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13 pages, 320 KiB

Open AccessArticle

Noncommutative Reduction of Nonlinear Schrödinger Equation on Lie Groups

by Alexander Breev, Alexander Shapovalov and Dmitry Gitman

Universe 2022, 8(9), 445; https://doi.org/10.3390/universe8090445 - 26 Aug 2022

Cited by 3 | Viewed by 1050

Abstract

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration [...] Read more.

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in the space of solutions of linear partial differential equations with a set of noncommuting symmetry operators. The approach is implemented for the generalized nonlinear Schrödinger equation on a Lie group in curved space with local cubic nonlinearity. General formalism is illustrated by the example of the noncommutative reduction of the nonstationary nonlinear Schrödinger equation on the motion group

E (2)

of the two-dimensional plane

R^{2}

. In this particular case, we come to the usual (

1 + 1

)-dimensional nonlinear Schrödinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schrödinger equation on the four-dimensional exponential solvable group. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

34 pages, 3563 KiB

Open AccessArticle

Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons

by Kayo Kinjo, Eriko Kaminishi, Takashi Mori, Jun Sato, Rina Kanamoto and Tetsuo Deguchi

Universe 2022, 8(1), 2; https://doi.org/10.3390/universe8010002 - 21 Dec 2021

Cited by 3 | Viewed by 2504

Abstract

We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the [...] Read more.

We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and

(N - 1)

-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

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11 pages, 281 KiB

Open AccessCommunication

Kinematics and Selection Rules for Light-by-Light Scattering in a Strong Magnetic Field

by Anatoly Shabad

Universe 2020, 6(11), 211; https://doi.org/10.3390/universe6110211 - 17 Nov 2020

Cited by 1 | Viewed by 1540

Abstract

Selection rules that follow from parity and four-momentum conservation are listed for head-on light-by-light scattering in a strong magnetic field taking into account nontrivial dispersion laws of different photon eigen-modes. The wave-length shifts occur for certain transitions between photon eigen-modes. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

30 pages, 1136 KiB

Open AccessArticle

Radiation Problems Accompanying Carrier Production by an Electric Field in the Graphene

by Sergei P. Gavrilov, Dmitry M. Gitman, Vadim V. Dmitriev, Anatolii D. Panferov and Stanislav A. Smolyansky

Universe 2020, 6(11), 205; https://doi.org/10.3390/universe6110205 - 6 Nov 2020

Cited by 5 | Viewed by 2098

Abstract

A number of physical processes that occur in a flat one-dimensional graphene structure under the action of strong time-dependent electric fields are considered. It is assumed that the Dirac model can be applied to the graphene as a subsystem of the general system [...] Read more.

A number of physical processes that occur in a flat one-dimensional graphene structure under the action of strong time-dependent electric fields are considered. It is assumed that the Dirac model can be applied to the graphene as a subsystem of the general system under consideration, which includes an interaction with quantized electromagnetic field. The Dirac model itself in the external electromagnetic field (in particular, the behavior of charged carriers) is treated nonperturbatively with respect to this field within the framework of strong-field QED with unstable vacuum. This treatment is combined with a kinetic description of the radiation of photons from the electron-hole plasma created from the vacuum under the action of the electric field. An interaction with quantized electromagnetic field is described perturbatively. A significant development of the kinetic equation formalism is presented. A number of specific results are derived in the course of analytical and numerical study of the equations. We believe that some of predicted effects and properties of considered processes may be verified experimentally. Among these effects, it should be noted a characteristic spectral composition anisotropy of the quantum radiation and a possible presence of even harmonics of the external field in the latter radiation. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

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15 pages, 513 KiB

Open AccessArticle

Saturation of Energy Levels of the Hydrogen Atom in Strong Magnetic Field

by Tiago C. Adorno, Dmitry M. Gitman and Anatoly E. Shabad

Universe 2020, 6(11), 204; https://doi.org/10.3390/universe6110204 - 5 Nov 2020

Viewed by 2759

Abstract

We demonstrate that the finiteness of the limiting values of the lower energy levels of a hydrogen atom under an unrestricted growth of the magnetic field, into which this atom is embedded, is achieved already when the vacuum polarization (VP) is calculated in [...] Read more.

We demonstrate that the finiteness of the limiting values of the lower energy levels of a hydrogen atom under an unrestricted growth of the magnetic field, into which this atom is embedded, is achieved already when the vacuum polarization (VP) is calculated in the magnetic field within the approximation of the local action of Euler–Heisenberg. We find that the mechanism for this saturation is different from the one acting, when VP is calculated via the Feynman diagram in the Furry picture. We study the effective potential that appears when the adiabatic (diagonal) approximation is exploited for solving the Schrödinger equation for the longitudinal degree of freedom of the electron on the lowest Landau level in the atom. We find that the (effective) potential of a point-like charge remains nonsingular thanks to the growing screening provided by VP. The regularizing length turns out to be

\sqrt{α / 3 π} {¯ λ}_{C}

, where

{¯ λ}_{C}

is the electron Compton length. The family of effective potentials, labeled by growing values of the magnetic field condenses towards a certain limiting, magnetic-field-independent potential-distance curve. The limiting values of even ground-state energies are determined for four magnetic quantum numbers using the Karnakov–Popov method. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

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14 pages, 341 KiB

Open AccessArticle

Conformal Anomaly in Yang-Mills Theory and Thermodynamics of Open Confining Strings

by Maxim N. Chernodub

Universe 2020, 6(11), 202; https://doi.org/10.3390/universe6110202 - 31 Oct 2020

Cited by 1 | Viewed by 2144

Abstract

We discuss thermodynamic properties of open confining strings introduced via static sources in the vacuum of Yang-Mills theory. We derive new sum rules for the chromoelectric and chromomagnetic condensates and use them to show that the presence of the confining string lowers the [...] Read more.

We discuss thermodynamic properties of open confining strings introduced via static sources in the vacuum of Yang-Mills theory. We derive new sum rules for the chromoelectric and chromomagnetic condensates and use them to show that the presence of the confining string lowers the gluonic pressure in the bulk of the system. The pressure deficit of the gluon plasma is related to the potential energy in the system of heavy quarks and anti-quarks in the plasma. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

Review

Jump to: Research

29 pages, 436 KiB

Open AccessReview

Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories

by Walter Felipe Wreszinski

Universe 2021, 7(7), 229; https://doi.org/10.3390/universe7070229 - 5 Jul 2021

Cited by 3 | Viewed by 2895

Abstract

We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the [...] Read more.

We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized field theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum fields. In a first part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as unstable particles. Various problems (mostly open) are finally discussed in connection with concrete models. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

32 pages, 589 KiB

Open AccessReview

Casimir and Casimir-Polder Forces in Graphene Systems: Quantum Field Theoretical Description and Thermodynamics

by Galina L. Klimchitskaya and Vladimir M. Mostepanenko

Universe 2020, 6(9), 150; https://doi.org/10.3390/universe6090150 - 9 Sep 2020

Cited by 25 | Viewed by 3312

Abstract

We review recent results on the low-temperature behaviors of the Casimir-Polder and Casimir free energy an entropy for a polarizable atom interacting with a graphene sheet and for two graphene sheets, respectively. These results are discussed in the wide context of problems arising [...] Read more.

We review recent results on the low-temperature behaviors of the Casimir-Polder and Casimir free energy an entropy for a polarizable atom interacting with a graphene sheet and for two graphene sheets, respectively. These results are discussed in the wide context of problems arising in the Lifshitz theory of van der Waals and Casimir forces when it is applied to metallic and dielectric bodies. After a brief treatment of different approaches to theoretical description of the electromagnetic response of graphene, we concentrate on the derivation of response function in the framework of thermal quantum field theory in the Matsubara formulation using the polarization tensor in (2 + 1)-dimensional space—time. The asymptotic expressions for the Casimir-Polder and Casimir free energy and entropy at low temperature, obtained with the polarization tensor, are presented for a pristine graphene as well as for graphene sheets possessing some nonzero energy gap

Δ

and chemical potential

μ

under different relationships between the values of

Δ

and

μ

. Along with reviewing the results obtained in the literature, we present some new findings concerning the case

μ \neq 0

,

Δ = 0

. The conclusion is made that the Lifshitz theory of the Casimir and Casimir-Polder forces in graphene systems using the quantum field theoretical description of a pristine graphene, as well as real graphene sheets with

Δ > 2 μ

or

Δ < 2 μ

, is consistent with the requirements of thermodynamics. The case of graphene with

Δ = 2 μ \neq 0

leads to an entropic anomaly, but is argued to be physically unrealistic. The way to a resolution of thermodynamic problems in the Lifshitz theory based on the results obtained for graphene is discussed. Full article

(This article belongs to the Special Issue Development of Modern Methods of QFT and Their Applications)

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