A commenter has pointed out some new battles between Joy Christian and some saner physicists that take place at the arXiv.org preprint server and at fqxi.org, a funding agency mainly for crackpots.
If you check Google Scholar or INSPIRE, you will find literally dozens of papers by Joy Christian (and his critics) about "disproofs of Bell's theorem", "illusion of entanglement" and similar things. Almost all and maybe all of the citations of these preprints are either selfcitations or a few papers arguing that Joy Christian is wrong.
Bell's theorem (claiming that local realist theories only predict correlations in certain experiments that belong to a certain interval, not sufficiently wide one to agree with the observations i.e. with the predictions of local nonrealistic theory called quantum mechanics) can't be disproved because it is easy to prove it; see e.g. this lecture of mine for a proof. Of course, Joy Christian pays no attention to the actual proof and doesn't even try to find any error in it. He just denies it.
Joy Christian's hidden variables in his wouldbe counterexample are basically built out of quaternions – a good enough feature for some superficial people to think that it's "cool". (He even prefers the terminology of the "Clifford algebra" for some trivial maths involving inner and cross products of 3D vectors you learned in high school math classes; this "Clifford algebra" terminology is even better to attract some lowbrow fans.) He multiplies them in various abstract and physically illdefined ways, obtaining correlations that are outside the interval allowed by Bell's inequality. Of course that it's irrelevant whether he uses quaternions; Bell's theorem doesn't allow you any exceptions for quaternions, octonions, or pigeongazillions. It works for any model of local hidden variables.
This selfciting network of crackpot papers has allowed him to keep an Oxford and/or Perimeter affiliation for years. I am amazed by it because he's just an unbelievably hardcore crackpot. Doesn't Oxford University find it painful to be connected with such a mad guy?
One year ago, Joy Christian made a tactical mistake, however. Aside from dozens of meaningless papers with dozens of pages of tedious, ambiguous, elementary and irrelevant algebra that no one wants to go through, he wrote a onepage summary of his wouldbe counterexample to Bell's theorem. It's short enough so that sensible people may actually decide to waste some time with it and precisely isolate the mistake.
In the past, various people have published preprints debunking Joy Christian's "disproof" – Florin Moldoveanu, Marc Holman,Philippe Grangier, Tung Ten Yong, and several others. I haven't read all of them, of course, but I found at least one of them unsatisfactory.
Finally, there's a preprint that makes complete sense to me, one by Richard Gill, a mathematician of Leiden. He explicitly shows what I've known for a few years (see e.g. Alpha Meme from June 2011): the main "trick" is a (deliberately?) ambiguous notation in Joy Christian's papers. He's using the symbol \(\beta_j\) in several ways, either as elements of a fixed basis of quaternions, or a particular basis of quaternions that depends on \(\lambda\). However, the algebra obeyed by quaternions doesn't specify the elements uniquely, it only specifies them up to an element of the \(SO(3)\) automorphism group. So if one only uses the algebra to "define" individual quaternions, there can always be a mistaken \(SO(3)\) transformation which may depend on the place in the equation.
It's a big difference whether or not you add the argument. Adding or not adding the argument \((\lambda)\) pretty much makes the difference between "knowing what's happening in the other lab" and "not knowing it". The \(SO(3)\) orientation may keep some information that would be inaccessible in a local realist theory.
At one point near the end of his "disproof", when he goes from equation 6 to equation 7, Joy Christian cleverly "forgets" the argument \((\lambda)\) of the vector \(\beta_j\) which is the reason why he gets a wrong result, one contradicting Bell's theorem. With details, he should have a \(\lambda^2=+1\) factor at the end; one \(\lambda\) from the algebra satisfied by \(\beta_j(\lambda)\) and another \(\lambda\) from converting \(\beta_j(\lambda)\) to \(\lambda\beta_j\). With the \(\lambda^2=1\) coefficient, the extra term doesn't go away and is inconvenient; however, when Joy Christian "hides" one of the \(\lambda\)'s, he may (incorrectly) claim that this inconvenient term averages out.
Moreover, he is very sloppy about things like division: \(a/b\) is ambiguous for noncommutative algebras such as quaternions because \(b^{1}a\) and \(ab^{1}\) are different and just two examples among many possible interpretations of \(a/b\).
Even if the trick with the missing \((\lambda)\) argument were avoided, Joy Christian's work isn't any counterexample to Bell's theorem because it doesn't really produce real values of the spin – the individual measurements have "quaternionic correlations" – so he violates the rules of Bell's game.
Gill spends some time at the end of the preprint with psychologically analyzing "Bell deniers" such as Joy Christian, their motivation, and their basic strategy to fool themselves and others. It's an interesting reading. Needless to say, Joy Christian has already published a preprint attacking Gill. He accuses Gill of not worshiping Joy Christian as the greatest genius from the Big Bang, and so on.
If you read the latest Joy Christian rant, it is very obvious that he hasn't read Gill's paper at all. He only responds – very emotionally – to some sentences in the abstract that make it clear that Gill doesn't consider Joy Christian a revolutionary. But Joy Christian doesn't address the actual content of Gill's paper which is the place that actually proves that Joy Christian is an idiot. Joy Christian doesn't address the nonreality of the objects that should be real at all, for example. And he doesn't address either of the two independent calculations by Gill that show that Joy Christian's result for the correlation is just wrong (i.e. the mistaken missing extra \(\lambda\)), either. So his numerous pages have nothing to say about the actual Gill's criticism.
Joy Christian says that Gill incorrectly uses the term "mistake" for what really is "innovative and original" about Joy Christian's ingenious revolution. Dear Joy, we're just talking about a trivial f*cking stupid kindergarten mistake with a missing factor of \(\lambda\). Only a lunatic could call this mistake an "innovative and original thing". Gill politely uses the term "mistake" for what is one of numerous explicit proofs of your staggering idiocy.
He's just a stupid and pathetic fraudster and crackpot obsessed with his own importance and I urge Perimeter and Oxford to prevent him from using the name of their institutions in his nonsensical papers.
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Luboš,
I don't think you should worry about the reputation of FQXi: if you had participated in one of their "essay competitions", you would have known that FQXi 'organization' is practically nonexistent. It's all charade. ;)
I listened to one of his talks and I don't think he is disproving Bell's theorem (he actually says himself that the theorem is not mathematically disproved). What he does claim is that if you introduce a different local topology for the local hidden variables then you can reproduce the Tsirelson bound. I haven't gone through his papers to know whether everything checks out or not, but I think his actual claims are not as controversial as they seem.
Don't be silly, Sebastian. He wrote half a dozen of papers and a book that have Disproof of Bell's inequality in the very title. If you haven't been able to see that he claims that he disproved Bell's inequality, maybe you should take English classes and learn English again.
I am just conveying what he said in the talk I listened to. I know it is in his titles, but I think he is just trying to be sensational. He even states in his papers (see say http://arxiv.org/abs/quantph/0703179) that Bell implicitly assumed commutativity for the local observables which does not hold in his model (see end of page 3 and beginning of page 4 of the paper I pasted above). So I would say he knows that he did not disprove the Bell's theorem within its own assumptions, but he is claiming that quantumness and local realism can both be recovered if some of the implicit assumption of commutativity is relaxed (which I think is not too controversial since in my view commutativity => quantumness).
Dear Sebastian, well, if he deliberately tries to be sensational, it makes things worse, not better.
At any rate, in your comment you are just parroting the illogical pseudoscientific gibberish by Joy Christian or Christian Joy himself.
It's complete nonsense to discuss "noncommutative" obsevables in the context of the theories excluded by Bell's theorem. Bell's theorem derives an inequality for local realist theories. Observables in local realist theories are inevitably ordinary commuting numbers. That's the rules of the game.
In quantum mechanics, observables are noncommuting observables. It's exactly the fact that different properties of the system (even projection operators corresponding to Yes/No properties) don't commute with each other that allow quantum mechanics to violate Bell's inequalities.
But Christian Joy isn't constructing quantum theories. His noncommuting objects are just equivalent to matrices. But this organization into matrices is a pure bookkeeping device. One may still divide the matrices into individual entries and they're commuting numbers just like in normal Bell's theorem and Bell's inequality applies.
There's no loophole. You have clearly been brainwashed into believing that this crank has found a loophole but he hasn't and he couldn't.
I'm not parroting anything. I never said I think Joy Christian is right. I would appreciate if you didn't jump to conclusions about what I do or do not believe.
I am saying that he is saying that he relaxes the commutativity assumption to get to his result. Which, in my view, just comes back to quantum mechanics.
Also, having just finished reading his summary, it is clear he makes some quite trivial mistakes in there, introducing extra hidden variables into his calculation. It's not entirely clear to me if that's identical to his previous calculations so I will reserve my judgment about the rest.
Dear Sebastian, please don't take it personally. I am just saying that the statements you are reproducing here are wrong.
And no, his model isn't "returned" to quantum mechanics; for example, his probabilities are never squared probability amplitudes, and so on. It's a classical model of a sort in which spins or their probabilities are expressed by "classical matrices" or other noncommuting objects. It's not possible because in local realist theories, the measured values must be commuting cnumber observables.
Yes, there are mistakes in arithmetics but the essence of this mistake *is* in the confusion what can be noncommuting about the local realist theories and whether the "basis of these noncommuting operators" may remember something.
Sure, the probabilities do not match with Born's rule. The point I was trying to make is that noncommutativity makes the observables behave in a nonclassical way. It's not identical to quantum mechanics.
As for local hidden variables, all you need is for the probability distribution to be of the form $\int \rho(\lambda)p(a,\lambda)p(b,\lambda)$. If you can show this matches the quantum distribution then you get both locality and realism. I have never actually seen Joy show explicitly the form of his $p$ and $\rho$.
You wrote: "Sure, the probabilities do not match with Born's rule. The point I was trying to make is that noncommutativity makes the observables behave in a nonclassical way. It's not identical to quantum mechanics."
You see that I understood very well what you were trying to say. I was trying to explain you that your point is wrong. There isn't any "third kind of a theory" which is neither classical nor fully quantum. The belief in this "third way" is what underlies Joy Christian's papers as well but this belief is just wrong.
One either talks about noncommuting but associative observables  operators in quantum mechanics  and then the full quantum mechanical framework is the only way how to connect this mathematical structure to observable probabilities; or one abandons quantum postulates such as the "probabilities are squared amplitudes", but if he does, then classical physics with elementary observables' being commuting cnumbers are the only way to formulate a physical theory. There is no other possibility.
" If you can show this matches the quantum distribution then you get both locality and realism."
This is a meaningless discussion. Bell's theorem shows that no local realist theory can violate the inequality. It doesn't matter whether you are imagining the hidden variables as being substituted to some noncommuting objects or whatever. They may always be written in terms of commuting variables.
Your comments make it clear that you share the elementary misconceptions with Joy Christian. There could have been doubts about it after your first comment but there's no doubt now. Can we stop with this fruitless exchange?
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