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 Stückelberg-Petermann renormalization group which describes the freedom in the perturbative construction with e-print archive: http://lanl.arXiv.org/abs/hep-th//0901.2038 R. BRUNETTI ET AL.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 one-parametric subfamily of the Stückelberg-Petermann group and that this subfamily is in general only a cocycle. Since the algebraic structure of the Stückelberg-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 β-function in a state independent way.

Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebraF (O) is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo-Ojima formalism; we define the BRSTtransformations as a graded derivation on the algebra of interacting fields and use the implementation ofs by the Kugo-Ojima operator Q int . Since our treatment is local, the operator Q int differs from the corresponding operator Q of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED.PACS. 11.15.-q Gauge field theories, 11.15.

In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann [42]. In our formalism the entries of the retarded products are local functionals of the off shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's 'Action Ward Identity ' [45]. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local * Work supported by the Deutsche Forschungsgemeinschaft.1 framework independently of the infrared behavior, as was first established by Hollands and Wald [32]. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A (n) of observables "up to n loops", where A (0) is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions.

In the framework of perturbative quantum field theory (QFT) we propose a new, universal (re)normalization condition (called 'master Ward identity') which expresses the symmetries of the underlying classical theory. It implies for example the field equations, energy-momentum, chargeand ghost-number conservation, renormalized equal-time commutation relations and BRST-symmetry.It seems that the master Ward identity can nearly always be satisfied, the only exceptions we know are the usual anomalies. We prove the compatibility of the master Ward identity with the other (re)normalization conditions of causal perturbation theory, and for pure massive theories we show that the 'central solution' of Epstein and Glaser fulfills the master Ward identity, if the UV-scaling behavior of its individual terms is not relatively lowered.Application of the master Ward identity to the BRST-current of nonAbelian gauge theories generates an identity (called 'master BRST-identity') which contains the information which is needed for a local construction of the algebra of observables, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction.