Foundations of Physics

, Volume 38, Issue 9, pp 783–795 | Cite as

A Representation of Quantum Measurement in Order-Unit Spaces



A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lüders-von Neumann quantum measurement as a probability conditionalization rule. A major result shows that the operator algebras must be replaced by order-unit spaces with some specific properties in the generalized approach, and it is analyzed under which conditions these order-unit spaces become Jordan algebras. An application of this result provides a characterization of the projection lattices in operator algebras.


Operator algebras Jordan algebras Convex sets Quantum measurement Quantum logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer, Berlin (1971) MATHGoogle Scholar
  2. 2.
    Alfsen, E.M., Shultz, F.W.: Non-commutative Spectral Theory for Affine Function Spaces on Convex Sets. Mem. Am. Math. Soc., vol. 172. Am. Math. Soc., Providence (1976) Google Scholar
  3. 3.
    Alfsen, E.M., Shultz, F.W.: State spaces of Jordan algebras. Acta Math. 140, 155–190 (1978) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alfsen, E.M., Shultz, F.W.: On non-commutative spectral theory and Jordan algebras. Proc. Lond. Math. Soc. 38, 497–516 (1979) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alfsen, E.M., Shultz, F.W., Størmer, E.: A Gelfand-Neumark theorem for Jordan algebras. Adv. Math. 28, 11–56 (1978) MATHCrossRefGoogle Scholar
  6. 6.
    Asimov, L., Ellis, A.J.: Convexity Theory and its Applications in Functional Analysis. Academic Press, New York (1980) Google Scholar
  7. 7.
    Bunce, L.J., Wright, J.D.M.: Quantum logic, state space geometry and operator algebras. Commun. Math. Phys. 96, 345–348 (1984) MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Bunce, L.J., Wright, J.D.M.: Quantum measures and states on Jordan algebras. Commun. Math. Phys. 98, 187–202 (1985) MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Bunce, L.J., Wright, J.D.M.: Continuity and linear extensions of quantum measures on Jordan operator algebras. Math. Scand. 64, 300–306 (1989) MATHMathSciNetGoogle Scholar
  10. 10.
    Christensen, E.: Non-commutative integration for monotone sequentially closed C*-algebras. Math. Scand. 31, 171–190 (1972) MATHMathSciNetGoogle Scholar
  11. 11.
    Christensen, E.: Measures on projections and physical states. Commun. Math. Phys. 86, 529–538 (1982) MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Edwards, C.M., Rüttimann, G.T.: On conditional probability in GL spaces. Found. Phys. 20, 859–872 (1990) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) MATHMathSciNetGoogle Scholar
  14. 14.
    Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitmann, Boston (1984) MATHGoogle Scholar
  15. 15.
    Iochum, B., Shultz, F.W.: Normal state spaces of Jordan and von Neumann algebras. J. Funct. Anal. 50, 317–328 (1983) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934) CrossRefGoogle Scholar
  17. 17.
    Kadison, R.V.: A Representation Theory for Commutative Topological Algebra. Mem. Am. Math. Soc., vol. 7. Am. Math. Soc., Providence (1951) Google Scholar
  18. 18.
    Kadison, R.V.: Unitary invariants for representation of operator algebras. Ann. Math. 66, 304–379 (1957) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kehlet, E.T.: On the monotone sequential closure of a C*-algebra. Math. Scand. 25, 59–70 (1969) MATHMathSciNetGoogle Scholar
  20. 20.
    Maeda, S.: Probability measures on projectors in von Neumann algebras. Rev. Math. Phys. 1, 235–290 (1990) CrossRefGoogle Scholar
  21. 21.
    Niestegge, G.: Non-Boolean probabilities and quantum measurement. J. Phys. A 34, 6031–6042 (2001) MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Niestegge, G.: Why do the quantum observables form a Jordan operator algebra? Int. J. Theor. Phys. 43, 35–46 (2004) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Niestegge, G.: An approach to quantum mechanics via conditional probabilities. Found. Phys. 38, 241–256 (2008) MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Pedersen, G.K.: Measure theory for C*-algebras III. Math. Scand. 25, 71–93 (1969) Google Scholar
  25. 25.
    Pulmannová, S.: Quantum logics and convex spaces. Int. J. Theor. Phys. 37, 2303–2332 (1998) CrossRefGoogle Scholar
  26. 26.
    Sakai, S.: C*-Algebras and W*-Algebras. Springer, Berlin (1971) Google Scholar
  27. 27.
    Schafer, R.: An Introduction to Nonassociative Algebras. Academic Press, New York (1966) MATHGoogle Scholar
  28. 28.
    Varadarajan, V.S.: Geometry of Quantum Theory I. Van Nostrand, New York (1968) MATHGoogle Scholar
  29. 29.
    Varadarajan, V.S.: Geometry of quantum theory II. Van Nostrand, New York (1970) MATHGoogle Scholar
  30. 30.
    Yeadon, F.J.: Measures on projections in W*-algebras of type II. Bull. Lond. Math. Soc. 15, 139–145 (1983) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Yeadon, F.J.: Finitely additive measures on projections in finite W*-algebras. Bull. Lond. Math. Soc. 16, 145–150 (1984) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.MuenchenGermany

Personalised recommendations