  We study finite-dimensional integrals in a way that elucidates the
mathematical meaning behind the formal manipulations of path integrals
occurring in quantum field theory. This involves a proper understanding of how
Wick's theorem allows one to evaluate integrals perturbatively, i.e., as a
series expansion in a formal parameter irrespective of convergence properties.
We establish invariance properties of such a Wick expansion under coordinate
changes and the action of a Lie group of symmetries, and we use this to study
essential features of path integral manipulations, including coordinate
changes, Ward identities, Schwinger-Dyson equations, Faddeev-Popov
gauge-fixing, and eliminating fields by their equation of motion. We also
discuss the asymptotic nature of the Wick expansion and the implications this
has for defining path integrals perturbatively and nonperturbatively.
