Spectral function ($\rho (\omega )$)

The spectral function records the weighted energy differences connected by an operator. With the convention $G_R=-i\theta ⟨[A,A]⟩$, it is proportional to $-\mathrm{Im}{G}_R$.

The fluctuation-dissipation theorem relates an equilibrium correlation function to the dissipative part of the corresponding linear response. If $\rho =-2\mathrm{Im}{G}_R$, one common convention is $S(\omega )=\rho (\omega )/(1-e^{-\omega /T})$, which becomes $S(\omega )\simeq T\rho (\omega )/\omega$ in the classical limit $|\omega |\ll T$.

A Matsubara correlator is obtained from the retarded spectral density by a Cauchy transform evaluated at an imaginary frequency.