open access

https://doi.org/10.14943/83343

https://hdl.handle.net/2115/68945

Quantization of angle - variables

英語

1993-05

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

199

2-31

Quantization of angle-variables 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 pseudo-differential operators of the Weyl type which may be singular. Some operator-theoretical aspects of the quantized angles are discussed. It is shown that the relation between a classical Hamiltonian and an angle-variable 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 n-1, 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.

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