  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.
