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

[Gordon Watson]

Abstract (given below): Also at http://vixra.org/abs/1406.0027 with clickable PDF download on same page.

"Generalizing Bell 1964:(15) to realizable experiments, CHSH (1969) coined the term “Bell's theorem”. Since the results of such experiments (eg, see Aspect 2002) contradict Bell's theorem: at least one step in his supposedly commonsense analysis must be false. Using undergraduate maths and logic, we find a mathematical error, a false equality, in Bell (1964). Uncorrected, and therefore continuing, this error undermines all of Bell's EPR-based analysis and many later variants, rendering them false. We can therefore predict with certainty that all loophole-free EPRB-style experiments will also give the lie to Bell's theorem."

Advancing the SPF cause: In that the Admins here are researchers too, how about we each exercise more self-discipline to stay on topic? And use some initiative to start new topics on unrelated matters arising?

To that end:

(1): The essay for discussion here is just 2 pages of text and equations: each equation [(1, 2, …)] and paragraph [#1, #2, …] is numbered; plus 1 page of Acks and Refs.

(2): The focus of the essay is Bell 1964: (15) --- and Bell's (1964) paper is available online (see the essay).

(3): Please note that the essay is based on undergrad maths and logic. So it should be possible for almost all of the discussion to begin with: RE eqn (1), or RE para #1. That way we might more easily track the common issues -- for the benefit of all.

(4): Please stay on topic: SO, given the above Abstract: there should no place in this thread for Bets, euros, QRC, debt-collection, spacetime, computer models/simulations, statistical-complexities, allegations of who knows nought, Joy Christian's more complex maths, etc.

(5): In other words: Let's all play by the rules and enjoy the unique facility that the Admins have provided. And while we're at it: Why not make our arguments and discussions so good that others want to sign up and join in too?

In a word: Let's have fun learning together in a disciplined fashion!

Thanks; Gordon

[gill1109]

Gordon, great initiative.

You say "Since the results of such experiments (eg, see Aspect 2002) contradict Bell's theorem: at least one step in his supposedly commonsense analysis must be false". It seems that you believe in local realism because Bell's theorem does not say such experimental results are impossible, it only says a certain kind of experimental results are highly unlikely under local realism (the bigger the sample size, the smaller the chance of seeing a violation of any particular size). Experimenters get to see and average just a finite number of numbers and always remember to report a standard error as well ... sampling error, statistical error. Likely error.

However, I don't quite see what you mean by saying that Bell's theorem is contradicted by experimental results. Bell's theorem does not prohibit any experimental results whatsoever.

Maybe you are talking about violation of Bell's inequality and you believe that Bell said that this was impossible (he didn't!).

Even then, there is another logical possibility: namely that the claim that experiments have violated Bell's inequality is actually a false (or at best: highly misleading) claim. Were the experiments which have been done experiments of the kind which Bell actually had in mind? Bell talks about an experiment carried out under a certain experimental protocol. If the experiments done to date used the wrong protocol, then they have proved nothing. Bell's logic simply does not apply.

And this is a true fact which has been well-known for years. See Caroline Thompson's webpage http://freespace.virgin.net/ch.thompson1/Papers/The%20Record/TheRecord.htm

Read Pearle (1970) on the detection loophole. Unfortunately not freely available on internet but anyone who wants a copy from me just has to send me an email.

Read Emilio Santos' wonderful paper: Bell's theorem and the experiments: Increasing empirical support to local realism http://arxiv.org/abs/quant-ph/0410193

The experiments still do not satisfy the experimental protocol which Bell had in his mind and many readers understood, but others did not. Read Bell's famous "Bertlmann's socks" paper (1980) and learn about "delayed-choice settings" and "event-ready detectors". In this paper, the discussions of the previous 15 years were summarized and in particular Bell takes care to put a stop to all the red-herring discussions which had been going on.

Better still read my own recent paper because now that another 35 years have gone by there are a lot more insights around and we can now give a shorter and more elementary proof of a yet stronger result.

[Joy Christian]

I disagree with both Gordon Watson and Richard Gill here.

There is no point in being concerned about Bell’s theorem without simultaneous being concerned about both spacetime and the physical space. The concept of local causality is at the heart of both Bell’s theorem and Einstein’s position. What is the point of talking about Bell’s theorem without also talking about local causality?

The strong quantum correlations are observed in nature all the time, in many areas of physics, not only in the EPRB type experiments. They are observed in solid state physics, and they are observed in elementary particle physics. To be sure, they are not subjected to the same scrutiny in these areas as they are in the context of the EPRB experiments. But that does not change the fact that they are observed in Nature, period. This fact cries out for explanation, whether you are a local realist or an adherent of the orthodox quantum ideology.

It is therefore pointless to simply argue that Bell’s theorem is wrong. So what if it is? That still does not explain why we see the strong correlations in Nature.

The only plausible explanation for their existence (at least in my opinion) is that they are properties of the physical space itself. This brings us back to spacetime (of which the physical space is naturally a part), and to the concerns of local causality of Einstein and Bell.

That is the real topic. Not a supposed error in Bell’s paper (which is flawed in my opinion too, but the error in that paper is much more subtle than what Gordon thinks it is).

[harry]

Hi Gordon,

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. But as others pointed out, that is wrong. Indeed, 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.

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. 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.

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.

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.
[...]

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.

[Ben6993]

Hello Gordon

I am quite rusty in using integration but I do not follow your refutation of AA = 1.

It all stems from an integration of a function F wrt λ. During the step of actually carrying out the integration, λ varies over all allowable values to enable the summation over all values. But, before the integration is executed, while still jiggling about with and reforming the function F into equation 14b [your naming], λ should remain constant. So the idea that A(a,λi) is not necessarily equal to A(a, λj) is not relevant as these two lambdas cannot [well, not without being explicit about it by using λi and λj in the functions, and they are not used] be used together during a playing around with function F.

AA can be calculated here as 1*1 = 1 or -1 * -1 = 1. So AA can be replaced by 1 in the function.

Or am I missing something?

[gill1109]

I agree with Ben and Harry.

[Heinera]

And so do I.

[Joy Christian]

Contrary to the above two unsurprising endorsements by the usual suspects, Bell's theorem is by no means correct. There is a fatal error in the very first equation of Bell's 1964 paper, as I have explained extensively on this page of my blog. As I mentioned above, Bell's error is rather subtle, and it is not what Gordon thinks it is.

[gill1109]

Please note that the essay is based on undergrad maths and logic.

In order to decode Bell's analysis, you also need (undergraduate?) probability and statistics - a good deal more than is typically included in undergraduate physics courses. You need undergraduate probability and statistics corresponding to fields where these topics are taken seriously. You need it at at least master's level. Econometricians and psychometricians understand Bell's analysis perfectly well, unfortunately most physicists simply don't have the training. This was already lamented by A. Fine in a landmark paper in the field 40 or so years ago. Caroline Thompson was both a physicist and a statistician. Her work was ignored and worse by the physics community for years, and she sadly died young of cancer, but nowadays experimentalists are actually adopting her ideas and recommendations. They are getting close to getting the experiment right, at last; there is intense competition, so now they all keep a very critical eye on one another. About time too.

In your essay I noticed an incorrect interpretation of the meaning of probabilistic calculations, and no accounting for statistical aspects.

[Gordon Watson]

Gordon, great initiative.

You say "Since the results of such experiments (eg, see Aspect 2002) contradict Bell's theorem: at least one step in his supposedly commonsense analysis must be false". It seems that you believe in local realism because Bell's theorem does not say such experimental results are impossible, it only says a certain kind of experimental results are highly unlikely under local realism (the bigger the sample size, the smaller the chance of seeing a violation of any particular size). Experimenters get to see and average just a finite number of numbers and always remember to report a standard error as well ... sampling error, statistical error. Likely error.

However, I don't quite see what you mean by saying that Bell's theorem is contradicted by experimental results. Bell's theorem does not prohibit any experimental results whatsoever.

Maybe you are talking about violation of Bell's inequality and you believe that Bell said that this was impossible (he didn't!).

Even then, there is another logical possibility: namely that the claim that experiments have violated Bell's inequality is actually a false (or at best: highly misleading) claim. Were the experiments which have been done experiments of the kind which Bell actually had in mind? Bell talks about an experiment carried out under a certain experimental protocol. If the experiments done to date used the wrong protocol, then they have proved nothing. Bell's logic simply does not apply.

And this is a true fact which has been well-known for years. See Caroline Thompson's webpage http://freespace.virgin.net/ch.thompson1/Papers/The%20Record/TheRecord.htm

Read Pearle (1970) on the detection loophole. Unfortunately not freely available on internet but anyone who wants a copy from me just has to send me an email.

Read Emilio Santos' wonderful paper: Bell's theorem and the experiments: Increasing empirical support to local realism http://arxiv.org/abs/quant-ph/0410193

The experiments still do not satisfy the experimental protocol which Bell had in his mind and many readers understood, but others did not. Read Bell's famous "Bertlmann's socks" paper (1980) and learn about "delayed-choice settings" and "event-ready detectors". In this paper, the discussions of the previous 15 years were summarized and in particular Bell takes care to put a stop to all the red-herring discussions which had been going on.

Better still read my own recent paper because now that another 35 years have gone by there are a lot more insights around and we can now give a shorter and more elementary proof of a yet stronger result.

Thanks Richard; with note to others: I'll cut out all the usual pleasantries from now on; they are implied! That to help me catch up with the early responses here.

Now, from ABSTRACT: I say that the results of realisable experiments contradict Bell's theorem (hereafter BT) which = Bell 1964:(15)! See Aspect (2002), eg.

Now: I need NO loophole-escape clauses (like Thompson, etc., as I recall) BECAUSE I know BT is false from undergrad math and logic (hereafter UMAL). So I expect the contradictions -- and there they are, all over the place. Bell's theorem refuted (hereafter BTR) -- experimentally and theoretically.

In short: I do not need loophole-free tests because I can predict their outcomes with certainty: BTR!

SO: I see no need YET to modify my essay.

[Gordon Watson]

Gordon, great initiative.


Better still read my own recent paper because now that another 35 years have gone by there are a lot more insights around and we can now give a shorter and more elementary proof of a yet stronger result.

I've opened a new thread for this: viewtopic.php?f=6&t=63

Please let's continue with your theory there.

[Gordon Watson]

Gordon, great initiative.


The experiments still do not satisfy the experimental protocol which Bell had in his mind and many readers understood, but others did not. Read Bell's famous "Bertlmann's socks" paper (1980) and learn about "delayed-choice settings" and "event-ready detectors". In this paper, the discussions of the previous 15 years were summarized and in particular Bell takes care to put a stop to all the red-herring discussions which had been going on.

To repeat: I need NO loopholes (as explained earlier).

Instead: My essay addresses a specific error in Bell's "Bertlmania" by equation number: Please bring it forward (with its location) and explain where it is ME that errs. Tx.

PS: I've given up renumbering all the unnumbered equations in the Bellian literature. From memory, you might want to bring something like Bell 1980:(18a), (19a)-(19e) HERE.

[Gordon Watson]

I disagree with both Gordon Watson and Richard Gill here.

There is no point in being concerned about Bell’s theorem without simultaneous being concerned about both spacetime and the physical space. The concept of local causality is at the heart of both Bell’s theorem and Einstein’s position. What is the point of talking about Bell’s theorem without also talking about local causality?

The strong quantum correlations are observed in nature all the time, in many areas of physics, not only in the EPRB type experiments. They are observed in solid state physics, and they are observed in elementary particle physics. To be sure, they are not subjected to the same scrutiny in these areas as they are in the context of the EPRB experiments. But that does not change the fact that they are observed in Nature, period. This fact cries out for explanation, whether you are a local realist or an adherent of the orthodox quantum ideology.

It is therefore pointless to simply argue that Bell’s theorem is wrong. So what if it is? That still does not explain why we see the strong correlations in Nature.

The only plausible explanation for their existence (at least in my opinion) is that they are properties of the physical space itself. This brings us back to spacetime (of which the physical space is naturally a part), and to the concerns of local causality of Einstein and Bell.

That is the real topic. Not a supposed error in Bell’s paper (which is flawed in my opinion too, but the error in that paper is much more subtle than what Gordon thinks it is).

"That is the real topic." Please: not in this thread; see OP.

Of course: if the "more subtle error" offsets the (14a)=(14b) error, then we need it here; so please bring it.

If not, it certainly warrants another (new) thread.

[gill1109]

To repeat: I need NO loopholes (as explained earlier).

I know, great. But I point out what appeared to me to be a failure of logic in the first sentence of your posting as well as a what seemed to me to be a major misconception as to what Bell actually was trying to do. You also make what seemed to me a hidden assumption and also seemed not to know the difference between Bell's theorem and Bell's inequality. So if we are going to have a sensible discussion about where Bell went wrong it might be good to clear the air of misconceptions at the start, as to what we are talking about. For instance, if we knew where Bell was headed that might change our understanding of what is going on in formula N or sentence X. You know, the context might provide some further clues.

Just an idea... maybe I am the one who is confused.

But I see now that your terminology is rather non-standard. You say "Bell's theorem = formula so and so from Bell 1964". That is not what the rest of the world called Bell's theorem. CHSH created endless confusion by writing down a slogan and giving it a name "Bell's theorem". On wikipedia you can see what most people nowadays understand by the words "Bell's theorem" but you won't find anything like that anywhere in Bell (1964) and if you look elsewhere in Bell's work you'll find rather more complex and subtle assertions.

On the other hand, one can do these things bottom up rather than top down. You want us to start with Bell (1964) formula (15) and work upwards and outwards ... ! Not a bad idea.

I understand that you want us to talk about what is usually called "Bell's three correlation inequality" or "Bell's original inequality".

I have created a Bell dropbox for papers which are hard to find. Send me an email if you'ld like to join.

Rules: feel free to put more relevant stuff in, but be careful not take stuff out. They then vanish on everyone else's computer. You can of course *copy* files to other places on your computer, but don't *move* them. You can't invite others to join it too, but you can ask me to do that. Please do not share the files further in an irresponsible way: this is a personal archive, a private library. I got these files in a legal way via my university which pays a heap of money to the publishers who have employees who need to pay the rent and buy dinner ... and shareholders who need a dividend from time to time. Probably my pension company is among the shareholders. So there are both moral and legal issues. However in the old days of real books and paper magazines one gave a photocopy to a friend or a poor student and one lent books to friends, and I don't plan to let DRM stop me from doing the same with electronic books ...

[gill1109]

Bell (1964) formula (15) is the famous three correlation inequality:

1 + P(b, c) >= |P(a, b) - P(a, c)|

He has previously made a number of definitions and assumptions.

We have to realize that an expression like P(a, b) can refer to at least three different things. His assumptions makes those two of these things equal, and the third likely to be close to equal to the other two.

On the side one thinks of P(a, b) as something accessible through experiment. We imagine repeatedly creating pairs of particles in a particular way, measuring them in a particular way depending on a and b, accuring a huge amount of data - two synchronized streams of numbers +/- 1, and then calculating the limit of their product as N goes to infinity. We imagine that that limit does exist and we call it the correlation.

On the another side we have a mathematical model for what goes on in one run of that experiment. It is a local hidden variables model. It is a probabilistic model. An element lambda is chosen by Nature at random from some mathematical set Lambda. It is chosen according to a probability distribution. Bell writes formulas in 60's physics style by sort of pretending that Lambda is a nice subset of R^p for some p and the probability distribution has a probability density rho(lambda) with respect to Lebesgue measure, so that the expectation value of some real function f of lambda can be calculated as int f(lambda) rho(lambda) d lambda. The model goes on to suppose that lambda gets carried by two particles to two measurement stations where what happens is that the experimenter gets to see the value of a function A(a, lambda) and the one station and B(b, lambda) at the other measurement station. The value if always +/- 1. Here, a and b are settings which the experimenter chose freely.

According to conventional probability theory, the expectation value of the product of A and B over many many independent runs all done exactly the same way would converge to the integral int A(a, lambda) B(b, lambda) rho(lambda) d lambda. We call it the correlation. (Actually, we shouldn't: it is an uncentered product moment. Another source of confusion.)

On the third side ... if we only do N runs and average the product we see something random, but if N is large, it will be close to the expectation value, with large probability. We call it the correlation.

It seems to me that Gordon has a problem with conventional probabilty theory, which is what (in my humble opinion) Bell is certainly using in his derivation of (15). In this part of the paper, we are doing mathematical physics. We have a mathematical model for some physical system. Our model is probabilistic. We are not doing an experiment. We are not talking about real experiments. We are talking perhaps about a conceptual experiment where the same thing is done infinitely many times ... quite a Gedankenexperiment, you see.

BTW I imagine that Bell was a frequentist and had a conventional frequentist understanding of "what probability means". Jaynes was a Bayesian and had a completely different understanding of "what probability means". If we are to discuss whether or not (15) is correct we might need to take account of what Bell's understanding of probability likely was.

[Gordon Watson]

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.

There's Para numbers and Eqn numbers; so no reason for lack of detail, to help us all.

But as others pointed out, that is wrong. Indeed, 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.

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.

After all, we're in the realms of EPRB and QM.

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. 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.

If you do not MATCH the λs you get nonsense in so far as EPRB is concerned.

BUT Bell was meant to be studying EPRB; and nothing else! So then it's NOT so fine!

Don't believe me alone: Think QM and the experiments that support QM on this very point.

Do you see that now?

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.

But surely the essay that we are discussing here is not too compact. And, in any case: with every Para and Eqn now numbered, there is no excuse: Why don't "most of you" bring your case to trial here now?

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?

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?)

So have you now found a good reason to abandon it?

If so, I'm sure the anti-Bellians will happily pay good $ for it!

PS: Try minkwe and Xray. From what I know, that's the very simple point (so fatal to Bell and his supporters) that they try to get across here repeatedly?

…..
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.

Next: One option [.] to be eliminated by the examinee; but proceed to C and D for now.

B: Bell's 1964:(14b) is [IS] [IS NOT] experimentally valid.

C: Bell's 1964:(14c) IS NOT experimentally valid -- because it equals his 1964:(15).

D: Bell's 1964:(15) --- BELL'S THEOREM, per CHSH --- IS NOT experimentally valid.

Please eliminate one option [.] from B above. Then give reasons for your answer.

HINT: From whence cometh the change from [IS] to [IS NOT]?
…..

With best regards; and thanks again; Gordon

[gill1109]

"experimentally valid" is not a relevant criterion

Bell is doing some elementary calculus and probability theory within a mathematical framework. There's a function A, a function rho, they have certain properties, and then we can derive some relations between some different functionals of A and rho.

Then we notice that the objects in that relation all correspond to things we can see in experiments.

Now we are in business.

This is physics! This is how physicists do physics! Create map from the real world *into* a mathematical world. Do math in the mathematical world. It's bigger (There are complex numbers there, not just real ones ... space can have any number of dimensions, we can even invent fractional dimension). Get exciting results in the math world. Make sure they concern things which are covered by the forwards map from real world to math world. Pull back to the real world. Get experimenter to do experiment and win Nobel prize.

[Gordon Watson]

"experimentally valid" is not a relevant criterion

After genius Bohm put the maths together for his EPRB experiment, Bohm's calculations were proven to be "experimentally valid."

After Bell put the maths together for Bohm's brilliant EPRB experiment, Bell's calculations were proven to be "experimentally invalid."

Take your pick if you wish: but whether building airplanes or putting maths together, BOTH are relevant criteria.

[gill1109]

"experimentally valid" is not a relevant criterion

After genius Bohm put the maths together for his EPRB experiment, Bohm's calculations were proven to be "experimentally valid."

After Bell put the maths together for Bohm's brilliant EPRB experiment, Bell's calculations were proven to be "experimentally invalid."

Take your pick if you wish: but whether building airplanes or putting maths together, BOTH are relevant criteria.

Gordon, I thought you were concerned about logical errors internal to a piece of mathematics.

A recent discussion in another thread has brought to my attention that Bell did indeed think he proved a mathematical theorem. His mathematical theorem (simplest form) might be the following.

Theorem. Let A and B be two functions from the product of a set of settings (a, b ...) and a set of values (lambda ....) of a so-called hidden variable to the set {-1, +1}. Let P be a probability measure on the space of values of lambda and let E denote expectation with respect to this probability distribution. Denote A(a), B(b) etc fas the random variables defined by the maps lambda -> A(a, lambda) etc etc. Then for any a, b, a', b'

E(A(a)B(b)) - E(A(a)B(b')) + E(A(a')B(b)) + E(A(a')B(b')) <= 2

Remark. We neglect measurability issues here.

Proof: Since all random variables here are bounded there are no issues in exchanging expectation values and summation. By elementary algebra we see that

A(a)B(b) - A(a)B(b') + A(a')B(b) + A(a')B(b') <= 2

Take the expectation left and right, and write expectation of a sum (and difference) of four terms as sum and difference of four expectation values.

QED.

According to Bell (as far as I can see) this is a prototypical statement and proof of Bell's theorem in a little bit more modern language than Bell used 50 years ago.

What's the problem with it? I mean, is there an internal problem? Do you just have problems with the applicability, ie the relevance, or do you believe that the tautology is wrong? It would be good to get that out of the way.

Bell like most sensible physicists was not bothered by technical mathematical niceties concerned with what exactly do we mean by a function, what exactly do we mean by integration, and so on. He could integrate anything he came across, no problem. Gerard 't Hooft has a very low opinion of mathematicians because when he did a course on measure theoretic integration he found out that mathematicians forbid him to integrate many things which he knew perfectly well how to integrate. Well who needs mathematicians then.


PS see http://arxiv.org/abs/1402.1972 for another mathematician's version of the theorem.

[Gordon Watson]

Hello Gordon

I am quite rusty in using integration but I do not follow your refutation of AA = 1.

It all stems from an integration of a function F wrt λ. During the step of actually carrying out the integration, λ varies over all allowable values to enable the summation over all values. But, before the integration is executed, while still jiggling about with and reforming the function F into equation 14b [your naming], λ should remain constant. So the idea that A(a,λi) is not necessarily equal to A(a, λj) is not relevant as these two lambdas cannot [well, not without being explicit about it by using λi and λj in the functions, and they are not used] be used together during a playing around with function F.

AA can be calculated here as 1*1 = 1 or -1 * -1 = 1. So AA can be replaced by 1 in the function.

Or am I missing something?

Hi Ben,

And Yes; it seems that you are missing a little something.

If you are studying EPRB, and if you are rusty using integration, you can always sum with and make sense of the world. That is, your results will be experimentally valid.

Such results [ie, calculations that are experimentally validated], usually help us to make sense of the world.

Now, in that my essay does that for you, I recommend that you try to figure the relevant summations for yourself: checking with my results as required.

Then you will be in a position to see that, if the two components of the product are randomly , then the outcome is NOT always .

Alternatively, if don't see that: bring back what you do see and we can take it from there.

NB: Even IF Bell was not rusty using integration -- his results are experimentally INVALID: such results tend to confuse.

For the question then arises: Was it a false assumption or just a maths error?

PS: You can see above, in my recent reply to Harry, how easy it is the locate where Bell went wrong.

Cheers; Gordon