How it works
Finally, we give a very crude view of what actually happens, how the                         whole thing really works, from what we now believe is the correct,                         quantum-dynamically accurate viewpoint, but of course only                         qualitatively described. In following the light from $A$ to $B$ in                         Fig. 26–3, we find that the light does not seem to be in                         the form of waves at all. Instead the rays seem to be made up of                         photons, and they actually produce clicks in a photon counter, if we                         are using one. The brightness of the light is proportional to the                         average number of photons that come in per second, and what we                         calculate is the chance that a photon gets from $A$ to $B$, say                         by hitting the mirror. The law for that chance is the following                         very strange one. Take any path and find the time for that path; then                         make a complex number, or draw a little complex vector, $\rho                         e^{i\theta}$, whose angle $\theta$ is proportional to the                             time. The number of turns per second is the frequency of the                         light. Now take another path; it has, for instance, a different time,                         so the vector for it is turned through a different angle—the angle                         being always proportional to the time. Take all the available                         paths and add on a little vector for each one; then the answer is that                         the chance of arrival of the photon is proportional to the square of                         the length of the final vector, from the beginning to the end! Now let us show how this implies the principle of least time for a mirror. We consider all rays, all                         possible paths $ADB$, $AEB$, $ACB$, etc., in Fig. 26–3. The                         path $ADB$ makes a certain small contribution, but the next path, $AEB$,                         takes a quite different time, so its angle $\theta$ is quite different.                         Let us say that point $C$ corresponds to minimum time, where if we                         change the paths the times do not change. So for awhile the times do                         change, and then they begin to change less and less as we get near                         point $C$ (Fig. 26–14). So the arrows which we have to add are                         coming almost exactly at the same angle for awhile near $C$, and then                         gradually the time begins to increase again, and the phases go around                         the other way, and so on. Eventually, we have quite a tight knot. The                         total probability is the distance from one end to the other, squared.                         Almost all of that accumulated probability occurs in the region                             where all the arrows are in the same direction (or in the same phase).                         All the contributions from the paths which have very different                         times as we change the path, cancel themselves out by pointing in                         different directions. That is why, if we hide the extreme parts of the                         mirror, it still reflects almost exactly the same, because all we did                         was to take out a piece of the diagram inside the spiral ends, and that                         makes only a very small change in the light. So this is the relationship                         between the ultimate picture of photons with a probability of arrival                         depending on an accumulation of arrows, and the principle of least time.                         