International Journal of Theoretical Physics

, Volume 54, Issue 12, pp 4601–4614 | Cite as

The Logic of Bundles

  • John Harding
  • Taewon Yang


Since the work of Crown (J. Natur. Sci. Math. 15(1–2), 11–25 1975) in the 1970’s, it has been known that the projections of a finite-dimensional vector bundle E form an orthomodular poset (omp) \(\mathcal {P}(E)\). This result lies in the intersection of a number of current topics, including the categorical quantum mechanics of Abramsky and Coecke (2004), and the approach via decompositions of Harding (Trans. Amer. Math. Soc. 348(5), 1839–1862 1996). Moreover, it provides a source of omps for the quantum logic program close to the Hilbert space setting, and admitting a version of tensor products, yet having important differences from the standard logics of Hilbert spaces. It is our purpose here to initiate a basic investigation of the quantum logic program in the vector bundle setting. This includes observations on the structure of the omps obtained as \(\mathcal {P}(E)\) for a vector bundle E, methods to obtain states on these omps, and automorphisms of these omps. Key theorems of quantum logic in the Hilbert setting, such as Gleason’s theorem and Wigner’s theorem, provide natural and quite challenging problems in the vector bundle setting.


Orthomodular poset Quantum logic Vector bundle Categorical quantum mechanics Decompositions Topological orthomodular poset State Automorphism 


  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425. IEEE Comput. Soc. Press, Los Alamitos (2004)CrossRefGoogle Scholar
  2. 2.
    Atiyah, M.: K-theory, Lecture Notes Series by W. A. Benjamin, Inc. (1967)Google Scholar
  3. 3.
    Amemiya, I., Araki, H: A remark on Pirons paper. Publ. Res. Inst. Math. Sci., Sect A 12, 423–427 (1966)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Birkhoff, G, von Neumann, J.: The logic of quantum mechanics. Ann. of Math. (2) 37(4), 823–843 (1936)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Crown, G: On some orthomodular posets of vector bundles. J. Natur. Sci. Math. 15(1–2), 11–25 (1975)MathSciNetMATHGoogle Scholar
  6. 6.
    Harding, J: Decompositions in quantum logic. Trans. Amer. Math. Soc 348(5), 1839–1862 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Harding, J.: A link between quantum logic and categorical quantum mechanics. Internat. J. Theoret. Phys. 48(3), 769–802 (2009)MathSciNetCrossRefADSMATHGoogle Scholar
  8. 8.
    Holland, S.S. Jr.: The current interest in orthomodular lattices. Trends in Lattice Theory (Sympos., U.S. Naval Academy, Annapolis, Md., 1966), pp. 41126. Van Nostrand Reinhold, New York (1970)Google Scholar
  9. 9.
    Kalmbach, G.: Orthomodular Lattices, London Mathematical Society Monographs, vol. 18. Academic Press Inc., London (1983)Google Scholar
  10. 10.
    Karoubi, M.: K-theory, Reprint of the 1978 edition, Classics in Mathematics. Springer-Verlag, Berlin (2008)Google Scholar
  11. 11.
    Mac Lane, S.: Categories for the working mathematician, 2nd edn. Springer, New York (1998)MATHGoogle Scholar
  12. 12.
    Mackey, G.W.: The Mathematical Foundations of Quantum Mechanics, A Lecture-Note Volume by W. A. Benjamin Inc. New York-Amsterdam (1963)Google Scholar
  13. 13.
    Milnor, J., Stasheff, J.: Characteristic Classes, Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)Google Scholar
  14. 14.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics, Fundamental Theories of Physics, vol. 44. Kluwer Academic Publishers Group, Dordrecht (1991)Google Scholar
  15. 15.
    Piron, C.: Foundations of Quantum Physics. Mathematical Physics Monograph Series 19. Benjamin-Cummings (1976)Google Scholar
  16. 16.
    Uhlhorn, U.: Representation of symmetry transformations in quantum mechanics. Arkiv Fysik 23, 307–340 (1962)Google Scholar
  17. 17.
    Wilce, A: Compact orthoalgebras. Proc. Amer. Math. Soc 133(10), 2911–2920 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yang, T.: The Logic of Bundles, Ph.D. Thesis, New Mexico State University (2013)Google Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.New Mexico State UniversityLas CrucesUSA

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