Covariant Integral Quantization of the Semi-Discrete SO(3)-Hypercylinder

Symmetry is an international, peer-reviewed, open access journal covering research on symmetry/asymmetry phenomena wherever they occur in all aspects of natural sciences. Symmetry is published monthly online by MDPI. On 10 November 2023, it published ‘Covariant Integral Quantization of the Semi-Discrete SO(3)-Hypercylinder’, an article by Jean-Pierre Gazeau and Romain Murenzi. Here is the abstract of the article:

Covariant integral quantization with rotational SO(3) symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a non-unitary operator that operates on the Hilbert space of square-integrable functions on SO(3). This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform.

You can read the full article here: https://www.mdpi.com/2073-8994/15/11/2044

Majorana stellar representation of twisted photons

Physical Review Research is a fully open access, peer-reviewed, international, multidisciplinary journal covering all research topics of interest to the physics community. Its July - September 2023 issue includes ‘Majorana stellar representation of twisted photons’, an article by Andrei B. Klimov, Romain Murenzi, Jean-Pierre Gazeau, and Luis L. Sánchez-Soto. As the article’s abstract explains,

Majorana stellar representation, which visualizes a quantum spin as points on the Bloch sphere, allows quantum mechanics to accommodate the concept of trajectory, the hallmark of classical physics. We extend this notion to the discrete cylinder, which is the phase space of the canonical pair angle and orbital angular momentum. We demonstrate that the geometrical properties of the ensuing constellations aptly encapsulate the quantumness of the state.

You can read the full article here https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.L032006

Majorana stellar representation of twisted photons. Authors: Nicolas Fabre, Andrei B. Klimov, Romain Murenzi, Jean-Pierre Gazeau, and Luis L. Sánchez-Soto. Learn more in Physical Review Research: https://t.co/d0o2LZBGQd. #Quantum pic.twitter.com/jkWYDTp4ko

— APS Physics (@APSphysics) July 13, 2023

TWAS’s 40th Anniversary

The World Academy of Sciences for the advancement of science in developing countries (TWAS) is an organization that has been at the forefront of promoting scientific excellence in the developing world.

Founded in 1983, TWAS has been a driving force in supporting scientists from diverse backgrounds and fostering global collaborations. This year marks the 40th anniversary of TWAS, and its President, Quarraisha Abdool Karim, has launched the celebrations to honor this momentous occasion.

On 25 May 2023, as the Executive Director of TWAS, I had the pleasure to open the anniversary event – a public lecture by TWAS President. I was joined onstage Stefano Fantoni, president of the Trieste International Foundation for the Progress and Freedom of Sciences (FIT); and Minister Plenipotentiary Giuseppe Pastorelli, director for integrated promotion and innovation at the Directorate General for Country Promotion of the ministry of Foreign Affairs and International Cooperation of Italy, a key supporter of TWAS.

From bridging the scientific divide to promoting sustainable development goals and empowering young scientists, TWAS has demonstrated its unwavering commitment to promoting scientific excellence worldwide. As we celebrate this milestone, let us remember that together we can forge a brighter and more equitable future through the power of scientific knowledge.

On the TWAS website https://twas.org/article/twas-president-launches-celebrations-academys-40th-anniversary you can read an article about the event.

With TWAS President Abdool Karim

 

Integral Quantization for the Discrete Cylinder

by 

Jean-Pierre Gazeau

 1,* and

Romain Murenzi

 2

1

CNRS, Astroparticule et Cosmologie, Université Paris Cité, 75013 Paris, France

2

Abstract: Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z × S^1 , i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z × SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L ^2 (S^1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.

Gazeau, J.-P.; Murenzi, R. Integral Quantization for the Discrete Cylinder. Quantum Rep. 2022, 4, 362–379. https://doi.org/10.3390/ quantum4040026.

https://www.mdpi.com/2624-960X/4/4/26.