Motivation

This book addresses perturbative Quantum Field Theory (pQFT). This may seem to be an old-fashioned subject. Looking at Schweber's wonderful account of its history [Schweber 1994], it is indeed clear that it goes back more than half a century. But, nevertheless, pQFT has dominated particle physics ever since. It is still the only successful tool we have to calculate everyday cross-sections required to compare our present understanding of theory with experiment. On the other hand, the quest to overcome pQFT has spurred a large amount of the recent developments in high-energy physics theory. This quest stems from the fact that pQFT has several features which are widely regarded as unsatisfactory. Amongst them is the problem of ultraviolet (UV)-divergences. From the very beginning, it was the most prominent problem of pQFT, and its solution was often regarded as technical and as lacking in elegance. The idea to overcome pQFT by something which was not based on seemingly ill-defined quantities from the very beginning is usually accepted as a motivation for other approaches to QFT. Thus, there are two major currents in present day approaches to QFT. On the one hand, there is the inelegant technical machinery of pQFT, pushed forward by the practitioners of multiloop calculations, testing and so far confirming the theory to higher and higher levels in the perturbative loop expansion of the Standard Model of elementary particle physics. On the other hand, at a conceptual level, pQFT is often considered insufficient.