Dear Joy,
Let me start with a disclaimer, please excuse the typos below as I did not write this in Word and I am addicted to Word's check spelling.
I am am still digesting 1101.1958 along with its introductory ideas from 0806.3078. However, before I fully understand those papers, I want to discuss your earlier disproof papers, the responses from the critics, and my prior incompletness claim. I'll start with the simplest topic, the critics. quant-ph/0703218 is obviously flawed. 0704.2038 is not an air tight counter argument. 0712.1637 is not valid as I do have myself a concrete counter argument for 0712.1637. However, I do mostly agree with Grangier. I would like to argue along the lines of this quote "So the proposed model [1] cannot be a local realistic model, it could at best be an alternative formulation of quantum mechanics [4], like Bohm's theory is." Now let me set the preparatory stage.
What captures best the spirit of QM? I can argue (following the lines of research of Emile Grgin) that the most important property of nature (and in particular of classical and quantum mechanics) is its invariance of the physical laws to composing two systems: put any 2 QM systems together and the composed system is still described by QM. In terms of abstract properties of QM, QM is described by two products: an anti-symmetric Lie algebra and a symmetric Jordan algebra. The rule of invariance under composition (the requirement to be able to construct a tensor product) demands three algebraic identities: Lie, Leibnitz (leading to the introduction of derivation) and a compatibility condition between Lie and Jordan algebras. Physically this corresponds to a 1-to-1 mapping between observables and generators, or in Alfsen and Shultz lingo, a "dynamic correspondence". Going from the algebraic approach to QM state spaces, one introduces an associative product by combinning the Lie and Jordan products using sqrt(-1). However, the 3 algebraic identities lead to a more general approach to QM than a simple standard C* algebra because there is no positivity condition associated with it. (Landsman for example works along similar lines when he talks about a Lie-Jordan algebra, but he incorporates the positivity condition by demanding an additional rule.) At this stage the plot one can look for example at the Cartan classification of Lie algebras and adding the 2 additional algebraic constraints would restrict the available realization of the usual Lie algebra classification. First one gets the usual su(N) associated with complex QM, but there are also exceptional solutions: so(1,2), so(3), so(6), so(1,5), so(2,4), so(3,3).
In general, from the classification of Jordan algebras, one get the standard NxN matrices over the division algebras, the spin factors, (and the Albert algebra). Geometrically, ignoring octonions, in state space this corresponds to cones and spheres (or hemisperes). All spheres can be easily described in terms of geometric algebras. Long story short, your counter-argument to Bell is based on one of the exceptional cases, the so(3), and this can actually be shown to be a limiting case of so(2,4)~su(2,2) corresponding to a "fermionization" of twistor theory when a twistor is considerd to represent the observable of the theory. (now some people became interested in the plain so(2,4) case-Bars, Segal). All the other exceptional cases can be easily cast in a geometric algebra formalism, but what you loose at this time is the physical justification for your topological argument. This does not make the math invalid, but so(3) was just a lucky coincidence which can be interpreted along your arguments for justifying bivectors because of the gimbal lock problem in the standard approach. In this sense your proof was incomplete. If there were no other exceptional solutions except so(3) allowing unrestricted composability (arguably a major requirement for any sensible physical theory), the argument against Bell would have been complete and irefutable. The presence of the other solutions opens the door for other interpretations as well. Because all the other cases are not fully investigated at this time, I cannot conclude with 100% certainty that your proof is right or wrong in imposing your particular topological interpretation.
However, I strongly feel that your topologial completness argument, while fully working in the so(3) case may not hold in general in QM based on other spin factors. (If I can prove it one way or the other, I would certantly publish it.) I am saying this because at least in one particular case based on so(2,4) one gets a different geometric phase which may kill any hope of natural justification of beables, unless the beables display a Yang-Mills gauge freedom-a completely non-classical behavior.
Moreover, your bivector beable interpretation does not work in the self-dual cone case (plain QM with no spin) =because this is not well suited for the geometric algebra formalism= but arguably there is no Bell inequality there, (and your example was intended as a counter example).
You may counter my incompletness claim by saying that your example is just a counter example and you only need one right? Not quite. If the aim was to mathematically disproof Bell's theorem, this is not achieved as you start with a different assumption. If the aim is to demolish the philosphical pedestal and the importance of Bell's theorem in justifying the opposition to local realism, then a simple counter eample is not enough. Either your interpretation is natural and the only one possible, or you find an actual mathematical flaw in Bell's proof. There are no flaws in proving Bell's theorem, and your interpretation IS natural. I am not convinced it is the only one possible.
Maybe you would not agree on imposing the requirement for a unique interpretation. Fine then. But I do know the origin of your counterexample: the spin factor state space geometry. As there are only a handful type of Jordan algebras, this limits the kind/type of possible beables. It is conceivable to be able to systematically categorize all beables and if there are cases where they only display non-classical behaviors, the argument against local realism still stands. You may claim then that Bell's theorem is incomplete in its broader aim of killing local classical realism and this hypothetical future results strenghtens it making it air tight. (You may actually claim that Bell's theorem is undeserving its reputation right now,but I am not clear on your stance on local realism.)
But although I am not convinced you can successfuly kill's Bell's theorem importance for QM, your approach has excellent merits. Your upper bound correlations interpretations based on maximum torsion is a wonderful result which I plan to understend in depth. And if this leads to a sensible octonionic QM, this would be a crowning victory and suddenly everyone will start paying attention. Let me read yor result some more, and I'll be back with questions and coments.
