Foundations of Physics

, Volume 19, Issue 11, pp 1299–1314 | Cite as

Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property

  • Christian Schindler
Article

Abstract

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections Δ of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of Δ. A section Δ is shown to have the JHDP if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results.

Keywords

Probability Measure Quantum Theory Physical Interpretation Geometrical Interpretation Measure Theory 

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Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Christian Schindler
    • 1
  1. 1.Institute of Mathematical StatisticsUniversity of BerneBerneSwitzerland

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