The Luttinger theorem is a cornerstone in the theory of condensed matter. As described qualitatively in Sec. 3.6, it requires that the volume enclosed by the Fermi surface is conserved independent of interactions, i.e., it is the same as for a system of non-interacting particles. Similarly, the Friedel sum rule is the requirement that the sum of phase shifts around an impurity is determined by charge neutrality, which was derived by Friedel [163] for non-interacting electrons. This section is devoted to a short summary of the original work of Luttinger and Ward and the extension of the arguments to the Freidel sum rule [166]. Here we explicitly indicate the chemical potential μ, since the variation from μ is essential to the arguments.

There are two key points: in the interacting system the wavevector in the Brillouin zone k is conserved so that excitations can be labeled k, and the self-energy ∑k(ω) is purely real at the Fermi energy ω = μ at temperature T = 0. The latter point is an essential feature of a Fermi liquid or a “normal metal,” which is justified by the argument that the phase space for scattering at T = 0 vanishes as ω → μ (see Sec. 7.5). Thus, at the Fermi energy the Green's function as a function of k is the same as for an independent-particle problem with eigenvalues. (Of course, for an interacting system at any other energy cannot be described by independent particles.) In an independent-particle system at T = 0 the occupation numbers jump from 1 to 0 as a function of k at the Fermi surface, and in the interacting system there is still a discontinuity in nk that defines the surface (Sec. 7.5).