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Fourier-Gauss transformation and quantization. (English) Zbl 0866.43003
Borisovich, Yu. G. (ed.) et al., Applied aspects of global analysis. Voronezh: Voronezh University Press. Nov. Global’nom Anal. 57-93 (1994).
The authors propose a new universal procedure for the quantization of classical mechanical systems based on the Fourier-Gauss transformation \[ f(x,\hbar)\to\widetilde f(x,p,\hbar)=Uf(x,\hbar)={1\over2^{n/2}(\pi\hbar)^{3n/4}}\int\overline G_{(x,p)}(x',\hbar)f(x',\hbar)dx', \] where \(\overline G_{(x,p)}(x',\hbar)=\exp\{{i\over\hbar} [p(x'-x)+(x'-x)^2]\}\). This transformation maps the function \(f(x,\hbar)\) defined on a configuration space \(\mathbb{R}^n\) to some subspace of functions \(\widetilde f(x,p,\hbar)\) on the phase space \(T^*\mathbb{R}^n\). Further, to any classical observable \(H(x,p)\) there is assigned a pseudodifferential operator \[ H(x,\widehat p)= U^{-1}H(x,p)U \quad(\text{mod}\sqrt\hbar). \] The method is based on the fact that the wave front of the wave function describes the localization in the phase space. In the article, properties of the method are investigated in detail. The procedure coincides with Schrödinger quantization for observables, with Maslov’s quantization for Lagrangian modules, and with Fock quantization for canonical transformations of the phase space.
For the entire collection see [Zbl 0843.00020].

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
22E70 Applications of Lie groups to the sciences; explicit representations
35S05 Pseudodifferential operators as generalizations of partial differential operators