My question is regarding apparent inconsistency between two expressions given in : https://arxiv.org/pdf/hep-th/0205051.pdf.
From what I gather, the naive GKPW prescription does not work for Minkowski signature since one can show that naively taking functional derivative of the on-shell action (for whatever probe field we have in the bulk) results in a quantity which is always purely real. Thus, there is no way it can capture dissipative dynamics of the boundary theory since dissipation must be encoded in the imaginary part of the the Green's function. Towards this, they propose a prescription as given in Eqn (3.15) and goes on to compute the retarded and Feynman propagator in Eqn (3.20) and Eqn (3.21). They claim that (3.21) which is the Feynman propagator can be obtained by Wick rotating the Euclidean propagator Eqn (3.22). However, they earlier also mention that Wick rotating Euclidean propagator gives the retarded propagator as they mention in Eqn (2.10). My central question are the following:
1) Wick rotating Euclidean propagators lead to what exactly---retarded propagators or Feynman propagators?
2) How does one perform the Wick rotation to obtain Eqn (3.21) from Eqn (3.22).