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

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 q...

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Main Authors: Hervé Bergeron, Jean-Pierre Gazeau
Format: Article
Language:English
Published: MDPI AG 2018-10-01
Series:Entropy
Subjects:
Online Access:http://www.mdpi.com/1099-4300/20/10/787
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spelling doaj-c8ac3819541346cca784285f750dbb6b2020-11-24T23:08:34ZengMDPI AGEntropy1099-43002018-10-01201078710.3390/e20100787e20100787Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-PlaneHervé Bergeron0Jean-Pierre Gazeau1ISMO, UMR 8214 CNRS, Université Paris-Sud, 91405 Orsay, FranceAPC, UMR 7164 CNRS, Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris, FranceAny 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.http://www.mdpi.com/1099-4300/20/10/787Weyl-Heisenberg groupaffine groupWeyl quantizationWigner functioncovariant integral quantization
collection DOAJ
language English
format Article
sources DOAJ
author Hervé Bergeron
Jean-Pierre Gazeau
spellingShingle Hervé Bergeron
Jean-Pierre Gazeau
Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
Entropy
Weyl-Heisenberg group
affine group
Weyl quantization
Wigner function
covariant integral quantization
author_facet Hervé Bergeron
Jean-Pierre Gazeau
author_sort Hervé Bergeron
title Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
title_short Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
title_full Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
title_fullStr Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
title_full_unstemmed Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
title_sort variations à la fourier-weyl-wigner on quantizations of the plane and the half-plane
publisher MDPI AG
series Entropy
issn 1099-4300
publishDate 2018-10-01
description 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.
topic Weyl-Heisenberg group
affine group
Weyl quantization
Wigner function
covariant integral quantization
url http://www.mdpi.com/1099-4300/20/10/787
work_keys_str_mv AT hervebergeron variationsalafourierweylwigneronquantizationsoftheplaneandthehalfplane
AT jeanpierregazeau variationsalafourierweylwigneronquantizationsoftheplaneandthehalfplane
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