Abstract
After a short review of results which we recently obtained on deformations of Lie algebras associated with symplectic manifolds, we discuss physical applications and treat some examples with deformed Poisson brackets. We make explicit a connection between classical and quantum mechanics, and the theory of Dirac brackets for second class constraints, from the viewpoint of deformation theory. Finally we discuss the general Dirac constraints formalism.
REFERENCES
- 1. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963). Google Scholar
- 2. M. Gerstenhaber, Ann. Math. 79, 59 (1964). Google ScholarCrossref
- 3. C. Chevalley and S. Eilenberg, Trans. Amer. Math. Soc. 63, 85 (1948). Google ScholarCrossref
- 4. J. Vey, “Déformation du crochet de Poisson sur une variété symplectique,” Comment. Helv. Math. (to be published). Google Scholar
- 5. A. Lichnerowicz, J. Math. Pures Appl. 53, 459 (1974) Google Scholar
A. Lichnerowicz, and C. R. Acad. Sci. Paris A 277, 215 (1973). Google Scholar - 6. M. Flato, A. Lichnerowicz, and D. Sternheimer, C. R. Acad. Sci. Paris A 279, 877 (1975); Google Scholar
Compositio Mathematica 31, 47 (1975). Google Scholar - 7. A. Nijenhuis, Indag. Math. 17, 390 (1955). Google ScholarCrossref
- 8. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956). Google Scholar
- 9. G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 (1970). Google ScholarCrossref
- 10. D. Arnal, “Symmetric non self‐adjoint operators in an enveloping algebra,” J. Funct. Anal. (to be published). Google Scholar
- 11. N. Bourbaki, Algèbres de Lie (Hermann, Paris, 1960). Google Scholar
- 12. R. Kerner, C. R. Acad. Sci. Paris A 269, 175 (1969). Google Scholar
- 13. M. Flato and D. Sternheimer, Phys. Rev. Letters 16, 1185 (1966); Google ScholarCrossref
M. Flato and D. Sternheimer, Commun. Math. Phys. 12, 296 (1969). , Google ScholarCrossref - 14. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Sciences Monograph Series No. 2 (Yeshiva University, New York, 1964). Google Scholar
- 15. J. Sniatycki, Ann. Inst. H. Poincaré 20, 365 (1974). Google Scholar
- 16. A. Lichnerowicz, C. R. Acad. Sci. Paris A 280, 523 (1974). Google Scholar
- 17. Y. Nambu, Phys. Rev. D 7, 2405 (1973). Google ScholarCrossref
- 18. F. Bayen and M. Flato, Phys. Rev. D 11, 3049 (1975). Google ScholarCrossref
- 19. N. Mukunda and E. C. G. Sudarshan, Phys. Rev. D 13, 2846 (1976). Google ScholarCrossref
- 20. A. Lichnerowicz, C. R. Acad. Sci. Paris A 280, 37 (1975); Google Scholar
A. Lichnerowicz, A 281 (1975); Google Scholar
Variétés symplectiques, variétés canoniques et systèmes dynamiques (E. T. Davies memorial volume, to be published); Google Scholar
Les variétés de Poisson et leurs algèbres de Lie associées (to be published). Google Scholar - 21. M. Flato, A. Lichnerowicz, and D. Sternheimer, “Algèbres de Lie attachées à une variété canonique,” J. Math. Pures Appl. 54, 445 (1975). Google Scholar
- © 1976 American Institute of Physics.
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
