Diagrammatic review and implications of the self-consistent field theory method

Some of the most intriguing and important phenomena in modern many-body physics are explainable in terms of self-consistent quantum mechanical field theory. This is the powerful theory developed by Umezawa and co-workers and modified by Benson and Hatch in applications to ferromagnetism. It is usually lengthy and involved mathematically. Thus, it is very helpful and meaningful to see its overall step-by-step progress in simple, diagrammatic flow starting from basic principles, with a ferromagnetic model as an example. As one immediately notes, there are two paths leading to very powerful physical conclusions and implications, and something most interesting is that each path implies the other. Many useful examples of applications of these methods are noted and some future possible applications are cited.