International Journal of Theoretical Physics

, Volume 37, Issue 9, pp 2303–2332 | Cite as

Quantum Logics and Convex Spaces

  • Sylvia Pulmannova
Article

Abstract

An orthomodular σ-lattice with rich set ofstates satisfying the property that every affinefunctional from the set of states into the unit intervalof the reals corresponds to an expectational functional of exactly one real observable (so-calledu-spectral logic) is compared with the noncommutativespectral theory of Alfsen and Shultz. Necessary andsufficient conditions are found under which these two approaches are in correspondence.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. Alfsen, E. M. (1971). Compact Convex Sets and Boundary Integrals, Springer-Verlag, Berlin.Google Scholar
  2. Alfsen, E. M., and Shultz, F. W. (1976). On non-commutative spectral theory for affine function spaces on convex sets, Memoirs of the American Mathematical Society, 6(172) (1976).Google Scholar
  3. Alfsen, E. M., Shultz, F. W., and Stormer, E. (1978). A Gelfand-Neumark theorem for Jordan algebras, Advances in Mathematics, 28, 11–56.Google Scholar
  4. Asimov, L., and Ellis, A. J. (1980). Convexity Theory and Its Applications in Functional Analysis, Academic Press, New York. (1981).Google Scholar
  5. Beltrametti, E. G., and Cassinelli, G. (1981). The Logic of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.Google Scholar
  6. Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics, Annals of Mathematics, 37, 823–843.Google Scholar
  7. Edwards, C. M., and Rüttimann, G. T. (1990). On conditional probability in GL spaces, Foundations of Physics, 20, 859–872.Google Scholar
  8. Foulis, D. J., and Bennett, M. K. (1994). Effect algebras and unsharp quantum logics, Foundations of Physics, 24, 1331–1352.Google Scholar
  9. Gudder, S. P. (1965). Spectral methods for a generalized probability theory, Transactions of the American Mathematical Society, 119, 428–442.Google Scholar
  10. Gudder, S. P. (1966) Uniqueness and existence properties of bounded observables, Pacific Journal of Mathematics, 19, 81–93; correction, ibid. 588–589.Google Scholar
  11. Guz, W., (1980). Conditional probability in quantum axiomatics, Annales de l' Institut Henri Poincaré A 33, 63–119.Google Scholar
  12. Hudson, R. L., and Pulmannová, S. (1993). Sum logics and tensor products, Foundations of Physics, 23, 999–1024.Google Scholar
  13. Iochum, B. (1984). Cônes Autopolaires et Algèbres de Jordan, Springer-Verlag, Berlin, 1984.Google Scholar
  14. Mackey, G. W. (1963). The Mathematical Foundations of Quantum Mechanics, Benjamin, New York.Google Scholar
  15. Navara, M. (1995). Uniqueness of bounded observables, Annales de l' Institut Henri Poincaré A, 63, 155–176.Google Scholar
  16. Pool, J. C. T. (1968a). Baer*-semigroups and the logic of quantum mechanics, Communications in Mathematical Physics, 9, 118–141.Google Scholar
  17. Pool, J. C. T. (1968b). Semimodularity and the logic of quantum mechanics, Communications in Mathematical Physics, 9, 212–229.Google Scholar
  18. Pták, P., and Pulmannová S. (1991). Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht. (1981).Google Scholar
  19. Rüttimann, G. (1981). Detectable properties and spectral quanatum logics, in Interpretations and Foundations of Quantum Theories, H. Neumann, ed., Mannheim, pp. 35–47.Google Scholar
  20. Schafer, R. (1966). An Introduction to Nonassociative Algebras, Academic Press, New York.Google Scholar
  21. Varadarajan, V. S. (1985). Geometry of Quantum Theory, Springer Verleg, Berlin.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • Sylvia Pulmannova

There are no affiliations available

Personalised recommendations