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Understanding Bell's logic 
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#1
Jun910, 11:55 PM

P: 375

I am hoping it may be helpful to separate Bell's logic from Bell's mathematics
http://www.physicsforums.com/showthread.php?t=406372. Understanding one may better help us understand the other. In the language that is evolving at "Understanding Bell's mathematics", http://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 EPRBell 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(GH,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(GH,a,λ,b) P2(G'H,b,λ,a) would avoid some wellknown 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(GH,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
Jun1010, 02:30 AM

P: 1,414

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. 


#3
Jun1010, 03:12 AM

P: 375

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. Much was learnt from the related experiments. Including that the logic was flawed. C'est la vie. That's what. 


#4
Jun1010, 04:21 AM

P: 1,414

Understanding Bell's logic
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. 


#5
Jun1010, 05:53 AM

P: 375

Ok. Thank you. No worries at all. And please ... Do not devalue your own efforts. You and your P(ABH) 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 


#6
Jun1010, 09:17 AM

Sci Advisor
PF Gold
P: 5,299

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 peerreviewed references in 100+ pages. Research on Hidden Variable Theories: a review of recent progresses, Marco Genovese (2005) http://arxiv.org/abs/quantph/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. 


#7
Jun1010, 10:17 AM

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#8
Jun1010, 10:42 AM

P: 678

(P(GG'H,a,b,λ) = P1(GH,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. 


#9
Jun1010, 12:25 PM

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#10
Jun1010, 06:09 PM

P: 375

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 Belltests show that Bell's supposition re Bell's lambda is false? Thank you. 


#11
Jun1010, 08:00 PM

P: 678

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(ABZ). 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(ABC) = P(AC)P(BC). It means that C screensoff 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(ABC) = P(AC)P(BC) are not accurate. For example if hidden element C is always present and C=42, then P(AB) is not different from P(ABC=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(ABC) = P(AC)P(BC) 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(ABC) = P(AC)P(BC), 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(ABC) 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? 


#12
Jun1010, 09:27 PM

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P: 8,470

c1: spinup on a, spinup on b, spinup on c c2: spinup on a, spinup on b, spindown on c c3: spinup on a, spindown on b, spinup on c c4: spinup on a, spindown on b, spindown on c c5: spindown on a, spinup on b, spinup on c c6: spindown on a, spinup on b, spindown on c c7: spindown on a, spindown on b, spinup on c c8: spindown on a, spindown on b, spindown on c (note that these are directly analogous to the eight possible hiddenfruit states on the cards in the scratch lotto card analogy) According to this type of hiddenvariables 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? 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 spacelikeseparated 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. 


#13
Jun1010, 09:50 PM

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#14
Jun1010, 10:12 PM

P: 678

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. So let us try again. 


#15
Jun1110, 08:52 AM

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P: 8,470

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(ABC)=P(AC)*P(BC) 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(GH,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 nonvariable 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. 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. 


#16
Jun1110, 09:18 AM

Sci Advisor
PF Gold
P: 5,299

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. 


#17
Jun1110, 05:59 PM

P: 678

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


#18
Jun1110, 08:14 PM

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P: 8,470

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. 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? 1. P(A1 and H1) = P(A1H1)*P(H1) 2. P(A1 and H2) = P(A1H2)*P(H2) 3. P(A2 and H1) = P(A2H1)*P(H1) 4. P(A2 and H2) = P(A2H2)*P(H2) Would you say that by using P(A and H)=P(AH)*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"? 


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