In a metal, can one electron decay into one lower-energy electron plus one phonon? (i.e., can the attached Feynman diagram occur?) If we replace phonons by photons and consider the process in a vacuum, I guess this is prohibited because you can always boost to a frame where the incoming and outgoing electron velocities are the same. Thus, the electron has no ability to transfer energy to the photon because there are no photons with finite q but zero energy. However, in a metal, Galilean and Lorentz invariance are broken by the crystal lattice, so it seems to me that the process ought to be allowed. Is my thinking here correct?