Bell’s Theorem Refuted: Bell’s 1964:(15) is False

[gill1109]

Dear Gordon

I admire your tenacity and your systematic approach.

The OP emphasises: that every Paragraph and Equation is numbered; that this thread is for focussed critiques and questions; etc. I could be mistaken, but I don't, at this moment, recall one Para # or one Eqn (.) cited by you?

You are not mistaken.

Here is a small data set of a CHSH experiment with N = 4. The format is setting Alice, setting Bob, outcome Alice, outcome Bob

1, 1, +1, +1
1, 2, +1, -1
2, 1, -1, -1
2, 2, -1, -1

The four empirical correlations are +1, -1, +1, +1
CHSH: S = 4
I have a LHV model (detection loophole) which gives exactly these 4 correlations however large N.

With a detection loophole? @%$

If you want a loophole-free LHV simulation model to generate an outcome like this (but with large N) you'll have to try many, many times ... but once in a long while it will happen. Probably you'll have to wait longer than the life of the universe if N is at all reasonably large.

My point: the existence of data-sets violating CHSH does not prove Bell's logic is wrong.
Yet that is exactly what you are saying ("experimental violation proves there must be an error")

I trust you mean "experimental data sets" -- and not loophole-based variants? And understand that science advances via both theory and practice working together?

To date, there has been no successful experiment which did not have a big loophole problem.
You don't have to tell me how science progresses.

I conclude that your logic is not mine.

A quoted Paragraph number # or Equation (.) from my 2-page-text would be still welcomed by me: with a direct response assured.

I have explained to you that the logic of the opening sentences of the OP (original post?) is (a) defective and (b) shows that you are not well informed concerning experiments which have been done to date: you said "experimental violation proves there must be an error". Moreover Bell explicitly wrote that he expected a loophole-free experiment violating his inequality to get done some day. So he both thought that he had proven a true theorem and he thought that experiment would violate his inequality. How do you explain this lapse in his "logic"? Or doesn't this suggest that you haven't actually grasped what the whole point must be?

Bell proved, given background assumptions, that if A then B. That was a little mathematical theorem. Nothing to do with experiments. Just pure math. But B had not been seen yet, and indeed, it still has't been seen to this day; at least, not when the background assumptions have also been enforced!

He expected that experiment with background assumptions enforced eventually would show "not B". He therefore perfectly logically expected that we were going to have to give up "A".

Actually A can be decomposed as (A1 & A2 & A3). He already knew this too. So if, with background assumptions imposed, experiment shows us that B is wrong, then at least one of A1, A2 and A3 must be false. And this is exactly what Bell has patiently explained time and time again and with great humour and wonderful illustrations.

Since Bell was a "realist" and since he thought "conspiracy" is stupid, he knew perfectly well that he had to give up "locality". So you are attacking Bell but clearly not well informed what Bell actually thought. There has been 50 years of discussion since Bell (1963). Bell himself has lucidly and patiently "solved" many of your issues. You are not the first one to have had these particular misconceptions.

To repeat: I see a lot of problems (a) with your logic (b) with your knowledge of relevant material, and I do also think that I have supported this "opinion" with quite a lot of pretty clear evidence. But sure, we can agree to differ...

Note: background assumptions = event-ready detectors, no non-detections, binary outcomes, ... See Chapters 13 and 16 of Speakable and Unspeakable

[Gordon Watson]

Hi Gordon

Aha ... , now I think I see why you are trying to do what you are doing.

Thanks Ben; I wish you well in your thinking: but, from what you write below, you seem to have so far gained little idea of what I have done.

As will be evident from my comments below: dipping into my essay to support your idea that AA = 1 has not worked well for you.

In my note about integration I used x instead of your λ. And my x is one dimension in Euclidean 3D space aka flatland. And I am sure that I am correct in flatland. Bell's Inequality is trivial to prove pictorally in flatland in a couple of lines based on a Venn diagram.

My example used your example. I make no mention of λ: I was proving the case against your AA = 1 via your own analogy.

What you seem to be doing is having a multidimensional x (i.e. λ) axis. That is your way of getting more dimensions into your equations. QM gets more dimensions by using a different maths (in Hilbert space). Joy gets more dimensions by using Clifford Algebra. You are getting more dimensions by saying x is multi-valued at what is only one point in flatland.

Ben: EPRB is an experiment in good-old fashioned 3-space; the space where we live, move and have our being. The vectors I use are the same as Bell. The λs I use are as specified by Bell in his final paper; please read Para. #1 of my essay.

So if I think of the multidimensions as a double cover (as I interpret it in Joy's model) at x, ie x1 and x2. A function which is apparently F in flatland is really an aggregate of F1 in cover 1 and an F2 in cover 2. And F1 and F2 can be quite different.

I see F1 and F2 as, in the total dimensionality, somehow contributing to the appropriate correlation, whereas F1 and F2 aggregate in flatland to an inappropriate correlation for particle behaviour.

Sorry; this looks beyond anything I would ever consider when analysing a real experiment; one that has clearly been conducted in 3-space. To be clear: I'm out to find and fix the error that must exist in Bell (1964); read the Abstract. I'm here to deliver Bell's hopes and expectations; consistent with the commonsense local realism of Einstein: use the link in my essay to another essay of mine.

I have only looked at a few equations in your paper (wrt AA = 1).

Ben: there are 12 equations in my essay; most are trivial; all within the bounds of undergraduate maths and logic.

Are you doing multidimensional calculations in flatland space?

Ben, a gentle suggestion that will be far more productive than what you have been doing: simply read my essay from the first word and try to get through to the last. (Has there ever been a shorter essay on Bell? Just 2-pages of text and equations?) Where you get stopped, have a think about it, then move on. Then go back and do it all again. If you get stopped on this second reading: bring the problem here and we'll sort it out; no worries.

Wishing you well, with good luck in all your studies; Gordon

[gill1109]

Gordon, your three page vixra pdf starts with:

Generalizing Bell 1964:(15) to realizable experiments, CHSH (1969) coined the term “Bell’s theorem”. Since the results of such experiments (eg, see Aspect 2002) contradict Bell’s theorem: at least one step in his supposedly commonsense analysis must be false.

CHSH coined the term "Bell's theorem" but by that term they did not mean the CHSH inequality.
So the first sentence exposes a major misunderstanding of the writer.
The second sentence further confirms that the writer doesn't know what he is writing about.

We had earlier tried your approach of going through your argument step by step and we earlier found out exactly where it derails. Apparently you did not understand several different persons' attempts to explain to you. So why should anyone else have a new try?

My conclusion was that you apparently also do not know enough undergraduate level probability and statistics in order to appreciate the difference between theoretical expectation value computed according to the rules of probability theory, and an average of repeated realizations observed in an experiment.

I think you need to read Bell's later papers, and the CHSH paper, and Caroline Thompson's webpages, and a basic text on statistics and probability, e.g. "Introduction to Mathematical Statistics and Data Analysis" by John A. Rice. Till then, no one is going to be able to convince you that you are wrong, and you are not going to be able to convince many people that you are right. Quite simply, you will change no-one's existing opinion.

[Ben6993]

Hi Gordon

I am not going to be able to get past the line where you disagree with AA=1. In normal everyday maths, an integration or a discrete summation is not done in the way that you are doing it.

Sorry, I can't raise the energy/motivation to do further checks knowing that one stage is impassable for me. I believe Michel did seem to support your logic, but I just cannot see it myself.

Best wishes

[Joy Christian]

I am not going to be able to get past the line where you disagree with AA=1. In normal everyday maths, an integration or a discrete summation is not done in the way that you are doing it.

Sorry, I can't raise the energy/motivation to do further checks knowing that one stage is impassable for me. I believe Michel did seem to support your logic, but I just cannot see it myself.

Ben,

There is nothing wrong with Bell's assumption that AA=1 at the point he makes that assumption. While Michel is not wrong in his logic, the issue is rather subtle.

Bell is completely justified in assuming AA=1, because that is what Einstein's "strongly objective" view of reality demands. Bell is simply implementing Einstein's point of view. So if there is anyone to be blamed for this assumption, it should be Einstein (and I say this despite being a staunch supporter of Einstein's view). Without Einstein's "strongly objective" view, the EPR argument for the incompleteness of quantum mechanics simply does not go through. This is what Bohr pointed out, or objected to, in the EPR argument. According to Einstein's view a measurement of a variable could be made in principle, but was not necessarily made in practice, to provide data for the prediction of a given attribute of Nature---for example, a value of the spin of an elementary particle. Unless EPR held this "strongly objective" view, they could not argue that an element of reality existed for all of the components of spin of the particle, those which could have been measured as well as the one that was actually measured.

Thus, in short, we simply cannot blame Bell for the assumption that AA=1. If we must, we must blame Einstein.

[minkwe]

I must disagree with both Ben and Joy here. Watson is correct. Bell starts his derivation by adding P(a,b|{λ}) + P(a,c|{λ})

Now what did he mean by that addition. It can be "Weakly objective", as in actual experiments and QM, is each correlation is from a different set of particles, or it can be "strongly objective",is each correlation is from the same set of particles. Both interpretations are fully consistent with Einstein's elements of reality, but they are not the same.

If I give you an average height, it could be the average of measuring the same persons height 100 times or it could be the average of measuring 100 different peoples heights each exactly once. Yet in both cases, the objective property of height exists.

In EPRB experiments you can never measure the same particles more than once so it should be the weakly objective view that applies to Bell's derivation. However his algebra implies that he is using the strongly objective view.

In the strongly objective view he has P(a,b|{λi}) + P(a,c|{λi}) (since he has the same set of particles in each correlation i=1..N) and the rest of his algebra works. However in the weakly objective view, the only one applicable to actual EPRB experiments, he has P(a,b|{λi}) + P(a,c|{λj}) where the two sets of particles i=1..N, j=1..M are not the same and the rest of his algebra does not work. This is what Watson has shown.

Therefore either:

1) Bell's derivation is based on the "strongly objective" view and is correct but can not be compared with QM predictions and actual experimental results which are weakly objective, or
2) Bell's derivation should be comparable to actual experiments (weakly objective) but is wrong as Watson shows.

[gill1109]

I must disagree with both Ben and Joy here. Watson is correct.

(1) Wonderful! We have a contradiction! Now we can hope to make some progress. (Niels Bohr, not a literal quote, but just what I am able to recall just at this moment).

(2) The opposite of a truth is a falsehood. The opposite of a fundamental truth however is often another fundamental truth. (Niels Bohr, not a literal quote, but just what I am able to recall just at this moment).

I think that Niels Bohr was actually rather wise; doubly wise in fact.

(1) when we have an apparent contradiction in science, we often do make progress ... two theories which give different predictions, means that we can do an experiment to find out which one is the good one.

(2) When contradictions turn up in "logical" reasoning they often actually show us that our categories of thought are deficient.

[Joy Christian]

Both interpretations are fully consistent with Einstein's elements of reality...

I am afraid not. Here is the famous and much analysed criterion of reality: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."

Now what does "can predict" mean here?

Does it mean that data are at hand for making the prediction, or does it mean that a measurement could be made in principle to provide data for the prediction?

Unless EPR assumed the latter, their argument simply does not go through. That is to say, without the latter meaning of "can predict" in their reality criterion they cannot conclude that the quantum mechanical description of reality is incomplete.

Bell is simply implementing the latter meaning of "can predict" when he adds probabilities in P(a,b|{λ}) + P(a,c|{λ}), because that is the meaning Einstein intended.

[minkwe]

As concerns AA = 1, P(a,a|{λ}) = 1 is true no matter the interpretation, since A(a,λi)B(a,λi) =1, and A(a,λj)B(a,λj) =1. However, Watson's point is that, A(a,λi)B(a,λj) =1 is not necessarily true even if A(a,λi)B(a,λi) =1, and A(a,λj)B(a,λj) =1 are true at the same time.

To see this, if you label the sides of coins +1 and -1 instead of H and T, then toss one coin (coin λi) on a glass table with Alice looking from below (multiplying her results by -1 always) and Bob from above. A(a,λi)B(a,λi) = 1 will always be true for every toss of coin λi. How ever we may toss two identical coins (λi,λj) instead. Even though A(a,λi)B(a,λi) =1, and A(a,λj)B(a,λj) =1will always be true, A(a,λi)B(a,λj) =1 will not necessarily be true.

This is the point, and it is fully consistent with EPR elements of reality which are present in both coins (λi and λj).

The problem arises because, when you insist on carrying out Bell's derivation under the weakly objective view as Gill does, you generate terms within the integral involving different subscripts. In other words, the crucial factorisation step can not proceed.

[Joy Christian]

The problem arises because, when you insist on carrying out Bell's derivation under the weakly objective view as Gill does...

I agree.

But Bell is not doing that. Therefore Watson's argument against Bell is misplaced. Bell was an Einsteinian, and implemented Einstein's views as strictly as possible.

[minkwe]

I am afraid not. Here is the famous and much analysed criterion of reality: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."

Now what does "can predict" mean here?

Does it mean that data are at hand for making the prediction, or does it mean that a measurement could be made in principle to provide data for the prediction?

Its a prediction period. It means IF we measure it, we will obtain such a result. It doesn't mean all such predictions are simultaneously measurable. Just because you have a weakly objective interpretation does not mean you can not predict with certainty and still have elements of reality corresponding to the prediction.

Bell is simply implementing the latter meaning of "can predict" when he adds probabilities in P(a,b|{λ}) + P(a,c|{λ}), because that is the meaning Einstein intended.

I can predict with certainty the results P(a,b|{λi}) for one set of particles and at the same time I can predict the results P(a,c|{λj}) for a different set of particles. According to EPR, that means elements of reality corresponding to both predictions exist. There is no conflict between EPR elements of reality and the weakly objective view. You can predict with certainty in both and have elements of reality corresponding to the predictions in both.

[gill1109]

The problem arises because, when you insist on carrying out Bell's derivation under the weakly objective view as Gill does, you generate terms within the integral involving different subscripts. In other words, the crucial factorisation step can not proceed.

Bell's derivation is carried out within a mathematical model, within which there is no issue of interpretation whatsoever. There are just two or three functions and some calculus.

The interpretation of probability is needed to make a bridge between probability concepts in the real world and how we do probability theory within mathematics.

So there is no problem at all. Gordon (and Michel) are (IMHO) confusing levels, confusing categories. Mixing up mathematics (including mathematical physics) and reality.

We can prove theorems about the real numbers by imagining them embedded in the complex number plane. We can apply those theorems to physical problems which only involve real numbers.

[gill1109]

I am afraid not. Here is the famous and much analysed criterion of reality: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."

Now what does "can predict" mean here?

Does it mean that data are at hand for making the prediction, or does it mean that a measurement could be made in principle to provide data for the prediction?

Its a prediction period. It means IF we measure it, we will obtain such a result. It doesn't mean all such predictions are simultaneously measurable. Just because you have a weakly objective interpretation does not mean you can not predict with certainty and still have elements of reality corresponding to the prediction.

Bell is simply implementing the latter meaning of "can predict" when he adds probabilities in P(a,b|{λ}) + P(a,c|{λ}), because that is the meaning Einstein intended.

I can predict with certainty the results P(a,b|{λi}) for one set of particles and at the same time I can predict the results P(a,c|{λj}) for a different set of particles. According to EPR, that means elements of reality corresponding to both predictions exist. There is no conflict between EPR elements of reality and the weakly objective view. You can predict with certainty in both and have elements of reality corresponding to the predictions in both.

Sorry Michel, Einstein and Christian are right here, and you and Gordon are wrong (IMHO).

Einstein in fact uses the predictions of quantum mechanics (and the assumption of locality) in order to deduce or derive CFD (counter-factual definiteness). And if things exist (within a model of reality) then even if we can't measure them we can still calculate expectation values of them ... within that model. What Michel calls "weakly objective" appears to be just the usual frequentist interpretation of probability, which is not something inside the models, but is a bridge connecting our experiences in the world of common sense with theoretical objects within a mathematical world (within the mathematical world of some particular mathematical-physical model of reality).

[minkwe]

The problem arises because, when you insist on carrying out Bell's derivation under the weakly objective view as Gill does...

I agree.

But Bell is not doing that. Therefore Watson's argument against Bell is misplaced. Bell was an Einsteinian, and implemented Einstein's views as strictly as possible.

Bell does not say what he is doing. We can infer from his factorisation that he is using the strongly objective view (factorisation is not possible otherwise). But he is discussing an experiment which can only ever produce weakly objective results (no matter what we imagine or predict) and then he compares his results with weakly objective predictions from QM. This is similar to Alice and Bob deriving a relationship from the symmetry of a single coin and expecting it to apply to results obtained from two separate coins. That is why Watson insists that A(a,λj)B(a,λj) =1 is not necessarily true. If you try to make Bell's derivation consistent with performable experiments, it doesn't work. This is the point.

[minkwe]

Richard, you clearly do not understand the issues so you can't contribute anything meaningful. Bell is not just doing abstract mathematics, he has an experiment in mind, the EPRB experiment. Not only that he goes on to compare his results with QM, which is not just an abstract mathematical theory, but a theory about results of performable experiments. Not only that, Bell's followers (including you) compare his results with outcomes of actual experiments and on the basis of those comparisons make claims about nature and the physical world.

It is not going to work to simply dodge the question of which interpretation Bell is using, by invoking mathematical insanity. Bell's derivation can only work in the strongly objective interpretation. This is a fact, as Gordon shows, and you admit in your LG paper. Yet the correlations from QM and experiments are clearly weakly objective. The remaining options are clear:

1) concede that the inequalities are irrelevant for physics
2) concede that if they should be relevant for the EPRB experiment then they are wrong

[gill1109]

Richard, you clearly do not understand the issues so you can't contribute anything meaningful. Bell is not just doing abstract mathematics, he has an experiment in mind, the EPRB experiment. Not only that he goes on to compare his results with QM, which is not just an abstract mathematical theory, but a theory about results of performable experiments. Not only that, Bell's followers (including you) compare his results with outcomes of actual experiments and on the basis of those comparisons make claims about nature and the physical world.

It is not going to work to simply dodge the question of which interpretation Bell is using, by invoking mathematical insanity. Bell's derivation can only work in the strongly objective interpretation. This is a fact, as Gordon shows, and you admit in your LG paper. Yet the correlations from QM and experiments are clearly weakly objective. The remaining options are clear:

1) concede that the inequalities are irrelevant for physics
2) concede that if they should be relevant for the EPRB experiment then they are wrong

I think your view of the relationship between mathematics and physics and experiment is too simplistic. (Gordon's too).

Oh well, never mind. We'll have to agree to differ.

[gill1109]

Bell does not say what he is doing. We can infer from his factorisation that he is using the strongly objective view (factorisation is not possible otherwise).

Michel Fodje makes this inference.

But in the mathematical model we are using (a LHV model) we do have CFD and factorization is possible.

You cannot infer Bell's interpretation of probability from how he does calculus within a mathematical world where there are functions A, B and rho. No interpretation of the functions is needed, in order to do some calculus on those functions. We are inside a mathematical world and we do mathematics. Free of any interpretation.

Fodje is saying that if I have x = 3 apples and y = 4 oranges, I am not allowed to calculate x - y = -1.

x = 3
y = 4
x - y = -1

That is mathematics

What it might or might not mean in the real world where x is a number of apples, and y is a number of oranges, depends on the context.

If Alice has x apples and Bob has y oranges and they want to do a swap where both apples and oranges are worth 1 Euro, then x - y = -1 certainly does have a meaning.

It doesn't mean that you can subtract apples from oranges.

[minkwe]

I think your view of the relationship between mathematics and physics and experiment is too simplistic. (Gordon's too).

Oh well, never mind. We'll have to agree to differ.

On the contrary it is your view that does not take into account all the subtleties involved in relating mathematics to experiments. When challenged you claim that Bell is not talking about experiments, contrary to the fact that he clearly is. I've asked you a simple question elsewhere and Gordon has been asking you a simple question many times which you are unable to answer yet you distract by making vain accusations.

[minkwe]

Bell does not say what he is doing. We can infer from his factorisation that he is using the strongly objective view (factorisation is not possible otherwise).

Michel Fodje makes this inference.

But in the mathematical model we are using (a LHV model) we do have CFD and factorization is possible.

Clearly you do not understand what you are talking about. I've clearly explained to you using a LHV model how CFD arises in a strongly objective interpretation, but does not arise in a weakly objective interpretation. You still haven't presented a clear view of how it might arise in a weakly objective view which you already claimed is the one you are using and Bell was using. Gordon has clearly shown that if Bell is using the weakly objective one as you have claimed, then his derivation fails woefully. You have no response to that but would rather make empty accusations.

You cannot infer Bell's interpretation of probability from how he does calculus within a mathematical world where there are functions A, B and rho. No interpretation of the functions is needed, in order to do some calculus on those functions. We are inside a mathematical world and we do mathematics. Free of any interpretation.

I surely can.

In the strongly objective view, I can factorize the expression A(a, λi)B(a, λi) - A(a, λi)B(c, λi) into A(a, λi)[B(a, λi) - B(c, λi)]. This is clearly what Bell does in his derivation. In a weakly objective interpretation, I can not factorize A(a, λi)B(a, λi) - A(a, λj)B(c, λj) into A(a, λi)[B(a, λi) - B(c, λj)]. Since A(a, λi) =/= A(a, λj) in the weakly objective view.

Therefore when Bell factorizes A(a, λ)B(a, λ) - A(a, λ)B(c, λ) into A(a, λi)[B(a, λ) - B(c, λ)], he MUST have been using the strongly objective view. That is the only view in which CFD enters his equations. This is very basic algebra. Your examples about oranges and apples completely miss the point and are designed to introduce murkiness rather than clarity.

The point Watson makes is very clear and it is revealing that you keep arguing against it when you must know or should know that it is correct. Bell's factorization does not work in the weakly objective interpretation which you have chosen.

[gill1109]

Clearly you do not understand what you are talking about. I've clearly explained to you using a LHV model how CFD arises in a strongly objective interpretation, but does not arise in a weakly objective interpretation. You still haven't presented a clear view of how it might arise in a weakly objective view which you already claimed is the one you are using and Bell was using. Gordon has clearly shown that if Bell is using the weakly objective one as you have claimed, then his derivation fails woefully. You have no response to that but would rather make empty accusations.

You use the word "clearly" a lot.

I could mirror a lot of what you have said. You clearly do not understand what you are talking about etc etc etc. (In fact, that is exactly what I think).

But then you also go on to make allegations about my motivations. You say that I would "rather make empty accusations". This is like a judge. You know I killed someone. In order to determine my sentence, you have to decide whether it is murder or manslaughter. This means that you have to somehow determine my motivation. Was it premeditated? Or did it happen on the spur of the moment, because I was so provoked that I went crazy and killed you "by accident" ...

You appear not only to understand the logical issues with crystal clear clarity, but also to be able to look inside my soul and know something about my conscience.

But you are not a judge! You and I are both scientists. And gentlemen. (On this forum, at least).

****

Suppose that what you say is not clear to me at all! Suppose that everything that I say, appears to be very clear to me! Then I am not "making empty accusations". I am then just a kind (but stupid, misguided, etc etc) person honestly trying to help you to see the error of your ways.

Gordon and me - that's exactly the same. He knows that he has sassed it all out and he is kindly trying to help me. I know that I have sassed it all out and I am kindly trying to help him. We both think that our arguments are crystal clear and that our logic is impeccable. We are both acting with the best of intentions, in perfect "good faith". We both are flumoxed that the other does not "see" the persuasive force of our arguments.

"C'est la vie". We will just have to go on respecting one another, even if we disagree with one another. We'll have to agree to disagree. Argue about it in the pub, and then drink another beer together.

Sorry for the lecture. I am past 60 so tend to pontificate ... I think you are a lot younger, so you can of course *think* all I say is s*** (probably that is a wise "default opinion"), but when you want to say so in public, you ought to be polite about it. It is fine that you think and even say (politely) that everything I say is s***! But please don't keep making inferences (in public) as to my motivations, character ... Keep it scientific. Keep it impersonal.

BTW: fantastic sophisticated forum software. If you write BS out in full (eight letters), it converts that to "s, triple asterisk". It has an automatic BS detector.