Functional integration

Warning As of August 21 2003, no completely rigorous definition has been made yet for functional integration in general.

Basically, functional integration is the functional analogue to integration over finite dimensional manifolds/measurable spaces.

For some kinds of functional integrals, if $\int {\mathcal {D}}\phi F[\phi ]\geq 0$ whenever $\phi \geq 0$, 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 guarentee this will give a good approximation or even converge.

Even simple Gaussian integrals like $\int D\phi e^{i\int dt{\frac {dx}{dt}}(t)^{2}}$ where $\phi :\mathbb {R} \rightarrow \mathbb {R}$ need renormalization to make sense and only ratios of such integrals can be defined in an invariant manner.