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18 pages, 436 KiB

Open AccessArticle

Landau Levels versus Hydrogen Atom

by Tekin Dereli, Philippe Nounahon and Todor Popov

Universe 2024, 10(4), 172; https://doi.org/10.3390/universe10040172 - 07 Apr 2024

Viewed by 1089

Abstract

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable

s o (2, 3)

representation. We show that the orthosymplectic algebra

o s p (1 | 4)

is [...] Read more.

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable

s o (2, 3)

representation. We show that the orthosymplectic algebra

o s p (1 | 4)

is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the

o s p (1 | 4)

symmetry broken down to the conformal symmetry

s o (2, 3)

. The even

s o (2, 3)

submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra

o s p (1 | 4)

creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd

s o (2, 3)

-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary

s o (2, 3)

representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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44 pages, 534 KiB

Open AccessArticle

Plea for Diagonals and Telescopers of Rational Functions

by Saoud Hassani, Jean-Marie Maillard and Nadjah Zenine

Universe 2024, 10(2), 71; https://doi.org/10.3390/universe10020071 - 02 Feb 2024

Viewed by 1118

Abstract

This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) [...] Read more.

This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form

P / Q^{k}

and

1 / Q

, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple

F_{1}^{2}

hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

14 pages, 386 KiB

Open AccessArticle

The Unruh Vacuum and the “In-Vacuum” in Reissner-Nordström Spacetime

by Roberto Balbinot and Alessandro Fabbri

Universe 2024, 10(1), 18; https://doi.org/10.3390/universe10010018 - 29 Dec 2023

Cited by 1 | Viewed by 896

Abstract

The Unruh vacuum is widely used as a quantum state to describe black hole evaporation since, near the horizon, it reproduces the physical state of a quantum field, the so-called “in-vacuum”, in the case where a black hole is formed by gravitational collapse. [...] Read more.

The Unruh vacuum is widely used as a quantum state to describe black hole evaporation since, near the horizon, it reproduces the physical state of a quantum field, the so-called “in-vacuum”, in the case where a black hole is formed by gravitational collapse. We examine the relation between these two quantum states in the background spacetime of a Reissner–Nordström black hole (both extremal and not), highlighting the similarities and striking differences. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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17 pages, 342 KiB

Open AccessArticle

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

by Viktor Abramov and Olga Liivapuu

Universe 2024, 10(1), 2; https://doi.org/10.3390/universe10010002 (registering DOI) - 21 Dec 2023

Viewed by 881

Abstract

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a

Z_{3}

-skew-symmetric covariant

SO (3)

-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry [...] Read more.

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a

Z_{3}

-skew-symmetric covariant

SO (3)

-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group

SO (3) ↪ SU (5)

. We find two

SO (3)

-invariants of

Z_{3}

-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by

K (z, z)

. We study the invariant properties of

K (z, z)

and find its stabilizer. Making use of these invariant properties, we define an

SO (3)

-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an

SO (3)

-irreducible geometric structure and find its curvature and torsion. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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17 pages, 355 KiB

Open AccessArticle

Trautman’s Description of Gravitational Radiation Is Universal: A “Pedestrian” Approach to Radiation Phenomena

by Jerzy Kijowski

Universe 2023, 9(12), 520; https://doi.org/10.3390/universe9120520 - 18 Dec 2023

Viewed by 890

Abstract

A simple approach to the Hamiltonian theory of radiation phenomena is proposed. It is shown that the so-called “Trautman-Bondi-mass”, known to a rather narrow circle of specialists in general relativity, appears naturally in any special relativistic field theory. The structure of the “radiation [...] Read more.

A simple approach to the Hamiltonian theory of radiation phenomena is proposed. It is shown that the so-called “Trautman-Bondi-mass”, known to a rather narrow circle of specialists in general relativity, appears naturally in any special relativistic field theory. The structure of the “radiation data phase space” for the field and its isomorphism with the “Cauchy data phase space” are thoroughly analyzed. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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11 pages, 367 KiB

Open AccessArticle

Dyonic Black Holes in Kaluza–Klein Theory with a Gauss–Bonnet Action

by Salvatore Mignemi

Universe 2023, 9(12), 509; https://doi.org/10.3390/universe9120509 - 08 Dec 2023

Viewed by 1009

Abstract

Kaluza–Klein theory attempts a unification of gravity and electromagnetism through the hypothesis that spacetime has five dimensions, of which only four are observed. The original model gives rise to the standard Einstein–Maxwell theory after dimensional reduction. However, in five dimensions, the Einstein–Hilbert action [...] Read more.

Kaluza–Klein theory attempts a unification of gravity and electromagnetism through the hypothesis that spacetime has five dimensions, of which only four are observed. The original model gives rise to the standard Einstein–Maxwell theory after dimensional reduction. However, in five dimensions, the Einstein–Hilbert action is not unique, and one can add to it a Gauss–Bonnet term, giving rise to nonlinear corrections in the dimensionally reduced action. We consider such a model, which reduces to Einstein gravity nonminimally coupled to nonlinear electrodynamics. The black hole solutions of the four-dimensional model modify the Reissner–Nordström solutions of general relativity. We show that in the modified solutions, the gravitational field presents the standard singularity at

r = 0

, while the electric field can be regular everywhere if the magnetic charge vanishes. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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14 pages, 3017 KiB

Open AccessArticle

Combined Screw and Wedge Dislocations

by Mikhail O. Katanaev and Alexander V. Mark

Universe 2023, 9(12), 500; https://doi.org/10.3390/universe9120500 - 29 Nov 2023

Viewed by 1143

Abstract

Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear parallel sources. It describes elastic media with parallel combined wedge [...] Read more.

Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear parallel sources. It describes elastic media with parallel combined wedge and screw dislocations. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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19 pages, 420 KiB

Open AccessArticle

Dynamic Aether as a Trigger for Spontaneous Spinorization in Early Universe

by Alexander Balakin and Anna Efremova

Universe 2023, 9(11), 481; https://doi.org/10.3390/universe9110481 - 14 Nov 2023

Cited by 1 | Viewed by 1027

Abstract

In the framework of the Einstein–Dirac-aether theory we consider a phenomenological model of the spontaneous growth of the fermion number, which is triggered by the dynamic aether. The trigger version of spinorization of the early Universe is associated with two mechanisms: the first [...] Read more.

In the framework of the Einstein–Dirac-aether theory we consider a phenomenological model of the spontaneous growth of the fermion number, which is triggered by the dynamic aether. The trigger version of spinorization of the early Universe is associated with two mechanisms: the first one is the aetheric regulation of behavior of the spinor field; the second mechanism can be related to a self-similarity of internal interactions in the spinor field. The dynamic aether is designed to switch on and switch off the self-similar mechanism of the spinor field evolution; from the mathematical point of view, the key of such a guidance is made of the scalar of expansion of the aether flow, proportional to the Hubble function in the isotropic cosmological model. Two phenomenological parameters of the presented model are shown to be considered as factors predetermining the total number of fermions born in the early Universe. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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Review

Jump to: Research

12 pages, 1512 KiB

Open AccessReview

Dynamics of Cosmological Scalar Fields Revisited

by Jan-Willem van Holten

Universe 2024, 10(5), 197; https://doi.org/10.3390/universe10050197 - 28 Apr 2024

Viewed by 375

Abstract

This paper reviews the dynamics of a single isotropic and homogeneous scalar field

φ (t)

in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution [...] Read more.

This paper reviews the dynamics of a single isotropic and homogeneous scalar field

φ (t)

in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution parameter for cosmic dynamics. General conclusions about the qualitative behaviour of the solutions can be drawn, and examples of how to obtain explicit solutions for some cosmological models of interest are given. For arbitrary potentials, analytical results can be obtained from the slow-roll approximation by using a series expansion for the Hubble parameter

H [φ]

, from which a quantitative estimate for the number of e-folds of expansion is obtained. This approach is illustrated with the examples of quadratic potentials and hilltop models, with special consideration of Higgs-type potentials. The GUT-scale is shown to come out of such a model quite naturally. Finally, it is discussed how to find scalar potentials giving rise to a predetermined scalar-field behaviour and the associated evolution of the scale factor. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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31 pages, 9256 KiB

Open AccessReview

The Language of Spheres in Physics

by Jean-Pierre Gazeau

Universe 2024, 10(3), 117; https://doi.org/10.3390/universe10030117 - 01 Mar 2024

Viewed by 1004

Abstract

Physical laws manifest themselves through the amalgamation of mathematical symbols, numbers, functions, geometries, and relationships. These intricate combinations unfold within a mathematical model devised to capture and represent the “objective reality” of the system under examination. In this symbiotic relationship between physics and [...] Read more.

Physical laws manifest themselves through the amalgamation of mathematical symbols, numbers, functions, geometries, and relationships. These intricate combinations unfold within a mathematical model devised to capture and represent the “objective reality” of the system under examination. In this symbiotic relationship between physics and mathematics, the language of mathematics becomes a powerful tool for describing and predicting the behavior of the physical world. The language used and the associated concepts are in a perpetual state of evolution, mirroring the ongoing expansion of the phenomena accessible to our scientific understanding. In this contribution, written in honor of Richard Kerner, we delve into fundamental, at times seemingly elementary, elements of the mathematical language inherent to the physical sciences, guided by the overarching principles of symmetry and group theory. Our focus turns to the captivating realm of spheres, those strikingly symmetric entities that manifest prominently within our geometric landscape. By exploring the interplay between mathematical abstraction and the tangible beauty of symmetry, we seek to deepen our understanding of the underlying structures that govern our interpretation of the physical world. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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17 pages, 535 KiB

Open AccessReview

Kerner Equation for Motion in a Non-Abelian Gauge Field

by Peter A. Horvathy and Pengming Zhang

Universe 2023, 9(12), 519; https://doi.org/10.3390/universe9120519 - 15 Dec 2023

Cited by 1 | Viewed by 992

Abstract

The equations of motion of an isospin-carrying particle in a Yang–Mills and gravitational field were first proposed in 1968 by Kerner, who considered geodesics in a Kaluza–Klein-type framework. Two years later, the flat space Kerner equations were completed by also considering the motion [...] Read more.

The equations of motion of an isospin-carrying particle in a Yang–Mills and gravitational field were first proposed in 1968 by Kerner, who considered geodesics in a Kaluza–Klein-type framework. Two years later, the flat space Kerner equations were completed by also considering the motion of the isospin by Wong, who used a field-theoretical approach. Their groundbreaking work was then followed by a long series of rediscoveries whose history is reviewed. The concept of isospin charge and the physical meaning of its motion are discussed. Conserved quantities are studied for Wu–Yang monopoles and diatomic molecules by using van Holten’s algorithm. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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