Meaning of an individual "Feynman path"

In this article we give an operational meaning to an individual "Feynman path." In other words, we describe a process of dense measurements, made in temporal sequence, which check whether the particle moves along any given trajectory in space-time. We show that in this process the two assumptions of the space-time formulation of quantum mechanics, are realized: (a) The weight that the particle moves along a trajectory that has been checked by this process is the same for all trajectories, and in fact, we show that the particle follows, with probability 1, the trajectory that is being checked. (b) A phase is systematically accumulated, so that, at the end of this process, the state is multiplied by the familiar factor $\exp \left[\right(\frac{i}{\hslash }\left)\int L\mathrm{dt}\right]$. As an immediate extension of the above formalism, we suggest a setup that measures the relative phase between any two trajectories. Finally, our approach points toward the possibility of extending the Feynman formalism in order to cover more general Hamiltonians.