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. (shrink)

Some of the physical implications involved in self-consistently selecting a superconducting (nonequivalent) representation for the BCS Hamiltonian are developed and discussed. This is done by comparing the phase symmetry of our system in original variables with that same symmetry when written in terms of physical variables. It is shown explicitly that Goldstone's theorem is satisfied and that dynamical rearrangement of symmetry has taken place in going from original to physical variables. Thus, it is found that the original phase symmetry transformation (...) is taken up by physical “massless” fields and that the Bose-Einstein condensations of these fields in the physical (superconducting) ground state produce the asymmetry by “printing” the quantum electron numbern on that state. (shrink)

Taking the BCS Hamiltonian written in second-quantized form, a modified form of Umezawa's self-consistent field theory method is applied, and a unitarily nonequivalent representation is selected in which the Hamiltonian obviously describes a superconducting system. This result is not at all obvious, since the original Hamiltonian is completely symmetric, and there is no reason a priori for expecting it to describe an asymmetric superconducting configuration. All higher order terms are accounted for, and in doing so, one finds the existence of (...) the energy-gap condition for Cooper pairs. The representation is picked out without using the adiabatic theorem or pair approximation, as is typically done in the self-consistent method for superconductivity. (shrink)