# Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

January 14, 2009

A new formalism for the perturbative construction of algebraic quantum field
theory is developed. The formalism allows the treatment of low dimensional
theories and of non-polynomial interactions. We discuss the connection between
the Stueckelberg-Petermann renormalization group which describes the freedom in
the perturbative construction with the Wilsonian idea of theories at different
scales . In particular we relate the approach to renormalization in terms of
Polchinski's Flow Equation to the Epstein-Glaser method. We also show that the
renormalization group in the sense of Gell-Mann-Low (which characterizes the
behaviour of the theory under the change of all scales) is a 1-parametric
subfamily of the Stueckelberg-Petermann group and that this subfamily is in
general only a cocycle. Since the algebraic structure of the
Stueckelberg-Petermann group does not depend on global quantities, this group
can be formulated in the (algebraic) adiabatic limit without meeting any
infrared divergencies. In particular we derive an algebraic version of the
Callan-Symanzik equation and define the beta-function in a state independent
way.

Keywords:

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