  We compare two non-perturbative techniques for calculating the
single-particle Green's function of interacting Fermi systems with dominant
forward scattering: our recently developed functional integral approach to
bosonization in arbitrary dimensions, and the eikonal expansion. In both
methods the Green's function is first calculated for a fixed configuration of a
background field, and then averaged with respect to a suitably defined
effective action. We show that, after linearization of the energy dispersion at
the Fermi surface, both methods yield for Fermi liquids exactly the same
non-perturbative expression for the quasi-particle residue. However, in the
case of non-Fermi liquid behavior the low-energy behavior of the Green's
function predicted by the eikonal method can be erroneous. In particular, for
the Tomonaga-Luttinger model the eikonal method neither reproduces the correct
scaling behavior of the spectral function, nor predicts the correct location of
its singularities.
