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Observables in the Logico-Algebraic Approach

  • Pavel Pták
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 111)

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.

Keywords

Boolean Algebra Quantum Logic Separable Banach Space Orthomodular Lattice Bounded Observable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Pavel Pták
    • 1
  1. 1.Department of MathematicsCzech Technical UniversityPrague 6Czech Republic

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