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

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.

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Schindler, C. Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property. Found Phys 19, 1299–1314 (1989). https://doi.org/10.1007/BF00732752

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Keywords

  • Probability Measure
  • Quantum Theory
  • Physical Interpretation
  • Geometrical Interpretation
  • Measure Theory