Particle with spin 1 in a magnetic field on the hyperbolic plane H_{2}

There are constructed exact solutions of the quantum-mechanical equation for a spin S=1 particle in 2-dimensional Riemannian space of constant negative curvature, hyperbolic plane, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels describing quantization of the motion of the vector particle in magnetic field on the 2-dimensional space H_{2} has been found, nonrelativistic and relativistic equations have been solved.Comment: 20 page

Some Consequences from the Dirac-Kahler Theory: on Intrinsic Spinor Sub-structure of the Different Boson Wave Functions

Properties of tensors equivalent to the direct product of two different 4-spinors are investigated. It is shown that the tensors obey additional 8 nonlinear restrictions, those are presented in Lorentz covariant form. In the context of the Dirac-K\"{a}hler theory, such a property can be interpreted as follows: if one wishes to consider the Dirac-K\"{a}hler field as consisting of two 4-spinor fields, one must impose additional restrictions on tensors of the Dirac-K\"{a}hler field, which leads to a non-linear wave equation for a complex boson field (composed on the base of two 4-spinor fields). Instead, the use of four bi-spinor fields gives possibility to construct the Dirac-K\"{a}hler tensor set of 16 independent components. However, the formulas relating the Dirac-K\"{a}hler boson to four fermion fields are completely different from those previously used in the literature. In explicit form, restrictions on four 4-spinor corresponding to separation of different simplest bosons with spin 0 or 1 and various intrinsic parities, are constructed.Comment: 16 pages, Report to: 16th International Conference and School "Foundation & Advances in Nonlinear Science", September 24-26, 2012, Mins

Quantum mechanics for a vector particle in the magnetic field on 4-dimensional sphere

Quantum-mechanical wave equation for a particle with spin 1 is investigated in presence of external magnetic field in spaces with non-Euclidean geometry with constant positive curvature. Separation of the variable is performed; differential equations in the variable r are solved in hypergeometric functions. T he study o f z-dependence of the wave function has been reduced to a system of three linked ordinary differential 2-nd order equations; till now the system in z variable is not solved.Comment: 27 pages; Nauchno-Technicheskie Vedomosti. Saint Petersburg State Polytechnical Universit

Spin 1/2 Particle in the Field of the Dirac String on the background of Anti de Sitter Space-Time

The Dirac monopole string is specified for anti de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of anti de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables radial equations have been solved exactly in terms of hypergeometric functions. The complete set of spinor wave solutions has been constructed, the most attention is given to treating the states of minimal values for total moment quantum number j_{min}. At all values of j, the energy spectrum is discrete.Comment: 21 page

Parameters of Lorentz Matrices and Transitivity in Polarization Optics

In the context of applying the Lorentz group theory to polarization optics in the frames of Stokes-Mueller formalism, some properties of the Lorentz group are investigated. We start with the factorized form of arbitrary Lorentz matrix as a product of two commuting and conjugate $4\times 4$-matrices, $L(q,q)= A(q_{a}) A^*(q_{a}); a= 0,1,2,3$. Mueller matrices of the Lorentzian type M=L are pointed out as a special sub-class i n the total set of $4\times 4$ matrices of the linear group GL(4,R). Any arbitrary Lorentz matrix is presented as a linear combination of 16 elements of the Dirac basis. On this ground, a method to construct parameters q_a by an explicitly given Lorentz matrix L is elaborated. It is shown that the factorized form of L=M matrices provides us with a number of simple transitivity equations relating couples of initial and final 4-vectors, which are defined in terms of parameters q_a of the Lorentz group. Some of these transitivity relations can be interpreted within polarization optics and can be applied to the group-theoretic analysis of the problem of measuring Mueller matrices in optical experiments.Comment: 15 pages. Report to The International Conference "Differential Geometry and Dynamical Systems - 2012". 29 August - 2 September 2012, Mangalia, Romani

General conditions for vanishing the current J^{z} for Dirac field on boundaries of the domain between two planes

In connection with the Casimir effect for a spinor field in presence of external magnetic field, of special interest are solutions of the Dirac equation in the domains restricted by two planes, which have vanishing the third projection of the conserved current J^{z} on two boundaries. General conditions for vanishing the current are formulated, they reduce to linear homogeneous algebraic system, for which solutions exist only when vanishing the determinant of the linear system, that is for the roots of a 4-th order algebraic equation with respect to the variable e^{2ik a}, where a is a half-distance between the planes, and k stands for the third projection of the Dirac particle momentum. All solutions of the equation have been found explicitly, each of them provides us in principle with a special possibility to get the quantization rules for parameter k; the most of produced expression for the roots can be solved with respect to parameter k only numerically. Generally, solutions e^{2ik a} depend on 4 arbitrary phase parameters which influence the appropriate wave functions with vanishing current: J^{z}(z=-z,+a)=0.Comment: 23 page

Spin 1/2 particle in the field of the Dirac string on the background of de Sitter space-time

The Dirac monopole string is specified for de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables, detailed analysis of the radial equations is performed; four types of solutions, singular, regular, in- and out- running waves, are constructed in terms of hypergeometric functions. The complete set of spinor wave solutions \Psi_{\epsilon, j,m, \lambda}(t,r, \theta, \phi) has been constructed, special attention is given to treating the states of minimal values of the total angular moment j_{\min}.Comment: 27 pages, 46 references; Report to 18 Conference NPCS-2011, May 17-20, 2011, Minsk, Belaru

Spinors, matrix structures, and projective geometry in polarization optics

The paper discusses the role played by Mueller and Jones formalisms in polarization optics, by addressing the following aspects: restriction to the SU(2) symmetry, non-relativistic Stokes 3-vectors; Cartan 2-spinors in polarization optics; Jones 4-spinors for partially polarized light; the linear group SL(4,R) and the classification of 1-parametric Mueller matrices; semi-group structure and classification of degenerate Mueller matrices.Comment: 13 pages, Report to: Finsler Extensions of Relativity Theory (FERT-2013), 26 - 30 September 2013. Debrecen, Hungar

Confluent Heun functions and the Coulomb problem for spin 1/2 particle in Minkowski space

In the paper, the well-known quantum mechanical problem of a spin 1/2 particle in external Coulomb potential, reduced to a system of two first-order differential equations, is studied from the point of view of possible applications of the Heun function theory to treat this system. It is shown that in addition to the standard way to solve the problem in terms of the confluent hypergeometric functions (proposed in 1928 by G. Darvin and W. Gordon), there are possible several other possibilities which rely on applying the confluent Heun functions. Namely, in the paper there are elaborated two combined possibilities to construct solutions: the first applies when one equation of the pair of relevant functions is expressed trough hypergeometric functions, and another constructed in terms of confluent Heun functions. In this respect, certain relations between the two classes of functions are established. It is shown that both functions of the system may be expressed in terms of confluent Heun functions. All the ways to study this problem lead us to a single energy spectrum, which indicates their correctness.Comment: 22 pages, submitted to: Nonlinear Phenomena in Complex System

Electromagnetic Field in de Sitter Expanding Universe: Majorana--Oppenheimer Formalism, Exact Solutions in non-Static Coordinates

Tetrad-based generalized complex formalism by Majorana--Oppenheimer is applied to treat electromagnetic field in extending de Sitter Universe in on-static spherically-symmetric coordinates. With the help of Wigner D-functions, we separate angular dependence in the complex vector field E_{j}(t,r)+i B_{j}(t,r) from (t,r)-dependence. The separation parameter arising here instead of frequency \omega in Minkowski space-time is quantized, non-static geometry of the de Sitter model leads to definite dependence of electromagnetic modes on the time variable. Relation of 3-vector complex approach to 10-dimensional Duffin-Kemmer-Petiau formalism is considered. On this base, the electromagnetic waves of magnetic and electric type have been constructed in both approaches. In Duffin-Kemmer-Petiau approach, there are constructed gradient-type solutions in Lorentz gauge.Comment: 35 page