Advertisement

International Journal of Theoretical Physics

, Volume 16, Issue 8, pp 615–633 | Cite as

On representation spaces in geometric quantization

  • Jedrzej Śniatycki
  • Stanley Toporowski
Article

Abstract

The structure of the space of wave functions in the representation given by a complete strongly admissible polarization of the phase space is investigated. The conditions that the wave functions should be covariant constant along the real part of the polarization define the Bohr-Sommerfeld set of the representation containing the supports of all wave functions. There is a natural scalar product for the wave functions defined on the Bohr-Sommerfeld set. It is shown, for a real polarization, that the resulting Hilbert space of wave functions is not trivial if and only if the Bohr-Sommerfeld set is not empty.

Keywords

Wave Function Hilbert Space Field Theory Phase Space Elementary Particle 
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. Auslander, L. and Kostant, B. (1971).Inventiones Mathematicae,14, 255.Google Scholar
  2. Bargmann, V. (1961).Communications in Pure and Applied Mathematics,14, 187.Google Scholar
  3. Blattner, R. J. (1973). “Quantization and Representation Theory,” inHarmonic Analysis on Homogeneous Spaces, Proceedings of Symposia in Pure Mathematics. American Mathematical Society, Providence, Rhode Island, Vol. 26, pp. 147–165.Google Scholar
  4. Kostant, B. (1970). “Quantization and Unitary Representations,” inLecture Notes in Mathematics. Springer, Berlin, Vol. 170, pp. 87–208.Google Scholar
  5. Nirenberg, L. (1957). “A complex Frobenius Theorem,” inSeries on Analytic Functions. Institute for Advanced Studies, Princeton, Vol. 1, pp. 1172–1189.Google Scholar
  6. Simms, D. J. (1974). “Geometric Quantization of Symplectic Manifolds,” inProceedings of the International Symposium on Mathematical Physics, Warsaw.Google Scholar
  7. Śniatycki, J. (1974). “Bohr-Sommerfeld Quantum Systems,” inProceedings of the Third International Colloquium on Group Theoretical Methods in Physics, Marseille, p. 42–51.Google Scholar
  8. Śniatycki, J. (1975).International Journal of Theoretical Physics,14, 277.Google Scholar
  9. Souriau, J.-M. (1970).Structure des Systèmes Dynamiques, Dunod, Paris.Google Scholar
  10. Toporowski, S. (1976). “Representation Spaces in Geometric Quantization for Complete Strongly Admissible Real Polarizations, MSc. Thesis, University of Calgary.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • Jedrzej Śniatycki
    • 1
  • Stanley Toporowski
    • 1
  1. 1.Department of Mathematics and StatisticsThe University of CalgaryCalgaryCanada

Personalised recommendations