Central to a mathematical understanding of the Bogolyubov transformation is the study and classification of linear lattice field theories. What follows might be familiar to many people, but I just want to set some terminology straight:
Symplectic-Linear Lattice Field Theories
By a linear lattice field theory, I mean a dynamical system where the phase space $(X,\omega)$ is precisely a symplectic vector space, on which the Hamiltonian is a quadratic form. For example, the discretized quantum mechanics of a particle on a one-dimensional lattice $\mathbb Z/N$ is such a theory, with
$$X=\mathbb C^{N}$$
In which case the symplectic form is defined as the imaginary part of the $\mathbb C$-inner-product. For linear lattice field theories, the infinitesimal time-evolution looks like a Schrodinger equation (but not necessarily with complex structure, as in the case of QM): $$\frac{\psi(t_{i+1})-\psi(t_{i})}{t_{i+1}-t_i}=\hat H\psi(t_i).$$ We'll call the $\mathbb R$-linear operator $\hat H$ in the equation above the linearized Hamiltonian, to distinguish it from the ordinary Hamiltonian $H$, which is a quadratic form. I've been able to reproduce, with some help from some online references, that linear lattice field theories generate symplectic time-evolution, i.e.
Theorem: The linearized Hamiltonian of a linear lattice field theory on $(X,\omega)$ lies in the Lie algebra of the symplectic transformations of its phase space, i.e. $$\hat H\in \mathfrak{sp}(X,\omega)$$ In other words, a linear lattice field theory generates symplectic dynamics.
Adding Compatible Complex-Linear Structure to a Lattice FT
In general, it is convenient to introduce a compatible complex structure $J^2=-1$ on the phase space, making it into a complex vector space, in which case the Hamiltonian may be rewritten as
$$H(z^i)=A^{ij}z_i\bar z_j+(B^{ij}z_iz_j+c.c.)$$
Where $A$ is a Hermitian matrix (notice that this theory, once quantized, potentially describes the QFT of a bosonic superfluid, like the cooper-pair condensate in a superconductor). I will now define a specific subclass of linear lattice field theories, called non-interacting, with respect to this complex structure, ones which obey
$$H(z^i)=A^{ij}z_i\bar z_j.$$
(notice that this theory, once quantized, becomes a non-interacting QFT, in the sense that the constituent bosons do not interact and have identical dynamics.) Notice how the definition of non-interacting is always a relative notion: to say that a system is non-interacting, requires the specification of a complex structure. I've also happened to reproduce the following result:
Theorem: The linearized Hamiltonian of a non-interacting linear lattice field theory on $(X,h)$ lies in the Lie algebra of the group of unitary transformations of its phase space, i.e. $$\hat H\in \mathfrak{u}(X,h)\subset \mathfrak{sp}(X,\omega)$$ In other words, a non-interacting linear lattice field theory generates $\mathbb C$-linear dynamics.
Now here's the question: is there a nice algorithm which assigns to a linear lattice field theory a complex structure $J^2=-1$ such that the dynamics is non-interacting, and most importantly, without completely diagonalizing the theory?
