Abstract
In this Letter, we construct nonequivalent star products on CPn by phase space reduction. It turns out that the nonequivalent star products occur very natural in the context of phase space reduction by deforming the momentum map of the U(1)-action on Cn+1\{0}; into a quantum momentum map and the corresponding momentum value into a quantum momentum value such that the level set, i.e. the ‘constraint surface’, of the quantum momentum map coincides with the classical one. All equivalence classes of star products on CPn are obtained by this construction.
Similar content being viewed by others
References
Abraham, R. and Marsden, J. E.: Foundations of Mechanics, 2nd edn, Addison Wesley, Reading, Mass., 1985.
Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory and quantization, Ann. Phys. 111(I) (1978), 61-110, part II: 111-151.
Bertelson, M., Cahen, M. and Gutt, S.: Equivalence of star products, Classical Quantum Gravity 14(1997), A93-A107.
Bordemann, M., Brischle, M., Emmrich, C. and Waldmann, S.: Phase space reduction for star products: an explicit construction for ℂℙn, Lett. Math. Phys. 36(1996), 357-371.
Bordemann, M., Brischle, M., Emmrich, C. and Waldmann, S.: Subalgebras with converging star products in deformation quantization: An algebraic construction for ℂℙn, J. Math. Phys. 37(1996), 6311-6323.
Bordemann, M. and Waldmann, S.: A Fedosov star product of the Wick type for Kähler manifolds, Lett. Math. Phys. 41(1997), 243-253.
De Wilde, M. and Lecomte, P. B. A.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett.Math. Phys. 7(1983), 487-496.
Fedosov, B.: A Simple geometrical construction of deformation quantization, J. Differential Geom. 40(1994), 213-238.
Fedosov, B.: Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1996.
Fedosov, B.: Non-Abelian reduction in deformation quantization, Preprint, 1997.
Gerstenhaber, M. and Schack, S.: Algebraic cohomology and deformation theory, in: M. Hazewinkel and M. Gerstenhaber (eds), Deformation Theory of Algebras and Structures and Applications, Kluwer Acad. Publ., Dordrecht, 1988.
Kontsevich, M.: Deformation quantization of Poisson manifolds, Preprint, September 1997, q-alg/9709040.
Nest, R. and Tsygan, B.: Algebraic index theorem, Comm. Math. Phys. 172(1995), 223-262.
Omori, H., Maeda, Y. and Yoshioka, A.: Weyl manifolds and deformation quantization, Adv. Math. 85(1991), 224-255.
Schirmer, J.: A Star product for complex Grassmann manifolds, Preprint, Freiburg, September 1997, q-alg/9709021.
Xu, P.: Fedosov *-Products and quantum momentum maps, Preprint, 1996, q-alg/9608006.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Waldmann, S. A Remark on Nonequivalent Star Products via Reduction for CPn . Letters in Mathematical Physics 44, 331–338 (1998). https://doi.org/10.1023/A:1007415208707
Issue Date:
DOI: https://doi.org/10.1023/A:1007415208707