Joy Christian wrote:Thanks, Jay. As you have asked, let me address the above claim by Stephen Parrott of my supposed mistake in some detail. To avoid any confusion, let me note that Jay’s equation (1) is my equation (D3) from the Appendix D of this paper:
https://arxiv.org/pdf/1501.03393v6.pdf.
In Gill’s notation, my equations (D3) says that the bounds on the CHSH sum of averages,
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) …. (1) ,
is in fact -4 to +4, and not -2 to +2 as claimed by the adherents of Bell’s theorem. This is not a mistake (or confusion, or misunderstanding) on my part at all, but an explicit claim of mine. Here, for simplicity, I am ignoring the fact that the actual local-realistic physical bounds on the above CHSH sum are in fact -2sqrt(2) to +2sqrt(2), provided we do not ignore the geometry and topology of the physical space we live in. Thus, it is my claim that the experimentally observed bounds of -2sqrt(2) to +2sqrt(2) “violating” the Bell-imposed limits of -2 to +2 has nothing to do with quantum entanglement, or non-locality, or non-reality, or lack of determinism, but has to do with the geometry and topology of the 3-sphere. Again, this is not a mistake on my part but an explicit claim, with abundance of evidence presented for it, sometimes with the kind help from others.
Now let me turn the argument presented in the Appendix D of my paper upside down to show, in a different way, where the bounds of -4 to +4 come from, even though physically only -2sqrt(2) to +2sqrt(2) is attainable for the reasons mentioned above. Let me now start with equation (D14), the last equation of my Appendix D. Again in Gill’s notation, it is the single average
E(A1B1 + A1B2 + A2B1 – A2B2) …. (2)
Everyone agrees that the bounds on this single average are from -2 to +2. However, as I have demonstrated in my Appendix D, physically this single average is an absurdity. It is an average of events that cannot possibly exist in any possible physical world, classical or quantum. Therefore anything derived or inferred from the above single average, such as the bounds of -2 to +2, are also an absurdity. Indeed it is not surprising that the bounds of -2 to +2 are “violated” in the actual experiments, and I have claimed that they will also be “violated” in my proposed classical experiment involving exploding toy balls.
Everyone also agrees that the equality between (1) and (2) is a mathematical identity:
E(A1B1 + A1B2 + A2B1 – A2B2) = E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) …. (3)
Therefore the adherents of Bell’s theorem claim that the bounds on (1) are also from -2 to +2. But how can that be? As I have demonstrated in my Appendix D the bounds of -2 to +2 are derived or inferred from considering an average of physical events that are impossible in any possible world. Even the God of Spinoza cannot make them possible. Therefore the claims of the bounds form -2 to +2 on the CHSH sum (1) are also absurd in the light of the identity (3), even if they are not mathematically wrong.
The question then is, what are the correct bounds on the CHSH sum (1) if not -2 to +2?
Well, I claim that if, for simplicity, we ignore the geometry and topology of the physical space, then the correct bounds are -4 to +4, which are in fact attainable at least in a computer simulation such as this one:
http://rpubs.com/jjc/84238. In fact it is quite easy to verify that the bounds are -4 to +4 by a simple observation, as done in the introduction of my Appendix D. All one needs for this is the assumption that the summation index used for the four separate sums in equation (1) above is not the same, as Jay verified earlier. Each of the four averages ranges from -1 to +1, and therefore CHSH sum can range from -4 to +4.
But what about the mathematical identity (3) that everyone (apart from me, at least) is so keen to exploit. Well, it is just a mathematical equality that leads to a physical absurdity. Once the summation indices on the four averages in the CHSH sum (1) are different from each other, the equality (3) no longer holds, as verified by Jay earlier.