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main theorem of perturbative renormalization

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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 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 𝒮\mathcal{S}, 𝒮\mathcal{S}' to the inductive construction of the perturbative S-matrix scheme as a function

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

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

𝒵:LocObs(E)[[,g,j]]g,jLocObs(E)[[,g,j]]g,j \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

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

The collection of these operations 𝒵\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 𝒮\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:

Review is in

Last revised on February 7, 2018 at 06:26:19. See the history of this page for a list of all contributions to it.