1 - Introduction

Summary

The structure of a quantum field theory often simplifies when one considers processes involving large momenta or short distances. These simplifications are important in improving one's ability to calculate predictions from the theory, and in essence form the subject of this book.

The first simplification to be considered involves the very existence of the theory. The problem is that there are usually ultra-violet divergences caused by large fluctuations of the field(s) on short distance scales. These manifest themselves in Feynman graphs as divergences when loop momenta go to infinity with the external momenta fixed. The simplification is that the divergences can be cancelled by renormalizations of the parameters of the action. Consequently our first task will be to treat the ultra-violet renormalizations. Renormalization is essential, for otherwise most field theories do not exist.

We will then expose the methods needed to handle high-energy/short-distance problems. The aim is to be able to make testable predictions from a strong interaction theory, or to improve the rate of convergence of the perturbation expansion in a weakly coupled theory. The simplifications generally take the form of a factorization of a cross-section or of an amplitude, each factor containing the dependence of the process on phenomena that happen on one particular distance scale.