Three paths toward the quantum angle operator |
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Authors: | Jean Pierre Gazeau Franciszek Hugon Szafraniec |
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Institution: | 1. Laboratoire APC, Univ Paris Diderot, Sorbonne Paris Cité, 75205 Paris, France;2. Instytut Matematyki, Uniwersytet Jagielloński, 30-348 Kraków, Poland |
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Abstract: | We examine mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements. We implement three methods of construction of quantum angle. The first one is based on operator theory and parallels the definition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are integral quantization generalizing in a certain sense the Berezin–Klauder approaches. One method pertains to Weyl–Heisenberg integral quantization of the plane viewed as the phase space of the motion on the line. It depends on a family of “weight” functions on the plane. The third method rests upon coherent state quantization of the cylinder viewed as the phase space of the motion on the circle. The construction of these coherent states depends on a family of probability distributions on the line. |
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Keywords: | Quantum angle Quantum phase Weyl&ndash Heisenberg Integral quantization Coherent states Self-adjointness |
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