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...

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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.

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Thank you for your extensive comments. Let me say from the outset that I do not accept Grangier’s criticism (or yours for that matter), for the reasons I give below.

1) It is quite clear from his critique that Grangier does not understand the first thing about Clifford algebra, or about my model based on it. Unfortunately his misguided critique has done grave damage to my program, not to...

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Thank you for your extensive comments. Let me say from the outset that I do not accept Grangier’s criticism (or yours for that matter), for the reasons I give below.

1) It is quite clear from his critique that Grangier does not understand the first thing about Clifford algebra, or about my model based on it. Unfortunately his misguided critique has done grave damage to my program, not to mention disservice to physics. So if you are basing your criticism on Grangier’s, then I am afraid you too have not understood my program. I think it is foolish to think of my counterexample as “alternative formulation of quantum mechanics.” The model is strictly local, evidently realistic (in every sense of the word), and manifestly complete (these are not just assertions by me---I have explicit and detailed demonstrations of each of these facts throughout my papers). Therefore my counterexamples cannot possibly be thought of as mere reformulation of quantum mechanics. I find your quoted assertion completely absurd.

2) Now it never ceases to amaze me how no one mentions the EPR argument when criticising my work. Unless you first understand Bell’s theorem within the context of EPR argument, you have not understood the theorem---and consequently you are unlikely to understand my work. The EPR argument is a logically impeccable argument which proves, once and for all, that QM is an incomplete theory of nature (provided you accept their premises). For this reason the SU(2) case discussed in my first paper is itself quite sufficient to prove the incompleteness of quantum mechanics, contrary to what you are asserting. It takes only one counterexample to Bell’s theorem to restore the EPR argument, and it takes only one example of incompleteness to demonstrate the incompleteness of the entire theory. It is an altogether different demand to ask for a local-realistic reproduction of all possible predictions of QM, but I am prepared to accept that challenge. In fact, that is precisely what I have been systematically doing for the past few years.

3) Contrary to what you assert, Bell’s theorem and all of its variants are fundamentally flawed. It is a myth promulgated by the followers of Bell that his theorem is a mathematically and logically impeccable theorem. It is neither of these, as I have demonstrated in several of my papers. I urge you to read the argument in my latest paper to appreciate how sloppy Bell’s reasoning is. His very first equation is fundamentally flawed. It is a non-starter.

4) Judging from your mathematical comments it is clear to me that you do not understand the hidden variable program. This program was systematized by von Neumann in 1930’s, in order to disprove it, and later Bell systematized it further to bring out the issue of locality. According to these systemizations, all that is required of any hidden variable program is to reproduce the expectation values predicted by quantum mechanics in a dispersion-free manner. So all of the mathematics-related arguments you have made are completely irrelevant to the real issue of the completeness of quantum mechanics. It is unfortunate that---because of the sociological successes of Bell’s theorem and the proliferation of the quantum information theory program---people have lost sight of the historical and logical origins of Bell’s theorem. To understand it fully (and to understand my work) it is inevitable to first understand the EPR argument, as well as the hidden variables program set forth by von Neumann. If you understand these two, then you would recognize that most of what you are saying is irrelevant to my program. More precisely, I believe that I can reproduce every expectation value predicted by quantum mechanics (at least in principle) within my local-realistic framework (in fact, I have done precisely that already to some considerable extent).

In summary, I am grateful for your efforts to understand my work (truly I am). But you have a long way to go before you actually understand my program.

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Thank you for your prompt reply. I wanted to give you some literature to back my claims, but if you accept my math comments, let's discuss your points instead. I did not say I completely agree with Grangier, but I agree with the quote I gave you from there. The degree to which Grangier understands Clifford algebra regarding my comments is irrelevant, because I do understand geometric algebra...

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Thank you for your prompt reply. I wanted to give you some literature to back my claims, but if you accept my math comments, let's discuss your points instead. I did not say I completely agree with Grangier, but I agree with the quote I gave you from there. The degree to which Grangier understands Clifford algebra regarding my comments is irrelevant, because I do understand geometric algebra myself. In fact, geometric algebra and bivectors are criticaly used in the work of Emile Grgin to discover a relativistic verrsion of QM ( actually a rediscovery of Hestene's work based on complex quaternions (even subalgebra of the space time algebra) and its link with SU(2)xU(1)). Geometric algebra is a very useful tool in analysing QM. (I do disagree however on you new paper on the need for a division algebra. Complex quaternions are a non-division algebra and QM based on this number system is doing just right, but this is a huge topic in itself).

I agree that in your example you achive a local realistic framework. I can say that your model makes QM intuitive in that case (a no easy feat). I would even agree that you have a good shot at shaking Bell's theorem social status (and this take a lot of courage). What I don't agree is that your model proves local realism. For that I feel you need much more, and your new paper is a step into the right direction to convince the critics. But this is my personal oppinion. As we are obviously working in different paradigms, for now, this is a matter of taste and I don't want to impose my views on you. Besides, it will be quite hard to do it in a few exchanges. If I would have an airtight mathematical proof, I will publish it as I said earlier.



Now I don't know what kind of damage Grangier did, I never met him and I have nothing in common with him. I feel his comments simply reflect the majority oppinion and the burden of proof is on you to convince the critics.

Now about EPR, my take on that is that I agree with the analasys done by Aerts http://www.vub.ac.be/CLEA/aerts/publications/1984MisElRealEP
R.pdf (And I feel that his approach may also lead to a way out of the measurement problem.) Now Aerts has some classical examples of how to illustrate EPR and the examples are non-local. In your example, you too have non-local beables which oddly enough can be undestood as local as well. Back to Grangier: "the central issue in Bell’s theorem, which is correlating clicks between detectors (corresponding to binary measurement results), and not correlating bivectors (which cannot be given any “local realistic meaning”)." Here may be the key to explain the gist of my argument: I disagree with Grangier on: "(which cannot be given any “local realistic meaning”" because this can be done and you actually prove it to my satisfaction (if my oppinion will actually count). What I contend is that this cannot ALWAYS be done in the geometric algebra settings. In other words, giving a local realistic meaning to QM in all spin factor cases MAY not be possible in general (I have strong heuristic reasons of why this may ultimately be the case. It is not yet a proof, and as we work in different paradigms we can agree to disagree at this point.).

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Dear Florin,

I better start with addressing you directly, because this thread is becoming very confusing otherwise.

Let me start with the damage Grangier’s critique has done to my program. It does not reflect the majority opinion; rather the majority opinion reflects his critique. Here is what I mean: Back in 2007 the majority did not immediately have an opinion. I had discussions...

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Dear Florin,

I better start with addressing you directly, because this thread is becoming very confusing otherwise.

Let me start with the damage Grangier’s critique has done to my program. It does not reflect the majority opinion; rather the majority opinion reflects his critique. Here is what I mean: Back in 2007 the majority did not immediately have an opinion. I had discussions with a great number of my colleagues, and they did not have a clear opinion on my work (mostly because they did not understand geometric algebra). Then came the critique of Grangier, which was immediately understood by many, because it reflects the traditional point of view, and this critique played a key role in setting a negative tone against my work. And once a negative tone sets in, it is very difficult to undo it within the current sociology of physics. As a wise person once said: A new idea can be killed by a sneer or a yawn; it can be stabbed to death by a joke, or worried to death by a frown. And Grangier’s critique managed to do just that---by misrepresenting my central idea. This is why I feel that he has done a disservice to physics. But you are right. Ultimately the burden of proof is on me. I personally feel, however, that I have provided the proof, in the seven papers on the subject I have written so far.

But enough of that. As for the rest of your comments, so far my goal has not been to directly “prove local realism”, but to disprove all Bell type theorems (thereby restore the original EPR argument). In this process geometric algebra has been an important tool for me, but only a tool. I am not committed to geometric algebra. So if it turns out not to be useful for some cases, then I am quite prepared to give it up. There is only one thing I am not prepared to give up, and that is local causality. And my reasons for that come from my work in quantum gravity (I am of course also not prepared to give up reality, but that is understood). In any case, I feel that you still haven’t grasped the central idea behind my work. Bivectors, geometric algebra, etc. etc. are only mathematical tools. The key to understand quantum correlations are the parallelizable spheres. All quantum correlations can be understood as local-realistic correlations among the four parallelized spheres (which are purely topological creatures). It is not important which mathematical tool one uses to capture this fact. What is important is the fact itself. So to say that my model has “non-local beables” is to miss this point. The bivectors are not non-local in any sense. They simply represent points in a different topological space than what is traditionally used within Bell’s theorem. There is absolutely nothing non-local going on in my models. This I have shown rigorously. In this light, even if you have an absolute proof that certain things cannot be done within geometric algebra, then so be it. That does not necessarily affect my program. The topological theorem I mentioned in my latest paper do not depend on geometric algebra.

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Dear Joy,

I feel that most criticism against your work is rooted in the unfamiliarity with geometric algebra and the prejudice for status quo. However, not Grangier' criticism. And you are right, geometric algebra is only a tool. In a pure strict mathematical sense, Bell killed von Neuman proof and you killed Bell's proof. But the real debate is on completeness of QM and on local realism in...

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Dear Joy,

I feel that most criticism against your work is rooted in the unfamiliarity with geometric algebra and the prejudice for status quo. However, not Grangier' criticism. And you are right, geometric algebra is only a tool. In a pure strict mathematical sense, Bell killed von Neuman proof and you killed Bell's proof. But the real debate is on completeness of QM and on local realism in the sense of Einstein. Let me explain.

In QM there are no go theorems for non-contextual hidden variable theories, but they do not apply for contextual HV. But this does not secure contextual HV theories the status of physical theories. Bell is credited for killing Neumann's proof not because of finding an objectionable assumption, but for carying that assumption to its logical conclusion and for discovering his inequality which in the end greatly strenghtened the difference between classical and QM. Should he nevered find the inequalities, his objection would have been ignored in the same way contextual HV theories are ignored today. So I guess that if your objection against Bell's theorem would lead to a strengthened argument either against or for local realism, then your result would replace Bell's result in its social status among physicists. Until then, your result, while mathematically correct, can be easily dismissed a la Grangier by saying that this is not normally what people understand while discussing the usual classical statistics. And he is right. The point of his criticism is on the implied ontology of your result and not on validity of the result itself. When Vietnam war ended, at the peace conference, the US general remarked to his Vietnamese counterpart: you never defeted us in any battle. The Vietnamese replied: true but irrelevant.

So on local classical realism, I think the battle is lost, there is no such thing as local classical realism. Your restoration of local classical realism is only based on the fortunate su(2)~so(3) isomorphism and may only be working in this and a few other isolated cases.

To restore local realism you need to prove that local classical realism is ALWAYS tennable. Only then you can claim you have successfuly killed Bell's theorem (or the ghost of Bell's theorem from physicists' collective psychology). My hunch is that a systematic analasys would reveal a kind of local realism which is far from any generally acceptable classical behaviour. Then this yet to be discovered result will replace Bell's theorem and your analasys would then become similar with Bohm's theory in the role it played. This is what Grangier ment by saying: "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."

On completness of QM, QM is most likely insufficient to describing nature as superselection rules may naturally occur and they have an impact on dynamics. But this is a big open problem at this point. So I guess Einstein was right after all about incompletness. If I am right in the paragraph above, he was right in the letter but not in the spirit.

Back to your original result, I became aware of it 2 years after it was published and I wanted to write a comment on it until I saw Grangier's comment which resonated strongly with my thinking. I was never influenced by him in any way and it was a rather pedantic excercise to write another comment lacking a proof of the ideas which I presented here.

If you may indulge me a bit more, I would make the following old-new comparison: Neumann's theorem-Bell's theorem. Bohm's theory - your result. Bell's theorem-hypothetical new no go theorem closing the loophole you discovered in Bell. GHZ-other exceptional cases besides so(3) for composability.

GHZ basically kills any hope for Bohm's quantum potential as a realistic model for QM as there is no interaction present there. The other exceptional cases in the search for non-unitary realizations of QM may kill the hope for a universal local realistic interpretation of QM, rendering the locally realistic physical interpretation based on so(3) as an isolated case and becoming an objection free no go theorem replacing Bell.

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Florin,

There are possible approaches to Christian's treatment of Bell's inequality. One approach, which you have beautifully illustrated above, is to bring all the mathematics at your command to the problem, and hope this answers the question.

Another is based upon physics and physical understanding. Because my theory is local-realistic and qualitatively explains many otherwise...

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Florin,

There are possible approaches to Christian's treatment of Bell's inequality. One approach, which you have beautifully illustrated above, is to bring all the mathematics at your command to the problem, and hope this answers the question.

Another is based upon physics and physical understanding. Because my theory is local-realistic and qualitatively explains many otherwise unexplained anomalies in today's physics, I have no problem accepting Christian's results, which make sense to me.

I say this knowing it will have no effect upon your approach, but simply to remind everyone tracking this conversation that yours is not the only approach that a physicist can take. Assume for a moment that QM is incomplete, as Einstein said, and as you seem to state: "So I guess Einstein was right after all about incompleteness. If I am right in the paragraph above, he was right in the letter but not in the spirit. "

This being the case, why should quantum mechanics be the be-all and end-all of the problem? If it is incomplete, it is incomplete, and it's century of successes are not to be discounted, but neither are they to be the only parameter by which we judge reality. And a quarter century of 'entanglement' if Bell's inequality is truly incorrect, led to much non-sense, based upon the false interpretation of measurement statistics leading to the conclusion that local realism did not exist.

There are consequences to approaches. Unquestioning acceptance of Bell's inequality has had (if Joy is correct) disastrous consequences. I dare say that these came from the side that respects mathematics above and beyond all physical reasoning. The 'social reality' discussed above has prevented my theory of local realism from being taken seriously by those committed to the non-locality that is the basis of the 'entanglement industry', an industry in which contracts, experiments, papers, publications, and professional status weigh heavily upon 'accepted' version of reality. [God bless fqxi.]

The known 120 orders of magnitude decrease in QED's vacuum energy and the apparent 31 orders of magnitude increase in the strength of gravito-magnetism combine to present physicists with 151 order of magnitude relative change between these energies and potential explanatory power. But have all of the QED calculations since 1947 been recalculated with a realistic vacuum energy? No. Old ideas of virtual particles, despite failure to find the expected 'sea of strange quarks' in the proton, despite the surprise of the 'perfect fluid' at RHIC and LHC when a 'quark gas' was expected, are well entrenched, and no one is being discomforted by the mere physical facts. QED cannot even come within 4 percent of the proton radius, for muonic hydrogen. And QCD has problems getting this close.

"Real anomalies, we don't need no stinkin' real anomalies." Instead, those who happily accept the non-real, non-local as "reality" have gone off into Multi-verses, extra dimensions, holographic extensions, qubits-as-virtual processors, and other fantastic but not-measureable and non-predictive physics. That 151 orders of relative change could actually mean a simplification of physics is not even resisted. It's ignored. No one, apparently, wants physics to be simpler. That a gravito-magnetic-based 'pilot wave' induced by every particle with momentum could actually be meaningful is ignored.

I'm not complaining. Planck said a century ago that "...theories are never abandoned until their proponents are all dead...science advances funeral by funeral." If true, we're in big trouble, since there are too many physicist proponents to all die off, and they are training their replacements.

And, Joy, perhaps you will find some joy in Einstein's statement: "I enjoy it that colleagues occupy themselves at all with the theory, although for the time being with the purpose of killing it..."

The mathematical battles are extremely important, but physics is still based on reality, and, it is my hope and belief that these 151 orders of magnitude changes imply a simpler, and more intuitive reality, one that I try to outline in my essay.

Edwin Eugene Klingman

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Dear Florin,

I am sorry, but I fundamentally disagree with almost everything you are saying. To begin with, I do not think you have read what I have written, especially in my latest paper. If you have, then you have not understood the fundamental problem with Bell’s very first equation. And since this equation is fundamentally flawed, both Grangier' criticism and the consensus view are also...

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Dear Florin,

I am sorry, but I fundamentally disagree with almost everything you are saying. To begin with, I do not think you have read what I have written, especially in my latest paper. If you have, then you have not understood the fundamental problem with Bell’s very first equation. And since this equation is fundamentally flawed, both Grangier' criticism and the consensus view are also fundamentally flawed. Because Grangier' criticism amounts to resorting back to the first equation of Bell---which means that he never understood my paper. But that is forgivable. Because one can argue that I did not make my position clear in my first paper. But your reliance on Grangier' criticism today is hardly forgivable, because since then I have written six more papers, explained my position many times over, and produced many more explicit calculations to support my claim. To understand my main objection (which also has far reaching implications), I urge you to reread the first two pages of my latest paper. This paper also takes us to the logical consequences of my refutation of Bell’s theorem. Namely, that a local-realistic theory must involve all four division algebras (without a devisor local causality cannot be maintained a la Bell), and there is no real distinction between the classical and the quantum apart from an accidental choice of a division algebra (S^0 and S^1 for the “classical” case and S^3 and S^7 for the “quantum” case). The dismissal a la Grangier of my work is thus fundamentally invalid, because it surreptitiously brings us back to the flawed first equation of Bell.

You also keep insisting on the fortunate su(2)—so(3) isomorphism, but that is not the only example I have worked out. That was the first example, and that is enough to restore the EPR conclusion that QM is an incomplete theory of nature. But I have also worked out examples involving the Hardy state and the GHZ states (i.e., 7-sphere). More importantly, my latest paper does not rely on any such special cases. It employs very general and powerful topological theorems to show that *all* quantum mechanical correlations can be understood as local-realistic correlations among the points of the four parallelizable spheres. So I do not accept your point about su(2)—so(3) isomorphism being a special case, on several grounds. In particular, my last two papers prove that local realism is *always* tenable. See especially section VI of 0904.4259. I have showed this two years ago, so I am puzzled about your statements. Moreover, I am writing yet another paper where this point will again be brought out in full generality. The key point again would be the inevitability of using all four division algebras, if the local causality is to be fully respected.



I also do not agree with your interpretation of Grangier’s statements. Neither do I agree with your comparison of my results with Bohm’s theory. I think such a comparison is red herring. I think you should read my papers 0904.4259 (especially section VI) and 1101.1958 in full before making any such comparison. Grangier made this comparison because he thought my results correspond to a theory that is local but “non-real”. This is of course silly. No one who knows geometric algebra would make such a statement. Thus Grangier’s critique is a travesty.

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Dear Joy,

I think I figured out what is going on. The EPR completness criterion simply means that one gets the entire state space, be it phase space, Hilbert space, or other spaces. By considering actuals al well as counterfactulas you are then in fact ignoring the time evolution and get the entire state space. In general, any state space will obey EPR's completness criterion. (Try your...

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Dear Joy,

I think I figured out what is going on. The EPR completness criterion simply means that one gets the entire state space, be it phase space, Hilbert space, or other spaces. By considering actuals al well as counterfactulas you are then in fact ignoring the time evolution and get the entire state space. In general, any state space will obey EPR's completness criterion. (Try your approach on classical mechanics for a hypothetical hidden variable theory.)

Local realism in your approach is only factorizability. Factorizability is the opposite of entangelment and this gives the approach its classical intuition. Factorizability is not local realism as locality was considered by Einstein and Bell to mean just that: spatial separation. So is Bell's theorem invalid? No, because a Hilbert space dimension is N^2 for N psi(s) and it is not always separable. Still, you manage to arrive at separability. But this is not done directly in the original Hilbert space. For example you need to embed Bloch sphere in S3. S3 becomes then a different kind of state space and in fact you are rewriting QM in a diferent state space with a different formalism. But wait a minute. Is QM not supposed to be uniquely written in Hilbert space formalism? How about Piron's result of recovering Hilbert space over division algebras from propositional logic? The answer is no as there is a counterexample to that: QM in phase space via Moyal bracket.

So your prescription for separability is embedding (if possible) the Hilbert space inside S0, S1, S3, or S7 to achieve parallelizability. Because this contradicts the nonseparability of the standard Hilbert space, this means that QM over S0..S7 is something qualitatively different than a standard Hilbert space formulation. (And indeed, in your formalism you use different mathematical objects.)

Also it is not clear if this formulation of QM goes beyond QM or not. In other words, can you always succeed in embedding any Hilbert space in S0..S7? Probably not based on dimensional analasys for higher dimensions.

Another issue. If Bloch sphere is embedded in S3, would not this mean that we still deal with traditional complex QM? Let's look at another example first. Real quantum mechanics can be embeded in complex QM, but the meaning of the wavefunctions is qualitatively different. Probably something along similar lines is happenning here, I don't know. To get a better grip, an analasys of time evolution might clarify things as time tends to dissapear from the picture as both actuals and counterfactuals are considered. Maybe this analasys will show that you are still in a traditional Hilbert space (the spin factor case), and that the Killing flow is what you traditionally obtain in the original embedded space.

If I were to venture a guess, in S3 the time flow is not the same as in Bloch sphere and the meaning of psi does not stay the same. For if they do stay the same, all possible time evolution in Bloch sphere would be enough to achieve EPR completness which is not the same as S3 is needed.

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