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Entropy 2018, 20(10), 787; https://doi.org/10.3390/e20100787

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

1
ISMO, UMR 8214 CNRS, Université Paris-Sud, 91405 Orsay, France
2
APC, UMR 7164 CNRS, Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris, France
*
Author to whom correspondence should be addressed.
Received: 1 September 2018 / Revised: 8 October 2018 / Accepted: 8 October 2018 / Published: 13 October 2018
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Abstract

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations. View Full-Text
Keywords: Weyl-Heisenberg group; affine group; Weyl quantization; Wigner function; covariant integral quantization Weyl-Heisenberg group; affine group; Weyl quantization; Wigner function; covariant integral quantization
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Bergeron, H.; Gazeau, J.-P. Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane. Entropy 2018, 20, 787.

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