Introduction

In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving. A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n(x, t) and a current density J(x, t) which satisfy the continuity equation, or a total momentum P(t) and a Lorentz force F(t) which is the time-derivative of P(t), or a total energy E(t) and a total power which is the time-derivative of E(t), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.