Advertisement

Communications in Mathematical Physics

, Volume 17, Issue 3, pp 210–232 | Cite as

Derivations of Lie brackets and canonical quantisation

  • A. Joseph
Article

Abstract

An extensive analysis of the Dirac problem of canonical quantisation is reported. In this a known solution [1] has been found to be unique to within a canonical transformation under a certain prescribed condition. This proves a conjecture due to Streater [2]. A further canonically inequivalent solution is obtained by relaxing this condition. The results obtained are discussed in terms of the derivation algebras pertaining to the Classical and Quantum Lie brackets. Applications to the study of higher symmetries and to realisations of Lie algebras as polynomial functions of canonical operators are pointed out.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Polynomial Function 
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

  1. 1.
    Souriau, J. M.: Commun. Math. Phys.1, 374 (1966).Google Scholar
  2. 2.
    Streater, R. F.: Commun. Math. Phys.2, 354 (1966).Google Scholar
  3. 3.
    Dirac, P. A. M.: Quantum mechanics, 4th Ed., Chap. IV, pp. 84–89. Oxford: Clarendon Press 1958.Google Scholar
  4. 4.
    Abraham, R.: Foundations of mechanics, Chap. IV, pp. 132–153. New York: W. A. Benjamin 1967.Google Scholar
  5. 5.
    Shewell, J. R.: Am. J. Phys.27, 16 (1959).Google Scholar
  6. 6.
    Groenewold, H. J.: Physica12, 405 (1946).Google Scholar
  7. 7.
    Hermann, R.: Lie groups for physicists, Chap. 16, pp. 137–149. New York: W. A. Benjamin 1966.Google Scholar
  8. 8.
    Fradkin, D. M.: Progr. Theoret. Phys.37, 798 (1967).Google Scholar
  9. 9.
    Mukunda, N.: Phys. Rev.155, 1383 (1967).Google Scholar
  10. 10.
    Simoni, A., Zaccaria, F.: Nuovo Cimento59A, 280 (1969).Google Scholar
  11. 11.
    Pauli, W.: Handbuch der Physik, Book V, Part I, p. 50. Berlin-Göttingen-Heidelberg: Springer 1958.Google Scholar
  12. 12.
    Rosenbaum, D. M.: J. Math. Phys.10, 1127 (1969).Google Scholar
  13. 13.
    Jacobson, N.: Lie algebras. New York: Interscience 1962.Google Scholar
  14. 14.
    Wollenberg, L. S.: Progress report no. 12, Wave mechanics group, Mathematical Institute, Oxford 1965–1966, p. 4.Google Scholar
  15. 15.
    —— Proc. Am. Math. Soc.20, 315 (1969).Google Scholar
  16. 16.
    Rosenbaum, D. M.: J. Math. Phys.8, 1973 (1967).Google Scholar
  17. 17.
    Van Hove, L.: Acad. Roy. Belg. Bull. Classe Sci. Mém. (5)37, 610 (1951).Google Scholar
  18. 18.
    Jauch, J. M.: Foundations of quantum mechanics, Chap. 12, pp. 195–205. London: Addison-Wesley Publ. Co. 1968.Google Scholar
  19. 19.
    Joseph, A., Solomon, A. I.: J. Math. Phys.11, 748 (1970).Google Scholar
  20. 20.
    Kurşunoğlu, B.: Modern quantum theory, Chap. XIII, pp. 364–367. London: W. H. Freeman and Co. 1962.Google Scholar
  21. 21.
    Goldstein, J. W.: Classical mechanics, Chap. 8, pp. 255–258. London: Addison-Wesley Publ. Co. 1962.Google Scholar
  22. 22.
    Eisenhart, L. P.: Continuous groups of transformations, Chap. VI, pp. 281–292. New York: Dover Publ. 1961.Google Scholar
  23. 23.
    Maiella, G., Vitale, B.: Nuovo Cimento57A, 330 (1967).Google Scholar
  24. 24.
    Duimio, F., Pauri, M.: Nuovo Cimento51A, 1141 (1967).Google Scholar
  25. 25.
    Cisneros, A., McIntosh, H. V.: J. Math. Phys.10, 277 (1969).Google Scholar
  26. 26.
    Pauri, M., Prosperi, G.: J. Math. Phys.7, 366 (1966).Google Scholar
  27. 27.
    Simoni, A., Zaccaria, F., Vitale, B.: Nuovo Cimento51A, 448 (1967).Google Scholar
  28. 28.
    Guest, P. B.: Nuovo Cimento61A, 593 (1969).Google Scholar
  29. 29.
    Chow, Y.: J. Math. Phys.10, 975 (1969).Google Scholar
  30. 30.
    Coleman, S., Wess, J., Zumino, B.: Phys. Rev.177, 2239 (1969).Google Scholar

Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • A. Joseph
    • 1
    • 2
  1. 1.Mathematical InstituteOxford
  2. 2.Physics DepartmentTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations