Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 31 August 2018 / Approved: 3 September 2018 / Online: 3 September 2018 (06:01:25 CEST)

A peer-reviewed article of this Preprint also exists.

Bergeron, H.; Gazeau, J.-P. Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane. Entropy 2018, 20, 787. Bergeron, H.; Gazeau, J.-P. Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane. Entropy 2018, 20, 787.

Journal reference: Entropy 2018, 20, 787
DOI: 10.3390/e20100787

## Abstract

Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with 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 ressources 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. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

## Keywords

Weyl-Heisenberg group; affine group; weyl quantization; wigner function; covariant integral quantization

## Subject

PHYSICAL SCIENCES, Mathematical Physics

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