  There has been some debate about the validity of quantum affine Toda field
theory at imaginary coupling, owing to the non-unitarity of the action, and
consequently of its usefulness as a model of perturbed conformal field theory.
Drawing on our recent work, we investigate the two simplest affine Toda
theories for which this is an issue - a2(1) and a2(2). By investigating the
S-matrices of these theories before RSOS restriction, we show that quantum Toda
theory, (with or without RSOS restriction), indeed has some fundamental
problems, but that these problems are of two different sorts. For a2(1), the
scattering of solitons and breathers is flawed in both classical and quantum
theories, and RSOS restriction cannot solve this problem. For a2(2) however,
while there are no problems with breather-soliton scattering there are instead
difficulties with soliton-excited soliton scattering in the unrestricted
theory. After RSOS restriction, the problems with kink-excited kink may be
cured or may remain, depending in part on the choice of gradation, as we found
in [12]. We comment on the importance of regradations, and also on the survival
of R-matrix unitarity and the S-matrix bootstrap in these circumstances.

High Energy Physics - Theory