3.5 DYSON Equation

By introducing the concept of self-energy the structure in Fig. 3.5 takes the form shown Fig. 3.6.

The corresponding analytic expression is given by

where the abbreviation and is used. The self-energy describes the renormalization of single-particle states due to the interaction with the surrounding many-particle system and the DYSON equation determines the renormalized GREEN's function.

Another important concept is the *proper self-energy insertion* which is a
self-energy insertion that can not be separated into two pieces by cutting a
single-particle line. By definition, the proper self-energy is the sum of all
proper self-energy insertions, and will be denoted by
. Using the
perturbation expansion, one can define the proper self-energy as an
irreducible part of the GREEN's function. Based on this definition
first-order proper self-energies, which are resulted from the first-order
expansion of the GREEN's function (see Section 3.4.2), are shown
in Fig. 3.7. These diagrams are irreducible parts
of Fig. 3.4-b and Fig. 3.4-c and are referred to as
the HARTREE (
) and the FOCK (
)
self-energies.

The self-energy can also in principle be introduced variationally [203]. A variational derivation of the self-energies for the electron-electron and electron-phonon interactions is presented in Appendices F.1 and F.2, respectively. It follows from these definitions that the self-energy consists of a sum of all possible repetitions of the proper self-energy

Correspondingly, the GREEN's function in (3.41) can be rewritten as

which can be summed formally to yield an

The corresponding analytic expression is given by

The validity of (3.44) can be verified by iterating the right hand-side, which reproduces (3.43) term by term. In a similar manner, one can show that the DYSON equation can be also written as