Causal perturbative QED and white noise. The inductive step

Abstract: We present the Bogoliubov's causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, \emph{i.e.} for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared -- and ultraviolet -- divergences in the transition to the adiabatic limit for interacting fields. Here we present the inductive step. We are especially concentrated with the analysis of the second order contribution to the scattering operator, and give explicit computations using the calculus of integral kernel operators, \emph{i.e} contractions of their vector-valued kernels, and splittings of the contractions into advanced and retarded parts. Explicit example of QED is provided, together with the detailed analysis of the key point in avoiding UV divergence.