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Title: Geometric momentum for a particle constrained on a curved hypersurface

The canonical quantization is a procedure for quantizing a classical theory while preserving the formal algebraic structure among observables in the classical theory to the extent possible. For a system without constraint, we have the so-called fundamental commutation relations (CRs) among positions and momenta, whose algebraic relations are the same as those given by the Poisson brackets in classical mechanics. For the constrained motion on a curved hypersurface, we need more fundamental CRs otherwise neither momentum nor kinetic energy can be properly quantized, and we propose an enlarged canonical quantization scheme with introduction of the second category of fundamental CRs between Hamiltonian and positions, and those between Hamiltonian and momenta, whereas the original ones are categorized into the first. As an Nā€‰āˆ’ā€‰1 (N ā©¾ 2) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the spherical surface, a long-standing problem in the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N, 1) symmetry for the motion on the sphere is demonstrated.
Authors:
 [1]
  1. School for Theoretical Physics, and Department of Applied Physics, Hunan University, Changsha 410082 (China)
Publication Date:
OSTI Identifier:
22250853
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 12; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMMUTATION RELATIONS; DIAGRAMS; DYNAMICAL GROUPS; EUCLIDEAN SPACE; GEOMETRY; HAMILTONIANS; KINETIC ENERGY; MECHANICS; POTENTIALS; QUANTIZATION; SPHERES; SPHERICAL CONFIGURATION; SURFACES