> So, is choosing the path that takes the shortest time like a standing wave?
Note that photons don't take the shortest path; they take a variationally stationary path (not necessarily the shortest one). My suspicion is that this is because all the possible photon states are coherent along and immediately nearby that path, whereas if the path is not variationally stationary then it interferes with nearby paths. I've done very little variationally physics, however, so take this with a grain of salt.
My thought is that, on a variationally stationary path, all nearby paths are of the same length. So if you have a photon leaving a point along many possible paths, near the variationally stationary path the waveforms of the photon at each path will be lined up, interfering constructively. On the other hand, if the path is not variationally stationary, the waveform will drift out of phase on nearby paths (because they have different lengths) until the waves interfere with each other.
I don't know enough about quantum FFTs to know if there is an analogy. However, I'm taking Aaronson's class next semester so I'll be able to tell you in a few months ;)
Note that photons don't take the shortest path; they take a variationally stationary path (not necessarily the shortest one). My suspicion is that this is because all the possible photon states are coherent along and immediately nearby that path, whereas if the path is not variationally stationary then it interferes with nearby paths. I've done very little variationally physics, however, so take this with a grain of salt.