Introduction

Abstract

The study of boson models and the problem of their solutions is as old as the free Bose gas model for which the celebrated phenomenon of Bose-Einstein condensation (BEC 1924) has been detected. This phenomenon puts in evidence a macroscopic, purely quantum phenomenon. Solving boson models means that we are interested in finding the ground and/or temperature states of the models, or that we are interested in deriving at least some of their properties from first quantum-mechanical principles. For a long time activities in this area belonged to the field of many-body physics, a field of high activity in theoretical physics. Green’s functions, series expansions, Feynman graphs and their summations, and much of numerics are the standard technologies. The ultimate aim is, as always, to understand the physical world from the point of view of its basic laws and constituents and, therefore, to derive as many exact results as possible having a large or universal range of validity for these systems.