With particular reference to the role of the renormalization group approach
and Ward identities, we start by recalling some old features of the
one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior.
Its dimensional crossover to the Landau normal Fermi liquid implies that a
non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature
superconductors, can be maintained in d>1, only in the presence of a
sufficiently singular effective interaction among the charge carriers. This is
the case when, nearby an instability, the interaction is mediated by
fluctuations. We are then led to introduce the specific case of
superconductivity in cuprates as an example of avoided quantum criticality. We
will disentangle the fluctuations which act as mediators of singular
electron-electron interaction, enlightening the possible order competing with
superconductivity and a mechanism for the non-Fermi-liquid behavior of the
metallic phase. This paper is not meant to be a comprehensive review. Many
important contributions will not be considered. We will also avoid using
extensive technicalities and making full calculations for which we refer to the
original papers and to the many good available reviews. We will here only
follow one line of reasoning which guided our research activity in this field.Comment: 23 pages, 10 figure
