Scaling algebras and pointlike fields. A nonperturbative approach to renormalization
Originally, renormalization was used in QFT to interpret infinite perturbative expressions. In the past thirty years an elegant powerful nonpertubative technique has been developed called Renormalization Group, pioneered by Wilson, Kadanoff and Fisher. It is a way to investigate the short distance behavior of a given field theoretical model and in particular has led to the discovery of asymptotic freedom in QCD. The present approach to renormalization is based on the results of Buchholz and Verch (BV) published during 1995-98 in the context of algebraic quantum field theory. The term \textit{algebraic} means that one starts out from a local net of \(C^*\) algebras, a theory that was initiated by Haag, Kastler, Doplicher and Roberts. BV obtained \textit{scaling limit states} as weak-* cluster points whose existence is guaranteed by the Alaoglu-Bourbaki Theorem. But now, in addition the existence of fields localized at spacetime points and associated to these algebras is assumed, in a way that was previously (2005) proposed and studied by Bostelmann under certain assumptions called \textit{phase space conditions}. For the fields as well as for their operator product expansions, a well-defined limiting procedure similar to that of BV is established. However, compared to the work of BV the approach chosen here is somewhat more general. The authors also study symmetry actions in the scaling limit. For suitable limit states, the group of scaling transformations leads to a dilation symmetry of the limit theory. It would be worthwhile to extend the results to charge carrying fields associated with a local net in the sense of Doplicher and Roberts.
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