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In Kamenev's book Field Theory of Non-Equilibrium Systems (he also has lecture notes online here, which contains the relevant statement on pg. 17), he states that the following equation $$G^K(\epsilon) = \coth\left(\frac{\epsilon-\mu}{2T}\right) \left[G^R(\epsilon) - G^A(\epsilon)\right]$$ is a statement of the fluctuation-dissipation theorem, where $G^{(K,R,A)}(\epsilon)$ are the Keldysh, retarded, and advanced propagators, respectively. I have only ever seen the FDT stated in terms of structure factors and susceptibilities. While I can see the superficial connection (since $G^A(\epsilon) = G^R(\epsilon)^\dagger$, the RHS should resemble something like $\text{Im}\chi$), I'm having difficulty rigorously connecting the two. Can someone help me understand the connection between these statements?

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The difference of the retarded and the advanced Green's functions in the right-had-side of this equation is actually the density-of-states, i.e. what you might call a structure factor, whereas $G^K$ tests the possibilities of adding/removing a particle, i.e. the susceptibility.

What makes me personally skeptical about interpreting this equation is that formulating it in terms of Keldysh formalism gives superficial illusion that FDT can be applied out of equilibrium (or at least that it has such a simple form out of equilibrium), whereas this is not the case.

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What you are referring to is the form of fluctuation-dissipation theorem (FDT) that relates the dynamical structure factor to some retarded susceptibilty. The equation you wrote down holds for bosonic systems, in which case the RHS can be interpreted as a susceptibility while the LHS is related to the dynamical structure factor through the relation $G^{<} = G^{K}+\frac{1}{2}\left(G^A-G^R\right)$. This leads to $G^{<}(\epsilon) = n_{B}(\epsilon)\mbox{Im}\left[G^R(\epsilon)\right]$, where $n_{B}(\epsilon)$ is the Bose distribution function.

For a fermionic system, however, $n_{B}(\epsilon)$ must be replaced by $n_{F}(\epsilon)$ - the Fermi distribution function - in the equation above. This gives a fermionic FDT. The familiar bosonic FDT can be recovered in this case by considering the two-particle excitations, which can be expressed as the product of single particle excitations using Wick's theorem.

$\Pi^{R}(t,t^{'}) = G^{R}(t,t^{'})G^{K}(t^{'},t) + G^{K}(t,t^{'})G^{A}(t^{'},t)$ is the retarded susceptibility, and similarly one can also write an expression for $\Pi^{<}$ in terms of $G^{R,A,K}$

At equilibrium, one can show that: $\Pi^{<}(\epsilon) = n_{B}(\epsilon)\mbox{Im}\left[\Pi^{R}(\epsilon)\right]$. This is the familiar form of FDT. You will find a detailed discussion in Kamenev's book ch. 9.

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