scholarly journals Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1867
Author(s):  
Alexander Breev ◽  
Alexander Shapovalov
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Homogeneous Space ◽  
Integration Method ◽  
Homogeneous Spaces ◽  
Separation Of Variables ◽  
General Structure ◽  
Three Dimensional ◽  
De Sitter ◽  
Elementary Functions

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

scholarly journals COULOMB SCATTERING OF THE DIRAC FERMIONS ON DE SITTER EXPANDING UNIVERSE

2007 ◽  
Vol 22 (34) ◽  
pp. 2573-2585 ◽  
Author(s):  
COSMIN CRUCEAN
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Cross Section ◽  
Dirac Field ◽  
Coulomb Scattering ◽  
De Sitter Spacetime ◽  
Dirac Fermions ◽  
Final States ◽  
De Sitter ◽  
Sitter Spacetime

The lowest order contribution of the amplitude of the Dirac–Coulomb scattering in de Sitter spacetime is calculated assuming that the initial and final states of the Dirac field are described by exact solutions of the free Dirac equation on de Sitter spacetime with a given momentum and helicity. One studies the difficulties that arises when one passes from the amplitude to cross section.

scholarly journals Direct integration method in three-dimensional elasticity and thermoelasticity problems for inhomogeneous transversely isotropic solids: governing equations in terms of stresses

2019 ◽  
pp. 102-105
Author(s):  
R. M. Kushnir ◽  
Y. V. Tokovyy ◽  
D. S. Boiko
Keyword(s):  
Stress Tensor ◽  
Integral Equations ◽  
Integration Method ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Technique ◽  
Direct Integration ◽  
Governing Equations ◽  
Direct Integration Method

An efficient technique for thermoelastic analysis of inhomogeneous anisotropic solids is suggested within the framework of three-dimensional formulation. By making use of the direct integration method, a system of governing equations is derived in order to solve three-dimensional problems of elasticity and thermoelasticity for transversely isotropic inhomogeneous solids with elastic and thermo-physical properties represented by differentiable functions of the variable in the direction that is transversal to the plane of isotropy. By implementing the relevant separation of variables, the obtained equations can be uncoupled and reduced to second-kind integral equations for individual stress-tensor components and the total stress, which represents the trace of the stress tensor. The latter equations can be attempted by any of the numerical, analyticalnumerical, or analytical means available for the solution of the second-kind integral equations. In order to construct the solutions in an explicit form, an advanced solution technique can be developed on the basis of the resolvent-kernel method implying the series representation by the recurring kernels, computed iteratively by the original kernel of an integral equation.

scholarly journals REDUCTION FORMALISM FOR DIRAC FERMIONS ON DE SITTER SPACE–TIME

2008 ◽  
Vol 23 (07) ◽  
pp. 1075-1087 ◽  
Author(s):  
COSMIN CRUCEAN ◽  
RADU RACOCEANU
Keyword(s):  
Perturbation Theory ◽  
Dirac Equation ◽  
Exact Solutions ◽  
Scattering Amplitude ◽  
De Sitter Space ◽  
Space Time ◽  
Green Functions ◽  
Dirac Fermions ◽  
De Sitter

The reduction formulas for Dirac fermions is derived using the exact solutions of free Dirac equation on de Sitter space–time. In the framework of the perturbation theory one studies the Green functions and derives the scattering amplitude in the first orders of perturbation theory.

scholarly journals Non-Hermitian Dirac Hamiltonian in Three-Dimensional Gravity and Pseudosupersymmetry

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Özlem Yeşiltaş
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Space Time ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
De Sitter ◽  
Metric Tensor ◽  
Curved Space ◽  
Dimensional Gravity ◽  
Dirac Hamiltonian

The Dirac Hamiltonian in the(2+1)-dimensional curved space-time has been studied with a metric for an expanding de Sitter space-time which is two spheres. The spectrum and the exact solutions of the time dependent non-Hermitian and angle dependent Hamiltonians are obtained in terms of the Jacobi and Romanovski polynomials. Hermitian equivalent of the Hamiltonian obtained from the Dirac equation is discussed in the frame of pseudo-Hermiticity. Furthermore, pseudosupersymmetric quantum mechanical techniques are expanded to a curved Dirac Hamiltonian and a partner curved Dirac Hamiltonian is generated. Usingη-pseudo-Hermiticity, the intertwining operator connecting the non-Hermitian Hamiltonians to the Hermitian counterparts is found. We have obtained a new metric tensor related to the new Hamiltonian.

scholarly journals EASY HOLOGRAPHY

2013 ◽  
Vol 22 (12) ◽  
pp. 1342013
Author(s):  
HEIKKI ARPONEN
Keyword(s):  
Homogeneous Spaces ◽  
Three Dimensional ◽  
De Sitter ◽  
Asymptotic Symmetry ◽  
Infinite Dimensional ◽  
Higher Dimensional ◽  
Black Hole Information ◽  
Holographic Description ◽  
Area Law

It is argued that the role of infinite-dimensional asymptotic symmetry groups in gravity theories are essential for a holographic description of gravity and possibly to a resolution of the black hole information paradox. I present a simple toy model in two-dimensional hyperbolic/anti-de Sitter (AdS) space and describe, by very elementary considerations, how the asymptotic symmetry group is responsible for the entropy area law. Similar results apply also in three-dimensional AdS space. The failure of the approach in higher-dimensional AdS spaces is explained and resolved by considering other asymptotically noncompact homogeneous spaces.

DIRAC EQUATION IN SOME HOMOGENEOUS SPACE-TIMES, SEPARATION OF VARIABLES AND EXACT SOLUTIONS

1993 ◽  
Vol 08 (25) ◽  
pp. 2351-2364 ◽  
Author(s):  
VÍCTOR M. VILLALBA
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Hypergeometric Functions ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Algebraic Method ◽  
Background Field ◽  
First Order ◽  
Klein Gordon ◽  
Complete Set

In this article, by the use of a further generalization of the algebraic method of separation of variables, the Dirac equation is separated in a family of space-times where it is not possible to find a complete set of first order commuting differential operators. After separation of variables, the Dirac equation is reduced to a set of coupled ordinary differential equations and some exact solutions corresponding to cosmological backgrounds and gravitational waves are computed in terms of hypergeometric functions. The Klein-Gordon equation in this background field is also discussed.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

scholarly journals Flowering plant embryos: How did we end up here?

2021 ◽  
Author(s):  
Stefan A. Rensing ◽  
Dolf Weijers
Keyword(s):  
Current Knowledge ◽  
General Structure ◽  
Three Dimensional ◽  
Flowering Plants ◽  
Flowering Plant ◽  
Adult Plant ◽  
Ancestral State ◽  
Evolutionary Trajectory ◽  
Current State ◽  
Plant Embryo

AbstractThe seeds of flowering plants are sexually produced propagules that ensure dispersal and resilience of the next generation. Seeds harbor embryos, three dimensional structures that are often miniatures of the adult plant in terms of general structure and primordial organs. In addition, embryos contain the meristems that give rise to post-embryonically generated structures. However common, flowering plant embryos are an evolutionary derived state. Flowering plants are part of a much larger group of embryo-bearing plants, aptly termed Embryophyta. A key question is what evolutionary trajectory led to the emergence of flowering plant embryos. In this opinion, we deconstruct the flowering plant embryo and describe the current state of knowledge of embryos in other plant lineages. While we are far yet from understanding the ancestral state of plant embryogenesis, we argue what current knowledge may suggest and how the knowledge gaps may be closed.

scholarly journals LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

2021 ◽  
Author(s):  
PETER SPACEK
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Laurent Polynomial ◽  
Complete Intersections ◽  
Laurent Polynomials ◽  
Young Diagrams ◽  
Similar Shape ◽  
Algebraic Torus ◽  
The One ◽  
Polynomial Potentials

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

Sign in / Sign up

Export Citation Format

Share Document