Assume we have a solid state hamiltonian, which is ellectrons plus nuclei plus all possible interactions:
H = T_{el} + V_{el-el} + T_{n} + V_{n-n} + V_{n-el}
Is it wrong to write down a self-consistent system of Green functions for electrons and phonons?
G_{el}=g + gSG_{el}
G_{ph}=g +...