main theorem of perturbative renormalization



A central result in the construction of perturbative quantum field theories via the method of causal perturbation theory is called the main theorem of perturbative renormalization theory. This says that any two renormalization schemes, hence any two solutions to the inductive construction of the S-matrix VS(V)V \mapsto S(V) for interaction terms VV, are related by a unique transformation Z:VVZ \colon V \mapsto V'

S=SZ. S' = S \circ Z \,.

The collection of these operations ZZ forms a group called the Stückelberg-Peterson renormalization group. Hence the space of renormalization schemes is a torsor over this group.

This theorem is a mathematical reflection of the idea that renormalization is about regarding a perturbative quantum field theory with interaction VV as a effective field theory at some energy scale and then discovering that at higher energy there is a more fundamental interaction Z(V)Z(V) which effectively looks like VV at lower energy.


The theorem is originally due to

  • G. Popineau and Raymond Stora, A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, unpublished preprint

In various variants it has been (re-)proved in the following articles:

Review is in

  • Katarzyna Rejzner, section 6.3 of Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

Revised on November 8, 2017 17:04:01 by Urs Schreiber (