The Statistics of Non-Boolean Event Structures

Abstract

The Kochen and Specker theory of partial Boolean algebras leads to the resolution of the core problem of interpretation of quantum mechanics, the problem of hidden variables. To recapitulate: Quantum mechanics incorporates an algorithm for assigning probabilities to ranges of values of the physical magnitudes: $${p_W}\left( {a \in S} \right) = Tr\left( {W{P_A}\left( S \right)} \right)$$ where W represents a statistical state of the theory, and P A (S) is the projection operator onto the subpsace in Hilbert space associated with the range S of the magnitude A. The statistical states generate all possible (generalized) probability measures on the partial Boolean algebra of subspaces of Hilbert space. Joint probabilities $$\begin{gathered} {p_W}\left( {{a_1} \in {S_1}\ {a_2} \in {S_2}\ \ldots \ {a_n} \in {S_n}} \right) = \hfill \\ = Tr\left( {W{P_{{A_1}}}\left( {{S_1}} \right){P_{{A_2}}}\left( {{S_2}} \right) \ldots {P_{{A_n}}}\left( {{S_n}} \right)} \right) \hfill \\ \end{gathered} $$ are defined only for compatible magnitudes A1, A2,…, A n , and there are no dispersion-free statistical states. The problem of hidden variables concerns the possibility of representing the statistical states of quantum mechanics by measures on a classical probability space in such a way that the algebraic structure of the magnitudes of the theory is preserved. This is the problem of imbedding the partial algebra of magnitudes into a commutative algebra or, equivalently, the problem of imbedding the partial Boolean algebra of idempotent magnitudes (properties, propositions) into a Boolean algebra.