It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest.
For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.
Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space.
In 1946, H. J. Groenewold considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space.
[1] This led him to discover the phase-space star-product of a pair of functions.
The modern theory of geometric quantization was developed by Bertram Kostant and Jean-Marie Souriau in the 1970s.
The geometric quantization procedure falls into the following three steps: prequantization, polarization, and metaplectic correction.
Prequantization produces a natural Hilbert space together with a quantization procedure for observables that exactly transforms Poisson brackets on the classical side into commutators on the quantum side.
Nevertheless, the prequantum Hilbert space is generally understood to be "too big".
[2] The idea is that one should then select a Poisson-commuting set of n variables on the 2n-dimensional phase space and consider functions (or, more properly, sections) that depend only on these n variables.
[a] A polarization is a coordinate-independent description of such a choice of n Poisson-commuting functions.
The metaplectic correction (also known as the half-form correction) is a technical modification of the above procedure that is necessary in the case of real polarizations and often convenient for complex polarizations.
is exact, meaning that there is a globally defined symplectic potential
We can consider the "prequantum Hilbert space" of square-integrable functions on
The prequantum operators satisfy for all smooth functions
The next step in the process of geometric quantization is the choice of a polarization.
[4][b] The idea is that in the quantum Hilbert space, the sections should be functions of only
is a function for which the associated Hamiltonian flow preserves the polarization, then
The half-form correction—also known as the metaplectic correction—is a technical modification to the above procedure that is necessary in the case of real polarizations to obtain a nonzero quantum Hilbert space; it is also often useful in the complex case.
with the square root of the canonical bundle of the polarization.
[6] In the case of a complex polarization on the plane, for example, the half-form correction allows the quantization of the harmonic oscillator to reproduce the standard quantum mechanical formula for the energies,
[7] Geometric quantization of Poisson manifolds and symplectic foliations also is developed.
For instance, this is the case of partially integrable and superintegrable Hamiltonian systems and non-autonomous mechanics.
In the case that the symplectic manifold is the 2-sphere, it can be realized as a coadjoint orbit in
, we can perform geometric quantization and the resulting Hilbert space carries an irreducible representation of SU(2).
More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold.
However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory.
For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2.
(This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom.
[8]) As a mere representation change, however, Weyl's map underlies the alternate phase-space formulation of conventional quantum mechanics.