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Given a Matsubara Green function $\mathscr{G}(i\omega)$, analytic continuation $i\omega \mapsto \omega+i0^+$ leads to the retarded/advanced Green functions $g^{r(a)}$. There is also an ansatz in equilibrium for the lesser Green function $g^<=f(\omega)(g^a-g^r)$ with $f(\omega)$ the Fermi distribution, which is found in some lecture notes (slide 31) without much further explanation.

Question: In equilibrium, does this ansatz always hold? Or if any other references mentioning this?

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Yes, it is a rather general statement known as the "fluctuation-dissipation theorem". It essentially follows from the "Kubo–Martin–Schwinger (KMS) condition". In terms of two-point functions, the latter can be written as \begin{equation} G^{>}(t,\mathbf{x};t',\mathbf{x}') = \pm \mathrm{e}^{-\beta\mu} G^{<}(t+\mathrm{i}\beta,\mathbf{x};t',\mathbf{x}'), \end{equation} where the plus (minus) sign corresponds to bosons (fermions).

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  • $\begingroup$ Thank you. So what you meant is that $g^<,g^>$ are related in equilibrium, and since they together are related to $g^a,g^r$ in general, we reach that $g^<$ is eventually not independent from $g(i\omega)$ in equilibrium as the formula in my post suggests? BTW, do you happen to know how $f_F(\omega)$ is replaced in bosonic case, just $f_B(\omega)$? $\endgroup$
    – xiaohuamao
    Commented Nov 11, 2022 at 7:22

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