Quantum Covers in Quantum Measure Theory

Sumati Surya, Petros Wallden

Research output: Contribution to journalArticlepeer-review

Abstract

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov sum rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers.
Original languageEnglish
Pages (from-to)585-606
Number of pages22
JournalFoundations of Physics
Volume40
Issue number6
DOIs
Publication statusPublished - Jun 2010

Keywords

  • Quantum interpretation
  • Quantum measure theory
  • Quantum topology
  • Lattice theory

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