Prior to 1986, research on strongly interacting electron systems was a fringe subject in solid state physics. The discovery of high-temperature superconductivity in the copperoxide ceramics changed this perception radically. The reason is simple. Undoped, the copper-oxide materials contain a partially filled d-band of electrons (A1987). Band theory tells us that partially filled bands conduct. However, the undoped cuprates are extremely good insulators. Although they become conducting after they are doped, they exhibit spectral weight redistributions over large energy scales that cannot be understood within the traditional theory of metals. It is this failure of band theory to predict insulating states in certain half-filled bands that Mott addressed in 1949 (M1949). Mott's analysis grew out of his study of NiO which contains two unpaire d electrons per unit cell but insulates nonetheless. Similar insulating states are found in most transitionmetal oxides, most notably VO2 and V2O3. Since band theory failed, Mott zeroed in on the electron correlations as the root cause of the insulating state in NiO. This line of inquiry has yielded some of the most subtle results and enigmatic problems in solid state physics. It is the physics of the Mott state and how it plays out in the cuprates and other systems that we describe in this chapter.

Any problem in solid state physics can be said to be solved once one has isolated the propagating degrees of freedom. These are the excitations that make the Lagrangian quadratic and give rise to pole-like singularities in the single-particle Green function.