Communications in Mathematical Physics

, Volume 291, Issue 1, pp 63–110 | Cite as

A Topos for Algebraic Quantum Theory

  • Chris Heunen
  • Nicolaas P. LandsmanEmail author
  • Bas Spitters
Open Access


The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos \({\mathcal{T}(A)}\) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra \({\underline{A}}\) . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum \({\underline{\Sigma}(\underline{A})}\) in \({\mathcal{T}(A)}\) , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on \({\underline{\Sigma}}\) , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from \({\underline{\Sigma}}\) to Scott’s interval domain. Noting that open subsets of \({\underline{\Sigma}(\underline{A})}\) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos \({\mathcal{T}(A)}\).

These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.


Topo Distributive Lattice Quantum Logic Riesz Space Heyting Algebra 
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.



The authors are indebted to Andreas Döring, Ieke Moerdijk, Chris Mulvey, Isar Stubbe, and Steve Vickers for guidance and useful feedback on talks and earlier drafts of this article. We are exceptionally grateful to the referee of this paper for unusually detailed and helpful comments.

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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© The Author(s) 2009

Authors and Affiliations

  • Chris Heunen
    • 1
  • Nicolaas P. Landsman
    • 1
    Email author
  • Bas Spitters
    • 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

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