Symmetry and Geometry in Physics II

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

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 16052

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Department of Mathematics, North Dakota State University, P.O.Box 6050, Fargo, ND 58108-6050, USA
Interests: hyperbolic geometry; mathematical physics
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Special Issue Information

Dear Colleagues,

Nature organizes itself using the language of symmetry. In particular, the symmetry group of special relativity theory is the Lorentz transformation group SO (1,3). A physical system has Lorentz symmetry if the relevant laws of physics are invariant under Lorentz transformations. Lorentz symmetry is one of the cornerstones of modern physics. However, entangled particles involve Lorentz symmetry violation. Understanding entanglement in relativistic settings has been a key question in quantum mechanics. Remarkably, a plausible candidate for the symmetry group of the spacetime of a system of m n-dimensional entangled particles is the Lorentz group SO (m, n) of signature (m, n), for any m, n ∈ ℕ.

Lorentz groups involve relativistically admissible velocities governed by hyperbolic geometry and controlled by Einstein velocity addition. The resulting Einstein addition is a binary operation which is neither commutative nor associative. As such, it is a non-group gyrogroup operation that gives rise to gyrocommutative gyrogroups and gyrovector spaces. The latter, in turn, form the algebraic setting for hyperbolic geometry, just as vector spaces form the algebraic setting for Euclidean geometry.

Papers that study any of the following topics are welcome:

  1. Differential or hyperbolic geometry associated with Einstein addition;
  2. Einstein and Einstein-related gyrogroups and gyrovector spaces and their hyperbolic geometry;
  3. Quantum entanglement that involves Lorentz violation; and
  4. Physical applications of any Lorentz group SO (m, n) of signature (m, n), m, n > 1.

Prof. Dr. Abraham A. Ungar
Guest Editor

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Keywords

  • Special relativity
  • Lorentz symmetry group SO (1, 3)
  • Lorentz symmetry groups SO (m, n), m, n > 1
  • SO (m, n), m, n > 1, and quantum entanglement
  • Lorentz symmetry violation in quantum entanglement
  • hyperbolic geometry approach to Einstein addition
  • differential geometry approach to Einstein addition
  • Einstein gyrogroups
  • Einstein gyrovector spaces

Published Papers (7 papers)

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Research

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13 pages, 405 KiB  
Article
On the Splitting Tensor of the Weak f-Contact Structure
by Vladimir Rovenski
Symmetry 2023, 15(6), 1215; https://doi.org/10.3390/sym15061215 - 07 Jun 2023
Cited by 1 | Viewed by 720
Abstract
A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it [...] Read more.
A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
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9 pages, 256 KiB  
Article
A Note on Generalized Solitons
by Amira Ishan and Sharief Deshmukh
Symmetry 2023, 15(4), 954; https://doi.org/10.3390/sym15040954 - 21 Apr 2023
Cited by 3 | Viewed by 914
Abstract
In this paper, we initiate the study of a generalized soliton on a Riemannian manifold, we find a characterization for the Euclidean space, and in the compact case, we find a sufficient condition under which it reduces to a quasi-Einstein manifold. We also [...] Read more.
In this paper, we initiate the study of a generalized soliton on a Riemannian manifold, we find a characterization for the Euclidean space, and in the compact case, we find a sufficient condition under which it reduces to a quasi-Einstein manifold. We also find sufficient conditions under which a compact generalized soliton reduces to an Einstein manifold. Note that Ricci solitons being self-similar solutions of the heat flow, this topic is related to the symmetry in the geometry of Riemannian manifolds. Moreover, generalized solitons being generalizations of Ricci solitons are naturally related to symmetry. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
14 pages, 342 KiB  
Article
Almost Riemann Solitons with Vertical Potential on Conformal Cosymplectic Contact Complex Riemannian Manifolds
by Mancho Manev
Symmetry 2023, 15(1), 104; https://doi.org/10.3390/sym15010104 - 30 Dec 2022
Cited by 1 | Viewed by 962
Abstract
Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual [...] Read more.
Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The potential of the studied soliton is assumed to be in the vertical distribution, i.e., it is collinear to the Reeb vector field. In this way, manifolds from the four main classes of the studied manifolds are obtained. The curvature properties of the resulting manifolds are derived. An explicit example of dimension five is constructed. The Bochner curvature tensor is used (for a dimension of at least seven) as a conformal invariant to obtain these properties and to construct an explicit example in relation to the obtained results. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
17 pages, 387 KiB  
Article
The Weak Field Approximation of General Relativity and the Problem of Precession of the Perihelion for Mercury
by Asher Yahalom
Symmetry 2023, 15(1), 39; https://doi.org/10.3390/sym15010039 - 23 Dec 2022
Viewed by 1971
Abstract
In this paper we represent a different approach to the calculation of the perihelion shift than the one presented in common text books. We do not rely on the Schwarzschild metric and the Hamilton–Jacobi technique to obtain our results. Instead we use a [...] Read more.
In this paper we represent a different approach to the calculation of the perihelion shift than the one presented in common text books. We do not rely on the Schwarzschild metric and the Hamilton–Jacobi technique to obtain our results. Instead we use a weak field approximation, with the advantage that we are not obliged to work with a definite static metric and can consider time dependent effects. Our results support the conclusion of Křížek regarding the significant influence of celestial parameters on the indeterminacy of the perihelion shift of Mercury’s orbit. This shift is thought to be one of the fundamental tests of the validity of the general theory of relativity. In the current astrophysical community, it is generally accepted that the additional relativistic perihelion shift of Mercury is the difference between its observed perihelion shift and the one predicted by Newtonian mechanics, and that this difference equals 43 per century. However, as it results from the subtraction of two inexact numbers of almost equal magnitude, it is subject to cancellation errors. As such, the above accepted value is highly uncertain and may not correspond to reality. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
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40 pages, 1450 KiB  
Article
Circuit Complexity in Z2 EEFT
by Kiran Adhikari, Sayantan Choudhury, Sourabh Kumar, Saptarshi Mandal, Nilesh Pandey, Abhishek Roy, Soumya Sarkar, Partha Sarker and Saadat Salman Shariff
Symmetry 2023, 15(1), 31; https://doi.org/10.3390/sym15010031 - 22 Dec 2022
Cited by 6 | Viewed by 1642
Abstract
Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in Z2 Even Effective Field Theories (Z2 EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such [...] Read more.
Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in Z2 Even Effective Field Theories (Z2 EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as ϕ4, ϕ6, and ϕ8. To facilitate our computation, we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to N oscillators. This study was carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path-ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators to the circuit complexity in our theory is discussed. We also explore the dependency of complexity on other parameters in our theory for various cases. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
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23 pages, 438 KiB  
Article
Newton’s Third Law in the Framework of Special Relativity for Charged Bodies
by Shailendra Rajput and Asher Yahalom
Symmetry 2021, 13(7), 1250; https://doi.org/10.3390/sym13071250 - 12 Jul 2021
Cited by 7 | Viewed by 7684
Abstract
Newton’s third law states that any action is countered by a reaction of equal magnitude but opposite direction. The total force in a system not affected by external forces is, therefore, zero. However, according to the principles of relativity, a signal cannot propagate [...] Read more.
Newton’s third law states that any action is countered by a reaction of equal magnitude but opposite direction. The total force in a system not affected by external forces is, therefore, zero. However, according to the principles of relativity, a signal cannot propagate at speeds exceeding the speed of light. Hence, the action and reaction cannot be generated at the same time due to the relativity of simultaneity. Thus, the total force cannot be null at a given time. In a previous paper, we showed that Newton’s third law cannot strictly hold in a distributed system where the different parts are at a finite distance from each other. This analysis led to the suggestion of a relativistic engine. As the system is affected by a total force for a finite period, the system acquires mechanical momentum and energy. The subject of momentum conversation was discussed in another previous paper, while energy conservation was discussed in additional previous papers. In those works, we relied on the fact that the bodies were macroscopically natural. Here, we relax this assumption and study charged bodies, thus analyzing the consequences on a possible electric relativistic engine. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
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Review

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21 pages, 650 KiB  
Review
Ptolemy’s Theorem in the Relativistic Model of Analytic Hyperbolic Geometry
by Abraham A. Ungar
Symmetry 2023, 15(3), 649; https://doi.org/10.3390/sym15030649 - 04 Mar 2023
Cited by 1 | Viewed by 1054
Abstract
Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of [...] Read more.
Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of hyperbolic geometry is based on the Einstein addition of relativistically admissible velocities and, as such, it coincides with the well-known Beltrami–Klein ball model of hyperbolic geometry. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the relativistic model of analytic hyperbolic geometry gives rise. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
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