## Perturbative algebraic quantum field theory and the renormalization groups.(English)Zbl 1201.81090

The renormalization group equations (RGE) arise from the freedom to reparametrize a Lagrangian density while preserving the physical content, e.g. the S-matrix. Commonly, the RGE are differential equations which express the change of all parameters in the Lagrangian. At the same time the RG method provides useful insight into the asymptotic behavior encountered in quantum quantum field theory. The method of RG was originally introduced by Gell-Mann and Low. It can also be used for calculations in statistical mechanics, a discovery due to Kenneth Wilson. However, no unique RG theory exists. There are at least three different formulations of the RG approach: the Gell-Mann-Low, the Wilson and the Stückelberg-Petermann formulation. It is the aim of the present paper to analyze the different concepts and to clarify the relation between various processes of renormalization: the BPHZ approach, the use of the flow equation for the effective potential, and the so-called causal perturbation theory of Epstein and Glaser. All this is done in the context of algebraic QFT. Section 2 after the Introduction provides definitions, algebraic structures and an outlook. Section 3 shows how to enlarge the space of observables in order to include non-trivial local interactions. Section 4 deals with the Epstein-Glaser method, while Section 5, the core of the article, compares this with the RG of Gell-Mann and Low (actually not a group but a cocycle in the massive case). The discussion continues in Section 6, where the focus is now on the adiabatic limit. Novel algebraic tools, termed generalized Lagrangian and algebraic Callan-Symanzik equation are defined. Section 7 deals with examples, $$\phi^4$$ in four dimensions and $$\phi^3$$ in six dimensions. Comparison with standard beta function treatments in standard textbooks show an agreement.

### MSC:

 81T17 Renormalization group methods applied to problems in quantum field theory 81R15 Operator algebra methods applied to problems in quantum theory 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T05 Axiomatic quantum field theory; operator algebras 81T10 Model quantum field theories
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