“ ... 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:

[video]

I am grateful to...

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“ ... 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:



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.

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I agree with Cristi. The issue is not philosophical (free will vs. determinism). The physical issue is whether nonlocal effects determine local outcomes.

Going back to Einstein's definition of "physically real" " ... independent in its physical properties, having a physical effect but not itself influenced by physical conditions ..." classical physics is all local. Communication among...

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I agree with Cristi. The issue is not philosophical (free will vs. determinism). The physical issue is whether nonlocal effects determine local outcomes.

Going back to Einstein's definition of "physically real" " ... independent in its physical properties, having a physical effect but not itself influenced by physical conditions ..." classical physics is all local. Communication among bodies is restricted by relativistic limits. "Local realism," then, applies to isolated systems -- and because Einstein knew (from Mach's principle) that there are no such isolated systems in the universe as a whole, general relativity describes a universe that is finite in time and unbounded in space.

As I understand it, the Joy Christian model is entirely consistent with GR's "finite but unbounded" result; i.e., the division algebras in 2, 4, 8 dimensions incorporating, respectively, the closed algebra of R, quaternion algebra (noncommutative) and octonion algebra (noncommutative and nonassociative), allow recurrence (i.e., finite, or bounded, time) by confining measured events to manifolds of parallelizable spheres S^7, S^3, S^1 (where "parallelizable" is a topological term describing spheres S^1 of 2 dimensions, S^3 of 4 dimensions and S^7 of 8 dimensions -- all smooth manifolds incorporating vector fields). So Christian foresees local field correlation of tangent vector properties in the same general way that Einstein did; i.e., finite in time and unbounded in space. He has exchanged the relativistic limit in the mathematical language of PDEs for a topological limit in the language of geometric algebra.

Joy's result may indeed subsume Bell's theorem (not "disprove" it); I reserve my opinion. Even on this basis, it is a truly great achievement.

However, I differ -- ironically, by using many of the same principles as Joy -- in the conclusion that local realism can be successfully described as finite in time and unbounded in space. I find the converse -- local realism is finite in space and unbounded in time.

Joy's among others' results aim to obviate quantum theory, and I don't see how that can work. After all, if Planck's constant were zero, we would have an ontologically classical world. As it is, we live with "fuzzy" measurement and nonlocal influence insofar as t = T = 1 (normalization). What must be established as "physically real" in a quantum mechanical model that is finite in space and unbounded in time, is information loss. We can then maintain "finite but unbounded" classical parameters, without losing any of the measurement characteristics of either classical or quantum mechanics. Such quantum-classical deterministic models are already reflected in the work of Gerard 't Hooft, Ted Jacobson and Erik Verlinde. Information loss, however slight, IS a nonlocal effect and "physically real."

In the next few days, I plan to post my own results, extended from my ICCS 2006 paper and "On breaking the time barrier" preprint.

Tom

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Computer Simulation of the Physical Reality:

Much has been made out of a possible demonstration of the validity of my arguments by means of a computer simulation. It has been suggested that if my ideas are correct, then one can write a computer program modelling them and obtain the Tsirelson bounds. While such a demonstration would indeed be sociologically important (especially considering...

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Computer Simulation of the Physical Reality:

Much has been made out of a possible demonstration of the validity of my arguments by means of a computer simulation. It has been suggested that if my ideas are correct, then one can write a computer program modelling them and obtain the Tsirelson bounds. While such a demonstration would indeed be sociologically important (especially considering the fact that Geometric Algebra remains an unfamiliar language for many), it is irrelevant as far as the logic of the EPR debate on physical reality is concerned. An analytical model such as mine cannot possibly be proved or disproved by its numerical simulation. A simulation of a model is a mere implementation of its analytical details, not an experiment that can either prove or disprove its validity. If reality can always be so simply simulated then there would be no need for the staggeringly expensive actual experiments. Reality is mathematically far richer and profounder than some of us are able to fathom, let alone a computer software. The EPR correlations are observed in nature in the actual, real-life experiments, not on a dead computer. I have produced impeccable analytical arguments to demonstrate that Bell’s theorem is wrong, both in its input and output. Moreover, I have produced impeccable and explicit analytical models to explain the correlations observed in the actual, real-life optical experiments performed at both Orsay and Innsbruck (cf. arXiv:1106.0748). Furthermore, I have proposed an actual real-life experiment that can indeed falsify my model (cf. arxiv:0806.3078). Therefore the additional demand of producing a computer simulation of my model is like demanding a flight-simulation from the Wright Brothers when they have in fact produced an actual, real-life flying machine!

There are also more fundamental reasons why the whole fashion of a computer simulation with regard to EPR correlations is completely wrong headed. It is based on a serious misconception of the true physical and mathematical reasons for the existence of the EPR correlations in nature. In all real-life demonstrations of the correlations, Alice and Bob always observe truly random outcomes of their measurements---A = +1 or -1, and B = +1 or -1. Therefore, any correlation between A and B themselves cannot possibly reproduce the observed quantum correlation, unless the topological non-trivialities of the physical space itself are also correctly simulated. In the language of my model this means that one has to model the physical space, not in terms of the R^3 points alone, but in terms of all of the subspaces of R^3 (namely, points, lines, areas, and volumes) within the even sub-algebra of the physical space (as conceived by Grassmann some 167 years ago). However, what I have seen so far in attempts to simulate the EPR correlations is a completely wrong headed approach. What is usually tried is based on an implicit assumption that the numbers A and B are only apparently, but not truly random, and if only we can somehow discover the correct functional dependence of A and B on the disposition of the apparatus and the hidden variables then the right correlation between A and B would emerge. I repeat: this is a completely wrong headed approach. One can never ever reproduce the EPR correlations this way. For the EPR correlations are what they are because of the topological non-trivialities of our physical space itself, NOT because of some hidden order in the randomness of the variables A and B.

Joy Christian

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For me, the question of whether Joy's theory is true or not, is settled. It is.

Knowing that in the larger world it will probably be controversial for a while longer, however, I'd like to get a dialogue started along the line that "If Joy Christian is right, these consequences follow ..."

Having found my "lost notebook" as I mentioned in Florin's blog "To be or not to be ..." I...

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For me, the question of whether Joy's theory is true or not, is settled. It is.

Knowing that in the larger world it will probably be controversial for a while longer, however, I'd like to get a dialogue started along the line that "If Joy Christian is right, these consequences follow ..."

Having found my "lost notebook" as I mentioned in Florin's blog "To be or not to be ..." I realize why I laid it aside a year ago. The conclusions seemed just too "out there." Now, though, Joy has shined considerable light on the paths of some leading edge problems. The most important, in my opinion, is Einstein's singularity problem; i.e., the inability to model physics with continuous functions in a relativistic domain without arbitrarily specifying boundary conditions ("Did Nature have a choice ...?")

In my notebook beginning 9 Aug 2010, I wrote the heading: "Kissing curves of S^n implies physical string dynamics."

Then I followed with the known definition of hyperbolic space (the space where string theory originates): "H^n is a maximally symmetric, simply connected n-dimensional Riemannian manifold with constant sectional curvature - 1."

The text:

1.0 The most fundamental aspect of string theory is the substitution of objects of nonzero length for point particles. More recent advances (ref Witten) favor 2-dimensional objects (branes) that imply a nonperturbative theory in 11 dimensions.

A background independent nonperturbative theory of this kind that unites quantum mechanics with general relativity is necessarily a continuous field theory -- i.e, the geometry of the field that includes a time continuum is smoothly connected to infinity.

The problem is finding the zero points. If particles are not described as points, but rather oscillating ends of hyperdimensional strings or branes, what does "zero" mean?

In standard particle physics, we understand the role of vacuum energy; virtual particles give us a zero place of origin -- the price, however, is a statistical inference that obviates time and requires infinite spatial extent. We renormalize to 1, when we record observations, but this "by hand" technique is unsatisfactory to most. Why should Nature demand a "screen" of sorts, between reality we can describe and an inaccessible singularity that jumps all over eternity until we ask where it is and how fast it is traveling?

Time dependence implies scale invariance(I have brackets around this statement with a big question mark, but I now believe the statement is true). And our time dependent measures are always local, so we shouldn't be surprised that solutions are infinite and that renormalization works.

The "screen" is a scale invariant field of infinitely self similar random fields. The localized measure, being time-dependent, is the least moment in which this complex network can match point for point between observer and point in the field.

(There is a figure of two superimposed quadrangles, the background object is labeled C*, and just inside the superimposed object a point is labeled P "hyperbolic projection" with a double headed arrow extending past the boundary of C* to a dot labeled "0" and the callout "R^n")

There is reference to Zaslow in Tim Gowers' Princeton Companion of Mathematics, p.163: "Every complex manifold of (complex) dimension n is also a real manifold of (real) dimension M = 2n." Also, pp. 163-164, "Complex manifolds always have real orientation." R_ij = 0.

[There's a lot more, but I am going to end my notebook text here, for now.]

It should be apparent that Joy's model fully accommodates pair correlation by orientation to all points of S^3. Why is this important? -- because it allows non-arbitrary boundary conditions for general relativity, in which "all physics is local." Joy's "Pacman screen" metaphor of two interchangeable antipodal points could never apply to Minkowski space-time which is finite in time and bounded in space. It does apply however, to a model which is finite in space and unbounded in time. Remarkably, this conversion does no damage to general relativity at all! -- it does give us access to antipodal points, however, that never disappear and at which paired events are correlated at arbitrarily far distance. As I showed in my remarks in Florin's "To Be or not to be ..." blog, the 4pi revolution in Joy's topological space vice the 2pi revolution in quantum configuration space (another Joy Christian result) orients all events to real (i.e., Lebesgue) measure between the two edges of.the "Pacman screen" which are imaginary.

As I implied, this formulation could very well open the door to real world string theory experiments.

Tom

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