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Dynamical equations for the quantum product on a symplectic space in affine coordinates. (English. Russian original) Zbl 1101.53059
Math. Notes 77, No. 1, 39-47 (2005); translation from Mat. Zametki 77, No. 1, 42-52 (2005).
From the introduction: Let $$X={\mathbb R}^{2n}$$ with symplectic form $$\omega$$ and symplectic connection $$\Gamma$$ without torsion. Then, following [F. Bayen et al., Ann. Phys. 111, 61–110 (1978; Zbl 0377.53024)], we can construct a star product $$\ast_{\hbar}$$ on $$X$$ which can be considered as the asymptotic expansion with respect to the deformation parameter $$\hbar$$ of a convergent product defined by an integral formula $(f \ast_{\hbar}g)(u)=\int_{X\times X}K_{x,y}^{\hbar} (u)f(x)g(y)\,dx\, dy.$ The problem is to find the integral kernel $$K^{\hbar}$$. The theory of the asymptotic quantization [M. V. Karasev and V. P. Maslov, Russ. Math. Surv. 39, No. 6, 133–205 (1984); translation from Usp. Mat. Nauk 39, No. 6(240), 115–173 (1984; Zbl 0588.58031)] aims to solve this problem in the semiclassical approximation. In the general Poisson case, the semiclassical asymptotics of the kernel $$K^{\hbar}$$ can be calculated by using the modified theory of Maslov’s canonical operator [see M. V. Karasev and V. P. Maslov, loc. cit.].
Here, according to the idea put forward in [M. Karasev, “Quantization and intrinsic dynamics”, in: Karasev, M.V. (ed.), Asymptotic methods for wave and quantum problems. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 208(53), 1–32 (2003; Zbl 1078.81044)], the authors use dynamical equations of the Schrödinger type for the kernel $$K^{\hbar}$$: $i\hbar \partial_y K_{x,y}^{\hbar} =H^{\hbar}_{y}\ast_{\hbar} K_{x,y}^{\hbar}$ with some Hamiltonian $$H^{\hbar}_{y}$$ depending on $$y\in X$$. The coefficients of $$H^{\hbar}$$ are regular functions of $$\hbar$$: $H^{\hbar}=H^{(0)}-i\hbar H^{(1)}+{\hbar}^2H^{(2)}+\dots,$ where $$H^{(1)}, H^{(2)},\dots$$ are real quantum corrections to the leading term $$H^{(0)}$$. It was shown in [M. Karasev, loc. cit.] under the additional assumption $$\operatorname{Im} H^{\hbar}=0$$ that the previous Schrödinger equation can be solved efficiently in geometrically invariant form using semiclassical approximation.
In the present paper, the authors calculate the Hamiltonian $$H^{\hbar}_{y}$$ in special $$\omega$$-affine coordinates on $$X$$ without assuming it to be real. The authors find the leading term of the WKB-asymptotics of the kernel $$K^{\hbar}$$, taking into account the imaginary quantum correction $$H^{(1)}$$. An example of such imaginary correction is discussed.
##### MSC:
 53D55 Deformation quantization, star products 46L65 Quantizations, deformations for selfadjoint operator algebras
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##### References:
 [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, ”Deformation theory and quantization,” Ann. Physics, 111 (1978), 61–151. · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5 [2] H. Omori, Y. Maeda, and A. Yoshioka, ”Weyl manifolds and deformation quantization,” J. Adv. Math., 85 (1997), 224–255. · Zbl 0734.58011 · doi:10.1016/0001-8708(91)90057-E [3] M. V. Karasev and V. P. Maslov, ”Asymptotic and geometrical quantization,” Uspekhi Mat. Nauk [Russian Math. Surveys], 39 (1984), no. 6, 115–173 [4] M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow, 1991; English transl.: Translations of Mathematical Monographs, vol. 119, Amer. Math. Soc., Providence, RI (1993). · Zbl 0731.58002 [5] M. V. Karasev, ”Analogs of objects of Lie group theory for nonlinear Poisson brackets,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 50 (1986), no. 3, 508–538. · Zbl 0608.58023 [6] A. Weinstein, ”Symplectic groupoids and Poisson manifolds,” Bull. Amer. Math. Soc., 16 (1987), 101–104. · Zbl 0618.58020 · doi:10.1090/S0273-0979-1987-15473-5 [7] M. V. Karasev, Quantization of Nonlinear Lie-Piosson Brackets in the Semiclassical Approximation [in Russian], Preprint no. 242 ITP-85-72P, Institute of Theoretical Physics, Kiev, 1985. [8] M. Karasev, ”Quantization and intrinsic dynamics,” in: Asymptotic Methods for Wave and Quantum Equations (M. Karasev, editor), Advances in Math. Sci., Amer. Math. Soc., Providence, RI, 2003. · Zbl 1078.81044 [9] O. N. Grigor’ev and M. V. Karasev, ”Intrinsic quantum dynamics and its operator representation over a plane with a nonstandard connection,” Russ. J. Math. Phys., 10 (2003), no.4, 422–435. · Zbl 1105.81304 [10] M. V. Karasev, ”Advances in quantization: quantum tensors, explicit star-products, and restriction to irreducible leaves,” Diff. Geometry Appl., 9 (1998), 89–134. · Zbl 0951.53059 · doi:10.1016/S0926-2245(98)00019-9 [11] M. V. Karasev, The Maslov Quantization Conditions in Higher Cohomology and Analogs of Notions Developed in Lie Theory for Canonical Fiber Bundles of Symplectic Manifolds (Manuscript deposited at VINITI on March 12, 1982; deposition nos. 242 1092-82, 1093-82), MIEM, Moscow, 1981. [12] V. P. Maslov, Operator Methods [in Russian], Nauka, Moskow, 1973. [13] F. A. Berezin, ”Quantization,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 38 (1974), no. 5, 1109–1165. · Zbl 0312.53049 [14] A. Weinstein, ”Traces and triangles in symmetric symplectic spaces,” Contemp. Math., 179 (1994), 262–270. · Zbl 0820.58024 [15] V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Izd. Moskov. Univ., Moscow, 1965. · Zbl 0653.35002 [16] G. Tuynman and P. Rios, Weyl Quantization from Geometric Quantization, Preprint, Univ. de Lille, Lille, France, 1999. · Zbl 1167.81423
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