On local equivalence of star-products on Poisson manifolds

TL;DR

This paper proves that all star-products on a Poisson manifold in a quantum canonical coordinate system are uniquely equivalent to a Moyal product, providing a systematic construction and explicit formulas up to fourth order in h.

Contribution

It establishes the uniqueness of the equivalence between star-products and Moyal products in quantum canonical coordinates on Poisson manifolds, with explicit construction methods.

Findings

Uniqueness of star-product equivalence to Moyal product in canonical coordinates.

Systematic construction method for the equivalence.

Explicit formula for the equivalence up to h^4.

Abstract

We present a proof that every star-product defined on a Poisson manifold and written in a given quantum canonical coordinate system is uniquely equivalent with a Moyal product associated with this coordinate system. The equivalence is assumed to satisfy some additional conditions which guarantee its uniqueness. Moreover, the systematic construction of such equivalence is presented and a formula for this equivalence in a case of a particular class of star-products is given, to the fourth order in $\hbar$.