Foundations of Physics

, Volume 20, Issue 7, pp 859–872 | Cite as

On conditional probability in GL spaces

  • C. Martin Edwards
  • Gottfried T. Rüttimann
Part III. Invited Papers Dedicated To The Memory Of Charles H. Randall (1928–1987)

Abstract

We investigate the notion of conditional probability and the quantum mechanical concept of state reduction in the context of GL spaces satisfying the Alfsen-Shultz condition.

Keywords

Conditional Probability State Reduction Mechanical Concept Quantum Mechanical Concept 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. M. Alfsen,Compact Convex Sets and Boundary Integrals (Springer, New York, 1971).Google Scholar
  2. 2.
    E. M. Alfsen and F. W. Shultz, “Non-commutative spectral theory for affine function spaces on convex sets,”Mem. Am. Math. Soc. Vol. 172, (1976).Google Scholar
  3. 3.
    E. M. Alfsen and F. W. Shultz, “On Non-commutative spectral theory and Jordan algebras,”Proc. London Math. Soc. 38, 497–516 (1979).Google Scholar
  4. 4.
    L. Asimow and A. J. Ellis,Convexity Theory and Its Applications in Functional Analysis (Academic Press, London, 1980).Google Scholar
  5. 5.
    G. Cassinelli and N. Zanghi, “Conditional Probabilities in Quantum Mechanics,”Nuovo Cimento B 73, 237–245 (1983).Google Scholar
  6. 6.
    C. M. Edwards, “The facialq-topology for compact convex sets,”Math. Ann. 230, 123–152 (1977).Google Scholar
  7. 7.
    C. M. Edwards and G. T. Rüttimann, “On the facial structure of the unit balls in a GL space and its dual,”Math. Proc. Cambridge Philos. Soc. 98, 305 322 (1985).Google Scholar
  8. 8.
    D. J. Foulis and C. H. Randall, “Conditioning maps on orthomodular lattices,”Glasgow Math. J. 12, 35–42 (1971).Google Scholar
  9. 9.
    D. J. Foulis and C. H. Randall, “Stability of pure weights under conditioning,”Glasgow Math. J. 15, 5–12 (1974).Google Scholar
  10. 10.
    H. Hanche-Olsen and E. Størmer,Jordan Operator Algebras (Pitman, Boston, 1984).Google Scholar
  11. 11.
    J. M. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts 1968).Google Scholar
  12. 12.
    P. Jordan, J. von Neumann, and E. Wigner, “On an algebraic generalization of the quantum mechanical formalism,”Ann. Math. 35, 29–64 (1934).Google Scholar
  13. 13.
    G. Lüders: Ueber die Zustandsänderung durch den Messprozess,Ann. Phys. (Leipzig) 8, 322–328 (1951).Google Scholar
  14. 14.
    W. Pauli, “Die allgemeinen Prinzipien der Wellenmechanik,” inHandbuch der Physik, Band V, Teil 1, S. Flügge, ed. (Springer, Berlin, 1958).Google Scholar
  15. 15.
    J. C. T. Pool, “Bear*-semigroups and the logic of quantum mechanics,”Commun. Math. Phys. 9, 118–141 (1968).Google Scholar
  16. 16.
    J. C. T. Pool, “Semimodularity and the logic of quantum mechanics,”Commun. Math. Phys. 9, 212–228 (1968).Google Scholar
  17. 17.
    C. H. Randall and D. J. Foulis,Operational Statistics and Tensor Products. InInterpretations and Foundations of Quantum Theory, H. Neumann, ed. (Bibliographisches Institut, Zürich, 1981), pp. 9–20.Google Scholar
  18. 18.
    C. H. Randall and D. J. Foulis, “Stochastics Entities,” inRecent Developments in Quantum Logic, P. Mittelstaedt and E.-W. Stachow, eds. (Bibliographisches Institut, Zürich, 1985), pp. 265–290.Google Scholar
  19. 19.
    G. T. Rüttimann, “Jauch-Piron States,”J. Math. Phys. 18, 189–193 (1977).Google Scholar
  20. 20.
    G. T. Rüttimann, “Facial sets of probability measures,”Prob. Math. Stat. 6, 187 (1985).Google Scholar
  21. 21.
    G. T. Rüttimann, “The approximate Jordan-Hahn decomposition,”Can. J. Math. 41, (1989).Google Scholar
  22. 22.
    D. M. Topping, “Jordan algebras of self-adjoint operators,”Mem. Am. Math. Soc. Vol. 53, (1965).Google Scholar
  23. 23.
    W. Wils, “The ideal center of partially ordered Banach spaces,”Acta Math. 127, 41–77 (1971).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • C. Martin Edwards
    • 1
  • Gottfried T. Rüttimann
    • 2
  1. 1.The Queen's CollegeOxfordUnited Kingdom
  2. 2.Institute for Mathematical StatisticsUniversity of BerneBerneSwitzerland

Personalised recommendations