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Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour...
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Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: 31 October 2026 | Viewed by 3398

Special Issue Editors

Dr. Hamed Pejhan

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Guest Editor
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Interests: mathematical physics; QFT in curved spacetime; group representation theory
Prof. Dr. Romain Murenzi

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Guest Editor
Department of Physics, Worcester Polytechnic Institute, Worcester, MA 01609, USA
Interests: wavelets; groups; coherent states; links between quantum mechanics; signal processing in one and more dimensions
Prof. Dr. Mariano A. del Olmo

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Guest Editor
Departamento de Física Teórica, Atómica y Optica and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain
Interests: symmetries in physics; group theory; special functions; differential equations

Special Issue Information

Dear Colleagues,

Symmetry remains a cornerstone concept in physics, guiding our understanding of nature across classical, quantum, and modern theoretical frameworks. As Paul Dirac once observed, “...as time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen”.

This Special Issue of Symmetry seeks to highlight innovative research and comprehensive reviews related to the role of symmetry in mathematical physics.

We welcome original contributions that address group and representation theory, symmetries in quantum field and conformal field theories, as well as the algebraic and geometric methods underlying these structures.

Special attention will be given to topics involving spacetime symmetries, especially in maximally symmetric spacetimes such as (anti-)de Sitter space, symmetry-breaking mechanisms, and integrable models. The goal is to foster interdisciplinary dialogue and bring together diverse perspectives that advance our understanding of symmetry in modern physics.

Dr. Hamed Pejhan
Prof. Dr. Romain Murenzi
Prof. Dr. Mariano A. del Olmo
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • symmetries in classical and quantum models
  • group representations
  • conformal field theory
  • integrable systems
  • gauge theories
  • symmetry breaking
  • quantum optics
  • coherent states
  • signal processing
  • aperiodic systems

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Published Papers (7 papers)

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Research

Jump to: Review

16 pages, 270 KB  
Article
Lie Symmetries and Invariants of General Time-Dependent Quadratic Hamiltonian System
by Kyu Hwang Yeon, Van Huy Pham and Keun Ho Ryu
Symmetry 2026, 18(6), 880; https://doi.org/10.3390/sym18060880 (registering DOI) - 22 May 2026
Abstract
Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which is obtained from a general time-dependent quadratic Hamiltonian. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three conserved quantities, which [...] Read more.
Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which is obtained from a general time-dependent quadratic Hamiltonian. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three conserved quantities, which are time-dependent quadratic forms in position and momentum, and trivial, 0. All obtained invariant quantities are represented by auxiliary conditions, which are two linearly independent solutions of a homogeneous differential equation of the equations of motion. Invariant variables associated with an invariant consisting of the linearity of x and p are defined. It shows that, if the motion of the system is oscillatory, the Poisson bracket of the two invariant variables is obtained as i, and in the case of monotonic motion, it is obtained as 1. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)
28 pages, 484 KB  
Article
Effective Potentials for de Sitter and Anti-de Sitter Quantum Fields
by Alfio Bonanno, Sergio Luigi Cacciatori and Ugo Moschella
Symmetry 2026, 18(5), 801; https://doi.org/10.3390/sym18050801 - 7 May 2026
Viewed by 155
Abstract
We derive a systematic treatment of one-loop effective potentials for interacting scalar fields in curved spacetimes, providing a general formula valid in arbitrary geometries and explicit results for de Sitter and anti-de Sitter backgrounds. We then compute the effective potential for a scalar [...] Read more.
We derive a systematic treatment of one-loop effective potentials for interacting scalar fields in curved spacetimes, providing a general formula valid in arbitrary geometries and explicit results for de Sitter and anti-de Sitter backgrounds. We then compute the effective potential for a scalar O(N) theory on a de Sitter space in any integer dimension. In d=3 and dimensional regularization, we extend the calculation up to two-loops and compute the β-function and the anomalous mass dimension. They coincide exactly with flat-space results, despite dramatic curvature modifications to physical masses/couplings. The flat limit R→∞ recovers Coleman–Weinberg, confirming consistency. Working in d=3 dimension, we repeat the calculation for AdS3 by using point-splitting regularization, obtaining analogue results for the β-function and anomalous mass dimension (Dedicated to Jean Pierre Gazeau on his 80th Birthday). Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)
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22 pages, 417 KB  
Article
Codings of B-Integers in Cantor Numeration Systems as Generators of Aperiodic Potentials
by Lubomíra Dvořáková, Zuzana Masáková and Edita Pelantová
Symmetry 2026, 18(3), 538; https://doi.org/10.3390/sym18030538 - 21 Mar 2026
Viewed by 303
Abstract
Cantor real numeration systems provide a natural algebraic source of self-similar aperiodic structures, extending the classical β-integers framework introduced in quasicrystal modeling by Gazeau. We study how the choice of algebraic parameters of the base [...] Read more.
Cantor real numeration systems provide a natural algebraic source of self-similar aperiodic structures, extending the classical β-integers framework introduced in quasicrystal modeling by Gazeau. We study how the choice of algebraic parameters of the base B=(βi)i∈Z influences the self-similarity and other combinatorial properties of the encoding symbolic sequence. These properties, namely repetitivity and palindromicity, are key features deciding the character of the spectrum of the underlying one-dimensional Schrödinger operator with aperiodic potential. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)
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18 pages, 4871 KB  
Article
From Quantum to Classical Within the Framework of Integral Quantization
by Ligia M. C. S. Rodrigues, Evaldo M. F. Curado, Diego Noguera and Alan C. Maioli
Symmetry 2026, 18(3), 403; https://doi.org/10.3390/sym18030403 - 25 Feb 2026
Viewed by 550
Abstract
Integral quantization is a powerful framework for mapping classical phase-space functions—defined on a symplectic manifold—onto quantum operators in a Hilbert space. It encompasses several quantization methods, such as coherent-state quantization, and inherently incorporates operator symmetrization. The formalism relies on a choice of weight [...] Read more.
Integral quantization is a powerful framework for mapping classical phase-space functions—defined on a symplectic manifold—onto quantum operators in a Hilbert space. It encompasses several quantization methods, such as coherent-state quantization, and inherently incorporates operator symmetrization. The formalism relies on a choice of weight function, whose flexibility allows for a family of possible quantizations. In this work, we address the inverse problem: given a quantum operator, how can one determine a classical phase-space function whose integral quantization reproduces exactly that operator? We propose a systematic method, within the integral quantization framework, to construct such a classical function, which depends on the chosen weight. We demonstrate that quantizing the resulting function recovers the original operator, thereby establishing a consistent two-way mapping between classical and quantum descriptions. The method is applied to several physically relevant operators: the projector, a mixed-state density operator, the annihilation operator, and an entangled state. We also analyze how quantum entanglement manifests in the structure of the corresponding classical phase-space function. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)
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15 pages, 333 KB  
Article
Twin Hamiltonians, Alternative Parametrizations of the Dyson Maps, and the Probabilistic Interpretation Problem in Quasi-Hermitian Quantum Mechanics
by Aritra Ghosh, Adam Miranowicz and Miloslav Znojil
Symmetry 2026, 18(1), 189; https://doi.org/10.3390/sym18010189 - 20 Jan 2026
Cited by 1 | Viewed by 522
Abstract
In quasi-Hermitian quantum mechanics (QHQM) of unitary systems, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, H≠H†. This makes its physical interpretation ambiguous. Without altering H, this ambiguity can be resolved either via a transformation of H [...] Read more.
In quasi-Hermitian quantum mechanics (QHQM) of unitary systems, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, H≠H†. This makes its physical interpretation ambiguous. Without altering H, this ambiguity can be resolved either via a transformation of H into its isospectral Hermitian form via a so-called Dyson map Ω:H→h, or via a (formally equivalent) specification of a nontrivial physical inner-product metric Θ in Hilbert space. Here, we focus on the former strategy. Our present construction of the Hermitian isospectral twins h of H is exhaustive. As a byproduct, it not only restores the conventional correspondence principle between quantum and classical physics, but it also provides a framework for a systematic classification of all of the admissible probabilistic interpretations of quantum systems using a preselected H in QHQM framework. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)

Review

Jump to: Research

25 pages, 1515 KB  
Review
Coherent-State Methods in Quantum Cosmology: Singularity Resolution, Semiclassical Dynamics, and Multiverse States
by Hervé Bergeron and Przemysław Małkiewicz
Symmetry 2026, 18(4), 637; https://doi.org/10.3390/sym18040637 - 10 Apr 2026
Viewed by 538
Abstract
We summarize our research program on the use of coherent states and covariant integral quantization in quantum cosmology. In particular, we present a recent development within this framework and include new results that shed light on some of its basic properties. Specifically, we [...] Read more.
We summarize our research program on the use of coherent states and covariant integral quantization in quantum cosmology. In particular, we present a recent development within this framework and include new results that shed light on some of its basic properties. Specifically, we investigate the quantum dynamics of a perturbed, fluid-filled Friedmann universe beyond the standard approximation in which the total state factorizes into background and perturbation wave functions. We assume the background geometry to be a superposition of two distinct coherent states—effectively a quantum cat state with no classical counterpart—each coupled to inhomogeneous perturbations. Starting from vacuum initial conditions, we analyze the evolution of a contracting universe through a bounce into the expanding phase. We find that an initially factorized state evolves into a biverse. This state consists of two distinct semiclassical branches, each described by a single coherent state and carrying enhanced perturbations in a slightly non-Gaussian state. We then explore how this dynamics depends on key model parameters, such as the perturbation wavelength and the choice of background solutions, and study their impact on the interaction between branches. The observed universe is assumed to correspond to one branch of this biverse state. This scenario illustrates how genuinely quantum properties of the background geometry may leave observable imprints in the early universe. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)
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45 pages, 1591 KB  
Review
Torsion-Induced Quantum Fluctuations in Metric-Affine Gravity Using the Stochastic Variational Method
by Tomoi Koide and Armin van de Venn
Symmetry 2026, 18(3), 525; https://doi.org/10.3390/sym18030525 - 18 Mar 2026
Viewed by 425
Abstract
This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of metric-affine gravity (MAG) and the stochastic variational method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, [...] Read more.
This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of metric-affine gravity (MAG) and the stochastic variational method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, and discuss the geometrical significance of torsion within this context. Subsequently, we summarize SVM, a powerful technique that facilitates quantization while effectively incorporating geometrical effects. By integrating these frameworks, we evaluate how the geometrical structures originating from torsion affect quantum fluctuations, demonstrating that they induce non-linearity in quantum mechanics. Notably, torsion, traditionally believed to influence only spin degrees of freedom, can also affect spinless degrees of freedom via quantum fluctuations. Furthermore, extending beyond the results of previous work [Koide and van de Venn, Phys. Rev. A112, 052217 (2025)], we investigate the competitive interplay between the Levi-Civita curvature and torsion within the non-linearity of the Schrödinger equation. Finally, we discuss the structural parallelism between SVM and information geometry, highlighting that the splitting of time derivatives in stochastic processes corresponds to the dual connections in statistical manifolds. These insights pave the way for future extensions to gravity theories involving non-metricity and are expected to deepen our understanding of unresolved cosmological problems. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Physics: Current Topics and Recent Advances—in Honour of Professor Jean-Pierre Gazeau on the Occasion of His 80th Birthday)
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