Orderings and non-formal deformation quantization. (English) Zbl 1136.53064

The authors study the non-formal deformation quantization of Fréchet-Poisson algebra. In the original sense, a deformation means a formal deformation of a Poisson algebra. The question of convergence naturally follows. This question has been studied later, first in the framework of \(C^*\)-algebras (by Rieffel) and then in the symplectic context (by Weinstein). Here, the authors concentrate on the deformation quantization of a Fréchet-Poisson algebra. The convergence problem as well as ordering problems are also considered. This work leads to a better understanding of star exponential functions.


53D55 Deformation quantization, star products
46L65 Quantizations, deformations for selfadjoint operator algebras
81S10 Geometry and quantization, symplectic methods
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[1] Andrews, G., Askey, R., Roy, R.: Special functions, vol. 71 Encyclopedia Mathematics and it Application, Cambridge (2000) · Zbl 1075.33500
[2] Arnal D., Cortet J.-C., Molin P. and Pinczon G. (1983). Covariance and geometrical invariance in * quantization. J. Math. Phys. 24(2): 276–283 · Zbl 0515.22015
[3] Bayen F., Flato M., Fronsdal C., Lichnerowicz A. and Sternheimer D. (1978). Deformation theory and quantization I. Ann. Phys. 111: 61–110 · Zbl 0377.53024
[4] Bayen F. and Maillard J-M. (1982). Star exponentials of the elements of the homogeneous symplectic Lie algebra. Lett. Math. Phys. 6: 491–497 · Zbl 0516.58018
[5] Bonneau P., Gerstenhaber M., Giaquinto A. and Sternheimer D. (2004). Quantum groups and deformation quantization: explicit approaches and implicit aspects. J. Math. Phys. 45: 3703–3741 · Zbl 1071.53052
[6] Bieliavsky P. and Maeda Y. (2002). Convergent star product algebras on ”ax + b” group. Lett. Math. Phys. 62: 233–243 · Zbl 1036.53067
[7] Bieliavsky P. and Massar M. (2001). Oscillatory integral formulae for left-invariant star products on a class of Lie groups. Lett. Math. Phys. 58(2): 115–128 · Zbl 0998.53059
[8] De Wilde M. and Lecomte P. (1983). Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7: 487–496 · Zbl 0526.58023
[9] Fedosov B. (1994). A simple geometrical construction of deformation quantization. J. Differ. Geom. 40: 213–238 · Zbl 0812.53034
[10] Flato M. and Sternheimer D. (1969). On an infinite-dimensional group. Comm. Math. Phys. 14: 5–12 · Zbl 0179.58501
[11] Flato M., Simon J., Snellman H. and Sternheimer D. (1972). Simple facts about analytic vectors and integrability. Ann. Sci. École Norm. Sup. 5(4): 423–434 · Zbl 0239.22019
[12] Flato M., Simon J. and Sternheimer D. (1973). Sur l’intégrabilité des représentations antisymétriques des algèbres de Lie compactes. C. R. Acad. Sci. Paris Sér. A-B 277: A939–A942 · Zbl 0269.17002
[13] Gel’fand I.M. and Shilov G.E. (1968). Generalized Functions, 2. Academic, London
[14] Kontsevich M. (2003). Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66: 157–216 · Zbl 1058.53065
[15] Maillard J.-M. (2004). Star exponentials for any ordering of the elements of the inhomogeneous symplectic Lie algebra. J. Math. Phys. 45(2): 785–794 · Zbl 1070.81077
[16] Maillard, J.-M., Sternheimer, D.: Sur certaines représentations non intégrables de l’algèbre de Lie \({\mathfrak{su}}(2)\) et leur contenu indécomposable. C. R. Acad. Sci. Paris Sér. A-B 280, Aii, A73–A75 (1975) · Zbl 0298.17006
[17] Moreno C. and da Silva A.P. (2000). Star products, spectral analysis and hyperfunctions. Math. Phys. Stud. 22: 211–214 · Zbl 1004.53066
[18] Natsume T. (2001). C*-algebraic deformation quantization and the index theorem. Math. Phys. Stud. 23: 142–150 · Zbl 1015.46040
[19] Natsume T., Nest R. and Ingo P. (2003). Strict quantizations of symplectic manifolds. Lett. Math. Phys. 66: 73–89 · Zbl 1064.53062
[20] Olver P.J. (1996). Non-associative local Lie groups. J. Lie Theory 6: 23–59 · Zbl 0862.22005
[21] Omori, H.: Infinite-dimensional Lie groups. Translated from the 1979 Japanese original and revised by the author, Translations of Mathematical Monographs, vol. 158. American Mathematical Society, Providence (1997) · Zbl 0871.58007
[22] Omori H. (2002). One must break symmetry in order to keep associativity. Banach Center Publ. 55: 153–163 · Zbl 1081.53080
[23] Omori H. (2004). Physics in Mathematics (in Japanese). Tokyo University Publication, Tokyo · Zbl 1059.92011
[24] Omori H. and Maeda Y. (2004). Quantum Theoretic Calculus (in Japanese). Springer, Tokyo
[25] Omori H., Maeda Y. and Yoshioka A. (1991). Weyl manifolds and deformation quantization. Adv. Math. 85(2): 224–255 · Zbl 0734.58011
[26] Omori H., Maeda Y., Miyazaki N. and Yoshioka A. (2000). Deformation quantization of Fréchet–Poisson algebras – Convergence of the Moyal product. Math. Phys. Stud. 22: 233–246 · Zbl 0987.53036
[27] Omori H., Maeda Y., Miyazaki N. and Yoshioka A. (2001). Singular systems of exponential functions. Math. Phys. Stud. 23: 169–187 · Zbl 1049.53063
[28] Omori, H., Maeda, Y., Miyazaki, N., Yoshioka, A.: Star exponential functions for quadratic forms and polar elements. Contemp. Math., 25–38 (2002) · Zbl 1047.53057
[29] Omori H., Maeda Y., Miyazaki N. and Yoshioka A. (2003). Strange phenomena related to ordering problems in quantizations. J. Lie Theory 13: 481–510 · Zbl 1046.53057
[30] Omori, H., Maeda, Y., Miyazaki, N., Yoshioka, A.: Star exponential functions as two-valued elements. Prog. Math. 232, 483–492 (2005) math.QA 0711.3668 · Zbl 1076.53105
[31] Omori, H., Maeda, Y., Miyazaki, N., Yoshioka, A.: Geometric objects in an approach to quantum geometry. Prog. Math. 252, 303–324 (2006) math.QA 0711.3665
[32] Omori H., Maeda Y., Miyazaki N. and Yoshioka A. (2007). Convergent star products on Fréchet linear Poisson algebras of Heisenberg type. Contemp. Math. 434: 99–123 · Zbl 1215.53081
[33] Omori, H., Maeda, Y., Miyazaki, N., Yoshioka, A.: Expressions of elements of algebras and transcendental noncommutative calculus. Noncommutative Geometry and Physcis 2005. In: Proceedings of the International Sendai-Beijing Joint Workshop, pp. 3–30 World Scientific Publication, Singapore (2007) math.QA 0711.2608
[34] Rieffel, M.: Deformation quantization for actions of \({\mathbb{R}}^n\) . Mem. Am. Math. Soc. 506 (1993) · Zbl 0798.46053
[35] Weinstein A. (1994). Traces and triangles in symmetric symplectic spaces. Contemp. Math. 179: 261–270 · Zbl 0820.58024
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