arXiv subjects
Publications and source records attributed to A. I. Breev.
The relationship between states obtained by the non-commutative integration method of the Schrödinger equation on Lie groups and generalized coherent states is investigated. It is shown that such solutions belong to the class of generalized coherent states when the corresponding λ-representation is real.
A method for constructing coherent states (CS) of finite-level systems with a given angular momentum is proposed. To this end we generalize the known spin equation (SE) to an infinite-dimensional Fock space. The equation describes a special quadratic system in the latter space. Its projections on $d$-dimensional subspaces, represent analogs of SE for $d$-dimensional systems in an external electromagnetic field which describe $d$-dimensional systems with a given angular moment. Using a modification of Malkin-Manko method developed in our earlier work, we construct the corresponding CS for the total quadratic system. Projections of the later CS on finite-dimensional subspaces we call angular moment CS (AMCS) of finite-level systems. The AMCS have a clear physical meaning, they obey the Schrödinger for a $d$-dimensional system with a given angular moment $j=\left(d-1\right)/2$ in an external electromagnetic field. Their possible exact solutions are constructed via exact solutions of the SE in $2$-dimensional space. The latter solutions can be found analytically and are completely described in our earlier works. A one subset of AMCS can be related to Perelomov spinning CS (PSCS). This reflects the fact that the set of possible AMCS is wider than the set of PSCS. AMCS states in a constant magnetic field are constructed. Some of them coincide with the Bloch CS.
We construct generalized coherent states (GCS) of a massive accelerated particle. This example is an important step in studying coherent states (CS) for systems with an unbounded motion and a continuous spectrum. First, we represent quantum states of the accelerated particle both known and new ones obtained by us using the method of non-commutative integration of linear differential equations. A complete set of non-stationary states for the accelerated particle is obtained. This set is expressed via elementary functions and is characterized by a continuous real parameter $η$, which corresponds to the initial momentum of the particle. A connection is obtained between these solutions and stationary states, which are determined by the Airy function. We solved the problem of constructing GCS, in particular, semiclassical states describing the accelerated particle, within the framework of the consistent method of integrals of motion. We have found different representations, coordinate one and in a Fock space, analyzing in detail all the parameters entering in these representations.
We present a new exactly solvable case in strong-field QED with one-dimensional step potential (x-step). The corresponding x-step is given by an analytic asymmetric with respect to the axis x reflection function. The step can be considered as a certain analytic "deformation" of the symmetric Sauter field. Moreover, it can be treated as a new regularization of the Klein step field. We study the vacuum instability caused by this x-step in the framework of a nonperturbative approach to strong-field QED. Exact solutions of the Dirac equation used in the corresponding nonperturbative calculations, are represented in the form of stationary plane waves with special left and right asymptotics and identified as components of initial and final wave packets of particles. We show that in spite of the fact that the symmetry with respect to positive and negative bands of energies is broken, distribution of created pairs and other physical quantities can be expressed via elementary functions. We consider the processes of transmission and reflection in the ranges of the stable vacuum and study physical quantities specifying the vacuum instability. We find the differential mean numbers of electron-positron pairs created from the vacuum, the components of current density and energy-momentum tensor of the created electrons and positrons leaving the area of the strong field under consideration. Besides, we study the particular case of the particle creation due to a weakly inhomogeneous electric field and obtain explicitly the total number, the current density and energy-momentum tensor of created particles. Unlike the symmetric case of the Sauter field the asymmetric form of the field under consideration causes the energy density and longitudinal pressure of created electrons to be not equal to the energy density and longitudinal pressure of created positrons.
In spite of the fact that photons do not interact with an external magnetic field, the latter field may indirectly affect photons in the presence of a charged environment. This opens up an interesting possibility to continuously control the entanglement of photon beams without using any crystalline devices. We study this possibility in the framework of an adequate QED model. In an approximation it was discovered that such entanglement has a resonant nature, namely, a peak behavior at certain magnetic field strengths, depending on characteristics of photon beams direction of the magnetic field and parameters of the charged medium. Numerical calculations illustrating the above-mentioned resonant behavior of the entanglement measure and some concluding remarks are presented.
We address the problem of the electromagnetic radiation produced by charge distributions in the framework of a semiclassical approach proposed in the work by Bagrov, Gitman, Shishmarev and Farias [J. Synchrotron Rad. (2020). 27, 902-911]. In this approach, currents, generating the radiation are considered classically, while the quantum nature of the radiation is kept exactly. Quantum states of the electromagnetic field are solutions of Schrödinger's equation and relevant quantities to the problem are evaluated with the aid of transition probabilities. This construction allows us to introduce the quantum transition time in physical quantities and assess its role in radiation problems by classical currents. We study radiated electromagnetic energies in detail and present a definition for the rate at which radiation is emitted from sources. In calculating the total energy and rate radiated by a pointlike charged particle accelerated by a constant and uniform electric field, we discover that our results are compatible with results obtained by other authors in the framework of the classical radiation theory under an appropriate limit. We also perform numerical and asymptotic analysis of the results.
In this work, using the noncommutative integration method of linear differential equations, we obtain a complete set of solutions to the Schrodinger equation for a quantum asymmetric top in Euler angles. It is shown that the noncommutative reduction of the Schrodinger equation leads to the Lame equation. The resulting set of solutions is determined by the Lame polynomials in a complex parameter, which is related to the geometry of the orbits of the coadjoint representation of the rotation group. The spectrum of an asymmetric top is obtained from the condition that the solutions are invariant with respect to a special irreducible $λ$-representation of the rotation group.
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution of a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.
In the framework of strong-field QED with $x$-steps, we study vacuum mean values of the current density and energy--momentum tensor of the quantized spinor field placed in the so-called $L$-constant electric background. The latter background can be, for example, understood as the electric field confined between capacitor plates, which are separated by a sufficiently large distance $L$. First, we reveal peculiarities of nonperturbative calculating of mean values in strong-field QED with $x$-steps in general and, in the $L$-constant electric field, in particular. We propose a new renormalization and volume regularization procedures that are adequate for these calculations. We find necessary representations for singular spinor functions in the external background under consideration. With their help, we calculate the above mentioned vacuum means. In the obtained expressions, we show how to separate global contributions due to the particle creation and local ones due to the vacuum polarization. We demonstrate how these contributions can be related to the renormalized effective Heisenberg-Euler Lagrangian.
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 noncommutative reduction of the nonstationary nonlinear Schrödinger equation on the motion group $E(2)$ of the two-dimensional plane $\mathbb{R}^{2}$. In the 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.
We consider a model for describing a QED system consisting of a photon beam interacting with quantized charged spinless particles. We restrict ourselves by a photon beam that consists of photons with two different momenta moving in the same direction. Photons with each moment may have two possible linear polarizations. The exact solutions correspond to two independent subsystems, one of which corresponds to the electron medium and another one is described by vectors in the photon Hilbert subspace and is representing a set of some quasi-photons that do not interact with each other. In addition, we find exact solution of the model that correspond to the same system placed in a constant magnetic field. As an example, of possible applications, we use the solutions of the model for calculating entanglement of the photon beam by quantized electron medium and by a constant magnetic field. Thus, we calculate the entanglement measures (the information and the Schmidt ones) of the photon beam as functions of the applied magnetic field and parameters of the electron medium.
We construct and study singular functions in strong-field $QED$ with two external electromagnetic fields that represent principally different types of external backgrounds, the first one belongs to the class of so-called $t$-potential electric steps (electric-like fields that are switched on and off at initial and final time instants), and the second one belongs to the class of so-called $x$-potential electric steps (time-independent electric-like fields of constant direction that are concentrated in a restricted spatial area). As the first background ($T$-constant electric field) is chosen an uniform electric field which acts during a finite time interval $T$ , whereas as the second background ($L$-constant electric field) is chosen a constant electric field confined between two capacitor plates separated by a large distance $L$. For the both cases we find \textrm{in}- and \textrm{out}-solutions of the Dirac equation in terms of light cone variables. With the help of these solutions, we construct Fock-Schwinger proper-time integral representations for all the singular functions that provide nonperturbative (with respect to the external backgrounds) calculations of any transition amplitudes and mean values of any physical quantities. Considering calculations in the $T$-constant field and in the $L$-constant field as different regularizations of the corresponding calculations in the constant uniform electric field, we have demonstrated their equivalence for sufficiently large $T$ and $L$.
A new exactly solvable case in strong-field quantum electrodynamics with a time-dependent external electric field is presented. The corresponding field is given by an analytic function, which is asymmetric (in contrast to Sauter-like electric field) with respect to the time instant, where it reaches its maximum value, that is why we call it the analytic asymmetric electric field. We managed to exactly solve the Dirac equation with such a field, which made it possible to calculate characteristics of the corresponding vacuum instability nonperturbatively. We construct the so-called in- and out-solutions and with their help calculate mean differential and total numbers of created charged particles, probability of the vacuum to remain a vacuum, vacuum mean values of current density and energy-momentum tensor of the particles. We study the vacuum instability in regimes of rapidly and slowly changing analytic asymmetric electric field, and compare the obtained results with corresponding ones obtained earlier for the case of the symmetric Sauter-like electric field. We also compare exact results in the regime of slowly changing field with corresponding results obtained within the slowly varying field approximation recently proposed by two of the authors, thus demonstrating the effectiveness of such an approximation.
We develop a noncommutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the noncommutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the noncommutative integration method. Also, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time $\mathrm{AdS_{3}}$ using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the noncommutative integration method.
In the present article, using a non-commutative integration method of linear differential equations, we, considering the Klein-Gordon equation with the $L$-constant electric field with large $L$ and using the light cone variables, find new complete sets of its exact solutions. These solutions can be related by integral transformations to previously known solutions that were found in Phys. Rev. D. $\textbf{93}$, 045033(2016). Then, using the general theory developed in Phys. Rev. D. $\textbf{93}$, 045002 (2016), we construct (in terms of the new solutions) the so-called in- and out-states of scalar particles confined between two capacitor plates.
We study the problem of the electron excitation spectrum in the presence of point-like and regularized Coulomb impurities in gapped graphene. To this end, we use the Dirac model and in the point-like case theory of self-adjoint extensions of symmetric operators. In the point-like case, we construct a family of self-adjoint Hamiltonians describing the excitations for any charge of an impurity. Spectra and (generalized) eigenfunctions for all such Hamiltonians are found. Then, we consider the spectral problem in the case of a regularized Coulomb potential of impurities for a special regularization. We study exact equations for charges of impurities that may generate bound states with energy that coincides with the upper boundary of the negative branch of the continuous spectrum (supercritical charges) and calculate these charges.
We consider the effects of vacuum polarization and particle creation of a scalar field on Lie groups with a non-stationary bi-invariant metric of the Robertson-Walker type. The vacuum expectation values of the energy momentum tensor for a scalar field determined by the group representation are found using the noncommutative integration method for the field equations instead of separation of variables. The results obtained are illustrated by the example of the three-dimensional rotation group.
We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.