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.
Highlights
In classical mechanics and in classical statistical physics as well as in electromagnetic field theory the observables are described by the functions of positions and momenta like energy of a particle or of the spatial coordinates and time like strength of electric or magnetic field
The Wigner functions and tomographic probability distributions of two-qubit states were discussed in [16], where the kernel of the map, which provides the expression of the state tomogram in terms of the discrete Wigner function of the two-qubit state and the kernel of the inverse map and the connection of the constructed maps with the star-product quantization scheme is obtained in an explicit form
We presented review of the map construction which provides the correspondence rule between the operators acting in a Hilbert space and the functions of some variables
Summary
In classical mechanics and in classical statistical physics as well as in electromagnetic field theory the observables are described by the functions of positions and momenta like energy of a particle or of the spatial coordinates and time like strength of electric or magnetic field. The aim of our work is to give the review of available different kinds of the starproduct of the functions and to construct a generic scheme of quantizer-dequantizer operators used to find the kernels of star-products.
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