Why not just take the equation at face value? Say that the object you're dealing with actually is this complex-valued self-interacting field that you get entangled with, and voila, no ontology issue. :p
You didn't answer the question. In an ontological sense, is a particle "really" definitely spinning either up or down even when the state vector says its both?
A theta greater than one corresponds to an angle greater than 60 or so degrees. (There are 2pi, not 2 radians in a circle.) Importantly, it means that S > L, which means your stated conclusion is wrong.
Your equations suggest that the shadow can never be faster than the hand, which is wrong. The first form of the equation also suggests that moving the screen further away won't make a difference, which is also wrong.
If my algebra is correct, then the equation you're actually looking for is...
That's because you're only interested in the position of the light source. I'm interested in how the FTL shadow looks like frm other positions, i.e. is it actually possible to transmit FTL events into someone else's lightcone?
Doing it with a point source and a planar screen means that the ends of the screen are further away from the source than the middle, which makes the calculations more difficult.:p
Doing the whole problem with a point source projected onto a curved screen makes the calculation easier, because all the points of the screen are equally close to the source. Doing it with a planar light source and a flat screen doesn't produce a shadow with the require properties.
So the lower beam splitter somehow magically knows that the situation is symmetrical?
Photons, being oscllating electromagnetic fields, have rotational phase. If you overlap two photons of the same frequency and amplitude 180 degrees out of phase with each other, they disappear completely, as...
Of course I won't take anyone's word for it. I'll take anyone's demonstration of it, though. :D
(I don't lack the ability to calculate it, just the time.)
It might look greater than c, but since the light itself is limited to c, and the shadow is a signal "in" the light, then there's something...