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The analytic continuation from bosonic imaginary frequency to a retarded frequency is $i{\omega }_n\mapsto \omega +i{0}^{+}$. ThusWriting $E_k=\sqrt{v^2{k}^2+m(T{)}^2}$ and using the Sokhotski–Plemelj formula gives, in this sign convention,The opposite overall convention for the retarded Green function reverses this sign; the corresponding spectral function is conventionally chosen positive at positive frequency.

Near the quantum critical point, $m/T$ is a function only of $\Delta /T$. The Green function has the scaling formwithThis is quantum critical scaling with dynamical critical exponent $z=1$ and leading large-$N$ anomalous dimension $\eta =0$.

With the convention $G_R(t)=-i\Theta (t)⟨[q(t),q(0)]⟩$, the retarded Green function isDefining the spectral function by $\varrho =-2\mathrm{Im}{G}_R$ givesOther common spectral-weight conventions differ by an overall factor.

For a damped ferromagnetic spin wave, shifting both poles into the lower half-plane preserves causality:Its spectral function isThe classical Fluctuation-dissipation theorem then givesHence the zero-frequency static structure factor iswhich has the stated proportionality after absorbing the normalization $4/s$.