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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(NOT quite correct thanks to "Squark"): 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.

Vague attempt for correction It seems we need to take into account Sommerfeld's 1/2 correction (it is related to Maslov's index). I mean we should consider H=r, and choose such "r" which satisfies Bohr-Sommerfeld quantization condition i.e. symplectic form integrated over interior of H=r should be integer +1/2. But we need somehow subtract this 1/2 from quantum eigenvalue to fit into the desired picture...


Background.

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. Lagrangian Submanifolds in Deformation Quantization


Coherent states

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 $<v|f>= ev_p(f)$. 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 Sergei Gukov


There are actually different generalizations of the coherent states. In particular group theoretical approach was developed by Soviet physicists Askold Perelomov who worked in ITEP:

A. M. Perelomov, Coherent states for arbitrary Lie groups, Commun. Math. Phys. 26 (1972) 222-236; arXiv: http://arxiv.org/abs/math-ph/0203002

A. Perelomov, Generalized coherent states and their applications, Springer, Berlin 1986.

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The example is not quite correct since the ground state energy of the quantum harmonic oscillator is positive, namely it's $\hbar \omega / 2$ (in your normalization $\omega = 2$) –  Squark Feb 8 '12 at 18:48
    
In case of a linear simple tic manifold your idea is simple to formalize. For example for the plane coherent states are eigenstates of $x+ip$ which makes clear the corresponding complex Lagrangian sub manifold and indeed it intersects the real cycle at one point –  Squark Feb 8 '12 at 18:53
    
It might be worthwhile to look for a generalization of x+ip to arbitrary Kahler manifolds –  Squark Feb 8 '12 at 18:56
    
@Squark thank you for the comment, Oops... you right ... somehow I missed $\omega=2$ ... –  Alexander Chervov Feb 8 '12 at 19:04
    
@Squark I tried to correct... –  Alexander Chervov Feb 8 '12 at 19:49
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1 Answer 1

This is a partial answer. I can understand the correspondence in the special case of coadjoint orbits $\mathcal{O}$ of compact Lie groups and Hamiltonians consisting of symbols of magnetic Laplacians on their cotangent bundlles. On one hand a solution of the Hamilton-Jacobi equation corresponding to these Hamiltonians define to a Lagrangian submanifold in $T^{*} \mathcal{O}$, and on the other hand the energy eigenspaces of the magnetic Laplacians are finite dimensional, thus the symbols of their orthogonal projectors in $L_2(\Gamma(L))$ (where, $L$ is the line bundle over $\mathcal{O}$ corresponding to the magnetic part of the symplectic form on $T^{*} \mathcal{O}$) are reproducing kernels which can be viewed as coherent states parametrized by points of the coadjoint orbit.

Now, for real coadjoint orbits, their complexifications are equivariently homeomorphic to their cotangent bundles, thus the correspondence can be passed to the complexification.

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@David Thank You very much ! What you write is very interesting for me, I have not thought from this direction while I should do it... –  Alexander Chervov Feb 8 '12 at 18:08
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