Description
Classical mechanics can be formulated in the setting of symplectic geometry, in which the algebra of observables of a physical system is the algebra of smooth functions on a symplectic manifold. In particular, the product of observables is a commutative operation. An additional structure called the Poisson bracket makes it a Poisson algebra.
In quantum mechanics, however, this commutativity is lost, but the anticommutator gives a Poisson algebra structure, which is related to Poisson algebra of the classical system in a certain way through "canonical quantization".
The approach of deformation quantization is to see the quantum algebra as a deformation of the classical algebra in the parameter of the Planck constant, thus staying in the setting of symplectic geometry to treat quantum systems.