Bell's inequality refuted via elementary algebra

[minkwe]

Please explain the physical significance of your labels.

PS: if you want to know why I ask, turn on PM and alert me.

Thanks.

Physical significance of labels is explained in the linked paper. It should be obvious by now but the essence is this -- it safeguards against comparing apples and oranges.

[Gordon Watson]

Please explain the physical significance of your labels.

PS: if you want to know why I ask, turn on PM and alert me.

Thanks.

Physical significance of labels is explained in the linked paper. It should be obvious by now but the essence is this -- it safeguards against comparing apples and oranges.

Thanks for this. Good to see that we agree re the need to 'safeguard against comparing apples and oranges' under EPRB, etc.

I define the related Bellian failures to do so as Bell's first error the flawed use of Bell 1964:(1).

To that end, and seeking certainty, please note that a boundary condition on ALL of my Bell-critiques is this:

VALID paired-results arise, both theoretically and physically, from outcomes generated in the SAME instance; and not otherwise. This condition is explicit in Bell (1964) in the sentence that introduces his eqn (1). So, when I need to make this point, I use "instance-trackers" to show where Bell/CHSH -- erroneously breaching instances -- deliver results which are readily refuted, both theoretically and experimentally.

I'd welcome your comments on this. And wonder if you think more is needed?

PS: What is the physical significance of your differing subscripts: ab versus abc?

Thanks again; G

[gill1109]

To that end, and seeking certainty, please note that a boundary condition on ALL of my Bell-critiques is this:

VALID paired-results arise, both theoretically and physically, from outcomes generated in the SAME instance; and not otherwise. This condition is explicit in Bell (1964) in the sentence that introduces his eqn (1).

You have entrapped yourself, Gordon! This condition is *not* explicit in Bell. You have badly mis-read Bell.

You set a boundary condition to all your critiques. But suppose your "boundary condition" is wrong? As long as you stick to your mis-reading, you remain stuck.

I am really glad that it now is perfectly clear for everyone where the divergence of opinions starts. *You* set a boundary condition. You didn't have to...

[Gordon Watson]

To that end, and seeking certainty, please note that a boundary condition on ALL of my Bell-critiques is this:

VALID paired-results arise, both theoretically and physically, from outcomes generated in the SAME instance; and not otherwise. This condition is explicit in Bell (1964) in the sentence that introduces his eqn (1).

You have entrapped yourself, Gordon! This condition is *not* explicit in Bell. You have badly mis-read Bell.

You set a boundary condition to all your critiques. But suppose your "boundary condition" is wrong? As long as you stick to your mis-reading, you remain stuck.

I am really glad that it now is perfectly clear for everyone where the divergence of opinions starts. *You* set a boundary condition. You didn't have to...

??? Please. Not so fast:

Studying EPRB, Bell sets this boundary (or qualifying) condition when introducing his eqn (1): ie, "... in the same instance ...".

SO I did NOT have to set this boundary (or qualifying) condition because, under EPRB, Bell HAD to (and DID)!

Alas, under EPRB, for you and for Bell, it is overlooked/ignored/neglected in Bell's move from his (14a) to his (14b).

Alas, too, for me. For, as a serious student of EPRB, I CANNOT overlook/ignore/neglect it.

However, with this discipline -- or, via this one FIX -- Bell's lifetime dilemma is resolved!

QED.

G

[gill1109]

Gordon, you are the one who goes too fast.

Forget the logic for a while. You don't convince anyone who doesn't want to be convinced with logic alone.

Collaborate with someone who can program computers and let's see a computer implementation of your ideas. Take an example from Joy! He initially stood by the point of view that his logic and his mathematics were self-sufficient. But he finally did get convinced that a computer simulation would make people wake up and take notice. He was right! He found people who were enthusiastic to work with him, and who did love programming and who were really good at it. This led to a lot of exciting developments. He also came up with a proposed lab experiment. This also raised lots of interest.

[Mikko]

VALID paired-results arise, both theoretically and physically, from outcomes generated in the SAME instance; and not otherwise. This condition is explicit in Bell (1964) in the sentence that introduces his eqn (1). So, when I need to make this point, I use "instance-trackers" to show where Bell/CHSH -- erroneously breaching instances -- deliver results which are readily refuted, both theoretically and experimentally.

The mentioned sentence is not alone in Bell's paper. It is part of the section "II. Formulation". In that section only one pair of particles is discussed. Therefore saying "in the same instance" is redundant and only serves as a remainder that only the two particles mentioned in the second sentence are discussed.
The section "IV. Contradiction", where the main result is derived, does not introduce more particles, either. It does not even discuss the two introduced in section II. It only discusses the functions that were introduced in section II.

[Gordon Watson]

VALID paired-results arise, both theoretically and physically, from outcomes generated in the SAME instance; and not otherwise. This condition is explicit in Bell (1964) in the sentence that introduces his eqn (1). So, when I need to make this point, I use "instance-trackers" to show where Bell/CHSH -- erroneously breaching instances -- deliver results which are readily refuted, both theoretically and experimentally.

The mentioned sentence is not alone in Bell's paper. It is part of the section "II. Formulation". In that section only one pair of particles is discussed. Therefore saying "in the same instance" is redundant and only serves as a remainder that only the two particles mentioned in the second sentence are discussed.
The section "IV. Contradiction", where the main result is derived, does not introduce more particles, either. It does not even discuss the two introduced in section II. It only discusses the functions that were introduced in section II.

You seem to be suggesting that Bell (1964) is studying EPRB with only one pair of particles.

Am I understanding your view correctly?

Further: it might help my understanding if you'd explain the significance of your "ab" "and abc" labels.

G

[Mikko]

You seem to be suggesting that Bell (1964) is studying EPRB with only one pair of particles.
Am I understanding your view correctly?

Yes, the one that is introduced in the second sentence of the section "II. Formulation". No other pair nor any unpaired particle is mentioned anywhere in Bell's paper except that the section "III. Illustration", which is not a part of Bell's argumentation, mentions a single particle without specifying whether it is one of the pair (for that is irrelevant to the point).

[Gordon Watson]

You seem to be suggesting that Bell (1964) is studying EPRB with only one pair of particles.
Am I understanding your view correctly?

Yes, the one that is introduced in the second sentence of the section "II. Formulation". No other pair nor any unpaired particle is mentioned anywhere in Bell's paper except that the section "III. Illustration", which is not a part of Bell's argumentation, mentions a single particle without specifying whether it is one of the pair (for that is irrelevant to the point).

A. Apologies first: I meant "minkwe's ab and abc labels."

B. I disagree with your reasoning:

1. Bell p.195 allows that the variable λ may denote a single variable. So you need more than one pair of particles to get the 3 expectations in his inequality.

2. Bell p.199: "The example considered [EPRB] above has the advantage that it requires little imagination to envisage the measurements involved actually being made."

3. #2 seems impossible to me with just one pair?

4. Re #3, Bell tests generally use more than one particle pair.

5. And Bell never complained about #4.

QED.

[Gordon Watson]

1. Bell (1964:195) sets out a clear intention: to provide a "more complete specification" of the EPRB experiment.

What makes you think so? In Bell's 1964 article there is no statement of intention nor any other support for this claim.

Here's what makes me know so:

1. Bell (1964:195) sets out a clear intention: "Let this more complete specification be effected by means of parameters λ."

That does not declare or imply any intent. It is merely the introduction of the symbol λ (for the "more complete specification" of the EPR argument).

It seems very clear to me that in 1964 Bell intends to provide a more complete specification of EPRB by means of parameters λ.

Then, until his death in 1990, via many papers -- indeed, as a result of his self-confessed dilemma [which you seem to overlook?] -- his work on quantum foundations is dominated by his inability to deliver on EPRB.

QED.

[Mikko]

It seems very clear to me that in 1964 Bell intends to provide a more complete specification of EPRB by means of parameters λ.

Your dreams, no matter how clear they are, are not evidence. You cannot infer this intent from the text of Bell's 1964 paper.

Then, until his death in 1990, via many papers -- indeed, as a result of his self-confessed dilemma [which you seem to overlook?] -- his work on quantum foundations is dominated by his inability to deliver on EPRB.

What Bell did or said later is by itself insufficient for determination of the intent of his 1964 article.

[Mikko]

1. Bell p.195 allows that the variable λ may denote a single variable. So you need more than one pair of particles to get the 3 expectations in his inequality.

That "So" is false: the statement does not follow from the preceding statement.
There indeed is something important in addition to the particles: the polarizer settings. As those settings are unknown at the time when expectations are calculated, there are many different expectations for the pair.

2. Bell p.199: "The example considered [EPRB] above has the advantage that it requires little imagination to envisage the measurements involved actually being made."
3. #2 seems impossible to me with just one pair?

That statement is not relevant to the main conclusion of the article, which is already presented in the previous paragraph. But the conclusion only is that either all local hidden variable theories are wrong or quantum mechanics is wrong, leaving unanswered which is wrong; which, although outside of the scope of the article, is an interesting question. Therefore the comment about possible experiments.

[minkwe]

VALID paired-results arise, both theoretically and physically, from outcomes generated in the SAME instance; and not otherwise. This condition is explicit in Bell (1964) in the sentence that introduces his eqn (1). So, when I need to make this point, I use "instance-trackers" to show where Bell/CHSH -- erroneously breaching instances -- deliver results which are readily refuted, both theoretically and experimentally.

PS: What is the physical significance of your differing subscripts: ab versus abc?

It distinguishes between paired outcomes measured as pairs, versus pairs of outcomes measured as triples. Unfortunately very many have gone astray by ignoring the difference and some noticing the difficulty have claimed (I believe falsely) to have resolved it with statistics. Let me illustrate in terms of an illustration I posted a few days ago in another thread:

Consider a universe which consists of only three elements F, J, G, which can have relationships/interact with each other. Let us represent a relationship/interaction with parenthesis and a distinct state with curly brackets. Now imagine all possible states of these individuals. There are 5 distinct states:



Where means isolated individuals, no interactions, means F and G are interacting but J is isolated from F and G, and all three individuals are interacting.

Let us propose a measurement Q we can make. Let Q be the question. "Are F and J related?" with the result being either "True" or "False".

Thus , and

In addition, let us propose a join operation we can perform on two different states. Let this operation be relationship-preserving such that any relationships which exist in the un-joined states must also be present in the joined state. Thus, a join between two states, simply combines all relationships between in the individual states. Therefore, and .

Consider however, the the following join. . Note that , but . In other words, F and J are not related in each of the two cases but after joining, they are related. That is, the (F,J) relationship in the joined state was generated by the join. But how does this apply to the Bell situation? The point is that in trying to understand relationships between three things, measuring pairs separately and then combining them is not the same as combining the three things and directly measuring the paired relationships.

Therefore, to answer your question: "ab", "ac", "bc" represents data that was measured separately in pairs and later combined in the same expression to deduce a relationship between "a", "b" and "c". While "abc" represents data that was measured together in triples from which pairs were then extracted. As illustrated in the example above, due to generative effects, you will not obtain the same results. In fact the relationships in the paired results appear to be stronger than expected.

[gill1109]

Consider a universe which consists of only three elements F, J, G, which can have relationships/interact with each other. Let us represent a relationship/interaction with parenthesis and a distinct state with curly brackets. Now imagine all possible states of these individuals. There are 5 distinct states:



Where means isolated individuals, no interactions, means F and G are interacting but J is isolated from F and G, and all three individuals are interacting.

Let us propose a measurement Q we can make. Let Q be the question. "Are F and J related?" with the result being either "True" or "False".

Thus , and

In addition, let us propose a join operation we can perform on two different states. Let this operation be relationship-preserving such that any relationships which exist in the un-joined states must also be present in the joined state. Thus, a join between two states, simply combines all relationships between in the individual states. Therefore, and .

Consider however, the the following join. . Note that , but . In other words, F and J are not related in each of the two cases but after joining, they are related. That is, the (F,J) relationship in the joined state was generated by the join. But how does this apply to the Bell situation? The point is that in trying to understand relationships between three things, measuring pairs separately and then combining them is not the same as combining the three things and directly measuring the paired relationships.

Therefore, to answer your question: "ab", "ac", "bc" represents data that was measured separately in pairs and later combined in the same expression to deduce a relationship between "a", "b" and "c". While "abc" represents data that was measured together in triples from which pairs were then extracted. As illustrated in the example above, due to generative effects, you will not obtain the same results. In fact, the relationships in the paired results appear to be stronger than expected.

This is very interesting, Michel. You brought it up in another thread and referred there to "category theory". Would you be so kind as to give a few literature references? I do know something about category theory already, living as I do in a mathematical department full of algebraists and number theorists and other pure, pure mathematicians. But I would like to consult your sources so I can better understand where you are coming from in order to better understand where you are going to.

I agreed with everything you said here, by the way, except that I disagree with your conclusion that Bell made a mistake (as you already know, of course). I think that Bell had some cogent physical arguments for making the assumptions that he did, and his purpose was to explore those physical arguments. His conclusion was that the physical assumptions needed reconsideration. I believe that his logic and his mathematics are both impeccable. [He is limited by his time and by language and his ideas also evolved over the years; he is a "moving target"]. He does rely on some ideas from probability and from statistics, which it seems, many people find difficult if not impossible to assimilate.

And he relies on the reader's ability to separate mathematics in the context of "a mathematical model" (self-contained, no connection to the real world at all) from mathematics as a language for talking physics.

[minkwe]

This is very interesting, Michel. You brought it up in another thread and referred there to "category theory". Would you be so kind as to give a few literature references?

Hi Richard, there is no specific reference for the example above although you can deduce it from any discussion of "generative effects" or "cascade effects" in any book on category theory. As far as I'm aware, nobody else has ever applied category theory to Bell's analysis. You can consult this video https://youtu.be/UusLtx9fIjs. Generative effects are discussed starting around 12:00.

I do know something about category theory already, living as I do in a mathematical department full of algebraists and number theorists and other pure, pure mathematicians. But I would like to consult your sources so I can better understand where you are coming from in order to better understand where you are going to.

I think my example is pretty clear and self-contained.

I agreed with everything you said here, by the way, except that I disagree with your conclusion that Bell made a mistake (as you already know, of course). I think that Bell had some cogent physical arguments for making the assumptions that he did, and his purpose was to explore those physical arguments. His conclusion was that the physical assumptions needed reconsideration. I believe that his logic and his mathematics are both impeccable. [He is limited by his time and by language and his ideas also evolved over the years; he is a "moving target"]. He does rely on some ideas from probability and from statistics, which it seems, many people find difficult if not impossible to assimilate.

And he relies on the reader's ability to separate mathematics in the context of "a mathematical model" (self-contained, no connection to the real world at all) from mathematics as a language for talking physics.

As you already know, I believe Bell made a mistakes and his mistakes have nothing to do with statistics or probability but with logic. The problem is that, as a purely mathematical exercise, it is possible to argue successfully that his math is accurate. The problem is that the issue he is addressing is not merely a mathematical problem, and requires very careful consideration of the physics of the experiments. It is convenient for Bell proponents to advocate separating the mathematics from the experiment because it is easier to defend Bell that way. However, it is only possible to appreciate the error once everything is evaluated together.

[gill1109]

As you already know, I believe Bell made mistakes and his mistakes have nothing to do with statistics or probability but with logic. The problem is that, as a purely mathematical exercise, it is possible to argue successfully that his math is accurate. The problem is that the issue he is addressing is not merely a mathematical problem, and requires very careful consideration of the physics of the experiments. It is convenient for Bell proponents to advocate separating the mathematics from the experiment because it is easier to defend Bell that way. However, it is only possible to appreciate the error once everything is evaluated together.

Thanks very much, I am having fun catching up on this part of category theory. There's an interesting PhD thesis going into the details http://www.mit.edu/~eadam/eadam_PhDThesis.pdf. Spivak is a very good lecturer, but the lecture course goes too slowly for my taste, with too much extraneous noise.

I think we *must* carefully separate the mathematics from the physics and from the experiment. We must separate and carefully look at all three. We next need to very carefully think about the connection between the physics and the mathematics.

My understanding is that Bell did very carefully motivate his "local realist" picture (which is a purely mathematical model). Einstein would definitely have agreed with Bell's motivation for the model. The motivation did come from cherished pre-existing fundamental principles of physics. So fundamental that many would consider them just "common sense" (Caroline Thompson's point of view), or perhaps "the bare minimum of assumptions that we need to make in order to make physics possible at all!" (the point of view of Alexei Nikulov, famous Russian guy, you can find him on ResearchGate).

Thus the local-realism mathematical model has fundamental physical principles or assumptions built into it.

If experiments apparently do not fit to the model, then we learn that the physical assumptions are apparently false.

As I said many times before, there are now many different standpoints which can logically be taken. At least five.

Bell said that Bohr would just have said "I told you so". I think that this is essentially your standpoint, and it is also expressed by many people such as Andrei Khrennikov, Marian Kupczynski (he's a Polish guy from Canada), Theo Nieuwenhuizen (Dutch guy from Amsterdam). This is a vociferous minority. But they run the Växjö conferences, and many people of many different persuasions come to the conferences.

I think that Bell's theorem should be thought of as a theorem from theoretical computer science, belonging to the field of distributed (classical) computing. Boris Tsirelson agrees. There are elegant proofs of this abstract theorem (Steve Gull's proof - an old exam question in the Cambridge master programme in theoretical physics) which do not have anything to do with statistics or with real experiments. You could say that the theorem is an easy corollary of standard results from Fourier analysis. Certain functions cannot be represented in certain ways. One does, of course, need some classical calculus (ordinary Riemann integration is enough), to formulate and prove the theorem in this way. In modern terminology, one could call it a pretty elementary theorem in functional analysis or in approximation theory. It would be very interesting to look at it from a category theory point of view.

[gill1109]

By the way, almost nobody ever refers to it, but at the very end of Bell's first paper on Bell's inequality, the one with the "original" Bell three correlations inequality, he does actually move to approximation theory. Remember that to derive Bell's original three correlations inequality, Bell assumes, as well as local realism, that there is perfect anti-correlation when the same measurement is made on each particle. If you would experimentally explore the inequality, you would find that the anti-correlation is actually not *exactly* perfect. Hence Bell's whole argument would be irrelevant. But Bell realises this and has an epsilon-delta argument in which he shows that if the anti-correlation at equal settings is almost perfect, then (under his other assumptions) the other correlations can't be far from what his perfect-anti-correlation assumption would imply. In principle, one can compute (or find usable bounds) such that a deviation of at most epsilon from perfect anti-correlation leads to a maximal deviation delta, function of epsilon, which the other correlations could then have from three correlations which do satisfy the original three correlations bound.

[Heinera]

I agree, but I would also say that CHSH had a much more elegant solution to this problem of experimentally perfect anti-correlations

[gill1109]

I agree, but I would also say that CHSH had a much more elegant solution to this problem of experimentally perfect anti-correlations

Sure. Which Bell himself adopted.

And indeed, you can deduce the Bell three correlation epsilon-delta result immediately from CHSH!

We finally learnt from A. Fine that the 8 one-sided Bell-CHSH inequalities are in fact necessary and sufficient for local realism, in a very precise mathematical sense, under the assumption of no-signalling (Alice's statistics give no information about Bob's setting and vice-versa). And when we are talking about an experiment with 2 parties, 2 settings per party, 2 outcomes per setting per party.

Thus Eberhard and Clauser-Horne are actually equivalent, theoretically, to CHSH. Because CHSH is all there is! But not statistically (with finite statistics) since the no-signalling assumption might be true in the large N limit, but for finite N there will be statistical fluctuations to take account of.

[Heinera]

I actually think that pedagocically the CHSH inequalities are in fact better than Bell's original inequality. In the CHSH there is a pleasing symmerty between Alice and Bob. What Bell proved is equivalent to the (simple) theorem that the CHSH urn model (slips with four outcomes, two for Alice and two for Bob) has the upper bound of 2 for the CHSH expression, no matter the distribution of the different slips.

I find it easy to make people understand this, since there are only 16 different slips, and they can simply go through all the 16 extreme cases (where the urn only contains one kind of slip), check that noen of them gives a CHSH value larger than 2, and then realize that all other distributions will only give CHSH values that are convex linear combinations of these. When it comes to pedagogical arguments, I like proofs with a finite number of cases that can be understood by simple enumeration.

Next step is then to argue that more than 16 values for the hidden variable is unnecessary, since any model can anyway only produce at most 16 different predictions. The advanced person will understand that I am talking about 16 different equivalence classes of values of the hidden variable, and that two different values in the same class must have the same empirical consequences, so there is no point in distinguishing between them.

And the final step is to make them understand than any LHV model can be cast in this way (the urn model), since all fancy mathematical stuff that are supposed to be in the functions A and B can just as well be moved into the mathematics of the source without changing any predictions of the model, so the hidden variable can simply be reduced to a slip exactly like the slip in the urn model.