The Nature of Bell\'s Hidden Constraints by Edwin Eugene Klingman

Edwin,

I am an empiricist; observations always trump hypotheses. Since all the actual experiments attempting to test Bell's ideas, have been carried out with decisions, rather than measurements, any experiment that purports to get a different result, when compared to the actual, existing experimental results, must correlate the same thing; decisions, not measurements. Otherwise, they are not comparable - of course one can get a different result, when one measures an entirely different thing.

Note that your first quote from Bell, begins with an "if" clause. My point is, that the clause is false. Classical statistical mechanics never deals with entities encoding only a single bit of information. That is what makes the quantum case so peculiar, in comparison. When there is only one bit of information in a message, there is nothing to average over, there are no better defined states, precisely because there are no other states at all, by definition of what is meant, by a single bit of information. Since such entities are never encountered in the classical realm, we have no intuitive understanding of how such things behave. But we seem to be observing such behavior, in the quantum case.

Rob McEachern

Gentlemen,

This has been an interesting and informative exchange, as polite differences generally are. Thank-you. Bell's Theorem seems to be the one topic which concentrates attention on the elusive characteristic of spin. And I say characteristic because it is only because of characteristic behavior both of electromagnetic and particle-like phenomenon that suggests some fundamental physical property. Yet I've found nothing anywhere that seems definitive of what that might be.

It isn't physical rotation, though its treated that way. As a purely classical puzzle it seems to me to be as much about the question of what is it in a field that exhibits apparent motion, as whether there is an induced angular motion in a particle or waveform. It intuitively seems that Spin is more a measure of a physical property that doesn't undergo a coherent rotation. It's weird! :) jrc

Thanks Doc,

That summation directs to good reading and really helped me connect dots in your argument. :-) jrc

Edwin,

I am not ignoring "the QM assumption of a 3-component vector", I am disputing it, as a misinterpretation of reality. The problem is not that the quantum world behaves oddly, but that the classical world behaves much more oddly than people suppose.

The reason people suppose they understand classical behavior, is simply because they have never, ever encountered the one type of classical behavior that they do not understand at all; an object encoding only a single bit of information. Like highly unstable, radioactive atoms, such objects do not exist in the natural world. Hence, they have never been observed; but they can be created. And they do not obey the triangular correlation function, so often discussed in regards to Bell's theorem, in order to claim that quantum and classical behaviors differ.

Furthermore, the hallmark, the signature, the fingerprint of such an entity, is that it will exhibit only two states, when one attempts to observe it, and it MUST obey the uncertainty principle (which, contrary to popular belief, has nothing to do with QM, but is a purely mathematical consequence of Fourier analysis). This is easily demonstrated if one considers the very poorly understood meaning of Shannon's capacity theorem, and the resulting uncertainty principle.

Shannon's capacity theorem is virtually always derived and discussed in such a way as to completely obscure its simple meaning:

The maximum number of bits of information that can be recovered from a signal, cannot exceed the number of bits of digitized data, required to completely reconstruct the continuous signal, to an arbitrary highly degree of accuracy. The latter number is simply equal to the number of samples, multiplied by the number of bits per sample, needed to reconstruct the continuous signal. The number of bits per sample is determined by the signal-to-noise ratio; that is the log-base-two-2 in the expression for Shannon's capacity. The number of samples is the product of the time-duration and the bandwidth; that reduces to the uncertainty principle, in the following special case:

Consider a signal in which the bandwidth is so restricted, that all samples within the time duration of the signal, have become so highly correlated, that there is only one independent sample. Then suppose that the signal-to-noise ratio is equal to 1.0, so that the single independent sample has only one significant bit. That is the origin of all observations that obey the uncertainty principle, and exhibit only two states; an entity encoding only a single bit of information.

Here is how to construct such a signal classically:

1) Create a polarized coin such that one semi-circle of one side is red, and the other semi-circle of the same side is green.

2) Represent each pixel in the image of the coin by +1 for red, -1 for green.

3) Correlate the coin against rotated versions of itself, to "decide" if the correlation is better aligned with green of red.

4) Now compute the decision correlation statistics versus rotation angle; you will get a triangular function

5) Now add noise and blur the image, such that only a single bit of information remains.

6) You will not be able to see the polarization visibly, the image is too noisy and blurry.

7) But you can correlate a clean image against the blurry one and compute the correlation statistics

8) But you will not get a triangular function.

Rob McEachern

Dear Edwin,

There are many claims in your paper that need to be discussed, but perhaps it is best to start with the main one. You claim to have produced a local theory that nonetheless predicts violations of Bell's inequality. But the theory simply does not appear to be local in Bell's sense. Of course, if one enforces certain global conservation restrictions on a system, that will have consequences for what is observed. The perfect anti-correlation between results for spin measurements in the same direction on particles prepared in the singlet state, for example, is predicted by enforcing 0 net spin for the system. But if each particle is not in a state which predetermines the outcome of the experiment, and is completely unaffected by whatever distant experiment is carried out, then enforcing the global conservation means that theory is not local in Bell's sense.

Let's put is more directly. Suppose that a system has two, widely separated parts and I carry out experiments on the parts at space-like separation. And suppose that as a consequence of carrying out an experiment on one part, the energy of that part changes. If I now enforce global energy conservation, so the energy that disappears from one side must appear on the other space-like-separated side, then the theory is not local in Bell's or Einstein's sense. Calling this non-local interaction between the sides "energy exchange" does not change this: energy exchange between space-like separated subsystems is a violation of locality. So it is not a surprise that your model can generate violations of Bell's inequality: it is not a local model.

The claim that your model is local is critical part of the paper: if it were true, then you wold have shown that there is some flaw in Bell's reasoning. As for your remarks on Bell's argument, you seem to have mistaken an illustrative example that he gives for part of the theorem itself, which it is not. The theorem is about any theory--whether the theory uses quantum-mechanical formalism or a completely different formalism in which there is no talk at all of eigenstates or eigenvalues--that makes certain predictions about correlations between outcomes of distant experiments. Because of this complete generality, it is not even correct to us the term "hidden variables" to describe the theorem, since that term itself is used only in connection with quantum theory. Peres, who you cite, has it right here. Since Bell's theorem only refers to the results of experiments and their correlations, he makes no assumptions at all of any kind about the theory predicting those results, save that it is local. His theorem does not apply to "energy exchange physics" because in this setting the energy exchange would not be a local process, and the theory would not be local. It is therefore not a counterexample to Bell's theorem.

Tim,

"If I now enforce global energy conservation, so that the energy that disappears from one side must appear on the other space-like-separated side..."

Where does he say that? A particle need not refer to a point exterior of itself to know its initial orientation to its state of motion. That is clearly implied by our definition of inertia globally, regardless of relative local energy transfers. If you are in a closed spacecraft intergalactically and the interior appears to be tumbling around you, is it? Gently, jrc

Edwin,

It is true that 1-bit events do not agree with the standard quantum mechanical interpretations. But they do agree with both classical and QM observations. That could be just a marvelous coincidence, but I think not.

Best Regards,

Rob

Dear Edwin Eugene Klingman

I hope you will forgive me for not commenting until now after your comment on my essay.

Bell's theorem, EPR, Bohm's suggested experiment, the 1982 Aspect experiment, all seem very interesting to me. Your essay indicated there are interesting conceptual issues still outstanding regarding entanglement. Although I have some familiarity with Korzybski's general semantics (you refer to him on page 1), I do not feel I know enough about QM to comment on the math in your article.

You write: (p. 2:) One must apply the right map at the right place. You quote: "Complex problems have simple, easy to understand, wrong answers." (P.2).

My comment: Maybe (and I wonder if this might be the case) entanglement would be a simpler problem with an easier to understand solution in a different conceptual reference frame. That seems to be an implicit possibility raised by your article.

Thank you for your article, and all your comments on the various essays.

Best wishes.

Bob Shour

Dear Sir,

You have rightly quoted Korzybski to say that math is the map and the physical world is the territory. But the map is not "any" territory - it describes the physical boundaries of a specified territory. This is what we say mathematics is not the sole languages of Nature, but only exhibits its quantitative aspect. We may have many "maps" of the universe, but each represents different features. Geometry, the mathematics of maps, always relates to two or three dimensional fields or structures, where the mathematics is always non-linear (distance is linear and its calculation is not geometry), even though both lead to perception of relations and patterns. Problems arise when we treat fields to represent integers. Fields are always analog, whereas eigenvalues are always discrete. The processes are dynamical, because all mathematical operations involve dynamics of the constituents. You have also said "the eigenvalues are generally taken to be truly representative of the system". Analog fields cannot be the sole representation of integers, as numbers are discrete and linear. The unit makes them non-linear. You also imply the same thing when you say: "Once a counter produces a number, another machine can add (subtract) this number to a different number to yield a new number".

We prefer mathematically simple theories to complex ones because Nature is economical. All thermo-dynamical processes lead to entropy to finally reach equilibrium. Each step takes the minimum energy to evolve in time subject to what you call as eigenvalue maps. Two spin eigenvalue maps differ because they are different or as you quote Messiah: "the initial states are statistically distributed over a somewhat extended domain".

You are right that a local model does produce correlation, based on energy-exchange physics. But when we analyze the underlying physics, some assumptions of quantum mechanics become questionable. A statistical model cannot ensure that all relevant parameters have been woven into it simply because our measurement processes are unitary - we measure limited aspects of a system over limited time. Generalizing the result of such measurement is fraught with the dangers of embracing uncertainty. As we have often said, uncertainty is not a law of Nature. It is the result of natural laws relating to measurement related to causality that reveal a kind of granularity at certain levels of existence. Since time evolution is not uniform, but conditional on interactions, we do not see each step from the flapping of the wings of the butterfly till it turns into tempest elsewhere. The creation is highly ordered and there is no randomness or chaos. We fault Nature to hide our inability to know.

For example, contrary to general belief (especially with reference to EPR), entanglement does not extend infinitely, but breaks down after some distance like a rubber string. Or it may remain exclusively like a pair of socks, though used only in pairs. Energies behave like a pair of socks - they co-exist. Interdependence of every system in the universe with all other systems makes one 'energy' to act, when a 'related energy' acts. This is not truly energy exchange. This principle also applies to your model, which Bell suppressed. Because of interdependence, no local model could reproduce the quantum mechanical prediction based on limited data over limited space and time. This is what Bell tells the hidden variables. Your constant field shows local equilibrium. The inhomogeneous field shows interdependence, which, as you say, can cause transitions. When Bell says: "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics", he ignored this interdependence.

Congratulations for presenting a complex model in fairly simple manner. We have clarified your comments in our post.

Regards,

basudeba

Dear Edwin,

To say that it is "de facto true" that Bell's example about spin is just illustrative and no part of the theorem does not address the point. The entire discussion of the detailed model makes no contact with the theorem. The theorem holds of any theory at all that is local (in the sense Bell articulates) and makes certain statistical predictions. That these are predictions about anything called "spin" or anything treated quantum-mechanically is no part of the theorem at all. All one needs are the conditional probabilities for outcomes of certain experiments, which need not be described in any more detail than "Instrument 1 is set to setting A" and "the outcome is outcome 1" or "outcome 2" One can use "spin measurements" in quantum theory as instances of this sort of thing, where the setting is the orientation of the Stern Gerlach magnet and the outcome is a spot on a screen appearing in one place or another. Clearly any theory at all might make such predictions. Since the theorem is only about these sorts of conditional probabilities, it is in no way "about" quantum theory.

Your initial characterization of the question Bell was asking is not accurate. He was not asking whether one could somehow find a theory that predicts the outcomes of experiments deterministically, he was interested rather in whether any local theory at all (deterministic or probabilistic) could recover a certain set of predictions. He insisted on this many times, and complained that his point had been almost universally missed. In fact, the paper relies on an understanding of the EPR argument, which had already established that locality can only be recovered in a situation with perfect EPR correlations if the theory is deterministic, but, as Bell says, "It is important to note that the limited degree to which determinism plays a role in the EPR argument, it is not assumed but inferred. What is held sacred is the principle of 'local causality'-or 'n o action at a distance'. Since the EPR correlation are recoverable by a local theory only if it is also deterministic, one can then ask about constraints on such theories. Bell demonstrates such constraints.

On p. 4, you list what you call "Bell's key physical assumptions". None of these are assumptions or premises of his theorem. The theorem applies to any situation in which the outcomes of certain experiments can be categorized as, e.g., "outcome 1" or "outcome 2", and correlations between the outcomes on different sides predicted. The theorem, which is not particularly about spin, has none of these assumptions as premises, so no discussion of them can have any significance for the theorem.

What is particularly odd about your presentation is that you claim that Bell has a "hidden constraint" in his proof, but nowhere actually discuss the proof itself, but rather only the illustrative example. It would help if you would actually point out where in the proof the supposed constraint appears. Your rather extensive discussion of the toy model makes no direct contact with the theorem itself.

As for your own model, let me try to understand the claim that you make. Your equation 4 has the consequence, as you say, that the deflections produced by Stern-Gerlach magnets will not be quantized, that is, that we cannot, as a practical matter, distinguish the outcomes into two classes, usually denominated "spin-up" and "spin down", determined by the location of the detected particle. If that is correct, then your model certainly does not reproduce the actual phenomenology reported in the lab, nor the predictions of quantum theory. Since the correlations discussed by Bell are correlations between the outcomes on the two sides, which are taken to always be either "spin-up" or "spin-down", and since these are also the predictions of quantum theory, then it would appear that your model actually makes no contact with Bell's topic. You do not explain how the top graph on p. 7 was created, or even what it means. Here is a key sentence from that page: "If I throw away this θ -information by truncating the measurement data, i.e., setting the results to A, B = ±1 , my constrained model cannot produce the correct correlations." The obvious reading of this sentence is that in your model, the outcomes of the experiments are not categorized into two classes, spin-up and spin-down outcomes, and that if one requires such a categorization of the outcomes then you get Bell's result. But it is an observed fact that the outcomes do sort into these two classes, and it is a prediction of the quantum theory that they will, and furthermore if they do not then it is not at all clear what the meaning of "correlation" in your theory is, since the predicted correlations are between these binary results. So you it would help if you could do these things:

1) point out where the supposed "hidden constraint" actually appears in Bell's theorem.

2) Explain, if your model does not predict quantized outcomes for spin experiments, what bearing it has on quantum theory, or Bell's theorem, and what you even mean by a "correlation" between the results on the two sides.

Regards,

Tim

Hi Edwin,

I congratulate your professionally written essay.

In another thread you said:

"I built first in my mind and then built a theory around. I believe modeling physics in your mind and then describing it mathematically is to be preferred to studying math and trying to guess what physics it describes. I believe that much math does not describe 'reality' in the same sense that much fantasy and fiction do not describe reality."

I agree with this statement completely. I followed this procedure to formulate Model Mechanics. Although we have different models of reality, but that is to be expected.

I believe that there are many assumptions in relativity are wrong. Specifically the idea of Relativity of Simultaneity (RoS). Why? Because it is in conflict with the idea that the speed of light is isotropic in all frames.

Regards,

Ken Seto