Connection between the renormalization groups of Stückelberg–Petermann and Wilson
Confluentes Mathematici, Volume 4 (2012) no. 1.

The Stückelberg–Petermann renormalization group is the group of finite renormalizations of the S-matrix in the framework of causal perturbation theory. The renormalization group in the sense of Wilson relies usually on a functional integral formalism, it describes the dependence of the theory on a UV-cutoff Λ; a widespread procedure is to construct the theory by solving Polchinski's flow equation for the effective potential.

To clarify the connection between these different approaches we proceed as follows: in the framework of causal perturbation theory we introduce an UV-cutoff Λ, define an effective potential VΛ, prove a pertinent flow equation and compare with the corresponding terms in the functional integral formalism. The flow of VΛ is a version of Wilson's renormalization group. The restriction of these operators to local interactions can be approximated by a subfamily of the Stückelberg–Petermann renormalization group.

Published online:
DOI: 10.1142/S1793744212400014

Michael Dütsch 1

1
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Michael Dütsch. Connection between the renormalization groups of Stückelberg–Petermann and Wilson. Confluentes Mathematici, Volume 4 (2012) no. 1. doi : 10.1142/S1793744212400014. https://cml.centre-mersenne.org/articles/10.1142/S1793744212400014/

[1] F. Brennecke and M. Dütsch, Removal of violations of the Master Ward Identity in perturbative QFT, Rev. Math. Phys. 20 (2008) 119–172.

[2] R. Brunetti, M. Dütsch and K. Fredenhagen, Perturbative algebraic quantum field theory and the renormalization groups, Adv. Theor. Math. Phys. 13 (2009) 1541–1599.

[3] R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208 (2000) 623.

[4] M. Dütsch and K. Fredenhagen, Algebraic quantum field theory, perturbation theory, and the loop expansion, Commun. Math. Phys. 219 (2001) 5.

[5] M. Dütsch and K. Fredenhagen, Perturbative algebraic field theory, and deformation quantization, Fields Inst. Commun. 30 (2001) 151–160.

[6] M. Dütsch and K. Fredenhagen, Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity, Rev. Math. Phys. 16 (2004) 1291–1348.

[7] M. Dütsch and K. Fredenhagen, Action Ward Identity and the Stückelberg– Petermann renormalization group, in Rigorous Quantum Field Theory, eds. A. Boutet de Monvel, D. Buchholz, D. Iagolnitzer and U. Moschella (Birkhäuser, 2006), pp. 113–123.

[8] H. Epstein and V. Glaser, The role of locality in perturbation theory, Ann. Inst. H. Poincaré A 19 (1973) 211.

[9] S. Hollands and R. M. Wald, On the renormalization group in curved spacetime, Commun. Math. Phys. 237 (2003) 123–160.

[10] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, 2nd edn. (Springer, 1990).

[11] G. Keller, C. Kopper and C. Schophaus, Perturbative renormalization with flow equations in Minkowski space, Helv. Phys. Acta 70 (1997) 247–274.

[12] J. Polchinski, Renormalization and effective Lagrangians, Nucl. Phys. B 231 (1984) 269–295.

[13] G. Popineau and R. Stora, A pedagogical remark on the main theorem of perturbative renormalization theory, unpublished preprint (1982).

[14] M. Salmhofer, Renormalization. An Introduction (Springer, 1999).

[15] R. Stora, Differential algebras in Lagrangean field theory, ETH-Zürich Lectures, January–February 1993.

[16] E. C. G. Stückelberg and A. Petermann, La normalisation des constantes dans la théorie des quanta, Helv. Phys. Acta 26 (1953) 499–520.

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