In the previous sections helpful forms of the electron-phonon interaction are
derived. One can link up with the many-particle theory and introduce phonon
contributions into the quantum kinetic equations. As shown in the derivation of the
general result (F.37), one has to add the phonon induced
contribution to the electronically screened interaction. Together
with (F.14), this defines the phonon self-energies which enters
the quantum kinetic equations. As the phonon induced interaction is not a
functional of the single-particle GREEN's function, the problem is slightly
less complicated than the electron-electron interaction. As in the purely
electronic case, one is dependent on approximation schemes. An expansion of the
self-energy in powers of the phonon-induced interaction is easily generated by
means of (F.14). The contribution linear in the phonon induced
interaction, i.e. the single-phonon self-energy takes the form

(F.49)

By performing eigen-function expansion (see Appendix F.2.1) one obtains

(F.50)

where

(F.51)

(F.52)

(F.53)

The phonon GREEN's functions entering (F.51) are discussed
in Appendix F.2.1.