Foundations of Physics

, Volume 10, Issue 9–10, pp 705–722 | Cite as

The algebraization of quantum mechanics and the implicate order

  • F. A. M. Frescura
  • B. J. Hiley


It has been proposed that the implicate order can be given mathematical expression in terms of an algebra and that this algebra is similar to that used in quantum theory. In this paper we bring out in a simple way those aspects of the algebraic formulation of quantum theory that are most relevant to the implicate order. By using the properties of the standard ket introduced by Dirac we describe in detail how the Heisenberg algebra can be generalized to produce an algebraic structure in which it is possible to describe space translations in a way that is analogous to the description of rotations in a Clifford algebra. This approach opens up the possibility of going beyond the limits of the present quantum formalism and we discuss briefly some of the new implications.


Quantum Mechanic Quantum Theory Algebraic Structure Mathematical Expression Clifford Algebra 
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.
    F. A. M. Frescura and B. J. Hiley,Found. Phys. 10, 7 (1980).Google Scholar
  2. 2.
    D. J. Bohm,Found. Phys. 1, 359 (1971);3, 139 (1973).Google Scholar
  3. 3.
    N. Bohr,Atomic Physics and Human Knowledge (New York, 1961).Google Scholar
  4. 4.
    B. D'Espagnat,Conceptual Foundations of Quantum Mechanics (Menlo Park, California, 1971).Google Scholar
  5. 5.
    D. J. Bohm and B. J. Hiley,Found. Phys. 5, 93 (1975).Google Scholar
  6. 6.
    P. Jordan, J. von Neumann, and E. P. Wigner,Ann. Math. 35, 29 (1934).Google Scholar
  7. 7.
    I. E. Segal,Ann. Math. 48, 930 (1947).Google Scholar
  8. 8.
    G. G. Emch,Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley-Interscience, London, 1972).Google Scholar
  9. 9.
    P. A. M. Dirac,Quantum Mechanics (Oxford, 1966).Google Scholar
  10. 10.
    G. Birkhoff and J. von Neumann,Ann. Math. 37, 823 (1936).Google Scholar
  11. 11.
    N. N. Bogoliubov, A. A. Logunov, and I. T. Todorov,Introduction to Axiomatic Quantum Field Theory (Benjamin, 1975).Google Scholar
  12. 12.
    V. Fock,Z. Physik 75, 622 (1932).Google Scholar
  13. 13.
    P. A. M. Dirac,Phys. Rev. 139 B, 684 (1965).Google Scholar
  14. 14.
    M. Schönberg,Nuovo Cim. (Suppl. 6, 356 (1957).Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • F. A. M. Frescura
    • 1
  • B. J. Hiley
    • 1
  1. 1.Department of Physics, Birkbeck CollegeUniversity of LondonLondonEngland

Personalised recommendations