  We consider a scalar quantum field theory, in which the interaction takes the
form of a field cutoff; the energy diverges to infinity whenever the value of
the field at some point falls outside a finite interval. In a simple
(1+1)-dimensional version of this theory, we may calculate the results of
certain scattering processes exactly. The main feature of the nontrivial
solutions is the appearance of shock fronts, whose time development is
irreversible. The resulting nonunitarity implies that these theories are, at a
minimum, radically different from conventional quantum field theories.
