Why do we use retarded Green's functions in response theory? When computing the response of a system to an external perturbation, we usually use the retarded Green's function to describe the response. On the other hand, from scattering theory in Quantum Mechanics, the perturbation series is given in terms of time-ordered Green's functions. If I would use time-ordered perturbation theory to calculate the response of my system, I would probably get a very different answer. How do I know which one I need to use?
For example, if I'm shining a laser on my material, is that scattering or is that more like equilibrium?
 A: Response of a system is usually described by a retarded Green's function, which reflects the causality (response follows the perturbation, rathe rthan preceeds it). Indeed an approximate evolution of an operator, under Hamiltonian $H=H_0 + V(t)$ is described by:
$$
\langle A_H(t)\rangle=\langle S^\dagger(t)A_I(t)S(t)\rangle\approx \langle A_I(t)\rangle - \frac{i}{\hbar}\int_{-\infty}^tdt_1\langle\left[A_I(t), V_I(t_1)\right]\rangle =\\
\langle A_I(t)\rangle - \frac{i}{\hbar}\int_{-\infty}^{+\infty}dt_1\langle\left[A_I(t), V_I(t_1)\right]\rangle\theta(t-t_1),\\
S(t)=T\exp\left[-\frac{i}{\hbar}\int_{-\infty}^tdt_1V_I(t_1) \right],
$$
where $A_H(t)$ and $A_I(t)$ are the operators in Heisenberg and the interaction representations. The retarded Green's functions is manifest in the second line.
The problem is that calculation of the retarded function is usually rather cumbersome, whereas for time-ordered Green's functions one may obtain nice closed expressions (e.g., the very intutive ones represented by the Feynmann diagrams). One therefore usually calculates the time-ordered Green's function and then passes to the retarded Green's function using Lehmann representation.
This logic is taken one step further in Keldysh formalism, where one calculates contour-ordered Green's function, which has time-ordered components alongside the retarded and advanced ones.
Detailed discussion of calculating retarded response function from a time-ordered one can be found in discussions of Kubo formula or dielectric response in most many-body texts, such as Fetter&Walecka or Mahan (AGD has it all too, but in rather cryptic form).
