Skip to main content
SpringerLink
Log in

Operational Galois Adjunctions

  • Chapter
Current Research in Operational Quantum Logic

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 111))

Abstract

We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order structures together with their applications in operational quantum theory. First, after reviewing the notion of residuation and its implementation at the level of quantaloids we consider some standard universal constructions and the extension of adjunctions to weak morphisms. Second, we present the categorical formulation of closure operators and introduce a hierarchy of contextual enrichments of the quantaloid of complete join lattices. Third, we briefly survey physical state-property duality and the categorical analysis of derived notions such as causal assignment and the propagation of properties.

The author is Post-Doctoral Researcher at Flanders’ Fund for Scientific Research.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AdAmek, J., Herrlich, H. and Strecker, G.E. (1990) Abstract and Concrete Categories, John Wiley and Sons, New York.

    MATH  Google Scholar 

  2. Aerts, D. (1982) Description of many separated physical entities without the paradoxes encountered in quantum mechanics, Foundations of Physics 12, 1131–1170.

    Article  MathSciNet  ADS  Google Scholar 

  3. Aerts, D. (1994) Quantum structures, separated physical entities and proba-bility, Foundations of Physics 24, 1227–1259.

    Article  MathSciNet  ADS  Google Scholar 

  4. Amira, H., Coecke, B. and Stubbe, I. (1998) How quantales emerge by in-troducing induction within the operational approach, Helvetica Physica Acta 71, 554–572.

    MathSciNet  MATH  Google Scholar 

  5. Bennett, M.K. and Foulis, D.J. (1998) A generalized Sasaki projection for effect algebras, Tatra Mountains Mathematical Publications 15, 55–66.

    MathSciNet  MATH  Google Scholar 

  6. Blyth, T.S. and Janowitz, M.F. (1972) Residuation Theory, Pergamon Press, New York.

    MATH  Google Scholar 

  7. Borceux, F. (1994) Handbook of Categorical Algebra, Cambridge University Press, Cambridge.

    Book  Google Scholar 

  8. Borceux, F. and Stubbe, I. (2000) Short introduction to enriched categories, This volume.

    Google Scholar 

  9. Coecke, B. (2000) Structural characterization of compoundness, International Journal of Theoretical Physics 39, To appear.

    Google Scholar 

  10. Coecke, B., Moore, D.J. and Stubbe, I. (2000) Quantaloids describing causation and propagation for physical properties, Journal of Pure and Applied Algebra, To appear.

    Google Scholar 

  11. Coecke, B. and Smets, S. (2000) Logical description for perfect measurements, International Journal of Theoretical Physics 39, To appear.

    Google Scholar 

  12. Coecke, B. and Stubbe, I. (1999a) On a duality of quantales emerging from an operational resolution, International Journal of Theoretical Physics 38, 3269–3281.

    Article  MathSciNet  MATH  Google Scholar 

  13. Coecke, B. and Stubbe, I. (1999b) Operational resolutions and state transi- tions in a categorical setting, Foundations of Physics Letters 12, 29–49.

    Article  MathSciNet  ADS  Google Scholar 

  14. Coecke, B. and Stubbe, I. (2000) State transitions as morphisms for complete lattices, International Journal of Theoretical Physics 39, To appear.

    Google Scholar 

  15. Daniel, W. (1982) On a non-unitary evolution of quantum systems, Helvetica Physica Acta 55, 330–338.

    MathSciNet  Google Scholar 

  16. Daniel, W. (1989) Axiomatic description of irreversible and reversible evolution of a physical system, Helvetica Physica Acta 62, 941–968.

    MathSciNet  Google Scholar 

  17. Derdérian, J.-C. (1967) Residuated mappings, Pacific Journal of Mathematics 20, 35–43.

    Article  MathSciNet  MATH  Google Scholar 

  18. Faure, Cl.-A. (1994) Categories of closure spaces and corresponding lattices, Cahiers de Topologie et Géométrie Différentielle Catégoriques 35, 309–319.

    MathSciNet  MATH  Google Scholar 

  19. Faure, Cl.-A. and Frölicher, A. (1993) Morphisms of projective geometries and of corresponding lattices, Geometriae Dedicata 47, 25–40.

    Article  MathSciNet  MATH  Google Scholar 

  20. Faure, Cl.-A. and Frölicher, A. (1994) Morphisms of projective geometries and semilinear maps, Geometriae Dedicata 53, 237–269.

    Article  MathSciNet  MATH  Google Scholar 

  21. Faure, Cl.-A. and Frölicher, A. (1995) Dualities for infinite-dimensional projective geometries, Geometriae Dedicata 56, 225–236.

    Article  MathSciNet  MATH  Google Scholar 

  22. Faure, Cl.-A. and Frölicher, A. (1996) The dimension theorem in axiomatic geometry, Geometriae Dedicata 60, 207–218.

    Article  MathSciNet  MATH  Google Scholar 

  23. Faure, Cl.-A. and Frölicher, A. (1998) Categorical aspects in projective geometry, Applied Categorical Structures 6, 87–103.

    Article  MathSciNet  MATH  Google Scholar 

  24. Faure, Cl.-A., Moore, D.J. and Piron, C. (1995) Deterministic evolutions and Schrödinger flows, Helvetica Physica Acta 68, 150–157.

    MathSciNet  MATH  Google Scholar 

  25. Foulis, D.J. (1960) Baer *-semigroups, Proceedings of the American Mathematical Society 11, 648–654.

    MathSciNet  MATH  Google Scholar 

  26. Foulis, D.J. (1962) A note on orthomodular lattices, Portugaliae Mathematica 21, 65–72.

    MathSciNet  MATH  Google Scholar 

  27. Foulis, D.J. and Randall, C.H. (1971) Conditioning maps on orthomodular spaces, Glasgow Mathematical Journal 12, 35–42.

    Article  MathSciNet  MATH  Google Scholar 

  28. Foulis, D.J. and Randall, C.H. (1974) The stability of pure weights under conditioning, Glasgow Mathematical Journal 15, 5–12.

    Article  MathSciNet  MATH  Google Scholar 

  29. Frazer, P.J., Foulis, D.J. and Randall, C.H. (1980) Weight functions on exten-sions of the compound manual, Glasgow Mathematical Journal 21, 97–101.

    MathSciNet  MATH  Google Scholar 

  30. Jauch, J.M. and Piron, C. (1969) On the structure of quantal proposition systems, Helvetica Physica Acta 42, 842–848.

    MathSciNet  MATH  Google Scholar 

  31. Mac Lane, S. (1971) Categories for the Working Mathematician, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  32. Moore, D.J. (1995) Categories of representations of physical systems, Hel-vetica Physica Acta 68, 658–678.

    MATH  Google Scholar 

  33. Moore, D.J. (1997) Closure categories, International Journal of Theoretical Physics 36, 2707–2723.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Moore, D.J. (1999) On state spaces and property lattices, Studies in History and Philosophy of Modern Physics 30, 61–83.

    Article  MATH  Google Scholar 

  35. Moore, D.J. (2000) Fundamental constructions in physics: A categorical ap-proach, International Journal of Theoretical Physics 39, To appear.

    Google Scholar 

  36. Paseka, J. and RosickY, J. (2000) Quantales, This volume.

    Google Scholar 

  37. Piron, C. (1964) Axiomatique quantique, Helvetica Physica Acta 37, 439–468.

    MathSciNet  MATH  Google Scholar 

  38. Piron, C. (1976) Foundations of Quantum Mechanics, W.A. Benjamin Inc., Reading.

    Google Scholar 

  39. Piron, C. (1990) Mécanique quantique bases et applications, Presses polytech-niques et universitaires romandes, Lausanne (Second corrected edition, 1998 ).

    MATH  Google Scholar 

  40. Rosenthal, K.I. (1991) Free quantaloids, Journal of Pure and Applied Algebra 77, 67–82.

    Article  Google Scholar 

  41. Valckenborgh, F. (2000) Operational axiomatics and compound systems, This volume.

    Google Scholar 

  42. Wilce, A. (2000) Generalized Sasaki projections, International Journal of Theoretical Physics 39, To appear.

    Google Scholar 

Download references

Author information

Author notes

  1. David Moore

    Present address: Department of Physics and Astronomy, University of Canterbury, PO Box 4800, Christchurch, New Zealand

Authors and Affiliations

  1. Department of Mathematics, Free University of Brussels, Pleinlaan 2, B-1050, Brussels, Belgium

    Bob Coecke

  2. Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, CH-1211, Geneva 4, Switzerland

    David Moore

Authors
  1. Bob Coecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, FUND, Free University of Brussels, Brussels, Belgium

    Bob Coecke

  2. Department of Theoretical Physics, University of Geneva, Geneva, Switzerland

    David Moore

  3. Department of Mathematics and Computer Science, Juniata College, Huntingdon, Pennsylvania, USA

    Alexander Wilce

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Coecke, B., Moore, D. (2000). Operational Galois Adjunctions. In: Coecke, B., Moore, D., Wilce, A. (eds) Current Research in Operational Quantum Logic. Fundamental Theories of Physics, vol 111. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1201-9_8

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1201-9_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5437-1

  • Online ISBN: 978-94-017-1201-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation