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Quantization of angle  variables
Title:  Quantization of angle  variables 
Authors:  Arai, A. Browse this author  Tominaga, N. Browse this author 
Issue Date:  May1993 
Publisher:  Department of Mathematics, Hokkaido University 
Journal Title:  Hokkaido University Preprint Series in Mathematics 
Volume:  199 
Start Page:  2 
End Page:  31 
Abstract:  Quantization of anglevariables in the classical Hamilton mechanics with one degree of freedom is considered in a mathematically rigorous way. The method taken in this paper is that of the Weyl quantization, so that the quantized angles are given by pseudodifferential operators of the Weyl type which may be singular. Some operatortheoretical aspects of the quantized angles are discussed. It is shown that the relation between a classical Hamiltonian and an anglevariable of it, which is given by a Poisson bracket relation, is not preserved in general to the quantized version where the Poisson bracket relation is replaced by a commutation relation; an "anomaly" may occur in the commutation relation of the quantized Hamiltonian with the quantized angle. The anomaly may be regarded as a quantum effect. Special attention is paid to the case of the harmonic oscillator. It is proven that the quantized angle eh of the harmonic oscillator, where 1i > 0 is a parameter denoting the Planck constant divided by 211", is represented as an integral operator of the Carleman type. The integral operator representation enables one to expand eh as a power series of n1, which describes the aymptotic behavior of eh as 1i + oo. Also it is shown that the classical limit 1i+ 0 of eh is given by a modified Hilbert transform. 
Type:  bulletin (article) 
URI:  http://hdl.handle.net/2115/68945 
Appears in Collections:  理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

Submitter: 数学紀要登録作業用
