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

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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    For a discussion about contradiction and superposition states see [11].

  2. 2.

    For the construction of a lattice using convex sets instead of rays as states, see [16].

  3. 3.

    In this line, we have built in a previous paper a QL that arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of \({\mathcal H}\) [14]. To do so, we defined a valuation that respects contextuality (first translating the Kochen–Specker (KS) theorem to topological terms [14, Theorem 4.3]) and a frame for the Kripke model of the language. As frames are complete Heyting algebras, the resulting logic is an intuitionistic one—with restrictions on the allowed valuations arising from the KS theorem—, thus it has “good” properties as the distributive lattice structure and a nice definition of the implication as a residue of the conjunction.

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)

    ADS  Article  MATH  MathSciNet  Google 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)

    Google 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)

    Article  MathSciNet  Google Scholar 

  4. 4.

    Aerts, D.: Construction of the tensor product of lattices of properties of physical entities. J. Math. Phys. 25, 1434–1441 (1984)

    ADS  Article  MathSciNet  Google 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)

    ADS  Article  Google Scholar 

  7. 7.

    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)

    Article  Google Scholar 

  8. 8.

    Balbes, R., Dwinger, Ph: Distributive Lattices. University of Missouri Press, Columbia (1974)

    Google Scholar 

  9. 9.

    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Text in Mathematics, vol. 78. Springer, New York (1981)

    Google 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)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Dalla Chiara, M.L., Giuntini, R., Greechhie, R.: Reasoning in Quantum Theory. Kluwer, Dordrecht (2004)

  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)

  14. 14.

    Domenech, G., Freytes, H.: Contextual logic for quantum systems. J. Math. Phys. 46, 012102 (2005)

    ADS  Article  MathSciNet  Google Scholar 

  15. 15.

    Domenech, G., Freytes, H., de Ronde, C.: Modal type othomodular logic. Math. Logic Q. 55, 287–299 (2009)

    Article  Google 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)

    ADS  Article  MathSciNet  Google 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)

    ADS  Article  MathSciNet  Google 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.

  19. 19.

    Dvurečenskij, A.: Tensor product of difference posets and effect algebras. Int. J. Theor. Phys. 34, 1337–1348 (1995)

    Article  MATH  Google Scholar 

  20. 20.

    Freyd, P.J.: Aspects of Topoi. Bull. Austral. Math. Soc. 7, 1–76 (1972)

    Article  MATH  MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Heunen, C., Landsman, N., Spitters, B.: A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Heyting, A.: Die formalen Regeln der intuitionistischen Logik. Die Preussische Akademie der Wissenschaften. Sitzungsberichte. Physikalische-Mathematische Klasse, pp. 42–56 (1930)

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  31. 31.

    Maeda, F., Maeda, S.: Theory of Symetric Lattices. Springer, Berlin (1970)

    Google Scholar 

  32. 32.

    Pulmannová, S.: Tensor product of quantum logics. J. Math. Phys. 26, 1–5 (1985)

    ADS  Article  MATH  MathSciNet  Google 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)

    ADS  Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the following grants: VUB Project GOA67; FWO-research community W0.030.06; CONICET RES. 4541-12 (2013-2014) and PIP 112-201101-00636, CONICET.

Author information

Affiliations

Authors

Corresponding author

Correspondence to G. Domenech.

Additional information

H. Freytes and C. de Ronde—Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Freytes, H., Domenech, G. & de Ronde, C. Physical Properties as Modal Operators in the Topos Approach to Quantum Mechanics. Found Phys 44, 1357–1368 (2014). https://doi.org/10.1007/s10701-014-9842-9

Download citation

Keywords

  • Intuitionistic quantum logic
  • Quantum phase spaces
  • Modal operators