  The S-matrix for each chiral sector of Liouville theory on a cylinder is
computed from the loop expansion of correlation functions of a one-dimensional
field theory on a circle with a non-local kinetic energy and an exponential
potential. This action is the Legendre transform of the generating function of
semiclassical scattering amplitudes. It is derived from the relation between
asymptotic in- and out-fields. Its relevance for the quantum scattering process
is demonstrated by comparing explicit loop diagrams computed from this action
with other methods of computing the S-matrix, which are also developed.
