Star products made (somewhat) easier

TL;DR

This paper introduces a new iterative method for deformation quantization on the real plane with arbitrary Poisson structures, simplifying the calculation of star products up to the fourth order.

Contribution

It presents a novel, effective iterative approach based on Weyl ordering and polydifferential representation to compute star products explicitly.

Findings

Successfully computed the star product up to fourth order

Provided a straightforward method potentially extendable to fifth order

Offered an explicit, physics-friendly description of deformation quantization

Abstract

We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which based on Weyl symmetrically ordered operator products. By using a polydifferential representation for deformed coordinates $\hat x^j$ we are able to formulate a simple and effective iterative procedure which allowed us to calculate the fourth order star product (and may be extended to the fifth order at the expense of tedious but otherwise straightforward calculations). Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products.