Symmetry and Geometry in Physics

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

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 27693

Special Issue Editor

Department of Mathematics, North Dakota State University, P.O.Box 6050, Fargo, ND 58108-6050, USA
Interests: hyperbolic geometry; mathematical physics
Special Issues, Collections and Topics in MDPI journals

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. Abraham A. Ungar
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.

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 (16 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Editorial

Jump to: Research, Review

4 pages, 200 KiB  
Editorial
Special Issue Editorial: “Symmetry and Geometry in Physics”
by Abraham A. Ungar
Symmetry 2022, 14(8), 1533; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081533 - 26 Jul 2022
Viewed by 753
Abstract
Nature organizes itself using the language of symmetries [...] Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)

Research

Jump to: Editorial, Review

14 pages, 308 KiB  
Article
Group Structure and Geometric Interpretation of the Embedded Scator Space
by Jan L. Cieśliński and Artur Kobus
Symmetry 2021, 13(8), 1504; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081504 - 17 Aug 2021
Cited by 3 | Viewed by 1070
Abstract
The set of scators was introduced by Fernández-Guasti and Zaldívar in the context of special relativity and the deformed Lorentz metric. In this paper, the scator space of dimension 1+n (for n=2 and n=3) is interpreted [...] Read more.
The set of scators was introduced by Fernández-Guasti and Zaldívar in the context of special relativity and the deformed Lorentz metric. In this paper, the scator space of dimension 1+n (for n=2 and n=3) is interpreted as an intersection of some quadrics in the pseudo-Euclidean space of dimension 2n with zero signature. The scator product, nondistributive and rather counterintuitive in its original formulation, is represented as a natural commutative product in this extended space. What is more, the set of invertible embedded scators is a commutative group. This group is isomorphic to the group of all symmetries of the embedded scator space, i.e., isometries (in the space of dimension 2n) preserving the scator quadrics. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
10 pages, 277 KiB  
Article
Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces T1,1 and Yp,q
by Mihai Visinescu
Symmetry 2021, 13(4), 591; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13040591 - 02 Apr 2021
Cited by 2 | Viewed by 1269
Abstract
In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry. We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form. We examine the [...] Read more.
In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry. We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form. We examine the modifications of the action–angle coordinates by the Sasaki–Ricci flow. We then pass to the particular cases of the contact structures of the five-dimensional Sasaki–Einstein manifolds T1,1 and Yp,q. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
56 pages, 725 KiB  
Article
Lorentz Symmetry Group, Retardation and Energy Transformations in a Relativistic Engine
by Shailendra Rajput, Asher Yahalom and Hong Qin
Symmetry 2021, 13(3), 420; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030420 - 05 Mar 2021
Cited by 12 | Viewed by 1786
Abstract
In a previous paper, we have shown that Newton’s third law cannot strictly hold in a distributed system of which the different parts are at a finite distance from each other. This is due to the finite speed of signal propagation which cannot [...] Read more.
In a previous paper, we have shown that Newton’s third law cannot strictly hold in a distributed system of which the different parts are at a finite distance from each other. This is due to the finite speed of signal propagation which cannot exceed the speed of light in vacuum, which in turn means that when summing the total force in the system the force does not add up to zero. This was demonstrated in a specific example of two current loops with time dependent currents, the above analysis led to suggestion of a relativistic engine. Since the system is effected by a total force for a finite period of time this means that the system acquires mechanical momentum and energy, the question then arises how can we accommodate the law of momentum and energy conservation. The subject of momentum conservation was discussed in a pervious paper, while preliminary results regarding energy conservation where discussed in some additional papers. Here we give a complete analysis of the exchange of energy between the mechanical part of the relativistic engine and the field part, the energy radiated from the relativistic engine is also discussed. We show that the relativistic engine effect on the energy is 4th-order in 1c and no lower order relativistic engine effect on the energy exists. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
Show Figures

Figure 1

25 pages, 406 KiB  
Article
Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces
by Vesselin G. Gueorguiev and Andre Maeder
Symmetry 2021, 13(3), 379; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030379 - 26 Feb 2021
Cited by 12 | Viewed by 2283
Abstract
Based on the principle of reparametrization invariance, the general structure of physically relevant classical matter systems is illuminated within the Lagrangian framework. In a straightforward way, the matter Lagrangian contains background interaction fields, such as a 1-form field analogous to the electromagnetic vector [...] Read more.
Based on the principle of reparametrization invariance, the general structure of physically relevant classical matter systems is illuminated within the Lagrangian framework. In a straightforward way, the matter Lagrangian contains background interaction fields, such as a 1-form field analogous to the electromagnetic vector potential and symmetric tensor for gravity. The geometric justification of the interaction field Lagrangians for the electromagnetic and gravitational interactions are emphasized. The generalization to E-dimensional extended objects (p-branes) embedded in a bulk space M is also discussed within the light of some familiar examples. The concept of fictitious accelerations due to un-proper time parametrization is introduced, and its implications are discussed. The framework naturally suggests new classical interaction fields beyond electromagnetism and gravity. The simplest model with such fields is analyzed and its relevance to dark matter and dark energy phenomena on large/cosmological scales is inferred. Unusual pathological behavior in the Newtonian limit is suggested to be a precursor of quantum effects and of inflation-like processes at microscopic scales. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
12 pages, 302 KiB  
Article
Construction of 2-Gyrogroups in Which Every Proper Subgyrogroup Is Either a Cyclic or a Dihedral Group
by Soheila Mahdavi, Ali Reza Ashrafi, Mohammad Ali Salahshour and Abraham Albert Ungar
Symmetry 2021, 13(2), 316; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020316 - 14 Feb 2021
Cited by 6 | Viewed by 1647
Abstract
In this paper, a 2-gyrogroup G(n) of order 2n, n3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup [...] Read more.
In this paper, a 2-gyrogroup G(n) of order 2n, n3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup lattice of G(n) are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order 2n, which causes us to use the name dihedral gyrogroup for this class of gyrogroups of order 2n. Moreover, all proper subgyrogroups of G(n) are subgroups. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
Show Figures

Figure 1

7 pages, 237 KiB  
Article
Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant
by Valerio Faraoni and Farah Atieh
Symmetry 2021, 13(2), 200; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020200 - 26 Jan 2021
Cited by 5 | Viewed by 1742
Abstract
Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues of the Friedmann equation describing a spatially homogeneous and isotropic cosmology in general relativity. These analogies are presented together with their Lagrangian and Hamiltonian formulations and with an invariant of the [...] Read more.
Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues of the Friedmann equation describing a spatially homogeneous and isotropic cosmology in general relativity. These analogies are presented together with their Lagrangian and Hamiltonian formulations and with an invariant of the Fibonacci sequence. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
10 pages, 277 KiB  
Article
On Quasi Gyrolinear Maps between Möbius Gyrovector Spaces Induced from Finite Matrices
by Keiichi Watanabe
Symmetry 2021, 13(1), 76; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13010076 - 04 Jan 2021
Cited by 3 | Viewed by 1267
Abstract
We present some fundamental results concerning to continuous quasi gyrolinear operators between Möbius gyrovector spaces induced by finite matrices. Such mappings are significant like as operators induced by matrices between finite dimensional Hilbert spaces. This gives a novel approach to the study of [...] Read more.
We present some fundamental results concerning to continuous quasi gyrolinear operators between Möbius gyrovector spaces induced by finite matrices. Such mappings are significant like as operators induced by matrices between finite dimensional Hilbert spaces. This gives a novel approach to the study of mappings between Möbius gyrovector spaces that should correspond to bounded linear operators on real Hilbert spaces. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
23 pages, 373 KiB  
Article
Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups
by Jaturon Wattanapan, Watchareepan Atiponrat and Teerapong Suksumran
Symmetry 2020, 12(11), 1817; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111817 - 02 Nov 2020
Cited by 5 | Viewed by 1263
Abstract
A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup G such that G [...] Read more.
A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup G such that G contains a gyro-isomorphic copy of G. We then prove that every strongly topological gyrogroup G can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup G. We also study several properties shared by G and G, including gyrocommutativity, first countability and metrizability. As an application of these results, we prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
30 pages, 389 KiB  
Article
Isomorphism of Binary Operations in Differential Geometry
by Nikita E. Barabanov
Symmetry 2020, 12(10), 1634; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12101634 - 03 Oct 2020
Cited by 2 | Viewed by 1623
Abstract
We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from [...] Read more.
We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
36 pages, 360 KiB  
Article
Differential Geometry and Binary Operations
by Nikita E. Barabanov and Abraham A. Ungar
Symmetry 2020, 12(9), 1525; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091525 - 16 Sep 2020
Cited by 5 | Viewed by 1651
Abstract
We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic [...] Read more.
We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
26 pages, 350 KiB  
Article
A Gyrogeometric Mean in the Einstein Gyrogroup
by Takuro Honma and Osamu Hatori
Symmetry 2020, 12(8), 1333; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12081333 - 10 Aug 2020
Cited by 5 | Viewed by 1434
Abstract
In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We [...] Read more.
In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
44 pages, 455 KiB  
Article
Binary Operations in the Unit Ball: A Differential Geometry Approach
by Nikita E. Barabanov and Abraham A. Ungar
Symmetry 2020, 12(7), 1178; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12071178 - 16 Jul 2020
Cited by 6 | Viewed by 2151
Abstract
Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n N , and discover the properties that qualify these operations to the title addition despite the fact that, [...] Read more.
Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
13 pages, 244 KiB  
Article
Ordered Gyrovector Spaces
by Sejong Kim
Symmetry 2020, 12(6), 1041; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12061041 - 22 Jun 2020
Cited by 8 | Viewed by 1830
Abstract
The well-known construction scheme to define a partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space we construct the partial order, called a gyro-order. We also give several inequalities of gyrolines and [...] Read more.
The well-known construction scheme to define a partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space we construct the partial order, called a gyro-order. We also give several inequalities of gyrolines and cogyrolines in terms of the gyro-order. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
37 pages, 1366 KiB  
Article
Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups
by Milton Ferreira and Teerapong Suksumran
Symmetry 2020, 12(6), 941; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12060941 - 03 Jun 2020
Cited by 13 | Viewed by 2035
Abstract
In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner [...] Read more.
In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
Show Figures

Figure 1

Review

Jump to: Editorial, Research

40 pages, 500 KiB  
Review
A Spacetime Symmetry Approach to Relativistic Quantum Multi-Particle Entanglement
by Abraham A. Ungar
Symmetry 2020, 12(8), 1259; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12081259 - 30 Jul 2020
Cited by 5 | Viewed by 2546
Abstract
A Lorentz transformation group SO(m, n) of signature (m, n), m, n  N, in m time and n space dimensions, is the group of pseudo-rotations of a [...] Read more.
A Lorentz transformation group SO(m, n) of signature (m, n), m, n  N, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature (m, n). Accordingly, the Lorentz group SO(1, 3) is the common Lorentz transformation group from which special relativity theory stems. It is widely acknowledged that special relativity and quantum theories are at odds. In particular, it is known that entangled particles involve Lorentz symmetry violation. We, therefore, review studies that led to the discovery that the Lorentz group SO(m, n) forms the symmetry group by which a multi-particle system of m entangled n-dimensional particles can be understood in an extended sense of relativistic settings. Consequently, we enrich special relativity by incorporating the Lorentz transformation groups of signature (m, 3) for all m  2. The resulting enriched special relativity provides the common symmetry group SO(1, 3) of the (1 + 3)-dimensional spacetime of individual particles, along with the symmetry group SO(m, 3) of the (m + 3)-dimensional spacetime of multi-particle systems of m entangled 3-dimensional particles, for all m  2. A unified parametrization of the Lorentz groups SO(m, n) for all m, n  N, shakes down the underlying matrix algebra into elegant and transparent results. The special case when (m, n) = (1, 3) is supported experimentally by special relativity. It is hoped that this review article will stimulate the search for experimental support when (m, n) = (m, 3) for all m  2. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
Back to TopTop