Isn’t Bell’s probability density for hidden variables too restrictive?

[nekkert llup]

J.S. Bell (Physics Vol.1, No. 3, 1964) excludes from consideration any distribution $$\rho$$ of the hidden variable $$\lambda$$ that formally depends on the vectors $$a$$ and $$b$$, except if $$\rho ( \lambda ,a,b) = \rho ' ( \lambda ,a) \rho ' ( \lambda ,b)$$ i.e. if the distribution can be factored in a part depending on $$a$$ and not on $$b$$ and another part depending on $$b$$ and not on $$a$$. Otherwise he could not derive (22). On precisely which grounds did Bell introduce this restriction? For example, does the locality requirement lead to this restriction? How?Are other principles involved?

Edit: I made an unforgivable error: According to Bell, neither $$\lambda$$ itself nor its density distribution $$\rho ( \lambda )$$ may depend on $$a$$ and $$b$$. The question is still the same: why not?

 

[Nugatory]

Bell makes this no-dependency assumption because he's deriving a result about theories that have this property. It's analogous to drawing conclusions about the properties of rational numbers starting from the assumption that a rational number can be written as the ratio of two integers; the assumption is "too restrictive" in the sense that there are numbers that can't be written in that form, but that doesn't mean that the assumption is not justified, it means that the conclusion only applies to numbers that can be written in that form.

If a theory contains no such dependency on $a$ and $b$, then it must obey his inequality. That's Bell's Theorem.

What makes this an interesting result is the contrapositive: if a theory predicts violations of the inequality, then it must include such a dependency. Quantum mechanics makes such a prediction, therefore any theory that agrees with QM must include that dependency.

 

[Demystifier]

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