Imagine an open quantum system interacting with an environment that admits a density matrix (Markovian) description in terms of Lindbladians ($c$ and $c^\dagger$). Is there a meaningful way to define a single particle Green function for this system up to the time the steady state is achieved, but without knowing any more details about the environment-system interaction?
Finally, how do steady state Green functions eventually become meaningful quantities to look at within a reduced density matrix description? For example, see here.
Background confusion: Single particle Green functions could be thought of as propagation amplitudes of particles/holes from one time to another. In a closed quantum system, one would sum all amplitudes leading from the initial to the final state, which is indeed the right thing to do. However, in an open quantum system with just a reduced density matrix (Lindbladian) description and one that exchanges matter, it feels like there is no way to sensibly decide how to coherently sum up amplitudes for different processes (eg. a process in which particle vanishes, then comes back from the environment etc.), esp. when starting from some initial random density matrix. But then to have a meaningful steady state Green function, some things must be magically falling into place in the steady state $-$ what am I missing? Any pointers would be greatly appreciated.