Basic Conceptual and Mathematical Problems of QFT

In this talk we discuss some of the most basic conceptual and mathematical difficulties that arise in the standard physics analysis of QFT. In particular we shall discuss the origin of UV divergences in QFT—pointing out that there is both a kinematic and a dynamic aspect to this problem, and that the standard physics explanation (’new physics’) only considers the latter—and suggest that despite the notoriety of the problem, UV divergences are essentially under control. Secondly we discuss Haag’s theorem—which ensures the nonexistence of the interaction picture and the triviality of the perturbative S-Matrix—and indicate how this is the most elementary manifestation of a series of infrared problems in QFT. Finally we will outline why the rigorous construction of path-integral measures is difficult. If we have time, we may discuss some difficulties associated with gauge theories such as the infraparticle problem of QED and the mass-gap problem of Yang-Mills theory.