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Operational Axiomatics and Compound Systems

  • Frank Valckenborgh
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 111)

Abstract

I present an overview of the Geneva approach to the foundations of physics, its basic postulates and the representation of its objects. Next, the classical results of Aerts and Daubechies on the propositional representation of compound physical systems are analyzed from a categorical point of view. The problems of the basic postulates in the case of the separated product lead to the construction of a category consisting of objects with somewhat less mathematical structure and associated morphisms. The subsystem recognition problem then yields a subcategory for which arbitrary coproducts exist.

Keywords

Physical System Property Lattice Projective Geometry Coupling Condition Division Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Aerts, D. (1980) Subsystems in physics described by bilinear maps between the corresponding vector spaces, Journal of Mathematical Physics 21, 778–788.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2]
    Aerts, D. (1981) The One and the Many. Towards a Unification of the Quantum and the Classical Description of One and Many Physical entities, Doctoral Dissertation, Vrije Universiteit Brussel.Google Scholar
  3. [3]
    Aerts, D. (1982) Description of many separated physical entities without the paradoxes encountered in quantum mechanics, Foundations of Physics 12, 1131–1170.MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    Aerts, D. (1984) Construction of a structure which enables to describe the joint system of a classical system and a quantum system, Reports on Mathematical Physics 20, 117–129.MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. [5]
    Aerts, D. (1994) Quantum structures, separated physical entities and proba-bility, Foundations of Physics 24, 1227–1259.MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    Aerts, D. and Daubechies, I. (1978a) About the structure-preserving maps of a quantum mechanical propositional system, Helvetica Physica Acta 51, 637–660.MathSciNetGoogle Scholar
  7. [7]
    Aerts, D. and Daubechies, I. (1978b) Physical justification for using the ten-sor product to describe two quantum systems as one joint system, Helvetica Physica Acta 51, 661–675.MathSciNetGoogle Scholar
  8. [8]
    Aerts, D. and Daubechies, I. (1983) Simple proof that the structure preserving maps between quantum mechanical propositional systems conserve the angles, Helvetica Physica Acta 56, 1187–1190.MathSciNetGoogle Scholar
  9. [9]
    Amemiya, I. and Araki, H. (1967) A remark on Piron’s paper, Publications of the Research Institute of Mathematical Science of Kyoto University, Series A 2, 423–427.MathSciNetGoogle Scholar
  10. [10]
    Birkhoff, G. and von Neumann, J. (1936) The logic of quantum mechanics, Annals of Mathematics 37, 823–843.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Borceux, F. and Stubbe, I. (2000) Short introduction to enriched categories, This volume.Google Scholar
  12. [12]
    Bruns, G. and Harding J. (2000) Algebraic aspects of orthomodular lattices, This volume.Google Scholar
  13. [13]
    Coecke, B. and Moore, D.J. (2000) Operational Galois adjunctions, This vol-ume.Google Scholar
  14. [14]
    Dacey, J.R. (1968) Orthomodular Spaces, Doctoral Dissertation, University of Massachusetts, Amherst.Google Scholar
  15. [15]
    Einstein, A. (1950) Physics, philosophy and scientific progress, Journal of the International College of Surgeons 14, 755–758.Google Scholar
  16. [16]
    Faure, C1.-A. and Frölicher, A. (1993) Morphisms of projective geometries and of corresponding lattices, Geometriae Dedicata, 47, 25–40.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Faure, C1.-A. and Frölicher, A. (1994) Morphisms of projective geometries and semilinear maps, Geometriae Dedicata, 53, 237–262.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Gell-Mann, M. and Ne’eman, Y. (1964) The Eightfold Way, Benjamin, New York.Google Scholar
  19. [19]
    Gleason, A.M. (1957) Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics 6, 885–893.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Holland, S.S. (1995) Orthomodularity in infinite dimensions: a theorem of M. Solèr, Bulletin of the American Mathematical Society, 32, 205–234.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Jauch, J.M. (1968) Foundations of quantum mechanics, Addison-Wesley, Reading, Massachusetts.Google Scholar
  22. [22]
    Kadison, R.V. and Ringrose, J.R. (1997) Fundamentals of the Theory of Op-erator Algebras, Volume I: Elementary Theory, American Mathematical So-ciety, Providence, Rhode Island.Google Scholar
  23. [23]
    Maeda, F. and Maeda, S. (1970) Theory of Symmetric Lattices, Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  24. [24]
    Moore, D.J. (1999) On state spaces and property lattices, Studies in the History and Philosophy of Modern Physics 30, 61–83.zbMATHCrossRefGoogle Scholar
  25. [25]
    Piron, C. (1964) Axiomatique quantique, Helvetica Physica Acta 37, 439–468.MathSciNetzbMATHGoogle Scholar
  26. [26]
    Piron, C. (1976) Foundations of Quantum Physics, W.A. Benjamin, Reading, Massachusetts.Google Scholar
  27. [27]
    Piron, C. (1990) Mécanique quantique, bases et applications, Presses poly-techniques et universitaires romandes, Lausanne.Google Scholar
  28. [28]
    Reignier, J. (1994) The principles of classical mechanics and their actuality in contemporary microphysics, in Van der Merwe, A. and Garuccio, A. (ed.), Waves and Particles in Light and Matter, 583–601, Plenum Press, New York.CrossRefGoogle Scholar
  29. [29]
    Solèr, M.P. (1995) Characterization of Hilbert spaces with orthomodular spaces, Communications in Algebra 23, 219–243.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Valckenborgh, F. (1997) Closure structures and the theorem of decomposition in classical components, Tatra Mountains Mathematical Publications 10, 75–86.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Valckenborgh, F. (2000) On subsystem recognition in compound physical sys-tems, International Journal of Theoretical Physics 39, To appear.Google Scholar
  32. [32]
    van Fraassen, B.C. (1991) Quantum Mechanics: An Empiricist View, Oxford University Press.Google Scholar
  33. [33]
    Varadarajan, V.S. (1968) Geometry of Quantum Theory, Van Nostrand Com-pany, Princeton, New Jersey.Google Scholar
  34. [34]
    Wick, J.C., Wightman, A.S. and Wigner, E.H. (1952) The intrinsic parity of elementary particles, Physical Review 88, 101–105.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Frank Valckenborgh
    • 1
  1. 1.Department of MathematicsFree University of Brussels, BelgiumBrusselsBelgium

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