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

TL;DR

This paper explores the relationship between Moyal and Kontsevich star products by explicitly computing the deformation quantization up to third order, revealing how coordinate changes affect the structure and associativity.

Contribution

It provides an explicit third-order deformation quantization formula under coordinate changes, illustrating the dependence on 44 and the resulting properties of the Poisson bi-vector.

Findings

Poisson bi-vector depends on 44 and does not satisfy Jacobi identity

Coefficients follow from star product associativity

Explicit third-order deformation quantization formula derived

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.