 Gaussian field theory

A Gaussian field theory has a free energy quadratic in its field modes. If $H_0={\sum }_{\mathbf{q}}^{+}J(q)|{\varphi }_{\mathbf{q}}{|}^2$ for a real field, then $⟨|{\varphi }_{\mathbf{q}}{|}^2⟩=J(q{)}^{-1}$ and its partition function is a product of elementary Gaussian integrals.