  We present some partial results on the general infrared behavior of
bulk-critical 1-d quantum systems with boundary. We investigate whether the
boundary entropy, s(T), is always bounded below as the temperature T decreases
towards 0, and whether the boundary always becomes critical in the IR limit. We
show that failure of these properties is equivalent to certain seemingly
pathological behaviors far from the boundary. One of our approaches uses real
time methods, in which locality at the boundary is expressed by analyticity in
the frequency. As a preliminary, we use real time methods to prove again that
the boundary beta-function is the gradient of the boundary entropy, which
implies that s(T) decreases with T. The metric on the space of boundary
couplings is interpreted as the renormalized susceptibility matrix of the
boundary, made finite by a natural subtraction.
