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 relation always hold? Or if any other references mentioning this?

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?

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 (7:51) without much further explanation.

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

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 relation always hold? Or if any other references mentioning this?