International Journal of Theoretical Physics

, Volume 4, Issue 4, pp 247–265

Phase space, fibre bundles and current algebras

  • Basil J. Hiley
  • Allan E. G. Stuart
Article

Abstract

The purpose of this paper is to extend into phase space the cellular description introduced by Bohmet al. (1970) and to show how this may help to give an understanding of the current algebra approach to elementary particle phenomena. We investigate this cellular structure in phase space in some detail and show how certain features of the structure may be described in terms of the mathematics of fibre bundle theory. The frame bundle is discussed and compared with the Yang-Mills theory. As a result of this discussion we are able to introduce generalised currents which are related to the duals of the curvature forms, and these are shown to span the Lie algebra of a sub-group of the structure group of the frame bundle. We then discuss the implications of these results in terms of our cell structure. By assuming that the de Rahm cohomology, defined by the curvature forms and their duals, reflect a cohomology on the integers defined on the original cell structure, we show that the currents and ‘curvature’ can be given a meaning in terms of a discrete structure. In this case the currents only span a Lie algebra in some suitable limit, implying that a description using Lie algebras is only an approximation.

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References

  1. Ambrose, W. and Singer, I. M. (1953).Transactions of the American Mathematical Society,75, 428.Google Scholar
  2. Bishop, R. L. and Crittenden, R. J. (1964).Geometry of Manifolds. Academic Press.Google Scholar
  3. Bohm, D. (1969).Proceedings of the Illinois Symposium of the Philosophy of Science. Google Scholar
  4. Bohm, D., Hiley, B. J. and Stuart, A. E. G. (1970).International Journal of Theoretical Physics, Vol. 3, No. 3, pp. 171–183.Google Scholar
  5. Herman, R. (1966).Lie Groups for Physicists. Benjamin.Google Scholar
  6. Hiley, B. J. (1968).Quantum Theory and Beyond, p. 181. Cambridge.Google Scholar
  7. Ishiwara, J. (1915).Tokyo Sugaku Buturigakkawi Kizi,8, 106.Google Scholar
  8. Loos, H. G. (1965).Nuclear Physics,72, 677.Google Scholar
  9. Loos, H. G. (1966).Annals of Physics,36, 486.Google Scholar
  10. Lubkin, E. (1963).Annals of Physics,23, 233.Google Scholar
  11. Mackey, G. W. (1963).The Mathematical Foundations of Quantum Mechanics. Benjamin.Google Scholar
  12. Mayer, M. E. (1966).Non-Compact Groups in Particle Physics. Benjamin.Google Scholar
  13. Ne'eman, Y. (1961).Nuclear Physics,26, 222.Google Scholar
  14. Olson, D. N., Schopper, H. F. and Wilson, R. R. (1961).Physical Review Letters,6, 286.Google Scholar
  15. O'Raifeartaigh, L. (1965).Physical Review,139, B.1052.Google Scholar
  16. Schild, A. (1949).Canadian Journal of Mathematics,1, 29.Google Scholar
  17. Takabayasi, T. (1965).Progress of Theoretical Physics,34, 124.Google Scholar
  18. Takabayasi, T. (1970).Progress of Theoretical Physics,43, 1117.Google Scholar
  19. Utiyama, R. (1956).Physical Review,101, 1597.Google Scholar
  20. Weyl, H. (1922).Space-Time-Matter. Dover.Google Scholar
  21. Wheeler, J. A. (1963).Relativity, Groups and Topology, p. 317. Blackie.Google Scholar
  22. Yukawa, H. (1965).Proceedings of the International Conference on Elementary Particles (Kyoto), 139.Google Scholar

Copyright information

© Plenum Publishing Company Limited 1971

Authors and Affiliations

  • Basil J. Hiley
    • 1
  • Allan E. G. Stuart
    • 2
  1. 1.Department of Physics, Birkbeck CollegeUniversity of LondonLondon W.C.1
  2. 2.Department of MathematicsThe City UniversityLondon

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