The Statistics of Non-Boolean Event Structures

Abstract

Quantum mechanics incorporates an algorithm for assigning probabilities to ranges of values of the physical magnitudes of a mechanical system: \( Pw(a \in S) = Tr(WPa(S)) \) where W represents a statistical state, and P _A (S) is the projection operator onto the subspace in the Hilbert space of the system associated with the range S of the magnitude A. (I denote values of A by a.) The statistical states (represented by the statistical operators in Hilbert space) generate all possible (generalized) probability measures on the partial Boolean algebra of subspaces of the Hilbert space.¹ Joint probabilities \( Pw(a_1 \in S_1 \& a_2 \in S_2 \& \ldots \& a_n \in S_n ) = \)\( {\text{ = Tr(WP}}_{A_1 } {\text{(S}}_1 {\text{)P}}_{A_2 } {\text{(S}}_2 {\text{)}} \ldots {\text{P}}_{A_n } {\text{(S}}_n {\text{))}} \) are defined only for compatible² magnitudes A₁, A₂,… A _n , and there are no dispersion-free statistical states.