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Bell's theorem - for or against Hidden Variables?
In http://arxiv.org/pdf/1501.03393v6.pdf Christian seems to get confused in eqns 11, 12, 13. As eqn 11 shows script-E is simply a (finite) sum of products of script-A and script-B terms. BTW there seems no need for introducing these script A and B symbols, elsewhere the exact same equation uses regular A and B. Anyway, eqn 11 then introduces a new symbology: expectation value of script-A-sub-A times script-B-sub-B, which is simply another way of writing the script-E summation. Eqn 12 shows CHSH equation written with this new symbology. Then he says Gill's mistake is to take the expectation brackets outside of the four terms, as shown in eqn 13.

With this new expectation-value symbol it looks, superficially, like he has a point. It looks exactly like a typical mistake one can make in QM. There, you can't in general treat expectations that way because of non-commuting observables. However if we just replace the expectation-value symbols with their equivalent summations from eqn 11, it's clear that the step, Gill's step, from 12 to 13 is not a mistake. We're simply removing the (identical) summations outside of the four terms which are combined by addition and subtraction - perfectly legal.

So it seems a deliberate attempt to obfuscate a very simple point, to rebut Gill. It's hard to accept that Christian, obviously an intelligent person, who knows his geometric algebra, would do this. But that's the way it looks.

Earlier he makes an interesting point about psuedo-vectors, which change upon reflection. Therefore it could be said that O(3) is the correct group for them, not SO(3). (BTW I'm sure this whole topic is very well understood by some people but can't find a good reference.) Now both O(3) and SU(2) are "double covers" of SO(3). So - this is pure conjecture on my part - is Christian trying to say that pseudo-vectors are properly represented as SU(2), or quaternions? But of course O(3) and SU(2) are not isomorphic, even though they have similar elements (two copies of SO(3)). O(3)'s covers are not connected like SU(2)'s so they are definitely very different groups. Maybe you could make something of this idea, however - if, indeed, that's what he's trying to do. I'll think about it.

In conclusion my opinion is unchanged although, certainly, there are subtleties here I haven't grasped to my satisfaction. Gill is right, the CHSH proof is simple algebra - according to Christian's own equations for script-E. The fact that he substitutes this new expectation-value symbology (nothing like it in the original paper), in the crucial eqns 11-13, might mean there is an important point that he simply is not expressing well. He's making a mistake; but if I understood what he's really getting at, perhaps there's real meat here. Or, it might just mean he's deliberately obfuscating. Either way Gill is right, based on what Christian actually wrote. Perhaps if he could write clearly, without mistakes, what he's really thinking, Gill's objection would be answered. I wish Christian himself would comment.
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RE: Bell's theorem - for or against Hidden Variables? - by secur - 08-25-2016, 01:17 AM

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