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arXiv:0902.4559 · Physica Scripta

Star-product quantization and symplectic tomography

Olga V. Man’ko

Published Jul 1, 2009• 1 citation
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Abstract

A review of the symplectic tomographic approaches within the framework of star-product quantization is presented. The classical statistical mechanics within the framework of the tomographic representation is considered. The kernels of the star product of functions–symbols of operators in classical and quantum mechanics, are presented.
DOI
10.1088/0031-8949/2009/t135/014004

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Abstract This paper presents a review of star-product formalism. This formalism provides a description for quantum states and observables by means of the functions called’ symbols of operators’. Those functions are obtained via bijective maps of the operators acting in Hilbert space. Examples of the Wigner-Weyl symbols (Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified for the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators required for construction of bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relationship between structure constants of associative star-product of operator symbols and quantizer-dequantizer operators is reviewed.
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