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## Main Question or Discussion Point

Feynman's first topic in his second lecture on QED is the nature by which light reflects off of a mirror. We work in ##\mathbb{R}^2##. Suppose we have a light source sitting at ##(-1,1)## and a photomultiplier sitting at ##(1,1)##, with a mirror along the x-axis. We also place a block between the source and photomultiplier. Let's say it extends from ##(0,+\infty)## to ##(0,1/2)##.

Suppose the following event occurs: the photomultiplier clicks. We want to find the expected path of the photon (I assume we'd want the path of minimal mean square). Feynman's heuristic argument, roughly stated, is that if a path allows many nearby paths of drastic time difference, these wave functions will cancel out. Since the path of minimal length (or minimal time elapse) will have nearby paths of similar length, the wave functions will constructively add. This heuristic leads us to conclude that the expected path will be approximately the path of shortest length.

In this example, Feynman only considers straight line paths that reflect upon the mirror. Later, he says that light doesn't have to follow straight paths, that we could apply the above over some set of reasonable paths. So if we revisit the example, with the same event (photomultiplier clicks) but with more (reasonable) sample paths, we should arrive at the same conclusion. However, the path connecting the source to the bottom of the block at ##(0,1/2)##, and then to the photomultiplier is clearly the path of shortest distance. Furthermore, the heuristic of nearby paths having similar distance is even more valid than the reflected path.

Following the same outline as Feynman, I'm left to conclude that QED predicts that the "classical" path of light would be to travel to the tip of the block, and then to the photomultiplier, and not to be reflected by the mirror.

So, something is wrong here, and I'm not sure what.

Suppose the following event occurs: the photomultiplier clicks. We want to find the expected path of the photon (I assume we'd want the path of minimal mean square). Feynman's heuristic argument, roughly stated, is that if a path allows many nearby paths of drastic time difference, these wave functions will cancel out. Since the path of minimal length (or minimal time elapse) will have nearby paths of similar length, the wave functions will constructively add. This heuristic leads us to conclude that the expected path will be approximately the path of shortest length.

In this example, Feynman only considers straight line paths that reflect upon the mirror. Later, he says that light doesn't have to follow straight paths, that we could apply the above over some set of reasonable paths. So if we revisit the example, with the same event (photomultiplier clicks) but with more (reasonable) sample paths, we should arrive at the same conclusion. However, the path connecting the source to the bottom of the block at ##(0,1/2)##, and then to the photomultiplier is clearly the path of shortest distance. Furthermore, the heuristic of nearby paths having similar distance is even more valid than the reflected path.

Following the same outline as Feynman, I'm left to conclude that QED predicts that the "classical" path of light would be to travel to the tip of the block, and then to the photomultiplier, and not to be reflected by the mirror.

So, something is wrong here, and I'm not sure what.