Abstract
We introduce the notion of a strict quantization of a symplectic manifold and show its existence under a topological condition.
Similar content being viewed by others
References
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory and quantization, I. II, Ann. Phys. 110 (1978), 62–110, 111-151.
Blanchard, E.: Subtriviality of continuous fields of nuclear C*-algebras (with Appendix by E. Kirchberg), J. reine angnew. Math. 489 (1997), 133–149.
Bott, R. and Tu, L.: Differential Forms in Algebraic Topology, Springer-Verlag, New York, Heidelberg, 1982.
De Wilde, M. and Lecomte, P. B. A.: Existence of star-product and of formal deformations in Poisson Lie algebra of arbitrary symplectic manifold, Lett. Math. Phys. 7 (1983), 487–496.
Dixmier, J.: C*-algebras, North-Holland, Amsterdam, 1977.
Fedosov, B. V.: Deformation Quantization and Index Theorem, Akademie-Verlag, Berlin,1996.
Gerstenhaber, M.: On the deformation of rings and algebras, Ann. Math. 79 (1964),59–103.
Gompf, R.: A new construction of symplectic manifolds, Ann. Math. 142 (1995), 527–595.
Kirchberg, E. and Wassermann, S.: Operations on continuous bundles of C*-algebras,Math. Ann. 303 (1995), 677–697.
Kirchberg, E. and Wassermann, S.: Exact groups and continuous bundles of C*-algebras,Math. Ann. 315 (1999), 169–203.
Klimek, S. and Lesniewski, A.: Quantum Riemann surfaces: II. The discrete series, Lett. Math. Phys. 24 (1992), 125–139.
Moser, J. K.: On the volume elements on manifolds, Trans. Amer. Math. Soc. 120 (1965),280–296.
Natsume, T.: C*-algebraic deformation quantization of closed Riemann surfaces, In: J. Cuntz and S. Echterhodd (eds), Proc. SNB-Workshopon C*-algebras, Münster,Germany, 1999, pp. 142–150.
Natsume, T. and Nest, R.: Topological approach to quantum surfaces, Comm. Math. Phys. 202 (1999), 65–87.
Omori, H., Maeda, Y. and Yoshioka, A.: Weyl manifolds and deformation quantization,Adv. Math. 85 (1991), 224–255.
Ozawa, N.: Amenable actions and exactness for discrete groups, C.R. Acad. Sci. Paris Sér. I Math. 330 (2000), 691–695, math.OA/0002185.
Podleś, P.: Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.
Rieffel, M. A.: Continuous fields of C*-algebras coming from group cocycles and actions,Math. Ann. 283 (1989), 631–643.
Rieffel, M. A.: Deformation quantization for Heisenberg manifolds, Comm. Math. Phys. 122 (1989), 531–562.
Sheu, A. J.-L.: Quantization of the Poisson SU(2) and its Poisson homogeneous space-the 2-sphere, Comm. Math. Phys. 135 (1991), 217–232.
Shubin, M. A.: Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Natsume, T., Nest, R. & Peter, I. Strict Quantizations of Symplectic Manifolds. Letters in Mathematical Physics 66, 73–89 (2003). https://doi.org/10.1023/B:MATH.0000017652.90999.d8
Issue Date:
DOI: https://doi.org/10.1023/B:MATH.0000017652.90999.d8