  We combine the recently developed many-body Green's function theory for
electrons and nuclei with the exact factorization of the wave function. The
existing Born-Oppenheimer Green's functions are shown to be special cases of
our exact approach. We consider the limitations of the laboratory frame
formulation of the Green's function theory and discuss why the body-fixed frame
formulation is needed in order to go beyond the Born-Oppenheimer theory. We
give exact forms of the electronic and nuclear Green's functions written in
terms of the exact factorized states, providing a systematic approach beyond
the Born-Oppenheimer approximation. The lowest order approximation to the exact
electronic Green's function is found to be an expected value of the
Born-Oppenheimer electronic Green's function with respect to the nuclear
density.
