On representations of star product algebras over cotangent spaces on Hermitian line bundles. (English) Zbl 1038.53087

From the authors’ abstract: For every formal power series \(B\) of closed two-forms on a manifold \(Q\) and every value of an ordering parameter from \([0,1]\) we construct a concrete star product \(B \) on the cotangent bundle \(T^{*}Q\). The star product \(B\) is associated to the symplectic form on \(T^{*}Q\) given by the sum of the canonical symplectic form and the pull back of \(B\) to \(T^{*}Q\). Deligne’s characteristic class of \(B\) is calculated and shown to coincide with the formal de Rham cohomology class of \(^{*}B \) divided by \(i\). Therefore, every star product on \(T^{*}Q\) corresponding to the canonical Poisson bracket is equivalent to some \(B\). It turns out that every \(B\) is strongly closed.
In this paper, we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on \(Q\). Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion.


53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
Full Text: DOI arXiv


[1] S. Bates, A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, Vol. 8, Amer. Math. Soc., Center for Pure and Applied Mathematics at the University of California, Berkeley, 1997. · Zbl 1049.53061
[2] Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. phys., 111, 61-151, (1978) · Zbl 0377.53025
[3] Bertelson, M.; Cahen, M.; Gutt, S., Equivalence of star products, Classical quantum gravity, 14, A93-A107, (1997) · Zbl 0881.58021
[4] Bordemann, M.; Neumaier, N.; Waldmann, S., Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Comm. math. phys., 198, 363-396, (1998) · Zbl 0968.53056
[5] Bordemann, M.; Neumaier, N.; Waldmann, S., Homogeneous Fedosov star products on cotangent bundles II: GNS representations, the WKB expansion, traces, and applications, J. geom. phys., 29, 199-234, (1999) · Zbl 0989.53060
[6] Bordemann, M.; Römer, H.; Waldmann, S., A remark on formal KMS states in deformation quantization, Lett. math. phys., 45, 49-61, (1998) · Zbl 0951.53057
[7] M. Bordemann, S. Waldmann, Formal GNS construction and WKB expansion in deformation quantization, in: D. Sternheimer, J. Rawnsley, S. Gutt (Eds.) Deformation Theory and Symplectic Geometry, Mathematical Physics Studies, Vol. 20, Kluwer Academic Publisher, Dordrecht, Boston, London, 1997, pp. 315-319. · Zbl 1166.53321
[8] Bordemann, M.; Waldmann, S., Formal GNS construction and states in deformation quantization, Commun. math. phys., 195, 549-583, (1998) · Zbl 0989.53057
[9] Connes, A.; Flato, M.; Sternheimer, D., Closed star products and cyclic cohomology, Lett. math. phys., 24, 1-12, (1992) · Zbl 0767.55005
[10] Deligne, P., Déformations de l’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et dewilde, lecomte, Sel. math. new series, 1, 4, 667-697, (1995) · Zbl 0852.58033
[11] DeWilde, M.; Lecomte, P.B.A., Star-products on cotangent bundles, Lett. math. phys., 7, 235-241, (1983) · Zbl 0514.53031
[12] Fedosov, B., Deformation quantization and index theory, (1996), Akademie Verlag Berlin · Zbl 0867.58061
[13] Guillemin, V.; Sternberg, S., Symplecic techniques in physics, (1984), Cambridge University Press Cambridge
[14] Gutt, S.; Rawnsley, J., Equivalence of star products on a symplectic manifold: an introduction to Deligne’s čech cohomology classes, J. geom. phys., 29, 347-392, (1999) · Zbl 1024.53057
[15] Hirsch, M.W., Differential topology, Graduate texts in mathematics, Vol. 33, (1976), Springer Berlin, Heidelberg, New York · Zbl 0121.18004
[16] Hörmander, L., The analysis of linear partial differential operators III, Grundlehren, Vol. 274, (1985), Springer Berlin
[17] M. Kontsevich, Deformation Quantization of Poisson Manifolds, I, preprint q-alg/9709040, 1997.
[18] Nest, R.; Tsygan, B., Algebraic index theorem, Commun. math. phys., 172, 223-262, (1995) · Zbl 0887.58050
[19] Nest, R.; Tsygan, B., Algebraic index theorem for families, Adv. math., 113, 151-205, (1995) · Zbl 0837.58029
[20] Pflaum, M., The normal symbol on Riemannian manifolds, New York J. math., 4, 95-123, (1998)
[21] Pflaum, M., A deformation theoretical approach to Weyl quantization on Riemannian manifolds, Lett. math. phys., 45, 277-294, (1998) · Zbl 0995.53057
[22] Safarov, Y., Pseudodifferential operators and linear connections, Proc. lond. math. soc., III. ser., 74, 2, 379-416, (1997) · Zbl 0872.35140
[23] T. Voronov, The complete symbol calculus for pseudodifferential operators, unpublished notes, 1992.
[24] A. Weinstein, P. Xu, Hochschild cohomology and characteristic classes for star-products, in: A. Khovanskij et al. (Eds.), Geometry of differential equations. Dedicated to V.I. Arnol’d on the occasion of his 60th birthday. Trans. Amer. Math. Soc. Ser. 2, 186 (39) (1998) 177-194.
[25] Wells, R.O., Differential analysis on complex manifolds, Graduate texts in mathematics, Vol. 65, (1980), Springer New York, Berlin, Heidelberg · Zbl 0435.32004
[26] Widom, H., Families of pseudodifferential operators, (), 345-395
[27] Widom, H., A complete symbol calculus for pseudodifferential operators, Bull. sci. math., 104, 2, 19-63, (1980) · Zbl 0434.35092
[28] Yano, K.; Ishihara, S., Tangent and cotangent bundles, differential geometry, (1973), Dekker New York · Zbl 0262.53024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.