In many-body theory we know that to find the retarded Green function in frequency space $G^R(\omega)$ we first find the Matsubara time-ordered Green function $\mathcal{G}(i\omega_n)$ and then replace $i\omega_n$ with $\omega+i\eta$.

First question: is this operation well-defined? Since, for example, if I multipy $\mathcal{G}$ by $1=e^{i\beta \omega_n}$ (for the bosonic case) the replacement $i\omega_n\rightarrow \omega+i\eta$ does not give the same answer as before. I know that both of these functions are related to the Hilbert transform of the spectral weight as

$$G(z)=\int_{-\infty}^{\infty} \frac{dx}{2\pi}\frac{\rho(x)}{x-z} $$

and that makes sense to me because in that case we have a function of real variable $z$ and the replacement (analytic continuation) $z\rightarrow i\omega_n$ gives a unique answer. However the reverse operation doesn't seem to give a unique answer.

Second question: is this replacement an actual analytic continuation?


In order to go from the Matsubara Green function to the retarded one using analytical continuation, you need to express the Matsubara Green function as a sum of simple pole terms. This is what results from the Lehmann representation.

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