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Title: Quantum motion on a torus as a submanifold problem in a generalized Dirac’s theory of second-class constraints

A generalization of Dirac’s canonical quantization scheme for a system with second-class constraints is proposed, in which the fundamental commutation relations are constituted by all commutators between positions, momenta and Hamiltonian, so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta which leads to ambiguous forms of the Hamiltonian and the momenta. The application of the generalized scheme to the quantum motion on a torus leads to a remarkable result: the quantum theory is inconsistent if built up in an intrinsic geometric manner, whereas it becomes consistent within an extrinsic examination of the torus as a submanifold in three dimensional flat space with the use of the Cartesian coordinate system. The geometric momentum and potential are then reasonably reproduced. -- Highlights: •A generalization of Dirac’s canonical quantization is proposed for a system with second-class constraints. •Quantum motion on torus surface is explicitly treated to show how Schrödinger formalism is complementary to the Dirac one. •The embedding effect in quantum mechanics is originated from the quantization.
Authors:
 [1] ;  [1] ;  [2]
  1. School for Theoretical Physics, and Department of Applied Physics, Hunan University, Changsha, 410082 (China)
  2. School of Science, Xidian University, Xi’an, 710071 (China)
Publication Date:
OSTI Identifier:
22224230
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics (New York); Journal Volume: 338; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CARTESIAN COORDINATES; COMMUTATION RELATIONS; COMMUTATORS; GEOMETRY; HAMILTONIANS; LIMITING VALUES; POTENTIALS; QUANTIZATION; QUANTUM MECHANICS; SPACE-TIME; SURFACES