Matsubara frequencies and analytic continuation - Domain and uniqueness

Question

In the Matsubara formalism, there is a commonly-made statement that an imaginary-time correlation function is related to a retarded correlation function via the replacement $i\omega_n \rightarrow \omega + i0^+$, also called analytic continuation.

In a mathematical sense, on the other hand, analytic continuation means to find an analytic function whose values match the "input data", which in this case are the values at the discrete Matsubara frequencies $i\omega_n$.

For analytic continuation to retarded correlation functions, I have two questions:

1. What is the domain of analytic continuation and why? That is to say, do we only care about analytically continuing in the upper half plane, excluding all frequencies $\omega_n \leq 0$ as part of our input data to analytic continuation? Do we also exclude the real line, so that we ignore the zeroth bosonic frequency? And why is that the "physical" choice?
2. In what sense is analytic continuation unique? Does one require additional "physical" conditions other than analyticity? There is a strong statement on the uniqueness of complex-analytic functions via the identity theorem; does the infinite Matsubara frequency count as an accumulation point and thus be sufficient for uniqueness?

Answer 1

I hope am not confusing something, the issue is that the retarted propagator $G^R$ is regular in the upper half plane, so by the second statement from your question, two functions equal on the set, with an accumulation point at infinity are equal at every point in the domain of analycity. The same statement also holds in lower plane, with $G^A$ instead of the retarded and $\omega - i 0$.