algebra of "operators" is an -deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over of the commutator expressed in the phase space formulation is . (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,.) Now, one looks for unitary representations 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 supermanifolds. Geometric quantization In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold , one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle over . On this space, one can map all classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of . One then proceeds by choosing a polarization, that is (roughly), a choice of variables on the -dimensional phase space. The quantum Hilbert space is then the space of sections that depend only on the chosen variables, in the sense that they are covariantly constant in the other directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like the Segal–Bargmann space. See also Correspondence principle Creation and annihilation operators Dirac bracket Moyal bracket Phase space formulation (of quantum mechanics) Geometric quantization References Historical References Silvan S. Schweber: QED and the men who made it, Princeton Univ. Press, 1994, General Technical References Alexander Altland,