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

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## Abstract

**:**

## 1. Introduction: A Historical Overview

## 2. Covariant Integral Quantization: A Summary

#### 2.1. General Settings

- (i)
- To $f=1$ there corresponds ${A}_{f}={I}_{\mathcal{H}}$, where ${I}_{\mathcal{H}}$ is the identity in $\mathcal{H}$,
- (ii)
- To a real function $f\in \mathcal{C}(X)$ there corresponds a(n) (essentially) self-adjoint operator ${A}_{f}$ in $\mathcal{H}$.

#### 2.2. Semi-Classical Framework With Probabilistic Interpretation

#### 2.3. Semi-Classical Picture Without Probabilistic Interpretation

## 3. Quantization of the Plane: Generalizations of the Wigner-Weyl Transform

#### 3.1. The Group Background

#### 3.2. Hyperbolic W-H Covariant Integral Quantization

#### 3.2.1. General Settings

#### 3.2.2. Resolution of the Identity

#### 3.2.3. Covariant Quantization and Properties

**Remark**

**1.**

#### 3.2.4. Trace Formula

#### 3.3. Invertible W-H Covariant Integral Quantization: Generalization of the Wigner-Weyl Transform

#### 3.3.1. General Settings

#### 3.3.2. Generalized Wigner Functions

**Remark**

**2.**

- The function $\mathsf{\Lambda}(F)$ only depends on the variable $qp$. Therefore it cannot belong to some ${L}^{r}$ space on the plane. Hence, the convolution product involved in (52) should be understood in general in the distribution sense.
- The function $\tilde{F}$ is defined as an integral only if F belongs to ${L}^{1}{(\mathbb{R},|\alpha |}^{-1}\mathrm{d}\alpha )$. In other cases an extension in the distribution framework is needed.
- An interesting question concerns the positiveness of ${\mathcal{W}}_{\psi}^{(F)}$. In the genuine Wigner-Weyl case ($F=\delta $), Hudson theorem [36] asserts that only gaussian states ψ lead to positive Wigner functions ${\mathcal{W}}_{\psi}^{(\delta )}(q,p)$, and so the latter can be interpreted as probability densities on phase space. Beyond the pure Gaussian case, see for instance [37]. The problem now is to formulate a generalized version of the Hudson theorem (involving maybe a different family of states) for the generalized Wigner function ${W}_{\psi}^{(F)}$). In other words, for a given state ψ, is it possible to “build” a function F such that the corresponding Wigner function ${\mathcal{W}}_{\psi}^{(F)}$ is positive?

#### 3.3.3. Examples of Invertible Map

## 4. Quantization of the Half-Plane With the Affine Group: Wigner-Weyl-Like Scheme

#### 4.1. The Group Background

#### 4.2. Wigner-Weyl-Like Covariant Affine Quantization

#### General Settings

**Remark**

**3.**

#### 4.3. Resolution of the Identity

#### 4.4. Affine Covariant Quantization and Properties

#### Trace Formula

#### 4.5. Invertible W-H-like Affine Covariant Quantization

#### 4.6. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Quantization of The Plane: Boundedness Of ${\mathcal{P}}_{\mathbf{0}}^{(\mathit{F})}$

## Appendix B. Quantization of The Half-Plane: Boundedness of ${\mathcal{P}}_{\mathit{q},\mathit{p}}^{(\mathit{F})}$

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**MDPI and ACS Style**

Bergeron, H.; Gazeau, J.-P.
Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane. *Entropy* **2018**, *20*, 787.
https://doi.org/10.3390/e20100787

**AMA Style**

Bergeron H, Gazeau J-P.
Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane. *Entropy*. 2018; 20(10):787.
https://doi.org/10.3390/e20100787

**Chicago/Turabian Style**

Bergeron, Hervé, and Jean-Pierre Gazeau.
2018. "Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane" *Entropy* 20, no. 10: 787.
https://doi.org/10.3390/e20100787