Klein Gordon Fock Research Articles

Solutions of Maxwell equations for admissible electromagnetic fields, in spaces with simply transitive four-parameter groups of motions

All non-equivalent solutions of vacuum Maxwell equations are found for the case when space-time manifolds admit simply transitive four-parameter groups of motions [Formula: see text]. The potentials of the admissible electromagnetic fields admit the existence of the algebra of motion integrals of the Hamilton–Jacobi and Klein–Gordon–Fock equations which is isomorphic to the algebra of the group operators for the same group [Formula: see text].

Algebra of the Symmetry Operators of the Klein–Gordon–Fock Equation for the Case When Groups of Motions G3 Act Transitively on Null Subsurfaces of Spacetime

The algebras of the symmetry operators for the Hamilton–Jacobi and Klein–Gordon–Fock equations are found for a charged test particle, moving in an external electromagnetic field in a spacetime manifold on the isotropic (null) hypersurface, of which a three-parameter groups of motions acts transitively. We have found all admissible electromagnetic fields for which such algebras exist. We have proved that an admissible field does not deform the algebra of symmetry operators for the free Hamilton–Jacobi and Klein–Gordon–Fock equations. The results complete the classification of admissible electromagnetic fields, in which the Hamilton–Jacobi and Klein–Gordon–Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of the r-parametric groups of motions of spacetime manifolds if (r≤4).

Algebras of integrals of motion for the Hamilton–Jacobi and Klein–Gordon–Fock equations in spacetime with four-parameter groups of motions in the presence of an external electromagnetic field

The algebras of the integrals of motion of the Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle moving in an external electromagnetic field in a spacetime manifold are found. The manifold admits four-parameter groups of motions that act nontransitively on the spacetime. All admissible electromagnetic fields for which such algebras exist are found. In the case of an arbitrary n-dimensional Riemannian space on which the groups of motions act, it is proved that the admissible field does not deform the algebra of symmetry operators of the free Hamilton–Jacobi and Klein–Gordon–Fock equations. In addition, the system of differential equations, which must be satisfied by the potentials of the admissible electromagnetic field, has been investigated for compatibility.

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Stability of a Solution to One Combined Mixed Problem for the Klein–Gordon–Fock Equation with a Variable Coefficient

Stability of a Solution to One Combined Mixed Problem for the Klein–Gordon–Fock Equation with a Variable Coefficient

Separation of variables in Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle in the stackel spaces of type (1.1)

All equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces provided that Hamilton–Jacobi equation and Klein–Gordon–Fock equation for a charged test particle can be integrated by the method of complete separation of variables are found. The separation is carried out using the complete sets of mutually commuting integrals of motion of type (1.1) whereby in a privileged coordinate system, the given equations turn into parabolic type equations. Hence, these metrics can be used as models for describing plane gravitational waves.

Scalar field radiation emitted by an accelerated scalar point source: A classical field theory approach

In this paper we will use classical field theory to address the interaction of an accelerated point source with a non-massive Klein-Gordon-Fock (KGF) field in Minkowski spacetime. For this, initially, we obtain the KGF equation for the non-massive scalar field via lagrangian formalism and the scalar potential through Green's function formalism. Finally, we reach the expression of the power radiated by a point scalar source under the influence of this field and its covariant generalization.

Solving the mixed problem for the Klein–Gordon–Fock type equation with integral conditions in the case of the inhomogeneous matching conditions

The classical solution of the mixed problem with integral conditions for the Klein–Gordon–Fock type equation in the half strip is considered when inhomogeneous matching conditions are fulfilled. An equivalent conjugation problem is formulated where conjugation conditions are set on characteristics. Constructed inhomogeneous conditions uniquely define gaps of the solution or its derivatives on characteristics and given gaps can be either remained or smoothed while the time argument increases depending on the kernel of the integral operator in unlocal conditions. The solution of this problem is reduced to solving the second-type Volterra integral equations and their systems. The unique solution of these equations in the class of the twice continuously differentiable functions exists when the initial functions are smooth enough. While considering the given problem the method of characteristics is used to construct both an analytical solution, when the solution of the integral equation can be found explicitly, and an approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When the numerical solution is constructed, matching conditions are significant and need to be considered while developing numerical methods.

Dirac and Klein–Gordon–Fock equations in Grumiller’s spacetime

We study the Dirac and the chargeless Klein–Gordon–Fock equations in the geometry of Grumiller’s spacetime that describes a model for gravity of a central object at large distances. The Dirac equation is separated into radial and angular equations by adopting the Newman–Penrose formalism. The angular part of the both wave equations are analytically solved. For the radial equations, we managed to reduce them to one dimensional Schrödinger-type wave equations with their corresponding effective potentials. Fermions’s potentials are numerically analyzed by serving their some characteristic plots. We also compute the quasinormal frequencies of the chargeless and massive scalar waves. With the aid of those quasinormal frequencies, Bekenstein’s area conjecture is tested for the Grumiller black hole. Thus, the effects of the Rindler acceleration on the waves of fermions and scalars are thoroughly analyzed.

A Time–Space Symmetry Based Cylindrical Model for Quantum Mechanical Interpretations

Following a bi-cylindrical model of geometrical dynamics, our study shows that a 6D-gravitational equation leads to geodesic description in an extended symmetrical time–space, which fits Hubble-like expansion on a microscopic scale. As a duality, the geodesic solution is mathematically equivalent to the basic Klein–Gordon–Fock equations of free massive elementary particles, in particular, the squared Dirac equations of leptons. The quantum indeterminism is proved to have originated from space–time curvatures. Interpretation of some important issues of quantum mechanical reality is carried out in comparison with the 5D space–time–matter theory. A solution of lepton mass hierarchy is proposed by extending to higher dimensional curvatures of time-like hyper-spherical surfaces than one of the cylindrical dynamical geometry. In a result, the reasonable charged lepton mass ratios have been calculated, which would be tested experimentally.

Coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry: Symmetries and conservation laws

A complete symmetry group classification of a coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is carried out. This system is a natural two-components generalization of the Klein–Gordon–Fock system with central symmetry. Moreover, conservation laws for the system are obtained. Some interesting physical conclusions related to the derived conserved vectors are discussed.

Group classification and conservation laws of the generalized Klein–Gordon–Fock equation

In the present paper, we perform Lie and Noether symmetries of the generalized Klein–Gordon–Fock equation. It is shown that the principal Lie algebra, which is one-dimensional, has several possible extensions. It is further shown that several cases arise for which Noether symmetries exist. Exact solutions for some cases are also obtained from the invariant solutions of the investigated equation.

The Klein–Gordon–Fock equation in the curved spacetime of the Kerr–Newman (anti) de Sitter black hole

Exact solutions of the Klein–Gordon–Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr–Newman–(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr–Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the solutions of the Heun equation in terms of generalised hypergeometric functions of Lauricella–Appell.

Massless Majorana-Like Charged Carriers in Two-Dimensional Semimetals

The band structure of strongly correlated two-dimensional (2D) semimetal systems is found to be significantly affected by the spin-orbit coupling (SOC), resulting in SOC-induced Fermi surfaces. Dirac, Weyl and Majorana representations are used for the description of different semimetals, though the band structures of all these systems are very similar. We develop a theoretical approach to the band theory of two-dimensional semimetals within the Dirac–Hartree–Fock self-consistent field approximation. It reveals partially breaking symmetry of the Dirac cone affected by quasi-relativistic exchange interactions for 2D crystals with hexagonal symmetry. Fermi velocity becomes an operator within this approach, and elementary excitations have been calculated in the tight-binding approximation when taking into account the exchange interaction of π ( p z ) -electron with its three nearest π ( p z ) -electrons. These excitations are described by the massless Majorana equation instead of the Dirac one. The squared equation for this field is of the Klein–Gordon–Fock type. Such a feature of the band structure of 2D semimetals as the appearance of four pairs of nodes is shown to be described naturally within the developed formalism. Numerical simulation of band structure has been performed for the proposed 2D-model of graphene and a monolayer of Pb atoms.

Spin-Zero Cox's particle With Intrinsic Structure: General Analysis in External Electromagnetic and Gravitational Fields

The relativistic theory of Cox’s scalar non-point particle with intrinsic structure in the Proca approach in external uniform magnetic and electric fields in the Minkowski space is developed. A generalized Klein–Gordon–Fock equation is derived and is detailed in the presence of uniform magnetic and electric fields. The extension of this formalism to the arbitrary Riemannian space-time background is given. For a special class of curved metrics allowing for the existence of nonrelativistic wave equations, a generalized Schr¨odinger-type quantum mechanical equation for Cox’s particle is derived. This generally covariant formalism is suitable in the presence of external magnetic and electric fields. It is shown that, in the most general form, the extended first-order Proca-like system of tensor equations contains non-minimal interaction terms through the electromagnetic tensor FaB and the Ricci tensor RaB.

On astigmatic exponentially localized solutions for the wave and the Klein–Gordon–Fock equations

Highly localized explicit solutions to the wave and Klein–Gordon–Fock multidimensional linear equations are presented. Their Fourier transforms are also found explicitly. Solutions depend on a set of parameters and demonstrate astigmatic properties. Asymptotic analysis for large and moderate time shows that constructed particle-like solutions have Gaussian localization near a point moving with the group speed.

Nonlinear self-adjointness and conservation laws of Klein–Gordon–Fock equation with central symmetry

The concept of nonlinear self-adjointness, introduced by Ibragimov, has significantly extends approaches to constructing conservation laws associated with symmetries since it incorporates the strict self-adjointness, the quasi self-adjointness as well as the usual linear self-adjointness. Using this concept, the nonlinear self-adjointness condition for the Klein–Gordon–Fock equation was established and subsequently used to construct simplified but infinitely many nontrivial and independent conserved vectors. The Noether’s theorem was further applied to the Klein–Gordon–Fock equation to explore more distinct first integrals, result shows that conservation laws constructed through this approach are exactly the same as those obtained under strict self-adjointness of Ibragimov’s method.

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.

Некоторые уравнения на основе одномерных хаотических динамик

Modified Klein–Gordon–Fock equations were obtained on the basis of one-dimensional chaotic dynamics and the original Lagrangians were found. The concepts of $m$-exponential map and groups with broken symmetry are introduced. A system of bitrial orthogonal functions is considered.

"SQUARE ROOT" OF THE PROCA EQUATION: SPIN-3/2 FIELD EQUATION

New equations describing particles with spin-3/2 are derived. The nonlocal equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein–Gordon–Fock equation. The local equation describes spin-3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multispin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing nonminimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.