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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