Configuration-space path integral ($K(q_f,q_i)={\int }_{q_i}^{q_f}\mathcal{D}q{e}^{iS[q]}$)

A configuration-space path integral integrates histories of the coordinates with weight $e^{iS}$. Fixed endpoint conditions specify a transition amplitude. For a quadratic kinetic energy it follows by Gaussian momentum integration from the phase-space path integral. Time-slicing fixes the normalization, and a regulator gives meaning to the otherwise formal functional measure.