  We discuss the mechanism of spontaneous symmetry breaking and the elementary
excitations for a weakly-interacting Bose gas at finite temperature. We
consider both the non-relativistic case, described by the Gross-Pitaevskii
equation, and the relativistic one, described by the cubic nonlinear
Klein-Gordon equation. We analyze similarities and differences in the two
equations and, in particular, in the phase and amplitude modes (i.e. Goldstone
and Higgs modes) of the bosonic matter field. We show that the coupling between
phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in
the non-relativistic case. Instead, in the relativistic case the spectrum has
two branches: one is gapless and the other is gapped. In the non-relativistic
limit we find that the relativistic spectrum reduces to the Bogoliubov one.
Finally, as an application of the above analysis, we consider the Bose-Hubbard
model close to the superfluid-Mott quantum phase transition and we investigate
the elementary excitations of its effective action, which contains both
non-relativistic and relativistic terms.
