# nLab main theorem of perturbative renormalization

## Concepts

Lagrangian field theory

quantum mechanical system

quantization

# Contents

## Idea

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 $V \mapsto S(V)$ for interaction terms $V$, are related by a unique transformation $Z \colon V \mapsto V'$

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

The collection of these operations $Z$ 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 $V$ as a effective field theory at some energy scale and then discovering that at higher energy there is a more fundamental interaction $Z(V)$ which effectively looks like $V$ at lower energy.

## References

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• G. Popineau and Raymond Stora, A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, unpublished preprint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 (46.183.103.8)