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

[minkwe]

Gordon apparently believes that in order to derive the CHSH inequality you have to imagine "testing a single pristine particle-pair four times" and therefore CHSH is wrong. I say that you don't have to imagine anything at all in the real world.

Really, you just said earlier:

It would concern an EPR-B type experiment in which we imagine Nature choosing a hidden variable lambda and sending it, within two particles, to two measurement stations

Now you want us to "forget about imaginations"

CHSH is calculus which you can do which tells you something useful to know. In order to derive CHSH you have to imagine three functions with certain properties.

Or rather, let us imagine functions but don't imagine those functions meaning anything in nature or reality. As this discussion goes on, Richard is being pushed further and further from any relevance to physics and there is a reason for that.

IF we have functions

A : {settings} x { values of a "hidden variable lambda} -> {-1, +1}
B : {settings} x { values of a "hidden variable lambda} -> {-1, +1}

rho : { values of a "hidden variable lambda} -> [0, infty) such that integral rho = 1

THEN for any a, a', b, b'

S = integral [ A(a, lambda)B(b, lambda) - A(a, lambda)B(b', lambda) + A(a', lambda)B(b, lambda) + A(a', lambda)B(b', lambda) ] rho(lambda) d lambda

is less than or equal to 2

PROOF: the expression in square brackets is less than or equal to 2 because ... and therefore ...

This is just calculus.

The reason is there is no physical assumption required to derive those inequalities, it is just calculus, therefore their violation by anything points just to a silly mistake in calculus, rather than non-locality or non-realism.

you can imagine it stands for what happens when we run Michel's programs with some lines added

Gill's caricature of my programs is of course not my model (viewtopic.php?f=6&t=61&start=10).

They don't correspond to something that happens in the real experiment but we can still imagine it.

Correct, anyone can imagine anything they like, including non-physical and nonsensical abstract ideas. But we are discussing physics here.

If we want to apply these calculus trivialities to physics (or to computer simulations) we need a bridge to the real world and to experiment

That bridging will be more difficult if we have focused the whole time on nonsensical non-physical imaginations.

(or computer simulation models).

That bridging can only happen if the computer simulation models we are working with are themselves not nonsensical non-physical imaginations. In case you didn't know, it is possible the very easy to simulate nonsense on computers. In fact, you have to take special care to make sure you are not simulating garbage.

On the other side of the bridge there is also something which we sometimes write E(a, b) but it stands for something in the real world. In the mathematical world there are apples, in the real world there are pears. Some of the pears correspond to some of the apples. But not every mathematical feature has a real world counterpart, and vice versa.

Even in the mathematical world, there are pears and apples. Not every E(a,b) that you see in the mathematical world, is the same as every other E(a,b) in the mathematical world. Weakly objective E(a,b) in the mathematical world is not the same as Strongly objective E(a,b) in the mathematical world, just like weakly objective <AB> in the real world is not the same as strongly objective <AB> in the real world. Mathematically, or in the real world, the average height of 100 different people, is a different thing from the average height of the same person measured 100 different times. The apples that you get from QM and Experiment are different things from the pears that you used in your calculus to derive your inequalities. This is the mathematical error underpinning violations of any inequalities. Not non-realism as you claim in your papers.

I'm sure you did not discuss the above in your talk.

[gill1109]

On the other side of the bridge there is also something which we sometimes write E(a, b) but it stands for something in the real world. In the mathematical world there are apples, in the real world there are pears. Some of the pears correspond to some of the apples. But not every mathematical feature has a real world counterpart, and vice versa.

Even in the mathematical world, there are pears and apples. Not every E(a,b) that you see in the mathematical world, is the same as every other E(a,b) in the mathematical world. Weakly objective E(a,b) in the mathematical world is not the same as Strongly objective E(a,b) in the mathematical world, just like weakly objective <AB> in the real world is not the same as strongly objective <AB> in the real world. Mathematically, or in the real world, the average height of 100 different people, is a different thing from the average height of the same person measured 100 different times. The apples that you get from QM and Experiment are different things from the pears that you used in your calculus to derive your inequalities. This is the mathematical error underpinning violations of any inequalities. Not non-realism as you claim in your papers.

I'm sure you did not discuss the above in your talk.

I did discuss this in my talk. "strongly objective" and "weakly objective" are interpretations of probability, hence metaphysics. They are possible bridges between the real world and the mathematical world.

Note that we are all the time talking about things. "Real world", "mathematical world", and "bridges between them" are all words. They are all concepts. They are all an idealization, they are part of a "model" of "doing science".

When people confuse the names of things with the things themselves, or in computer science, the name of a variable with the value stored in the variable, we get paradoxes and computer errors. In C, pointers are so confusing! What is foo and what is foo*? This kind of mistake is called a category error in philosophy. I believe that many people who believe that Bell was wrong etc etc etc are making category errors. I talked about this extensively in the talk, and I tried very hard to explain to a couple of quantum c****pot that they were making category errors.

In fact, Hans de Raedt, Marian Kupczynski and I got into a very heated discussion in which Hans took my side and agreed that Marian was making category errors. He also at one point got almost angry and told Marian to stop talking and start listening, for a change.

Maybe we should do a skype conference instead of posting messages to forums. It's so nineteen nineties. In the twenty tens people had other technology.

[Joy Christian]

I tried very hard to explain to a couple of quantum c****pot that they were making category errors.

Really? You were looking into mirror and talking to yourself? How funny!

[FrediFizzx]

That would be at least two mirrors.

[gill1109]

I had a nice conversation with Hans de Raedt about Christian's use in public of quotations from private emails between Karl Hess and Joy Christian. Karl Hess is a good friend of Hans de Raedt. I knew Walter Philip a bit, but never had the fortune to meet Karl.

However since the conversation was private I won't quote from it here.

[Joy Christian]

I had a nice conversation with Hans de Raedt about Christian's use in public of quotations from private emails between Karl Hess and Joy Christian. Karl Hess is a good friend of Hans de Raedt. I knew Walter Philip a bit, but never had the fortune to meet Karl.

However since the conversation was private I won't quote from it here.

I am having nice conversations with several distinguished colleagues of mine about Gill's use of public forums to promulgate misleading propaganda and outright lies about the works and characters of many scholars for selfish and political purposes other than science.

However since these on-going conversations are private, I won't quote from them here.

[gill1109]

I had a nice conversation with Hans de Raedt about Christian's use in public of quotations from private emails between Karl Hess and Joy Christian. Karl Hess is a good friend of Hans de Raedt. I knew Walter Philip a bit, but never had the fortune to meet Karl.

However since the conversation was private I won't quote from it here.

I am having nice conversations with several distinguished colleagues of mine about Gill's use of public forums to promulgate misleading propaganda and outright lies about the works and characters of many scholars for selfish and political purposes other than science.

However since these on-going conversations are private, I won't quote from them here.

[Joy Christian]

I had a nice conversation with Hans de Raedt about Christian's use in public of quotations from private emails between Karl Hess and Joy Christian. Karl Hess is a good friend of Hans de Raedt. I knew Walter Philip a bit, but never had the fortune to meet Karl.

However since the conversation was private I won't quote from it here.

I am having nice conversations with several distinguished colleagues of mine about Gill's use of public forums to promulgate misleading propaganda and outright lies about the works and characters of many scholars for selfish and political purposes other than science.

However since these on-going conversations are private, I won't quote from them here.

[FrediFizzx]

Guys, let's get back on topic. Thanks.

[Gordon Watson]

Guys, let's get back on topic. Thanks.

Thanks Fred,

And to assist in that direction I've provided an update at viXra, a 3-page pdf: http://vixra.org/pdf/1406.0027v2.pdf

This version provides less wriggle-room for Bell's supporters and adds some further erroneous Bellian examples from well-known physicists.

They include 't Hooft 2014:(8.22)-(8.23) and Mermin 2005:(2)-(3); re the latter, see the pre-publication version at http://tx.technion.ac.il/~peres/mermin.pdf

This latter error is of special relevance here as Mermin (2005:p.2076) thanks Richard GILL "for providing reassuring reality checks …".

I'd therefore be pleased if Richard and others could explain how Mermin 2005:eqn (2) = Mermin 2005:eqn (3)?

To phrase my question in exemplary form, compatible with the intention of this thread:

In the context of http://vixra.org/pdf/1406.0027v2.pdf eqn. (12), could Richard Gill and others please explain --- from http://tx.technion.ac.il/~peres/mermin.pdf (or from the published paper) --- how Mermin 2005:eqn (2) = Mermin 2005:eqn (3)?

PS: There is NO need to address any other matter; and certainly NOT the "presence of time-correlated hidden variables in the detectors" because such are irrelevant when defining Bell's repeated errors!

[gill1109]

Guys, let's get back on topic. Thanks.

Thanks Fred,

And to assist in that direction I've provided an update at viXra, a 3-page pdf: http://vixra.org/pdf/1406.0027v2.pdf

This version provides less wriggle-room for Bell's supporters and adds some further erroneous Bellian examples from well-known physicists.

They include 't Hooft 2014:(8.22)-(8.23) and Mermin 2005:(2)-(3); re the latter, see the pre-publication version at http://tx.technion.ac.il/~peres/mermin.pdf

This latter error is of special relevance here as Mermin (2005:p.2076) thanks Richard GILL "for providing reassuring reality checks …".

I'd therefore be pleased if Richard and others could explain how Mermin 2005:eqn (2) = Mermin 2005:eqn (3)?

To phrase my question in exemplary form, compatible with the intention of this thread:

In the context of http://vixra.org/pdf/1406.0027v2.pdf eqn. (12), could Richard Gill and others please explain --- from http://tx.technion.ac.il/~peres/mermin.pdf (or from the published paper) --- how Mermin 2005:eqn (2) = Mermin 2005:eqn (3)?

PS: There is NO need to address any other matter; and certainly NOT the "presence of time-correlated hidden variables in the detectors" because such are irrelevant when defining Bell's repeated errors!

Dear Gordon

A month of so ago several of your opponents to you explained where they thought *you* were going wrong. You didn't understand us. We explained how you were confusing a list of values of hidden variables pertaining to N separate runs in an actual experiment, to the list of all possible value of a hidden variable in a probabilistic model of "what is going on" in any single run.

So, as far as I am concerned, there is no need to address any other matter at all.

I might be interested if you would show me where I am making an error in the proof of Theorem 1 in Section 2 of my paper Statistics, Causality and Bell's Theorem http://arxiv.org/abs/1207.5103.

Richard

PS alternatively I am interested in formulating a computer challenge or bet between the two of us. If you are so sure that you are right, maybe you can write a suite of computer programs which provide a counter-example to Bell's theorem in the context of a delayed-choice random settings, event-ready detectors (or pulsed) Bell-CHSH type experiment. I hope you agree to the logic here: if a certain theorem is false, a theorem of the type "if ... then ... is impossible", then there has to be a counter-example, ie there has to exist something which satisfies the conditions specified after the "if", for which the fact specified after "then" does actually occur.

[Gordon Watson]

Guys, let's get back on topic. Thanks.

Thanks Fred,

And to assist in that direction I've provided an update at viXra, a 3-page pdf: http://vixra.org/pdf/1406.0027v2.pdf

This version provides less wriggle-room for Bell's supporters and adds some further erroneous Bellian examples from well-known physicists.

They include 't Hooft 2014:(8.22)-(8.23) and Mermin 2005:(2)-(3); re the latter, see the pre-publication version at http://tx.technion.ac.il/~peres/mermin.pdf

This latter error is of special relevance here as Mermin (2005:p.2076) thanks Richard GILL "for providing reassuring reality checks …".

I'd therefore be pleased if Richard and others could explain how Mermin 2005:eqn (2) = Mermin 2005:eqn (3)?

To phrase my question in exemplary form, compatible with the intention of this thread:

In the context of http://vixra.org/pdf/1406.0027v2.pdf eqn. (12), could Richard Gill and others please explain --- from http://tx.technion.ac.il/~peres/mermin.pdf (or from the published paper) --- how Mermin 2005:eqn (2) = Mermin 2005:eqn (3)?

PS: There is NO need to address any other matter; and certainly NOT the "presence of time-correlated hidden variables in the detectors" because such are irrelevant when defining Bell's repeated errors!

Dear Gordon

A month of so ago several of your opponents to you explained where they thought *you* were going wrong. You didn't understand us. We explained how you were confusing a list of values of hidden variables pertaining to N separate runs in an actual experiment, to the list of all possible value of a hidden variable in a probabilistic model of "what is going on" in any single run.

So, as far as I am concerned, there is no need to address any other matter at all.

I might be interested if you would show me where I am making an error in the proof of Theorem 1 in Section 2 of my paper Statistics, Causality and Bell's Theorem http://arxiv.org/abs/1207.5103.

Richard

PS alternatively I am interested in formulating a computer challenge or bet between the two of us. If you are so sure that you are right, maybe you can write a suite of computer programs which provide a counter-example to Bell's theorem in the context of a delayed-choice random settings, event-ready detectors (or pulsed) Bell-CHSH type experiment. I hope you agree to the logic here: if a certain theorem is false, a theorem of the type "if ... then ... is impossible", then there has to be a counter-example, ie there has to exist something which satisfies the conditions specified after the "if", for which the fact specified after "then" does actually occur.

Dear Richard,

Re your Statistics, Causality and Bell's Theorem http://arxiv.org/abs/1207.5103

1. To save me possibly copying your theorem here inaccurately, please copy (at least) the last paragraph on page 4 and equation (3) into this thread. I understand that (3) defines your theorem so please be sure that the requested reproduction leaves no loopholes.

2. Please explain in detail the mathematical and/or physical significance and relevance of your use of .

3. Excusing any hasty confusion on my part (I'm in a meeting), please explain why your theorem appears to allege the following mathematically irrelevant certainties:

(i) after one observation:

(ii) after two observations:

(iii) after 50 observations:

4. Please explain why your theorem has anything to do with my request that you mathematically justify the advice that you gave in Mermin (2005) re the equality of his equations (2) and (3).

5. Given your claimed connection (p.3) to Bell-CHSH inequalities: please provide and explain the four Bell-CHSH experimental settings that are represented by the "experimentally observed averages" that you derive. Am I correct in noting that your derivations come from what amounts to N/16 16x4 sub-spreadsheets built from equiprobable distributions of A, A', B, B' = ±1?

Sincerely; with best regards; Gordon

[gill1109]

Dear Richard,

Re your Statistics, Causality and Bell's Theorem http://arxiv.org/abs/1207.5103

1. To save me possibly copying your theorem here inaccurately, please copy (at least) the last paragraph on page 4 and equation (3) into this thread. I understand that (3) defines your theorem so please be sure that the requested reproduction leaves no loopholes.

2. Please explain in detail the mathematical and/or physical significance and relevance of your use of .

3. Excusing any hasty confusion on my part (I'm in a meeting), please explain why your theorem appears to allege the following mathematically irrelevant certainties:

(i) after one observation:

(ii) after two observations:

(iii) after 50 observations:

4. Please explain why your theorem has anything to do with my request that you mathematically justify the advice that you gave in Mermin (2005) re the equality of his equations (2) and (3).

5. Given your claimed connection (p.3) to Bell-CHSH inequalities: please provide and explain the four Bell-CHSH experimental settings that are represented by the "experimentally observed averages" that you derive. Am I correct in noting that your derivations come from what amounts to N/16 16x4 sub-spreadsheets built from equiprobable distributions of A, A', B, B' = ±1?

Sincerely; with best regards; Gordon

Re Question 1: Here is the *first* paragraph of page 4, which defines some notation, and the theorem.

Suppose that for each row of the spreadsheet, two fair coins are tossed independently of one another, independently over all the rows. Suppose that depending on the outcomes of the two coins, we either get to see the value of A or A′, and either the value of B or B′. We can therefore determine the value of just one of the four products AB, AB′, A′B, and A′B′, each with equal probability 1/4 for each row of the table. Denote by ⟨AB⟩_obs the average of the observed products of A and B (“undefined” if the sample size is zero). Define ⟨AB′⟩_obs, ⟨A′B⟩_obs and ⟨A′B′⟩_obs similarly.

Theorem 1. Given an N × 4 spreadsheet of numbers ±1 with columns A, A′, B and B′, suppose that, completely at random, just one of A and A′ is observed and just one of B and B′ are observed in every row. Then, for any η ≥ 0,

Prob( ⟨AB⟩_obs +⟨AB′⟩_obs +⟨A′B⟩_obs −⟨A′B′⟩_obs ≤ 2+η ) ≥ 1 − 8 e^(−N(η/16)^2). (3)

It may be useful to define S_obs = ⟨AB⟩_obs +⟨AB′⟩_obs +⟨A′B⟩_obs −⟨A′B′⟩_obs

It may be convenient to drop the subscript "obs" as long as we remember what we are talking about. Given is an Nx4 table of numbers +/-1/ It's fixed. For any such table we can now toss 2N fair coins and pick either A or A', and either B or B' from each row, and then calculate the average of the observed products AB, AB', A'B, A'B'.

With this agreed I will try to answer your remaining questions.

Re Question 2: The theorem says that if you pick any number eta >= 0, and then ask yourself "what is the probability that S is smaller than 2 + eta", then this probability is at least 1 − 8 e^(−N(η/16)^2). You can test the theorem by simulation. Think of an Nx4 table. Keep it fixed from now on. Repeatedly, toss 2N fair coins and calculate S. How often does it exceed, say, 2.1? the answer is that it won't be larger than 2.1, more than a fraction 1 − 8 e^(−N(η/16)^2) of the times, with eta = 0.1.

Re Question 3. It is not clear to me what value of eta you took in parts (i), (ii), and (iii). Anyway, for small eta and for small N equation (3) though true is pretty uninteresting. For instance, with eta = 0, it is always totally uninteresting. But try, for instance, N = 10 000 and eta = 0.4. I find that the probability on the right hand side of (3) is 1 minus 0.01544363. Thus 98.5% of the time we repeat this experiment, S will be smaller than 2.4.

I find the theorem useful for setting up bets with anti-Bellists. I want to bet on the outcome of a LHV simulation of a CHSH type experiment (no loopholes; in particular, no detection loophole, no memory loophole). Take N = 1 000 000 and eta = 0.1. I can clearly bet quite heavily on S being smaller than 2.1

Re Question 4. Discussing my theorem here has a pedagogical purpose. We need to start distinguishing concepts which, it seems, so far you do not even have words to express. There are phenomena to which you are blind. So you need to be exposed to them in a safe, playful, arena where they do not constitute any kind of threat.

Re Question 5. You asked me to provide and explain the four Bell-CHSH experimental settings which correspond to experimentally observed averages. The answer is it doesn't matter. Read Bertlmann's socks where he explains what his "theorem" (he means his derivation of the CHSH inequality for a LHV model described by probability density rho(lambda), and measurement functions A(a, lambda) and B(b, lambda)) is about. It's about some macroscopic experimental layout. Alice repeatedly chooses between settings a and a', Bob between b and b'. In other words, they each have a binary setting choice, and make it, repeatedly, completely at random.

[harry]

Hello Gordon, I'm back.

Hi Gordon,

Hi Harry, and thanks,

As you decided to restart - but with more focus - the topic "Commonsense local realism refutes Bell's theorem", here's a summary of my latest opinion as expressed in that thread:

Apparently you assume that Bell grouped his derivation corresponding to experimental sequence N.

No; not at all: and I don't see why this seems so to you?

A sum can be done in any sequence. But the tests are sequential, numbered like mine or some other way.

So, to help clear this up, please bring these claims into the context of the essay that we are focussing on here now.

OK. The point was not the ordering of the sequence (indeed you are free to change the order) but the sequence itself: the data. As I stressed next by means of an illustration, Bell did not calculate according to experimental data collection.

[..]

[..] Bell's integral is not over N or t, but over λ. Bell keeps λ constant over each integration step: on purpose one whole line corresponds to a single λ - and not a λi and a different λn+i which have different outcomes.

I'm not following this, at all.

That's a pity, as it is absolutely necessary to follow it!
Once more, do you understand my illustration here below? If you can follow my illustration then you will likely also follow what we have been saying to you!

Please explain this to me:

What is the physical significance (when studying EPRB) of "Bell keeps λ constant over each integration step: on purpose one whole line corresponds to a single λ"?

NB: Bell allows me to use discrete λ, so I do just that. [..]

I will answer that after you first answer the same about my ilustration here below:

Here's an illustration. A carpenter determines the average length of two similar beams as follows: He places them on top of each other, puts a mark halfway between the ends of the two beams as follows:

-------------- . . . . x
---------------------------------

Next he measures the length upto the mark of the top beam. I see him do that, and happen to know the lengths of the two beams.
So I calculate (230+240) / 2 = 235 cm and shout out that number to him. He shouts back: "Right - how did you know?"

My calculation should in theory give the same result as the measurement, despite the fact that there is not a 1-to-1 correspondence between the two.

Thus: what is the physical significance of my average length calculation (a+b)/2 ? Does it match in any way what the carpenter actually did?

Bell did similarly not stick to the experimental procedure for his derivation of what may be predicted as experimental outcomes. That doesn't mean that Bell didn't make a mistake of course; but he did not mix up the lambda's.

But the mistake is that Bell does not MATCH his λs: Please think about the experiments in Paris and Peru. Please consider my (12). Please think about your old married-couples analogy.

No, we were mistaken, as others as well as I explained. Bell certainly matched his lambda's, as can be seen from his math notation. Any criticism on Bell's derivation must start from that fact.
[...]

As a reminder, here's a copy of my elaboration of my interpretation of your version of that part of Bell's derivation; that may come handy for a detailed discussion as it fills up a few blanks in both Bell's and your paper. Most of us had difficulty following your argumentation which is overly compact. [..]
I added in red the corresponding (or seemingly corresponding) equations in Bell's paper.

Please correct it where I misinterpret your argument, and take it from there!

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

[...]

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
[note: that section is similar to the corresponding section in the paper under discussion]

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.
[Note: Watson chose i and n+i, suggesting two subsequent measurement series.]

<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 [Watson apparently argues that] Bell's (14b) should mean, in discrete form and with added precision.

To this point looks OK. But I question what follows:

From this follows (see (14)) :

Sorry, but does (14) support your next step?

<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)]

Is this correct?

Trying to keep it simple here, to show possible error.

In general: 1/n ΣXk [1-Yk] ≠ 1/n ΣXk [1/n Σ(1-Yk)] .

Isn't that, effectively, what you've done?

No, not exactly. As we are here multiplying averages it is not so simple; and I just tried to guess how Bell went from 14b to 15.
However you make an interesting point! Perhaps Bell was making unwarranted assumptions going from 14b to 15. We should scrutinize every move. Anyway, you did not critizise that aspect of his derivation in your papers so one should not linger over it in an expansion of your argument.

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.
[...]

And next Bell claims, [or so it seems] 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 Bell's equation (15) etc. do not follow, according to Watson.

When it comes to QM and BT, I'm not much into analogies.

However, this one looks to be a beauty!

You've nicely identified exactly what Bell does! (Haven't you?) [..]

Once more: no, the above detailed reconstruction does not correctly reproduce Bell. We overlooked the fact that of course he held lambda constant when he integrated over lambda (it's an unforgettable sin not to do so). And because of that he had to include the relative frequency (probability) of each lambda - something that is glaringly missing in the above reconstruction, because that reconstruction ignores what he really tried to do.

[..]

…..
Finally, to help you return to your prior very-clear thinking.

Ignoring the many essays in which similar equations are unnumbered in the Bellian literature, please consider this:

A: Bell's 1964:(14a) IS mathematically and experimentally valid.

I hope that it is clear now that Bell's (14) does not directly relate to experiment; the terms are grouped per the same recurring lambda instead of per observed particle pairs.

Thus, your question about "experimentally valid" is too ambiguous. Compare once more: my average calculation about the carpenter's experiment gives the correct experimental result but does not match experiment. Do you call that "experimentally invalid"? Depending on your answer, in your wording we then get that Bell's 1964:(14a) IS / IS NOT experimentally valid. And the same for all what follows.

[minkwe]

Once more: no, the above detailed reconstruction does not correctly reproduce Bell. We overlooked the fact that of course he held lambda constant when he integrated over lambda (it's an unforgettable sin not to do so). And because of that he had to include the relative frequency (probability) of each lambda - something that is glaringly missing in the above reconstruction, because that reconstruction ignores what he really tried to do.

Harry,

Do you believe the following integrals misrepresent what Bell was trying to do?





Note that Bell did not specify a range for his definite integral. So we supply ranges for him, consistent with what he was doing, and with what makes sense, given the thought experiment he had in mind. The reasons being: since in such experiments we have no control over lambda, and we are measuring each correlation on a different set of particles, it would be foolish to assume the exact same integration range (or set of lambdas for each correlation). If you have a different choice of range, I would like to hear your justifications for choosing that range and evaluate if they make sense. Now Watson's point is that with the sensible choice we just picked above, it is impossible to complete Bell's derivation, unless some rather unphysical and unreasonable assumptions are made about the integration range.

One of the other posters have suggested that we can assume that the three sets are the same since according to him "Nature is the one picking lambda, and we can assume that nature picks the same set of lambda everytime". Of course, such an assumption will allow the derivation to proceed but is such an assumption reasonable? Definitely not. In any case, even if you disagree that such an assumption is silly, it is one more candidate for rejection when the inequality so-derived is violated. There is nothing about local hidden variable theories that implies nature must pick the exact same set of lambdas every time -- none whatsoever. Some may be tempted at this point to invoke "law of large numbers". However, a deflating question to squash that thought is: How many particle pairs should be measured, to make sure that we have sampled enough distinct values of lambda to obtain the same probability distribution in each of the correlations? 100000, 1000000, .... ??? As soon as anyone tries to answer this question, they immediately realize that they will have to make an additional assumption about the number of distinct values of lambda which exist! More candidates for rejection when the inequality is violated.

To pre-empt "non-replies" which claim contrary to fact that Bell was not talking about experiments, here is what Bell said:

With the example advocated by Bohm and Aharonov [6], the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins σ1 and σ2. If measurement of the component σ1.a, where a is some unit vector, yields the value +1 then, according to quantum mechanics, measurement of σ2.a must yield the value -1 and vice versa. Now we make the hypothesis [1], and it seems one at least worth considering, that if the two measurements are made at places remote from one another, the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wavefunction does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.
Let this more complete specification be affected by means of parameters λ.

[gill1109]

One of the other posters have suggested that we can assume that the three sets are the same since according to him "Nature is the one picking lambda, and we can assume that nature picks the same set of lambda everytime". Of course, such an assumption will allow the derivation to proceed but is such an assumption reasonable? Definitely not. In any case, even if you disagree that such an assumption is silly, it is one more candidate for rejection when the inequality so-derived is violated. There is nothing about local hidden variable theories that implies nature must pick the exact same set of lambdas every time -- none whatsoever.

Bell's derivation indeed assumes that Nature is picking from the same set of lambda every time. Not only does he assume that the three sets are the same, he also assumes that the probability distribution rho(lambda) is the same too.

So if you you want to violate the inequality by a local hidden variables theory, you have to change the sets, or at least, change the distribution.

This is why in the best CHSH experiments, we choose settings anew independently and at random for each new pair of particles. Also the best proofs of CHSH take account of this. For instance, my derivation of CHSH does not assume that the distribution of the hidden variables stays the same ... instead it assumes the independence and randomness of the settings. My results are stronger than Bell's.

The detection loophole is a clever way to make the distribution of lambda depend on the settings. Particles are rejected in either wing of the experiment depending on which setting is locally in force *and* on the hidden variable (the variable which determines the outcome when they are not rejected). For each setting pair, conditional on both particles being accepted, one gets a different distribution of values of the hidden variable.

This is called biased sampling in statistics.

Similarly the coincidence loophole allows selection of different populations of hidden variable values depending on the setting pair by having the measurement times of the particles depend on setting and hidden variable. Pairs are only kept in the analysis when the difference between their measurement times is small.

This is called biased sampling in statistics.

In statistics, we use randomization in order to force the validity of needed assumptions. Think of double blind randomized clinical trials. The patients getting treatment A and the patients getting treatment B come from the *same* population because we pick them in any which way we like but then assign their treatment completely at random.

[Joy Christian]

The detection loophole is a clever way to make the distribution of lambda depend on the settings. Particles are rejected in either wing of the experiment depending on which setting is locally in force *and* on the hidden variable (the variable which determines the outcome when they are not rejected). For each setting pair, conditional on both particles being accepted, one gets a different distribution of values of the hidden variable.

This is called biased sampling in statistics.

This is a traditional but misguided point of view. It stems, on the one hand, from a total lack of understanding of the actual physics involved in the EPRB experiments, and, on the other hand, from the failure to recognize the elementary topological error Bell made in specifying his measurement functions. Once the correct topology of the physical space is taken into account, with the correct understanding of the physics involved in the EPRB scenario, it becomes evident that even the distribution of lambda need not depend on the settings, and no particles are rejected in either wing. Unfortunately understanding these facts require considerable knowledge of topology, which statisticians like Richard Gill lack. As a result, most practitioners like him continue to misinterpret the actual state of affairs in the EPRB experiments.

[gill1109]

Topology versus statistics.

[Joy Christian]

Topology versus statistics.

Truth versus lies.

[Gordon Watson]

Hello Gordon, I'm back.

Hi Gordon,

Hi Harry, and thanks,

As you decided to restart - but with more focus - the topic "Commonsense local realism refutes Bell's theorem", here's a summary of my latest opinion as expressed in that thread:

Apparently you assume that Bell grouped his derivation corresponding to experimental sequence N.

No; not at all: and I don't see why this seems so to you?

A sum can be done in any sequence. But the tests are sequential, numbered like mine or some other way.

So, to help clear this up, please bring these claims into the context of the essay that we are focussing on here now.

OK. The point was not the ordering of the sequence (indeed you are free to change the order) but the sequence itself: the data. As I stressed next by means of an illustration, Bell did not calculate according to experimental data collection.

Harry; with a BIG welcome back!

Now, since experimental data can be collected any way you like, what do you mean: "Bell did not calculate according to experimental data collection"?

Are you trying to avoid the following simple fact?

Bell's (14a) delivers the correct experimental results; his (14b)-(15) do NOT.

So: Does this not tell you where he went wrong? Like: (14b) is wrong!?

NB: I make no reference here (on this occasion) to how "Bell did not calculate according to experimental data collection."

What is relevant is that Bell did not calculate according to the reality of EPRB! If YOU start with an equation representing the experimental reality for EPRB, and YOU finish with one that does NOT: What do YOU conclude?

Time to announce a theorem? Or time to check for errors?
.

[..]

[..] Bell's integral is not over N or t, but over λ. Bell keeps λ constant over each integration step: on purpose one whole line corresponds to a single λ - and not a λi and a different λn+i which have different outcomes.

I'm not following this, at all.

That's a pity, as it is absolutely necessary to follow it!
Once more, do you understand my illustration here below? If you can follow my illustration then you will likely also follow what we have been saying to you!

So let me follow your illustration IN THE CONTEXT of my essay! You are playing the bisexual role of Alice and Bob combined, right? For YOU know both measurement outcomes; A = 230 cm and B = 240 cm, right? So you THEN get the average as 235 cm, right? So bisexual you has been able to act as coincidence-counter and expectation/average calculator, right?

And here's your MATHS! (A + B)/2 = 235 cm.

But the carpenter is ALSO the (duplicate/backup) coincidence-counter (placing - NB my accuracy here - the short beam on top of the long; and think voltages), duplicating your effort. As well, the carpenter is also the (duplicate/backup) expectation/average calculator in his bi-role, too, right?

So he goes, correctly, as programmed in trade-school: A + (B-A)/2 = (A + B)/2; in readiness for measurement.

Then, given that his coincidence-count and measurement is conducted AFTER the outputs (PLURAL) that you have previously accessed, we have this scene:

As he is about to measure the average directly and correctly, you interrupt with a shouted 235 cm; thereby disrupting the scientific process of having your result independently confirmed!

Does this accurately represent your illustration, IN THE CONTEXT of my essay?
.

Please explain this to me:

What is the physical significance (when studying EPRB) of "Bell keeps λ constant over each integration step: on purpose one whole line corresponds to a single λ"?

NB: Bell allows me to use discrete λ, so I do just that. [..]

I will answer that after you first answer the same about my ilustration here below:

I've been interrupted. So maybe you will get to answer before I get back. I'll try to slip some more meaningful stuff in below, as time permits right now.

Thanks; Gordon

Here's an illustration. A carpenter determines the average length of two similar beams as follows: He places them on top of each other, puts a mark halfway between the ends of the two beams as follows:

-------------- . . . . x
---------------------------------

Next he measures the length upto the mark of the top beam. I see him do that, and happen to know the lengths of the two beams.
So I calculate (230+240) / 2 = 235 cm and shout out that number to him. He shouts back: "Right - how did you know?"

My calculation should in theory give the same result as the measurement, despite the fact that there is not a 1-to-1 correspondence between the two.

Thus: what is the physical significance of my average length calculation (a+b)/2 ? Does it match in any way what the carpenter actually did?

See above. I look forward you answering your own question here!

How about doing so in the context of my essay? To see where we differ; or what I'm missing?
.

Bell did similarly not stick to the experimental procedure for his derivation of what may be predicted as experimental outcomes. That doesn't mean that Bell didn't make a mistake of course; but he did not mix up the lambda's.

But the mistake is that Bell does not MATCH his λs: Please think about the experiments in Paris and Peru. Please consider my (12). Please think about your old married-couples analogy.

No, we were mistaken, as others as well as I explained. Bell certainly matched his lambda's, as can be seen from his math notation. Any criticism on Bell's derivation must start from that fact.

It is not clear (to me) what fact we must start from?

Perhaps, if you told me where you'd like to start, then I could start there and get the same result: Bell's theorem refuted.

I could start where Bell starts: His (1) and (2) and prove (3) to hold?

Or at his next start, his (14a): and prove (15) not to hold?
.

[...]

As a reminder, here's a copy of my elaboration of my interpretation of your version of that part of Bell's derivation; that may come handy for a detailed discussion as it fills up a few blanks in both Bell's and your paper. Most of us had difficulty following your argumentation which is overly compact. [..]
I added in red the corresponding (or seemingly corresponding) equations in Bell's paper.

Please correct it where I misinterpret your argument, and take it from there!

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

[...]

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
[note: that section is similar to the corresponding section in the paper under discussion]

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.
[Note: Watson chose i and n+i, suggesting two subsequent measurement series.]

<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 [Watson apparently argues that] Bell's (14b) should mean, in discrete form and with added precision.

To this point looks OK. But I question what follows:

From this follows (see (14)) :

Sorry, but does (14) support your next step?

<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)]

Is this correct?

Trying to keep it simple here, to show possible error.

In general: 1/n ΣXk [1-Yk] ≠ 1/n ΣXk [1/n Σ(1-Yk)] .

Isn't that, effectively, what you've done?

No, not exactly. As we are here multiplying averages it is not so simple; and I just tried to guess how Bell went from 14b to 15.
However you make an interesting point! Perhaps Bell was making unwarranted assumptions going from 14b to 15. We should scrutinize every move. Anyway, you did not critizise that aspect of his derivation in your papers so one should not linger over it in an expansion of your argument.

So we let his slide for the moment?
.

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.
[...]

And next Bell claims, [or so it seems] 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 Bell's equation (15) etc. do not follow, according to Watson.

When it comes to QM and BT, I'm not much into analogies.

However, this one looks to be a beauty!

You've nicely identified exactly what Bell does! (Haven't you?) [..]

Once more: no, the above detailed reconstruction does not correctly reproduce Bell. We overlooked the fact that of course he held lambda constant when he integrated over lambda (it's an unforgettable sin not to do so). And because of that he had to include the relative frequency (probability) of each lambda - something that is glaringly missing in the above reconstruction, because that reconstruction ignores what he really tried to do.

How, exactly, does he hold lambda constant when he integrates over it? Don't constants come out of the integral as constants?

What am I missing here, please?

Please explain this, which should not be beyond me but is: (it's an unforgettable sin not to do so).

Soin? Not to do what? Maybe give me an example of the sin in practice, IN THE CONTEXT of my essay?

[..]

…..
Finally, to help you return to your prior very-clear thinking.

Ignoring the many essays in which similar equations are unnumbered in the Bellian literature, please consider this:

A: Bell's 1964:(14a) IS mathematically and experimentally valid.

I hope that it is clear now that Bell's (14) does not directly relate to experiment; the terms are grouped per the same recurring lambda instead of per observed particle pairs.

Recurring lambda when lambda is taken from a continuum? P = 0.

Recurring lambda from an infinite discrete set? P= 0.

Please explain: How do lambda recur in EPRB? Except by mistakenly thinking of them as being some beable to do with Bertlmann's finite number of socks?

Thus, your question about "experimentally valid" is too ambiguous. Compare once more: my average calculation about the carpenter's experiment gives the correct experimental result but does not match experiment. Do you call that "experimentally invalid"? Depending on your answer, in your wording we then get that Bell's 1964:(14a) IS / IS NOT experimentally valid. And the same for all what follows.

Harry, surely: There's a BIG misleading TYPO here!!

Bell's (14a) is FINE! It IS experimentally valid! Surely we ALL agree on that?

!! (14b) is INVALID IN EVERY WAY if you are studying EPRB!

Of course: it is quite OK for Bertlmann's smelly socks. They can be non-destructively washed and re-tested many times!

HTH? What would help me (for sure) is shorter passages upon which to comment. Any number of same; no problem.

PS: I still have no idea as to where my analysis is wrong! Especially as I can "do Bell" with lambda continuous or discrete -- and get the same result.

Glad to have you back; with best regards; Gordon