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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 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.

In spite of the fact that photons do not interact with an external magnetic field unless nonlinearity of QED is taken into account, 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 propose an identification of the free parameter in the model of nonlinear electrodynamics proposed in Gaete and Helayël-Neto (Eur Phys J C 74:2816, 2014 ) by equating the second term in the power expansion of its Lagrangian with that in the expansion of the Heiseberg–Euler Lagrangian. The resulting value of the field-energy of a point-like charge makes 0.988 of the electron mass, if the charge is that of the electron.

The quantum description of relativistic orientable objects by a scalar field on the Poincaré group is considered. The position of the relativistic orientable object in Minkowski space is completely determined by the position of a body-fixed reference frame with respect to the position of the space-fixed reference frame, so that all the positions can be specified by elements q of the Poincaré group. Relativistic orientable objects are described by scalar wave functions f(q), where the arguments q=(x,z) consist of space–time points x and of orientation variables z from SL(2,C) matrices. We introduce and study the double-sided representation T(g)f(q)=f(gl−1qgr), g=(gl,gr)∈M, of the group M. Here, the left multiplication by gl−1 corresponds to a change in a space-fixed reference frame, whereas the right multiplication by gr corresponds to a change in a body-fixed reference frame. On this basis, we develop a classification of orientable objects and draw attention to the possibility of connecting these results with particle phenomenology. In particular, we demonstrate how one may identify fields described by polynomials in z with known elementary particles of spins 0, 12, and 1. The developed classification does not contradict the phenomenology of elementary particles and, in some cases, even provides a group-theoretic explanation for it.

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 prove corresponding completeness and orthogonality relations. Conditions for minimizing uncertainty relations were studied, and the set of the corresponding parameters was determined. From the GCS, a family of states is isolated, which usually is called the CS. This family of states is parameterized by a real parameter σ q , which has the meaning of the standard deviation of the coordinate at the initial time instant. The CS minimize the Robertson–Schrödinger uncertainty relation at all the time instants and the Heisenberg uncertainty relation at the initial time. The probability density is given by a Gaussian distribution with the standard deviations σ q ( τ ) at the time τ . Coordinate mean values are moving along classical trajectories of the accelerated particles and coincide with trajectories of the maximum of the wave packets. We prove the completeness and orthogonality relations for the obtained GCS and CS. Standard deviations for the GCS and CS are calculated. On this base, and considering the change in the shape of wave packets with time, we define general conditions of the semi-classicality and a class of the CS that can be identified with semiclassical states. As follows from these conditions, in contrast to a free particle case, where CS can be considered as semiclassical states if the Compton wavelength of the particle is much less than the coordinate standard deviation at the initial time moment, after a sufficiently long time period, the CS of the accelerated particle can be always considered as semiclassical ones. It is interesting that this conclusion is matched with the one obtained in a recent work by Sazonov, in studying the Caldirola–Kanai model. Namely, there were demonstrated that the force of resistance and viscous friction prevent the spreading of a quasi-classical wave packet. Thus, the resistance force suppresses the quantum properties of the particle, increasingly highlighting the classical features in its movement over the time.

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 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 R2. 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.

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.

The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In this monograph, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.

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.