Foundations of Physics Letters

, Volume 17, Issue 5, pp 403–432 | Cite as

On Quantum Event Structures. Part III: Object of Truth Values

  • Elias Zafiris


In this work we expand the foundational perspective of category theory on quantum event structures by showing the existence of an object of truth values in the category of quantum event algebras, characterized as subobject classifier. This object plays the corresponking role that the two-valued Boolean truth values object plays in a classical event structure. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of quantum systems.

Boolean localizations topos adjunction quantum logic subobject classifier quantum truth values 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. S. Varadarajan, Geome'ry of Quantum Mechanics, Vol. 1 (Van Nostrand, Princeton, New Jersey, 1968).Google Scholar
  2. 2.
    G. Birkhoff and J. von Neumann, “The logic of quantum mechanics.” Ann. Math. 37, 823 (1936).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    E. Zafiris, “On quantum event structures. Part I: The categorical scheme,” Found. Phys. Lett. 14(2), 147 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    E. Zafiris, “On quantum event structures. Part II: Interpretational aspects,” Found. Phys. Lett. 14(2), 167 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. Zafiris, “Quantum event structures from the perspective of Grothendieck topoi,” to appear in Found. Phys. Google Scholar
  6. 6.
    6._ S. Kochen and E. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. and Mech. 17, 59 (1967).MathSciNetzbMATHGoogle Scholar
  7. 7.
    F. W. Lawvere and S. H. Schanuel, Conceptual Mathematics (University Press, Cambridge, 1997).zbMATHGoogle Scholar
  8. 8.
    S. MacLane, Categories for the Working Mathematician (Springer, New York, 1971).zbMATHGoogle Scholar
  9. 9.
    S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic (Springer, New York, 1992).CrossRefGoogle Scholar
  10. 10.
    J. L. Bell, Toposes and Local Set Theories (University Press, Oxford, 1988).zbMATHGoogle Scholar
  11. 11.
    M. Artin, A. Grothendieck, and J. L. Verdier, Theorie de topos et cohomologie etale des schemas (Springer LNM 269 and 270) (Springer, Berlin, 1972).zbMATHGoogle Scholar
  12. 12.
    J. L. Bell, “From absolute to local mathematics,” Synthese 69 (1986).Google Scholar
  13. 13.
    J. L. Bell, “Categories, toposes and sets,” Synthese 51(3) (1982).Google Scholar
  14. 14.
    J. Butterfield and C. J. Isham, “A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations,” Intern. J. Theoret. Phys. 37, 2669 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. Butterfield and C. J. Isham, “A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects and classical analogues,” Intern. J. Theoret. Phys. 38, 827 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J. P. Rawling and S. A. Selesnick:, “Orthologic and quantum logic, models and computational elements,” J. Assoc. Computing Machinery 47, 721 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    I. Raptis, “Presheaves, sheaves, and their topoi in quantum gravity and quantum logic,” gr-qc/0l10064.Google Scholar
  18. 18.
    J. Butterfield and C. J. Isham, “Some possible roles for topos theory in quantum theory and quantum gravity,” Found. Phys. 30, 1707 (2000).MathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Takeuti, “Two applications of logic to mathematics,” Mathematical Society of Japan 13, Kano Memorial Lectures 3 (1978).Google Scholar
  20. 20.
    M. Davis, “A relativity principle in quantum mechanics,” Internat. J. Theoret. Phys. 16, 867 (1977).ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Elias Zafiris
    • 1
    • 2
  1. 1.University of Sofia, Faculty of Mathematics and InformaticsSofiaBulgaria
  2. 2.University of Athens, Institute of Mathematics, PanepistimiopolisAthensGreece

Personalised recommendations