Deformation quantization

Summary

Deformation quantization equips the smooth functions on a Poisson manifold with a noncommutative star product. The star product is a formal power series in a parameter $\hbar$. The commutator of the star product reproduces the Poisson bracket at first order in $\hbar$. The Moyal product on $\R^{2n}$ is the first example. Kontsevich's formality theorem shows that every Poisson manifold admits a star product.