On the Diagonalizability and Factorizability of Quadratic Boson Fields

We provide a necessary and sufficient condition on the coefficient matrices $A, C$ for the diagonalizability of quadratic fields of the form, $X=\sum_{i,j=1}^n \left(A_{ij}a^{\dagger}_i a^{\dagger}_j \bar{A}_{ij}a_ia_j+C_{ij}a^{\dagger}_ia_j\right),$ where the $a$'s and $a^\dagger$'s are the generators of the multi-dimensional Schrödinger Lie algebra. We also consider the Fock vacuum characteristic function $\left\langle \Phi , e^{i\,s\,X}\,\Phi\right\rangle,$ of $X$ and study its factorizability/decomposability and how it relates to the commutativity of the simple quadratic components of $X$.