Microscopic Quantum Hydrodynamics of Systems of Fermions: Part I

The fundamental equations of the microscopic quantum hydrodynamics of fermions in an external electromagnetic field (i.e., the particle balance equation, the momentum balance equation, the energy balance equation, and the magnetic moment balance equation) are derived using the Schrödinger equation. The form of the spin–spin interaction Hamiltonian is specified. To close the system of the balance equations for a multiparticle fermion system, the effective one-particle Schrödinger equation must be introduced.     

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 517–528, June, 1999.     

Pseudoclassical Foldy-Wouthuysen transformation and canonical quantization of (D=2n)-dimensional relativistic particle with spin in an external electromagnetic field

The canonical quantization of a (D=2n)-dimensional Dirac particle with spin in an arbitrary external electromagnetic field is performed in a gauge that makes it possible to describe simultaneously particles and antiparticles (both massive and massiess) already at the classical level. A pseudoclassical Foldy-Wouthuysen transformation is used to find the canonical (Newton-Wigner) coordinates. The connection between this quantization scheme and Blount's picture describing the behavior of a Dirac particle in an external electromagnetic field is discussed.Erevan Physics Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 378–383, March, 1995.     

Solution of the Kemmer and Breit equations for a plane-polarized wave

All solutions of the Kemmer and Breit equations for a plane-polarized wave are presented. In the limit as the particle mass tends to zero, some of the solutions coincide with the electromagnetic field of the wave acting on the particle. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 282–294, May, 1999.     

Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure

In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure. In our model the gravitational and electromagnetic fields are identified as geometrical objects of the structure.We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term ${{\bf A} \wedge {\it d}{\bf A}}$ that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic field plus the gravitational field, together with a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.     

Equation of motion for a radiating charged particle in classical electrodynamics

We propose classical equations of motion for charged particles in an electromagnetic field. These are general formulas for the particle acceleration that take the radiation-induced deceleration into account and contain no second derivatives of the particle velocity. In several particular cases considered, the new equations yield results coinciding with those known in the literature and experimentally verified. We show that in the range of ultrahigh energies, classical electrodynamics does not lead to inherent inconsistencies and in principle allows particle motion with energies exceeding the Pomeranchuk limit.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 112–130, April, 2005.     

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.     

Formalism of two potentials for the numerical solution of Maxwell’s equations

A new formulation of Maxwell’s equations based on the introduction of two vector and two scalar potentials is proposed. As a result, the electromagnetic field equations are written as a hyperbolic system that contains, in contrast to the original Maxwell system, only evolution equations and does not involve equations in the form of differential constraints. This makes the new equations especially convenient for the numerical simulation of electromagnetic processes. Specifically, they can be solved by applying powerful modern shock-capturing methods based on the approximation of spatial derivatives by upwind differences. The cases of an electromagnetic field in a vacuum and an inhomogeneous material are considered. Examples are given in which electromagnetic wave propagation is simulated by solving the formulated system of equations with the help of modern high-order accurate schemes.     

Simulation of collisionless ultrarelativistic electron–proton plasma dynamics in a self-consistent electromagnetic field

The evolution of a collisionless electron–proton plasma in the self-consistent approximation is investigated. The plasma is assumed to move initially as a whole in a vacuum with the Lorentz factor. The behavior of the dynamical system is analyzed by applying a three-dimensional model based on the Vlasov–Maxwell equations with allowance for retarded potentials. It is shown that the analysis of the solution to the problem is not valid in the “center-of-mass frame” of the plasmoid (since it cannot be correctly defined for a relativistic plasma interacting via an electromagnetic field) and the transition to a laboratory frame of reference is required. In the course of problem solving, a chaotic electromagnetic field is generated by the plasma particles. As a result, the particle distribution functions in the phase space change substantially and differ from their Maxwell–Juttner form. Computations show that the kinetic energies of the electron and proton components and the energy of the self-consistent electromagnetic field become identical. A tendency to the isotropization of the particle momentum distribution in the direction of the initial plasmoid motion is observed.     

Effect of interconvertibility of electromagnetic and gravitational waves in strong external electromagnetic fields and the propagation of waves in the field of a charged “black hole” : PMM vol. 38, n 6, 1974, pp. 1122–1129

It is shown that the behavior of an arbitrary wave propagating in the field of a nonrotating charged black hole is defined (with the use of quadratures) by four functions. Each of these functions obeys its second order equation of the wave kind. Short electromagnetic waves falling onto a black hole are reflected by its field in the form of gravitational and electromagnetic waves whose amplitude was explicitly determined. In the case of the wave carrying rays winding around the limit cycle the reflection and transmission coefficients were obtained in the form of analytic expressions.Various physical processes taking place inside, as well as outside a collapsing star, may induce perturbations of the gravitational, electromagnetic and other fields, and lead to the appearance in the surrounding space of waves of various kinds which propagate over a distorted background and are dissipated along its inhomogeneities.In the absence of rotation and charge in a star, the analysis of small perturbations of the gravitational fields is based on the system of Einstein equations linearized around the Schwarzschild solution. In [1, 2] this system of equations, after expansion of perturbations in spherical harmonics and Fourier transformation with respect to time, was reduced to two independent linear ordinary differential equations of second order of the form of the stationary Schrödinger equation for a particle in a potential force field. Each of these equations defines one of two possible independent perturbation kinds: “even” and “odd” (the different behavior of spherical tensor harmonics at coordinate inversion is the deciding factor in the determination of the kind of perturbation [1, 2]). Although these equations were derived with the superposition on the perturbations of the metric of specific coordinate conditions, they define, as shown in [4], the behavior of invariants of the perturbed gravitational field, which imparts to the potential barriers appearing in these equations an invariant meaning.The system of Maxwell equations on the background of Schwarzschild solution also reduces to similar equations, which differ from the above only by the form of potential barriers appearing in these [5].In the presence in the unperturbed solution of a strong electromagnetic field the gravitational and electromagnetic waves interact with each other, and transmutation takes place. The train of short periodic electromagnetic waves generates the accompanying train of gravitational waves. This phenomenon was first analyzed in [6] on and arbitrary background. It was shown in [7, 8] that dense stars surrounded by hot plasma may acquire a charge owing to splitting of charges by radiation pressure and the “sweeping out” of positrons nascent in vapors in strong electrostatic fields. The interaction of waves becomes particularly clearly evident in the neighborhood of black holes which may serve as “valves” by maintaining equilibrium between the relict electromagnetic and gravitational radiation in the Universe. Rotation of black holes intensifies this effect [6].If a nonrotating star possesses an electrostatic charge, the definition of perturbations of the electromagnetic and gravitational fields must be based on the complete system of Einstein-Maxwell equations linearized around the Nordström-Reissner solution. (Small perturbations of electromagnetic field outside a charged black hole were considered in [9, 10] on the basis of the system of Maxwell equations on a “rigid” background of the Nordström-Reissner solution, without taking into account the interconvertibility of gravitational and electromagnetic waves, which materially affects their behavior in the neighborhood of a charged black hole). Here this system of equations which define the interacting gravitational and electromagnetic perturbations are reduced to four independent second order differential equations, two for each kind of perturbations (an importsnt part is played here by the coordinate conditions imposed on the perturbations of the metric, proposed by the authors in [4]). Perturbation components of the metric and of the electromagnetic field are determined in quadratures by the solutions of these equations. If the charge of a star tends to vanish, two of the derived equations convert to equations for gravitational waves on the background of the Schwarzschild solution [1, 2], while the twoothers become equations which are equivalent to Maxwell solutions on the same background. The short-wave asymptotics of derived equations is determined throughout including the neighborhood of the limit cycle for the wave carrying rays. These solutions far away from the point of turn coincide with those obtained in [6] for any arbitrary background. Approximation of geometric optics does not provide correct asymptotics for impact parameters of rays which are close to critical for which the Isotropie and geodesic parameters wind around the limit cycle. This case is investigated below.A similar situation in the Schwarzschild field was analyzed in [11], where analytic expressions for the wave reflection and transmission coefficients were determined, and the integral radiation stream trapped by a black hole produced by another radiation component of the dual system was calculated.     

Generalized Electrodynamics With Ternary Internal Structure

In Refs. [2]–[7] we suggested generalized dynamic equations of motion of relativistic charged particles inside electromagnetic fields. The dynamic equations had been formulated in terms of external as well as internal momenta. Evolution equations for external momenta, the Lorentz-force equations, had been derived from evolution equations for internal momenta. In this paper, along with relativistic dynamics we generalize electromagnetic fields within the scope of ternary algebras. The full theory is constructed in 4D euclidean space. This space possesses an advantage to build ternary mappings from three vectors onto one. The dynamics is given by non-linear evolution equations with cubic characteristic polynomial. In polar representation the internal momenta obey the Jacobi equations whereas external momenta obey the Weierstrass equations for elliptic functions. The generalized electromagnetic fields are defined by the triple fields where the first one has properties of the electric field and the other two have properties of the magnetic field. The field equations for the triple fields analogous to the Maxwell equations are suggested.     

Method of Integral Equations for the Three-Dimensional Problem of Wave Reflection from an Irregular Surface

The problem on the reflection of the field of a plane H-polarized three-dimensional electromagnetic wave from a perfectly conducting interface between media which contains a local perfectly conducting inhomogeneity is considered. To construct a numerical algorithm, the boundary value problem for the system of Maxwell equations in an infinite domain with irregular boundary is reduced to a system of singular integral equations, which is solved by the approximation–collocation method. The elements of the resulting complex matrix are calculated by a specially developed algorithm. The solution of the system of singular integral equations is used to obtain an integral representation for the reflected electromagnetic field and computational formulas for the directional diagram of the reflected electromagnetic field in the far region.     

A numerical method with particle conservation for the Maxwell-Dirac system

A numerical method is presented for solving the Maxwell-Dirac systems. The Maxwell equations with particle and current densities as the source terms are discretized explicitly. To guarantee the particle conservation, the Dirac equations coupled electromagnetic potentials are discretized by the time-splitting method and implicit finite difference. These numerical schemes are conservative in particle density and have second-order accuracy in time and space. One-dimensional numerical results are given to validate the accuracy and the conservation and three-dimensional examples are presented to describe dynamical behaviors of the Maxwell-Dirac system with several external potentials.     

Smoothed Particle ElectroMagnetics: A mesh-free solver for transients

In this paper an advanced mesh-free particle method for electromagnetic transient analysis, is presented. The aim is to obtain efficient simulations by avoiding the use of a mesh such as in the most popular grid-based numerical methods. The basic idea is to obtain numerical solutions for partial differential equations describing the electromagnetic problem by using a set of particles arbitrarily placed in the problem domain. The mesh-free smoothed particle hydrodynamics method has been adopted to obtain numerical solution of time domain Maxwell's curl equations. An explicit finite difference scheme has been employed for time integration. Details about the numerical treatment of electromagnetic vector fields components are discussed. Two case studies in one and in two dimensions are reported. In order to validate the new proposed methodology, named as Smoothed Particle ElectroMagnetics, a comparison with the standard finite difference time domain method results is performed. The intrinsic adaptive capability of the proposed method, has been exploited by introducing irregular particles distribution.     

Solutions of the classical equation of motion for a spin in electromagnetic fields

The evolution of a charged particle spin in electromagnetic fields of a special form is investigated based on the Bargmann-Michel-Telegdi equation. For some types of nonhomogeneous fields, an exact analytic form of the dependence of the spin vector on the four-dimensional velocity of the particle is found. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 509–518, December, 1999.     

Threshold Phenomena in Intense Electromagnetic Fields

The influence of intense electromagnetic fields on the formation and decay of quasistationary states of different quantum systems is investigated based on exact solutions of quantum equations for charged particle motion. The method allows examining systems where a spontaneous decay may occur as well as phenomena that occur only under the action of the field. Different values of the total magnetic moment of the system are taken into account in this consideration. A consistent use of the analytic continuation method allows obtaining nonlinear equations that determine complex energies in an external field. The asymptotic expansions for real and imaginary energy values under the action of weak and strong electromagnetic fields are investigated. The developed approach allows establishing the characteristic values for the length parameters that determine the formation of the processes in superstrong fields. We note that a significant decrease of distances in strong fields may lead to effects with a new characteristic length scale, characterizing a modified quantum electrodynamics (QED) formalism, namely, the QED with the fundamental mass formalism.     

Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings

We consider interaction of charged particles with an electromagnetic (electrostatic) low frequency wave propagating perpendicular to a uniform background magnetic field. The effects of particle trapping by the wave and further acceleration of a surfatron type are discussed in details. Method for this analysis based on the adiabatic theory of separatrix crossing is used. It is shown that particle can unlimitedly accelerate in the trapping in electromagnetic waves and energy of particle does not increase for the system with electrostatic wave.     

Heisenberg representation of second-quantized fields in stationary external fields and nonlinear dielectric media

The Heisenberg formalism for the creation and annihilation operators of quantized fields in stationary external fields is developed. Fields with spin 0, 1/2, 1 are considered in external electromagnetic and scalar fields and in a field of stationary dielectric properties of a nonlinear medium. An elliptic operator that depends on the time as a parameter and whose eigenfunctions can be used to expand the field variables in the Heisenberg representation is constructed. The connection between the creation and annihilation Heisenberg operators and the operators found by diagonalizing the Hamiltonian by Bogolyubov transformations is established. Heisenberg equations of motion are obtained for external fields of arbitrary form. The phenomenological Hamiltonian that is widely used to describe parametric generation of light is derived in the framework of the quantum field theory, and the limits of applicability of the Hamiltonian are established.Technological Institute, St. Petersburg. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 431–451, December, 1993.     

Many-Particle and Semiclassical Limit Transitions for Nonrelativistic Bosons in a Quantized Electromagnetic Field

Using the method of the analytic germ, we obtain a system of equations for the amplitudes of one-particle phase densities of a system of several species of classical particles with electromagnetic interaction. The corresponding equations result from an extremely complicated limit transition in the theory of bosons interacting with a quantized electromagnetic field rather than in the classical equations for N particles in a magnetic field. This transition implies a double limit: first, the limit of large numbers of particles and photons and, second, the semiclassical limit. Moreover, in the first of these limits under some additional assumptions, we obtain the equations that are the steady-state conditions for an action functional considered in a recent paper by Faddeev and Niemi.