Fractional Fourier transform and geometric quantization

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• 作者：Witold Chmielowiec ; ^{wchmiel@cft.edu.pl} ; Jerzy Kijowski ^{kijowski@cft.edu.pl}
• 关键词：81S10 ; 53D50 ; 35Q41 ; 43A32
• 刊名：Journal of Geometry and Physics
• 出版年：2012
• 期刊代码：29_03930440
• 类别：ph
• 出版时间：June, 2012
• 卷：62
• 期：6
• 页码：1433-1450
• 文件大小：337 K

摘要

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the 鈥渇ractional Fourier transform鈥?provides a simple example of this construction. As an application of this technique we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schr枚dinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.