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arXiv:1302.2800 · Annals of Physics

Generalized Weyl quantization on the cylinder and the quantum phase

Maciej Przanowski, Przemysław Brzykcy

Published Jun 28, 2013• 5 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2013.06.013

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This work presents a generalized Weyl quantization for the time of arrival function for a system consisting of a free particle confined to move on a line segment.The generalized Weyl quantization relies crucially on the phase space of the classical system being quantized and on the choice of the group of symmetries of this space, which is non-unique.The Hilbert space of the corresponding quantum system is the representation Hilbert space of the unitary irreducible representations of the symmetry group.For the system considered here, the phase space of our time of arrival function is the cylinder × ℝ and the symmetry group (2) is the Euclidean motion group of the plane.We consider two coordinate systems leading to two distinct quantization.
The Weyl–Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalizations. In this paper, we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time–frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.