Copied verbatim from my Facebook:
Let me try to write a clear derivation of the correlation of the two-channel optical Bell test, WITHOUT using quantum mechanics. I will use only probability theory.
If I succeed, this means the 2022 Nobel Prize in Physics was awarded in error. That, in fact, the work on which it was based should be retracted as uncorrectably erroneous.
Before I present my derivation—it is rather simple—let me make two observations that themselves should settle the matter.
The first is that consistency of mathematics requires that any two methods MUST reach the same result. Therefore, if I try to solve for the Bell test correlation using probability theory and reach a result OTHER than the so-called "quantum" correlation (for instance "CHSH inequalities"), obviously I have made a mathematical error. That this has not occurred to physicists is astounding.
The second observation is that I can replace polarized photons with equatorially striped golf balls, the light source with a zany two-ball putter, and polarizing beam splitters with weird miniature golf obstacles. The problem remains exactly equivalent and yet there is nothing "quantum" about it. Even the quantum mechanics of the problem is identical. To argue that you cannot use quantum mechanics because the objects are "macroscopic" is nothing less than to argue that very small objects are magical and have a special magical incantation tongue.
Now, here is the experiment.
We have a light source. It emits two light pulses, which we will call "photons", but about which we will assume nothing except that they have either horizontal or vertical polarization. The two photons will have opposite polarization, but which is horizontal and which vertical is decided at random.
Each photon will impinge upon a polarizing beam splitter (PBS). A PBS set to an angle x behaves as follows. If the impinging photon is polarized horizontally, then, with probability cos² x, a photodetector is triggered whose value is +1 for the correlation calculation. With probability sin² x, a photodetector whose value is -1 is triggered. If the photon is vertically polarized, the probabilities are reversed. The two PBS have angular settings a and b, respectively.
These photodetector values make the correlation equal to the covariance and thus simplify the calculation.
Let us solve first the simplest case, where b = 0 and a is some general angle, say a = a'. In that case it is ALWAYS the +1 photodetector that is triggered if the b photon is horizontally polarized, but the -1 photodetector if the b photon is vertically polarized. Thus we can ignore the b PBS's photodetectors in our calculations. Their value contains no information not already available.
Using some probability theory notation, here is a small chart of probabilities and conditional probabilities. Let t equal the value of the a PBS's photodetectors and k the polarization of the impinging photon.
P(t = +1 | k = horiz) = cos² a'
P(t = -1 | k = vert) = cos² a'
P(t = +1 | k = vert) = sin² a'
P(t = -1 | k = horiz) = sin² a'
Thus joint probabilities can be calculated, as in this example:
= P(t = +1 | k = horiz) P(k= horiz)
The following table can be constructed.
P(t = +1 and k = horiz) = ½ cos² a'
P(t = -1 and k = vert) = ½ cos² a'
P(t = +1 and k = vert) = ½ sin² a'
P(t = -1 and k = horiz) = ½ sin² a'
The covariance and thus the correlation can be calculated from this table. Keep in mind that k = horiz corresponds to a -1 value from the b PBS and k = vert corresponds to a +1 value. Thus
= (+1)(-1)P(t = +1 and k = horiz)
+ (-1)(+1)P(t = -1 and k = vert)
+ (+1)(+1)P(t = +1 and k = vert)
+ (-1)(-1)P(t = -1 and k = horiz)
where the last step is an identity ("double-angle relation") that can be found in the CRC Handbook of Mathematical Sciences, among other places.
This result has the desired form:
The sole remaining difficulty is how to extend the result to values of b other than zero.
Fortunately, this is not difficult at all. Notice that -cos 2(a - b) is invariant under in-unison rotation of the angles a and b. That is, one can add ANY angle c to both a and b, without changing the value of the correlation. Thus the solution derived above is, in fact, the general one:
How is it that physicists have missed this fact and not seen that it leads to a solution of the Bell test WITHOUT quantum mechanics? You tell me. I have no explanation other than that, as a profession, they are no good at physics. I was able to see it because I am NOT a physicist, but a disabled computer programmer with a digital signal processing diploma. But you tell me, if you have a better explanation.
In any case, this means that all the stuff about "entanglement" and "non-locality" is fit for nothing but retraction. Every existing physics text that deals with this stuff should be withdrawn and pulped.
