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International Journal of Theoretical Physics

, Volume 58, Issue 12, pp 4208–4234 | Cite as

Category-Theoretic Interpretative Framework of the Complementarity Principle in Quantum Mechanics

  • Elias ZafirisEmail author
  • Vassilios Karakostas
Article

Abstract

This study aims to provide an analysis of the complementarity principle in quantum theory through the establishment of partial structural congruence relations between the quantum and Boolean kinds of event structure. Specifically, on the basis of the existence of a categorical adjunction between the category of quantum event algebras and the category of presheaves of variable Boolean event algebras, we establish a twofold complementarity scheme consisting of a generalized/global and a restricted/local conceptual dimension, where the latter conception is subordinate to and constrained by the former. In this respect, complementarity is not only understood as a relation between mutually exclusive experimental arrangements or contexts of comeasurable observables, as envisaged by the original conception, but it is primarily comprehended as a reciprocal relation concerning information transfer between two hierarchically different structural kinds of event structure that can be brought into partial congruence by means of the established adjunction. It is further argued that the proposed category-theoretic framework of complementarity naturally advances a contextual realist conceptual stance towards our deeper understanding of the microphysical nature of reality.

Keywords

Quantum event structures Complementarity Adjoint functors Kochen-Specker theorem Boolean frames Local-global relation Realist account 

Notes

References

  1. 1.
    Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036 (2011)ADSGoogle Scholar
  2. 2.
    Awodey, S.: Category theory, 2nd edn. Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  3. 3.
    Bell, J.L.: Toposes and local set theories. Dover, New York (1988/2008)zbMATHGoogle Scholar
  4. 4.
    Beller, M.: Quantum dialogue: The making of a revolution. University of Chicago Press, Chicago (1999)zbMATHGoogle Scholar
  5. 5.
    Bohr, N.: The quantum postulate and the recent development of atomic theory. Nature Suppl. 121, 580–590 (1928). Reprinted in J. Kalckar (ed.) Niels Bohr Collected Works, Vol. 6: Foundations of Quantum Physics I (1926-1932), pp. 109–136. North-Holland, Amsterdam (1985)ADSzbMATHGoogle Scholar
  6. 6.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev. 38, 696–702 (1935)ADSzbMATHGoogle Scholar
  7. 7.
    Bohr, N.: Causality and complementarity. Phil. Sci. 4, 289–298 (1937)Google Scholar
  8. 8.
    Bohr, N.: The causality problem in atomic physics. In: New Theories in Physics, pp. 11-30. International Institute of Intellectual Co-operation, Paris (1939). Reprinted in J. Kalckar (1996) (ed.) Niels Bohr Collected Works, Vol. 7: Foundations of Quantum Physics II (1933-1958), pp. 303–322. North-Holland, Amsterdam (1996)Google Scholar
  9. 9.
    Bohr, N.: On the notion of causality and complementarity. Dialectica 2, 312–319 (1948)zbMATHGoogle Scholar
  10. 10.
    Bohr, N.: Light and life revisited. In: Niels Bohr, Essays 1958-1962 on atomic physics and human knowledge, pp 23–29. Interscience, London (1963)Google Scholar
  11. 11.
    Borceux, F.: Handbook of categorical algebra 3: Categories of sheaves. Encyclopedia of mathematics and its applications 52. Cambridge University Press, Cambridge (1994/2008)zbMATHGoogle Scholar
  12. 12.
    Bramon, A., Garbarino, G., Hiesmayr, B. C.: Quantitative complementarity in two path interferometry. Phys. Rev. A 69, 022112 (2004)ADSGoogle Scholar
  13. 13.
    Busch, P., Grabowski, M., Lahti, P.: Operational quantum physics. Springer, Berlin (1995)zbMATHGoogle Scholar
  14. 14.
    Busch, P., Shilladay, C.: Complementarity and uncertainty in Mach-Zehnder interferometry and beyond. Phys. Rep. 435, 1–31 (2006)ADSGoogle Scholar
  15. 15.
    Chiara, M. D., Giuntini, R., Greechie, R.: Reasoning in quantum theory: Sharp and unsharp quantum logics. Kluwer, Dordrecht (2004)zbMATHGoogle Scholar
  16. 16.
    Coecke, B.: Quantum picturalism. Contemp. Phys. 51(1), 59–83 (2010)ADSGoogle Scholar
  17. 17.
    Coles, P. J.: Role of complementarity in superdense coding. Phys. Rev. A 88, 062317 (2013)ADSGoogle Scholar
  18. 18.
    Dieks, D.: Niels Bohr and the formalism of quantum mechanics. In: Folse, H., Faye, J (eds.) Niels bohr and philosophy of physics: Twenty-first century perspectives, pp 303–334. Bloomsbury Academic, London (2017)zbMATHGoogle Scholar
  19. 19.
    Domenech, G., Freytes, H.: Contextual logic for quantum systems. J. Math. Phys. 46, 012102 (2005)ADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Döring, A., Isham, C.J.: “What is a thing?”: Topos theory in the foundations of physics. In: Lecture notes in physics 813, pp 753–937. Springer, Berlin (2011)zbMATHGoogle Scholar
  21. 21.
    Epperson, M., Zafiris, E.: Foundations of relational realism: A topological approach to quantum mechanics and the philosophy of nature. Book series: Contemporary Whitehead Studies. Lexington Books, New York (2013)Google Scholar
  22. 22.
    Faye, J., Folse, H. (eds.): Niels Bohr and contemporary philosophy. Kluwer, Dordrecht (1994)Google Scholar
  23. 23.
    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)ADSzbMATHGoogle Scholar
  24. 24.
    Heunen, C., Landsman, N. P., Spitters, B.: Bohrification. In: Halvorson, H (ed.) Deep beauty, pp 271–313. Cambridge University Press, New York (2011)Google Scholar
  25. 25.
    Heunen, C.: Complementarity in categorical quantum mechanics. Found. Phys. 42, 856–873 (2012)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    Hilgevoord, J., Uffink, J.: The uncertainty principle. In: Zalta, E. (ed.) The stanford encyclopedia of philosophy. https://plato.stanford.edu/archives/win2016/entries/qt-uncertainty/ (2016)
  27. 27.
    Howard, D.: Who invented the ‘Copenhagen interpretation’? A study in mythology. Phil. Sc. 71, 669–682 (2004)MathSciNetGoogle Scholar
  28. 28.
    Isham, C. J., Butterfield, J.: A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalised valuations. I.t. J. Theor. Phys. 37, 2669–2733 (1998)zbMATHGoogle Scholar
  29. 29.
    Isham, C. J., Butterfield, J.: A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects, and classical analogues. Int. J. Theor. Phys. 38, 827–859 (1999)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Johnstone, P. T.: Sketches of an elephant: A topos theory compendium Vols, vol. 1-2. Clarendon Press, London (2002)zbMATHGoogle Scholar
  31. 31.
    Karakostas, V.: Realism and objectivism in quantum mechanics. J. Gen. Phil. Sci. 43, 45–65 (2012)MathSciNetGoogle Scholar
  32. 32.
    Karakostas, V.: Correspondence truth and quantum mechanics. Axiomathes 24, 343–358 (2014)Google Scholar
  33. 33.
    Karakostas, V., Zafiris, E.: Contextual semantics in quantum mechanics from a categorical point of view. Synthese 194, 847–886 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Karakostas, V., Zafiris, E.: On the structure and function of scientific perspectivism in quantum mechanics. arXiv:1806.08788 [quant-ph] (2018)
  35. 35.
    Kochen, S., Specker, E. P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lahti, P. J.: Uncertainty and complementarity in axiomatic quantum mechanics. Int. J. Theor. Phys. 19, 789–842 (1980)MathSciNetGoogle Scholar
  37. 37.
    Lam, T. X.: Lectures on modules and rings. Springer, New York (1999)zbMATHGoogle Scholar
  38. 38.
    Ma, X., Kofler, J., Zeilinger, A.: Delayed-choice gedanken experiments and their realizations. Rev. Mod. Phys. 88, 015005 (2016)ADSGoogle Scholar
  39. 39.
    MacLane, S., Moerdijk, I.: Sheaves in geometry and logic: A first introduction to topos theory. Springer, New York (1992)Google Scholar
  40. 40.
    Murdoch, D.: Niels Bohr’s philosophy of physics. Cambridge University Press, Cambridge (1987)Google Scholar
  41. 41.
    Scheibe, E.: The logical analysis of quantum mechanics. Pergamon Press, Oxford (1973)Google Scholar
  42. 42.
    Selesnick, S.: Correspondence principle for the quantum net. Int. J. Theor. Phys. 30(10), 1273–1292 (1991)MathSciNetGoogle Scholar
  43. 43.
    Svozil, K.: Physical (A)Causality. Fundamental Theories of Physics 192 Springer Open (2018)zbMATHGoogle Scholar
  44. 44.
    Zafiris, E.: Probing quantum structure with Boolean localization systems. Int. J. Theor. Phys. 39(12), 2761–2778 (2000)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Zafiris, E.: Boolean coverings of quantum observable structure: A setting for an abstract differential geometric mechanism. J. Geom. Phys. 50, 99–114 (2004)ADSMathSciNetzbMATHGoogle Scholar
  46. 46.
    Zafiris, E.: Interpreting observables in a quantum world from the categorial standpoint. Int. J. Theor. Phys. 43(1), 265–298 (2004)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Zafiris, E.: Sheaf-theoretic representation of quantum measure algebras. J. Math. Phys. 47, 092103 (2006)ADSMathSciNetzbMATHGoogle Scholar
  48. 48.
    Zafiris, E.: Generalized topological covering systems on quantum events structures. J. Phys. A, Math. Gen. 39, 1485–1505 (2006)ADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    Zafiris, E: Quantum observables algebras and abstract differential geometry: The topos-theoretic dynamics of diagrams of commutative algebraic localizations. Int. J. Theor. Phys. 46(2), 319–382 (2007)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Zafiris, E., Karakostas, V.: A categorial semantic representation of quantum event structures. Found. Phys. 43, 1090–1123 (2013)ADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    Zhu, H.: Information complementarity: A new paradigm for decoding quantum incompatibility. Scient. Rep. 5, 14317 (2015)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Parmenides FoundationCenter for the Conceptual Foundations of ScienceMunichGermany
  2. 2.Department of MathematicsUniversity of AthensAthensGreece
  3. 3.Department of Philosophy and History of Science, Faculty of ScienceUniversity of AthensAthensGreece

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