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Hi Everyone,
The dispute over Bell's theorem seems to have been finally settled, or at least a precise point of disagreement has been pinpointed. Below I am reproducing my recent exchange with Richard Gill on Retraction Watch, following my post there I have reproduced just above. Let me summarize the essence of this exchange, bringing out the exact point of remaining disagreement. It should now be a simple Yes / No answer for anyone interested to decide for themselves who is right and who is wrong:
In the notations used in my posts above, let X = A(a)*[B(b)+B(b’)] be the random variable of interest and let p(X) be the probability of X taking the values +2, 0, or -2.
Since the random variable X(lambda) can only take the values +2, 0, or -2, the average or expectation value of X is given by
E(X) = Sum_i [ X_i p(X_i) ] = +2 p(X = +2) – 2 p(X = -2).
The remaining point of disagreement is then the following:
Richard Gill claims that the three probabilities, p(X = +2), p(X = 0) and p(X = -2), are non-negative and add up to +1.
I, on the other hand, claim that the three probabilities, p(X = +2), p(X = 0) and p(X = -2), are all identically zero:
p(X = +2) = 0, p(X = 0) = 0, and p(X = -2) = 0.So, identically zero, or not identically zero, that is the question: Yes / No.
If what I claim is true, then the conclusion of my initial post in this thread holds, and Bell's theorem is finally put to rest.
Richard Gill wrote:
There are just three possible events and just three probabilities associated with the random variable X = A(a)*[B(b)+B(b’)].
According to local hidden variables, A(a)*[B(b)+B(b’)] is just some function of the hidden variable lambda. We don’t get to observe lambda and we don’t get to observe X. But if local hidden variables are true, then lambda and X(lambda) do exist. And X can take the values -2, 0 and 2. The set of all possible values of lambda can therefore be split into three disjoint subsets: the subset of all lambda where X(lambda) = -2, the subset where X(lambda) = 0, and the subset where X(lambda) = +2.
Those three subsets can be called “events”. The event where X = 2, etc. Which of the three events actually happens in any single trial is not observed by Alice and Bob. But when nature does pick a value of lambda, it will fall in just one of these three subsets, and it does that with probabilities which I’ll denote p(+2), p(0) and p(-2) respectively.
Those three probabilities are equal to the integrals of rho(lambda) d lambda over each of the three subsets. The three probabilities are nonnegative and add up to +1.
Finally, E(X) = 2 p(2) – 2 p(-2).
Christian seems to confuse “event” and “random variable” and still does not have the good formula for “expectation value”.
Joy Christian wrote:
I disagree with Gill on several counts. In a local hidden variable theory lambdas of course do exist by definition, but the function X(lambda) = A(a, lambda)*[B(b, lambda)+B(b’, lambda)] does not have any physical meaning. Nor can X take the values -2, 0 and +2 in a physically meaningful sense.
There are only two particles at disposal to Alice and Bob for each run of the experiment, not three. Such pairs of particles can be identify with the hidden variable lambda^k, or just with the index k as I have done in my Appendix D. Now X(lambda) defined by Gill is a function of three possible detection events in spacetime, or of three clicks recorded by the two detectors detecting the two particles at disposal to Alice and Bob, for each lambda (or k). These clicks of the two detectors are recorded as the numbers A(a), B(b) and B(b’), where B(b) and B(b’) are only counterfactually possible numbers. Therefore there is no physical sense in which X can take the values -2, 0 and +2. Another way of saying the same thing is that the probabilities of the function X(lambda) taking the values -2, 0, and +2 are identically zero: p(+2) = 0, p(0) = 0 and p(-2) = 0, unless of course Alice and Bob belong to a world in which the probability of finding a single dice landing on both 3 and 6 at the same time is non-vanishing. But since all three of these probabilities must be identically zero, the average or expectation value of X is also identically zero. Consequently the CHSH correlator must lie between -0 and +0.
I should stress that I have no problem with what Gill has written if X(lambda) is defined by involving only two detection events, such as x(lambda) = A(a, lambda)*B(b, lambda) or x(lambda) = A(a, lambda)*B(b’, lambda). These two definitions have perfectly reasonable physical meanings. But an expression like X(lambda) = A(a, lambda)*[B(b, lambda)+B(b’, lambda)] is physically self-contradictory even for any local hidden variable theory. It is not demanded by either Einstein’s or Bell’s conception of local realism.
Richard Gill wrote:
Nobody is saying that X has some physical meaning. But if we believe in local hidden variables, then lambda exists, the function X exists, and X(lambda) exists. Once nature has chosen lambda, X(lambda) is determined. Even if Alice and Bob have gone home and switched off their detectors.
Joy Christian wrote:
I am sure X(lambda) can be assumed to exist as a function. But the probabilities of it taking the values -2, 0, and +2 are identically zero: p(+2) = 0, p(0) = 0 and p(-2) = 0. Unless of course Alice and Bob belong to a world in which the probability of finding a tossed coin on heads and tails at the same time is non-vanishing.
Richard Gill wrote:
A(a, lambda), B(b, lambda) and B(b’, lambda) all exist, and are all equal to +/-1, whatever Alice and Bob actually measure. Clearly X(lambda) can only be -2, 0, or +2.
Joy Christian wrote:
I agree entirely: “A(a, lambda), B(b, lambda) and B(b’, lambda) all exist [at least counterfactually], and are all equal to +/-1 [by construction], whatever Alice and Bob actually measure. Clearly X(lambda) can only be -2, 0, or +2.”
But that is not going to rescue Bell’s theorem. Because the probabilities p(-2), p(0) and p(+2) of X(lambda) taking the possible values -2, 0, and +2 are all identically zero: p(+2) = 0, p(0) = 0 and p(-2) = 0. Unless of course the probability of my being in New York and Miami at exactly the same time can be non-vanishing in the local-realistic world we live in.
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