Foundations of Physics

, Volume 44, Issue 12, pp 1357–1368 | Cite as

Physical Properties as Modal Operators in the Topos Approach to Quantum Mechanics

Article

Abstract

In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.

Keywords

Intuitionistic quantum logic Quantum phase spaces Modal operators 

References

  1. 1.
    Aerts, D., Daubechies, I.: Mathematical condition for a sub-lattice of a propositional system to represent a physical subsystem with a physical interpretation. Lett. Math. Phys. 3, 19–27 (1979)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aerts, D.: Description of compound physical systems and logical interaction of physical systems. In: Beltrameti, E., van Fraassen, B. (eds.) Current Issues in Quantum Logic, pp. 381–405. Plenum, New York (1981)CrossRefGoogle Scholar
  3. 3.
    Aerts, D.: Construction of a structure which enables to describe the join system of a classical and a quantum system. Rep. Math. Phys 20, 421–428 (1984)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Aerts, D.: Construction of the tensor product of lattices of properties of physical entities. J. Math. Phys. 25, 1434–1441 (1984)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Abramsky, S., Coecke, B.: Categorical quantum mechanics. Handbook of Quantum Logic and Quantum Structures, vol. II. Elsevier, Amsterdam (2008)Google Scholar
  6. 6.
    Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)CrossRefGoogle Scholar
  8. 8.
    Balbes, R., Dwinger, Ph: Distributive Lattices. University of Missouri Press, Columbia (1974)MATHGoogle Scholar
  9. 9.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Text in Mathematics, vol. 78. Springer, New York (1981)MATHGoogle Scholar
  10. 10.
    Coecke, B., C. Heunen, C., Kissinger, A.: Compositional quantum logic. arXiv:1302.4900
  11. 11.
    da Costa, N., de Ronde, C.: The paraconsisten logic of quantum superpositions. Found. Phys. 43, 845–858 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Dalla Chiara, M.L., Giuntini, R., Greechhie, R.: Reasoning in Quantum Theory. Kluwer, Dordrecht (2004)Google Scholar
  13. 13.
    de Ronde, C., Freytes, H., Domenech, G.: Interpreting the modal Kochen-Specker Theorem: possibility and many worlds in quantum mechanics. Stud. Hist. Phil. Mod. Phys. 45, 11–18 (2014)Google Scholar
  14. 14.
    Domenech, G., Freytes, H.: Contextual logic for quantum systems. J. Math. Phys. 46, 012102 (2005)ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Domenech, G., Freytes, H., de Ronde, C.: Modal type othomodular logic. Math. Logic Q. 55, 287–299 (2009)CrossRefGoogle Scholar
  16. 16.
    Domenech, G., Holik, F., Massri, C.: A quantum logical and geometrical approach to the study of improper mixtures. J. Math. Phys. 51, 052108 (2010)ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    Döring, A., Isham, C.J.: A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory. J. Math. Phys. 49, 053516 (2008)ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    Döring, A., Isham, C.J.: What is a thing? Topos theory in the foundations of physics. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes for Physics, vol. 813. Springer, Berlin, 2010, 753–937.Google Scholar
  19. 19.
    Dvurečenskij, A.: Tensor product of difference posets and effect algebras. Int. J. Theor. Phys. 34, 1337–1348 (1995)CrossRefMATHGoogle Scholar
  20. 20.
    Freyd, P.J.: Aspects of Topoi. Bull. Austral. Math. Soc. 7, 1–76 (1972)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Gelfand, I.M., Naimark, M.A.: On the imbedding of normed rings into the ring of operators in Hilbert space. Mat. Sbornik 12, 197–213 (1943)Google Scholar
  22. 22.
    Gudder, S.P.: Some unresolved problems in quantum logic. In: Marlow, A.R. (ed.) Mathematical Foundations of Quantum Theory. Academic, New York (1978)Google Scholar
  23. 23.
    Heunen, C., Landsman, N., Spitters, B., Wolters, S.: The Gelfand spectrum of a noncommutative C\(^{*}\)-algebra: a topos theoretic approach. J. Aust. Math. Soc. 90, 39–52 (2011)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Heunen, C., Landsman, N., Spitters, B.: A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Heyting, A.: Die formalen Regeln der intuitionistischen Logik. Die Preussische Akademie der Wissenschaften. Sitzungsberichte. Physikalische-Mathematische Klasse, pp. 42–56 (1930)Google Scholar
  26. 26.
    Heyting, A.: Die formalen Regeln der intuitionistischen Mathematik II, III. Die Preussische Akademie der Wissenschaften. Sitzungsberichte. Physikalische-Mathematische Klasse, pp. 57–71, 158–169 (1930)Google Scholar
  27. 27.
    Isham, C.: Topos methods in the foundations of physics. In: Halvorson, H. (ed.) Deep Beauty: Understanding the Quantum World Through Mathematical Innovation, pp. 187–206. Cambridge University Press, Cambridge (2010)Google Scholar
  28. 28.
    Johnstone, P.T.: Stone spaces, Cambridge Studies. Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1982)Google Scholar
  29. 29.
    Lawvere, F.W.: Quantifiers and sheaves. Actes Congres Intern. Math. 1, 329–334 (1970)Google Scholar
  30. 30.
    Macnab, D.S.: Modal operators on Heyting algebras. Algebra Univ. 12, 5–29 (1981)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Maeda, F., Maeda, S.: Theory of Symetric Lattices. Springer, Berlin (1970)CrossRefGoogle Scholar
  32. 32.
    Pulmannová, S.: Tensor product of quantum logics. J. Math. Phys. 26, 1–5 (1985)ADSCrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Randall, C.H., Foulis, D.J.: Empirical logic and tensor products. In: Neumann, H. (ed.) Interpretation and Foundations of Quantum Theory, pp. 21–28. Bibliographisches Institute, Mannheim (1981)Google Scholar
  34. 34.
    Zafiris, E., Karakostas, V.: A categorial semantic representation of quantum event structures. Found. Phys. 43, 1090–1123 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Dipartimento di FilosofiaUniversità di CagliariCagliariItaly
  2. 2.Departamento de MatemáticaUNR-CONICETRosarioArgentina
  3. 3.Instituto de Filosofía “Dr. Alejandro Korn”UBA-CONICETBuenos AiresArgentina
  4. 4.Center Leo Apostel (CLEA) - Brussels Free UniversityBrusselsBelgium
  5. 5.Foundations of The Exact Sciences (FUND)Brussels Free UniversityBrusselsBelgium

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