gill1109 wrote:
Michel,
There are many ways to derive a mathematical formula.
That's your problem. The same mathematical formulas can represent completely different things in physics. One can represent a strongly objective correlation on the same pair of particles as the others, while another similar-looking formula can represent weakly objective correlations on disjoint sets of particle pairs. The formulas may look the same but the *values* may be completely different. Just because a mathematical formula shows up in an equation does not mean it *represents* the same thing as a similar-looking formula from somewhere else.
Bell first of all writes down some physical hypotheses which tell him about existence of functions A, B and rho and lead him to the formula P(a, b) = integral….. Observe that B must be the negative of A because P(a, a) = -1 for all a. Now pick any a, b, c and write down formulas for P(a,b), P(a,c), and P(b,c). The same formula three times with different arguments filled in. Now do some simple algebra and calculus. Get his original three correlations inequality. That little derivation is a piece of mathematical formula manipulation. Its validity does not depend on any physical interpretation of any of the intermediate expressions.
All true but completely irrelevant. The validity of Bell's inequality is uncontested. What is contested is the physical interpretation of its meaning and application.
Now notice that the inequality you got is violated by QM predictions. Notice that the QM predictions are confirmed in experiments. Conclusion: those physical hypotheses are invalid.
Wrong! For two of the terms, QM gives you a prediction. For the third term in the inequality, you have used the wrong QM prediction to claim violation. You are making an error in your application of your albeit valid inequality to the physical situation described by QM and observed in experiments. You are doing a bait and switch.
The three expressions P(a,b), P(a,c), and P(b,c) are not *mathematically independent* because they are built up out of some common ingredients: a probability distribution over a set containing all values of some unobserved physical variable lambda, which does not depend on the settings, and a measurement function.
Again, this is a subterfuge. P(a,b) and P(a,c) being weakly objective (as Bell claims) are independent in ways in which P(b,c) is not relative to P(a,b) and P(a,c). This much is very obvious. Bell starts with P(a,b) and P(b,c) and from those two he derives P(b,c). P(b,c) in the inequality is more restricted than P(a,b) and P(a,c) due to the cyclic nature of the terms. This is the core ingredient of all variants of Bell's theorem. They include a hidden reduction of independence that is simply ignored when comparing the results with experimental data where no such reduction in independence is present.
Why must the terms be cyclical? Why has nobody been able to "prove" Bell's theorem without relying on a cyclicity? What manner of "non-realism" or "non-locality" is this that only shows up when terms are cyclically related? I would have expected more curiosity from you on such topics.
Computer simulations which violate the inequality do so by violating the assumptions. They could in principle correspond to a physically reasonable explanation of the results of some experiments.
Oh so now you agree with me that it is very important to know "What" statistics converge to 2? You admit that there are local realistic statistics that converge to 2 and non-local statistics that do not converge to 2?
Justo is right. Moreover Bell is very clear what he is doing. The objections people raise today were also raised in the early years, and Bell answered his critics, in my opinion quite adequately,
He did not. Here is what he did. Critics argued that he was using a single 3xN spreadsheet instead of 3 disjoint 2xN spreadsheets. He replied that he was using 3 disjoint 2xN spreadsheets. But based on his equations, that is false. He is using 2 2xN spreadsheets and from each of them he separates one column, recombines them to generate a third 2xN spreadsheet that is not independent of the other two in the same way the first two are independent from each other. While it appears to diffuse the criticism, it does not eliminate it. Perhaps he did not understand the root of the problem.