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19 pages, 410 KB

Open AccessArticle

Asymptotic Non-Hermitian Degeneracy Phenomenon and Its Exactly Solvable Simulation

by Miloslav Znojil

Symmetry 2026, 18(3), 506; https://doi.org/10.3390/sym18030506 - 16 Mar 2026

Viewed by 223

A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential

V^{(I C O)} (x) = i x^{3}

, which is by itself not acceptable as a meaningful quantum model due [...] Read more.

A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential

V^{(I C O)} (x) = i x^{3}

, which is by itself not acceptable as a meaningful quantum model due to a combination of its non-Hermiticity, unboundedness, and most of all the Riesz-basis non-diagonalizability of the Hamiltonian, known as its intrinsic exceptional point (IEP) feature. For the purposes of a perturbation-theory-based simulation of the emergence of such a singular system, a simplified (though not too strictly related) toy-model Hamiltonian is proposed. It combines an

N -

point discretization of the real line of coordinates with an ad hoc interaction in a two-parametric N-by-N-matrix Hamiltonian

H = H^{(N)} (A, B)

. After such a simplification, one can still encounter a somewhat weaker form of non-diagonalizability at the conventional Kato’s exceptional-point (EP) limit of parameters

(A, B) \to (A^{(E P)}, B^{(E P)})

. The IEP-non-diagonalizability phenomenon itself appears mimicked by the less enigmatic EP degeneracy of the discrete toy model, especially at large

N ≫ 1

. What we gain is that, in contrast to the IEP case, the regularization of the simplified toy model in vicinity to the black conventional EP becomes feasible. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

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28 pages, 376 KB

Open AccessArticle

The Validity of the Ehrenfest Theorem in Quantum Gravity Theory

by Claudio Cremaschini, Cooper K. Watson, Ramesh Radhakrishnan and Gerald Cleaver

Symmetry 2026, 18(1), 182; https://doi.org/10.3390/sym18010182 - 19 Jan 2026

Viewed by 1133

Abstract

The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of [...] Read more.

The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

16 pages, 3533 KB

Open AccessArticle

The Three-Body Problem: The Ramsey Approach and Symmetry Considerations in the Classical and Quantum Field Theories

by Edward Bormashenko and Mark Frenkel

Symmetry 2025, 17(9), 1404; https://doi.org/10.3390/sym17091404 - 28 Aug 2025

Viewed by 1268

Abstract

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting [...] Read more.

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the “momenta graph”. According to the Ramsey theorem, this graph contains at least one mono-chromatic triangle. This is true even for chaotic motion of three bodies; thus, illustrating the idea supplied by the Ramsey theory, total chaos is impossible. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. Shannon entropy of the momenta graph is calculated. The particular configurations of bodies are addressed, including the Lagrange configuration and the figure eight-shaped motion. The suggested approach is generalized for the quantum field theory with the Pauli–Lubanski pseudo-vector. The suggested coloring procedure is the Lorenz invariant. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

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25 pages, 665 KB

Open AccessArticle

About a Classical Gravitational Interaction in a General Non-Inertial Reference Frame: Applications on Celestial Mechanics and Astrodynamics

by Daniel Condurache, Mihail Cojocari and Ionuț Popa

Symmetry 2025, 17(3), 368; https://doi.org/10.3390/sym17030368 - 28 Feb 2025

Viewed by 1154

Abstract

This paper offers new insights into gravitational interactions within a general non-inertial reference frame. By utilizing symbolic tensor calculus, the study establishes a unified framework that connects time derivatives in non-inertial frames to those in inertial frames. The research introduces new first integrals [...] Read more.

This paper offers new insights into gravitational interactions within a general non-inertial reference frame. By utilizing symbolic tensor calculus, the study establishes a unified framework that connects time derivatives in non-inertial frames to those in inertial frames. The research introduces new first integrals of motion for a system of many particles in arbitrary non-inertial and barycentric rotating reference frames. These first integrals provide a kinematic and geometric visualization of motion in non-inertial frames. Additionally, a generalized potential energy function is presented for broader applicability. For the gravitational two-body problem, the paper delivers a closed-form, coordinate-free solution for the motion of each body relative to the original frame. Consequently, sufficient conditions for stability against collisions are established within the context of the two-body problem in a non-inertial reference frame. Furthermore, the paper examines the relative orbital motion of spacecraft, presenting a closed-form and coordinate-free solution in the local vertical local horizontal (LVLH) non-inertial frame, which is centered on the center of mass of the main spacecraft. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

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21 pages, 337 KB

Open AccessArticle

Planck Length Emerging as the Invariant Quantum Minimum Effective Length Determined by the Heisenberg Uncertainty Principle in Manifestly Covariant Quantum Gravity Theory

by Claudio Cremaschini and Massimo Tessarotto

Symmetry 2024, 16(8), 1042; https://doi.org/10.3390/sym16081042 - 14 Aug 2024

Cited by 1 | Viewed by 3480

Abstract

The meaning of the quantum minimum effective length that should distinguish the quantum nature of a gravitational field is investigated in the context of manifestly covariant quantum gravity theory (CQG-theory). In such a framework, the possible occurrence of a non-vanishing minimum length requires [...] Read more.

The meaning of the quantum minimum effective length that should distinguish the quantum nature of a gravitational field is investigated in the context of manifestly covariant quantum gravity theory (CQG-theory). In such a framework, the possible occurrence of a non-vanishing minimum length requires one to identify it necessarily with a 4-scalar proper length s.It is shown that the latter must be treated in a statistical way and associated with a lower bound in the error measurement of distance, namely to be identified with a standard deviation. In this reference, the existence of a minimum length is proven based on a canonical form of Heisenberg inequality that is peculiar to CQG-theory in predicting massive quantum gravitons with finite path-length trajectories. As a notable outcome, it is found that, apart from a numerical factor of

O (1)

, the invariant minimum length is realized by the Planck length, which, therefore, arises as a constitutive element of quantum gravity phenomenology. This theoretical result permits one to establish the intrinsic minimum-length character of CQG-theory, which emerges consistently with manifest covariance as one of its foundational properties and is rooted both on the mathematical structure of canonical Hamiltonian quantization, as well as on the logic underlying the Heisenberg uncertainty principle. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

25 pages, 309 KB

Open AccessArticle

Anisotropic Generalization of the ΛCDM Universe Model with Application to the Hubble Tension

by Øyvind G. Grøn

Symmetry 2024, 16(5), 564; https://doi.org/10.3390/sym16050564 - 5 May 2024

Cited by 4 | Viewed by 1744

Abstract

I deduce an exact and analytic Bianchi type I solution of Einstein’s field equations, which generalizes the isotropic ΛCDM universe model to a corresponding model with anisotropic expansion. The main point of the article is to present the anisotropic generalization of the ΛCDM [...] Read more.

I deduce an exact and analytic Bianchi type I solution of Einstein’s field equations, which generalizes the isotropic ΛCDM universe model to a corresponding model with anisotropic expansion. The main point of the article is to present the anisotropic generalization of the ΛCDM universe model in a way suitable for investigating how anisotropic expansion modifies observable properties of the ΛCDM universe model. Although such generalizations of the isotropic ΛCDM universe model have been considered earlier, they have never been presented in this form before. Several physical properties of the model are pointed out and compared with properties of special cases, such as the isotropic ΛCDM universe model. The solution is then used to investigate the Hubble tension. It has recently been suggested that the cosmic large-scale anisotropy may solve the Hubble tension. I consider those earlier suggestions and find that the formulae of these papers lead to the result that the anisotropy of the cosmic expansion is too small to solve the Hubble tension. Then, I investigate the problem in a new way, using the exact solution of the field equations. This gives the result that the cosmic expansion anisotropy is still too small to solve the Hubble tension in the general Bianchi type I universe with dust and LIVE (Lorentz Invariant Vacuum Energy with a constant energy density, which is represented by the cosmological constant) and anisotropic expansion in all three directions—even if one neglects the constraints coming from the requirement that the anisotropy should be sufficiently small so that it does not have any significant effect upon the results coming from the calculations of the comic nucleosynthesis during the first ten minutes of the universe. If this constraint is taken into account, the cosmic expansion anisotropy is much too small to solve the Hubble tension. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

38 pages, 579 KB

Open AccessArticle

Groups of Coordinate Transformations between Accelerated Frames

by Georgy I. Burde

Symmetry 2023, 15(6), 1226; https://doi.org/10.3390/sym15061226 - 8 Jun 2023

Viewed by 3293

Abstract

The analysis of the present paper reveals that, besides the relativistic symmetry expressed by the Lorentz group of coordinate transformations which leave invariant the Minkowski metric of space-time of inertial frames, there exists one more relativistic symmetry expressed by a group of coordinate [...] Read more.

The analysis of the present paper reveals that, besides the relativistic symmetry expressed by the Lorentz group of coordinate transformations which leave invariant the Minkowski metric of space-time of inertial frames, there exists one more relativistic symmetry expressed by a group of coordinate transformations leaving invariant the space-time metric of the frames with a constant proper-acceleration. It is remarkable that, in the flat space-time, only those two relativistic symmetries, corresponding to groups of continuous transformations leaving invariant the metric of space-time of extended rigid reference frames, exist. Therefore, the new relativistic symmetry should be considered on an equal footing with the Lorentz symmetry. The groups of transformations leaving invariant the metric of the space-time of constant proper-acceleration are determined using the Lie group analysis, supplemented by the requirement that the group include transformations to or from an inertial to an accelerated frame. Two-parameter groups of two-dimensional (1 + 1), three-dimensional (2 + 1), and four-dimensional (3 + 1) transformations, with the group parameters related to the ratio of accelerations of the frames and the relative velocity of the frame space origins at the initial moment, can be considered as counterparts of the Lorentz group of corresponding dimensions. Defining the form of the interval and the groups of coordinate transformations satisfying the relativity principle paves the way to defining the invariant forms of the laws of dynamics and electrodynamics in accelerated frames. Thus, the problem of extending the relativity principle from inertial to uniformly accelerated frames has been resolved without use of the equivalence principle and/or the general relativity equations. As an application of the transformations to purely kinematic phenomena, the problem of differential aging between accelerated twins is treated. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

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12 pages, 407 KB

Open AccessArticle

Gravitational Refraction of Compact Objects with Quadrupoles

by Nurzada Beissen, Daniya Utepova, Medeu Abishev, Hernando Quevedo, Manas Khassanov and Saken Toktarbay

Symmetry 2023, 15(3), 614; https://doi.org/10.3390/sym15030614 - 28 Feb 2023

Cited by 11 | Viewed by 4444

Abstract

We use the material medium approach to derive the refractive index that can be associated with the gravitational field of a compact object with a quadrupole moment. We consider both a static deformation and a stationary rotation of the gravitational source as the [...] Read more.

We use the material medium approach to derive the refractive index that can be associated with the gravitational field of a compact object with a quadrupole moment. We consider both a static deformation and a stationary rotation of the gravitational source as the source of the quadrupole. We show that up to the first order in the quadrupole, the refractive indices of both configurations are equivalent such that from the point of view of refraction, a mimicking effect can occur. This also holds at the level of the deflection angle. We argue that it is possible to use the refractive indices and the parameters of the trajectories of light rays that propagate in a gravitational field to determine the physical parameters of the source. Full article

(This article belongs to the Special Issue Symmetry in Classical and Quantum Gravity and Field Theory)

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