The Green’s function method is a very convenient formalism in condensed matter physics, and many progresses have been achieved in the last fifty years. When applied to interacting systems, such an approach is usually based on the hypothesis that the interaction among the particles is weak and can be treated in the framework of some perturbation schemes. In this line of thinking a consolidated scheme has been constructed, mostly based on diagrammatic expansions, Wick’s theorem, Dyson equation, and so on. However, in the last few decades new materials with unconventional properties have been discovered. It is believed that the origin of such anomalous behaviors is generally due to strong electronic correlations in narrow conduction bands [1]. In this line of thinking many analytical methods have been developed for the study of strongly correlated electron systems [2]. The main difficulties are connected with the absence of any obvious small parameter in the strong coupling regime and with the simultaneous presence of itinerant and atomic aspects. The concept that breaks down is the existence of the electrons as particles with some well-defined and intrinsic properties. The presence of interaction modifies the properties of the particles: what are observed are new particles with new peculiar properties entirely determined by the dynamics and by the boundary conditions. These new objects appear as the final result of the modifications imposed by the