What is the right time for path integrals?

The Feynman path integral formulation of quantum mechanics has proven to be a powerful tool for calculations in matter optics. It is natural to introduce the path integral in the context of Young's double slit experiment for matter waves as Feynman did,\footnote{R. P. Feynman, Rev. Mod. Phys. \textbf{20}, 367-387 (1948).} perhaps after discussing the analogous situation for optics. While intuitive, this approach can lead to a \textit{pedagogical} misrepresentation of the theory, namely in the phase accumulated along single free-particle trajectories. How is the use of the accumulated phase, $2\pi L/\lambda_{dB} $, along a path of length $L$ justified? The free-particle action gives a phase that differs by a factor of two. The guiding principle that interference occurs only for two paths that are indistinguishable from one another provides a correct solution: interfering paths must originate and terminate at equal times. We will present several simple thought experiments to illustrate incorrect and correct methods for determining phase shifts.