# Feynman path integrals for mathematicians

${\displaystyle Z=\int Dx\,e^{i{\mathcal {S}}[x]/\hbar }}$
where  ${\displaystyle {\mathcal {S}}[x]=\int _{0}^{T}\mathrm {d} tL[x(t)]}$
is the classical action, x(t) is a path, i.e. a map from the interval [0,T] to space-time,${\displaystyle \hbar }$ is the Planck constant divided by ${\displaystyle 2\pi }$, and here is the rub: Dx is the measure for integration over the infinite-dimensional space of "all" paths.