Regarding Larry’s question about how close the photons have to be before they merge --
The solution to that problem comes from the fact that Eliezer’s experiment is (necessarily) simplifying things. I’m sure he’ll get to this in a later post so you might be better off waiting for a better explanation (or reading Feynman’s QED: The Strange Theory of Light and Matter, which I think is a fantastically clear explanation of this stuff.) But if you’re willing to put up with a poor explanation just to get it quicker...
In reality, you don’t have just one initial amplitude of a photon at exactly time T. To get the full solution, you have to add in the amplitude of the photon arriving a little earlier, or a little later, and with a little smaller or a little larger wavelength, and even travelling faster or slower or not in a straight line, and possibly interacting with some stray electron along the way, and so on for a ridiculously intractable set of complications. Each variation or interaction shows up as a small multiplier to your initial amplitude.
But fortunately, most of these interactions cancel out over long distances or long times, just like the case of the two photons hitting opposite detectors, so in this experiment you can treat the photon as just having arrived at a certain time and you’ll get very close to the right answer.
Or in other words—easier to visualize but perhaps misleading—the amplitude of the photons is “smeared out” a little in space and time, so it’s not too hard to get them to overlap enough for the experiment to work.
Regarding Larry’s question about how close the photons have to be before they merge --
The solution to that problem comes from the fact that Eliezer’s experiment is (necessarily) simplifying things. I’m sure he’ll get to this in a later post so you might be better off waiting for a better explanation (or reading Feynman’s QED: The Strange Theory of Light and Matter, which I think is a fantastically clear explanation of this stuff.) But if you’re willing to put up with a poor explanation just to get it quicker...
In reality, you don’t have just one initial amplitude of a photon at exactly time T. To get the full solution, you have to add in the amplitude of the photon arriving a little earlier, or a little later, and with a little smaller or a little larger wavelength, and even travelling faster or slower or not in a straight line, and possibly interacting with some stray electron along the way, and so on for a ridiculously intractable set of complications. Each variation or interaction shows up as a small multiplier to your initial amplitude.
But fortunately, most of these interactions cancel out over long distances or long times, just like the case of the two photons hitting opposite detectors, so in this experiment you can treat the photon as just having arrived at a certain time and you’ll get very close to the right answer.
Or in other words—easier to visualize but perhaps misleading—the amplitude of the photons is “smeared out” a little in space and time, so it’s not too hard to get them to overlap enough for the experiment to work.
--Jeff