Causal Perturbative QFT and Space-time Geometry

Abstract: This work is devoted to the causal perturbative Quantum Field Theory (QFT) due to Bogoliubov, including QED and other realistic QFT. It is given the white noise formulation of this theory. The white noise analysis and the Hida operators as the creation and annihilation operators for free fields are used. The whole Bogoliubov method is unchanged. Causal axioms of such QFT make sense on any globally causal space-times. Perturbative QFT with Hida operators is analysed on the flat Minkowski space-time and on the static Einstein Universe (EU). On the flat Minkowski space-time this allowed us to go a step further in the analysis of the adiabatic limit problem, the existence of which has a much wider scope than in the theory based on Wightman's operator valued distributions. The natural condition of existence of the adiabatic limit (as generalized integral kernel operators of white noise calculus) for higher-order contributions to the interacting fields, together with the remaining assumptions of causal QFT (including the natural invariance conditions, e.g. gauge invariance), allowed us to reduce the freedom in choice of renormalization and imposed nontrivial conditions on the masses of elementary particles. Obtained results on the flat space-time are confirmed by the analogous results obtained for QFT on UE, where, moreover, the operators of the interacting fields are much more regular, so, e.g. a consequent treatment of the bound state problem becomes possible on EU. Our approach also enables the analysis of the infrared asymptotics of QED on flat Minkowski spacetime, which we also present.