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Understanding Bell's logic

  1. Jun 9, 2010 #1
    I am hoping it may be helpful to separate Bell's logic from Bell's mathematics
    https://www.physicsforums.com/showthread.php?t=406372.

    Understanding one may better help us understand the other.

    Thank you Bill.

    In the language that is evolving at "Understanding Bell's mathematics", https://www.physicsforums.com/showthread.php?t=406372, we have Alice with outcomes G or R (detector oriented a), Bob with outcomes G' or R' (detector oriented b).

    H specifies an EPR-Bell experiment.

    λ represents Bell's supposed [page 13] variables "which, if only we knew them, would allow decoupling ... " [of the outcomes].

    Question: Why would Bell want to decouple outcomes which are correlated? Is he too focussed on separating variables?

    Bell's λ would allow Bell to write -- consistent with with his (11) --

    (11a) (P(GG'|H,a,b,λ) = P1(G|H,a,λ) P2(G'|H,b,λ).

    So Bell's logic, as cited above in bold, leads him to suggest that

    (11b) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a)

    would avoid some well-known inequalities.

    I do not follow Bell's logic. I do not see that his move avoids any inequalities.

    Note 1: a and b are not signals.

    Note 2: Probability theory, widely seen as the logic of science, would have --

    (11c) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G).

    So, by comparison [Bell's (11b) with (11c)], Bell's (11b) and his logic is equivalent to dropping G from the conditionals on G'.

    Which is equivalent to saying that G and G' are not correlated?
     
  2. jcsd
  3. Jun 10, 2010 #2
    Ok, so Bell's logic was flawed. This was demonstated in the thread "Understanding Bell's Mathematics".

    This has been known, and demonstrated, years ago.

    Bottom line, few people care. If Bell's logic was flawed and if violations of Bell inequalities don't tell us anything about nature then ... so what.
     
  4. Jun 10, 2010 #3
    Do doubters understand the full implication of Bell's lambda, and therefore the full implication of his logic?

    Do supporters?

    Probabilistic refutations do not impress his myriad supporters.

    The difference might be in how one views the logic attached to Bell's lambda.


    Ether-logic was flawed. Stomach-ulcer logic was flawed. ...

    Much was learnt from the related experiments.

    Including that the logic was flawed.

    C'est la vie.

    That's what.
     
  5. Jun 10, 2010 #4
    Not to devalue your efforts, but my apprehension of the view of the physics community at large (garnered from conversations with dozens of working physicists over the years) is that Bell's theorem just isn't important.

    If Bell was right then we have nonlocal or ftl influences that can't be detected or used for any conceivable purpose. If Bell was wrong, well, then he was just wrong. Nothing is affected either way (except wrt the agendas of a very small minority of physics professionals).

    Nevertheless, it is satisfying to periodically revisit and dispell myths. And, I think that you and billschnieder have done a nice job in that regard.

    I sensed that there was something not quite right about Bell's LR ansatz from the first time I saw it. But, lacking the requisite skills to communicate this clearly, I was only able to talk about my apprehension of it in rather vague terms.

    So, I thank you. And don't let my previous post in this thread tarnish your efforts, or diminish the admiration I have wrt your ability to elucidate something which I intuitively saw but was unable communicate.
     
  6. Jun 10, 2010 #5
    Dear Thomas,

    Ok. Thank you. No worries at all. And please ...

    Do not devalue your own efforts.

    You and your P(AB|H) are the catalysts that prompted me to present my similar intuition, backed by some knowledge of probability theory, etc.

    So thank you again,

    and prepare for the storm,

    Jenni
     
  7. Jun 10, 2010 #6

    DrChinese

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    No, there is no known flaw in his logic. Please provide a peer reviewed reference that states this if you believe I am wrong. As I say over and over again, you must read it in the context he wrote it. If you don't like his derivation, there are plenty of other peer reviewed versions of it available. For example, Mermin. Or Aspect. Or Zeilinger.

    Or even better, derive it for yourself. You will see that you can do it a variety of ways. You always use some variation of the following:

    a) The setting at a does not affect the outcome at B, and vice versa.
    b) P(A)+P(~A)=100%, and all variations of this with A, B and C simultaneously.
    c) The QM prediction is cos^(theta).

    Folks, please get a grip on this subject. Genovese does a review of Bell tests periodically, and his last review had over 500 peer-reviewed references in 100+ pages.

    Research on Hidden Variable Theories: a review of recent progresses,
    Marco Genovese (2005)
    http://arxiv.org/abs/quant-ph/0701071

    Do you seriously think that they just happened to overlook these "flaws" in Bell? If you do, publish a paper on it. Otherwise, I am going to point you back to Forum guidelines on personal theories. If you have a question, ask it. But quit making statements that are your pet opinions.
     
  8. Jun 10, 2010 #7

    JesseM

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    No, it's not equivalent to saying G and G' are not correlated in a general sense, it just means that if you already know λ then learning G will give you no further information about the probability of G'. For example, if G and G' are correlated, but this correlation is explained entirely by a common cause that lies in the region where the past light cones of the two measurements overlap (the common cause might be that the two particles sent to either experimenter are always created with an identical set of hidden variables by the source), then if you already have the information about the common cause contained in λ (which could detail the set of hidden variables associated with each particle, for example) then in this case it will be true that learning G won't change your estimate of the probability of G'. This is precisely the sort of physical logic that Bell was using, and my argument in which λ was made to stand for all facts in the past light cones of the measurement events was an attempt to make this a bit more rigorous.
     
  9. Jun 10, 2010 #8
    You are wrong, and JenniT is correct dropping G from
    (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G)
    means clearly that in the probability space defined by (H, a,b,λ) G and G' are not correlated. In other words under a given set of specific conditions ("H", "a","b","λ"), there will be no correlation between G and G'. It is a simple exercise to see if this is consistent with the EPR situation Bell was attempting to model. Your misunderstanding is fueled by a confusion between functional notation and probability notation. P(G'|H,b,λ,a,G) does not mean P2 is a function of (H,b,λ,a,G). It simply means the specific conditions (H,b,λ,a,G) define the probability space in which P(G') is calculated.
     
  10. Jun 10, 2010 #9

    JesseM

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    Yes, but there can still be a correlation in the total probability space even if there is no correlation in any subset of trials where ("H", "a","b","λ") all have some fixed value. You already showed that you understood this distinction in this post on your old thread when you said:
    In the same way, P(GG') may be different than P(G)*P(G') in our probability space (so the 'marginal probabilities' of G and G' are correlated), while at the same time P(GG'|H,a,b,λ) = P(G|H,a,b,λ)*P(G'|H,a,b,λ).
    In the EPR situation only the marginal probabilities, along with conditional probabilities which condition on observable conditions like the detector settings, are actually measurable. The λ is defined to represent hidden-variable states so conditional probabilities involving that term cannot be directly observed, although we can reason theoretically about some general properties of these conditional probabilities that must be true under the theoretical assumption of local realism.
    What misunderstanding would that be? You disagree that Bell's equation allows there to be a (marginal) correlation between G and G'? If not, that's all I was saying, and it should have been quite obvious from the context that I was talking about a marginal correlation and not a correlation conditioned on λ.
    I never used any words like "is a function of", so I have no idea what this criticism is referring to. And I don't know that "function of" has some precise definition in probability theory that forbids you from saying that the expression P(A|B) "is a function of" A and B (even if there was this would be more of a semantic quibble than a substantive critique). Also, I think it is legitimate to say that the probability space used to calculate P(G'), which includes events where H,b,λ,a,G take different values, is the same as the probability space used to calculated P(G'|H,b,λ,a,G), where we assume H,b,λ,a,G all have some known values. It's just that the expression P(G'|H,b,λ,a,G) indicates we must look at a subset of events in the larger sample space where H,b,λ,a,G take these known values, and look at the the frequency of G' within that subset.
     
  11. Jun 10, 2010 #10

    Thank you JesseM. I appreciate this detail. I have some basic questions.

    1. Could you define for me (briefly) and distinguish Bell's use of the words observable and beable? Is Bell's lambda an observable or a beable or something else -- like what? What size set might it be?


    2. If Bell's lambda were an infinite set of spinors (because we want a realistic general "Bell" vector that applies to both bosons and fermions), then wouldn't we need aG to define the infinite subset of spinors that were relevant to the applicable conditional? You seem to require that we would know a priori which of that infinite set satisfied this subset aG conditional? This a priori subset being the lambda you would require here?


    3. Beside which, if aG were implicit in your lambda, its restatement/extraction by me would be superfluous and not change the outcome that attaches to the disputed conditional? Note that you seem to require lambda to be an undefined infinite set, perhaps not recognizing that it is an infinite subset (selected by the condition aG, out of your undefined infinite set) which is relevant here?

    4. As with the ether experiments and their outcome, don't Bell-tests show that Bell's supposition re Bell's lambda is false?

    Thank you.
     
  12. Jun 10, 2010 #11
    I'm not sure you understand the point at all.

    if A and B are correlated marginally, then P(AB) > P(A)P(B)

    If you collect data such that your data samples the entire probability space (that is what marginal probability is) , then the above expression is true. It is no different that defining "Z = All possible facts in the universe", and writing P(AB|Z). You are still dealing with a marginal probability.

    Now, if there exists a certain factor C within Z such that the set (C, notC) is the same as Z, then if we say C is the cause of the marginal correlaction between A and B, it means within C, under certain circumstances it maybe correct to write P(AB|C) = P(A|C)P(B|C). It means that C screens-off the marginal correlation between A and B.

    However, and please pay attention to this part, this means if data is collected in the full universe fairly sampling both situations where C is true and situations where C is not true (or notC is true), a correlation will be observed in the data, and if data is collected only under situations where C is True, there will be no correlation in the data.

    1) You see therefore why it makes no sense to define C as vaguely as you are defining it
    2) If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data. So no matter how hard you try, all your observed frequencies will always be conditioned on the actual contexts that created the data, whether you like it or not.
    3) Following from (2), if hidden elements of reality exist, then the correlations observed in experiments exist even when conditioned on C. Because it is impossible to not condition the results on C. Therefore equations such as P(AB|C) = P(A|C)P(B|C) are not accurate. For example if hidden element C is always present and C=42, then P(AB) is not different from P(AB|C=42)
    4) For the type of situation Bell is modelling, where he is assuming that hidden elements of reality exist. Marginal probabilities do not come into the picture because the existence of hidden elements of reality MUST always be a conditioning factors.
    5) Therefore I hope it is clear to you now why it makes no sense to say the observed EPR correlations are caused by the hidden variables and yet write an equation such as P(AB|C) = P(A|C)P(B|C) in which means if the hidden elements of reality C are realized, no correlation between will be observed between A and B.

    Again, just in case it wasn't clear the first time, by writing P(AB|C) = P(A|C)P(B|C), you are saying if the hidden elements of reality C exist, then no correlation will be observed between A and B. Yet Bell starts out by assuming that hidden elements of reality exists. Just because you drop C from the LHS of P(AB|C) does not enable you to escape this trap. The only escape is for you to show how it is possible in a real experiment to collect data fairly for situations where C is true and also for situations where C is not True.

    In case you still insist on your approach, could you answer one simple question.

    Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?
     
  13. Jun 10, 2010 #12

    JesseM

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    No, I don't. If local realism is true, then the random variable representing "the state of all local physical variables in the past light cones of the measurements" will have a perfectly well-defined value on each measurement. Are you planning on answering the question I asked you in my most recent post to you (post #80) on the "Understanding Bell's Mathematics" thread? Again:
    Of course, if you want a simpler example of a "C" you could also consider post #18 from the thread where we first got into the Bell discussion, either the scratch lotto card analogy or the flashlight analogy. Do you disagree that in both those examples, there would be a correlation in the marginal probabilities of different measurement outcomes, but if C represented the value of the "hidden" facts on each trial (the hidden fruits behind the cards in the lotto analogy, the fact about whether Alice got flashlight X or flashlight Y in the flashlight example), then conditioned on C there would be no correlation in measurement outcomes?
    C is a random variable which can take multiple values on different trials. The simplest type of hidden-variables theory would just say that on each trial, the particles have hidden variables that predetermine their spins on each of the measurement settings. For example, if there are three measurement settings a=0 degrees, b=120 degrees, and c=240 degrees, then on each trial the random variable C might take any one of the 8 values c1, c2, c3, c4, c5, c6, c7, c8, defined as:

    c1: spin-up on a, spin-up on b, spin-up on c
    c2: spin-up on a, spin-up on b, spin-down on c
    c3: spin-up on a, spin-down on b, spin-up on c
    c4: spin-up on a, spin-down on b, spin-down on c
    c5: spin-down on a, spin-up on b, spin-up on c
    c6: spin-down on a, spin-up on b, spin-down on c
    c7: spin-down on a, spin-down on b, spin-up on c
    c8: spin-down on a, spin-down on b, spin-down on c

    (note that these are directly analogous to the eight possible hidden-fruit states on the cards in the scratch lotto card analogy)

    According to this type of hidden-variables theory, do you deny that on each trial C would have one of these values, and the complete sample space would include trials with all possible values of C?
    Do the "actual contexts" include hidden variables? For example, consider again the flashlight analogy:
    So, would "C" include the fact about whether H1 or H2 obtain on each trial? If we do define it this way, do you agree that P(AB|C) = P(A|C)*P(B|C), even though P(AB) is not equal to P(A)*P(B)?
    I think you've confused yourself with purely verbal, nonmathematical arguments. If you actually examine one of my examples that involve elements hidden from the experimenters (and which help determine the measurement outcomes), you'll see that your general verbal arguments are giving you incorrect conclusions when applied to these examples.
    Again, the whole idea is that the variable can take different values on different trials, like in the flashlight example where the random variable H could take value H1 or H2 on different trials? Do you disagree that this was meant to be true of Bell's λ, since he actually integrated over all possible values of λ in equation (2) in his paper?
    C can take different values, and for any specific value, if you look only at the subset of trials where C took that trial, there will be no correlation between A and B, but if you look at the total collection of trials, there will be a correlation. Of course this is a theoretical conclusion based on the assumption that the universe obeys local hidden variables, since C represents hidden variables, even if such a theory was correct there would be no way for us to actually know the value of C on each trial (which is why it is helpful to think of all equations involving hidden variables as having precise values that would be known by an imaginary omniscient observer).
    Nope, a marginal correlation will be observed between A and B. By writing that equation I'm only saying that if hidden variables exist, then there would be no correlation between A and B in any subset of trials where the hidden variables all took the same value.
    In reality, or under the assumption that we live in a universe with local realist laws? Bell's whole approach is to derive certain inequalities from the assumption of local realism, show these inequalities conflict with actual quantum-mechanical results, and therefore conclude (proof by contradiction) that real-world quantum physics is inconsistent with local realism.

    If you're talking about reality, I think Bell's reasoning is correct and quantum mechanics rules out local realism, so I don't think there are any real local hidden variables you can condition measurement outcomes on to make the correlations disappear. If you're talking about what would be true theoretically in a universe obeying local realist laws, then in that case all correlations between spacelike-separated measurements could only be marginal ones, and conditioned on a sufficiently large set of local physical facts in the past light cones of the measurements there could be no correlations between measurements.
     
  14. Jun 10, 2010 #13

    JesseM

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    beables represent local hidden variables that are supposed to explain correlations seen in observables, and observables are just facts we can actually measure, like whether a particle gives result "spin-up" or "spin-down" when passed through a Stern-Gerlach device oriented at some angle. Take a look at my scratch lotto card analogy in this post (beginning with the paragraph that starts 'Suppose we have a machine that generates pairs...')--the observables would be the cherries or lemons that Alice and Bob actually see when they pick a single square to scratch, the beables would be the complete set of hidden fruits behind all three squares, which are used to explain why it is that they always find the same fruit whenever they scratch the same box (the assumption being that on each trial, the two cards have the same set of hidden fruits).
    I don't know much about relativistic quantum theory which is where I think "spinors" appear--my question here would be, are spinors actually local variables associated with a single point in spacetime, or are they defined in some more abstract "space" like Hilbert space?
    Not clear what you mean by "this subset aG conditional", can you elaborate?
    What do you mean by "implicit in"? Do you mean that the measurement a and the result G can be determined from the value of lambda? If so, I'm not sure why you think that, the measurement can be random and I told you in post #82 on this thread that the probabilities of different outcomes may be other than 0 or 1 in a probabilistic local realist theory.
    Nothing "undefined" about it, as I said to billschnieder:
    Like I said to billschnieder in the last post, the basic logic of Bell's argument is a proof-by-contradiction. He starts only by assuming that the universe obeys local realist laws, and then shows that they produce predictions about the statistics of Aspect-type experiments that contradict the predictions (and experimental results) of QM, and so concludes that QM is incompatible with local realism (so if QM's predictions hold up to experimental tests, our own universe must not obey local realist laws).
     
  15. Jun 10, 2010 #14
    JesseM,
    With all due respect, I do not take you seriously because you completely ignore everything I say. You keep repeating points I have debunked and expect me to keep repeating myself. You keep dragging tangential discussions from thread to thread and I don't bother going down that rabbit trail because it hijacks the thread. You redefine everything I say so that it means something different and then you use the strawman to purport to be arguing against what I said. The recent one is your claim that C is a random variable. It is NOT.


    My simple response to everything in your last post is that C is NOT a random variable so you are arguing against yourself. At best, A and B may be considered random variables but C is definitely positively NOT a random variable. It is a specific conditioning factor. You keep repeating the fautly idea that C has multiple values. C as it appears in the equation I wrote, is a specific set of elements of reality. C is NOT all possible sets of elements of reality. It can not be because some of those sets will be mutually exclusive and you can not condition a probability on mutually exclusive factors. As I have explained, in calculating a conditional probability everything after the "|" is assumed to be true simultaneously. It is therefore fallacious to suggest that a probability can be conditioned on mutually exclusive factors at the same time. Until you understand this simple point, you will be totally confused by everything I'm saying.

    You did not answer. Clearly correlations are calculated in Aspect type experiments otherwise there will be nothing to compare to Bell's inequalities. Are those correlations marginal or conditional on the hidden elements of reality causing them. The answer should be simple, although it is a trick question.

    Clearly you seem to be confused about what it means to make an assumption. Once you make the assumption, that the universe is local realistic, there is (and should) no longer any be any distinction between reality and local reality in all your equations. If you continue to make a distinction and have one equation for reality and a different one for local reality, then your claim of having assumed local reality is false.

    So let us try again.
     
  16. Jun 11, 2010 #15

    JesseM

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    You never gave a clear definition of what C is. You did say "If hidden elements of reality C exist" which suggests it should be a random variable, since the value of the hidden variables would differ from one trial to another. But then you said "C will define the actual context of the data" which perhaps suggests you intended C to be a mere specification of the sample space of possible combinations of values that could obtain on any trial, similar to the way you were defining "z" in posts 55, 70, and 70 on the other thread. If you want C not to be a random variable, but simply a specification of the sample space which should be the same on every trial, then it's exceedingly weird notation to actually include that as a symbol in your equations, in any standard probability equation the sample space will be defined beforehand and it'll then be implicit in all the equations rather than represented using a symbol. And if your C is not a random variable, then it has nothing to do with Bell's λ or with the symbols that have played a similar role in my equations like H, since those symbols were random variables--do you disagree?

    If you agree those were defined as random variables that could take different values on different trials, then perhaps you can see why your whole discussion becomes a totally irrelevant tangent: you triumphantly declared that P(AB|C)=P(A|C)*P(B|C) also implies P(AB)=P(A)*P(B) as if this somehow discredited my earlier arguments about there being a marginal correlation but no conditioned correlation, but while this might be true under your definition of C, it in no way shows there is anything wrong with my argument that P(GG'|H,a,b,λ)=P(G|H,a,λ,b)*P(G'|H,b,λ,a,G) and yet P(GG') is not equal to P(G)*P(G') (i.e. G and G' are marginally correlated but uncorrelated conditioned on H,a,b,λ), since here λ is obviously meant to be a random variable. Nor does it show there is anything wrong with Bell's equation (2) in his original paper, where λ was also a random variable. So sure, I agree with your statement that if C is a non-variable that simply represents the sample space, then your arguments in post #11 are correct, but it would be completely incoherent to use those arguments to try to discredit my arguments or Bell's, since your C is defined in a completely different way than the λ and H that appeared in the equations.
    "Faulty" only under your definition of C, which you did not actually make clear in your previous post. So, now I agree that your C cannot have multiple values, but hopefully you agree that Bell's λ is a random variable that does take multiple values (as made clear by the fact that he is integrating over all values of λ in equation 2), and likewise that when I and other defenders of Bell write equations like P(AB|H)=P(A|H)*P(B|H), the symbols playing a similar role there like H are also intended to be random variables that can take different values on different trials (different points in the sample space). Do you disagree with this?
    Hold on, are you saying that regardless of your own personal definition of C, there is something incorrect in general about writing a conditional probability equation where the conditioning factor is a random variable that can take different values on different trials? For example, if H is a random variable that can take values H1 and H2 (as in my flashlight example), and A is another random variable that can take values A1 and A2, are you claiming it would then be incorrect to write the equation P(A and H) = P(A|H)*P(H)? If so you are badly confused, when a probability equation is written with random variables, all that means is that the equation should hold for each possible combination of specific values of the random variables--for example, P(A and H) = P(A|H)*P(H) is true as long as it's true that P(A1 and H1)=P(A1|H1)*P(H1) and P(A1 and H2)=P(A1|H2)*P(H2) and P(A2 and H1)=P(A2|H1)*P(H1) and P(A2 and H2)=P(A2|H2)*P(H2). If all four of those equations involving all possible combinations of specific values of A and H are true, that means the general equation P(A and H) = P(A|H)*P(H) is also true.
    I made clear that the question wasn't sufficiently well-defined when I asked for the clarification "In reality, or under the assumption that we live in a universe with local realist laws?"
    Yes, this would be "reality".
    The correlations measured in real experiments may be conditional on detector settings, but they are not conditional on any hidden elements of reality, regardless of whether such hidden elements exist or not (since even if they do exist we don't know their value on each trial so we can't measure a frequency which is conditioned on them).

    Bell's approach is to start from the theoretical assumption of local realism, then use that to derive predictions about what correlations should be seen when you don't condition on the hidden variables. Since these predictions differ from the correlations predicted by QM and also from those observed in real experiments, this is taken as evidence that local realism is false in our universe.
    None of the equations in Bell's derivation of the Bell inequality involve statements about what is true in our reality, they are only about what should theoretically be true in a universe obeying local realist laws. Some of these equations actually involve the hidden local variables, while others involve only things that would be observed by theoretical experimenters in such a theoretical universe (which can then be compared with the observations of real experimenters in the real universe), but either way we are assuming a universe whose laws are locally realistic. If you ever thought I was saying anything different, it's you who was confused about the basic logic of Bell's proof.
     
    Last edited: Jun 11, 2010
  17. Jun 11, 2010 #16

    DrChinese

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    Well said, JesseM! I wish more people would listen to these words.

    The local realist is making an "extra" assumption (or two). If the term local realist is to mean anything, then such assumption(s) should be spelled out. It is then subject to verification or rejection... or in this case to be shown to be incompatible with something else (QM).

    I think any reasonable local realist can come up with a mathematical constraint or requirement that models locality and realism. Once that is agreed upon, I think the Bell program can be applied and the conclusion will simply match Bell. On the other hand, failure to provide such constraints for locality and realism would be tantamount to accepting the result prima facie.
     
  18. Jun 11, 2010 #17
    Did you completely ignore my statement that the set ("C", "notC") is equivalent to Z. Since you insist that C must have multiple values, can you explain what "notC" will represent according to your understanding of what C is supposed to entail. Give a short example of the different values you think could represent C and at the same time specify clearly what "notC" represents in your example -- please no 15-page scratch lotto cards examples that will require me to respond to every sentence because I will just ignore it.


    Yes I disagree.

    Go back and read it. I said no such thing.

    λ in Bell's paper is supposed to represent the EPR "elements of reality" which cause in the observed correlations. EPR elements of reality are not random variables no matter how loudly you shout that they are.
    That is circular reasoning and I can use the same point against you -- each term in the integral can only have a single value of λ so in fact by integrating you are adding P(AB|a,b,λ1) + P(AB|a,b,λ2) + ... + P(AB|a,b,λn) where n represents the number of possible realizations of your λ. You still can not escape the fact that the conditioning elements can never be as broad as λ. In each case in which a joint conditional probability is calculated, λn is specific and definitely not a random variable. That is why I told you repeatedly that in calculating a conditional probability you can not condition on a vague concept such as λ with multiple values. This is the same reason why in such a case, the LHS of Bell's equation (2) can not be a probability conditioned on the vaquely defined λ. It clearly looks like a marginal probability.

    So the claim that his inequalities derived from such an expression are based on the assumption that hidden variables exist is specious.

    Absolutely disagree, see my response above, H can not have multiple values in that expression. If you want to write it as P(AB)=P(A|H1)*P(B|H1) + P(A|H2)*P(B|H2) + ... + P(A|Hn)*P(B|Hn) go ahead, but don't deceive yourself and others that you are calculating P(AB|H).

    BTW The sample space for H1 is different from the sample space for H2 etc. They are not part of the same sample space. If H1 and H2 are mutually exclusive, your so-called H-sample space is undefined.

    I am definitely saying saying if H can take on multiple values H1 and H2 it is OK to write P(AB|H1) and P(AB|H2), but when you write P(AB|H) the only meaning here is that H is a placeholder for a specific value of H not all possible values of H simultaneously. ie, each concrete value of P(AB|H) you could ever calculate can only be valid for a specific H not the vague concept of being conditioned on "the H variable" or all values of H. For example if H represents the face of a coin and has two possible realizations in a toss "heads" or "tails", writing P(...|H) in which H includes all possible values is no different than writing P(...|heads, tails) But since heads and tails are mutually exclusive, your probability is undefined and meaningless if you insist on that definition. If H1 and H2 are mutually exclusive P(AB|H1H2) is undefined and meaningless. Therefore P(AB|H) can not imply that H is a variable with multiple values in a single expression.

    Can you explain to me how Aspect and others made sure in their experiments that IF hidden elements of reality exist, then the measured data will not depend on the their presence.

    In other words, is it possible for hidden elements of reality to exist and not exist at the same time? Isn't it obvious that IF hidden elements of reality exist, then they govern the results observed in Aspect type experiments?

    IF hidden elements of reality exist, then it is impossible for Aspect et al to collect data under circumstances in which hidden variables do not exist. Therefore your statement that the correlations they observed is "regardless of whether such hidden elements exist or not" is far off base.

    In other words you are saying the correlations Bell is calculating are those that should be seen in our universe if experiments are performed such that those variables (which we have assumed exist), should not affect the results.
    Now can you point me to an experiment in which the experimenters made sure that IF hidden elements of reality exist, they should not affect the data measured? By your own admission, those are the only data that are comparable to Bell's inequalities.

    Could you explain how Aspect et al made sure their data was collected in such a way that IF hidden elements of reality exist, they should not influence the results.

    Once the assumption is made that our universe is local realistic, the distinction you are trying to make is artificial.
    Until and unless you can demonstrate that the "theoretical experiments" are comparable to actual experiments performed in our universe, you can not use Bell's equations to say anything about our universe.
     
    Last edited: Jun 11, 2010
  19. Jun 11, 2010 #18

    JesseM

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    I guess I did miss that when reviewing your post, but since you didn't clearly define C I have to guess at your meaning. You did say "If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data", which suggested you believed "notC" would be some impossible situation that would never hold on any possible trial, so focusing on that statement, I assumed that C represented some general facts which would be true on every possible trial (for example, a statement about all combinations of values for the hidden variables allowed by the laws of physics, without specifying which combination is found on any particular trial). I guess if this were the case "notC" would represent some logical possibilities which are ruled out as impossible by the actual laws of physics, like all the combinations of values for the hidden variables which were logically possible but not physically possible given whatever laws of physics govern these variables.

    But it's true, this interpretation of your words doesn't really allow me to make sense of your claim that if Z="All possible facts in the universe" and that "if there exists a certain factor C within Z such that the set (C, notC) is the same as Z" and "C is the cause of the marginal correlaction between A and B". How can a set of opposite possibilities which can't both be simultaneously true be equivalent to "all possible facts in the universe"? I can't really make sense of this, please either define your terms more clearly, or give me some simple "toy model" of a universe where very few facts determine some outcomes A and B, like the 8 possible combinations of hidden fruits on each trial in the lotto card analogy, and explain precisely what C and notC and Z would represent in this toy model. You don't have to use any of the analogies I've already come up with for the toy model, but some specific example would certainly help in making your terms more intelligible to me.
    The scratch lotto analogy was only a few paragraphs and would be even shorter if I didn't explain the details of how to derive the conclusion that the probability of identical results when different boxes were scratched should be greater than 1/3, in which case it reduces to this:
    Is that too long for you? If you just have a weird aversion to this example (or are refusing to address it just because I have asked you a few times and you just want to be contrary), I suggest you come up with your own toy model since I don't know what would satisfy you. On the other hand, if you are willing to reconsider, then I can certainly explain what my hypothesis about what you mean by the symbols "C" and "notC" would say about the meaning of these symbols in this example.
    Which part do you disagree with? You disagree that λ in Bell's equations and H in mine were supposed to represent variables that could take multiple values? Or do you agree with that part, but then disagree that the fact that your C is not a random variable implies that it's not relevant to a discussion of Bell's proof?
    You said:
    Since you said "it is impossible to collect data under situations where C is not True", I interpreted that to mean you're saying C is "always present", So P(AB) "is not different from" P(AB|C), and therefore if P(AB) is not equal to P(A)P(B) then that also implies that "equations such as P(AB|C) = P(A|C)P(B|C) are not accurate" (i.e. you're saying that because C is present and has the same value in all trials, then any probability which is conditioned on C will be the same as the marginal probability, so if P(AB|C)=P(A|C)P(B|C) that would automatically imply P(AB)=P(A)P(B)). Perhaps I misunderstood you, but if so you certainly aren't expressing yourself very clearly, I can't see how the above quote would be compatible with the idea that the value of P(AB) could be different from P(AB|C), or that P(B) could be different from P(B|C).
    Perhaps you are focusing on the word "random"--as I said, I accept that 0 and 1 are still valid probabilities, so even if the value of the hidden variables λ on each trial was generated by a completely deterministic process I would still refer to λ as a "random variable" if its value could differ from one trial to another. So let's focus on the "variable" part--do you disagree that Bell was defining λ as a variable whose value could differ from one trial to another, with each possible value of λ expressing some combination of values for all the hidden variables? (for example λ=1 might be defined to mean "spin-up on 0-degree axis, spin-down on 120-degree axis, spin-up on 240-degree axis" while λ=2 might be defined to mean "spin-down on 0-degree axis, spin-up on 120-degree axis, spin-up on 240-degree axis")

    If you disagree with the basic premise that λ is intended to be a variable whose value could differ from one trial to another, can you explain why you think Bell wrote equation (2) as an integral with respect to λ? Isn't it basic to the notion of an integral that the "variable of integration" is allowed to vary?
    Well, no, it doesn't remotely resemble "circular reasoning" since I am not arriving at any conclusion by taking the conclusion as a premise.
    How is that using the same point against me?? I 100% agree with the above, and in fact I have tried to say exactly the same thing in a number of my previous posts to you. For example, in post #75 on the other recent thread I said:
    Well yes, that's exactly what "random variable" means, something that takes different specific values on each trial. For example, if I am flipping coins, I can define the random variable R to have value 1 if the coin comes up heads and 0 if it comes up tails...this would be a "discrete random variable". See wikipedia's random variable page:
    Do you agree that my definition of R above constitutes a perfectly good random variable in experiments where a coin is flipped on each trial, with the value of R differing on different trials? If so, what's wrong with defining λ as a more complex random variable whose value also differs on different trials, depending on the specific values of whatever hidden variables exist?
    Do you think in a coinflip experiment there would be something wrong with conditioning on R, which also takes multiple values depending on whether the coin comes up heads or tails on each trial? For example, if S is some other random variable representing some other set of mutually exclusive events which can happen on each trial, do you think it would be incorrect to write the equation P(R and S) = P(S|R)*P(R) ?
    OK, take a look at section 13.1 of this book, titled "Conditioning on a random variable", where the author writes:
    Do you think the author is making an error in saying that expressions like P(A|X), where X is a random variable, have a well-defined meaning in probability theory?
    Huh? There is nothing stopping us from considering a set of trials which includes both trials where H1 was true and trials where H2 was true, even though they are "mutually exclusive" in the sense they can't both be true on any single trial. For example, if H1 represented the event of my coin coming up heads, and H2 represented the event of my coin coming up tails, then I could consider a sample space including instances of trials where H1 was true and H2 false, as well as instances of trials where H1 was false and H2 was true, but no trials where they were both simultaneously true or both simultaneously false.
    I have no idea what it would mean to say H stands for "all possible values of H simultaneously," you'll have to give me a definition or example. All I am saying is that if you write some equality or inequality involving random variables like A and H, like my example of P(A and H) = P(A|H)*P(H), then such an equation is understood to be equivalent to the statement that the equation holds for all possible combinations of specific values of A and H. If A can take only two values A1 and A2, and H can take only two values H1 and H2, then writing P(A and H)=P(A|H)*P(H) is simply a shorthand for the statement that all four of the following equations are true:

    1. P(A1 and H1) = P(A1|H1)*P(H1)
    2. P(A1 and H2) = P(A1|H2)*P(H2)
    3. P(A2 and H1) = P(A2|H1)*P(H1)
    4. P(A2 and H2) = P(A2|H2)*P(H2)

    Would you say that by using P(A and H)=P(A|H)*P(H) as a shorthand for the idea that all for of these more specific equations are true, I am illegally using H to represent "all possible values of H simultaneously"?
    Yes, it is different. As noted in the textbook, P(...|H) would itself represent a random variable which can take different values on different trials. And if you write an equality involving random variables, like P(...|H) = P(... and H)/P(H), that means that even though the values of each side individually can vary from one trial to another, it must be true on every trial that the specific value of the left side in that trial works out to be equal to the specific value of the right side in that same trial.
     
  20. Jun 11, 2010 #19

    JesseM

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    (continued from previous post)
    Uh, why should they need to make sure of that? Saying the correlations "are not conditional on any hidden elements of reality" does not mean they are not causally influenced by hidden elements of reality, it just means the correlation that's calculated is not a conditional one that controls for those elements. For example, suppose I have a large population of people, each of whom is either a smoker or nonsmoker, each of whom either has lung cancer or doesn't, and each of whom either has yellow teeth or doesn't. I can certainly calculate the correlation between yellow teeth and lung cancer alone, i.e. find the fraction of people who satisfy (yellow teeth AND lung cancer) and compare it to the product of the fraction that satisfy (yellow teeth) and the fraction that satisfy (lung cancer), even if it happens to be true that the correlation can be explained causally by the fact that smoking increases the chances of both. That's all it means to say that the correlation I'm calculating is not "conditioned on" the smoking variable, that I'm just not bothering to include it in my calculations, not that it isn't causally influencing the correlation I do see between yellow teeth and lung cancer.
    They don't need to! I can collect data on the marginal probability a random member of a population will have yellow teeth or will have lung cancer, and calculate the marginal correlation between these two variables, even if it happens to be true that each person either is or isn't a smoker and that this "hidden" variable is having a causal influence on the two variables I am measuring/correlating.
    Nope, you're just confused about the difference between saying a calculated correlation is not "conditioned on" some variable and saying it's not causally affected by that variable.
    No one is assuming our universe is local realist. They are calculating what would be observed by experimenters in a theoretical local realist universe, then comparing that with actual observations by actual real experimenters in our own real universe. If they differ, that means the predictions of local realism are falsified, therefore the theory that our universe is local realist is falsified. That's how theory-testing works in all of physics--you do a theoretical analysis to figure out what would be observed if a certain theory were true, then you compare that with real-world observations.
    What do you mean by "comparable"? You can show theoretically that, under local realism, if two experimenters each have a choice of three detector settings which they make randomly, and both their choices and measurements are made at a spacelike separation from one another, then that implies certain statistical conclusions about the results of their measurements, regardless of exactly what it is they are measuring. So if you set up some quantum-mechanical experiments satisfying these basic conditions, and you find that the statistical conclusions are false in these experiments, then you've falsified local realism.
     
  21. Jun 11, 2010 #20
    Exactly.
    Yes, you can define C like that. But No, if you define C like that, you can not use it as a condition in calculating a conditional probability because those values could be mutually exclusive. Any such probability will be an impossibility.
    The point is that once you define C as such, there is no difference between C and Z. In other words the P(notC) is zero and P(C) = 1. Do you see now why it makes no sense to talk of a conditional probability while defining the condition the way you do?
    I gave that example specifically to show you how absurd the implications of your approach are. I am happy you now see it.
    I disagree with the idea that such variables could represent the EPR hidden elements of reality. I disagree with the idea that any probabilities calculated with such constructions could be compared to actual experiments IF hidden variables do exist.

    Yes you did. If hidden elements of reality exist, and C represents those hidden elements of reality
    then P(AB|C) = P(A|C)P(B|C) implies that there is no correlation between the A and B. Therefore such an equation can not model the EPR situation in which correlations are in fact observed between A and B. You claimed earlier that the correlation exists marginally even if it does not exist conditionally. But I just showed you that by defining C the way you do, there is no difference between marginal and conditional. So then IF hidden elements of reality exist, and correlations are still observed in their presence, the equation
    P(AB|C) = P(A|C)P(B|C) will not appropriately model the situation.

    When you write a term such as P(S|R), where R = ("heads","tails"), R can not represent "all possible" values in a single term. R can only be a place-holder for one of the possible values of R. Sure, you can add up many separate probability terms involving the different instances, Rn, of R but the result you get can not said to be conditioned on R, the so-called "random variable".
    So in a universe in which only one of those instances of R are actually realized, whatever probability you obtain in your summation calculation can not be compared to anything measurable in such a universe. Because their marginal probability is NOT defined on the same probability space as your marginal probability. In other words, if in their universe only R="tails" is possible, then their marginal probabilities are the same as your P(...|tails). Just because their universe was one of the terms in your probability does not mean your probability is conditioned on the existence of their universe.
     
  22. Jun 12, 2010 #21

    JesseM

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    If P(C)=1, and C is a fixed statement of facts rather than a variable that can take different values on different trials, then presumably whatever facts are referred to by "C" are true on every possible trial. Is that correct?
    No, because your definition of C (assuming my understanding above is right) has nothing to do with the hidden-variable terms that appeared in either my or Bell's equations.
    But it isn't my approach, since none of the terms in my equations are like your C. All you've shown is that if you define your terms in silly ways it won't be very useful.
    You're not answering my question. Do you agree or disagree with my claim that, unlike your C which takes the same value in every trial (assuming I got that part right), the λ in Bell's equations and H in mine were supposed to represent variables that could different values on different trials?
    Yes, you "disagree", but so far on this thread you haven't actually presented any argument as to what's wrong with Bell's reasoning, instead you've just come up with symbols and definitions of your own that have nothing to do with how Bell defined the symbols that appear in his own equations, and acted as though this somehow scores points against Bell's arguments.

    Also, you refuse to engage with the simple numerical examples I give that show exactly how we can reason about unseen hidden variables (like hidden fruits imagined to exist behind the three boxes of the lotto cards) to draw conclusions about the statistics that should be seen in experiments (observations about the frequencies that Alice and Bob get the same fruit depending on whether they choose to scratch the same box or different boxes) under the assumption that the results of the experiments are determined by the hidden variables. It seems to me like you aren't really making a good-faith effort to understand and engage with the arguments of people you disagree with, but are just trying to use rhetorical strategies to "win" and make the opposing side look bad.
    But if your C is defined to be a non-variable that must hold on every single trial, then the meaning of the equation P(AB|C)=P(A|C)P(B|C) is completely different from any equation that appeared in Bell's paper, or any equation I have written down during the course of our discussion like P(AB|H)=P(A|H)P(B|H). So, it would simply be a strawman to claim that with your definition of C, either I or Bell would assert that P(AB|C)=P(A|C)P(B|C) in the first place.
    Er, what? I don't define C any way, I'm just trying to understand your definition, which is still rather unclear to me. But if my basic understanding is correct that you are defining C so that it is a non-variable which is true on every single trial, then I would certainly not assert that the correlation conditioned on C is any different than the marginal correlation. When I said that the correlation conditioned on some other symbol like H or λ could be different than the marginal correlation, I was always using a symbol that was supposed to represent a variable which could take different values on different trials, like a variable H in the lotto card example that would take the value H=1 if the hidden fruits were cherry-cherry-cherry, and would take the value H=2 if the hidden fruits were cherry-cherry-lemon, etc. If A and B represent some observable measurements, it's certainly possible to come up with an example where P(AB|H)=P(A|H)*P(B|H) is true for every possible value of H (i.e. where it's true that P(AB|H=1) = P(A|H=1)*P(B|H=1), and also true that P(AB|H=2) = P(A|H=2)*P(B|H=2), and so on for all possible specific values of H), and yet where P(AB) is not equal to P(A)*P(B) -- do you disagree that this is possible?
    I don't know the meaning of the phrase "represent all possible values in a single term" (a phrase I never used), and I suspect you don't either and are just making lofty-sounding assertions which have no clear meaning. If you think the phrase has a well-defined meaning, so that it is meaningful to assert that R cannot "represent all possible values in a single term", then please give me a precise definition.
    When I say that there is no correlation between A and B when "conditioned on R", all I mean is that for any possible specific value Rn that R can take, it will be true that P(AB|Rn) = P(A|Rn)*P(B|Rn). For example, if R can take only two values, R=1 and R=2, then saying there's no correlation between A and B conditioned on R would just be shorthand for the claim that both the following equations hold:

    1. P(AB|R=1) = P(A|R=1)*P(B|R=1)
    2. P(AB|R=2) = P(A|R=2)*P(B|R=2)

    Hopefully you agree that the claim that both these specific equations hold is perfectly well-defined as statistical claims go? If so, then even if you don't like using the phrase "no correlation between A and B conditioned on R" to describe this claim, that is merely a semantic quibble, now you hopefully understand what I mean even if you don't like the vocabulary I use to describe it (and hopefully you agree that the above two equations can hold even in a situation where the marginal probability P(AB) is different from the product of the marginal probabilities P(A)*P(B)). In terms of pure semantics, I think the statistics community would side with me on this, not you; after all, I just linked to a textbook which explicitly talks about conditioning on a random variable.
    But I'm not talking about a universe where only one value of the random variable λ is actually realized, and neither was Bell. Again, in a local hidden variables theory it is quite possible that if you do multiple trials with different pairs of particles, the hidden variables associated with the pair on one trial may be different than the hidden variables associated with a different pair on a different trial. For example, the simplest local-hidden-variables theory to try to explain quantum experiments would just say that if the particles always have the same spin whenever they're measured on the same axis, that must mean each pair is created with the same predetermined answers to what spin they'll give if measured on a given axis, so λ=1 might represent the hidden variable state "spin-up on axis 1, spin-up on axis 2, spin-up on axis 3" while λ=2 could represent the hidden variable state "spin-up on axis 1, spin-up on axis 2, spin-down on axis 3", and so on and so forth. If on some trials an experimenter picks axis 3 and gets spin-up, while on other trials the experimenter picks axis 3 and gets spin-down, then in this simple hidden-variables theory it must be true that the value of λ differs from one trial to another.
     
  23. Jun 12, 2010 #22
    Actually, this isn't Bell's approach. Bell does not speak anywhere of 'local realism'. He speaks only of local causality as his starting theoretical assumption. As evidence of this, see Bell's papers entitled, La Nouvelle Cuisine, Free Variables and Local Causality, The Theory of Local Beables, and Bertlemann's Socks and the Nature of Reality. Also have a look at Norsen's paper,

    Against `Realism'
    Authors: Travis Norsen
    Journal reference: Foundations of Physics, Vol. 37 No. 3, 311-340 (March 2007)
    http://arxiv.org/abs/quant-ph/0607057

    What Norsen essentially points out is that there isn't some additional notion of realism in Bell's theorem, over and above the notion of realism already implicit in Bell's definition of locality. This is a crucially important point to recognize in any discussion about what Bell actually said and did.
     
  24. Jun 12, 2010 #23

    DrChinese

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    Maaneli, I disagree strongly with your point; Norsen is wrong too. I can point out the exact spot in Bell in which realism is introduced, as it is explicit:

    After (14): It follows that c is another unit vector... [in addition to the a and b of Bell (2)]

    Bell did not hightlight this as the introduction of hidden variables, realism, counterfactuality or whatever one may choose to call it. But there it is, and it is quite impossible to derive the Bell result without dear old c. Please don't cheat c of her 15 minutes...

    And as you are undoubtedly aware, there are many who feel that - and with substantial justification from other arguments against realism I might add - that there is no possibility of a realistic (non-contextual) theory under any circumstances. I realize that you tend towards the non-local side of things and don't follow that line of thinking.
     
    Last edited: Jun 12, 2010
  25. Jun 12, 2010 #24
    (continued from previous post)
    You do not understand the difference between a theoretical exercise and an actual experiment. If I am trying to study the relative effectiveness of two possible treatments T = (A, B) against a kidney stone disease. Theoretically it is okay to say that you randomly select two groups of people from the population of people with the disease, give treatment A to one and B to the second group and then calculate the the relative frequencies of those who recovered in group 1 after taking treatment A. Theoretically speaking, you can then compare that value with relative frequency of those who recovered in group 2 after taking treatment B. This is fine as a theoretical exercise.

    Now fast-forward to an actual experiment in which the experimenters do not know about all the hidden factors. What does "select groups at random" mean in a real experiment? Say the experimenters select the two groups according to their best understanding of what may be random. And then after calculating their relative frequencies, they find that Treatment B is effective in 280 of the 350 people (83%), but treatment A is only effective in 273 of the 350 people (78%). So they conclude that Treatment B is more effective than treatment A. Is this a reasonable conclusion according to you?

    Now suppose the omniscient being, knowing fully well that the size of the kidney stones is a factor and after looking at the data he finds that if he divides the groups according to the size of kidney stones the patients had the groups break down as follows

    Group 1 (those who received treatment A): (87-small stones, 263-large stones)
    Group 2( those who received treatment B): (270-small stones, 80-large stones)

    He now finds that of the the 81 of the 87 (93%) in group 1 who had small stones were cured by treatment A, and 192 of the 263 (73%) of those with large stones in group 1 were cured by treatment A.
    For group 2, he finds that 234 of the 270 (87%) with small stones were cured and 55 of the 80 (69%) with large stones were cured.

    Clearly, when all the hidden factors are considered, Treatment A is more effective than than treatment B contrary to results obtained by the experimenters. Does this then mean there is some spooky business happening? This is called Simpson's paradox and I believe I have pointed this to you not too long ago.

    As you can hopefully see here, not knowing about all the hidden factors at play, the experiments can not possibly collect a fair sample, therefore their results are not comparable to the theoretical situation in which all possible causal factors are included. That is why I have repeatedly pointed out to you that in order to collect a fair sample comparable with Bell's inequalities, experimenters in Aspect type experiments must design their experiments such that, not only should all possible "values" of the hidden elements of reality are realized, but they should be realized fairly. In the kidney stone example above, all possible values were realized but not fairly. As a result, the conclusions were are odds with what is known by the omniscient being and therefore not comparable. In other words, all values observed by the experimenters is conditioned on their assumptions about what is causing the results. Their definition of random in this case was flawed.

    So again, do you have a reference to any Aspect type experiment in which they ensured randomness with respect to all possible hidden elements of reality causing the results? By comparing observed correlations to Bell's inequalities, you are claiming that they are in fact comparable.

    Huh? The break down of a conclusion can only be taken to imply a failure of one of the premises of that conclusion. The argument usually goes as follows:
    (1) Bell's inequalities accurately model local realistic universes
    (2) Our universe is locally realistic
    (3) Therefore actual experiments in our universe must obey Bell's inequalities.

    However, we now know that actual experiments in our universe do not obey Bell's inequalities. It therefore follows that either (1) is false or (2) is false. If your claim now is that (2) is not a premise in that argument, then you are admitting that (1) is false. There is no escape here.
     
  26. Jun 12, 2010 #25
    Sorry, but you are the one who is wrong. The introduction of the unit vector c is not where realism is initially introduced, nor does c contain within it some independent and additional assumption of 'realism', over and above the notion of realism that is already implicitly introduced by Bell's condition of local causality. In other words, all the realism in Bell's theorem is introduced as part of Bell's definition and application of his local causality condition. And the introduction of the unit vector, c, follows from the use of the local causality condition. Indeed, in La Nouvelle Cuisine (particularly section 9 entitled 'Locally explicable correlations'), Bell explicitly discusses the relation of c to the hidden variables, lambda, and the polarizer settings, a and b, and explicitly shows how they follow from the local causality condition. To summarize it, Bell first defines the 'principle of local causality' as follows:

    "The direct causes (and effects) of events are near by, and even the indirect causes (and effects) are no further away than permitted by the velocity of light."

    In fact, this definition is equivalent to the definition of relativistic causality, and one can readily see that it implicitly requires the usual notion of realism in special relativity (namely, spacetime events, and their causes and effects) in its very formulation. Without any such notion of realism, I hope you can agree that there can be no principle of local causality.

    Bell then defines a locally causal theory as follows:

    "A theory will be said to be locally causal if the probabilities attached to values of 'local beables' ['beables' he defines as those entities in a theory which are, at least, tentatively, taken seriously, as corresponding to something real, and 'local beables' he defines as beables which are definitely associated with particular spacetime regions] in a spacetime region 1 are unaltered by specification of values of local beables in a spacelike separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a spacetime region 3 [he then gives a figure illustrating this]."

    You can clearly see that the local causality principle cannot apply to a theory without local beables. To spell it out, this means that the principle of local causality is not applicable to nonlocal beables, nor a theory without beables of any kind.

    Bell then shows how one might try to embed quantum mechanics into a locally causal theory. To do this, he starts with the description of a spacetime diagram (figure 6) in which region 1 contains the output counter A (=+1 or -1), along with the polarizer rotated to some angle a from some standard position, while region 2 contains the output counter B (=+1 or -1), along with the polarizer rotated to some angle b from some standard position which is parallel to the standard position of the polarizer rotated to a in region 1. He then continues:

    "We consider a slice of space-time 3 earlier than the regions 1 and 2 and crossing both their backward light cones where they no longer overlap. In region 3 let c stand for the values of any number of other variables describing the experimental set-up, as admitted by ordinary quantum mechanics. And let lambda denote any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR. Suppose that the c and lambda together give a complete specification of at least those parts of 3 blocking the two backward light cones."

    From this consideration, he writes the joint probability for particular values A and B as follows:


    {A, B|a, b, c, lambda} = {A|B, a, b, c, lambda} {B|a, b, c, lambda} ​

    He then says, "Invoking local causality, and the assumed completeness of c and lambda in the relevant parts of region 3, we declare redundant certain of the conditional variables in the last expression, because they are at spacelike separation from the result in question. Then we have


    {A, B|a, b, c, lambda} = {A|a, c, lambda} {B|b, c, lambda}. ​

    Bell then states that this formula has the following interpretation: "It exhibits A and B as having no dependence on one another, nor on the settings of the remote polarizers (b and a respectively), but only on the local polarizers (a and b respectively) and on the past causes, c and lambda. We can clearly refer to correlations which permit such factorization as 'locally explicable'. Very often such factorizability is taken as the starting point of the analysis. Here we have preferred to see it not as the formulation of 'local causality', but as a consequence thereof."

    Bell then shows that this is the same local causality condition used in the derivation of the CSHS inequality, and which the predictions of quantum mechanics clearly violate. Hence, Bell concludes that quantum mechanics cannot be embedded in a locally causal theory.

    And again, the variable c here is nothing but part of the specification of the experimental set-up (as allowed for by 'ordinary quantum mechanics'), just as are the polarizer settings a and b (in other words, a, b, and c are all local beables); and the introduction of c in the joint probability formula follows from the local causality condition, as part of the complete specification of causes of the events in regions 1 and 2. So, again, there is no notion of realism in c that is any different than in a and b and what already follows from Bell's application of his principle of local causality.

    So there you go, straight from the horses mouth. I hope you will have taken the time to carefully read through what I presented above, and to corroborate it for yourself by also reading (or re-reading) La Nouvelle Cuisine. It's really important, at least for the sake of intellectual honesty, to understand this point about what Bell said, and to not misrepresent what he claimed and what he actually proved.

    No, you misrepresent what I think. I am well aware that there is no plausible possibility of a realistic non-contextual theory, and I have even stated so many times on this forum before. But (A) this is not relevant to my point of disagreement with you regarding what Bell actually said and proved, and (B) the inability to retain contextuality in an empirically adequate realistic theory is not a 'problem' for realistic theories, in any scientifically meaningful sense. It may be a problem for some people's naive intuitions about how a realistic theory of the physical world should work, but that's completely subjective, and in any case, I myself have never found contextuality to be a counter-intuitive or paradoxical or inelegant notion. Also, I have strong reasons to think that contextuality is already a property of measurements in classical nonequilibrium statistical mechanics, in which case, the usual assumption that non-contextuality is a fundamental property of measurements in classical realistic physical theories, is just wrong.
     
    Last edited: Jun 12, 2010
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