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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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