Fractional Fourier transform and geometric quantization
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Publication:414524
DOI10.1016/J.GEOMPHYS.2012.02.008zbMATH Open1239.81052arXiv1002.3908OpenAlexW1964318540MaRDI QIDQ414524
Author name not available (Why is that?)
Publication date: 11 May 2012
Published in: (Search for Journal in Brave)
Abstract: 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 "fractional Fourier transform" provides a simple example of this construction. As an application of this techniques 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"odinger 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.
Full work available at URL: https://arxiv.org/abs/1002.3908
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