On a third S-matrix in the theory of quantized fields on curved spacetimes

Wightman functions for interacting quantum fields on curved space times are cal- culated via the perturbation theory of the Yang-Feldman equations, where the incoming field is a free field in a quasifree representation. We show that these Wightman functions that are ob- tained as a sum over extended Feynman graphs fulfill the basic axioms of hermiticity, invariance, spectrality (on stationary spacetimes), perturbative positivity and locality (the latter property is shown up to second order in the loop expansion). In the case of non-stationary spacetimes, the outgoing field in general is in a non-quasifree representation of the CCR. This makes it necessary to develop a method to calculate the unitary transformation between a non quasifree representa- tion and a quasifree one. This is carried out using ⋆-calculus on the dual of the Borchers algebra with a combinatorial co-product. Given that preferred quasifree representations for early and late times exist, we thus obtain a complete scattering description using three S-matrices: The first is determined by vacuum expectation values between incoming and outgoing fields. The second is a unitary transformation between the non-quasifree representation for the "out"-fields and the quasifree representation for the "in"-field. The last one is the Bogoliubov transformation between the preferred representation at early times (i.e. the "in"-field representation) and the preferred representation at late times.