On classical models of spin

Abstract

The reason for recalling this old paper is the ongoing discussion on the attempts of circumventing certain assumptions leading to the Bell theorem (Hess-Philipp, Accardi). If I correctly understand the intentions of these Authors, the idea is to make use of the following logical loophole inherent in the proof of the Bell theorem: Probabilities of counterfactual events A and A' do not have to coincide with actually measured probabilities if measurements of A and A' disturb each other, or for any other fundamental reason cannot be performed simulaneously. It is generally believed that in the context of classical probability theory (i.e. realistic hidden variables) probabilities of counterfactual events can be identified with those of actually measured events. In the paper I give an explicit counterexample to this belief. The "first variation" on the Aerts model shows that counterfactual and actual problems formulated for the same classical system may be unrelated. In the model the first probability does not violate any classical inequality whereas the second does. Pecularity of the Bell inequality is that on the basis of an in principle unobservable probability one derives probabilities of jointly measurable random variables, the fact additionally obscuring the logical meaning of the construction. The existence of the loophole does not change the fact that I was not able to construct a local model violating the inequality with all the other loopholes eliminated.