Abstract
B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov’s operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally, some remarks are made on the implications for deformation quantization of Fedosov’s index theorem on general symplectic manifolds.
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References
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Deformation theory and quantization, I and II, Ann. Phys. 111, (1977), 61–151.
Berezin, F.A., Quantization, Math USSR Izv. 8 (1974), 1109–1165.
Bohm, D., Quantum Theory, Prentice-Hall, New York, 1951.
Boutet de Monvel, L. and Guillemin, V., The spectral theory of Toeplitz operators, Annals. of Math. Studies 99, Princeton University Press, Princeton, 1981.
De Wilde, M. and Lecomte, P., Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), 487–496.
Donin, J., On the quantization of Poisson brackets, Advances in Math. (to appear).
Fedosov, B., Formal quantization, Some Topics of Modern Mathematics and their Applications to Problems of Mathematical Physics (in Russian), Moscow (1985), 129–136.
Fedosov, B., Index theorem in the algebra of quantum observables, Sov. Phys. Dokl. 34 (1989), 318–321.
Fedosov, B.V., Reduction and eigenstates in deformation quantization, Advances in Partial Differential Equations, Akademie Verlag, Berlin (to appear).
Fedosov, B., A simple geometrical construction of deformation quantization, J. Diff. Geom. 30 (1994), 327–333.
Flato, M., and Sternheimer, D., Closedness of star products and cohomologies, this volume, pp. 241–259.
Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Interscience, New York, 1963.
Krantz, S.G. and Parks, H.R., A Primer of Real Algebraic Functions, Basler Lehrbücher vol. 4, Birkhäuser, Basel, 1992.
Nest, R. and Tsygan, B., Algebraic index theorem for families, Advances in Math., (to appear).
Omori, H., Maeda, Y., and Yoshioka, A., Weyl manifolds and deformation quantization, Advances in Math. 85 (1991), 224–255.
Weinstein, A., Symplectic manifolds and their lagrangian submanifolds, Advances in Math. 6 (1971), 329–346.
Weinstein, A., Classical theta functions and quantum tori, Publ. RIMS Kyoto Univ. 30 (1994), 327–333.
Weinstein, A., Deformation quantization, Séminaire Bourbaki, 46ème année, 1993–94, n° 789, juin 1994 (to appear in Astérisque).
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Research supported by Deutsche Forschungsgemeinschaft, Az.: Em 47/1–1.
Research partially supports by NSF Grants DMS-90–01089 and DMS-93–09653.
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Emmrich, C., Weinstein, A. (1994). The Differential Geometry of Fedosov’s Quantization. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics, vol 123. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0261-5_7
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DOI: https://doi.org/10.1007/978-1-4612-0261-5_7
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