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A Generalisation of Stone Duality to Orthomodular Lattices

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Reality and Measurement in Algebraic Quantum Theory (NWW 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 261))

Abstract

With each orthomodular lattice L we associate a spectral presheaf \(\underline{\varSigma }^{L}\), generalising the Stone space of a Boolean algebra, and show that (a) the assignment \(L\mapsto \underline{\varSigma }^{L}\) is contravariantly functorial, (b) \(\underline{\varSigma }^{L}\) is a complete invariant of L, and (c) for complete orthomodular lattices there is a generalisation of Stone representation in the sense that L is mapped into the clopen subobjects of the spectral presheaf \(\underline{\varSigma }^{L}\). The clopen subobjects form a complete bi-Heyting algebra, and by taking suitable equivalence classes of clopen subobjects, one can regain a complete orthomodular lattice isomorphic to L. We interpret our results in the light of quantum logic and in the light of the topos approach to quantum theory.

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Notes

  1. 1.

    In quantum theory, for most states a given proposition “\(A\,\varepsilon \,\varDelta \)” is neither true nor false, but can only be assigned a probability, which is usually interpreted as the probability that when measuring the physical quantity A in the given state, a measurement outcome in \(\varDelta \) is obtained.

  2. 2.

    For those not familiar with this example: if you go to the quantum hotel and they offer you eggs and (bacon or sausage), you cannot expect to get (eggs and bacon) or (eggs and sausage) due to nondistributivity of ‘and’ and ‘or’. As a formula, \(e\wedge (b\vee s)\ne (e\wedge s)\vee (e\wedge s)\) in general in an orthomodular lattice.

  3. 3.

    On this topic, there is some ongoing work with Masanao Ozawa and Benjamin Eva.

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Acknowledgements

We are very grateful to Masanao Ozawa for giving us the opportunity to contribute an article (without page limit!) to the proceedings of the Nagoya Winter Workshop 2015. Moreover, AD thanks Masanao for his continued generosity and friendship over the years and for many discussions. We also thank Chris Isham, Rui Soares Barbosa, Carmen Constantin, Boris Zilber, and Benjamin Eva for discussions.

Author information

Authors and Affiliations

  1. College of Computing, Georgia Institute of Technology, North Avenue, Atlanta, GA, 30332, USA

    Sarah Cannon

  2. Independent researcher, Frankfurt, Germany

    Andreas Döring

Authors
  1. Sarah Cannon
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  2. Andreas Döring
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Corresponding author

Correspondence to Andreas Döring .

Editor information

Editors and Affiliations

  1. Graduate School of Informatics, Nagoya University, Nagoya, Japan

    Masanao Ozawa

  2. Trinity College, Cambridge University, Cambridge, UK

    Jeremy Butterfield

  3. Department of Philosophy, Princeton University, Princeton, NJ, USA

    Hans Halvorson

  4. Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science, London, UK

    Miklós Rédei

  5. College of Industrial Technology, Nihon University, Narashino, Japan

    Yuichiro Kitajima

  6. Graduate School of Informatics, Nagoya University, Nagoya, Japan

    Francesco Buscemi

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Cannon, S., Döring, A. (2018). A Generalisation of Stone Duality to Orthomodular Lattices. In: Ozawa, M., Butterfield, J., Halvorson, H., Rédei, M., Kitajima, Y., Buscemi, F. (eds) Reality and Measurement in Algebraic Quantum Theory. NWW 2015. Springer Proceedings in Mathematics & Statistics, vol 261. Springer, Singapore. https://doi.org/10.1007/978-981-13-2487-1_1

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