  We address the problem of the bosonization of finite fermionic systems with
two different approaches.
  First we work in the path integral formalism, showing how a truly bosonic
effective action can be derived from a generic fermionic one with a quartic
interaction. We then apply our scheme to the pairing hamiltonian in the
degenerate case proving that, in this instance, several of the features
characterizing the spontaneous breaking of the global gauge symmetry U(1)
occurring in the infinite system persist in the finite system as well.
  Accordingly we interpret the excitations associated with the addition and
removal of pairs of fermions as a quasi-Goldstone boson and the excitations
corresponding to the breaking of a pair (seniority one states in the language
of the pairing hamiltonian) as Higgs modes.
  Second, we face the more involved problem of a non-degenerate single particle
spectrum, where one more kind of excitations arises, corresponding to the
promotion of pairs to higher levels. This we do by solving directly the
Richardson equations. From this analysis the existence emerges of critical
values of the coupling constant, which signal the transition between two
regimes, one dominated by the mean field physics, the other by the pairing
interaction.
