Editing Canonical quantization (section)

===Deformation quantization===
The classical theory is described using a [[spacelike]]  [[foliation]] of [[spacetime]] with the state at each slice being described by an element of a [[symplectic manifold]] with the time evolution given by the [[symplectomorphism]] generated by a [[Hamiltonian mechanics|Hamiltonian]] function over the symplectic manifold. The ''quantum algebra'' of "operators" is an {{mvar|ħ}}-[[deformation quantization|deformation of the algebra of smooth functions]] over the symplectic space such that the '''leading term''' in the Taylor expansion over {{mvar|ħ}} of the [[commutator]] {{math| [''A'', ''B'']}}  expressed in the [[phase space formulation]] is {{math|''iħ''{''A'', ''B''} }}.  (Here, the curly braces denote the [[Poisson bracket]]. The subleading terms are all encoded in the [[Moyal bracket]], the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved,
and providing the arguments of such brackets,  ''ħ''-deformations are highly nonunique—quantization is an "art", and is specified by the physical context.
(Two ''different'' quantum systems may represent two different, inequivalent, deformations of the same [[classical limit]], {{math| ''ħ'' → 0}}.)

Now, one looks for [[unitary representation]]s of this quantum algebra. With respect to such a unitary representation, a symplectomorphism in the classical theory would now deform to a (metaplectic) [[unitary transformation]]. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian.

A further generalization is to consider a [[Poisson manifold]] instead of a symplectic space for the classical theory and perform an ''ħ''-deformation of the corresponding [[Poisson algebra]] or even [[Poisson supermanifold]]s.