Superconductivity, nonadiabaticity and strong correlation in the light of recent experiments

Recently a series of experiments in Scanning Tunneling Spectroscopy, angle-resolved photoemission spectroscopy (ARPES) and isotope effects have provided a great deal of information about the nature of the superconducting state, even thought many questions are still open. In this paper we present a discussion of the various theoretical frameworks in the light of these new experiments. Strong correlations started as the possible path to an entirely novel physics but, in many recent versions, they are reduced to the modest role of just raising the coupling in a BCS gap equation with phonon mediators. We argue instead that a conceptual generalization of the Migdal-Eliashberg (ME) theory is necessary and we discuss the consequences arising from the breakdown of the adiabatic assumption. The Nonadiabatic Theory of Superconductivity represents a complex generalization of the ME theory which is natural and, in many respects, unavoidable. In fact Migdal's theorem is strongly violated in these materials for a variety of reasons. The quantification of these effects is technically complicated and it is rather model dependent. In this framework strong correlations can have an important role in locating the system in a favourable range of parameters (forward scattering). We present a critical discussion of these developments in the light of the recent experiments and we also propose some crucial tests.