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Bell's derivation; socks and Jaynesby harrylin
Tags: bell socks jaynes 
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#1
Feb2512, 10:31 AM

P: 3,184

Hello,
For this little discussion I base myself on Bell's paper on Bertlmann's socks: http://cdsweb.cern.ch/record/142461 Although I have participated in a number of discussions about Bell's theorem, I always had the uneasy feeling not to fully understand the definitions of symbols and the notation  in particular how to account for lambda in probability calculations. So, although I intend to discuss here the validity (or not) of Jayne's criticism of Bell's equation no.11, I'll start very much more basic. Using Bell's example of socks, I think that we could write for example: P1(pink) = 0.5 Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations. As the colour depends on Bertlmann's mood, we can then account for that mood as an unknown variable "lambda" (here I will just put X, for unknown). However, any local realistic theory that proposes such an unknown variable as explanation, still must predict the same observed result. Therefore, I suppose that if we include X as causal factor, we must still write: P1(pinkX) = 0.5 Thus far correct? 


#2
Feb2512, 04:12 PM

P: 177




#3
Feb2512, 05:50 PM

P: 3,184

P1(pinkX)=P1(pink)/P(X) = P1(pink)/1 =0.5 It's exactly the same as for a fair coin: P(head  fair coin) = 0.5. I can imagine that someone would like to split the probability estimation up into unknown "knowns": then we can separate it into the cases that Bertlmann decides to put a pink sock on his left leg, and the cases that he decides to put another colour on his left leg. However, what we are interested in the result over many times, and then we are necessarily back at where we were here above. Thus, I don't see any use for that. 


#4
Feb2512, 08:11 PM

P: 177

Bell's derivation; socks and Jaynes
Edit: A couple of papers that may be relevent to this discussion: Jaynes' view of EPR, a critque of Jaynes' view 


#5
Feb2612, 03:02 AM

P: 3,184

@IsometricPion: thanks for the links! I suspect that our own discussion here, which is based on http://cdsweb.cern.ch/record/142461, will show that the Arxiv paper misses the point; we'll see!
instead of running to eq.11, I will first work out the example that Bell gave in his introduction, as he did not do so himself. Note that in Bell's paper the pictures come after the text. I'll start with a partial retake. Elaborating on Bell's example of Bertlmann's socks, we could write for example: P1(pink) = 0.5 Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations. As the colour depends on Bertlmann's mood, we can account for that mood as an unknown function "lambda" (here I will just put X, for unknown). However, any "classical" theory that proposes such a physical model, still must predict the same observed result. Therefore, if we include X as invisible cause for the outcome, we must still write: P1(pinkX) = 0.5 (Compare: P(head  fair coin) = 0.5) Similarly we can write for the right leg: P2(pinkX) = 0.5 Bell remarks: P(L,RX) =/= P1(LX) P2(RX) Here L stands for "pink on left leg", and R stands for "pink on right leg". Ok so far? 


#6
Feb2612, 09:28 AM

P: 5

pink = 1, not pink = 1
then: P(LR) = 0, because L and R are always different; formally: P(LR) = P(LR)*P(R); P(R) = P(L) = 1/2; but: P(LR) = 0 <> P(L); (both socks have never the same colour) corr = 0  1 = 1, full anticorrelation. And using Bell reasoning: P(LR) = P(L)*P(R) = 1/2 * 1/2 = 1/4; corr = 2*1/4  2*1/4 = 0. Two random socks, and completely independent, of course. 


#7
Feb2612, 09:58 AM

P: 3,184

@ alsor: it appears that in this matter we both agree with Jaynes.
However, again you ran far ahead of me and I'm not sure if everyone who, so far, didn't "see" this point of Jaynes etc., could follow you. So, I'll continue my slow pace to make sure that everyone who watches this topic can follow me and that we all agree on the basic facts as well as notation. I'll catch up with you later. 


#8
Feb2612, 02:00 PM

P: 177

If instead one does not know his mood (or anything about it other than it can take on one of two sets of values), P(L)=P(LRP)P(RP)+P(LR¬P)P(R¬P)=0*0.5+1*0.5=0.5, by exchangeability. The problem Jaynes sees with Bell's reasoning is not his statistical or mathematical procedure/ability, rather he thinks Bell is to restrictive in what he (Bell) consideres to be valid variables for the probability distributions for a theory upholding local realism. 


#9
Feb2712, 06:00 AM

P: 3,184

Anyway, we're not there yet: the problem with the illustration of Bertlmann's socks is that it by far doesn't catch the complexity of the problem at hand. If the observations would always be perfectly anticorrelated, there wouldn't be a riddle. Now, I'm afraid that his next illustration of Lille and Lyon matches it even less well; thus, for this discussion I have been trying to come up with a variant of Bertlmann's socks that addresses the fact that the local conditions affect the observed correlation, but I didn't come up with a good looking one (I thought of observation of white or yellow socks in daylight/artificial light, as well as mud on his socks, but I'm not satisfied). Any better suggestion? If not, we should perhaps move on to the introduction of eq.11. 


#10
Feb2712, 06:18 AM

P: 1,414




#11
Feb2712, 06:31 AM

P: 3,184




#12
Feb2712, 09:00 AM

Mentor
P: 11,623

http://www.physicsforums.com/showthread.php?t=283519 This was before you joined PF, so you may not have seen this. It may or may not fit in with the direction you were planning to go. It was split off from another thread, by the way, which is why it appears to start in the middle of a discussion. 


#13
Feb2712, 09:34 AM

P: 3,184




#14
Feb2812, 02:05 AM

P: 3,184

Still, I started reading it and I notice some disagreement about what Bell claimed to prove. There is no use getting into arguments about the meaning of "local realism" and philosophy. What the "local realist" Einstein insisted on, and what Bell claimed to be incompatible with QM, was "no spooky action at a distance". Or, as Bell put it in his first paper: Bell puts it this way in his Bertlmann's socks paper: 


#15
Mar112, 02:06 PM

P: 3,184

If Jaynes' criticism focuses on Bell's equation no.11 in his "socks" paper, it was perhaps due to a misunderstanding about what Bell meant (his comments were based on an earlier paper). P(ABa,b,x) = P(Aa,x) P(Bb,x) (Bel 11) Here x stands for Bell's lambda, which corresponds to the circumstances that lead to a single pair correlation (in contrast to my earlier X, which causes the overall correlation for many pairs). According to Jaynes it should be instead, for example: P(ABa,b,x) = P(AB,a,b,x) P(Ba,b,x) Perhaps Jaynes thought that Bell meant: P(ABa,b,X) = P(Aa,X) P(Bb,X) in which case Jaynes claimed that: P(ABa,b,X) = P(AB,a,b,X) P(Ba,b,X) This is really tricky. However, he really was disagreeing with the integral equation. According to him, it should not be: P(ABa,b) = ∫ P(Aa,x) P(Bb,x) p(x) dx but: P(ABa,b) = ∫ P(ABa,b,x) P(xa,b) dx and thus: P(ABa,b) = ∫ P(ABa,b,x) p(x) dx = ∫ P(AB,a,b,x) P(Ba,b,x) p(x) dx Is my summary of the disagreement correct? What is the significance of little p(x) instead of P(x)? 


#16
Mar112, 03:24 PM

Sci Advisor
PF Gold
P: 5,299

As I read it, this is one of Jaynes's arguments. However, I think it is attacking a straw man. The essence of Bell's argument does not require the factorization so much as a definition of what realism is. For a SINGLE photon, not a pair: does it have a welldefined polarization at 0, 120, and 240 degrees independent of the act of observation? Once you answer this in the affirmative, as any local realist must, the Bell conclusion (a contradiction between the assumption and QM's predictions) follows quickly. If you answer as no, then you already deny local realism so it is moot. So I really don't see the significance here of Jaynes' argument. The only people that take it seriously are local realists looking for support for their position. The vast majority of scientists see it for what it is, something of a technicality with no serious implications for the Theorem whatsoever. In other words, it would be helpful to see an example that somehow related specifically to photon polarization rather than urns (which does not seem to be much of an analogy). 


#17
Mar112, 04:11 PM

P: 1,583




#18
Mar112, 04:21 PM

P: 177

Jaynes refers to probabilities essentially as logical statements of uncertain truth value. His P(yY) correspond to logical statements where Y is the predicate and y is the antecedent the truth value of which one is uncertain (the amount of (rational) belief one has that y has a value between u and v is P(u≤y≤vY)=∫_{u}^{v} P(yY)dy). He refers to any probability not of this form as p(y), since one cannot ascribe a logical statement to such a probability without more information regarding its context. Since the context here is clear and consistently applied, I think it is just a matter of formalism (i.e., there is no substantial difference). (Jaynes defines what he means by these symbols in Appendix B of Probability Theory: The Logic of Science.) Jaynes states what he thinks are Bell's hidden assumptions: I think a key point to this discussion is how to define local realism in terms of the functional dependence of probability distributions of outcomes of the EPR (thought) experiment. Once this is agreed upon (i.e., all the variables and symbols we are using are welldefined) the rest should just be a matter of mathematics (about which I think we all should be able to agree). 


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