# Probability representation of quantum mechanics and star product quantization

## TL;DR

This paper reviews the star-product formalism in quantum mechanics, illustrating how quantum states and observables can be represented by functions called symbols, with examples like Wigner distributions and tomograms, and discusses the mathematical structure behind these representations.

## Contribution

It provides a comprehensive review of the star-product formalism, including the properties of quantizer-dequantizer operators and their role in operator-symbol mappings in quantum mechanics.

## Key findings

- Analysis of Wigner-Weyl symbols and tomographic distributions
- Discussion of quantizer-dequantizer operator properties
- Relation between star-product structure constants and operators

## Abstract

The review of star-product formalism providing the possibility to describe quantum states and quantum observables by means of the functions called symbols of operators which are obtained by means of bijective maps of the operators acting in Hilbert space onto these functions is presented. Examples of the Wigner-Weyl symbols (like Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified with the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators which are needed to construct the bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relation of the structure constants of the associative star-product of the operator symbols to the quantizer-dequantizer operators is reviewed.