skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Motion Caused by Magnetic Field in Lobachevsky Space

Abstract

We study motion of a relativistic particle in the 3-dimensional Lobachevsky space in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motion are found and equations of motion are solved exactly in the special cylindrical coordinates. Motion on surface of the cylinder of constant radius is considered in detail.

Authors:
; ; ;  [1]
  1. Institute of Physics, National Academy of Sciences of Belarus, 68 Nezavisimosti Ave., 220072, Minsk (Belarus)
Publication Date:
OSTI Identifier:
21371316
Resource Type:
Journal Article
Resource Relation:
Journal Name: AIP Conference Proceedings; Journal Volume: 1205; Journal Issue: 1; Conference: International conference in honor of Ya.B. Zeldovich's 95. anniversary on the sun, the stars, the Universe and general relativity, Minsk (Belarus), 20-23 Apr 2009; Other Information: DOI: 10.1063/1.3382314; (c) 2010 American Institute of Physics
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANGULAR VELOCITY; COORDINATES; EQUATIONS OF MOTION; EUCLIDEAN SPACE; INTEGRAL EQUATIONS; LAGRANGIAN FIELD THEORY; LOBACHEVSKY GEOMETRY; MAGNETIC FIELDS; RELATIVISTIC RANGE; THREE-DIMENSIONAL CALCULATIONS; DIFFERENTIAL EQUATIONS; ENERGY RANGE; EQUATIONS; FIELD THEORIES; GEOMETRY; MATHEMATICAL SPACE; MATHEMATICS; PARTIAL DIFFERENTIAL EQUATIONS; QUANTUM FIELD THEORY; RIEMANN SPACE; SPACE; VELOCITY

Citation Formats

Kudryashov, V. V., Kurochkin, Yu. A., Ovsiyuk, E. M., and Red'kov, V. M.. Motion Caused by Magnetic Field in Lobachevsky Space. United States: N. p., 2010. Web. doi:10.1063/1.3382314.
Kudryashov, V. V., Kurochkin, Yu. A., Ovsiyuk, E. M., & Red'kov, V. M.. Motion Caused by Magnetic Field in Lobachevsky Space. United States. doi:10.1063/1.3382314.
Kudryashov, V. V., Kurochkin, Yu. A., Ovsiyuk, E. M., and Red'kov, V. M.. Wed . "Motion Caused by Magnetic Field in Lobachevsky Space". United States. doi:10.1063/1.3382314.
@article{osti_21371316,
title = {Motion Caused by Magnetic Field in Lobachevsky Space},
author = {Kudryashov, V. V. and Kurochkin, Yu. A. and Ovsiyuk, E. M. and Red'kov, V. M.},
abstractNote = {We study motion of a relativistic particle in the 3-dimensional Lobachevsky space in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motion are found and equations of motion are solved exactly in the special cylindrical coordinates. Motion on surface of the cylinder of constant radius is considered in detail.},
doi = {10.1063/1.3382314},
journal = {AIP Conference Proceedings},
number = 1,
volume = 1205,
place = {United States},
year = {Wed Mar 24 00:00:00 EDT 2010},
month = {Wed Mar 24 00:00:00 EDT 2010}
}
  • Various possibilities to define analogs of the uniform magnetic field in the Lobachevsky space are considered using different coordinate systems in this space. Quantum mechanical problem of motion in the defined fields is also treated. Variables in the Schroedinger equation are separated and diagonal operators are found. For some cases, exact solutions are obtained.
  • We study symmetry breaking and gap generation for fermions in the 2D space of constant negative curvature (the Lobachevsky plane) in an external covariantly constant magnetic field in a four-fermion model. It is shown that due to the magnetic and negative curvature catalyses phenomena the critical coupling constant is zero and there is a symmetry breaking condensate in the chiral limit even in free theory. We analyze solutions of the gap equation in the cases of zero, weak, and strong magnetic fields. As a byproduct, we calculate the density of states and the Hall conductivity for noninteracting fermions that maymore » be relevant for studies of graphene.« less
  • Reactions and decays can be described in such a way that upon change of coordinate system the spin does not undergo a Lorentz transformation but a translation on a hyperboloid surface in veiocity space. In this case the spin remains a three-dimensional vector and the whole theory of spin effects (addition of spins and their change during scattering) reduces to ordinary nonrelativistic theory. The only correction arises as a result of relativistic spin-orbit coupling (Thomas precession) and can be calcuiated on basis of simple geometrical considerations. (auth)
  • The Inoenue-Wigner contraction from the SO(2, 1) group to the E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the corresponding two-dimensional homogeneous spaces: two-dimensional one sheeted hyperboloid and two-dimensional pseudo-Euclidean space. Here we consider the contraction limits of some basis functions for the subgroup coordinates only.