Propagator for free point-particle

For the free point particle the action is
\[S = \int_{t_{I}}^{t_{F}} dt \, \left[-\dot{q} \cdot p + \frac{1}{2m}p^{2} \right] \]
We will solve the equations of motion and determine the integration constants in terms of the boundary coordinates:
\[\frac{d^{2} q_{cl}}{d t^{2}} = 0 \qquad q_{cl} (t) = q_{I} + \frac{q_{F} - q_{I}}{\Delta t} \left(t - t_{I}\right) \qquad S_{cl} = -\frac{m}{2} \frac{\left(q_{F} - q_{I}\right)^{2}}{\Delta t} \qquad \Delta t \equiv t_{F} - t_{I}\]
The Van Vleck "matrix" reads
\[\mathcal{V} \equiv -i\frac{\partial^{2} S_{cl}}{\partial q_{F} \partial q_{I}} = -i\frac{m}{\Delta t}\]
This leads to a transition amplitude:
\[\left\langle q_{F}, \, t_{F} | q_{I}, \, t_{I} \right\rangle = \sqrt{-i\frac{m}{\Delta t}} \, \exp{\left(\frac{i m}{2} \frac{\left(q_{F} - q_{I}\right)^{2}}{\Delta t}\right)}\]
Note that when $\Delta t \rightarrow 0$ we obtain the usual orthonormality condition of the position basis,
\[\left\langle q_{F}, \, t_{I} | q_{I}, \, t_{I} \right\rangle = \sqrt{2 \pi} \delta \left(q_{F} - q_{I}\right)\]