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Bell's theorem - for or against Hidden Variables?
(06-05-2016, 09:26 PM)Schmelzer Wrote: Depends on what you consider.  If you consider a local-realistic theory which fulfills the conditions of the theorem, and talk about the sum
\[ S=E(a,b)-E(a,b^{\prime })+E(a^{\prime },b)+E(a^{\prime },b^{\prime }),\]
then S=4 is not possible. If you talk about the sum +1 -(-1) + 1 + 1, then it is not only possible, but holds always.  If you talk not about computing the sum for the same \(\lambda\), but for four different particular values, without any averaging, then it is possible.  If you average enough so that you have
\[E(a,b) = \int A(a,\lambda)B(b,\lambda) \rho(\lambda) d\lambda\]
with sufficient accuracy and the same functions \(A, B, \rho\) for all four expressions, then not.

So, a typical trick question, you do not give enough specification for a unique answer and hope nonetheless for an answer.

Ok, you say it "depends" so no real answer to the question.  But I did perfectly specify what I meant.  All that I specified was the CHSH string of expectation terms that can range from -1 to +1.  Nothing else was specified.  Now, it is pretty easy to see from what I did specify that the answer has to be yes that a result of +1 - (1) + 1 +1 = 4 is possible.  What this boils down to is that if the CHSH string of terms are independent from each other, then we can have an inequality,
\[ E(a,b)-E(a,b^{\prime })+E(a^{\prime },b)+E(a^{\prime },b^{\prime }) \leq 4.\]
And that is the inequality that both QM and all experiments to date have used and have never violated.  All one has to do is to look and see that all experiments and QM have always used this inequality with independent expectation terms.  I have never seen a counter example to this simple demonstration by mathematical inspection.  If what Bell says is right, then there ought to be one.  And note that local realistic theories don't even need to enter into this demonstration.  Just show how QM or the experiments have ever violated one of Bell's inequalities.  You can't; it is impossible.

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RE: Bell's theorem - for or against Hidden Variables? - by FrediFizzx - 06-05-2016, 10:54 PM

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