Functional integration techniques in physics were pioneered by Richard Feynman, who successfully applied his "path integral" to problems in quantum mechanics and quantum field theory, as well as classical and quantum statistical mechanics.
As of August 2003, no rigorous definition of functional integration has been given which is applicable to all instances where it arises heuristically. Another way to say this is that important problems whose solutions are obtained by heuristic methods involving functional integrals have eluded formulation in terms of any of the existing rigorous definitions of functional integration.
The problem of functional integration is to make sense of expressions such as
When , a functional measure might be possible and we have a Wiener integral. Otherwise, we might have something which looks very fishy, like the use of summing of nonconvergent infinite series and the use of infinitesimals before the introduction of concepts like ε-δ, uniform convergence, etc..
Functional integrals over manifolds are sometimes approximated by a lattice, but there is no guarantee this will give a good approximation or even converge. This is related to statistical field theory. The so-called renormalization group methods allow a rigorous continuum limit if the lattice theory has an ultraviolet fixed point. In fact, the direct naïve approximation by a lattice can have its pitfalls, because, for example, the fermion doubling problem, among other things.
Even simple Gaussian integrals like where need renormalization to make sense and only ratios of such integrals can be defined in an invariant manner.Functional integration in probability theory
The Feynman integral
where is "Lebesgue measure" on an infinite dimensional space where φ is valued.