Observables in the Logico-Algebraic Approach

Abstract

The notion of observable is a quantum mechanical variant of that of a random variable (see [32], [69], [20], [29], etc.). It is supposed to help in treating noncompatible events. In the logico-algebraic approach an observable is defined as a σ-additive measure with values in a “quantum logic” (= with values in an abstract σ-complete orthomodular poset). Though an individual observable ranges in a Boolean σ-algebra, non-classical phenomena occur when one develops “noncomutative” probability theory. First, the range Boolean σ-algebra does not have to be set-representable (even in the fundamental case of the Hilbert space logic!). Second, when one treats collections of observables, the respective range Boolean σ-algebras vary and often do not allow for an umbrella Boolean σ-algebra. In this article we want to survey a few lines of recent research on observables. We also want to formulate — in the main text and in the notes at the end (referred to as ) — some open questions which seem related to quantum axiomatics and noncommutative probability theory. An extensive bibliography of papers dealing with observables is also provided.