We investigate bosonic Gaussian quantum states on an infinite cubic lattice
in arbitrary spatial dimensions. We derive general properties of such states as
ground states of quadratic Hamiltonians for both critical and non-critical
cases. Tight analytic relations between the decay of the interaction and the
correlation functions are proven and the dependence of the correlation length
on band gap and effective mass is derived. We show that properties of critical
ground states depend on the gap of the point-symmetrized rather than on that of
the original Hamiltonian. For critical systems with polynomially decaying
interactions logarithmic deviations from polynomially decaying correlation
functions are found. Moreover, we provide a generalization of the matrix
product state representation for Gaussian states and show that properties hold
analogously to the case of finite dimensional spin systems.