  We derive an expression for the relation between two scattering transition
amplitudes which reflect the same dynamics, but which differ in the description
of their initial and final state vectors. In one version, the incident and
scattered states are elements of a perturbative Fock space, and solve the
eigenvalue problem for the `free' part of the Hamiltonian --- the part that
remains after the interactions between particle excitations have been `switched
off'. Alternatively, the incident and scattered states may be coherent states
that are transforms of these Fock states. In earlier work, we reported on the
scattering amplitudes for QED, in which a unitary transformation relates
perturbative and non-perturbative sets of incident and scattered states. In
this work, we generalize this earlier result to the case of transformations
that are not necessarily unitary and that may not have unique inverses. We
discuss the implication of this relationship for Abelian and non-Abelian gauge
theories in which the `transformed', non-perturbative states implement
constraints, such as Gauss's law.
