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.

A classical path joining prescribed endpoints is a stationary point of the action, satisfying the Euler-Lagrange equation and the endpoint boundary conditions. Expanding the path integral about each such path separates its classical phase from its quadratic fluctuation determinant. There may be multiple classical paths or none within a chosen real saddle class; the semiclassical propagator must respect that saddle structure rather than assuming a unique trajectory for every potential.

For $H=p^2/(2m)+V(q)$, time slicing gives a Euclidean path integral over paths with the specified endpoints and action $S_E=\int [m{\dot{q}}^2/2+V(q)]dt$. Each step contributes normalization $(m/(2\pi \hslash \delta ){)}^{1/2}$. This normalization fixes the units and endpoint factors of the kernel, which differ from amplitudes between normalized localized wavepackets.