Foundations of Physics

, Volume 2, Issue 4, pp 287–314 | Cite as

Survey of general quantum physics

  • C. Piron


The abstract description of a physical system is developed, along lines originally suggested by Birkhoff and von Neumann, in terms of the complete lattice of propositions associated with that system, and the distinction between classical and quantum systems is made precise. With the help of the notion of state, a propositional system is defined: it is remarked that every irreducible propositional system (of more than three dimensions) is isomorphic to the lattice of all closed subspaces of a Hilbert space constructed on some division ring with involution. The propositional system consisting of a family of separable complex Hilbert spaces is treated as a particular case which is sufficiently general to include both classical and quantum mechanics. The theory of the Galilean particle without spin is given as an illustration. Finally, the basis for the statistical interpretation of wave mechanics is developed with the help of Gleason's theorem. In an appendix, a proof of essentially the first part of Gleason's theorem is given which is a little different (perhaps more geometric) from that originally given by Gleason.


Hilbert Space Quantum Mechanic Quantum System Physical System Abstract Description 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. G. Bade,Pacific J. Math. 4, 393 (1954).Google Scholar
  2. 2.
    G. Birkhoff and J. von Neumann,Ann. Math. 37, 823 (1936).Google Scholar
  3. 3.
    J. Dixmier,Les algèbres d'opérateurs dans l'espace Hilbertien, 2nd ed. (Gauthier-Villars, Paris, 1969) exercise 3f, Chapter I, paragraph 7, p. 120.Google Scholar
  4. 4.
    A. Einstein, P. Podolsky, and N. Rosen,Phys. Rev. 47, 777 (1935).Google Scholar
  5. 5.
    A. M. Gleason,J. Rat. Mech. Anal. 6, 885 (1957).Google Scholar
  6. 6.
    P. R. Halmos,Lectures on Boolean Algebra (1963), Chapter 9, p. 35.Google Scholar
  7. 7.
    J. M. Jauch,Helv. Phys. Acta 37, 284 (1964).Google Scholar
  8. 8.
    L. W. Loomis,Mem. Amer. Math. Soc. 1955, No. 18.Google Scholar
  9. 9.
    G. Ludwig,Lecture Notes in Physics 4 (Springer-Verlag, Berlin, 1970).Google Scholar
  10. 10.
    G. W. Mackey,Induced Representations of Groups and Quantum Mechanics (Benjamin, New York, 1968).Google Scholar
  11. 11.
    J. von Neumann,Math. Ann. 102, 370 (1929).Google Scholar
  12. 12.
    C. Piron,Helv. Phys. Acta 37, 439 (1964).Google Scholar
  13. 13.
    C. Piron, “Observables in general quantum theory,” delivered at International School of Physics “Enrico Fermi,” Foundations of Quantum Mechanics, 29 June–11 July, 1970, to be published.Google Scholar
  14. 14.
    D. Ballentine, Rev. Mod. Phys.42, 358 (1970).Google Scholar

Copyright information

© Plenum Publishing Corporation 1972

Authors and Affiliations

  • C. Piron
    • 1
  1. 1.University of DenverDenver

Personalised recommendations