Generally relativistical Tetrode-Weyl-Fock-Ivanenko formalism and behaviour of quantum-mechanical particles of spin 1/2 in the Abelian monopole field

Some attention in the literature has been given to the case of a particle of spin 1/2 on the background of the external monopole potential. Some aspects of this problem are reexamined here. The primary technical novelty is that the tetrad generally relativistic method of Tetrode-Weyl-Fock-Ivanenko for describing a spinor particle is exploited. The choice of the formalism has turned out to be of great fruitfulness for examining the system. It is matter that, as known, the use of a special spherical tetrad in the theory of a spin 1/2 particle had led Schrodinger to a basis of remarkable features. The basis has been used with great efficiency by Pauli in his investigation on the pro- blem of allowed spherically symmetric wave functions in quantum mechanics. For our purposes, just several simple rules extracted from the much more com- prehensive Pauli's analysis will be quite sufficient; those are almost mnemo- nic working regulations. So, one may remember some very primary facts of D- functions theory and then produce automatically proper wave functions. It seems rather likely, that there may exist a generalized analog of such a re- presentation for J(i)-operators, that might be successfully used whenever in a linear problem there exists a spherical symmetry, irrespective of the con- crete embodiment of such a symmetry. In particular, the case of electron in the external Abelian monopole field completely come under the Sch-Pau method.Comment: 27 pages, Latex20

Generally relativistical Daffin-Kemmer formalis and behaviour of quantm-mechanical particle of spin 1 in the Abelian monopole field

It is shown that the manner of introducing theinteraction between a spin 1 particle and external classical gravitational field can be successfully uni- fied with the approach that occurred with regard to a spin 1/2 particle and was first developed by Tetrode, Weyl, Fock, Ivanenko. On that way a general- ly relativistical Duffin-Kemmer equation is costructed. So, the manner of extending the flat space Dirac equation to general relativity case indicates clearly that the Lorentz group underlies equally both these theories. In other words, the Lorentz group retains its importance and significance at changing the Minkowski space model to an arbitrary curved space-time. In contrast to this, at generalizing the Proca formulation, we automatically destroy any relations to the Lorentz group, although the definition itself for a spin 1 particle as an elementary object was based on just this group. Such a gravity's sensitiveness to the fermion-boson division might appear rather strange and unattractive asymmetry, being subjected to the criticism. Moreover, just this feature has brought about a plenty of speculation on this matter. In any case, this peculiarity of particle-gravity field inter- action is recorded almost in every handbook. In the paper, on the base of the Duffin-Kemmer formalism developed, the problem of a vector particle in the Abelian monopole potential is considered.Comment: 17 pages, Latex20

On solutions of Schrodinger and Dirac equations in spaces of constant curvature, spherical and elliptical models

Exact solutions of the Schrodinger and Dirac equations in generalized cylindrical coordinates of the 3-dimensional space of positive constant curvature, spherical model, have been obtained. It is shown that all basis Schrodinger's and Dirac's wave functions are finite, single-valued, and continuous everywhere in spherical space model S_{3}. The used coordinates (\rho, \phi,z) are simply referred to Eiler's angle variables (\alpha, \beta, \gamma), parameters on the unitary group SU(2), which permits to express the constructed wave solutions \Psi(\rho, phi,z) in terms of Wigner's functions $D_{mm'}^{j}(\alpha, \beta, \gamma)$. Specification of the analysis to the case of elliptic, SO(3.R) group space, model has been done. In so doing, the results substantially depend upon the spin of the particle. In scalar case, the part of the Schrodinger wave solutions must be excluded by continuity considerations, remaining functions are continuous everywhere in the elliptical 3-space. The latter is in agrement with the known statement: the Wigner functions D_{mm'}^{j}(\alpha, \beta, \gamma) at j =0,1,2,... make up a correct basis in SO(3.R) group space. For the fermion case, it is shown that no Dirac solutions, continuous everywhere in elliptical space, do exist. Description of the Dirac particle in elliptical space of positive constant curvature cannot be correctly in the sense of continuity adjusted with its topological structure.Comment: 54 pages. Report to 13th International School & Conference "Foundation & Advances in Nonlinear Science", Eds.: Kuvshinov V.I., Krylov G.G., Minsk, 200

On WKB-Quantization for Kepler Problem in Euclide, Riemann and Lobachevsky 3-Space

Quantum mechanical WKB-method is elaborated for the known quantum Kepler problem in curved 3-space models Euclide, Riemann and Lobachevsky in the framework of the complex variable function theory. Generalized Schr\"{o}dinger, Klein-Fock hydrogen atoms are considered. Exact energy levels are found and their exactness is proved on the base of exploration into $n$-degree terms of the WKB-series. Dirac equation is solved too, but only approximate energy spectrum is established.Comment: 18 page

The doublet of Dirac fermions in the field of the non-Abelia monopole and parity selection rules

The paper concerns a problem of Dirac fermion doublet in the external monopole potential arisen out of embedding the Abelian monopole solution in the non-Abe- lian scheme. In this particular case, the Hamiltonian is invariant under some symmetry operations consisting of an Abelian subgroup in the complex rotational group SO(3.C). This symmetry results in a certain (A)-freedom in choosing a discrete operator entering the complete set {H, j^{2}, j_{3}, N(A), K} . The same complex number A represents a parameter of the wave functions constructed. The generalized inversion-like operator N(A) implies its own (A-dependent) de- finition for scalar and pseudoscalar, and further affords some generalized N(A)-parity selection rules. It is shown that all different sets of basis func- tions Psi(A) determine the same Hilbert space. In particular, the functions Psi(A) decompose into linear combinations of Psi(A=0). However, the bases con- sidered turn out to be nonorthogonal ones when A is not real number; the latter correlates with the non-self-conjugacy property of the operator N(A) at those A-s. (This is a shortened version of the paper).Comment: 15 pages, latex20

Particle with spin S=3/2 in Riemannian space-time

Equations for 16-component vector-bispinor field, originated from Rarita-Schwinger Lagrangian for spin 3/2 field extended to Riemannian space-time are investigated. Additional general covariant constrains for the field are produced, which for some space-time models greatly simplify original wave equation. Peculiarities in description of the massless spin 3/2 field are specified. In the flat Minkowski space for massless case there exist gauge invariance of the main wave equation, which reduces to possibility to produce a whole class of trivial solutions in the the form of 4-gradient of arbitrary (gauge) bispinor function, \Psi ^{0}_{c} = \partial_{c} \psi. Generalization of that property for Riemannian model is performed; it is shown that in general covariant case solutions of the gradient type \Psi^{0}_{\beta} = (\nabla_{\beta} + \Gamma_{\beta})\Psi exist in space-time regions where the Ricci tensor obeys an identity R_{\alpha \beta} - {1 \over 2} R g_{\alpha \beta} = 0.Comment: 12 pages, 61 references, Chapter 7 in: V.M. Red'kov, Fields in Riemannian space and the Lorentz group (in Russian). Publishing House "Belarusian Science", Minsk, 200

Monopole BPS-Solutions of the Yang-Mills Equations in Space of Euclid, Riemann, and Lobachevski

Procedure of finding of the Bogomolny-Prasad-Sommerfield monopole solutions in the Georgi-Glashow model is investigated in detail on the backgrounds of three space models of constant curvature: Euclid, Riemann, Lobachevski's. Classification of possible solutions is given. It is shown that among all solutions there exist just three ones which reasonably and in a one-to-one correspondence can be associated with respective geometries. It is pointed out that the known non-singular BPS-solution in the flat Minkowski space can be understood as a result of somewhat artificial combining the Minkowski space background with a possibility naturally linked up with the Lobachewski geometry. The standpoint is brought forth that of primary interest should be regarded only three specifically distinctive solutions -- one for every curved space background. In the framework of those arguments the generally accepted status of the known monopole BPS-solution should be critically reconsidered and even might be given away.Comment: 7 page

Geometry of 3-Spaces with Spinor Structure

A special approach to examine spinor structure of 3-space is proposed. It is based on the use of the concept of a spatial spinor defined through taking the square root of a real-valued 3-vector. Two sorts of spatial spinor according to P-orientation of an initial 3-space are introduced: properly vector or pseudo vector one. These spinors, \eta and \xi, turned out to be different functions of Cartesian coordinates. To have a spinor space model, you ought to use a doubling vector space. The main idea is to develop some mathematical technique to work with such extended models. Two sorts of spatial spinors are examined with the use of curvilinear coordinates (y_{1},y_{2},y_{3}): cylindrical parabolic, spherical and parabolic ones. Transition from vector to spinor models is achieved by doubling initial parameterizing domain G(y_{1},y_{2},y_{3}) \Longrightarrow \tilde{G}(y_{1},y_{2},y_{3}) with new identification rules on the boundaries. Different spinor space models are built on explicitly different spinor fields \xi(y) and \eta(y). Explicit form of the mapping spinor field \eta(y) of pseudo vector model into spinor \xi(y) of properly vector one is given, it contains explicitly complex conjugation.Comment: 36 pages, 31 figure

Spinor Structure of P-Oriented Space, Kustaanheimo-Stifel and Hopf Bundle - Connection between Formalisms

In the work some relations between three techniques, Hopf's bundle, Kustaanheimo-Stiefel's bundle, 3-space with spinor structure have been examined. The spinor space is viewed as a real space that is minimally (twice as much) extended in comparison with an ordinary vector 3-space: at this instead of 2\pi-rotation now only 4\pi-rotation is taken to be the identity transformation in the geometrical space. With respect to a given P-orientation of an initial unextended manyfold, vector or pseudovector one, there may be constructed two different spatial spinors, $\xi$ and $\eta$, respectively. By definition, those spinors provide us with points of the extended space odels, each spinor is in the correspondence $2 \longrightarrow 1 with points of a vector space. For both models an explicit parametrization of the spinors \xi and \eta by spherical and parabolic coordinates is given, the parabolic system turns out to be the most convenient for simple defining spacial spinors. Fours of real-valued coordinates by Kustaanheimo-Stiefel, U_{a} and V_{a}, real and imaginary parts of complex spinors \xi and \eta respectively, obey two quadratic constraints. So that in both cases, there exists a Hopf's mapping from the part of 3-sphere S_{3} into the entire 2-sphere S_{2}. Relation between two spacial spinor is found: \eta = (\xi - i\sigma ^{2}\xi ^{*})/\sqrt{2}, in terms of Kustaanheimo-Stiefel variables U_{a} and V_{a} it is a linear transformation from SO(4.R), which does not enter its sub-group generated by SU(2)-rotation over spinors.Comment: 28 page

On Concept of Parity for a Fermion

The known problem of fermion parity is considered on the base of investigating possible linear single-valued representations of spinor coverings of the extended Lorentz group. It is shown that in the frame of this theory does not exist, as separate concepts, P-parity and T-parity for a fermion, instead only some unified concept of (PT)-parity can be determined in a group-theoretical language.Comment: 8 page