Generalizing Functional Integration for Piecewise-Smooth Manifolds

[|Psi>]

I'm intrigued by the fact that apparently no general theory of functional integration has been developed - something along the lines of Riemann, Cauchy, and Lebesgue. Feynman developed an approach to evaluating functional integrals for paths in spacetime - but I'm wondering whether it is possible to generally decompose the set of all piecewise-differentiable paths into infinitesimal line segments - and whether our choice of decomposition affects the value of the integral. Might the axiom of choice lead to complications? And how might one generalize this to general functions defined on piecewise-smooth manifolds in arbitrary dimension and with arbitrary signature?

 

[Kummer]

Well in complex analysis the notion of "a path integral" existed. Cauchy new about it.

 

[Klaus_Hoffmann]

Although is not the case of knowing exact functional integral formulae, there are many results that can be generalized to Infinite-dimensional integrals as for example Poisson sum-formula.

$$\sum_{m=-\infty}^{\infty}F[x_{0}(t)+m\delta (t-t')]= \int \mathcal D[x(t)]\sum_{m=-\infty}^{\infty}exp(2\pi m\int_{a}^{b} dt x(t))F[x(t)+x_{0} (t)]$$