Commonsense local realism refutes Bell's theorem

[gill1109]

Suppose that there is a theory that the mean difference in height between married men and women within US Catholic married couples is at most 3 centimeters. Say that Alice is married to Bob, Ann is married to Ben, and so on. Then a theorist might propose to measure married US Catholic couples and compute their average heights. He asserts that the use of commonsense assumptions leads to a prediction of more than 4 cm difference. However in order to reach that conclusion he compares Alice to Ben, Ann to Bob (and so on); none of the men in his set are married to any of the women in his set. Next an experimentalist samples 1000 couples and finds an average difference of say 2.8 cm, reports the standard deviation, and claims that commonsense has been resoundingly disproved.

Do you agree with me that in such a case the mathematics of the theorist is erroneous?

Consider the set of pairs of integers {(u,v) = (n, n+1), for some n = ...-2, -1, 0, 1, 2, ...}
ie pairs of integers such that the second is one larger than the first.

Put some probability distribution over this set, ie there are probabilities p_n; nonnegative numbers adding to 1.
This defines a pair of random integers (U,V) with a joint distribution such that V = U+1 with probability 1.

Take a sample of size N from this distribution and average the U's.
Do the same again and average the V's.

Provided that the mean value of U is finite, ie sum p_n abs(n) < infty, then for large N the difference between the averages will be close to +1.
Obviously if one takes a sample of any size of pairs (U,V) and averages V-U the answer is exactly 1, whatever N.

Now if all p_n are positive the theorist can argue for for every v there is a u which is larger by 2 (namely u = v+2). ie he can pair off all the married couples such that all the women are 2 units taller than all the men, instead of all being 1 unit smaller.

Well in the Catholic church this is not allowed but perhaps the Pope will allow us to make this immoral thought experiment in our thoughts, only.

I would not say that the argument of the theorist is erroneous. I would say it is irrelevant.

[harry]

[..]
Now if all p_n are positive the theorist can argue for for every v there is a u which is larger by 2 (namely u = v+2). ie he can pair off all the married couples such that all the women are 2 units taller than all the men, instead of all being 1 unit smaller.

Well in the Catholic church this is not allowed but perhaps the Pope will allow us to make this immoral thought experiment in our thoughts, only.

I would not say that the argument of the theorist is erroneous. I would say it is irrelevant.

In my book (and no doubt also for most math and physics teachers), the faulty conclusion of a theorist based on his inappropriate math is not just irrelevant, but erroneous. If a student calculates the stiffness of a long, narrow beam by mixing up the width and the length, he or she will get a wrong answer and I call that faulty math (yes I've seen that several times, regretfully). In any case, no matter how we call such errors, "wrong" or "irrelevant", the resulting conclusion is unsupported.

I recall that the issue of the necessity of correct grouping in pairs has been brought up a number of times before in the peer reviewed literature (perhaps De Raedt in the paper on Boole and Bell).
Now please have another look at section 4 of watson's paper, and check that Bell doesn't mix up the elements between pairs in the same way as mixing up marriage mates, so that what follows becomes irrelevant. I did not find an error in Watson's derivation, although I redid it more elaborately for myself. But it would be surprising if thousands of statisticians overlooked such a basic mistake, so maybe I overlooked something!

[gill1109]

I recall that the issue of the necessity of correct grouping in pairs has been brought up a number of times before in the peer reviewed literature (perhaps De Raedt in the paper on Boole and Bell).
Now please have another look at section 4 of watson's paper, and check that Bell doesn't mix up the elements between pairs in the same way as mixing up marriage mates, so that what follows becomes irrelevant. I did not find an error in Watson's derivation, although I redid it more elaborately for myself. But it would be surprising if thousands of statisticians overlooked such a basic mistake, so maybe I overlooked something!

Well, in my opinion, quite a few non-statisticians (Adenauer, Watson, de Raedt, Fodje, Hess ...) are the ones who are making a basic mistake. Namely the mistake of confusing a specific sequence of values of the local hidden variable which are realized in N runs of a real experiment, with the huge list of all possible *different* values, together with their probabilities (long run relative frequencies). In other words, the distinction between sample average and population mean.

This distinction was neglected or even ignored in the earlier literature. It seems that physisicts' training in statistics and probability is very inhomogenous and even downright disastrously lacking, on the whole. Arthur Fine mentioned in an early and milestone paper on Bell inequalities that the level of discourse concerning matters of statistics and causality in the physics literature is Kindergarten level compared to that in psychometrics and econometris.

Fortunately Bell's "Bertlman's socks" paper paid special attention to this "loophole" in many physicist's thinking. So no-one has any excuse not to understand the transition from theory to experiment any more.

So in conclusion: Watson simply doesn't understand the difference between sample average and population mean. He clearly hasn't read "Speakable and unspeakable" and instead, he's trapped within his own beginner's limitation of thought.

[harry]

Well, in my opinion, quite a few non-statisticians (Adenauer, Watson, de Raedt, Fodje, Hess ...) are the ones who are making a basic mistake. Namely the mistake of confusing a specific sequence of values of the local hidden variable which are realized in N runs of a real experiment, with the huge list of all possible *different* values, together with their probabilities (long run relative frequencies). In other words, the distinction between sample average and population mean.

This distinction was neglected or even ignored in the earlier literature. It seems that physisicts' training in statistics and probability is very inhomogenous and even downright disastrously lacking, on the whole. Arthur Fine mentioned in an early and milestone paper on Bell inequalities that the level of discourse concerning matters of statistics and causality in the physics literature is Kindergarten level compared to that in psychometrics and econometris.

Fortunately Bell's "Bertlman's socks" paper paid special attention to this "loophole" in many physicist's thinking. So no-one has any excuse not to understand the transition from theory to experiment any more.

So in conclusion: Watson simply doesn't understand the difference between sample average and population mean. He clearly hasn't read "Speakable and unspeakable" and instead, he's trapped within his own beginner's limitation of thought.

For sure most if not all the people you refer to know the difference between sample average and population mean - and the criticism in section 4 has nothing to do with that distinction. Also, I know Bell's socks paper, and as far as I can see it doesn't address the issue that Watson brought up. It appears that Bell mixed up terms with different lambda's as if they belonged to the same lambda and such that averaging doesn't help (not a problem if he assumed lambda to be constant, but that he denied). By now I find it hard to believe that you actually looked at Watson's section 4. Although it sounds similar, this particular issue may be more pertinent than what others have brought up. In case you really found the error in his refutation there, please explain: which statement or what equation on which line is wrong, and why?

If someone is interested then I'll copy here my expansion of Watson's section 4, which is itself an expansion of a section of Bell's derivation. That may lead to a more useful discussion than the general comments so far!

[Heinera]

If someone is interested then I'll copy here my expansion of Watson's section 4, which is itself an expansion of a section of Bell's derivation. That may lead to a more useful discussion than the general comments so far!

Please, go ahead with that. I found Watson's paper (and style) pretty much unreadable, so any expansion could only be an improvement.

[Xray]

If someone is interested then I'll copy here my expansion of Watson's section 4, which is itself an expansion of a section of Bell's derivation. That may lead to a more useful discussion than the general comments so far!

Please, go ahead with that. I found Watson's paper (and style) pretty much unreadable, so any expansion could only be an improvement.

Harry, please copy your expansion here. Maybe Gordon Watson erred by not bringing his request here, from vixra http://vixra.org/abs/1405.0020, where his draft is posted. There he says,

"An earlier Draft version of my theory is located at http://vixra.org/abs/1403.0089. I am seeking critical comments, discussion, etc, on these Drafts so that I know what issues need clarification or correction as I work towards an improved essay on the subject.

To make my position clear: Bell's theorem is based on an easily identified error; a false "equality". Bell's famous "inequality" is, thus, a consequent derivative false (compensatory) inequality. Had Bell's first derivation been correct, his false inequality would not have arisen.

NB: Naive-realistic (classical) experiments agree with Bell's theorem because Bell's mathematical error eliminates the generality required for his work to agree with EPRB, the experiment that he set out to analyse.

Sincerely; Gordon"

Heinera, if you "found Watson's paper (and style) pretty much unreadable, so any expansion could only be an improvement", why not send him (or post here) some advice, suggestions, corrections, etc? That is what he is looking for. And now that you know, surely that's what this forum is about?

Dedicated to the sci.physics.* UseNet groups of yesteryear!

Xray

[Xray]

Suppose that there is a theory that the mean difference in height between married men and women within US Catholic married couples is at most 3 centimeters. Say that Alice is married to Bob, Ann is married to Ben, and so on. Then a theorist might propose to measure married US Catholic couples and compute their average heights. He asserts that the use of commonsense assumptions leads to a prediction of more than 4 cm difference. However in order to reach that conclusion he compares Alice to Ben, Ann to Bob (and so on); none of the men in his set are married to any of the women in his set. Next an experimentalist samples 1000 couples and finds an average difference of say 2.8 cm, reports the standard deviation, and claims that commonsense has been resoundingly disproved.

Do you agree with me that in such a case the mathematics of the theorist is erroneous?

xxx

Well in the Catholic church this is not allowed but perhaps the Pope will allow us to make this immoral thought experiment in our thoughts, only.

I would not say that the argument of the theorist is erroneous. I would say it is irrelevant.

Gill, with respect, your jokes are sometimes as crazy as your stats! A Papal Nuncio assures me that the Pope sees nothing immoral in such thoughts, nor in such experiments.

He added (as an aside) that the Pope believes that your "pope" is that mere mortal -- John Stewart Bell -- and that it is the argument of your pope that you HERE assert to be IRRELEVANT!

He thought this was "a cute tactical avoidance strategy" on your part. In his view, the word ERRONEOUS is more direct and certainly more CORRECT!

Cheers,

Xray

[menoma]

In the macroworld in which we live -- i.e., the domain of "commonsense local realism" -- Bell's theorem is not violated in physical experiment with everyday classical objects or when employing classical statistical analysis. Read Bell's "Bertlmann's Socks and the Nature of Reality"

http://cds.cern.ch/record/142461/files/198009299.pdf

and it's impossible not to understand why that's the case. Nor would you want the world in which you spend your life to be otherwise.

The open question is whether or not BT (a pure statement of local realism) is truly violated in quantum experiments -- i.e., the issue is whether or not the quantum domain is in fact plausibly either nonlocal or nonrealistic. This question won't be answered in a manner approaching definitiveness for at least another five years when all identified loopholes have been closed simultaneously in a single experiment, according to Richard.

[gill1109]

In the macroworld in which we live -- i.e., the domain of "commonsense local realism" -- Bell's theorem is not violated in physical experiment with everyday classical objects or when employing classical statistical analysis. Read Bell's "Bertlmann's Socks and the Nature of Reality"

http://cds.cern.ch/record/142461/files/198009299.pdf

and it's impossible not to understand why that's the case. Nor would you want the world in which you spend your life to be otherwise.

The open question is whether or not BT (a pure statement of local realism) is truly violated in quantum experiments -- i.e., the issue is whether or not the quantum domain is in fact plausibly either nonlocal or nonrealistic. This question won't be answered in a manner approaching definitiveness for at least another five years when all identified loopholes have been closed simultaneously in a single experiment, according to Richard.

Thank you Menoma!

[FrediFizzx]

In the macroworld in which we live -- i.e., the domain of "commonsense local realism" -- Bell's theorem is not violated in physical experiment with everyday classical objects or when employing classical statistical analysis. Read Bell's "Bertlmann's Socks and the Nature of Reality".

Actually, that is not known since a proper test has never been done. Joy's experiment (or something similar) needs to be done to find out for sure.

[gill1109]

In the macroworld in which we live -- i.e., the domain of "commonsense local realism" -- Bell's theorem is not violated in physical experiment with everyday classical objects or when employing classical statistical analysis. Read Bell's "Bertlmann's Socks and the Nature of Reality".

Actually, that is not known since a proper test has never been done. Joy's experiment (or something similar) needs to be done to find out for sure.

That depends to some extent as to whether or not you see Bell's theorem as a tautology ie true by definition of "classical", or actually having non-trivial content, ie being in principle refutable by experiment.

In the latter case we have to operationalize "refutable".

(1) We have to take account of statistics. As Michel Fodje has pointed out, the only certain bound on the empirical counterpart of the famous CHSH theoretical quantity S is "4".

(2) We have to remove all ambiguity from our description of the experimental protocol. It seems that page 4 of Christian's experimental paper (pages 1 to 3 are theory; page 4 describes an experiment) is not understood by all readers in the same way. Despite weeks and weeks of effort to remove all ambuguity by formulating a decisive bet and later a decisive challenge, still not all readers understand the terms of the experimental paper / bet / challenge in the same way.

Hence the challenge is presently "on hold" pending clarification by supporters of Christian as to exactly what the experimenter is supposed to do in the final phase. Maybe Fred Diether or Michel Fodje or Hugh Matlock can help out here. I want an unambiguous description of the format of the data files (the files containing observed directions of angular momentum) and I want Mathematica code, or Python code, or R code, which processes those files and computes correlations E(a, b) for any independently chosen a and b. Fred and/or Michel and/or Hugh can collaborate with Joy on this enterprise. Once they are done, and agreed, they can publish their code on this forum for the world to see. Then I can see whether or not I want to propose some kind of bet or challenge around that code.

There is no way at all to run a public scientific bet or challenge about the content of a paper whose supporters do not even agree how it should be interpreted, and whose author comes up with a new story every week as to what it actually always obviously did mean. A kind of shape-shifter, or slippery eel. We end up with the interpretation: the paper is correct because it's correct, it's unambiguous because it's unambuguous, and it means only and exactly what Christian knows it means. All other interpretations are illegitimate. You might as well hold a bet with the so-called Pope of Rome concerning the question of the so-called Catholic church's infallibility.

(I was brought up in the One True Catholic Church: also known as the CofE).

[harry]

If someone is interested then I'll copy here my expansion of Watson's section 4, which is itself an expansion of a section of Bell's derivation. That may lead to a more useful discussion than the general comments so far!

Please, go ahead with that. I found Watson's paper (and style) pretty much unreadable, so any expansion could only be an improvement.

[..] Harry, please copy your expansion here. Maybe Gordon Watson erred by not bringing his request here, from vixra http://vixra.org/abs/1405.0020, where his draft is posted. [..]

OK then.

Watson's section [4] "Bell's 1964 analysis refuted":

Already at "A, B, C are discrete" my mind went blank (is C an angle at A or at B?); my mind came back on track with the use of discrete variables which I find more appropriate than the use of integrals, but next I found the fragmentary references to Bell's derivation a bit confusing. I read on to what I suppose to be the essential point of that section:
Following the good example of some others, Watson corrects Bell's λ to λ(i). He finds then (equations (6) and (7)) that Bell produced an erroneous simplification by mixing up λ(i) and λ(n+i) - no problem if those lambda's were the same, but Bell denied that!
Such a striking allegation demands verification and here's my attempt at it, expanding both on Bell and Watson, and modifying notations to my taste.

Expansion based on Bells 1964 paper eq. 13-15, in http://www.drchinese.com/David/Bell_Compact.pdf
and Gordon Watson's recent Vixra paper section 4, in http://vixra.org/pdf/1403.0089v3.pdf

A, B are observation outcomes of Alice and Bob respectively and a, b, c, are three corresponding angles in the two systems of Alice and Bob.
Bells derivation of eq.(13) - (15) in discrete notation:
A(a,λi) = - B(a,λi) [ and thus also: A(b,λi) = - B(b,λi) , A(c,λi) = - B(c,λi) ] . . . (13)

The average of the products of a series of subsequently measured related pairs at angles (a, b) will be:
<A(a) B(b)> = 1/n Σ [A(a,λi) B(b,λi) ] (sum of i = 1 to n; n -> ∞)
From (13) => <A(a) B(b)> = -1/n Σ [A(a,λi) A(b,λi) ] . . . (14)

Observations with angles (a, c) cannot be done at the same time and on the same particles as the observations with angles (a, b).
Often these are measured sequentially and at random. → require indices i, j.

<A(a) B(b)> − <A(a) B(c)> = - 1/n Σ [A(a,λi) A(b,λi) − A(a, λj) A(c,λj)] . . . (14a) }
A(a,λi) = +/-1 => A(a,λi) A(a,λi) = 1 and similar A(b,λi) A(b,λi) = 1 . . . . . . . . . . }
=>
<A(a) B(b)> − <A(a) B(c)> = 1/n Σ A(a,λi) A(b,λi) [A(a,λi) A(b,λi) A(a,λj) A(c,λj) − 1] . . . (14b)

This is what Bell's (14b) should mean, in discrete form and with added precision.
From this follows (see (14)) :
<A(a) B(b)> − <A(a) B(c)> = <A(a) B(b)> 1/n Σ [1 - A(a,λi) A(b,λi) A(a, λj) A(c,λj)]
and thus also:
|<A(a) B(b)> − <A(a) B(c)>| = |<A(a) B(b)>| 1/n Σ [1 - A(a,λi) A(b,λi) A(a, λj) A(c,λj)]
And as the multiplication factor |<A(a) B(b)>| <= 1, we also find:
|<A(a) B(b)> − <A(a) B(c)>| <= 1/n Σ [1 - A(a,λi) A(b,λi) A(a,λj) A(c,λj)]

Assuming that his A(a,λ) A(a,λ) = (+1)*(+1) or (-1)*(-1), Bell finds (without the i and j):
|<A(a) B(b)> − <A(a) B(c)>| <= 1/n Σ [1 - A(b,λi) A(c,λj)] . . . (14c)

However, in general A(a,λi) A(a,λj) = +/-1, as i and j refer to unrelated events.
(Note: I think that the consequences of that error should be more thoroughly examined).

And next Bell claims that the second term on the right in (14c) is <A(b) B(c)>, so that:
1 + <A(b) B(c)> >= |<A(a) B(b)> − <A(a) B(c)>| . . . (15)

However, the second term on the right in (14c) is in general NOT <A(b) B(c)>:
1/n Σ -A(b,λi) A(c,λj) = 1/n Σ A(b,λi) B(c,λj).
That is not about the measurements of related (entangled) pairs but of unrelated events!
In that population sampling procedure, no sample contains a married couple at all.
Thus equation (15) etc. do not follow.

My first impression is therefore that Watson is right: Bell's equation (15) and following are based on faulty math.
Comments and corrections are welcome!

[gill1109]

Observations with angles (a, c) cannot be done at the same time and on the same particles as the observations with angles (a, b).
Often these are measured sequentially and at random. → require indices i, j.

This is where the misunderstanding starts.
If you want to talk about an actual experiment with N repetitions and just one measurement done on each particle at each repetition, then you might introduce an index i = 1, ... N.

But Bell is not talking about N runs of an experiment. He does not even imagine an experiment at all, when he derives his famous inequality. It is not about an experiment. It is a theoretical calculation done within a particular model, i.c. a local hidden variables model. The model says: Nature picks lambda at random according to the probability distribution rho. If Experimenter would choose to measure A, she would get to see A(lambda). If Experimenter would choose to measure B, she would get to see B(lambda). But she doesn't have to do anything. The experiment doesn't need to be done: not once, not 100 times, not infinitely often.

The set Lambda is the set of all possible values of the hidden variable. Maybe it is a continuum. Maybe it consists just of 16 elements. Each element has its own probability. Consider one particular element l0. Suppose it occurs 10% of the time. If we always measure A and B then this gives a contribution 1/10 A(l0)B(l0) to the mean value of A times B. If we always measure A and C then that same value l0 would give a contribution 1/10 times A(l0)C(l0) to the mean value of A times C.

Now we can compare mathematically the mean value of A times B and the mean value of A times C. Still no experiment has been done.

Now we can think of an experiment where 100 times we measure A and B. In roughly 1/10 of the runs lambda will equal l0 and this will give a contribution roughly equal to 1/10 times A(l0)B(l0) to the average value of A times B. Now do another 200 runs, measuring A and C. In roughly 1/10 of the runs lambda will equal l0 and this will give a contribution roughly equal to 1/10 times A(l0)C(l0) to the average value of A times C.

Sample averages are close to theoretical mean values if sample sizes are large etc etc.

Read Bertlmann's socks, carefully. Forget Bell's first paper on Bell's inequality. He got better at explaining things as the years went by. A lot of people had the same common minor misunderstandings so he expanded the more delicate parts of his explanation. He made other parts easier to understand.

[harry]

Observations with angles (a, c) cannot be done at the same time and on the same particles as the observations with angles (a, b).
Often these are measured sequentially and at random. → require indices i, j.

This is where the misunderstanding starts.
If you want to talk about an actual experiment with N repetitions and just one measurement done on each particle at each repetition, then you might introduce an index i = 1, ... N.

But Bell is not talking about N runs of an experiment. He does not even imagine an experiment at all, when he derives his famous inequality. It is not about an experiment. It is a theoretical calculation done within a particular model, i.c. a local hidden variables model. The model says: Nature picks lambda at random according to the probability distribution rho. If Experimenter would choose to measure A, she would get to see A(lambda). If Experimenter would choose to measure B, she would get to see B(lambda). But she doesn't have to do anything. The experiment doesn't need to be done: not once, not 100 times, not infinitely often.

In the end Bell is definitely talking about experiments - classical physics is about explaining experiments, and quantum mechanics is only about predicting experimental results. And how nature chooses unknown local hidden variables isn't known. For that reason Bell's unproven starting assumptions are suspect, and it's much more solid to leave those out as done by Watson. However, I suspect that you are mistaken to think that that disagreement is relevant for the essential point that we discuss here. Surely you agree that lambda is not assumed to be a constant, so that it is less sloppy to write lambda(i) (or lambda(rho) or whatever you like) and keep track of the indices - see next.

The set Lambda is the set of all possible values of the hidden variable. Maybe it is a continuum. Maybe it consists just of 16 elements. Each element has its own probability. Consider one particular element l0. Suppose it occurs 10% of the time. If we always measure A and B then this gives a contribution 1/10 A(l0)B(l0) to the mean value of A times B. If we always measure A and C then that same value l0 would give a contribution 1/10 times A(l0)C(l0) to the mean value of A times C.

Now we can compare mathematically the mean value of A times B and the mean value of A times C. Still no experiment has been done.

Now we can think of an experiment where 100 times we measure A and B. In roughly 1/10 of the runs lambda will equal l0 and this will give a contribution roughly equal to 1/10 times A(l0)B(l0) to the average value of A times B. Now do another 200 runs, measuring A and C. In roughly 1/10 of the runs lambda will equal l0 and this will give a contribution roughly equal to 1/10 times A(l0)C(l0) to the average value of A times C.

Sample averages are close to theoretical mean values if sample sizes are large etc etc.

That argument sounds plausible, but it would be more convincing if it can be shown to work on your catholic couples example in a way that corresponds to the matter at hand.
Let's assume that men are on the average 10 cm longer than women, and for simplicity's sake all men and women are married (100% yield). Now a theory has that on the average, a husband is 2 cm longer than his wife due to a set of hidden variables in their heads. So if we will now apply the averaging method as you describe here, how does that work out? Following your protocol:
We can think of an experiment where 100 times we measure of couples the husband's length A and his wife's length B. In roughly 1/10 of the runs lambda will equal l0 and this will give a contribution roughly equal to 1/10 times [B(λ0) - A(λ0)] to the average value of A minus B. Likely that will work correctly. But what happens if instead we calculate by mistake B[A(λ0) - A(λ1)], etc? Intuitively I thought that we will simply end up with the average population difference of 10 cm. However, it may depend on how lambda works. and in fact that example is not needed, it's sufficient to get back to your A(λ0)B(λ0). The derived allegation here is that Bell by mistake takes terms like A(λ0)B(λ1) and pretends those to be terms like A(λ0)B(λ0). Now your point is probably that you don't doubt that, but that λ0 must be equal to λ1 for the same A because of the (nearly) 100% match for equal settings, or at least that the expectation values must be the same. Correct? That is certainly worth discussing!

Read Bertlmann's socks, carefully. Forget Bell's first paper on Bell's inequality. He got better at explaining things as the years went by. A lot of people had the same common minor misunderstandings so he expanded the more delicate parts of his explanation. He made other parts easier to understand.

I did read Bertlmann's socks carefully and I fully agree with your preference for that paper; which section of it are you referring to? For the derivation part Bell's 1964 paper is just fine for this (admission: one or two lines of my elaboration are based on Bertlmann's socks).

Let's wait a moment for comments by other participants.

[Heinera]

Observations with angles (a, c) cannot be done at the same time and on the same particles as the observations with angles (a, b).

But that is exactly what can be done (and which Bell does), as long as you understand that Bell's theorem is not a theorem about experiments (which would be meaningless anyway), but a theorem about a certain class of models (or computer programs, as I prefer to call them). It says that a certain class of models (the class of LHV models) cannot produce outputs that exhibit certain correlations. Another way to state Bell's theorem is to say that it is impossible to win the QRC challenge. No need to perform any experiment there.

With a particular LHV model (or computer program) in mind, there is no problem in asking the question "what happens if I change the setting (a, c) to (a, b) but keep everything else fixed, including the hidden variable?" You just run the program again, taking great care to ensure that the only thing that changes are the angle inputs.

You should read Richard Gill's paper on a version of Bell's theorem, the CHSH inequality. There, a table is explicitly constructed where Alice's and Bob's outcomes are pairwise matched. You will see that nowhere in the proof are Alice's outcomes mixed together with Bob's outcomes from another row in the table. (Also, be sure to read the last section of that paper, where a procedure for constructing such a table given any LHV model is devised). So to say that Bell mixes together unrelated events in his proof is simply not correct.

[harry]

Observations with angles (a, c) cannot be done at the same time and on the same particles as the observations with angles (a, b).

[..] No need to perform any experiment there.

With a particular LHV model (or computer program) in mind, there is no problem in asking the question "what happens if I change the setting (a, c) to (a, b) but keep everything else fixed, including the hidden variable?" You just run the program again, taking great care to ensure that the only thing that changes are the angle inputs.

While that is obviously true, it is just as obviously irrelevant to QM's predictions which only concern possible experimental outcomes - only those are the ones that a physical model is required to predict.

You should read Richard Gill's paper on a version of Bell's theorem, the CHSH inequality. There, a table is explicitly constructed where Alice's and Bob's outcomes are pairwise matched. You will see that nowhere in the proof are Alice's outcomes mixed together with Bob's outcomes from another row in the table. (Also, be sure to read the last section of that paper, where a procedure for constructing such a table given any LHV model is devised). So to say that Bell mixes together unrelated events in his proof is simply not correct.

We are here discussing a criticism of Bell's derivations, and I expanded on the part related to Bells' derivation of his original inequality. When I find the time I will gladly also look at Gill's paper; however it would be strange (although not impossible) if Bell's first derivation were wrong and Gill's later derivation were right, and the question what is wrong with Gill's paper is a different one from the question what is wrong with Watson's paper.

[Heinera]

With a particular LHV model (or computer program) in mind, there is no problem in asking the question "what happens if I change the setting (a, c) to (a, b) but keep everything else fixed, including the hidden variable?" You just run the program again, taking great care to ensure that the only thing that changes are the angle inputs.

While that is obviously true, it is just as obviously irrelevant to QM's predictions which only concern possible experimental outcomes - only those are the ones that a physical model is required to predict.

And Bell proves that no LHV computer program can reproduce the QM correlations (still no mention of experiment here; just a comparison of two mathematical models). In the process of constructing his proof, he asks the perfectly legal question "what happens with the program output if I change the setting (a, c) to (a, b) but keep everything else fixed, including the hidden variable?"

You should read Richard Gill's paper on a version of Bell's theorem, the CHSH inequality. There, a table is explicitly constructed where Alice's and Bob's outcomes are pairwise matched. You will see that nowhere in the proof are Alice's outcomes mixed together with Bob's outcomes from another row in the table. (Also, be sure to read the last section of that paper, where a procedure for constructing such a table given any LHV model is devised). So to say that Bell mixes together unrelated events in his proof is simply not correct.

We are here discussing a criticism of Bell's derivations, and I expanded on the part related to Bells' derivation of his original inequality. When I find the time I will gladly also look at Gill's paper; however it would be strange (although not impossible) if Bell's first derivation were wrong and Gill's later derivation were right.

Indeed. Fortunately they are both right, and I mentioned this paper because Gill's derivation might help you to better understand Bell's derivation, especially since you think Bell mixed up Alice's outcomes with Bob's outcomes from different pairs of particles.

[gill1109]

Bell's theorem is about a class of physical models. LHV models. Think of them as a class of computer programs, a class of simulation models. He shows what correlation structures can occur, according to that class of models.

According to QM, a larger set of correlation structures is possible.

If you are perfectly happy with QM there is no problem for you. In fact you should be happy: QM allows more things to be possible than those silly LHV models.

However, if you think that the class of LHV models should be "everything", then you haves problem with QM.

If moreover an experiment shows that *Nature* allows correlation structures which are allowed by QM but disallowed by LHV, you have a big problem.

[gill1109]

PS Indeed there is nothing terribly wrong with Bell's classic paper but many things could have been said better, and later were said better. The Bertlmann's socks explicitly addresses a lot of the common concerns of the first generation of readers.

My recent version is a *strengthening* of Bell's since I prove finite N probability inequalities about sample averages. Let N tend to infinity and you get Bell's version as a corollary.

I also take account of many of the insights which have been gained over the last 50 years. Time has not stood still...

[harry]

[..] I mentioned this paper because Gill's derivation might help you to better understand Bell's derivation, especially since you think Bell mixed up Alice's outcomes with Bob's outcomes from different pairs of particles.

I apprecaite that. However, I don't just think so, in my elaboration of Bell's derivation in Watson's manner I found indeed a mix-up between different pairs of particles. If I understood it correctly, Gill suggested that the mix-up is innocent because it averages out, but you suggest that the mix-up doesn't occur in Bell's derivation.

Thus, once more, Gill is not Bell, and either:
a) the outcomes of different pairs of particles are mixed up in Bell's 1964 derivation just as Watson suggested and I reproduced with more detail; but in that case these mix-ups should be shown to be innocent (or "nothing terribly wrong");
or
b) no such mix-up occurs in Bell's derivation, but then the question is where is the mistake in Watson's critique of Bell's derivation as I elaborated. Also X-ray didn't find the error in Watson's paper, maybe you want to point it out to him? I have reconstructed the whole derivation in full detail, making it easy to point out mistakes.

So, which is it? We can't have both.

PS: I do hope that Watson will comment on my post of 28 May (what no post numbers?!), to tell if my free interpretation of that part of his paper is essentially correct.