scholarly journals Proper Dirac quantization of a free particle on a D-dimensional sphere

1997 ◽  
Vol 232 (5) ◽  
pp. 327-332 ◽  
Author(s):  
Hagen Kleinert ◽  
Sergei V. Shabanov
Keyword(s):  
Free Particle ◽  
Dimensional Sphere ◽  
Dirac Quantization

scholarly journals IMPROVED DIRAC QUANTIZATION OF A FREE PARTICLE

2000 ◽  
Vol 15 (31) ◽  
pp. 1915-1922 ◽  
Author(s):  
SOON-TAE HONG ◽  
WON TAE KIM ◽  
YOUNG-JAI PARK
Keyword(s):  
Energy Spectrum ◽  
Free Particle ◽  
Dimensional Sphere ◽  
Class Constraint ◽  
Dirac Quantization ◽  
Physical Consequences

In the framework of Dirac quantization with second-class constraints, a free particle moving on the surface of a (d-1)-dimensional sphere has an ambiguity in the energy spectrum due to the arbitrary shift of canonical momenta. We explicitly show that this spectrum obtained by the Dirac method is consistent with the result of the Batalin–Fradkin–Tyutin formalism, which is an improved Dirac method, at the level of the first-class constraint by fixing the ambiguity, and discuss its physical consequences.

DISCRETIZED BRST TRANSFORMATIONS AND QUANTUM MECHANICAL POTENTIAL OF A FREE PARTICLE ON SD

2005 ◽  
Vol 20 (09) ◽  
pp. 699-706 ◽  
Author(s):  
KATSUTARO SHIMIZU
Keyword(s):  
Path Integral ◽  
Free Particle ◽  
Quantum Mechanical ◽  
Dimensional Sphere ◽  
Brst Transformation ◽  
Canonical Variables

We evaluated a quantum mechanical potential of a free particle on D-dimensional sphere. This system has the second-class constraints. We change the second-class constraints into first-class ones with new canonical variables. A BRST transformation is induced by the first-class constraints and is discretized. A discretized BRST invariant path integral is considered and the quantum mechanical potential is evaluated as R/12.

scholarly journals DIRAC QUANTIZATION OF FREE MOTION ON CURVED SURFACES

2006 ◽  
Vol 03 (04) ◽  
pp. 655-666 ◽  
Author(s):  
ALEXEY V. GOLOVNEV
Keyword(s):  
Particle Motion ◽  
Free Particle ◽  
Quantum Potential ◽  
Free Motion ◽  
Curved Surfaces ◽  
Dirac Quantization ◽  
Operator Realization ◽  
Natural Way

We give an explicit operator realization of Dirac quantization of free particle motion on a surface of codimension 1. It is shown that the Dirac recipe is ambiguous and a natural way of fixing this problem is proposed. We also introduce a modification of Dirac procedure which yields zero quantum potential. Some problems of Abelian conversion quantization are pointed out.

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.

EXCITON INSULATOR STATES IN MATERIALS WITH ‘MEXICAN HAT’ BAND STRUCTURE DISPERSION AND IN GRAPHENE

2015 ◽  
Vol 11 (1) ◽  
pp. 2927-2949
Author(s):  
Lyubov E. Lokot
Keyword(s):  
Band Structure ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Electron Hole ◽  
Dinger Equation ◽  
Fermi Sphere ◽  
Zinc Blende ◽  
Bulk Gan ◽  
Structure Dispersion ◽  
Excitonic States

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in grapheneand in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensionalSchrödinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection isexactly solved. The solutions of Schrödinger equation in momentum space in studied materials by projection the twodimensionalspace of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensionalSchrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. Instudied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and lightabsorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole areseparated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materialsallows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone andhence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a widthof band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of theequation of motion for a microscopic dipole of graphene where >1 s r . The absorption spectrums for the zinc-blendeGaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with availableexperimental data shows good agreement.

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

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