Lorentz group theory and polarization of the light

Some facts of the theory of the Lorentz group are specified for looking at the problems of light polarization optics in the frames of vector Stokes-Mueller and spinor Jones formalism. In view of great differences between properties of isotropic and time-like vectors in Special Relativity we should expect principal differences in describing completely polarized and partly polarized light. In particular, substantial differences are revealed when turning to spinor techniques in the context of the polarized light. Because Jones complex formalism has close relation to spinor objects of the Lorentz group, within the field of the light polarization we could have physical realizations on the optical desk of some subtle topological distinctions between orthogonal L_{+}^{\uparrow} =SO_{0}(3.1) and spinor SL(2.C) groups. These topological differences of the groups find their corollaries in the problem of the so-called spinor structure of physical space-time, some new points are considered.Comment: 17 pages. Talk given at 16 International Seminar: NCPS, May 19-22, 2009, Minsk. A shorter vertion published as a journal pape

Geometrization of the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime

The geometrical Kosambi–Cartan–Chern approach has been applied to study the systems of differential equations which arise in quantum-mechanical problems of a particle on the background of non-Euclidean geometry. We calculate the geometrical invariants for the radial system of differential equations arising for electromagnetic and spinor fields on the background of the Schwarzschild spacetime. Because the second invariant is associated with the Jacobi field for geodesics deviation, we analyze its behavior in the vicinity of physically meaningful singular points r = M, ∞. We demonstrate that near the Schwarzschild horizon r = M the Jacobi instability exists and geodesics diverge for both considered problems.The geometrical Kosambi–Cartan–Chern approach has been applied to study the systems of differential equations which arise in quantum-mechanical problems of a particle on the background of non-Euclidean geometry. We calculate the geometrical invariants for the radial system of differential equations arising for electromagnetic and spinor fields on the background of the Schwarzschild spacetime. Because the second invariant is associated with the Jacobi field for geodesics deviation, we analyze its behavior in the vicinity of physically meaningful singular points r = M, ∞. We demonstrate that near the Schwarzschild horizon r = M the Jacobi instability exists and geodesics diverge for both considered problems

Maxwell Equations in Complex Form of Majorana - Oppenheimer, Solutions with Cylindric Symmetry in Riemann S_{3} and Lobachevsky H_{3} Spaces

Complex formalism of Riemann - Silberstein - Majorana - Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space - time in accordance with the tetrad recipe of Tetrode - Weyl - Fock - Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.Comment: 39 page

Geometrization of the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime

The geometrical Kosambi–Cartan–Chern approach has been applied to study the systems of differential equations which arise in quantum-mechanical problems of a particle on the background of non-Euclidean geometry. We calculate the geometrical invariants for the radial system of differential equations arising for electromagnetic and spinor fields on the background of the Schwarzschild spacetime. Because the second invariant is associated with the Jacobi field for geodesics deviation, we analyze its behavior in the vicinity of physically meaningful singular points r = M, ∞. We demonstrate that near the Schwarzschild horizon r = M the Jacobi instability exists and geodesics diverge for both considered problems

Spin ½ Particle with Anomalous Magnetic Moment in Presence of External Magnetic Field, Exact Solutions

We examine a generalize Dirac equation for spin 1/2 particle with anomalous magnetic mo-ment in presence of the external uniform magnetic field. After separation of the variables, the problem is reduced to a 4-order ordinary differential equation, which is solved exactly with the use of the factorization method. A generalized formula for Landau energy levels is found. Solutions are expressed in terms of confluent hypergeometric functions

Dirac Particle in the Presence of a Magnetic Charge in De Sitter Universe: Exact Solutions and Transparency of the Cosmological Horizon

It has been shown that the monopole background does not a˙ect the transparency properties of the de Sitter cosmological horizon for quantum particles with spin 1/2. One can expect that this conclusion is due only to the geometry of the de Sitter space-time, and it does not depend on spin of the particles. PACS numbers: 02.30.Gp, 02.40.Ky, 03.65Ge, 04.62.+

Spin 2 Particle with Anomalous Magnetic Moment in Presence of Uniform Magnetic Field, Exact Solutions and Energy Spectra

The 50-component matrix equation for spin 2 particle with anomalous magnetic moment is studied in presence of external magnetic field. The matrix tetrad based form of equation in the cylindrical coordinates is used. By diagonalizing the operators of energy, of the third projection of the total angular momentum and the third projection of the linear momentum wa derive the system of 50 differential equations of the first order in polar coordinate. In accordance with the method by Fedorov – Gronskiy based on the use of projective operators, we express all the 50 variables trough 7 different functions? equations for them reduce to the confluent hypergeometric functions. In the result, we obtain a 50-component system algebraic equations which should determine the structure of the total wave function. After eliminating the variables related to 40 components of the third rank tensor we derive the homogeneous algebraic system of 10 equations. It is solved, giving 5 independent solutions. There arise 5 different energy spectra as solutions of the 2-nd and the third order equations. They are found in analytical form and studied numerically

Fradkin Equation for a Spin 3/2 Particle in Electromagnetic and Gravitational Fields

Many year ago, a special generalized equation for a spin 3/2 particle, different from Pauli–Fierz and Rarita–Schwinger model, was proposed by Fradkin. To the presenttime it is not clear which additional structure underlies this extended wave equation. We investigate this model systematically, applying the general Gel’fand–Yaglom formalism. Having used a standard set of requirements: relativistic invariance, singlenonzero mass and single spin S=3/2, P-symmetry, existence of Lagrangian formulation for the model, we derive a set of spinor equations. The 20-component wave function consists of bispinor and vector-bispinor. It is shown that in the free case the Fradkin equation reduces to minimal Pauli–Fierz and Rarita–Schwinger equation for a bisbinor. In presence of external electromagnetic fields, the minimal form the Fradkin equation for a bispinor function contains an additional interaction term governed electromagnetic tensor. Finally we take into account the external curved space-time background, in generally covariant case the Fradkin equation contains additional gravitational interaction through the Ricci tensor. If the electric charge of the particle is zero, the Fradkin model remains correct and describes a neutral spin 3/2 particle of Majorana type interacting additionally with geometrical background by means of the Ricci tensor

Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell electrodynamics in vacuum is investigated within the matrix formalism. The matrix form of electrodynamics includes three real 4 \times 4 matrices. Within the squaring procedure we construct four formal solutions of the Maxwell equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem of separating physical electromagnetic waves in the linear space \lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3} is investigated, several particular cases, plane waves and cylindrical waves, are considered in detail.Comment: 26 pages 16 International Seminar NCPC, May 19-22, 2009, Minsk, Belaru