Some authors build their Green's function from Hamiltonian using quasiclassical propagator. Sometimes this propagator is referred as Gor'kov propagator or Eilenberger propagator. I could not find literature explaining in details about usage of propagators to build a Green's function from Hamiltonian. Obviously, it is advanced algebra I am not familiar with. Is it build from Green's function that contains Pauli matrices and then propagator is used to "streamline" or "group" Pauli matrices in the Green's function? Any examples and suggestions on study material is appreciated. I need to understand and get a "picture" of what does the propagator do. I attached image taken from textbook to illustrate example of propagator I am talking about.