[This is the abstract of A Critical Re-examination of Bell’s Theorem. The abstracts of the other two articles are short. Too see them, please open the respective PDFs.]
Bell’s Theorem (Bell, 1964; hereafter, B64) is widely considered to be a definitive proof of “non-locality,” or, as Einstein called it, “spooky action at a distance.” In the present article it is shown that the derivation of Bell’s Inequality in B64 contains mathematical errors, and that therefore Bell’s Theorem is not a proof of non-locality. In this respect the present article is a critique of Bell’s Theorem principally on grounds of internal consistency; it does not attempt directly to resolve the foundational problem of non-locality. Moreover, it is argued here that even if Bell’s Theorem did not contain mathematical errors, it would still not be a definitive proof of non-locality because it addresses only deterministic hidden variables, a category which is not comprehensive of all potentially reasonable hidden-variable theories.
Bell’s Theorem is an analysis of Bohm’s (1951a) thought experiment, which concerns two spin-½ particles in the singlet state that become separated and move in opposite directions. The nominal program of B64 is as follows. To the assumptions of the quantum theory of the Bohm thought experiment, B64 adds additional assumptions concerning hypothetical local hidden variables. Using these additional assumptions, B64 derives an inequality (Bell’s Inequality) describing the predicted results of the experiment. The inequality is then compared to the predictions of the quantum theory. The derived inequality fails to agree with the quantum-theoretical prediction. The inequality also fails to agree with the results of experimental realizations of the Bohm thought experiment, which, with one accidental exception, occurred later than the publication of B64. The failure of the inequality to agree with the predictions of the quantum theory and with the results of experimental realizations − which agree with each other − is taken to refute the additional assumptions of B64 concerning local hidden variables, and therefore to constitute a proof of non-locality (spooky action).
The argument of Bell’s Theorem, however, fails in at least three independent respects.
(1) The derivation of Bell’s Inequality in B64 depends critically on an inadmissible algebraic substitution.
(2) The derivation of Bell’s Inequality in B64, even if it were algebraically valid, would exclude only hypothetical non-local hidden variables, not local ones. The apparent applicability of Bell’s Inequality to the case of local hidden variables is the result of confusion caused by a poor choice of notation.
(3) The schema of B64 addresses only the case of hypothetical deterministic hidden variables. Determinism (though not by that name) is one of the demands of Einstein, Podolsky and Rosen (1935). B64 does not address the case of hypothetical hidden variables that influence the local probability distributions for the outcomes of observations but do not uniquely determine the outcomes. Thus, even if it were otherwise valid, Bell’s Theorem would not constitute an airtight proof of non-locality simply because it does not cover this important class of cases.
The failure of Bell’s Inequality to agree with experimental realizations of the Bohm experiment has been taken as incontrovertible proof of the reality of spooky action. However, theoretical predictions may fail to agree with the results of experiment for reasons that do not signify anything about the truth or falsehood of their assumptions. In particular, they may fail because of errors in the logic that carries their assumptions into their predictions. This is the case with Bell’s Inequality.
There might be good reasons to accept non-locality, a.k.a. spooky action. However, if there are, Bell’s Theorem is not one of them.