On Relation Between Moyal and Kontsevich Quantum Products. Direct Evaluation up to the $\hbar^3$-Order

Abstract

In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Moyal product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bi-vector is shown to depend on \hbar and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product.