QM/QC : Busting Bell : Part Two
Before Bell, QM already knew that if you measure certain apparently rotational properties of a pair of entangled particles from complementary observational angles, particle a comes out +1 and particle b of the entangled pair comes out -1.
QM also had discovered that if you're off on your angle, if two observations diverge from a complementary angle, the odds, statistically in multiple observations, of clearly obtaining a clear and correct +1 or -1, was tied to the cosine of the divergence in observational angle.
Bell's Theorem ostensibly proves that the classical mathematical odds for such coincidence of observational result does not equal cos θ ("cosine theta" where θ "theta" is the angle of divergence between observational angles for the two observations).
However, in this, Bell appears to assume integrality of dimension as a property of quantum mechanical entities.
What Bell is really proving in his original paper "On the Einstein Podolsky Rosen Paradox" is that no vector can be applied uniformly to a particle that could explain the observational anomaly.
In plain technomerican, that means: Let's say you spin a cue ball on some axis and then go about measuring its spin as Left or Right depending on which way it seems to be going as you look at it. You can give it trick spins with lots of English if you like: a very crude observation is going to be made and the result will be simply "as far as pure Left or Right is concerned, it's spinning more to the Left (or Right)."
There is no spin you could give to that cue ball that would cause your L-or-R determinations to coincide from different observational angles in the proportions such observations do coincide in quantum physics experiments.
Now suppose the particles we are observing really "look" like the image at the left (a Hubble telescope shot of Planetary Nebula K 4-55). K 4-55 is, for our purposes, infintely recursively subconstituent. It's wispy stuff made of wispy stuff, mostly containing far more space than matter by volume at any and every scale of resolution as far down on the scale as we can imagine. Yet it can be measured as one entity.
Observed in motion faintly blurred over a time scale vastly exceeding ours, K 4-55 would appear to occupy something roughly like a sphere. That is, over a few million or billion years, a three-dimensional sphere is occupied by K 4-55 fractally. Not only that,but viewed as a whole, certainly K-455, and perhaps your typical subatomic particle, possess only fractal membership in any dimension they occupy.
So does Bell's Inequality hold for spin measurments applied to entities that only fractally occupy the dimensional space they nominally and for the purpose of measurement are assumed to encompass? This is a formal way of asking if the results of the Stern-Gerlach experiment are classically surprising if a silver atom is more like a fried-egg-shaped cloud of gas in Brownian motion internally than like a cue ball.