of Mathematical Physics_

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Cited by in corpus (4)

• Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace
• Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics
• On the coordinate (in)dependence of the formal path integral
• Quantizations of Momentum Maps and G-Systems