skip to main content

SciTech ConnectSciTech Connect

Title: Can Dirac quantization of constrained systems be fulfilled within the intrinsic geometry?

For particles constrained on a curved surface, how to perform quantization within Dirac’s canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper canonical quantization is attainable. In this paper, we take a minimum surface, helicoid, on which the motion is constrained, to explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that not only an inconsistency within Dirac theory occurs, but also an incompatibility with Schrödinger theory happens. In contrast, in three-dimensional Euclidean space, the Dirac quantization turns out to be satisfactory all around, and the resultant geometric momentum and potential are then in agreement with those given by the Schrödinger theory. -- Highlights: • Quantum motion on a minimummore » surface, helicoid, is examined within canonical quantization. • Both geometric momentum and geometric potential are embedding quantities. • No canonical quantization can be fulfilled within the intrinsic geometry.« less
Authors:
 [1] ;  [2] ;  [1]
  1. School for Theoretical Physics, and Department of Applied Physics, Hunan University, Changsha, 410082 (China)
  2. (China)
Publication Date:
OSTI Identifier:
22233548
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics (New York); Journal Volume: 341; 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:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CARTESIAN COORDINATES; EUCLIDEAN SPACE; GEOMETRY; HAMILTONIANS; POTENTIALS; QUANTIZATION; QUANTUM MECHANICS; SPACE-TIME; SURFACES; THREE-DIMENSIONAL CALCULATIONS