Coherent states http://en.wikipedia.org/wiki/Coherent_states
are vectors in the Hilbert space which in certain sense are strongly localized
and "corresponds" to points in classical phase space (see below).
On the other hand vectors in Hilbert space in classical limit corresponds to Lagrangian submanifolds (see below).
How these two ideas correspond to each other ?
Is the following reasonable/known: coherent states corresponds to Lagrangian submanifolds in the COMPLEXIFICATION of the phase space, which intersect with real points of the phase space by the just one point ?
Motivating example: consider R^2 and H=p^2+q^2. Consider p^2+q^2=0 - the real slice is just one point. On the other hand the in complexification we have 1-dimensional Lagrangian submanifold. The wave function $\psi$ corresponding to Lagrangian submanifold H=0,
is constructed in a simple way $\hat H \psi=0$. $\hat H$ is hamiltonian of the harmonic oscillator and its eigenfuction is well-known to be coherent state.
So in this example idea seems to work.
Quantization of Lagrangian submanifolds
Down-to-earth idea of the construction of the vector in Hilbert space from the Lagrangian submanifold.
Consider sumanifold defined by the equations $H_i=0$.
Consider "corresponding" quantum hamiltonians $\hat H_i $,
consider vector $\psi$ in the Hilber space such that $\hat H_i \psi = 0$.
This $\psi$ we are talking about.
Why it is important "Lagrangian" ? It is easy. If $A \psi =0$ and $B\psi = 0$
then it is true for commutator $[A,B]\psi = 0$.
In classical limit commutator correspond to Poisson bracket so we see
that even if we start from $H_i$ which is not close with respect to Poisson
bracket we must close it - so we get coisotropic submanifold.
Lagrangian - just restiction on the dimension - that it should be of minimal possible dimension - so after quantization we may expect finite dimensional subspace (in the best case 1-dimensional).
There are related by slightly different points of view.
Sometimes one can realize Hilbert space corresponding to classical symplectic manifold $M$,
as holomorphic functions on $M$.
Each point $p\in M$ defines a functional $ev_p: Fun(M) \to C$ just evaluation of the function at point $p$.
On the other hand in Hilbert space any functional corresponds to a vector $v$,
such that $= ev_p(f)$.
So $v$ is coherent state corresponding to $p$.
This approach is probably due to J. Rawnsley.
See Mauro Spera "On Kahlerian coherent states" http://www.emis.de/proceedings/Varna/vol1/GEOM19.pdf
The idea that complexification of the phase space is important
plays a role in "Brane Quantization" approach by Witten and Gukov.
See e.g. http://arxiv.org/abs/1011.2218 Quantization via Mirror Symmetry