Is action at a distance possible as envisaged by the EPR Paradox.

[charlylebeaugosse]

Not sure I would agree here that delayed choice experiments are not relevant. What is the meaning of EACP if you have the future affecting the past? And you cannot be certain that is not happening once you look at those experiments.

I personally cannot see that EACP is a "weaker" assumption than locality. I mean, it seems a subjective assessment.

DrChines, thanks for the answers: once more, I reply first to the last "question" (in fact you made a statement, but there is an implicit question, I presume). In what follow, I start from the view-point that Locality is an hypothesis stronger that Non-locality. The (or should I say "one" ?) proof that the EACP assumption is weaker than Locality lies in the fact that when assuming the EACP, one can ALSO either assuming Locality, or Non-Locality, or not make any assumption of that sort. If one assumes the EACP and non-locality, it becomes trivial not only to prove that the EACP is weaker than Locality, but also that most of the correlations that are easy to compute when assuming Locality are not any more eas to compute, and in fact cannot at all be evaluated from QM, weak realism, the EACP and Non-locality (for instance the quantity of the for <Y,X'> or <Y',X> where a prime means that tehnobservable only exist by realism and X, Y correspond to teh two observation stations (say Alice has X and X', Bob has Y and Y'). The paper that I have cited mention several comparisons of the EACP with Locality. For instance, in a universe without "realism",
the negation of the EACP permit Super Luminal Signaling while it is known that the contrary of Locality does not (or there would probably be more supporters of Einstein against Non-locality). I will not try to copy that paper here.

As for the story of delay, I meant delayed erasure (including thus the supposed realisation of Wheeler's experiment by Jacques et al which in fact uses delayed erasure). As I said, Cthugha has beautifully explained the delayed erasure of Kim et al: there erasure only permits the re-appearance of a structure in the coincidence between D2 OR D3 and D1.
It is a co-structure so to speak that one gets by erasing the marking of the paths. Indeed, if one would consider D2+D3 vs D1, the wavy structure in the coincidence count would be washed out. This couples with a weakness of the Copenhagen interpretation to provide a weird story, but you can even stick to Copenhagen and realize that all funny effects of delays are illusory (I mean, the funny-ness, so to speak, is illusory, the effects are there, but need to be clearly understood, and again, Cthugha did a great work on that). In fact, he, you and a few others convinced me by the quality of the posts, to come back to this Forum after trying it for one day or so some time ago and convincing myself then that it was useless. Now I understand that perhaps I got my first copies of Bell 1964 and EPR thanks to you (which is true if you did post them as the sources, initially with a very heavy copy of Bell's paper): if so, I have to (and I do) thank you very much as this is what convinced me that the reason why I initially decided to come back to my youth-dream -subject was, essentially at least,... a fraud. I have seen some pieces of you where you defend what I also consider as what needs to be defended, and feel confident that here were we seem to disagree, we will end up on the same side once I put my act/words together. As for Wheeler type delay, we need another thread as, contrary to what happens for delayed erasure, one has to slightly take on Copenhagen.
I do not know how to meet you on another thread that you or me would create.
b.t.w., there are some questions that I would like to post and that may be new threads.
I will now look into that as, indeed, for Wheeler's delay proper, there is much to say that I do find crucial for the overall story, and that would probably warrant another thread if such a thread is not there yet (assuming that details on Bell type theory do belong to (naive) questions on EPR such as "Is action at a distance possible as envisaged by the EPR Paradox?" (where I have added a "?" myself to have a bona-fide question, to which I have proposed my answer in segment I have just posted).

 

[DrChinese]

1. Now I understand that perhaps I got my first copies of Bell 1964 and EPR thanks to you (which is true if you did post them as the sources, initially with a very heavy copy of Bell's paper):

2. I do not know how to meet you on another thread that you or me would create.
b.t.w., there are some questions that I would like to post and that may be new threads.

1. You are welcome if so! I used to have a worse (darker) copy and then someone helped me get a better one.

2. We could discuss the Tresser paper and related in a new thread, no prob there. If you want I can start it. I tend to hold onto locality so the Tresser ideas are pretty interesting. His work has been on my radar for a while although I have not read closely.

 

[charlylebeaugosse]

1. I think the EPR & Bell source papers are wonderful, I have copies available on my site for those who wish to read them. I have read a bit here or there about Einstein on Bohm (I am sure you didn't mean Bell), but perhaps you are referring to some specific comnment? I am not sure I follow your point here.

2. Einstein gave us the "the moon is there when not looking at it" comment, so I am not sure I quite agree if you are saying that Einstein was not a "naive" realist. (Although I personally don't care for the use of the word naive as it comes off as an insult.) But I would be interested in a quote that clearly expresses a) what realism looks like which is NOT naive; and more importantly b) any evidence Einstein subscribed to that view. Given his "moon" comment, which is pretty clearly in the "naive" school.

In my mind: the HUP flies in the face of all versions of realism. I mean, the word just doesn't have much meaning if you reject "simultaneous elements of reality" as too naive.

3. Einstein's name was on the 1935 paper, not really sure why there would be a need to back away from it. It was a great paper, and is quite important even while being wrong (in its final conclusion).

On 1. (Just discovered that it is easier to answer first t the last question but that you can do that respecting there numbering in the way the answers are displayed. Indeed I should perhaps use WORD: any advice?) I was just meaning a link giving access to the paper of Bell and EPR, where originally at least (when I used that link), the copy of Bell was an heavy image from a scan with some markings. This was my own first access to the original versions of Bell and EPR (as I had no access to an university library), and I thank very much whoever posted these access (in a page telling about EPR, or EPR-Bohm, or Bell, I cannot remember).

On 2. Not sure if you believe that the moon is not there when you do not look a it: if, as I assume from what you write elsewhere, you do believe that the moon is here even when nobody looks at it BUT yet think that Einstein's question to Pais was naive, then I will explain. If you do not believe that the moon is here when... then, I give up (but again, I trut you have the sane point of view, like A.E. indeed). I need your help: what is HUP?
(sorry for he low level of this question, but I prefer to understand all that is written). In fact, even if I understood HUP, I would still have problems with "the HUP flies in the face of all versions of realism". Please remember that the effective language of science is not English, but broken English as many of us did not grow up in an English speaking country.

On 3. See the book of Fine and especially, read what Einstein wrote on that (starting in about 1933 when he already used the word paradox). If the paper is beautiful, it is hard and had hidden in it a "proof" that QM is false, causing part of the difficulty to understand Bohr's answer. Einstein's standards were such that he would not go public on his opinion of the EPR paper, but see Fins's "The Shaky Game" and use that to find other writings by S.E. himself. The paper EPR is interesting but Einstein's treatment of the completeness question is at an Einsteinian level. This being said, on time I expect to:
-(a) Explain why the main official thesis of EPR is right
- (b) Explain the relation of EPR with another paper (Einstein, Tolman, Podolsky 1931) so that in some sense, Podolsky is also right in his main agenda.
-(c) Explain why a proper use of the reality element (in the spirit of what is written before they are defined) would not permit a Bell-type result, while the way Podolsky uses the elements of reality allowed Richard Friedberg to get such inaqualities (at a time Bell's work was known by a minority: see the main book by Jammer on QM).
- (d) Defend that the completeness issue is no more relevant and try to explain why it was then, especially for Einstein.

All that I promise is a bit of history and a bit of physics. I would not like to spend too much time on history, so that I may take some time to deliver (a) to (d). I have not thought at the ordering f these points, hence do not know in which order I will answer, not if some other issues will have to be covered as well to make my exposition comprehensible (harder because English is not my native tong, as the English speakers will have caught).

Besides:

- I hate "your" statement about "Einstein's name was on the 1935 paper": it has happened to me to have a paper submitted without my imprimatur: it is VERY painful. I write "I hate" because it is not only your statement: many people have told me the
same thing, but nobody for whom a similar experience was painful.

- As for your "not really sure why there would be a need to back away from it", you'll see in what I have quoted above (probably a very incomplete list in fact) that this is worth the pain and the time.

- As for your "It was a great paper, and is quite important even while being wrong (in its final conclusion)", I'll defend the conclusion in the way I said, but the fact that it was a great paper cannot be (reasonably) disputed, be it by its impact on Bohr, but of course, there is much more as we know. (Note: You may know that in 1985, Rosen considered the proof in the paper correct but the paper incorrect because he had been convinced that physics is non-local, which is why I hate the way presented the whole story along the years, often keeping his hidden agenda close to his vest to the point that Wigner considered Bell 1964 as the nicest proof of the non-existence of HVs where Bell was laying a stone to build the "non-locality and realism" household. btw, the paper by Wigner on Bell's Theorem is worthwhile reading, be it only because the preferred proof on no HV of his close friend John von Neuman can probably only be found in that paper (in a note), i.e., in particular it was not the pseudo-proof proven to be wanting by Bell in the first paper that he wrote on HVs.

 

[DrChinese]

(harder because English is not my native tong, as the English speakers will have caught).

Your English is fine. HUP is my abbreviation for the Heisenberg Uncertainty Principle.

If you have any links to the material you are referencing, that would be helpful. Or alternately if you can give verbatim quotes with a little more context. Thanks!

I don't believe the moon is there when it is not being looked at, and I mean that strictly in the sense that I deny the existence of simultaneous elements of reality a la EPR. So if that puts me in the "hopeless" category for you, well, so be it.

I do believe the official conclusion of EPR is correct: the part about the Alice's reality is dependent on the nature of the observation by Bob. But they considered this unreasonable (an opinion which turned out to be experimentally incorrect a la Aspect).

 

[RUTA]

On 2. Not sure if you believe that the moon is not there when you do not look a it: if, as I assume from what you write elsewhere, you do believe that the moon is here even when nobody looks at it BUT yet think that Einstein's question to Pais was naive, then I will explain. If you do not believe that the moon is here when... then, I give up (but again, I trut you have the sane point of view, like A.E. indeed).

I'm with DrC, I also don't believe "the Moon is there when nobody looks." By "when nobody looks" I mean "when not interacting with anything." [Some people use the term "screened off" for this situation.] Perhaps those who believe otherwise have met Harvey

 

[DevilsAvocado]

Hey, look at my post count!

+3,000 posts! Respect and congratulations!

 

[charlylebeaugosse]

Your English is fine. HUP is my abbreviation for the Heisenberg Uncertainty Principle.

If you have any links to the material you are referencing, that would be helpful. Or alternately if you can give verbatim quotes with a little more context. Thanks!

I don't believe the moon is there when it is not being looked at, and I mean that strictly in the sense that I deny the existence of simultaneous elements of reality a la EPR. So if that puts me in the "hopeless" category for you, well, so be it.

I do believe the official conclusion of EPR is correct: the part about the Alice's reality is dependent on the nature of the observation by Bob. But they considered this unreasonable (an opinion which turned out to be experimentally incorrect a la Aspect).

I'll think about an answer that may be helpful for you and for people who believe that the moon pre-existed humans presence on earth. I'll hgave also to defend EPR. As for teh quotes, before I am organized, I can be asked for references when I am not precise enough giving them... But I'll need some time.

 

[DevilsAvocado]

Bell's words:

"-My theorem answers some of Einstein's questions in a way that Einstein would have liked the least."


responding to Einstein's:

"-On this I absolutely stand firm. The world is not like this."

Yes, this is very true. Einstein’s own argument boomeranged on him:

no action on a distance (polarisers parallel) ⇒ determinism
determinism (polarisers nonparallel) ⇒ action on a distance​

But to be fair we must say that also Niels Bohr was somewhat 'wrong'. If it turns out that nonlocality is fact, then QM must be considered 'incomplete'... or ...??

There is no doubt in my mind that Einstein, if he was alive, would have accepted the work of Bell as starting point for "something new", not a starting point for an old man to get 'grumpy'.

 

[DevilsAvocado]

Great post!

Thanks DrC.
(And don’t forget: I’ve learned mostly everything from you, RUTA, JesseM and the other very skilled people here on PF. )


P.S: very skilled <> billschnieder :zzz:

 

[DevilsAvocado]

Bill, give it up, I don't know where you're getting the ideas you espouse here, but JesseM is tearing them apart. I'll say it again, you can post in bulk, but it doesn't change that your posts are rambling and borderline-crackpot, whereas JesseM is sticking to the science.

I agree, of course, but trying to talk reasonable to Bill is a waste of time. He lives in his own little bubble; firmly convinced he represents the "universe", when the fact is that he’s totally lost and totally alone in his "reasoning".

As usual, Bill misunderstands everything about everything, and he for real thinks he has an undisputable right to do what he wants here at PF, and he apparently don’t understand simple English. When I informed him about the https://www.physicsforums.com/showthread.php?t=414380", and that: "Poorly formulated personal theories, unfounded challenges of mainstream science, and overt crackpottery will not be tolerated anywhere on the site."

His answer was, of course, "brilliant and out of this world":

Let us see what the document you linked to says:

When posting a new topic do not use the CAPS lock (all-CAPS), bold, oversized, or brightly colored fonts, or any combination thereof. They are hard to read and are considered yelling. When replying in an existing topic it is fine to use CAPS or bold to highlight main points.

What can one say? The "genius" has spoken. + +

But don’t worry nismaratwork, this is the fact:

Use of this Forum and your comments is not a right. It is a privilege granted you by Physics Forums under the terms of this agreement and can be revoked at any time without warning.

And sooner or later we will see this:

He’s getting cocky...

 

[DevilsAvocado]

The Effect After Cause Principle (EACP) states ONLY that:
For any Lorentz observer O, once an effect E of cause C is observed by observer O, no fiddling with C can change E.

Okay charly, thanks for info.

How do EACP handle more than one observer in different frames of reference and RoS?

 

[DevilsAvocado]

I'm with DrC, I also don't believe "the Moon is there when nobody looks."

I guess I agree... one thing that bothers me though... How does the Moon know if someone is looking??

I mean, it’s no problem for the Owl Nebula, it has eyes!

[PLAIN]http://seds.org/messier/Pics/More/m97rosse.jpg

 

[billschnieder]

Yes, Bill. Would you deny, for example, that a physical process that had P(++)=0.3, P(+-)=0.2, P(-+)=0.15, and P(--)=0.35 (with all of these numbers being the frequentist probabilities that would represent the fraction of trials with each value in the limit as the number of trials goes to infinity) could easily generate the following results on 4 trials?

I gave you an abstract list. No mention of anything such as trial. No mention of anything such a physical process. I asked you to give me the probability of one of the entries from the list, and you told me it was impossible despite the fact that this is what is done everyday in your favorite frequentist approach to probability. When ever you say the probability of Heads and Tails is 0.5 you are doing it, whenever you say the probability of one face of a die is 1/6, you are doing the exact same thing you now claim is impossible. Go figure.

I already gave you the answer which is 1/4. Now you want to ask me about a completely different question than the one I asked you.

Note that the wikipedia article says "close to the expected value", not "exactly equal to the expected value". And note that this is only said to be true in a large number of trials, the article does not suggest that if you have only four trials the average on those four trials should be anywhere near the expectation value.

First of all, you were the one arguing that Bell's equation (2) is not a definition of expectation value which according to you is defined according to the law of large numbers:

true probabilities are understood to be different from actual frequencies on a finite number of trials in the frequentist view, and I don't think there's any sensible way to interpret the probabilities that appear in Bell's proof in non-frequentist terms. An "expectation value" like E(a,b) would be interpreted in frequentist terms as the expected average result in the limit as the number of trials (on a run with detector settings a,b) goes to infinity, and likewise the ideal probability distribution ρ(λi) would in frequentist terms give the fraction of all trials where λ took the specific value λi, again in the limit as the number of trials goes to infinity. Then you can show theoretically that given Bell's physical assumptions, we can derive an inequality like this one:

So I see an admission that you were wrong here.

Secondly, I gave you an abstract list, no mention of trials. The context of the question is entirely within the list you were given. The list is the population. There is no need for any trials. This is the frequentist view. With a coin or a die, you can give a probability without any trials. You do not need a single trial. You are way way of base here.

Finally, note that in the forms section of the article they actually distinguish between the "sample average" and the "expected value", and say that the "sample average" only "converges to the expected value" in the limit as n (number of samples) approaches infinity. So, it seems pretty clear the wikipedia article is using the frequentist definition as well.

Again, nothing to see here. No mention about "samples" in my question to you, just an abstract list, I did not expect you to use any other definition than the frequentist one. My abstract list I gave you is the population just like (Heads, Tails) is the population for a coin, and (1,2,3,4,5,6) are the population for a die. The true probability is the one in the population. The law of large numbers is only meaningful for samples taken from the population. You can visualize it by thinking that if you would randomly pick an entry from the the list I gave you, the relative frequency in these trials will approach the true probability in the list as the number of trials tends towards infinity. But I gave you an abstract list and asked you for the true probability of the list, and you dodged it because you saw the impact of the question to your ridiculous position.

Here the wikipedia article is failing to adequately distinguish between the "mean" of a finite series of trials and the "mean" of a probability distribution. If you think the expectation value is exactly equal to the average of a finite series of trials, regardless of whether the number of trials is large or small, then you are disagreeing with the very wikipedia quote you posted earlier from the Law of Large Numbers page

Wah JesseM, who said anything about trials! Now according to you it is the wikipedia article that is "failing"? On the contrary, it is you who is failing to understand basic statistics and probability theory. The question I asked you was the following:

You are given a theoretical list of N pairs of real-valued numbers x and y. Write down the mathematical expression for the expectation value for the paired product.

and your answer was:

It's impossible to write down the correct objective/frequentist expectation value unless we know the sample space of possible results

The law of large numbers says if you would randomly pick a large number of pairs from our given abstract list, the average value will get close to the true expectation value as the number of pairs you pick tends towards infinity. It does not say the true expectation value for the list can not be known unless the size of the list is infinity. The true expectation value for the list I gave is <xy>. This is the value, you will approach if you were to randomly pick a very large number of pairs from our list and calculated their average. Capice?

 

[billschnieder]

According to you, would it be more correct to write "the average of the results obtained from any number of trials would be exactly equal to the expected value"?

No! You are not paying attention, and it is you who is confusing the theoretical and the empirical. You keep bringing up the word "trials" which is tripping you up. Just because I give you an abstract list does not mean the list represents "trials" in an experiment. If I wanted the list to represent results of trials, I would state that clearly. For a coin N = 2, for a die, N=6, yet you can still calculate the expectation value without any infinite trials! You still do not get the distinction between a sample from a population, and a population. In statistics, if you are given the population, you can calculate the true probabilities without any trials. It is done every day in the frequentist approach, which you claim to understand!

Your arguments may have been assuming the "finite frequentist" view, but as I said that's not what I'm talking about. I'm talking about the more common "frequentist" view that defines objective probabilities in terms of the limit as the number of trials goes to infinity.

So now you are no longer talking about "frequentist view" but some special version of the frequentist view? This is just plain wrong, as the section of Wikipedia I quoted to you clearly explains. The law of large numbers gives you an approximation to the "true" probability and that approximation get's more accurate as the number of trials increases. It does not define an objective probability. The objective probability is defined by the actual population content and the population size can be any number.

What do you mean by "true expectation value"? Are you using the same definition I have been using--the average in the limit as the number of trials goes to infinity--or something else?

Bah! Do you have short-term memory issues or something. What have we been discussing these past 5 pages of posts?

Did you not see this definition I posted from Wikipedia: http://en.wikipedia.org/wiki/Expected_value

In probability theory and statistics, the expected value (or expectation value, or mathematical expectation, or mean, or first moment) of a random variable is the integral of the random variable with respect to its probability measure.

For discrete random variables this is equivalent to the probability-weighted sum of the possible values.

For continuous random variables with a density function it is the probability density-weighted integral of the possible values.
...
The expected value may be intuitively understood by the law of large numbers: The expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity.

Also, when you say "it is their only hope", what is "it"?

In case you have forgotten, we are discussin Bell's inequality

|E(a,b) + E(a,c)| - E(b,c) <= 1

According to Bell, E(a,b), E(a,c) and E(b,c) are "true" expectation values with a uniform ρ(λ) for all three terms, or if you prefer "objective" expectation values for the paired product of outcomes at two stations. Those expectation values are defined by Bell in equation (2) of his paper, using the standard mathematical definition of expectation values for continuous variables.

Experimenters measure "something". From this "something" they calculate certain empirical expectation values E(a1,b1), E(a2,c2) and E(b3,c3) where the numbers corresponds to the run of the experiment. They then plug these empirical expectation values into the LHS of the above inequality and obtain a value which they compare with the RHS and notice that the inequality is violated.

In case you forgot, this whole discussion is about whether those empirical expectaion values are appropriate terms to be used in Bell's inequality. If those empirical expectation values are not good enough estimates of the true expectation values, they are not good enough to be used since Bell's inequality will not guaranteed to be obeyed. But without knowing anything about the λ's, it is not possible to design experiments which will ensure that the expectation values are good enough. So the only hope left for experimenters doing such experiments is to verify whether the data they obtained in their experiments at least provides a uniform ρ(λ) for all three terms, by re-sorting the datasets of pairs together. If it does not, Bell's inequality can not be applied to the data, if it can, Bell's inequality MUST be obeyed.

 

[billschnieder]

Are you saying their only hope is to make some assumptions about the values of λ in their experiment, such as the idea that the distribution of different λi's in their sample is similar to the one given by the probability distribution function

I'm not saying that, nor do I need to say that because they ALREADY make the assumption ("fair sampling assumption") that the sample of measured pairs is representative of the population of emitted pairs, which is exactly the assumption that the sample ρ(λ) is the same as the population ρ(λ).

And if you are saying physicists have to make some assumption about the values of λ in their experiment, why did you object so venomously at the end of post #1228 to a statement of mine which just suggested this was what you were saying, namely

Because you claimed I was saying, physicists need to know the values of λ, in order to calculate expectation values, even though the words I wrote actually said they don't.

Yes, but to get a "damn good estimate of the true expectation values", all that's necessary is that the actual frequencies of different measurement results were close to the "true probabilities" (in frequentist terms) in equations like this one

E(a,b) = (+1*+1)*P(detector with setting a gets result +1, detector with setting b gets result +1) + (+1*-1)*P(detector with setting a gets result +1, detector with setting b gets result -1) + (-1*+1)*P(detector with setting a gets result -1, detector with setting b gets result +1) + (-1*-1)*P(detector with setting a gets result -1, detector with setting b gets result -1)

False! The above equation does not appear in Bell's work and is not the expectation value he is calculating in equation (2). Furthermore, the probability distribution which is required to be uniform across all terms is ρ(λ).

As long as the fraction of trials where they got a given pair of results like (+1 on detector with setting a, +1 on detector with setting b) is close to the corresponding "true probability" P(detector with setting a gets result +1, detector with setting b gets result +1) then the sample average of all the products of measured pairs will be close to the expectation value. And the law of large numbers says that the measured fractions are likely to be close to the true probabilities for a reasonably large number of trials (a few thousand or whatever)

Since you continue to insist on this ridiculous idea, explain how they can know what the "true probability" is. And in case you are going to respond with the "large numbers" argument, make sure you also explain how they can be sure that the number of trials is large-enough. I remember asking you where you got "1000 or more" from and you did not answer, so remember to answer it this time. Explain what you mean for a sample to be reasonably large and make sure you exlain how to decide what is reasonably large and what is not. As you seem to have already decided that a few thousand is large enough, please back up that claim.

even if the number of trials is small compared to the number of possible values of λ so that the frequencies of different λi's in the particles they sampled were very different from the frequencies in the limit of an infinite number of trials, which is what is given by the probability distribution on λ. Do you disagree with that "even if"?

I absolutely disagree. In Bell's work, the expectation values are obtained by integrating over ρ(λ), the derivation of the inequalities relies on the fact that ρ(λ) is the same for all three terms. If this requirement is not met, the inequalities CAN NOT be derived. It is common sense to realize the fact that, if in any sample ρ(λ) is not the same for all three terms, that sample does not conform to the mathematical requirements inherent in the derivation of Bell's inequality.

The fair sampling assumption discussed on wikipedia doesn't say anything about the full set of all hidden variables associated with the particles, it just says the fair sampling assumption "states that the sample of detected pairs is representative of the pairs emitted"

Huh? What do you think the underlined statement means. It means the sample is not significantly different from the population when you consider the probability of parameters that are important to the calculation being performed. And I have already demonstrated that ρ(λ) is important. If P(λ3) in your sample is 0.2 and P(λ3) in your population is 0.6, your sample can not be representative of the population. If you are interested in studying the height distribution of people in Washington, and you pick a sample of people in which the sample height distribution is different from the true height distribution in Washington, your analysis will be useless because your sample is unrepresentative.

, i.e. if 2000 pairs were emitted but only 1000 pairs were detected and recorded, then if 320 of those pairs gave result (+1 on detector with setting a, -1 with detector with setting b), then the fair sampling assumption would say that about 640 of the pairs emitted would have been predetermined to give result (+1 on detector with setting a, -1 with detector on setting b). Aside from those two predetermined results, the fair sampling assumption doesn't assume anything else about the hidden variables in your sample being "representative" of all those emitted.

The underlined words are clearly an admission that the fair sampling assumption can not be divorced from the requirement that ρ(λ) not be significantly different between the population and the sample. How else will they be predetermined. Remember in Bell's notation, the outcomes are given by the functions:
A(a,λ)=+/-1, B(b,λ)=+/-1 with the understanding that the terms in parenthesis are deterministically resulting in the outcomes. So you are arguing with yourself here.

If ρ(λ) in the sample is not significantly different from ρ(λ) in the population, then the distribution of the outcomes will not be significantly different. However, just because the distribution of the outcomes is not significantly different is not proof that ρ(λ) is the same. It is a necessary but not a sufficient condition as you still must be able to resort the data.

 

[billschnieder]

So can you just tell me yes or no, was I right to think that by "resorting" you meant renumbering the iterations on each of the three runs, in such a way that if we look at the ith iteration of runs with settings (a,b) and (a,c) they both got the same result for setting a, if we look at the ith iteration of runs with settings (a,b) and (b,c) they both got the same result for setting b, and if we look at the ith iteration of runs with settings (b,c) and (a,c) they both got the same result for setting c?

Yes.

If this is correct, then I'll just note that even if you can do this resorting, it doesn't guarantee that the "hidden triples" associated with the ith iteration of all three runs were really the same, much less that the value of λ (which can encompass many more details than just three predetermined results for each setting) was really the same on all three.

I already answered this:

Every pair of outcomes at those angles is deterministically determined by the specific λ being realized for that iteration. So if for example we had only 5 possible λ's (λ1, λ2, λ3, λ4, λ5), the only possible outcomes are (++, +-, -+, --) which means some of the λ's must result in the same outcome. If say λ5 and λ3 each result in the same outcome (++) deterministically, and each of them was realized in the experiment exactly once, when you resort it, it doesn't matter whether the (++) at the top of the resorted list corresponds to λ5 or λ3 for the following reasons. If in your large number of iterations, λ5 and λ3 are fairly represented, you will still have the right number of (++)'s for both λ5 and λ3 and it doesn't matter if the specific (++) you got at the top is a λ5 ++ or a λ3 ++. Also, if for the three angles under consideration a,b,c a number of λ's deterministically resulted in the same outcomes for (a,b), (b,c) and (a,c) those lambdas are effectively equivalent as far as the experiment is concerned and you could combine them, updating the combined P(λ) appropriately. Finally as clearly explained in my posts #1211 and #1212, being able to sort the data is a test to see if the data meets the mathematical consistency required by Bell's derivation, in which the (b,c) term is derived by factoring out the b from the (a,b) term and factoring out the c from the (a,c) term and multiplying them together. Such factorization imposes a consistency requirement that unless you can do that, the inequality can not be derived and any data which can not be factored likewise, is mathematically incompatible with the inequality.

Of course if you can do such a resorting it shows that it is hypothetically possible that your dataset could have been generated by hidden variables which were the same for the ith iteration of all three runs, and if you can do such a resorting it also guarantees that your data will obey the inequality. Is that all you're claiming, or are you claiming something more about the significance of "resorting"?

I am claiming that the derivation of Bell's inequality and the factorization steps involved demand that any datasets of pairs used for calculating the terms for the inequalities, must be resortable as I explained and you understood, and if they can not, Bell's inequality is not guaranteed to be obeyed for no other reason than violation of mathematical consistency requirements. So being able to resort the data, is a necessary condition for the dataset to be usable as a source of terms for the inequality. This does not mean it is a sufficient condition, but it is necessary. You are right that being able to resort does not ensure that ρ(λ), is uniform. However, not being able to resort, is definite proof that ρ(λ) is not uniform. So resorting is a necessary but not necessarily a sufficient condition if ρ(λ) is uniform.

First of all, it's a physical assumption that the result A on Earth depends only on a and λ and can therefore be written A(a,λ)--if you allow "spooky" influences, why can't the result A on Earth depend on the setting b, so that if on Earth we have setting a1 and hidden variables in state λ5, and on the other planet the experimenter is choosing from settings b1 and b2, then it could be true that A(a1, λ5, b1)=+1 but A(a1, λ5, b2)=-1?

You did not understand. A(a,λ) is a function that maps a given value of a and λ to an outcome of +/-1. The function has nothing to do with probability. A(a,λ), "a" can depend on "b". All that the notation A(a,λ) = +/-1 means is that given a specific value of "a" and a specific value of "λ", we get a specific value of either +1 or -1. It is a function of two variables not three and there is nothing in the notation itself that should suggest to you that dependence on "b" is not allowed.

then it could be true that A(a1, λ5, b1)=+1 but A(a1, λ5, b2)=-1?

Wrong. "a1" depends on "b2" means that if you looked at how the values of "a" and "b" varied with time after the fact, it will not be random, but certain values of "a" will always be paired with certain values of "b" due to the instant communcation when the settings were being made. It doesn't mean a specific value of "a" will give a different result depending on which value of "b" existed on the opposite side.

It's also a physical assumption ...

You completely missed the point. Despite the fact that the "physical assumptions" in this example are completely contrary to Bell's, we still obtained the exact same expression for the expectation value of the paired product. Which means the "physical assumptions" are peripheral.

 

[JesseM]

I gave you an abstract list. No mention of anything such as trial. No mention of anything such a physical process. I asked you to give me the probability of one of the entries from the list, and you told me it was impossible despite the fact that this is what is done everyday in your favorite frequentist approach to probability.

Not if we are excluding "finite frequentism", which I already told you I was doing. If you want to quibble over terminology, I'll just bypass that by using the term "limit frequentist probability" to refer to the notion of probability I have been using consistently throughout this discussion, where a "limit frequentist" probability is understood to mean the frequency in the limit as the sample size goes to infinity. Does your list of four give us enough information to know the frequency of ++ in the limit as the sample size goes to infinity? If not, then there is not enough information to estimate the "limit frequentist probability".

When ever you say the probability of Heads and Tails is 0.5 you are doing it, whenever you say the probability of one face of a die is 1/6, you are doing the exact same thing you now claim is impossible. Go figure.

No, in those cases I am just using the physical symmetry of the object being flipped/rolled to make a theoretical prediction about what the limit frequency would be, perhaps along with the knowledge that empirical tests do show each option occurs with about equal frequency in large samples, and that the law of large numbers says there is only a small probability the frequency in large samples would differ much from the "limit frequentist probability"

I already gave you the answer which is 1/4.

Yes, and that answer is incorrect if we are talking about the "limit frequentist probability", as I already made clear I was doing.

Note that the wikipedia article says "close to the expected value", not "exactly equal to the expected value". And note that this is only said to be true in a large number of trials, the article does not suggest that if you have only four trials the average on those four trials should be anywhere near the expectation value.

First of all, you were the one arguing that Bell's equation (2) is not a definition of expectation value which according to you is defined according to the law of large numbers:

true probabilities are understood to be different from actual frequencies on a finite number of trials in the frequentist view, and I don't think there's any sensible way to interpret the probabilities that appear in Bell's proof in non-frequentist terms. An "expectation value" like E(a,b) would be interpreted in frequentist terms as the expected average result in the limit as the number of trials (on a run with detector settings a,b) goes to infinity, and likewise the ideal probability distribution ρ(λi) would in frequentist terms give the fraction of all trials where λ took the specific value λi, again in the limit as the number of trials goes to infinity. Then you can show theoretically that given Bell's physical assumptions, we can derive an inequality like this one

So I see an admission that you were wrong here.

Um, how do you figure? The two statements of mine are entirely compatible, obviously you are misunderstanding something here but if you don't explain your "logic" I have no idea why you think they are incompatible. Both statements are defining "expectation value" in terms of a sum of possible outcomes weighted by their "limit frequentist probabilities" (which is what I meant by 'true probabilities' in the first statement), which means they are defined in terms of the limit as the number of trials goes to infinity. The first statement is just pointing out that an average over a finite number of trials is never guaranteed to be equal to the "limit frequentist" expectation value which is defined in terms of the limit as the number of trials goes to infinity, although the larger the finite number, the higher the probability that it's close to the "limit frequentist" expectation value.

Secondly, I gave you an abstract list, no mention of trials. The context of the question is entirely within the list you were given.

Yes, and with that context there isn't enough information to estimate the limit frequentist probability, which is the only notion of probability I want to use because it's the only one I think is relevant to Bell's proof. Anyway, Bell's proof does not assume we have an abstract list, it assumes we have a physical process which can yield various outcomes on as many trials as we care to perform, so do you actually have a point relevant to the discussion of Bell's theorem here, or are you just on a quest to prove that I "don't understand probability"?

You can visualize it by thinking that if you would randomly pick an entry from the the list I gave you

Well, that's an entirely separate question, because then you are dealing with a process that can repeatedly pick entries "randomly" from the list for an arbitrarily large number of trials. But you didn't say anything about picking randomly from the list, you just presented a list of results and asked what P(++) was.

Here the wikipedia article is failing to adequately distinguish between the "mean" of a finite series of trials and the "mean" of a probability distribution.

Now according to you it is the wikipedia article that is "failing"?

Failing to do the specific thing I said it should do, yes. Do you deny that in probability theory it is commonly understood that there is a difference between the "sample mean" and the "population mean"? (the latter can also be referred to as the 'mean of the probability distribution, and either way it's usually denoted with the symbol \mu) You may have missed the references discussing this distinction which I added in an edit to my post:

(edit: See for example this book which distinguishes the 'sample mean' \bar X from the 'population mean' \mu, and says the sample mean 'may, or may not, be an accurate estimation of the true population mean \mu. Estimates from small samples are especially likely to be inaccurate, simply by chance.' You might also look at this book which says 'We use \mu, the symbol for the mean of a probability distribution, for the population mean', or this book which says 'The mean of a discrete probability distribution is simply a weighted average (discussed in Chapter 4) calculated using the following formula: \mu = \sum_{i=1}^n x_i P[x_i ]').

The law of large numbers says if you would randomly pick a large number of pairs from our given abstract list, the average value will get close to the true expectation value as the number of pairs you pick tends towards infinity.

Again, you said nothing about "randomly picking" from a list, you just gave a list itself and asked for the probabilities of one entry on that list. If you want to add a new condition about "randomly picking", with "randomly" meaning that you have an equal limit frequentist probability of picking any of the four entries on the list, then in that case of course I agree that P(++)=1/4...well duuuuh! But that wasn't the question you asked.

 

[JesseM]

No! You are not paying attention, and it is you who is confusing the theoretical and the empirical. You keep bringing up the word "trials" which is tripping you up. Just because I give you an abstract list does not mean the list represents "trials" in an experiment. If I wanted the list to represent results of trials, I would state that clearly.

Well, excuse me for thinking your question was supposed to have some relation to the topic we were discussing, namely Bell's theorem. It didn't occur to me to think that it had no relation at all to Bell's theorem (where the only "lists" we might deal with would be lists of results from repeated measurements of entangled particles), and that you were just on a quest to prove I "don't understand probabilities" by asking me a bizarre question of a kind that would never appear in any statistics textbook. Can I play this game too? Here's an "abstract list" of letters (or is it a list of words?):

The quick brown fox jumped over the lazy dog

Quick now, what's the probability of "ox"?

For a coin N = 2, for a die, N=6, yet you can still calculate the expectation value without any infinite trials!

Only if you assume by symmetry that it's a "fair" die or coin, in which case you have a reasonable theoretical basis for believing the "limit frequency" of each result would appear just as often as every other one. If you had an irregularly-shaped coin (say, one that had been partially melted) it wouldn't be very reasonable to just assume the limit frequency of "heads" is 0.5.

In statistics, if you are given the population, you can calculate the true probabilities without any trials. It is done every day in the frequentist approach, which you claim to understand!

Not in the "limit frequentist" approach where we are talking about frequencies in the limit as number of times the population is sampled approaches infinity (unless we make some auxiliary assumptions about how the population is being sampled, like the assumption we're using a process which has an equal probability of picking any member of the population)

Your arguments may have been assuming the "finite frequentist" view, but as I said that's not what I'm talking about. I'm talking about the more common "frequentist" view that defines objective probabilities in terms of the limit as the number of trials goes to infinity.

So now you are no longer talking about "frequentist view" but some special version of the frequentist view?

Yes, I think I have explained a bunch of times now that I am talking about "probability" defined as the frequency in the limit as the number of trials (or the 'sample size' if you prefer) goes to infinity. There is also such a thing as "finite frequentism" which just says if you have a finite set of N trials, and a given result occurred on m of those trials, then the "probability" is automatically defined as m/N (see frequency interpretations from the Stanford Encyclopedia article on probability for more on 'finite frequentism')...this is not a definition I have ever been using, but I thought perhaps you were, since you gave a list with 4 entries and said the "probability" of an entry that appeared once on the list was 1/4.

This is just plain wrong, as the section of Wikipedia I quoted to you clearly explains.

What is just plain wrong? That there are multiple meanings of "frequentism", and that there is such a thing as "finite frequentism" as distinct from what I'm here calling "limit frequentism"? If you think that's wrong, go read the Stanford Encyclopedia article. If it's some other thing you think I was claiming, can you be specific?

The law of large numbers gives you an approximation to the "true" probability and that approximation get's more accurate as the number of trials increases. It does not define an objective probability.

Sure, that's what I've been saying all along, again with the understanding that by "true" probability I mean the limit frequentist probability.

The objective probability is defined by the actual population content and the population size can be any number.

I don't know what you mean by "defined by the actual population". By "population" do you mean the sample space of possible outcomes, or do you mean a "population" of trials (or picks or whatever) from the sample space? (you may remember from a previous discussion that wikipedia does at times use 'population' to refer to a large set of trials, see here) If you're talking about a population of trials, then the limit frequentist probability would require us to consider the limit as the population size approaches infinity. If you're just talking about the outcomes in the sample space, are you claiming that if there were N outcomes the "objective probability" would automatically be 1/N?

What do you mean by "true expectation value"? Are you using the same definition I have been using--the average in the limit as the number of trials goes to infinity--or something else?

Bah!

Humbug!

Do you have short-term memory issues or something.

Not that I can recall!

What have we been discussing these past 5 pages of posts?

Bell's theorem, and your odd criticisms of it which seem to presuppose a notion of probability different from the limit frequentist notion (for example, at the end of post #1224 you acted as though my comment that we might not be able to 'resort' the data from a finite series of trials in the way you suggested was equivalent to an 'admission' that it is 'possible for ρ(λi) to be different'). Which is why I ask if you are "using the same definition I have been using--the average in the limit as the number of trials goes to infinity--or something else?" I have asked a few times if you are willing to use this definition at least for the sake of analyzing Bell's proof to see if it makes more sense that way, but you have never given me a clear answer. Can you take a quick break from venting hostility at me and just answer yes or no, is this the definition you've been using? And if not, would you be willing to use it for the sake of discussion, to see if the problems you have with the applicability of Bell's results might go away if we assume he was using this type of definition?

Did you not see this definition I posted from Wikipedia: http://en.wikipedia.org/wiki/Expected_value

I saw, but you have a tendency to interpret quotes from other sources in odd ways that differ from how I (or any physicist, I'd wager) would interpret them, like with much of your interpretation of Bell's paper. So can you please just answer the question: are you using (or are you willing to use for the sake of this discussion) the limit frequentist notion of probability, where "probability" is just the frequency in the limit as the number of trials goes to infinity?

Also, when you say "it is their only hope", what is "it"?

In case you have forgotten, we are discussin Bell's inequality

|E(a,b) + E(a,c)| - E(b,c) <= 1

According to Bell, E(a,b), E(a,c) and E(b,c) are "true" expectation values with a uniform ρ(λ) for all three terms, or if you prefer "objective" expectation values for the paired product of outcomes at two stations.

Yes.

Those expectation values are defined by Bell in equation (2) of his paper, using the standard mathematical definition of expectation values for continuous variables.

No, the "standard mathematical definition" of an expectation value involves only the variable whose value you want to find the expectation value for, in this case the product of the two measurement results. The standard definition is to take each possible value for this variable (not some other variable like λ), and multiply by the probability of that value, giving a weighted sum of the form \sum_{i=1}^N R_i P(R_i ). In the standard definition would give us:

E(a,b) = (+1)*P(detector with setting a gets result +1, detector with setting b gets result +1) + (-1)*P(detector with setting a gets result +1, detector with setting b gets result -1) + (-1)*P(detector with setting a gets result -1, detector with setting b gets result +1) + (+1)*P(detector with setting a gets result -1, detector with setting b gets result -1)

Bell is not trying to provide a totally new definition of "expectation value", instead he's just giving a physical argument that the expectation value as conventionally understood (i.e. the definition above) would be equal to the expression he's giving in equation (2). But that equation isn't how he "defines" the expectation value, it would just be silly to try to provide a new definition of such a commonly used term.

Experimenters measure "something". From this "something" they calculate certain empirical expectation values E(a1,b1), E(a2,c2) and E(b3,c3) where the numbers corresponds to the run of the experiment. They then plug these empirical expectation values into the LHS of the above inequality and obtain a value which they compare with the RHS and notice that the inequality is violated.

Yes, I agree (although I think 'empirical expectation value' is a confusing phrase, I would just use 'empirical average' or something like that).

In case you forgot, this whole discussion is about whether those empirical expectaion values are appropriate terms to be used in Bell's inequality.

Since the E(a,b), E(b,c) and E(a,c) in Bell's inequality are defined in the conventional way, i.e.

E(a,b) = (+1)*P(detector with setting a gets result +1, detector with setting b gets result +1) + (-1)*P(detector with setting a gets result +1, detector with setting b gets result -1) + (-1)*P(detector with setting a gets result -1, detector with setting b gets result +1) + (+1)*P(detector with setting a gets result -1, detector with setting b gets result -1)

...with the probabilities in that equation understood as the "limit frequentist" probabilities, it follows from the law of large numbers that the bigger the sample size, the smaller the probability that there will be any significant difference between the "limit frequentist" expectation values and the empirical averages of this form:

Avg(a,b) = (+1)*(fraction of trials in run where detector with setting a got result +1, detector with setting b got result +1) + (-1)*(fraction of trials in run where detector with setting a got result +1, detector with setting b got result -1) + (-1)*(fraction of trials in run where detector with setting a got result -1, detector with setting b got result +1) + (-1)*(fraction of trials in run where detector with setting a got result -1, detector with setting b got result -1)

Hopefully you at least agree that in the limit as the number of trials becomes large, the expression for the empirical average below should approach my definition (which I claim is of the standard form) for the expectation value above. In that case, does your whole argument hinge on the fact that you think Bell's equation (2) was giving an alternate definition of "expectation value", one which would actually differ from the one I give?

 

[JesseM]

Are you saying their only hope is to make some assumptions about the values of λ in their experiment, such as the idea that the distribution of different λi's in their sample is similar to the one given by the probability distribution function

I'm not saying that, nor do I need to say that because they ALREADY make the assumption ("fair sampling assumption") that the sample of measured pairs is representative of the population of emitted pairs, which is exactly the assumption that the sample ρ(λ) is the same as the population ρ(λ).

Assuming that "the sample ρ(λ) is the same as the population ρ(λ)" is an assumption that "the distribution of different λi's in their sample is similar to the one given by the probability distribution function", and therefore an "assumption about the values of λ in their experiment", at least it would be in my way of speaking. So there was really no need to jump down my throat (and compare me to a political pundit, ouch!) at the end of post #1228 for saying "If you think a physicists comparing experimental data to Bell's inequality would actually have to draw any conclusions about the values of λ on the experimental trials..."?

Because you claimed I was saying, physicists need to know the values of λ, in order to calculate expectation values

No, I said "draw any conclusions about the values of λ", which would include conclusions about the statistics of different values of λi in all three runs. Maybe you shouldn't rush to assume that the most negative interpretation of my words is the correct one.

Yes, but to get a "damn good estimate of the true expectation values", all that's necessary is that the actual frequencies of different measurement results were close to the "true probabilities" (in frequentist terms) in equations like this one

E(a,b) = (+1*+1)*P(detector with setting a gets result +1, detector with setting b gets result +1) + (+1*-1)*P(detector with setting a gets result +1, detector with setting b gets result -1) + (-1*+1)*P(detector with setting a gets result -1, detector with setting b gets result +1) + (-1*-1)*P(detector with setting a gets result -1, detector with setting b gets result -1)

False! The above equation does not appear in Bell's work and is not the expectation value he is calculating in equation (2).

True! Bell was writing for an audience of physicists, who would understand that he didn't mean for (2) to indicate he was totally rewriting the standard meaning of "expectation value", but was just making the argument that the expectation value as conventionally understood (i.e., equal to my equation above) would in this case be equal to the integral in (2).

In any case, even if you persist in the wrongheaded belief that (2) was meant to be a definition rather than a physical conclusion, it's not hard to see that (2) can be reduced to my expression above anyway. I showed this way back in post #855:

Remember, though, Bell is assuming that the value of A and B is completely determined by the values of a, b, and λ. So, the integral on the right of (2) is exactly equivalent to the following weighted sum of four integrals:

(+1)*(+1)\int P(A=+1, B=+1|a,b,\lambda)P(\lambda)\,d\lambda +
(+1)*(-1)\int P(A=+1, B=-1|a,b,\lambda)P(\lambda)\,d\lambda +
(-1)*(+1)\int P(A=-1, B=+1|a,b,\lambda)P(\lambda)\,d\lambda +
(-1)*(-1)\int P(A=+1, B=+1|a,b,\lambda)P(\lambda)\,d\lambda

The reason this works is because for any given value of λ, say λ=λ_i, three of the probabilities in the four integrals above will be equal to zero, while the other probability will be equal to 1. So by splitting up the single integral into the four above, you aren't overcounting or undercounting A*B*P(λ) for any specific value of λ, you're counting it exactly once. This is easier to see if you suppose λ can only take a discrete set of values from 0 to N, so the integral on the right side of (2) can be replaced by the sum \sum_{i=0}^N A(a,\lambda_i)*B(b,\lambda_i)*P(\lambda_i). Then if a,b,λ completely determine the values of A and B (which each take one of two values +1 or -1), that means the four-term sum (+1)*(+1)*P(A=+1,B=+1|a,b,λ_i) + (+1)*(-1)*P(A=+1,B=-1|a,b,λ_i) + (-1)*(+1)*P(A=-1,B=+1|a,b,λ_i) + (-1)*(-1)*P(A=-1,B=-1|a,b,λ_i) will always be equal to A(a,λ_i)B(b,λ_i) for each specific value of λ_i [for example, if a,b,λ_i determine that A=+1 and B=-1, then (+1)*(+1)*P(A=+1,B=+1|a,b,λ_i) + (+1)*(-1)*P(A=+1,B=-1|a,b,λ_i) + (-1)*(+1)*P(A=-1,B=+1|a,b,λ_i) + (-1)*(-1)*P(A=-1,B=-1|a,b,λ_i) = (+1)*(+1)*0 + (+1)*(-1)*1 + (-1)*(+1)*0 + (-1)*(-1)*0 = (+1)*(-1) = A(a,λ_i)B(b,λ_i)]. So, if we substitute the four-term sum in for the individual term A(a,λ_i)B(b,λ_i) in the sum over all possible values of λ I wrote above, we get:

\sum_{i=0}^N [(+1)*(+1)*P(A=+1,B=+1|a,b,\lambda_i)+\,(+1)*(-1)*P(A=+1,B=-1|a,b,\lambda_i)+ \,(-1)*(+1)*P(A=-1,B=+1|a,b,\lambda_i)+\,(-1)*(-1)*P(A=-1,B=-1|a,b,\lambda_i)]*P(\lambda_i)

Which can be split up into the following four sums:

\sum_{i=0}^N (+1)*(+1)*P(A=+1,B=+1|a,b,\lambda_i)*P(\lambda_i) +
\sum_{i=0}^N (+1)*(-1)*P(A=+1,B=-1|a,b,\lambda_i)*P(\lambda_i) +
\sum_{i=0}^N (-1)*(+1)*P(A=-1,B=+1|a,b,\lambda_i)*P(\lambda_i) +
\sum_{i=0}^N (-1)*(-1)*P(A=-1,B=-1|a,b,\lambda_i)*P(\lambda_i)

...which is just the discrete version of the four integrals I wrote before.

So, the left side is an expectation value which can be broken up into a weighted sum of four probabilities of the form P(AB|ab), and the right side can be broken up into a weighted sum of four integrals or sums over all possible values of λ of terms of the form P(AB|a,b,λ). For example, on the left side one of the four weighted probabilities is (+1)*(-1)*P(A=+1,B=-1|ab), and on the right side one of the four weighted integrals is (+1)*(-1)\int P(A=+1, B=-1|a,b,\lambda)P(\lambda)\,d\lambda. So if you take the marginalization equation P(A=+1,B=-1|a,b) = \int P(A=+1, B=-1|a,b,\lambda)P(\lambda)\,d\lambda and then multiply both sides by A*B=(+1)*(-1) and add this equation to three other marginalization equations where both sides have been multiplied by the corresponding value of A*B, you get something mathematically equivalent to equation (2) in Bell's proof.

As long as the fraction of trials where they got a given pair of results like (+1 on detector with setting a, +1 on detector with setting b) is close to the corresponding "true probability" P(detector with setting a gets result +1, detector with setting b gets result +1) then the sample average of all the products of measured pairs will be close to the expectation value. And the law of large numbers says that the measured fractions are likely to be close to the true probabilities for a reasonably large number of trials (a few thousand or whatever)

Since you continue to insist on this ridiculous idea, explain how they can know what the "true probability" is.

They can't, but they can know that whatever the true probability is, the law of large numbers says their measured frequencies become increasingly unlikely to differ significantly from the true probabilities as the number of trials gets larger and larger.

And in case you are going to respond with the "large numbers" argument, make sure you also explain how they can be sure that the number of trials is large-enough.

Large enough for what? It's up to them how statistically strong they want their result to be.
But if you do N trials and get a given result on m of those trials, you can find an epsilon and p such that P(number of positive results on N trials is m or greater | true probability is smaller than m/N by p or more) < epsilon, and the larger the number of trials the smaller you can make p and epsilon (and obviously you can write a similar equation to cover the possibility that the true probability is greater than m/N by p or more)

I remember asking you where you got "1000 or more" from and you did not answer, so remember to answer it this time.

Please phrase your requests in a more civil manner than "remember to answer it this time", I'll let this go but in future I will ignore bullying commands, it's not that hard to type the word "please". Anyway, 1000 is just an example (that's why I always used phrases like 'say, 1000 trials' or 'a few thousand trials or whatever'), it's a number of trials where the probability of significant differences between observed frequencies and actual probabilities tends to become pretty small, provided the frequencies aren't as small as 1/1000 or so. For example, say the actual probability is 0.3, what are the chances that the observed frequency will be more than 10% larger, i.e. more than 330? Well, in this case we can use the http://stattrek.com/Tables/Binomial.aspx, it shows that the probability of getting more than 330 in this case is only about 1.8%. So if you get a result of 331, or an observed frequency of 0.331 in your sample of 1000, you know the probability of getting that result in any case where the actual probability is 0.3 or lower must be 1.8% or lower. And if that's not statistically strong enough for you, you can always increase the number of trials.

even if the number of trials is small compared to the number of possible values of λ so that the frequencies of different λi's in the particles they sampled were very different from the frequencies in the limit of an infinite number of trials, which is what is given by the probability distribution on λ. Do you disagree with that "even if"?

I absolutely disagree. In Bell's work, the expectation values are obtained by integrating over ρ(λ), the derivation of the inequalities relies on the fact that ρ(λ) is the same for all three terms. If this requirement is not met, the inequalities CAN NOT be derived. It is common sense to realize the fact that, if in any sample ρ(λ) is not the same for all three terms, that sample does not conform to the mathematical requirements inherent in the derivation of Bell's inequality.

But when you write ρ(λ), is that intended to be a probability distribution where the probabilities are interpreted in "limit frequentist" terms? If so, it is trivial to see that two runs drawn from the same "limit frequentist probability distribution" may have a different actual frequency of different λi's, just like two runs of 10 flips of a fair coin with "limit frequentist probability distribution" P(heads)=0.5 and P(tails)=0.5 may end up having a different number of heads on each run.

The fair sampling assumption discussed on wikipedia doesn't say anything about the full set of all hidden variables associated with the particles, it just says the fair sampling assumption "states that the sample of detected pairs is representative of the pairs emitted"

Huh? What do you think the underlined statement means.

It means that the actual data observed in your sample is representative of the data you would have gotten if all emitted pairs had been detected. If you're doing a run with detectors set to a and b, then as long as the "emitted but undetected pairs" have the same statistics on their predetermined results for detectors a and b as the actual observed results for detected pairs with these settings, the fair sampling assumption is satisfied. It doesn't matter if other hidden variables had different statistics for the "emitted but undetected" group and the detected group, all that matters is the actual results for the detected group and the corresponding predetermined results for the undetected group (the results you would have gotten had you actually detected them).

The underlined words are clearly an admission that the fair sampling assumption can not be divorced from the requirement that ρ(λ) not be significantly different between the population and the sample.

And again, you are apparently not using "limit frequentist" definitions here, since it is quite possible for two finite samples of particles to have differing frequencies of different λi's despite the fact that both samples were drawn from the same "limit frequentist" probability distribution.

In any case, as I said, the fair sampling assumption only requires that the statistics of predetermined results in the "emitted but undetected" group match the actual results in the detected group. The number of possible values of λi may be vastly larger than the number of possible combinations of predetermined results on two axes (which just amounts to four combinations: a=+1,b=+1 and a=+1,b=-1 and a=-1,b=+1 and a=-1,b=-1), so it's quite possible for the statistics of results/predetermined results to match in the two groups while the statistics of λi's do not.

How else will they be predetermined. Remember in Bell's notation, the outcomes are given by the functions:
A(a,λ)=+/-1, B(b,λ)=+/-1 with the understanding that the terms in parenthesis are deterministically resulting in the outcomes. So you are arguing with yourself here.

What does the fact that the outcomes are predetermined by the detector setting and the value of λi have to do with the idea that the fair sampling assumption only requires that the statistics of predetermined results match the measured results, but that it doesn't otherwise require the statistics of values of λi match in the measured/unmeasured group?

If ρ(λ) in the sample is not significantly different from ρ(λ) in the population, then the distribution of the outcomes will not be significantly different. However, just because the distribution of the outcomes is not significantly different is not proof that ρ(λ) is the same. It is a necessary but not a sufficient condition as you still must be able to resort the data.

Please answer my question about whether you are willing to just use the "limit frequentist" notion of probability in this discussion--and if you are, do you see why with this understanding it doesn't make sense to say "ρ(λ) in the sample is not significantly different from ρ(λ) in the population" when you are really just talking about the frequencies of different values of λi in the finite sample, not the frequencies that would be found if we took an infinite sample under the same conditions?

 

[JesseM]

If this is correct, then I'll just note that even if you can do this resorting, it doesn't guarantee that the "hidden triples" associated with the ith iteration of all three runs were really the same, much less that the value of λ (which can encompass many more details than just three predetermined results for each setting) was really the same on all three.

I already answered this:

Your answer only seems to address the part that your ability to do this "resorting" doesn't guarantee the value of λ was really the same for all three (and you basically seemed to agree but say it doesn't matter), but you didn't address the point that even the "hidden triples" may be different than the imaginary triples you created via resorting. For example, suppose after resorting we find the 10th iteration of the first run is a=+1,b=-1, the 10th iteration of the second run is b=-1,c=-1 and the 10th iteration of the third is a=+1,c=-1. Then we are free to self-consistently imagine that each run had the same triple for iteration #10, namely a=+1,b=-1,c=-1. However, in reality this might not be the case--for example, the 10th iteration of the first run might actually have been generated from the triple a=+1,b=-1,c=+1. So, the statistics of the imaginary triples you come up with after resorting might not match the statistics of actual triples on each run, or on all three runs combined.

Maybe you'd agree with this point about the statistics but say it's irrelevant, that all you care about is the fact that "resorting" means Bell's inequality is mathematically guaranteed to be obeyed. I agree it means that, but of course I would say Bell's inequality doesn't require that the data be "resortable" in order to have a high probability of being satisfied in a local realist universe. The reason, once again, is that all the expectation values that appear in the inequality are understood to relate to the "limit frequentist probabilities" of different outcomes in the way I've described:

E(a,b) = (+1)*P(detector with setting a gets result +1, detector with setting b gets result +1) + (-1)*P(detector with setting a gets result +1, detector with setting b gets result -1) + (-1)*P(detector with setting a gets result -1, detector with setting b gets result +1) + (+1)*P(detector with setting a gets result -1, detector with setting b gets result -1)

So if these expectation values can be shown to satisfy the Bell inequality |E(a,b) + E(a,c)| - E(b,c) <= 1 in a local realist universe, then for a large sample size it's unlikely those ideal expectation values will differ significantly from the corresponding empirical sample averages, and so it's unlikely the sample averages will fail to satisfy the inequality in a local realist universe either (the larger the sample size, the more unlikely it becomes).

Is the only part of this you disagree with the claim that Bell's expectation values E(a,b), E(b,c) and E(a,c) were understood by him (and every other physicist) to be equal to the expression I write above? In other words, if you could be convinced that his expectation values are equal to that expression (along with the expression Bell wrote in equation (2), based on various physical assumptions), then would you agree if he successfully shows that expectation values defined this way should satisfy |E(a,b) + E(a,c)| - E(b,c) <= 1 in a local realist universe, that implies (by the law of large numbers) that empirical sample averages would also be highly unlikely to violate the inequality for a large sample in a local realist universe?

You are right that being able to resort does not ensure that ρ(λ), is uniform. However, not being able to resort, is definite proof that ρ(λ) is not uniform.

If we interpret ρ(λ) in "limit frequentist terms", it is quite possible all three samples (i.e. the three runs) could be drawn from the same ρ(λ) and yet the actual data on these finite samples could not be resorted--do you disagree?

First of all, it's a physical assumption that the result A on Earth depends only on a and λ and can therefore be written A(a,λ)--if you allow "spooky" influences, why can't the result A on Earth depend on the setting b, so that if on Earth we have setting a1 and hidden variables in state λ5, and on the other planet the experimenter is choosing from settings b1 and b2, then it could be true that A(a1, λ5, b1)=+1 but A(a1, λ5, b2)=-1?

You did not understand. A(a,λ) is a function that maps a given value of a and λ to an outcome of +/-1.

Are you just declaring that this is a starting assumption in your example, so we're not allowed to question it? I'm not interested in discussing your example unless it's supposed to be analogous to what Bell was doing--do you disagree that in Bell's equation, A just stood for the result of a measurement on a particle by one experimenter (let's call her Alice), so it was a physical assumption Bell made that A could be written as a deterministic function of only a and λ? Do you deny it's logically possible that if Alice is measuring a particle which always yields result +1 or -1, then her outcome might depend not only on her detector angle a and some variables associated with the particle λ, but also on the choice of detector angle b used by a distant experimenter Bob? If you agree this is logically possible, you should be able to see why Bell needs to invoke physical arguments (specifically, local realism and the fact that Alice and Bob are far apart) to justify the claim that Alice's result A depends only on a and λ but not on b.

The function has nothing to do with probability. A(a,λ), "a" can depend on "b". All that the notation A(a,λ) = +/-1 means is that given a specific value of "a" and a specific value of "λ", we get a specific value of either +1 or -1. It is a function of two variables not three and there is nothing in the notation itself that should suggest to you that dependence on "b" is not allowed.

Again, if this is supposed to apply to Bell's paper, then you can't treat Bell's equations as definitions, rather they represent physical conclusions he argues for, like the conclusion that the result A couldn't depend on a distant detector setting b in a local realist universe. If you just want to discuss your own example, I don't really see the point unless it is meant to be analogous to what Bell was doing.

then it could be true that A(a1, λ5, b1)=+1 but A(a1, λ5, b2)=-1?

Wrong. "a1" depends on "b2" means that if you looked at how the values of "a" and "b" varied with time after the fact, it will not be random, but certain values of "a" will always be paired with certain values of "b" due to the instant communcation when the settings were being made. It doesn't mean a specific value of "a" will give a different result depending on which value of "b" existed on the opposite side.

This would be a very cogent criticism if I had ever said that my equation was meant to describe a situation where "a1 depends on b2", but since I never claimed anything of the sort I have no idea what statement of mine this is supposed to be in response to. I just said that it's logically possible that the result A might depend on the distant setting b, in which case it could be true that A(a1, λ5, b1)=+1 but A(a1, λ5, b2)=-1...since this is logically possible, it means Bell has to invoke physical assumptions to justify the notion that the result A depends solely on a and λ.

You completely missed the point. Despite the fact that the "physical assumptions" in this example are completely contrary to Bell's, we still obtained the exact same expression for the expectation value of the paired product. Which means the "physical assumptions" are peripheral.

Sounds like bizarro logic to me. Your example seemed to consist of little more than you declaring that various conditions were true, like the condition that A was a function only of a and λ...how does this prove anything about whether Bell needs physical assumptions to actually justify similar equations in the experiment he's describing? If I can come up with some contrived example where the motion of a toy train on its track is described by the same equations that describe elliptical orbits of planets (because I designed the example to work that way, picking an elliptical track shape and carefully controlling the speed of the train), have I thereby proved that "physical assumptions are peripheral" in deriving the result that planets have elliptical orbits?

 

[zonde]

2. Einstein gave us the "the moon is there when not looking at it" comment, so I am not sure I quite agree if you are saying that Einstein was not a "naive" realist. (Although I personally don't care for the use of the word naive as it comes off as an insult.) But I would be interested in a quote that clearly expresses a) what realism looks like which is NOT naive; and more importantly b) any evidence Einstein subscribed to that view. Given his "moon" comment, which is pretty clearly in the "naive" school.

You can find something about this in Wikipedia http://en.wikipedia.org/wiki/Ensemble_Interpretation" :

Probably the most notable supporter of such an interpretation [Ensemble Interpretation] was Albert Einstein:
"The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems."

Although I think that reading EPR paper with that quote in mind can provide better understanding about the role of "reduction of wave packet" in EPR argument.

 

[DevilsAvocado]

If there is anyone who believes there is any relevance in billschnieder’s dopy posts, https://www.physicsforums.com/showpost.php?p=2833234&postcount=1241".

 

[RUTA]

I guess I agree... one thing that bothers me though... How does the Moon know if someone is looking??

In Relational Blockworld, if the entity "isn't there," i.e., is "screened off," it doesn't exist at all. So, the answer to your question is that there is no Moon to wonder

 

[DevilsAvocado]

In Relational Blockworld, if the entity "isn't there," i.e., is "screened off," it doesn't exist at all. So, the answer to your question is that there is no Moon to wonder

Okay, sounds fair... it must mean that if someone, let’s say billschnieder, was blinking his eyes rapidly towards the Moon... it would exist and non-exist quite rapidly...?

And we could say that Bill is actually creating the Moon... maybe he is The Man in the Moon...??

 

[DevilsAvocado]

Probably the most notable supporter of such an interpretation [Ensemble Interpretation] was Albert Einstein:

I love Einstein, he’s my hero. The question is – do you think that he would have rejected Bell's Theorem and EPR-Bell experiments?

And how does the Ensemble Interpretation explain if we decide to have very long intervals between every entangled pair in EPR-Bell experiments, let’s say weeks or months? Where is the "Global RAM" situated in a case like this? That fixes the experimentally proved QM statistics, for the whole 'spread out' ensemble??

 

[charlylebeaugosse]

a) Yes, this is very true. Einstein’s own argument boomeranged on him:

no action on a distance (polarisers parallel) ⇒ determinism
determinism (polarisers nonparallel) ⇒ action on a distance​

But to be fair we must say that also Niels Bohr was somewhat 'wrong'. If it turns out that nonlocality is fact, then QM must be considered 'incomplete'... or ...??

b) There is no doubt in my mind that Einstein, if he was alive, would have accepted the work of Bell as starting point for "something new", not a starting point for an old man to get 'grumpy'.

I have added a) and b) to the quote to answer point by point.
a) In a) it seems to me that you forget that in a classical pair, there would be more quantities being conserved than can be given simultaneous meaning in the QM case (and probably in the microscopic case. What is bizarre here from the historic point of view is that Einstein did NOT consider the values of measurement. His argument in Alice-Bob terms
was:

Assume Alice chooses a measurement n particle p1, say position or momentum while p2, the other particle of the pair, is spatially remote so that no action on p1 should affect p2. Then the Psi function (we now say wave function) of p2 would depend on that choice of measurement (no output mentioned) while such choice cannot have a remote action if one accept spatial separation. This one statement of the pair
(QM is complete, no influence on spatially remote objects)
has to be relinquished.

To the contrary of what Einstein wrote about completeness of QM in many places from 1933 to 1952 at least, in the EPR paper, written by Podolsky "because of language" (probably meaning the logic language as used by Godel in the proof of his Non-completeness Theorem since the only native English speaker was Rosen, and poor English shows up in the title of the EPR paper: see the book of Fine - and I am not an overall Fine fan, but his book is fan-tastic), values of coordinates are used so that in effect Podolsky "proves" non-completeness by first proving QM false as a particles end up having values for conjugate variables (to my own opinion, the structure of the EPR argument (again, see Fine's book:"The Shaky Game")
participates to the schizophrenic appearance of Bohr's reply since Bohr does take care of defending that QM is not false at the same time that he answer the completeness question.
Notice that Fine does not state that Podolsky used the falsity detour: he says that Podolsky thought he proved that the position and momentum coexist, and this is what I term "QM false" (but Podolsky's proof is shaky - Fine tells us and there is the issue of coexistence backward in time that makes things compliated). I have written in a previous post that I would explain (as part of a list (a to d) of self assigned items) why Podolsky may be right, but in a way that can be contested. But this will take time and I should perhaps respond to questions and posts before I choose my own items. Also, Einstein never used the elements of reality (EoR), but Podilsky used them without taking into account himself that they should be rooted in experience: if any observable being used must be measured, then at most 2 spin projections can be measured on a pair, and there is not enough room for Bell inequalities for such "complete EoR". But Podolsky used incomplete EoR's (i.e., without the need for anyone making sense to be measured) and with those, Richard Friedberg showed in an unpublished work quoted by Jammer in his book on the Philosophy of QM that EPR's
EoR lead to a Bell inequality (without the name as Friedberg did not know then about Belll's work. Unfortunately, no one seems to have pointed out the condition self imposed (but not used) by Podolsky who had his own agenda as he wanted to prove QM wrong (and wrote to the NY time about the fact that Einstein and co-worker had proven that, which infuriated Einstein, as documented by at least one of Jammer and Fine.

b) If Einstein would have live till 1964+ and had looked at this then obscure publication of Bell, he would probably have noticed that the realism (in teh form of Pr3edictive HV compatible with QM) being used was as heavy as in the work of de Broglie and Bohm that he was often making fun of (despite respect for the part of their works that he considered valuable: after all Einstein supported de Broglie when initially no one thought his thesis had value, but he through away the attempt he made himself in 1927 at a naive HV theory). Now if you start from something physically false (for Einstein) such as naive HVs, you can of course deduce anything idiotic that you can think about. But since Bell did not prove HV wrong (something that cannot be proved, perhaps, but physics is not the world of proofs)
he could not prove all one can think about, but "only" one consequence: non-locality. In 1955, when Einstein died, HVs had been disposed by the community of physicists despite a few dissidents, and I do not take von Neumann's "proofs" since that is not most physicist function. Consequently, proving that there cannot be local realism while the absence of realism AT THE MICROSCOPIC LEVEL had been essentially accepted by the profession could not be so exciting. But 9 years later, in fact a bit more as it took time for Bell's work to be noticed, people on both sides of the Iron Curtain were no more thinking about the foundation or interpretation of QM and concentrated on solving the puzzle of the 4 fundamental forces and the particles and resonances that where popping out from the new machines at the various particle labs in CERN , in the US etc. So someone as smart as Bell and from CERN, could certainly move people and generate passion where no grand master was paying attention: see the modest interest attached to Aspect's experiment by Gell Man in his popular book and the story circulating about Feyman throwing Clauser out of his office. Now these two people and Anton Zeilinger got together a Wolf Prize: one can document that the old masters knew better. The 1931 Einstein, Tolman, Podolsky paper show an UP backward in time, so that an observable value cannot pre-exist it's measurement (I have already mentioned that the argument applies to generic particles, but not to EPR (or in particular EPR-B) particles). So understanding that many spin projections cannot coexist, how could have Einstein be excited. How could he have not be upset by a paper that fully misrepresent the EPR paper. Assuming that he had seen his own version documented by others, and knowing that Bell could not ignore that, how could he not have been upset by Bell's attitude on Einstein supposed support for HV's? In 1935, Einstein stopped communication with Podolsky: using the words grumpy and old man in one sentence is a form of discrimination: better grumpy than naive when i comes to physics and its history. Demystifiers have to pay attention of not being carried by the massive flows of mystification such as what is going on about Bell. Bell assumed the false hypothesis that there is as much realism at the microscopic level as used in classical physics (whatever you think of the moon's dependence on your person) and locality that seems to belong to the spirit of relativity and that has never been challenged in itself by any experience to find a Boole inequality that means that said wrong hypothesis and locality form a pair that contains at least one false statement. The only virtue of that
(and an important one for me as it conditions my work for 5 years at least) is that it invites us to better found non-realism (whatever we think of it at the macroscopic level), and I mean the absence of classical realism at the microscopic level, so that Quantum-Compatible realism (in the form of he CQT of Griffith, Omnes, Gell-Man, Hartle (even if I am not a fan of that), or in the form envisioned but not spelled out because they did not know by Shrödinger and Einstein) is OK from this point of view at least. Perhaps the work of the hyper realist Bell (the champion of Bohm and de Broglie (after all)) will have the main virtue of making non-existence of microscopic realism one of those thing that went from the real of pure philosophy to that other sub-realm of philosophy called "physics". But there is still work to be done (with little reward to be expected).

 

[charlylebeaugosse]

If there is anyone who believes there is any relevance in billschnieder’s dopy posts, https://www.physicsforums.com/showpost.php?p=2833234&postcount=1241".

I looked there. This is factually exact (but see *), but does not reflect the push toward "QM non-local" that resulted from Bell's theorem: the summaries of what physics has lost and won from there is for the least incomplete.

Since realism was not believed in the progress brought by Bell's theorem is not that big. But, for me:
- (as I developed in a recent post) it invites at proving once and for all that realism is the villain and that we can continue to hold on to locality.
- it has motivated the work on techniques that may prove useful, even if the great value of Q crypto and Q computation remain to be established. I am personally sure that Q crypto will be fine, at least if one accept that detection of eavesdropper is enough.


*Bell's inequality, and the other forms such as CHSH is a form of Boole's inequality.
Bell's theorem is not Bell's Inequalities, it is the statement of incompatibility of QM with a pair of hypothesis as stated in the link mentioned in the quote. Thus, GHZ is another, perhaps better, form of Bell's theorem but there is no inequality in there. Altogether one needs to be very precise and without the historical context at the time Bell wrote his paper, and what believed the great masters of QM, I find the link rather misleading even if the main statement are not false. The global result of Bell's work over the years is that, contrary to what the great Sir Anthony Leggett things, people around Scully have tested (to their disarray) that more people believe that non-locality hold true tan people who relinquish realism! AND YOU CALL THAT A POSITIVE EFFECT ON PHYSICS? (especially when science is again assailed by the back powers of blind beliefs and anti-science powers!).

If indeed the glove is picked up and we prove non-realism at the microscopic level (I leave the moon level to others, but what if a chimp looks a the moon?) because it was doubted upon as a consequence of Bell's work, them I will admit that after all it was a good move to point out this weakness in the tool-box of physics.

 

[nismaratwork]

If there is anyone who believes there is any relevance in billschnieder’s dopy posts, https://www.physicsforums.com/showpost.php?p=2833234&postcount=1241".

Well, his last little rant aside, I'm just relieved that he's taking a break. Whatever he chooses to believe about this being some "JesseM vs. Him" or "[insert name] doesn't understand probability", I don't really care anymore... as long as he stops cluttering the thread with transparent crap.

 

[RUTA]

Okay, sounds fair... it must mean that if someone, let’s say billschnieder, was blinking his eyes rapidly towards the Moon... it would exist and non-exist quite rapidly...?

If bill was the ONLY thing interacting with the Moon, yes. Of course, the Moon is interacting with MANY things, even things that don't "see." It's not an issue of consciousness, the issue is interaction.

 

[charlylebeaugosse]

1. You are welcome if so! I used to have a worse (darker) copy and then someone helped me get a better one.

2. We could discuss the Tresser paper and related in a new thread, no prob there. If you want I can start it. I tend to hold onto locality so the Tresser ideas are pretty interesting. His work has been on my radar for a while although I have not read closely.

Seems you were right to wait as a new preprint clarifies the Bell part of that paper (leaving the GHZ mostly untouched).

DrChinese.
How do I start a new thread? I do not even know how to easily go to this thread on EPR, nor how to navigate without hopping each time for a miracle to lead me where I need to go: can someone provide a link to where I could learn the basics of Physics Forums? e.g., I just uploaded 2 files (or so did I think), but have no idea of what are the ways for others or me to access them (assuming that the upload functioned). Sorry for these beginner's questions that do not belong to a thread, but then where should I ask (even if that question does not belong either).

Besides, when you asked for more precise indications, was it about Wigner's paper or else?
And if I find a paying link, such as a journal, can I just give that (perhaps I must do that?) or do I have to (or should I) find a link to free access or provide a copy from my files when I can? Perhaps that's in the rules... Sorry to be such a schmuck and I apologize for cluttering the thread with that.