Feynman path integrals for mathematicians

Feynman path integral, or simply path integral, is a gadget whereby physicists calculate probability amplitudes for quantum transitions by "summing over all possible classical histories (or paths)". Naively, it makes no sense, but it is powerful and aesthetically pleasing.

The basic form of the path integral is as follows: (text taken and modified from path integral formulation)


 * $$Z = \int Dx\, e^{i\mathcal{S}[x]/\hbar}$$

where $$\mathcal{S}[x]=\int_0^T \mathrm{d}t L[x(t)]$$

is the classical action, x(t) is a path, i.e. a map from the interval [0,T] to space-time,$$\hbar$$ is the Planck constant divided by $$2\pi$$, and here is the rub: Dx is the measure for integration over the infinite-dimensional space of "all" paths.