The formulation of the Composite Operator Method(*COM*) and some general results are given in
Refs. [1,2,3,4,5,6,7,8,9].

The *COM* formulation is based on the following points:

- The old schemes, based on the perturbation expansion, are absolutely
inadequate to describe the new physics coming out from the recent
experiments. The appropriate description of a strongly correlated
system must be obtained in terms of some
*new particles*coming out by the interactions and whose properties should be fixed by a self-consistent procedure [1,4,5]. - The microscopic level, where we are concerned with the operators, and the macroscopic level, where we are concerned with the averages, are intimately connected [2,4,5,7].

The first point above is well known and has been crystallized in
many formulations (the projection method is one example
[10,11,12,13]; the dynamical map and
the rearrangement of symmetry, formulated by H. Umezawa and
collaborators, is another one [14]). The
latter point, instead, is genuine and really new. It appears as an
obvious statement, but we think that no much attention has been
paid to it in the literature. For instance, when we use the
Green's function formalism, we must specify the Hilbert space we
are considering. The field equations give relations among the
matrix elements, but they are not sufficient to uniquely determine
the Green's function. The relations among the operators, coming
from the algebra and the *boundary conditions*, must be
implemented to correctly choose the right Hilbert space. This can
be done by determining the parameters not bounded by the dynamics
in such a way that the Hilbert space has the right properties to
conserve the relations among matrix elements imposed by symmetry
conditions. In particular, the chosen representation should not
violate the symmetry required by the Pauli principle. By Pauli
principle we mean all the relations among operators dictated by
the algebra.

The issue of ergodicity [15,16,17] is also relevant. In general, in the analysis of the two-point Green's function the so-called ergodicity constants naturally appear. These constant are not present in the spectral functions and cannot be determined by the dynamics. While in the existing literature it is common to assume the ergodicity of the dynamics of all quantities, we have shown [7] that this is not always the case. In a fully self-consistent scheme the ergodicity constants must be calculated case by case. The non-ergodicity of the bosonic dynamics, in the bulk systems too, cannot be ruled out a priori. Again, the proper representation will fix the value of the ergodicity constants.

We have explicitly shown [7] that this formulation gives the exact answer when we consider solvable models like small clusters. In this latter case, the ergodicity issue appears really relevant as the non-ergodicity of the bosonic dynamics is a direct consequence of the discreteness of the energy levels.

The self-consistent scheme of the *COM* has been solved
by using two approximations: the polar approximation
[2,4] and the two-site
approximation [3,18]. The
relation with other approaches has been discussed in
Refs. [4,6]. A generalization
of the polar approximation has been presented in
Ref. [8].

We have also formulated an extension of the *COM* which
is able to resolve coherent low-energy features embedded in a
broad high-energy background [9]. This
approach, namely the Asymptotic Field Approach(*AFA*), in a problem with more than one energy scale,
which is typical of strongly correlated systems, succeed to
capture low-energy features. It extends and improves upon the
Roth's method by combining the advantages of the methods based on
the equations of motion and the slave boson techniques. It is
worth noting that when there is an expansion parameter such as the
size of the group, or the size of the representation, this
approach can be formulated so as to reduce to the correct solution
in the exactly solvable limit.