Henning Bostelmann

Scaling algebras and pointlike fields


The concept of scaling limits plays an important role in high energy physics. It is based on the expectation that at short distance, a quantum field theory can be approximated by a simpler theory, its scaling limit. This scaling limit theory may have very different properties from the original theory, e.g. it can be a free theory (asymptotic freedom). For a mathematical description of these limits, a very natural approach has been proposed by Buchholz and Verch in the framework of algebraic quantum field theory. It is however unclear how it relates to more usual approaches, in particular to the renormalization of quantum fields. In order to clear up this matter, we consider models of algebraic quantum field theory with associated pointlike Wightman fields. We find that the Buchholz-Verch scaling limit leads to a multiplicative renormalization scheme for the fields, as expected from perturbation theory. However, the renormalization factors, which are usually added to the model as a technical input, turn up as a consequence of our analysis; they can be computed rather than postulated. In particular, all n-point functions and OPE coefficients automatically converge in the scaling limit.