Notes on LSZ, i epsilon Prescriptions and Perturbation Theory, in QFT and Cosmology

We review the original argument of Lehmann et al (LSZ) that relates the flat space S-matrix to the correlation function of field operators and clarify some confusing issues. Next we discuss the origin of the $i\epsilon$ prescriptions following Weinberg, but without assuming that the vacuum is free at asymptotic times. Then we discuss the corresponding argument in inflationary cosmology, emphasizing that unitarity violating contour deformations are unnecessary.