F.3 Approximation of the Self-Energy

Starting such an iterative solution, one can first neglect vertex corrections in (F.23) and obtain an approximation for the self-energy by means of (F.14) together with (F.21) and (F.22). Making use of this approximation, one calculates , and includes vertex corrections in the next step. The iteration of such a procedure generates an expansion in terms of the screened interaction and the GREEN's function defined as a self-consistent solution of the DYSON equation.

For the iterative procedure, the sequence of steps can be defined by the vertex function (F.23), which yields by means of the chain rule the recurrence relation

One starts with the HARTREE-approximation, i.e. , which delivers , and the screened interaction . In the subsequent step one obtains , and and so forth. The effect of this interaction is two-fold. In the -th step, the GREEN's functions contributing to become dressed by an additional interaction line and additionally new types of diagrams are generated.

For the *Self-consistent approximations*, one selects a certain class of
self-energy diagrams . The DYSON equation becomes a non-linear
functional equation of the GREEN's functions, which has to be solved
self-consistently. The selection corresponds to the summation of a certain
class of diagrams up to infinite-order in the interaction, whereas others
which contribute even in lower order are neglected. The difficulty is in
finding the correct way to choose a subset of diagrams for each order. In
order to deliver physically meaningful results, any approximation should
guarantee certain macroscopic conservation laws. This condition can be imposed
by the postulate that all diagrams contributing to the self-energy are
obtained from the functional derivative of a functional with respect
to . Solving the DYSON equation self-consistently with a -*derivable self-energy* yields a GREEN's function which conserves particle
number, energy, and momentum [93].