International Journal of Theoretical Physics

, Volume 50, Issue 12, pp 3635–3645 | Cite as

Quantum Axiomatics: Topological and Classical Properties of State Property Systems



The definition of ‘classical state’ from (Aerts in K. Engesser, D. Gabbay and D. Lehmann (Eds.), Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009), used e.g. in Aerts et al. (, 2010) to prove a decomposition theorem internally in the language of State Property Systems, presupposes as an additional datum an orthocomplementation on the property lattice of a physical system. In this paper we argue on the basis of the (ε,d)-model on the Poincaré sphere that a notion of topologicity for states can be seen as an alternative (operationally foundable) classicality notion in the absence of an orthocomplementation, and compare it to the known and operationally founded concept of classicality.


State property system Orthocomplementation Property lattice Closure space Classical property 


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  1. 1.
    Aerts, D.: Classical theories and non-classical theories as a special case of a more general theory. J. Math. Phys. 24, 2441–2453 (1983) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Aerts, D.: A possible explanation for the probabilities of quantum mechanics. J. Math. Phys. 27, 202–210 (1986) MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Aerts, D.: Quantum axiomatics. In: Engesser, K., Gabbay, D., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam (2009) Google Scholar
  4. 4.
    Aerts, D., Aerts, S.: The hidden measurement formalism: quantum mechanics as a consequence of fluctuations on the measurement. In: Ferrero, M., van der Merwe, A. (eds.) New Developments on Fundamental Problems in Quantum Physics, pp. 1–6. Kluwer Academic, Dordrecht (1997) CrossRefGoogle Scholar
  5. 5.
    Aerts, D., Coecke, B., Durt, T., Valckenborgh, F.: Quantum, classical and intermediate II. The vanishing vectorspace structure. Tatra Mt. Math. Publ. 10, 241–266 (1997) MathSciNetMATHGoogle Scholar
  6. 6.
    Aerts, D., Colebunders, E., Van der Voorde, A., Van Steirteghem, B.: State property systems and closure spaces: a study of categorical equivalence. Int. J. Theor. Phys. 38, 359–385 (1999) MATHCrossRefGoogle Scholar
  7. 7.
    Aerts, D., Deses, D., D’Hooghe, B.: (2010). A decomposition theorem for state property systems.
  8. 8.
    Aerts, D., Durt, T.: Quantum, classical and intermediate, an illustrative example. Found. Phys. 24, 1353–1369 (1994) MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Aerts, D., Pulmannová, S.: Representation of state property systems. J. Math. Phys. 47, 1–18 (2006) CrossRefGoogle Scholar
  10. 10.
    Beltrametti, E.G., Cassinelli, G.: The Logic of Quantum Mechanics. Addison-Wesley, Reading (1981) MATHGoogle Scholar
  11. 11.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823–843 (1936) CrossRefGoogle Scholar
  12. 12.
    Birkhoff, G.: Lattice Theory. AMS, Providence (1967) MATHGoogle Scholar
  13. 13.
    Czachor, M.: On classical models of Spin. Found. Phys. Lett. 5(3), 249–264 (1992) MathSciNetCrossRefGoogle Scholar
  14. 14.
    D’Hooghe, B.: On the orthocomplementation of state-property-systems of contextual systems. Int. J. Theor. Phys. 49(12), 3069–3084 (2010) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Dorfer, G., Dvurečenskij, A., Länger, H.: Symmetric difference in orthomodular lattices. Math. Slovaca 46, 435–444 (1996) MathSciNetMATHGoogle Scholar
  16. 16.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic, Dordrecht (2000) MATHGoogle Scholar
  17. 17.
    Foulis, D.J., Randall, C.H.: Operational statistics I. Basic concepts. J. Math. Phys. 13, 1667–1675 (1974) MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Garola, C., Pykacz, J., Sozzo, S.: Quantum machine and semantic realism approach: a unified model. Found. Phys. 36(6), 862–882 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Greechie, R.J.: On the structure of orthomodular lattices satisfying the chain condition. J. Comb. Theory 4, 210–218 (1968) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Harding, J.: A link between quantum logic and categorical quantum mechanics. Int. J. Theor. Phys. 3, 769–802 (2009) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading (1968) MATHGoogle Scholar
  22. 22.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983) MATHGoogle Scholar
  23. 23.
    Mackey, G.W.: The Mathematical Foundations of Quantum Mechanics. Benjamin, New York (1963) MATHGoogle Scholar
  24. 24.
    Piron, C.: Axiomatique quantique. Helv. Phys. Acta 37, 439–468 (1964) MathSciNetMATHGoogle Scholar
  25. 25.
    Piron, C.: Foundations of quantum physics. Benjamin, Reading (1976) MATHGoogle Scholar
  26. 26.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic, Dordrecht (1991) MATHGoogle Scholar
  27. 27.
    Solèr, M.P.: Characterization of Hilbert spaces by orthomodular spaces. Commun. Algebra 23, 219–243 (1995) MATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centre Leo Apostel and Department of MathematicsVrije Universiteit BrusselBrusselsNetherlands
  2. 2.Department of MathematicsVrije Universiteit BrusselBrusselsNetherlands

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