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Title: Comment on 'On infinite walls in deformation quantization'

Abstract

We discuss a recent method proposed by Kryukov and Walton to address boundary-value problems in the context of deformation quantization. We compare their method with our own approach and establish a connection between the two formalisms.

Authors:
 [1];  [2]
  1. Departamento de Matematica, Universidade Lusofona de Humanidades e Tecnologias, Av. Campo Grande, 376, 1749-024 Lisbon (Portugal). E-mail: ncdias@mail.telepac.pt
  2. Departamento de Matematica, Universidade Lusofona de Humanidades e Tecnologias, Av. Campo Grande, 376, 1749-024 Lisbon (Portugal). E-mail: joao.prata@ulusofona.pt
Publication Date:
OSTI Identifier:
20766991
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics (New York); Journal Volume: 321; Journal Issue: 2; Other Information: DOI: 10.1016/j.aop.2005.10.003; PII: S0003-4916(05)00192-2; Copyright (c) 2005 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; BOUNDARY-VALUE PROBLEMS; DEFORMATION; QUANTIZATION; QUANTUM MECHANICS; QUANTUM OPERATORS

Citation Formats

Costa Dias, Nuno, and Prata, Joao Nuno. Comment on 'On infinite walls in deformation quantization'. United States: N. p., 2006. Web. doi:10.1016/j.aop.2005.10.003.
Costa Dias, Nuno, & Prata, Joao Nuno. Comment on 'On infinite walls in deformation quantization'. United States. doi:10.1016/j.aop.2005.10.003.
Costa Dias, Nuno, and Prata, Joao Nuno. Wed . "Comment on 'On infinite walls in deformation quantization'". United States. doi:10.1016/j.aop.2005.10.003.
@article{osti_20766991,
title = {Comment on 'On infinite walls in deformation quantization'},
author = {Costa Dias, Nuno and Prata, Joao Nuno},
abstractNote = {We discuss a recent method proposed by Kryukov and Walton to address boundary-value problems in the context of deformation quantization. We compare their method with our own approach and establish a connection between the two formalisms.},
doi = {10.1016/j.aop.2005.10.003},
journal = {Annals of Physics (New York)},
number = 2,
volume = 321,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2006},
month = {Wed Feb 15 00:00:00 EST 2006}
}
  • We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called *-genvalue ('stargenvalue') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of amore » solvable exponential potential. Before the limit is taken, the corresponding *-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schroedinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.« less
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  • Various problems of quantization in the infinite momentum frame are discussed. Naive application of the Dirac procedure is shown to yield inconsistent results. A rigorous method of proceeding from an action to a Hailtonian with consistent equal-time commutators is proposed. The method is applied to a free, massive scalar field theory, the non-linear sigma model in (1+1) dimensions and quantum electrodynamics in (3+1) dimensions.
  • A general approach to the problems of quantizing fields whlch have infinite-dimensional invariance groups ls given. Space and time are treated on a completely equal footing. A Poisson bracket is defined by means of Green's functions, independently of the discovery or recognition of canonical variables, and is shown to satisfy all the usual identities. In accordance with the measurement theoretical foundations of the quantum theory, the Polsson bracket (i.e., commutator) ls defined only for physically measurable group invariants. The Green's functions give a direct description of the propagation of small disturbances arising from a pair of mutually interfering measuremerts. Inmore » order to estabish a correspondence between this approach and conventional canonical theory, a motivation for the adopted definition of the Poisson bracket is outlined with the aid of the fundamental theorem of canonical transformatlon theory. The rest of the discussion is logically independent of this, however. The general theory of "wave operators" and their associated Green' s functions is briefly reviewed. Specific details connected with the group theoretlcal side of the theory are handled in such a way that problems of constraints are completely avoided. In the last section the general method is applied to the Yang-Mills field, as a nontrivial example. The problem of factor ordering is not studied. (auth)« less