Quantum mechanics and classical probability theory

TLDR: In this article, it was shown that the Copenhagen interpretation of the quantum mechanics cannot be interpreted in a physically acceptable way which is consistent with classical probability theory, which is a somewhat more general proof of this has been given by Nelson ([11], p. 117).

Abstract: The mathematical formalism of quantum mechanics, via its customary interpretation, leads to a density function for each observable. Using this formalism, there is also a natural way to calculate joint density functions for two observables. However, it has been known for some time that joint density functions, obtained in this way, may be incompatible with classical probability theory [1], [10], [15]. More recently, Cohen [4] has shown that there are density functions for non-commuting observables and functions of these observables, given by the quantum mechanical formalism, which have the following property. There is no joint density function (compatible with classical probability theory) from which all these density functions may be derived as marginal densities. A somewhat more general proof of this has been given by Nelson ([11], p. 117). These results are not too surprising when one realizes that the numbers which the Copenhagen Interpretation of the quantum mechanical formalism identifies as 'probabilities' actually have the mathematical structure of probability-like measures whose domain is an ortho-complemented lattice. The ortho-complemented lattice in this case is the partially ordered set of all closed sub-spaces of a separable, infinite dimensional, complex Hilbert space [7], [8], [2]. These probability-like measures on an ortho-complemented lattice have many similarities to ordinary, classical probability measures defined on a Boolean algebra. But there are some dissimilarities which, as Varadarajan [17] has shown, make it impossible to define something like a classical joint density function for these probability-like measures. These considerations have led many people, among them yon Neumann (Birkhoff and yon Neumann [2]), Mackey [7], [8], Suppes [15], [16], and Varadarajan [17] to speculate that the formalism of quantum mechanics cannot be interpreted in a physically acceptable way which is consistent with classical probability theory. This is to say that the customary