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.

For a real field, ${\varphi }_{-\mathbf{q}}={\varphi }_{\mathbf{q}}^{\ast }$. A sum ${\sum }_{\mathbf{q}}^{+}$ takes one representative from each nonzero pair $\{\mathbf{q},-\mathbf{q}\}$ so that every independent complex Fourier amplitude is counted once.

A Gaussian variational approximation chooses a quadratic trial kernel and minimizes a variational upper bound on the interacting free energy. The optimized kernel often equals the bare kernel plus a self-consistent, wavevector-independent mass shift.

For an exact dimensionless Hamiltonian $H$ and trial Hamiltonian $H_0$, the free energies obeywhere $F_0=-\log Z_0$ and the expectation uses the trial Gibbs distribution $Z_0^{-1}{e}^{-H_0}$.