  We consider the current correlation function for a three-dimensional system
of fermions embedded in a homogeneous background and mutually interacting via
an attractive short-range potential, below the (superconducting) critical
temperature. Diagrammatic contributions in the broken-symmetry phase are
identified, that yield for the (wave-vector and frequency dependent) current
correlation function the fermionic BCS approximation in the weak-coupling limit
and the bosonic Bogoliubov approximation in the strong-coupling limit (whereby
composite bosons form as bound-fermion pairs). The temperature dependence of
the superfluid density (from the BCS exponential behavior at weak coupling to a
power-law behavior at strong coupling) and the form of the Pippard-like kernel
at zero temperature are explicitly obtained from weak to strong coupling. Quite
generally, it is shown that the Pippard-like kernel is the sum of a local
(London-like) term and of a nonlocal component, the local term being dominant
in the strong-coupling limit and the nonlocal component in the BCS
(weak-coupling) limit. It is also shown that the range of the nonlocal
component is determined by the coherence length measuring the spatial
correlations of the amplitude of the order parameter, namely, the correlations
among different Cooper pairs (or composite bosons), rather than between the
fermionic partners of a given pair. In addition, a prescription is provided for
mapping the fermionic onto the bosonic diagrammatic theories in the
broken-symmetry phase, thus complementing what already done in the normal
phase.
