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Synthese

, Volume 186, Issue 3, pp 719–752 | Cite as

Bohrification of operator algebras and quantum logic

  • Chris Heunen
  • Nicolaas P. Landsman
  • Bas Spitters
Open Access
Article

Abstract

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set \({\mathcal{C}(A)}\) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that \({\mathcal{C}(A)}\) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” \({\underline A}\) of A, which is a commutative Rickart C*-algebra in the topos of functors from \({\mathcal{C}A}\) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of \({\underline A}\) for A = B(H).

Keywords

Quantum logic Intuitionistic logic C*-algebras Locales Bohrification 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Chris Heunen
    • 1
  • Nicolaas P. Landsman
    • 1
  • Bas Spitters
    • 3
    • 2
  1. 1.Institute for Mathematics, Astrophysics, and Particle PhysicsRadboud Universiteit NijmegenNijmegenThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.Institute for Computing and Information SciencesRadboud Universiteit NijmegenNijmegenThe Netherlands

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