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
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].