  Andreev scattering and the Josephson current through a one-dimensional
interacting electron liquid sandwiched between two superconductors are
re-examined. We first present some apparently new results on the
non-interacting case by studying an exactly solvable tight-binding model rather
than the usual continuum model. We show that perfect Andreev scattering (i.e.
zero normal scattering) at the Fermi energy can only be achieved by fine-tuning
junction parameters. We also obtain exact results for the Josephson current,
which is generally a smooth function of the superconducting phase difference
except when the junction parameters are adjusted to give perfect Andreev
scattering, in which case it becomes a sawtooth function. We then observe that,
even when interactions are included, all low energy properties of a junction
(E<<\Delta, the superconducting gap) can be obtained by "integrating out" the
superconducting electrons to obtain an effective Hamiltonian describing the
metallic electrons only with a boundary pairing interaction. This boundary
model provides a suitable starting point for bosonization/renormalization
group/boundary conformal field theory analysis. We argue that total normal
reflection and total Andreev reflection correspond to two fixed points of the
boundary renormalization group. For repulsive bulk interactions the Andreev
fixed point is unstable and the normal one stable. However, the reverse is true
for attractive interactions. This implies that a generic junction Hamiltonian
(without fine-tuned junction parameters) will renormalize to the normal fixed
point for repulsive interactions but to the Andreev one for attractive
interactions. An exact mapping of our tight-binding model to the Hubbard model
with a transverse magnetic field is used to help understand this behavior.
