  Using $\star$-calculus on the dual of the Borchers-Uhlmann algebra endowed
with a combinatorial co-product, we develop a method to calculate a unitary
transformation relating the GNS representations of a non-quasifree and a
quasifree state of the free hermitian scalar field. The motivation for such an
analysis and a further result is the fact that a unitary transformation of this
kind arises naturally in scattering theory on non-stationary backgrounds.
Indeed, employing the perturbation theory of the Yang-Feldman equations with a
free CCR field in a quasifree state as an initial condition and making use of
extended Feynman graphs, we are able to calculate the Wightman functions of the
interacting and outgoing fields in a $\phi^p$-theory on arbitrary curved
spacetimes. A further examination then reveals two major features of the
aforementioned theory: firstly, the interacting Wightman functions fulfil the
basic axioms of hermiticity, invariance, spectrality (on stationary
spacetimes), perturbative positivity, and locality. Secondly, the outgoing
field is free and fulfils the CCR, but is in general not in a quasifree state
in the case of a non-stationary spacetime. In order to obtain a sensible
particle picture for the outgoing field and, hence, a description of the
scattering process in terms of particles (in asymptotically flat spacetimes),
it is thus necessary to compute a unitary transformation of the abovementioned
type.
