The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches.

One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.