  We present an exactly solvable quantum field theory which allows
rearrangement collisions. We solve the model in the relevant sectors and
demonstrate the orthonormality and completeness of the solutions, and construct
the S-matrix. In the light of the exact solutions constructed, we discuss
various issues and assumptions in quantum scattering theory, including the
isometry of the M\"oller wave matrix, the normalization and completeness of
asymptotic states, and the non-orthogonality of basis states. We show that
these common assertions do not obtain in this model. We suggest a general
formalism for scattering theory which overcomes these, and other, shortcomings
and limitations of the existing formalisms in the literature.
