Abstract
Starting from work by F. A. Berezin, an earlier paper by the author obtained an invariant star product on every nonexceptional symmetric Kähler space. This would be a generalization to those spaces of the star product on ℝ2n corresponding to Wick quantization. In this Letter we consider, via geometric quantization, the unitary operators corresponding to geodesic symmetries, and we define a Weyl quantization (first defined by Berezin on rank 1 spaces) in a way similar to the way in which the Weyl quantization can be obtained from the Wick quantization on ℝ2n. We then calculate every Hochschild 2-cochain of another invariant star product, equivalent to the Wick one, which would be a generalization to those spaces of the Moyal star product on ℝ2n. M. Cahen and S. Gutt have already provided a theorem of existence and essential unicity of an invariant star product on every irreducible Kähler symmetric space.
Similar content being viewed by others
References
Flato, M., Lichnerowicz, A., and Sternheimer, D., Compt. Math. 31, 47–82 (1975).
Flato, M., Lichnerowicz, A., and Sternheimer, D., J. Math. Phys. 17, 1754–1762 (1976).
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sterneimer, D., Ann. Phys. 111, 61–110 (1978) and 111, 111–151 (1978).
Vey, J., Comm. Math. Helv. 50, 421–454 (1975).
Lichnerowicz, A., Ann. Di math. 123, 287–330 (1980).
Lichnerowicz, A., Ann. Inst. Fourier 32, 157–209 (1982).
Agarwal, G. and Wolf, E., Phys. Rev. 2, 2161 (1970).
Fronsdal, C., Rep. Math. Phys. 15, 11 (1978).
Moreno, C., ‘*-Products and Geometric Quantization’, (Preprint). ‘Journées d'Analyse Harmonique et Quantification Geométrique’ 22–24 novembre 1984, Lyon, France.
Moreno, C., Lett. Math. Phys. 11, 61 (1986).
Lichnerowicz, A., Lett. Math. Phys. 2, 133 (1977).
De Wilde, M., and Lecompte, P., Lett. Math. Phys. 7, 487 (1983).
Cahen, M. and Gutt, S., Lett. Math. Phys. 6 395 (1982)
Cahen, M. and Gutt, S., C. R. Acad. Sci. Paris 291A, 545 (1980)
Cahen, M. and Gutt, S., Lett. Math. Phys. 5, 219 (1981).
Cortet, J. C., ‘Déformations et alèbres d'opérateurs: Applications en Mécanique Statistique et Théorie des Groupes’, Thèse d'État, Université de Dijon (1983).
Molin, P., ‘Invariance et convariance de déformations de structures sur une variété symplectique. Applications au Groupe de Poincaré’, Thède de 3° cycle, Université de Dijon (1981).
Bayen, F. and Fronsdal, C., ‘Quantization on the Sphere’, Preprint (1978).
Kammerer, J. B., J. Math. Phys. 27, 529 (1986).
Bohnke, G., C. R. Acad. Sci. Paris. 3031, 729 (1986).
Wildberger, N.J., ‘On the Fourier Transform of a Compact, Semisimple, Lie Group’, Preprint, University of Toronto (1986).
Arnal, D. and Cortet, J. C., ‘La notion de ⋆-produit et ses applications aux représentations de groupes’. Dans les Actes des Journées Rélativistes 1979, Université de Angers, France.
Hazewinkel, M. (ed.), Deformations of Algebras and Applications, Nato-Asi, 1–14 June 1986, Il Ciocco, Italy. D. Reidel, Dordrecht, Holland (to appear).
Berezin, F. A., Math. U.S.S.R. Izvestija 9, 341 (1975).
Berezin, F. A., Math. U.S.S.R. Izvestija 3, 1109 (1974).
Grossmann, A. and Huguenin, P., ‘Group Theoretical Aspects of the Wigner-Weyle Isomorphism’, Preprint (1977).
Berezin, F. A., Commun. Math. Phys. 40, 153 (1975).
Moreno, C., ‘Produits ⋆ et analyse spectrale’, Preprint (1981).
Haris-Chandra, Amer. Math. 77, 743 (1955), 78 1 (1956); and 78, 564 (1956).
Knapp, A. W., ‘Bounded Symmetric Domains and Holomorphic Discrete Series’ in W. M.Boothby and G. L.Weiss (eds.), Symmetric Spaces, Pure and Applied Mathematics No. 8. Marcel Dekker, N.Y. (1972).
Moreno, C., Lett. Math. Phys. 12, 217 (1986).
Moreno, C. and Ortega-Navarro, P., Ann. Inst. Henri Poincaré 38, 215 (1983) and Lett. Math. Phys. 7, 181 (1983).
Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York (1978).
Onofri, E., J. Math. Phys. 16, 1087 (1975).
Rawnsley, J. H., Quart. J. Math. Oxford (2), 28, 403 (1977).
Perelomov, A. M., Commun. Math. Phys. 26, 222 (1972).
Combet, E., Intégrales exponentielles, Lectures Notes in Math. No. 937, Springer-Verlag, Berlin (1982).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Moreno, C. Geodesic symmetries and invariant star products on Kähler symmetric spaces. Lett Math Phys 13, 245–257 (1987). https://doi.org/10.1007/BF00423452
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00423452