Generalizations
For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to integrable ("most") functions, not just polynomial functions), see Weyl quantization.
Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.
Read more about this topic: Weyl Algebra