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Symmetry
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Symmetry in Mathematical and Theoretical Physics
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Symmetry in Mathematical and Theoretical Physics

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

Deadline for manuscript submissions: closed (15 May 2023) | Viewed by 9044

Special Issue Editor

Dr. Vladimir Al. Osipov

E-Mail Website
Guest Editor
Department of Chemistry, University of California Irvine, Irvine, CA 92697, USA

Special Issue Information

Dear Colleagues,

Mathematical apparatuses used as tools for the solution of physical problems and formulation of physical theories constitute the subject of mathematical physics. The methods of mathematical physics have found applications in a wide variety of theoretical physics problems, such as quantum field theory and theory of elementary particles, fundamental problems of nuclear physics, condensed matter physics, nonrelativistic quantum mechanics, gravitation theory, and theory of chaos. The efforts to put physical theories on a mathematically rigorous basis not only proved useful for physics but also gave rise to new mathematical methods and mathematical objects in such areas as functional analysis, operator algebras, representation theory, analysis of ultrametric spaces, and the random matrix theory. Over the years, with the growth of scientific knowledge and human practical activity, the number of applications in the focus of mathematical physics is continuously expanding, and their complexity is also increasing. These challenges require researchers to develop new mathematical approaches, improve the existing methods, extend the area of their applications, and also expand the communication chnnales between the research groups. This Special Issue of Symmetry, entitled "Symmetry in Mathematical and Theoretical Physics", provides a platform for publication of recent research and review articles in the field of mathematical physics whith the emphasis on the problems in which various kind  symmetries play a special role. We kindly invite all researchers working in the area to contribute to this Special Issue.

Dr. Vladimir Al. Osipov
Guest Editor

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 100 words) can be sent to the Editorial Office for announcement on this website.

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.

Published Papers (5 papers)

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Research

Jump to: Review

22 pages, 777 KiB  
Article
Topological Quantum Statistical Mechanics and Topological Quantum Field Theories
by Zhidong Zhang
Symmetry 2022, 14(2), 323; https://doi.org/10.3390/sym14020323 - 4 Feb 2022
Cited by 4 | Viewed by 2274
Abstract
The Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z2 symmetry, the phenomena revealed in Ising [...] Read more.
The Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z2 symmetry, the phenomena revealed in Ising systems may afford us lessons for other types of interactions in nature. In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan–von Neumann–Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan–von Neumann–Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry. Full article
(This article belongs to the Special Issue Symmetry in Mathematical and Theoretical Physics)
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11 pages, 304 KiB  
Article
Fourier Transform for Locally Integrable Functions with Rotational and Dilation Symmetry
by Jerzy A. Przeszowski, Elżbieta Dzimida-Chmielewska and Jan L. Cieśliński
Symmetry 2022, 14(2), 241; https://doi.org/10.3390/sym14020241 - 26 Jan 2022
Cited by 3 | Viewed by 1722
Abstract
The Fourier transform for slowly increasing functions is defined by the Parseval equation for tempered distributions. This definition was supplemented by a novel method of performing practical calculations by computing the Fourier transform for a suitably tempered function and then by integration by [...] Read more.
The Fourier transform for slowly increasing functions is defined by the Parseval equation for tempered distributions. This definition was supplemented by a novel method of performing practical calculations by computing the Fourier transform for a suitably tempered function and then by integration by parts. The application of this method is illustrated both for the toy case, in which the function is integrable, so its Fourier transform can also be computed using the standard formula, and for the case of Coulomb-like potentials, which are only locally integrable functions. All of them have spherical symmetry, and two of them additionally have dilation symmetry. The proposed novel method does not violate these symmetries at any stage of the calculation. Full article
(This article belongs to the Special Issue Symmetry in Mathematical and Theoretical Physics)
44 pages, 2594 KiB  
Article
Lie Group Classification of Generalized Variable Coefficient Korteweg-de Vries Equation with Dual Power-Law Nonlinearities with Linear Damping and Dispersion in Quantum Field Theory
by Oke Davies Adeyemo and Chaudry Masood Khalique
Symmetry 2022, 14(1), 83; https://doi.org/10.3390/sym14010083 - 5 Jan 2022
Cited by 6 | Viewed by 1260
Abstract
Many physical phenomena in fields of studies such as optical fibre, solid-state physics, quantum field theory and so on are represented using nonlinear evolution equations with variable coefficients due to the fact that the majority of nonlinear conditions involve variable coefficients. In consequence, [...] Read more.
Many physical phenomena in fields of studies such as optical fibre, solid-state physics, quantum field theory and so on are represented using nonlinear evolution equations with variable coefficients due to the fact that the majority of nonlinear conditions involve variable coefficients. In consequence, this article presents a complete Lie group analysis of a generalized variable coefficient damped wave equation in quantum field theory with time-dependent coefficients having dual power-law nonlinearities. Lie group classification of two distinct cases of the equation was performed to obtain its kernel algebra. Thereafter, symmetry reductions and invariant solutions of the equation were obtained. We also investigate various soliton solutions and their dynamical wave behaviours. Further, each class of general solutions found is invoked to construct conserved quantities for the equation with damping term via direct technique and homotopy formula. In addition, Noether’s theorem is engaged to furnish more conserved currents of the equation under some classifications. Full article
(This article belongs to the Special Issue Symmetry in Mathematical and Theoretical Physics)
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14 pages, 343 KiB  
Article
A p-Adic Matter in a Closed Universe
by Branko Dragovich
Symmetry 2022, 14(1), 73; https://doi.org/10.3390/sym14010073 - 4 Jan 2022
Cited by 7 | Viewed by 1225
Abstract
In this paper, we introduce a new type of matter that has origin in p-adic strings, i.e., strings with a p-adic worldsheet. We investigate some properties of this p-adic matter, in particular its cosmological aspects. We start with crossing symmetric [...] Read more.
In this paper, we introduce a new type of matter that has origin in p-adic strings, i.e., strings with a p-adic worldsheet. We investigate some properties of this p-adic matter, in particular its cosmological aspects. We start with crossing symmetric scattering amplitudes for p-adic open strings and related effective nonlocal and nonlinear Lagrangian which describes tachyon dynamics at the tree level. Then, we make a slight modification of this Lagrangian and obtain a new Lagrangian for non-tachyonic scalar field. Using this new Lagrangian in the weak field approximation as a matter in Einstein gravity with the cosmological constant, one obtains an exponentially expanding FLRW closed universe. At the end, we discuss the obtained results, i.e., computed mass of the scalar p-adic particle, estimated radius of related closed universe and noted p-adic matter as a possible candidate for dark matter. Full article
(This article belongs to the Special Issue Symmetry in Mathematical and Theoretical Physics)
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Review

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53 pages, 616 KiB  
Review
Local Zeta Functions and Koba–Nielsen String Amplitudes
by Miriam Bocardo-Gaspar, Hugo García-Compeán, Edgar Y. López and Wilson A. Zúñiga-Galindo
Symmetry 2021, 13(6), 967; https://doi.org/10.3390/sym13060967 - 29 May 2021
Cited by 4 | Viewed by 1685
Abstract
This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic [...] Read more.
This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points. Full article
(This article belongs to the Special Issue Symmetry in Mathematical and Theoretical Physics)
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