Advertisement

The Hilbert Space Axiom in Quantum Mechanics

  • Stanley P. Gudder

Abstract

The Hilbert space axiom is the basic postulate of conventional quantum mechanics. In essence it maintains that the observables for an isolated quantum mechanical system S can be represented by self-adjoint operators on a complex Hilbert space H; the states for S can be represented by density operators on H; and the dynamics can be represented by a one-parameter unitary group on H. The power of this axiom is indisputable, but what is its justification? Where does the Hilbert space come from? And if it is necessary to use Hilbert spaces, why must they be complex?

Keywords

Quantum Mechanics Hide Variable Division Ring Definite Function Complex Hilbert Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Notes

  1. 1.
    G. Birkhoff and J. von Neumann, “The Logic of Quantum Mechanics,” Ann. Math. 37, 823–843 (1936).CrossRefGoogle Scholar
  2. 2.
    D. Bohm and J. Bub, “A Refutation of the Proof of Jauch and Piron that Hidden Variables Can Be Excluded in Quantum Mechanics,” Rev. Mod. Phys. 38, 470–475 (1966).CrossRefGoogle Scholar
  3. 3.
    A. Clifford and G. Preston, The Algebraic Theory of Semigroups, Vol. 1 (Amer. Math. Soc. Providence, Rhode Island, 1961).Google Scholar
  4. 4.
    E. B. Davies, Quantum Theory of Open Systems (Academic Press, New York, 1976).Google Scholar
  5. 5.
    E. B. Davies and J. Lewis, “An Operational Approach to Quantum Probability,” Commun. Math. Phys. 17, 239–260 (1970).CrossRefGoogle Scholar
  6. 6.
    P. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, England, 1930).Google Scholar
  7. 7.
    N. Dunford and J. Schwartz, Linear Operators ,Parts 1 and 2 (Wiley Interscience, New York, 1958 and 1963).Google Scholar
  8. 8.
    G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley Interscience, New York, 1972).Google Scholar
  9. 9.
    D. Foulis, “Baer *-Semigroups,” Proc. Am. Math. Soc. 11, 648–654 (1960).Google Scholar
  10. 10.
    D. Foulis, “A Note on Orthomodular Lattices,” Portugal Math. 21, 65–72 (1962).Google Scholar
  11. 11.
    A. Gleason, “Measures on Closed Subspaces of a Hilbert Space,” J. Rat. Mech. Anal. 6, 885–893 (1957).Google Scholar
  12. 12.
    R. Greechie, “An Orthomodular Lattice with a Full Set of States not Embeddable in Hilbert Space,” Carïb. J. Sci. Math. 2, 15–26 (1969).Google Scholar
  13. 13.
    R. Greechie, “Orthomodular Lattices Admitting no States,” J. Comb. Theory 10, 119–132 (1971).CrossRefGoogle Scholar
  14. 14.
    S. Gudder, “Elementary Length Topologies in Physics,” SIAM J. Appl. Math. 16, 1011–1019 (1968).CrossRefGoogle Scholar
  15. 15.
    S. Gudder, “Quantum Probability Spaces,” Proc. Am. Math. Soc. 21, 296–302 (1969).CrossRefGoogle Scholar
  16. 16.
    S. Gudder, “Quantum Logics, Physical Space, Position Observables and Symmetries,” Rep. Math. Phys. 3, 193–202 (1972).CrossRefGoogle Scholar
  17. 17.
    S. Gudder, “Generalized Measure Theory,” Found. Phys. 3, 399–411 (1973).CrossRefGoogle Scholar
  18. 18.
    S. Gudder, Stochastic Methods in Quantum Mechanics (North-Holland, New York, 1979).Google Scholar
  19. 19.
    S. Gudder and R. Hudson, “A Noncommutative Probability Theory,” Trans. Am. Math. Soc. 245, 1–41 (1978).CrossRefGoogle Scholar
  20. 20.
    S. Gudder and J.-P. Marchand, “A Coarse-Grained Measure Theory,” Bull. Acad. Polon. Sci. 28, 557–564 (1980).Google Scholar
  21. 21.
    S. Gudder and J. Michel, “Representations of Baer * Semigroups” Proc. Amer. Math. Soc. 81, 157–163 (1981).Google Scholar
  22. 22.
    S. Gudder and C. Piron, “Observables and the Field in Quantum Mechanics,” J. Math. Phys. 12, 1583–1588 (1971).CrossRefGoogle Scholar
  23. 23.
    R. Haag and D. Kastler, “An Algebraic Approach to Quantum Field Theory,” J. Math. Phys. 5, 848–861 (1964).CrossRefGoogle Scholar
  24. 24.
    E. Hewitt and H. Zuckerman, “The l1-Algebra of a Commutative Semigroup,” Trans. Am. Math. Soc. 83, 70–97 (1956).Google Scholar
  25. 25.
    J. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
  26. 26.
    R. Lindahl and P. Maserick, “Positive-Definite Functions on Involution Semigroups,” Duke Math. J. 39, 771–782 (1971).CrossRefGoogle Scholar
  27. 27.
    G. Mackey, The Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963).Google Scholar
  28. 28.
    M. Naimark, Normed Algebras (Noordhoff, Gröningen, The Netherlands 1972).Google Scholar
  29. 29.
    C. Piron, “Axiomatique quantique,” Helv. Phys. Acta. 37, 439–468 (1964).Google Scholar
  30. 30.
    C. Piron, Foundations of Quantum Mechanics (Benjamin, New York, 1976).Google Scholar
  31. 31.
    J. Pool, “Baer *-Semigroups and the Logic of Quantum Mechanics,” Commun. Math. Phys. 9, 118–141 (1968).CrossRefGoogle Scholar
  32. 32.
    J. Pool, “Semi-Modularity and the Logic of Quantum Mechanics,” Commun. Math. Phys. 9, 212–228 (1968).CrossRefGoogle Scholar
  33. 33.
    R. Powers, “Self-Adjoint Algebras of Unbounded Operators,” Commun. Math. Phys. 21, 85–124 (1971).CrossRefGoogle Scholar
  34. 34.
    G. Rüttiman, “On the Logical Structure of Quantum Mechanics,” Found. Phys. 1, 173–182 (1972).CrossRefGoogle Scholar
  35. 35.
    I. Segal, “Postulates for General Quantum Mechanics,” Ann. Math. 48, 930–948 (1947).CrossRefGoogle Scholar
  36. 36.
    I. Segal, Mathematical Problems of Relativistic Physics (Amer. Math. Soc, Providence, Rhode Island, 1963).Google Scholar
  37. 37.
    M. Stone, “On One-Parameter Unitary Groups in Hilbert Space,” Ann. Math. 33, 643–648 (1932).CrossRefGoogle Scholar
  38. 38.
    V. Varadarajan, Geometiy of Quantum Theory ,Vol. 1 (Van Nostrand Reinhold, Princeton, New Jersey, 1968).Google Scholar
  39. 39.
    J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey, 1955).Google Scholar
  40. 40.
    J. Zerbe, “Generalized Measure Theory,” Dissertation, Univ. of Denver, Denver, Colorado (1979).Google Scholar
  41. 41.
    N. Zierler, “Axioms for Non-relativistic Quantum Mechanics,” Pac. J. Math. 11, 1151–1169 (1961).Google Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • Stanley P. Gudder
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of DenverDenverUSA

Personalised recommendations