The formal path integral and quantum mechanics

TL;DR

This paper develops a formal Feynman path integral framework for quantum mechanics on Riemannian manifolds, demonstrating invariance under coordinate changes and conditions for divergence cancellation, thereby connecting path integrals with quantum evolution laws.

Contribution

It provides a rigorous diagrammatic construction of the path integral for particles on manifolds, including divergence analysis and invariance properties, extending quantum mechanics foundations.

Findings

Formal path integral is invariant under volume-preserving coordinate changes.

Divergences cancel for inhomogeneous-quadratic Lagrangians with constant determinant matrices.

Constructs a Feynman-diagrammatic path integral for particles in electromagnetic fields on manifolds.

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.