Studia Logica

, Volume 71, Issue 1, pp 47–56 | Cite as

Disjunctive Quantum Logic in Dynamic Perspective

  • Bob Coecke
Article
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Abstract

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application “towards dynamic quantum logic”, namely in describing the temporal indeterministic propagation of actual properties of physical systems. This paper can as such by conceived as an addendum to “Quantum Logic in Intuitionistic Perspective” that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement.

Quantum logic dynamic logic intuitionistic logic property lattice operational resolution quantaloid 

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References

  1. Abramsky, S., and S. Vickers, (1993) ‘Quantales, observational logic and process semantics’, Mathematical Structures in Computer Science3, 161.Google Scholar
  2. Amira, H., B. Coecke, and I. Stubbe, (1998) ‘How quantales emerge by introducing induction within the operational approach’, Helvetica Physica Acta71, 554.Google Scholar
  3. Bruns, G., and H. Lakser, (1970) ‘Injective hulls of semilattices’, Canadian Mathematical Bulletin13, 115.Google Scholar
  4. Borceux, F., (1994) Handbook of Categorical Algebra I & II, Cambridge University Press.Google Scholar
  5. Borceux, F., and I. Stubbe, (2000) ‘Short introduction to enriched categories’, in: B. Coecke, D. J. Moore and A. Wilce, (Eds.), Current Research in Operational Quantum Logic: Algebras, Categories and Languages, pp. 167-194, Kluwer Academic Publishers.Google Scholar
  6. Coecke, B., and I. Stubbe, (1999) ‘Operational resolutions and state transitions in a categorical setting’, Foundations of Physics Letters12, 29; arXiv: quant-ph/0008020.Google Scholar
  7. Coecke, B., (2000) ‘Structural characterization of compoundness’, International Journal of Theoretical Physics39, 581; arXiv: quant-ph/0008054.Google Scholar
  8. Coecke, B., and D. J. Moore, (2000) ‘Operational Galois adjunctions’, in: B. Coecke, D. J. Moore and A. Wilce, (Eds.), Current Research in Operational Quantum Logic: Algebras, Categories and Languages, pp. 195-218, Kluwer Academic Publishers; arXiv: quant-ph/0008021.Google Scholar
  9. Coecke, B., D. J. Moore, I. and Stubbe, (2001) ‘Quantaloids describing causation and propagation for physical properties’, Foundations of Physics Letters14, 133; arXiv: quant-ph/0009100.Google Scholar
  10. Coecke, B., and S. Smets, (2000) ‘A logical description for perfect measurements’, International Journal of Theoretical Physics39, 591; arXiv: quant-ph/0008017.Google Scholar
  11. Coecke, B., (2001a) ‘Do we have to retain Cartesianity in topos-approaches to quantum theory, and, quantum gravity’, preprint.Google Scholar
  12. Coecke, B., (2001b) ‘The Sasaki-Hook is not a static implicative connective but induces a backward (in time) dynamic one that assigns causes of truth/actuality’, paper submitted to International Journal of Theoretical Physicsfor the proceedings of IQSA V, Cesena, Italy, April 2001.Google Scholar
  13. Coecke, B., (2002) ‘Quantum logic in intuitionistic perspective’, Studia Logica70, 411-440; arXiv: math.LO/0011208.Google Scholar
  14. Daniel, W., (1989) ‘Axiomatic descrition of irreversable and reversable evolution of a physical system’, Helvetica Physica Acta62, 941.Google Scholar
  15. Faure, Cl.-A., and A. Frölicher, (1993) ‘Morphisms of projective geometries and of corresponding lattices’, Geometriœ Dedicata47, 25.Google Scholar
  16. Faure, Cl.-A., and A. Frölicher, (1994) ‘Morphisms of projective geometries and semilinear maps’, Geometriœ Dedicata53, 237.Google Scholar
  17. Faure, Cl.-A., D.J. Moore, and C. Piron, (1995) ‘Deterministic evolutions and Schrödinger flows’, Helvetica Physica Acta68, 150.Google Scholar
  18. Harding, J., (1999) Private communication.Google Scholar
  19. Johnstone, P. T., (1982) Stone Spaces, Cambridge University Press.Google Scholar
  20. Kalmbach, G., (1983) Orthomodular Lattices, Academic Press.Google Scholar
  21. Piron, C., (1976) Foundations of Quantum Physics, W. A. Benjamin, Inc.Google Scholar
  22. Pool, J. C. T., (1968) ‘Baer *-semigroups and the logic of quantum mechanics’, Communications in Mathematical Physics9, 118.Google Scholar
  23. Resende, P., (2000) ‘Quantales and observational semantics’, in: B. Coecke, D. J. Moore and A. Wilce, (Eds.), Current Research in Operational Quantum Logic: Algebras, Categories and Languages, pp. 263-288, Kluwer Academic Publishers.Google Scholar
  24. Rosenthal, K. I., (1991) ‘Free quantaloids’, Journal of Pure and Applied Algebra77, 67.Google Scholar
  25. Smets, S. (2001): ‘The Logic of physical properties in static and dynamic perspective’, PhD-thesis, Free University of Brussels.Google Scholar
  26. Sourbron, S., (2000) ‘A note on causal duality’, Foundations of Physics Letters13, 357.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Bob Coecke
    • 1
  1. 1.ComlabOxford UniversityOxfordUK

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