  We show that bosonic and fermionic Gaussian states (also known as "squeezed
coherent states") can be uniquely characterized by their linear complex
structure $J$ which is a linear map on the classical phase space. This extends
conventional Gaussian methods based on covariance matrices and provides a
unified framework to treat bosons and fermions simultaneously. Pure Gaussian
states can be identified with the triple $(G,\Omega,J)$ of compatible K\"ahler
structures, consisting of a positive definite metric $G$, a symplectic form
$\Omega$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian
states can also be identified with such a triple, but with $J^2\neq -1\!\!1$.
We apply these methods to show how computations involving Gaussian states can
be reduced to algebraic operations of these objects, leading to many known and
some unknown identities. We apply these methods to the study of (A)
entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of
driven systems. From this, we compile a comprehensive list of mathematical
structures and formulas to compare bosonic and fermionic Gaussian states
side-by-side.

Quantum Physics;Statistical Mechanics;High Energy Physics - Theory