nLab main theorem of perturbative renormalization

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 given a suitable free field vacuum to be perturbed (this def.), then any two renormalization schemes for perturbative quantum field theory around this free field theory, hence any two solutions $\mathcal{S}$, $\mathcal{S}'$ to the inductive construction of the perturbative S-matrix scheme as a function

$$\mathcal{S} \;\colon\; LocObs(E)[ [\hbar , g ] ]\langle g , j\rangle \longrightarrow PolyObs(E)_{mc}((\hbar))[ [ g, j] ]$$

from local observables $g S_{int} + j A$, regarded as adiabatically switched interaction action functionals, to scattering amplitude Wick algebra elements $S\mathcal{S}(\mathbf{L}_{int})$, are related by a unique perturbative transformation

$$\mathcal{Z} \;\colon\; LocObs(E)[ [\hbar, g, j] ]\langle g, j\rangle \longrightarrow LocObs(E)[ [\hbar, g, j] ]\langle g, j\rangle$$

of the space of local interaction action functionals via precomposition

$$\mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z} \,.$$

The collection of these operations $\mathcal{Z}$ forms a group, called the Stückelberg-Petermann renormalization group. Hence the space of renormalization schemes is a torsor over this group.

The precise nature of this group depends on which set of renormalization conditions one imposes. The larger this set, the smaller the corresponding renormalization group (Dütsch 18, remark 3.102).

Beware the terminology: Contrary to common practice, the construction of a single $\mathcal{S}$ is more properly called a choice of normalization rather a “re”-normalization (e. g. Scharf 95, section 4.3), but the “main theorem” above says that the elements in the Stückelberg-Petermann renormalization group are precisely that: re-normalizations, passing from one choice of normalization to another.

Details

See at renormalization this theorem.

References

The theorem is originally due to

• G. Popineau, Raymond Stora, A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, Nucl. Phys. B 912 (2016), 70–78, preprint: LAPP–TH, Lyon (1982)

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

• G. Pinter, section 6 of The Action Principle in Epstein Glaser Renormalization and Renormalization of the S-Matrix in $\phi^4$-Theory, Annalen Phys. 10 (2001) 333-363 (arXiv:hep-th/9911063)

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, section 4.1 of Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics (arXiv:0901.2038)

• Michael Dütsch, Klaus Fredenhagen, section 4.2 of Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity, Rev. Math. Phys. 16, 1291 (2004) (arXiv:hep-th/0403213)

• Stefan Hollands, theorem 2 in Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys. 20:1033-1172, 2008 (arXiv:0705.3340)

Review is in

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

• Michael Dütsch, section 3.5.2 of From classical field theory to perturbative quantum field theory, 2018