Using the physics laid out in this book, the answer to this riddle is that photons never strike the mirror. The pilot wave that accompanies every photon acts like a cushion, and it is off of this cushion that the photon bounces.
Photon with pilot wave striking a reflective surface of atoms
While photons are tiny, the pilot waves surrounding photons are big relative to atoms. They can easily even out a tolerably smooth surface without upsetting their host particle. In this way, each photon sees a perfectly smooth cushion. It bounces off of this, unaffected by any underlying irregularity in the surface of the mirror.
The phenomenon of reflection can in this way be seen as supporting evidence for the existence of pilot waves.
Polarization through reflection
Light reflecting off a mirror at an angle will end up polarized. This means that every photon must have some sort of axis along which it is oriented. Otherwise, no polarization could be possible.
Combining this fact with what we have so far concluded about photons, we must further conclude that the pilot wave has the ability to orient photons when compressed against a reflecting surface.
The simplest possible explanation for this is that photons are like little sticks. When hit against the compressed cushions of their pilot waves, they end up aligning in parallel with the underlying surface.
Note that the orientation of the aligned photons is random when polarized in this way. On average, there are just as many photons oriented left to right as right to left.
Photons, passing from left to right, being polarized on reflection
This fits well with what we have thus far concluded about the photon, namely that it is an assembly corresponding to an electron and a positron. Assuming that the arrangements of particle quanta in electrons and positrons are inherited directly from photons, we end up with a two orb model of the photon, making them in essence tiny sticks.
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