" ... what is proved by impossibility proofs is lack of imagination."--John S. Bell
This post is in response to Florin Moldoveanu's, "To Be Or Not To Be (a Local Realist)," which discusses my research. You can read his post for some background to the topic, and also view my own talk on my work at the 2009 FQXi conference in the Azores, below:
[youtube: J2Sc0ZvNMe4, 560, 340]
I am grateful to Florin Moldoveanu for trying to understand my work on Bell's theorem. My arguments use a rather unfamiliar language of Geometric Algebra, and for this reason many have found them less than comprehensible. I am relieved to note that geometric algebra has not been an obstacle for Florin. I am, however, quite surprised by his conclusions. I do not agree with most of the assertions he has made, both here and elsewhere. In particular, contrary to what Florin asserts, I do not start with mathematical assumptions different from those of Bell, and my local-realistic framework is fully capable of providing a strictly local and manifestly realistic understanding of all possible quantum mechanical correlations.
Let me begin by pointing out a number of alarming confusions in Florin's reasoning---not only concerning my own work, but also concerning the significance of Bell's theorem itself. The worse of these confusions occurs in his understanding of the notion of contextuality. Florin has painted this notion as something to be avoided at all cost. But that sharply differs from how Bell himself viewed the notion. Bell vigorously defended contextuality in his very first paper. He famously stressed that "the result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the disposition of the apparatus." I could not agree more! He goes on in the paper to strongly criticise the theorems by von Neumann, Jauch and Piron, and Gleason for neglecting contextuality, and argues that the demand of strict non-contextuality implicit in these theorems is "quite unreasonable" from the physical point of view. The idea of contextuality, he argues, should not be confused with the idea of realism. These are two completely separate notions.
Indeed, some of our best classical theories are deeply contextual. For example, Einstein's theory of gravity---in which the results of observations are unavoidably dictated by the disposition of the apparatus---is a profoundly contextual theory. But this certainly does not make general relativity any less realistic, or non-local. Thus Florin is simply wrong to assert that "contextual realism is almost a contradiction in terms." Bell would have strongly disagreed, and so do I, because that would imply that Einstein's theory of gravity is almost a contradiction in terms! Worse still, Florin claims that contextual ontology is "at least just as strange as quantum mechanics itself." This is quite wrong too. Contextual hidden variable theories do not admit any of the quantum mechanical oddities like superposition, entanglement, objective chance, or objective indefiniteness at the fundamental level. There is no unsharpness of any kind in these theories. Thus quantum mechanics is ontologically quite distinct from the contextual hidden variable theories. The crucial point of the latter theories is not that they are any less realistic (or less local), but that the reality they accommodate is far less rigid than what we would normally expect from our theories. This is analogous to how Einstein's dynamic theory of space-time is far less rigid compared to Newton's static theory of space-time. Thus, what we need to exorcise from physics is not contextuality in general, but *remote* contextuality (or "non-locality"). And that is exactly what I have set out to do.
In fact, the issue of contextuality brought up by Florin is largely irrelevant for my program, because the framework I am using happens to be non-contextual at heart. Florin has been misled to think otherwise, because he has largely relied on my older, preliminary papers, and neglected my latest preprints. Had he studied my latest preprint (arXiv:1106.0748v3) he would have realized that even the local non-commutativity of the local beables in my models---which he claims makes contextuality inevitable in general---is only an intermediate calculational tool. The actual eventualities, A, B, etc., in the models (i.e., the actual measurement results) always commute with each other, because they are simply scalar numbers, +1 or -1. In statistical terms, these actual measurement results are raw scores, and the corresponding non-commuting local beables---i.e., the bivectors---are standard scores. The standard scores---or the standardized variables---in my models indeed do not commute in general, but they are simply intermediate mathematical constructs, not something that is actually observed in the experiments. Therefore Florin is quite wrong to build his case around the non-commutativity of these intermediate constructs (although I must admit that my earlier papers do not make this distinction between the raw scores and standard scores very clear). Moreover, the mystique of the classical non-commutativity within my models completely evaporates when one realizes that it can always be understood as a vector addition in a higher-dimensional space. Thus non-commutativity within my models does not have the ontological significance Florin seems to think it has.
Furthermore, the actual eventualities in the models (i.e., the actual measurement results A, B, etc.) are not contextual at all. Although they may refer to the disposition of the measuring apparatus, they do not actually depend on it. One can see this most clearly in the explicit example constructed in this preprint: arXiv:1103.1879v1. This fact may seem to be at odds with the well known theorems like those of Gleason and Kochen and Specker, but these theorems are not directly applicable to my framework. This is because none of them even remotely address the topological concerns I have raised within the context of Bell's theorem. Thus it is not appropriate to apply these theorems to my framework without due consideration of the fact that I am working at the level of probability amplitudes within a fundamentally different geometrical and topological framework, whereas these theorems are formulated at the level of the probabilities themselves within the traditional models of the physical space. Florin's analysis of my work does not respect this crucial distinction, and therefore it cannot be taken seriously.
Moreover, I feel that it is not appropriate to try to fit my framework in any pre-existing conceptual box, not the least because it uses the unusual tools of Geometric Algebra. Therefore, instead of worrying about Florin's analysis (or what implications the formalistic "impossibility theorems" may or may not have for my framework) let me spell out exactly how and why it is capable of reproducing quantum mechanical correlations in a purely local-realistic and non-contextual manner. This can be summarized by the following theorem.
Theorema Egregium: Every quantum mechanical correlation among a set of measurement results can be understood as a purely classical, local-realistic correlation among a set of non-contextual scalar points within a parallelized 7-sphere.
Here "local" means that the measurement results among which the correlations are exhibited are determined by a "common cause" (i.e., a common event), which has originated in their common past; "realistic" means that all such results have a definite value, predetermined by this common cause; "a non-contextual scalar point" means a measurement result that refers to but does not depend on the disposition of the apparatus; and "parallelized" means that the 7-sphere we are considering has a zero curvature but a non-vanishing torsion that may have different values for different points of the manifold.
The claim I am making here is much stronger than what has appeared in some of my papers. The main ingredients of the proof of the above theorem are already there in several of my papers, but I will try to put it together in one place in a future publication.
Now 7-sphere has a very rich topological structure. To begin with, it happens to be homeomorphic to the space of unit octonions. And as is well known, octonions form the most general possible division algebras. In the language of fiber bundles one can view 7-sphere simply as a 4-sphere worth of 3-spheres. In other words, in the language of fiber bundles each fiber of the 7-sphere is a 3-sphere, and each one of these 3-spheres is a 2-sphere worth of circles. Thus the four parallelizable spheres---S7, S3, S1, and S0---can all be viewed as nested into a 7-sphere, much like the Russian dolls. The EPR-Bohm correlations, for example, can then be understood as correlations among the equatorial points of one of the fibers of this 7-sphere; namely, a parallelized 3-sphere. Thus, in my picture, the EPR-Bohm correlations are what they are because of a nontrivial twist in the Hopf fibration of the 3-sphere. They have nothing to do with entanglement or non-locality per se.
Alternatively, the 7-sphere can be thought of as a 6-sphere worth of circles. Thus the above theorem can be framed entirely in terms of circles, each of which can be described by a classical, octonionic spinor, with a well-defined sense of rotation (i.e., whether it describes a clockwise rotation within the 7-sphere or a counter-clockwise rotation). This sense of rotation in turn defines a definite handedness about every point of the 7-sphere. Let us designate this handedness by a random number, +1 or -1. The local measurement results for a give physical scenario, A, B, etc., can then be shown to depend only on this random handedness (or orientation) of the 7-sphere, as explained in the latest two of my papers: arXiv:1103.1879v1 and this one arXiv:1106.0748. The correlations among all such local results (which are simply numbers that may refer to but do not depend on the disposition of the apparatus) are then determined purely by the topological subtleties of where they are situated within the 7-sphere. That is to say, the devil is in the details of the 7-sphere. I have shown how this works only in the simple cases of EPR, GHZ, and Hardy states, but the same logic works (at least formally) for any arbitrary quantum mechanical state. In practice, however, the calculations for more involved states are horrendously complicated, because the topology of the 7-sphere is far from straightforward.
What is important to note here is that, since the eventualities A, B, etc., are pure numbers, there is no non-commutativity involved. Thus contrary to Florin's assertions, I do not start with assumptions any different from those of Bell as far as the eventualities themselves are concerned. Moreover, as already mentioned, there is no contextuality involved either, because all local eventualities are predetermined entirely by the orientation of the 7-sphere. Only the correlations among these eventualities depend on the dispositions of various apparatus, as they must. Thus the reason for the existence of the quantum correlations is not the local variables themselves, but the global twists and turns within the 7-sphere. The local variables are, as always, purely random, and hence they by themselves cannot possibly reproduce the quantum mechanical correlations, as correctly recognized by Bell. But what Bell failed to recognize is that this conclusion depends on the topology of the co-domain of the variables A, B, etc. In my view this topology, in general, is that of a parallelized 7-sphere.
In conclusion, Florin is simply wrong to think that local-realistic models of quantum correlations may not always be possible within my framework. He is also wrong to paint contextuality so negatively; and wrong again to think that I start with assumptions different from those of Bell. Both Bell and I start only with commuting numbers, A = +1 or -1, B = +1 or -1, etc. Both Bell and I defend local contextuality as a natural physical concept fully compatible with realism. The only difference between our assumptions is that I take measurement results to be the points of a parallelized 7-sphere, whereas Bell incorrectly assumed that they are points of the real line. More importantly, my framework is entirely consistent with the EPR criteria of locality, reality, and completeness, and it also fully respects Bell's own criterion of factorizability. Therefore, in my opinion---although Bell's theorem may well retain some of its relevance for the information and communication engineering as Florin has argued---it has no relevance for the future "theory of everything." Indeed, in the light of my results I believe that the successful "theory of everything" will be a *locally causal* theory in which contextuality will play a role no different from the role it plays in general relativity. In other words, in my opinion all attempts to build a non-local theory of quantum gravity will fail miserably, just as anticipated by Einstein.
Finally, let me thank Florin again for his serious efforts to understand my work. His is the most civilized, honest, and thoughtful reaction to my work I have encountered so far.