Special Issue Editorial: “Symmetry and Geometry in Physics”

1. Introduction

Nature organizes itself using the language of symmetries. Thus, in particular, the underlying symmetry group by which Einstein’s special relativity theory can be understood is the Lorentz group ${\mathrm{SO}}_{\mathrm{c}}(1,3)$ of signature $(1,3)$ in one temporal dimension and three spatial dimensions. A physical system obeys Lorentz symmetry if the relevant laws of physics are invariant under Lorentz transformations. Lorentz symmetry is one of the cornerstones of modern physics. However, it is known that entangled particles in relativistic quantum mechanics involve the violation of Lorentz symmetry [1,2,3,4,5]. Indeed, several explorers exploit entangled particles to observe Lorentz symmetry violation; see, for instance, [6,7,8,9,10,11]. Accordingly, a natural question arises: What is the symmetry group that underlies quantum multi-particle entanglement? The answer, supported by mathematical patterns and analogies with special relativity theory, is proposed in [12].

A Lorentz transformation without rotation is called a boost. In general, the composition of two successive boosts is not a boost but, rather, a boost preceded (or followed) by a rotation, called a Thomas precession. The relativistic effect known as Thomas precession is extended by abstraction into automorphisms called gyrations. Boosts are parametrized by a velocity parameter, so that boost compositions involve the Einstein addition of relativistically admissible velocities along with gyrations. Gyrations, in turn, give rise to the prefix “gyro” that is extensively used to emphasize analogies with classical terms. Thus, one prefixes a gyro to a classical term in algebra and in Euclidean geometry to mean the analogous term in nonassociative algebra and in hyperbolic geometry. As an example, a gyrogroup operation is gyroassociative, just as a group operation is associative.

Seemingly structureless, Einstein addition is neither commutative nor associative. Surprisingly, however, Einstein addition possesses a rich, elegant structure regulated by hyperbolic geometry. It is both gyrocommutative and gyroassociative, thus encoding the concepts of the gyrogroup and the gyrovector space, which are studied in several papers in this Special Issue of Symmetry. The concept of gyrogroups is a natural generalization of the concept of groups, reviewed in [12]. Some gyrocommutative gyrogroups admit scalar multiplication, turning themselves into gyrovector spaces, studied in [13,14]. Interestingly, gyrovector spaces form the algebraic setting for various models of the hyperbolic geometry of Lobachevsky and Bolyai, just as vector spaces form the algebraic setting for the standard model of Euclidean geometry. Gyrogroups and gyrovector spaces are generalized in [12] into bi-gyrogroups and bi-gyrovector spaces, along with applications of symmetry in theoretical physics.

In this Special Issue invited researchers elaborate on recent advances in the theory of gyrogroups and the theory of gyrovector spaces and, more generally, in topics of symmetry and geometry in physics.

2. Symmetry and Geometry in Physics—The Papers

The following manuscripts were selected for publication. The articles were prepared by scientists working in leading universities and research centers in Bulgaria, Canada, Iran, Israel, Japan, Korea, Poland, Portugal, Romania, Switzerland, and the USA.

Jaturon Wattanapan, Watchareepan Atiponrat, and Teerapong Suksumran, in the paper “Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups” [15], show how gyrogroups are modeled on the space of relativistically admissible velocities endowed with Einstein addition, and embed strongly topological gyrogroups in path and locally path-connected gyrogroups. Let G be a given gyrogroup; they offer a new way to construct a new gyrogroup $G^{*}$ that contains a gyro-isomorphic copy of G. Then, they prove that every strongly topological gyrogroup G can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup $G^{*}$. They also study several properties shared by G and $G^{*}$, including gyrocommutativity, first countability, and metrizability. As an application, they prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups.

Nikita E. Barabanov and Abraham A. Ungar, in the paper “Binary Operations in the Unit Ball: A Differential Geometry Approach” [16], study binary operations in the open, unit ball of the Euclidean n-space, within the framework of differential geometry. They discover the properties that qualify these binary 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 turns out to be the Einstein addition of relativistically admissible velocities in special relativity theory, and, similarly, the binary operation of the Beltrami–Poincaré ball model of hyperbolic geometry turns out to be Möbius addition. These and other binary operations are derived from metric tensors, giving rise to various gyrocommutative gyrogroups and gyrovector spaces.

Nikita E. Barabanov and Abraham A. Ungar, in the paper “Differential Geometry and Binary Operations” [17], derive a set of binary operations that are isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as Einstein addition. They prove that each of these binary operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup. They also prove that a set of cogyrolines for Einstein addition is the same as the set of gyrolines of another binary operation, which turns out to be commutative. Similar results are obtained for the binary operation of the Beltrami–Poincaré ball model of hyperbolic geometry, known as Möbius addition. Furthermore, they discover a canonical representation of metric tensors of binary operations isomorphic to Einstein addition. Interestingly, they derive a formula for Gaussian curvature spaces with canonical metric tensors, obtaining necessary and sufficient conditions for the Gaussian curvature to vanish.

Nikita E. Barabanov, in the paper “Isomorphism of Binary Operations in Differential Geometry” [18], presents and studies within the framework of differential geometry smooth binary operations that are invariant with respect to unitary transformations, which generalize the binary operations of Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry known, respectively, as Einstein addition and Möbius addition. He provides necessary and sufficient conditions for canonical metric tensors to generate binary operations, and necessary and sufficient conditions for binary operations to be isomorphic. Furthermore, he proves that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups.

Soheila Mahdavi, Ali Reza Ashrafi, Mohammad Ali Salahshour, and Abraham A. Ungar, in the paper “Construction of 2-Gyrogroups in Which Every Proper Subgyrogroup is Either a Cyclic or a Dihedral Group” [19], construct interesting gyrogroups. Specifically, they construct a 2-gyrogroup $G\left(n\right)$ of order $2^n$, $n\ge 3$, in which every proper subgyrogroup is either a cyclic or a dihedral group. Then, they prove that the subgyrogroup lattice and normal subgyrogroup lattice of $G\left(n\right)$ are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order $2^n$. They find that, moreover, all proper subgyrogroups of $G\left(n\right)$ are subgroups.

Milton Ferreira and Teerapong Suksumran, in the paper “Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups” [20], prove an orthogonal decomposition theorem for real inner-product gyrogroups.

Sejong Kim, in the paper “Ordered Gyrovector Spaces” [13], introduces order into gyrovector spaces. The well-known construction scheme to define partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space, he constructs the partial order that he calls the gyro-order.

Takuro Honma and Osamu Hatori, in the paper “A Gyrogeometric Mean in the Einstein Gyrogroup” [21], introduce the definition of gyrogeometric mean into Einstein gyrovector spaces. An Einstein gyrovector space is an Einstein gyrogroup into which scalar multiplication has been introduced.

Keiichi Watanabe, in the paper “On Quasi Gyrolinear Maps between Möbius Gyrovector Spaces Induced from Finite Matrices” [14], studies gyrolinear maps between Möbius gyrovector spaces that correspond to bounded linear operators on real Hilbert spaces.

Abraham A. Ungar, in the paper “A Spacetime Symmetry Approach to Relativistic Quantum Multi-Particle Entanglement” [12], demonstrates that the symmetry group that underlies quantum multi-particle entanglement of m n-dimensional particles is the Lorentz group ${\mathrm{SO}}_{\mathrm{c}}(\mathrm{m},\mathrm{n})$ of signature $(m,n)$ in m temporal dimensions and n spatial dimensions for any $m,n\in \mathbb{N}$ ($n=3$ in physical application, but $n\in \mathbb{N}$ in geometry). The demonstration is justified by means of mathematical patterns and analogies with Einstein’s special theory of relativity.

Valerio Faraoni and Farah Atieh, in the paper “Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant” [22], present continuous generalizations of the Fibonacci sequence that describe a spatially homogeneous and isotropic cosmology in general relativity.

Vesselin G. Gueorguiev and Andre Maeder, in the paper “Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces” [23], illuminate within the Lagrangian framework the general structure of physically relevant classical matter systems, based on the principle of reparametrization invariance. Furthermore, they emphasize the geometric justification of the interaction field Lagrangians for the electromagnetic and gravitational interactions and infer relevance to dark matter and dark energy phenomena on large/cosmological scales. Unusual pathological behavior in the Newtonian limit is suggested to be a precursor of quantum effects and of inflation-like processes at microscopic scales.

Shailendra Rajput, Asher Yahalom, and Hong Qin, in the paper “Lorentz Symmetry Group, Retardation and Energy Transformations in a Relativistic Engine” [24], indicate analysis that led to suggestion of a relativistic engine, raising the question of how can we accommodate the law of momentum and energy conservation? They give a complete analysis of the exchange of energy between the mechanical part of the relativistic engine and the field part and discuss the energy radiated from the relativistic engine. They show that the relativistic engine effect on the energy is 4th-order in $(1/c)$.

Mihai Visinescu, in the paper “Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces $T^{1,1}$ and $Y^{p,q}$” [25], is concerned with completely integrable Hamiltonian systems and generalized action-angle coordinates in the setting of contact geometry. He investigates the deformations of the Sasaki–Einstein structures.

Jan L. Cieśliński and Artur Kobus, in the paper “Group Structure and Geometric Interpretation of the Embedded Scator Space” [26], interpret and study the scator space of dimension 1 + n, for n = 2 and n = 3, as an intersection of some quadrics in the pseudo-Euclidean space of dimension 2n with zero signature. The set of scators was introduced by Fernández-Guasti and Zaldívar in the context of special relativity and the deformed Lorentz metric.