In the dual-slit experiment, can the time it takes for a photon to arrive reveal the path it took? It seems like this experiment is possible:


*

*Fire a photon, and start a timer

*The photon travels through the slit(s)

*The photon strikes a plate -- the timer is stopped.


Based on where on the plate the photon struck, the time it took, and the speed of light, it seems possible to deduce through which slit the photon passed.
Or, would the times correspond with a direct line from the source to destination?
 A: It is possible. But it doesn't change anything about the "we can't know through which slit it went and still get an interference pattern" issue. (Surprise!)
In order to do this you need photons that can be timed very precisely, i.e. $\Delta t$ needs to be small: smaller than $c/d$ where $d$ is the slit seperation. According to Heisenberg uncertainty, this means that the energy uncertainty must be big for $\Delta E\cdot\Delta t>\hbar$ to remain satisfied. If $\Delta E$ is big, then the wavelength uncertainty will also be big; namely
$$
  \Delta\lambda \approx \frac{\Delta\nu}{c}
  = \frac{\Delta E}{2\pi\hbar c}
  > \frac{d}{2\pi}
$$
But if the wavelength uncertainty is in the order of the slit seperation, there won't be a proper interference pattern as this requires the light to be coherent on the length scales of the experiment!
A: To explain in a more simple way, to be more easily understood by non-physicists: Two waves can obviously only interfere if they arrive at the same time. If the photon wave packet (which is the interval in which the photon might be found when measured) is so narrow that the wave packets going through the left and the right slit don't overlap, then for this reason no interference is possible. However if they do overlap, then in the overlapping region you cannot know from which wave packet you measured the photon, that is, you cannot know which way the photon went.
