This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, of course), we can take the classical phase space with polarization as a high-dimensional Kahler manifold $K$ (A priori, one would get $K=\mathbb C^N$, but physical considerations can yield more interesting parameter spaces, e.g. modding-out by gauge transformations).
I am interested in seeing the local picture in all of this: take the classical field Hamiltonian $H$, which is a real-valued $C^\infty$-function on $K$. The canonical quantization of the Hessian of this function at a point $x\in K$ (roughly) represents the mean-field approximation for the statistical dynamics of a field configuration localized near $x\in K$. How do I provide the following?:
- A proof that $d^2H$ is pointwise diagonalizable by a symplectic transformation.
- An algorithm/formula for said symplectic diagonalization.
- A formula for the resulting change in complex structure on the tangent space.
Any help would be appreciated, including references (Kahler geometry is new to me).