The first investigations of a spin 3/2 field were performed by Pauli and Fierz. Later, within the general approach of Gelfand and Yaglom to the theory of relativistic wave equations, a more general equation for spin 3/2 particle was proposed by Fradkin. This equation contains one additional parameter, to present time its physical interpretation is not understood, also no solutions for this equation are not known as well. In the present paper, the Fradkin equation has been solved in presence of the uniform magnetic field. We apply the covariant tetrad formalism and use the system of cylindrical coordinates, the wave function transforms as a vector-bispinor under the local Lorentz group. On searched solutions, we diagonalize operators of the energy, the third projection of lineal momentum, and the third projection of the total angular momentum. After separating the variables, we derive the system of 16 differential equations in polar coordinate r. To resolve this system, we apply the method by Fedorov - Gronskiy, which is based on four projective operator constructed from the 16 x 16 spin matrix S3 for vector-bispinor wave function. According to this approach, each projective constituent is determined by only one corresponding function F-i(r), i = 1, 2, 3, 4. Solutions for these basic variables F(i )are found in terms of the confluent hypergeometric functions; due to the presence of the external magnetic field there arises the definite quantization rule for basic spectral parameter, which will be related with the possible values of energy of the particle. Within the used approach, there exist possibility to transform the system of 16 differential equations to homogeneous system of algebraic equations. From vanishing its determinant, we derive an equation of sixth order with respect to energy parameter E2. Its solutions are studied numerically; in this way we arrive at four physically interpretable positive roots, and two complex-valued conjugate roots which relate to anomalous solutions.

In the present paper we develop the theory of the massive spin 2 particle in presence of an external uniform magnetic field. We apply the matrix equation for spin 2 particle in Minkowski space-time, specifying it in cylindrical coordinates t, r, phi, z and tetrad formalism. By diagonalizing operators of the energy, the third projection of the total angular momentum, and the third projection of the linear momentum, we derive the system of 39 differential equations in polar coordinate r. In order to resolve this system we apply the method by Fedorov-Gronskiy based on the projective operator method. In accordance with this method, the dependance of all 39 functions is determined by only five different functions of the polar variable r, which belong to the hypergeometric type. We find in the explicit form five independent solutions of the basic matrix equation. For the energy values, we derived a 7-th order algebraic equation, it has been studied by numerical method; the physically interpretable energy values were separated.

The theory of the graviton field in linear approximation is studied. We apply the matrix equation in Minkowski space-time, specifying it the cylindrical coordinates t, r, phi, z and tetrad. By diagonalizing the operators of the energy, the third projection of the total angular momentum, and the third projection of the linear momentum, we derive the system of 39 differential equations in polar coordinate r . It is resolved with the use of the method by Fedorov-Gronskiy. In accordance with this, dependance of all 39 functions is determined through only five different functions of the variable r , in the case under consideration they are expressed in terms of Bessel functions. We have constructed six linearly independent solutions of the basic equation. In order to eliminate the gauge degrees of freedom, we use the general definition for gauge solutions according to the Pauli-Fierz approach, now adjusted to tetrad formalism. These gauge solutions are constructed with the use the exact solutions for massless spin 1 field. In this way, we find explicit form of four independent gauge solutions for spin 2 field. In the end, we find a explicit form of two gauge-free solutions for spin 2 field, as it should be expected by physical reason.

A spin 3/2 particle is considered in the presence of an external uniform magnetic field. The covariant representation of the Rarita - Schwinger first order equation for vector-bispinor wave function in cylindrical coordinates and tetrad is used.On searching solutions we diagonalize the operators of the energy, the third projection of the linear momentum, and the third projection of the total angular momentum, as a result we derive the system of 16 first order differential equations in the variable r. To resolve this system of equations we apply the Fedorov-Gronskiy method which is based on the use of the projective operators related to 16-dimensional generator J12 for vectorbispinor. Within this approach we decompose the complete wave function into the sum of four projective constituents, each of them is determined by only one corresponding function fi(r), r = 1, 2, 3, 4. For these four basic functions we have constructed the exact solutions in terms of confluent hypergeometric functions. In accordance with the general Fedorov-Gronskiy approach we transform the differential first order system of 16 equations into algebraic homogenous system. From vanishing its determinant we derive and algebraic equation of the fourth order with respect to the squared energy, its solutions give possible values for the energy of the particle. In this way, we find 4 series of real-valued and physically interpretable energy spectra, all remaining ones provide us with complex-valued energies and they should be ignored (they are the so called anomalous solutions).

Our treatment will be with definite accents: the main attention is given to classical electrodynamics in material media, focusing on the structure of Minkowski constitutive relations, matrix complex form of Maxwell theory in the form of Riemann-SilbersteinMajorana-Oppenheimer, and the theory of complex rotation group SO(3.C), isomorphic to the Lorentz group. This review includes the topics: introduction; matrix complex form of Maxwell theory in a vacuum; modified Lorentz symmetry in electrodynamics; Minkowski electrodynamics in moving bodies; Minkowski constitutive relations in the complex 3-vector form; symmetry properties of the matrix equation in any linear media; Dirac matrices and electromagnetic field.

In the present paper, we have developed the theory of a massless spin 2 particle. We apply the matrix equation in Minkowski space-time, specifying it in cylindrical coordinates t , r , φ, z and tetrad. By diagonalizing energy operators, the third projection of total angular momentum, and the third projection of linear momentum, we derive the system of 39 differential equations in a polar coordinate r . In order to resolve this system, we apply the Fedorov–Gronskiy method based on the projective operator method. In accordance with this method, the dependence of all 39 functions is determined only by five different functions of the polar variable r that in the considered case are expressed in terms of Bessel functions. We find the explicit form of six independent solutions of the basic matrix equation. In order to eliminate gauge degrees of freedom, we use the general structure of gauge solutions according to the Pauli-Fierz approach, when the gauge solutions for the spin 2 field are constructed on the basis of the exact solution for a massless spin 1 field (in Bessel functions as well). In this way, we find the explicit form of two independent gauge solutions for the spin 2 field. In the end, we derive the explicit form of two gauge-free solutions for the massless spin 2 field, as should be expected by physical reason.

The most of studies in the theory of spin 2 field were performed with the use of the 2-nd order equations. The spin 2 particle theory proposed by F.I. Fedorov is based on the first order equations requires a 30-component set of tensors. Besides, by him and coauthors was elaborated a more general theory, which is based on 50-component set of tensors. In the present paper, we consider this more general theory in presence of arbitrary electromagnetic fields and Riemannian space-time backgrounds. First we study the 50-component theory for a massive particle. In this case, there arises the non-minimal interaction with the curved space-time background through the Ricci and Riemann tensors. It is important that the theory under consideration allows for a new massless limit for the spin 2 field. This fact is of special interest, because the conventional Pauli - Fierz theory for the massless field does not possess gauge symmetry in the curved space-time, in particular, in models with the vanishing Ricci tensor. We show that the generalized theory possesses such a gauge symmetry in all space-time models for which the Ricci tensor vanishes.

It is Petras who first developed the P-symmetric theory for a spin 1/2 particle with an anomalous magnetic moment within the general Gel’fand – Yaglom approach. Recently, similarly it was introduced a P-asymmetric wave equation for a spin 1/2 particle which describes a particle with an electric dipole moment. In this paper, we study solutions of the equation for the P-asymmetric particle in presence of external magnetic fields. It turns out that the energy spectra are the same for P-asymmetric and P-symmetric particles. To clarify this coincidence, we demonstrate that there exists a simple transformation relating these two models, by which one wave equation can be reduced to the form of the other. Meanwhile, expressions for wave functions and P-reflection operators are different in these two theories. We extend this approach to the model in which both P-symmetric and P-asymmetric sectors are presented. The main result is the same, namely, there exists a simple, more general as compared with the mentioned above transformation relating the P-symmetric model and the model with two sectors, and expressions for wave functions and P-reflection operators are different in these two bases. We demonstrate that in the presence of an external uniform magnetic field, the energy spectra in the model with two sectors coincide with those in the P-symmetric theory. Thus, we develop a general theory for the P-asymmetric model and the model with two sectors within the Petras approach.

The well-known relativistic wave equation for a spin 3/2 particle proposed by Pauli and Fierz is based on the use of the wave function with the transformation properties of vector-bispinor. Less known is the Fradkin theory based on the vector-bispinor wave function as well. At the vanishing Fradkin parameter Λ, this equation reduces to the Pauli – Fierz equation. To clarify the physical meaning of the additional parameter, in the present paper the nonrelativistic approximation in the Fradkin equation is studied, at this we take into account the presence of external electromagnetic fields. With the use of the technique of projective operators, we decompose the wave function into big and small constituents, and then derive a generalized nonrelativistic equation for a 16-component wave function. It is shown that when preserving only the terms of first order in the Fradkin parameter Λ after transition to 4 independent components of the nonrelativistic wave function there arises the ordinary nonrelativistic equation for the Pauli – Fierz theory without any additional interaction with electromagnetic fields. When preserving the terms of second order in parameter Λ, we obtain a 4-component nonrelativistic equation with additional interaction; however, only with the magnetic field. This interaction is quadratic in magnetic field components and governed by six 4-dimensional matrices. So the Fradkin theory may be understood as relevant to a particle with magnetic quadrupole moment.

Relativistic system for a vector-bispinior describing a massless spin 3/2 field is studied in the spherical coordinates of Minkowski space. Presentation of the equation with the use of the covariant Levi-Civita tensor exhibits existence of the gauge solutions in the form of the covariant 4-gradient of an arbitrary bispinor. Substitution for 16-component field function is based on the use of Wigner functions, it assumes diagonalization of the operators of energy, square and third projection of the total angular momentum, and space reflection. We derive radial system for eight independent functions. General structure of the spherical gauge solutions is specified, and it is demonstrated that the gauge radial functions satisfy the derived system. It is proved that the general system reduces to two couples of independent 2-nd order and nonhomogeneous differential equations, their particular solutions may be found with the use of the gauge solutions. The corresponding homogeneous equations have one the same form, they have three regular singularities and one irregular of the rank 2. Frobenius types solutions for this equation have been constructed, and the structure of the involved power series with 4-term recurrent relations sre studied. Six remaining radial functions may be straightforwardly found by means of the simple algebraic relations. Thus, we have constructed two types of solutions with opposite parities which do not contain gauge constituents.

It is known that vacuum Maxwell equations being considered on the background of any pseudo-Riemannin space-time may be interpreted as Maxwell equations in Minkowski space but specified in some effective medium, which constitutive relations are determined by metric of the curved space-time. In that context, we have considered de Sitter, anti de Sitter, and Schwarzschild models. Also we have studied hyperbolic Lobachevsky and spherical Riemann models, parameterized by coordinates with spherical or cylindric symmetry. We have proved that in all the examined cases, effective tensors and of electric permittivity epsilon(ij)(x) and magnetic permeability mu(ij)(x) obey one the same condition: epsilon(ij)(x) mu(jk)(x) = delta(ik). Expressions for tensors epsilon(ij)(x) and mu(jk)(x) are simple, but this simplicity is misleading. For each curved space-time model we are to solve Maxwell equations separately and anew. We have constructed the solutions, applying Maxwell equations in spinor form.

Generalized Klein–Fock–Gordon equation for a spinless particle with the Darwin–Cox structure, which takes into account distribution of the electric charge of a particle inside a finite spherical region is studied in presence of an external Coulomb field. There have been constructed exact Frobenius type solutions of the derived equations, convergence of the relevant power series with 8-term recurrent relations has been studied. As an analytical quantization rule is taken the so-called transcendency conditions. It provides us with a 4-th order algebraic equation with respect to energy values, which has four sets of roots. One set of roots, 0 < En;k < 1, depending on the angular momentum n = 0; 1; 2; : : : and the main quantum number n = 0; 1; 2; : : : may be interpreted as corresponding to some bound states of the particle in a Coulomb field. In the same manner, a generalized nonrelativistic Schr¨odinger equation for such a particle is studied, the final results are similar.

In the frame of the general Gel’fand – Yaglom formalism, the Fradkin theory for a spin 3/2 particle in presence of external fields is investigated. Applying the standard requirements of relativistic invariance, P-symmetry, existence of a Lagrangian for the model, we derive a set of spinor equations, first in absence of external fields. The wave function consists of a bispinor and a vector-bispinor. It is shown that in absence of external fields the Fradkin model is reduced to the Pauli – Fierz theory. Taking into account the presence of external electromagnetic fields, the Fradkin theory can be turned to the minimal form of the equation for the main bispinor. This equation contains an additional interaction term governed by the electromagnetic tensor F αβ . Meanwhile, there appears a parameter in the Fradkin equation related to any characteristic of the particle additional to its charge. The theory is generalized for taking into account the pseudo-Riemannian space-time geometry. In this case, the Fradkin equation contains an additional interaction term, governed by the Ricci tensor R αβ . If the electric charge of the particle is zero, the Fradkin model remains correct and describes a neutral spin 3/2 particle of the Majorana type interacting nonminimally with the geometrical background through the Ricci tensor. To clarify the meaning of the additional physical characteristics underlying the Fradkin model in contrast to the Pauli – Fierz one we have considered nonrelativistic approximation for both theories in presence of an external uniform magnetic field, and found respective energy spectra. The structure of the ninrelativistic Fradkin equation permits to consider such an additional parameter as polarizability.

The relativistic wave equation is well-known for a spin 3/2 particle proposed by W. E. Pauli and M. E. Fierz and based on the 16-component wave function with the transformation properties of the vector-bispinor. In this paper, we investigated the nonrelativistic approximation in this theory. Starting with the first-order equation formalism and representation of Pauli – Fierz equation in the Petras basis, also applying the method of generalized Kronecker symbols and elements of the complete matrix algebras, and decomposing the wave function into large and small nonrelativistic constituents with the help of projective operators, we have derived a Pauli-like equation for the 4-component wave function describing the non-relativistic particle with a 3/2 spin.

The relativistic wave equation is well-known for a spin 3/2 particle proposed by W. E. Pauli and M. E. Fierz and based on the 16-component wave function with the transformation properties of the vector-bispinor. In this paper, we investigated the nonrelativistic approximation in this theory. Starting with the first-order equation formalism and representation of Pauli – Fierz equation in the Petras basis, also applying the method of generalized Kronecker symbols and elements of the complete matrix algebras, and decomposing the wave function into large and small nonrelativistic constituents with the help of projective operators, we have derived a Pauli-like equation for the 4-component wave function describing the non-relativistic particle with a 3/2 spin.

Within the theory of relativistic wave equations with extended sets of Lorentz group representations, a new P-noninvariant 20-component wave equation for spin 1/2 particle is proposed. The presence of an external electromagnetic field and Riemannian space-time background are taken into account. Due to internal structure of the particle, additional interaction terms appears, it relates to anomalous magnetic moment of the particle. Exact solutions of the equation in the presence of an external Coulomb field have been constructed, radial wave functions are expressed in terms of the confluent Heun functions.

Fradkin’s model for a spin-3/2 particle in the presence of external fields is investigated. Applying the general Gel’fand–Yaglom formalism, we develop this model on the base of a set of six irreducible representations of the proper Lorentz group, making up a 20-component wave function. Applying the standard requirements such as the relativistic invariance, single nonzero mass, spin S =3/2, P-symmetry, and existence of a Lagrangian for the model, we derive a set of spinor equations, firstly in the absence of external fields. The 20-component wave function consists of a bispinor and a vector-bispinor. In the absence of external fields, the Fradkin model reduces to the minimal Pauli–Fierz (or Rarita–Schwinger) theory. Details of this equivalence are given. Then we take the presence of external electromagnetic fields into account. It turns out that the Fradkin equation in the minimal form contains an additional interaction term governed by electromagnetic tensor Fab. In addition, we consider the external curved space-time background. In the generally covariant case, the Fradkin equation contains the additional gravitational interaction term governed by the Ricci tensor Rab. If the electric charge of a particle is zero, the Fradkin model remains correct and describes a neutral Majorana-type spin-3/2 particle interacting additionally with the geometric background through the Ricci tensor.

The wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In the frame of the Pauli–Fierz approach, the complete equation is split into the main equation and two additional constraints, algebraic and differential. The solutions are constructed, on which 4 operators are diagonalized: energy, square and third projection of the total angular momentum, and spatial reflection, these correspond to quantum numbers {ε, j, m, P}. After separating the variables, we have derived the radial system of 8 first-order equations and 4 additional constraints. Solutions of the radial equations are constructed as linear combinations of the Bessel functions. With the use of the known properties of the Bessel functions, the system of differential equations is transformed to the form of purely algebraic equations with respect to three quantities a1, a2, a3. Its solutions may be chosen in various ways by solving the simple linear equation A1a1 + A2a2 + A3a3 = 0 where the coefficients Ai are expressed trough the quantum numbers ε, j. Two most simple and symmetric solutions have been chosen. Thus, at fixed quantum numbers {ε, j, m, P} there exists double-degeneration of the quantum states.