# Spectral density and Green's function

this is a basic question but from what I can see it has not been asked before. I am reading Nolting's "Fundamentals of Many-Body Physics".

He speaks about the spectral density in characterising the intensity of a given spectra. He then states that the spectral density can be related to arbitrary operators and is thus closely related to retarded Green functions.

$S_{AB}(t,t^{\prime})=\frac{1}{2\pi}\langle [A(t),B(t^{\prime})] \rangle$

From what I understand the spectral density is related to the properties of a system of particles. However the Green's function describes the propagation of a single particle. But they are equivalent. I think I am deeply misunderstanding this. Any clarification would be appreciated.

• I don't have much time for a detailed answer right now, but this is a good start. Very shortly, basically a spectral function is the imaginary part of the retarded Green function. Having the spectral function is enough to reconstruct the Green function because they are related by a Hilbert transform. en.wikipedia.org/wiki/Green%27s_function_(many-body_theory) May 13, 2016 at 9:53