In the previous chapters we set up renormalization theory in momentum space. In this chapter, we will give a treatment in coordinate space. Now, the utility of a momentum-space description, such as we gave in the earlier chapters, comes from the translation in variance of a problem. However, the momentum-space formulation rather obscures the fact that UV divergences arise from purely short-distance phenomena. Thus a coordinate-space treatment is useful from a fundamental point of view. There are also a number of situations, essentially external field problems, where a coordinate-space treatment is the most appropriate from a more practical point of view. A particular advantage is that the coordinate-space method makes it easy to see that the counterterms are the same as with no external field.

An important case, which we will treat in detail in this chapter, is that of thermal field theory at temperature T (Fetter & Walecka (1971), Bernard (1974), and references therein). There one works with imaginary time using periodic boundary conditions (period 1/T).

It is first necessary to work out the short-distance singularities of the free propagator. This we will do in Section 11.1. A number of forms of the propagator will be given, whose usefulness will become apparent when we treat some examples in Section 11.2.