Formal differential star products and first-order normalization

Let $(M,\pi )$ be a real Poisson manifold as in General Poisson manifolds from bivector fields, with its well-defined bracket from Coordinate Jacobi and Schouten self-bracket criterion for Poisson bivectors, and set $A=C^{\mathrm{\infty }}(M,\mathbb{R})$. A formal associative deformation $\star$ of $A$ from Coefficientwise formal associative deformations is a differential star product quantizing $\pi$ if every coefficient $B_r$ is bidifferential and $B_1(f,g)-B_1(g,f)=\{f,g{\}}_{\pi }(f,g\in A)\mathrm{.}$ Here bidifferential means that in every coordinate chart each $B_r$ is a finite sum of terms $c_{\alpha ,\beta }(x){\mathrm{\partial }}^{\alpha }f{\mathrm{\partial }}^{\beta }g$, with smooth coefficients and finite differential order in each input. The normalization fixes the leading commutator to be $[f,g{]}_{\star }=\mathrm{\hslash }\{f,g{\}}_{\pi }$ modulo ${\mathrm{\hslash }}^2$; no factor of $1\mathrm{/}2$ or $i$ is implicit. A convention using $i\mathrm{\hslash }$ instead requires complexified coefficients and is not used here.

The product is interpreted only coefficientwise in the $\mathrm{\hslash }$-adic topology. This definition asserts neither convergence for a numerical $\mathrm{\hslash }$ nor existence, uniqueness, classification, or an operator quantization for a given Poisson manifold.