Renormalization

The character of relativistic quantum field theory will continue to be illustrated by the simple example of scalar field theory with the λϕ 4 interaction. Previously we have computed a few basic processes by working out the perturbative development provided by the functional integral representation. Based upon this experience, we can now systematize what has been done in the form of Feynman rules that enable one to compute an amplitude in momentum space directly from its graphical representation. With this in hand, we then turn to describe the character of the divergencies which appear in the theory and how these divergencies are removed by renormalization. We shall present only a very rough outline of how the renormalization process works. A detailed description and proof of the renormalizability of the theory is a non-trivial task whose methods are of little use in other contexts, and so this we will not do. We shall, however, describe in some detail the nature of the parameter renormalization in the minimally subtracted, dimensionally regulated scheme, for this leads directly to the renormalization group which is a very useful tool. The behavior that the renormalization group implies for a general field theory — a theory which could have a fixed point of the renormalization group or a theory such as quantum chromodynamics which is asymptotically free — will be described. Composite operators are introduced, and the relationship of the stress-energy tensor to the renormalization group is described. The chapter concludes with a sketch of the operator product expansion.