Bell's theorem supposedly demonstrates an irreconcilable conflict between quantum mechanics and local realistic hidden-variable theories. Most proofs of Bell's theorem, are based on inequalities. In this paper we present an alternative proof which does not involve inequalities, but only a direct comparison between correlation functions calculated using quantum mechanics on the one hand, and those calculated according to local realistic hidden-variable theories on the other. Our proof is based on a physically motivated use of Fourier series for periodic functions, and confirms that local realistic hidden-variable theories are incompatible with quantum mechanics.
This article takes up a suggestion that the reason we cannot find certain hidden variable theories for quantum mechanics, as in Bell’s theorem, is that we require them to assign joint probability distributions on incompatible observables. These joint distributions are problematic because they are empirically meaningless on one standard interpretation of quantum mechanics. Some have proposed getting around this problem by using generalized probability spaces. I present a theorem to show a sense in which generalized probability spaces can’t serve as hidden variable theories for quantum mechanics, so the proposal for getting around Bell’s theorem fails. 1 Introduction2 Bell’s Theorem and Classical Probability Spaces 2.1 Bell’s derivation of the Bell inequalities 2.2 Pitowsky’s derivation of the Bell inequalities3 Incompatible Observables4 Generalized Probability Spaces5 A ‘No-Go’ Theorem6 Conclusions