Advances in Theoretical and Mathematical Physics

Perturbative algebraic quantum field theory and the renormalization groups

R. Brunetti, M. Dütsch, and K. Fredenhagen

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Abstract

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 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.

Article information

Source
Adv. Theor. Math. Phys., Volume 13, Number 5 (2009), 1541-1599.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1282054101

Mathematical Reviews number (MathSciNet)
MR2672469

Zentralblatt MATH identifier
1201.81090

Citation

Brunetti, R.; Dütsch, M.; Fredenhagen, K. Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13 (2009), no. 5, 1541--1599. https://projecteuclid.org/euclid.atmp/1282054101


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