The'small print' in Bell's theorem.
In 1935, Einstein, Podolsky and Rosen (EPR) published their famous 'thought experiment'. It revealed certain inconsistencies of Quantum Mechanics (QM). In short, it argued that QM is not correct in stating that particles, or photons, have no definite state until measured.
The argument (for photons): the angle of polarization is fixed at emission but unknown (EPR) or, not fixed at that stage and therefore unknown (QM). Experiments showed a difference in favour for QM, but made it necessary to assume a mysterious co-operation between the photons, and that regardless of the distance separating them.
Only some form of telepathy, and the ability to act upon the 'signals', explains the results. Bell proved, some thirty years later, that no ordinary reality could adequately explain the quantum results. Experiments by Aspect confirmed this. He not only showed the telepathic character of the results, but also that the transmission velocity of the 'signals' is greater than the velocity of light.
This is not, and can not be physics. Not when the explanation can only be found in the super natural. It is therefore very difficult to accept for people who are convinced that the fundamentals of physics are simple, natural and understandable. There is another way of looking at this, it can also be considered as proof that there is something wrong with the arguments and interpretation.
The Aspect results are accepted. That leaves Bell's theorem. There is nothing wrong with the mathematics, but that theorem is of course no better than the assumptions it is based on. The fact that certain quantum properties are dual in character forms the basis for this theorem: up or down, vertical (V) or horizontal (H) etc.
It seems therefore reasonable to assume a fifty-fifty distribution for the probabilities, however, measuring V and H, the instrument also has to stop and pass the light at that rate.
I have been unable to find a basis for that assumption, but to have at least some idea of what is going on, a thought experiment with a reasonable model is worked out. It shows, that when light with the H property is stopped, light with the V property is passed at a rate that is clearly different from the fifty-fifty assumption.
Light must have a property that gives it, at right angles to the line of travel, a definite direction. For the purpose of this argument assume this property as a length -d-. An easier to follow simulation avoids then the difficulties with the measurement of light.
To simulate the measuring of the polarization of light, take for -d- a thin needle 100-mm long. For the two 'filters', take two pieces of clear plastic with parallel lines at ~70-mm centres (100 / SQR 2)).
The figure, BELL-I, shows one of the 'filters'. With d' = d / sqr 2 . For an angle -a- greater than 45° there is no pass, and for -a- less than 45° always a pass. This is the assumed fifty-fifty distribution of the probabilities.
But, when the lines are not to be touched, we have instead of a pass-rate of fifty percent, only an average of twenty-five percent. For an angle of e.g. 30° degrees, p equals 0.29 in stead of the 0.50 assumed for the Bell theorem.
The dissidents believe that emission fixes the angle of polarization. In the simulation, that means checking passes of the needle with two identical filters put on top of each other.
Touching the lines on the plastic means no-pass and vice versa. In the figure BELL-II, the dotted lines show the Bell assumption for the probabilities and the triangular shaped curves the probabilities for the simulation.
That probability distribution, for the angles from -90° to +90° shown, is definitely not proportional with the enclosed angle.
The next figure shows the angle b = 2 . a, (a as in Bell II). From the figure it is clear that with the Bell assumption the probability p is equal to that shown. For 'Bell' a 30° angle -a- gives a probability p of 120/180 or 0.67 and for the simulation (cos 30) 2 or 0.75 ( equal to Malus)!
Bell-III shows that this is the general result for the simulation. The angle -b- is twice the angle -a-, of the previous figure.
The area - I -, is not a triangle, but a circle segment with radius -d-, the difference is small and varies, but is never more than a few percentage points of the QM value.
This simulation shows clearly that the probability distribution is not specific for quantum mechanics. The needle and the plastic 'filters' proof that. Ordinary reality can produce the QM results. Common every day nature is sufficient, without recourse to telepathy or other extra-ordinary phenomena.
To maintain that light behaves in exactly the same way may go too far. Comparing the quantity of polarized light with the quantity presented, will shed some light on this. The simulation suggests that the polarized photons number about 25%, in stead of the, assumed, 50% of the light before filtering.
Whether or not light behaves in this way is not essential for the argument that if EPR results are obtainable in the real world, it at least casts some doubt on the necessity to accept the extreme consequences of the extra-ordinary QM world.