nLab
renormalizable interaction

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

Definition

In the construction of interacting perturbative quantum field theory the specification of

  1. the gauge-fixed free field vacuum (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H) that the perturbation is about (this def.);

  2. the local interaction action functional gS intLocObs(E BV-BRST)[[,g]]gg S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g\rangle which prescribes the nature of the perturbation (this def.)

still leaves, in general, a countable set of finite dimensional affine spaces of choices {c k α} k𝕟\{c^\alpha_k \in \mathbb{C}\}_{k \in \mathbb{n}} to fully specify the corresponding interacting perturbative QFT (this prop.). This is called the choice of ("re"-)normalization constants of the perturbative QFT.

One says that the interaction gS intg S_{int} is renormalizable if this potentially countably infinite dimensional space of choices happens to be a finite dimensional space, hence if there is nn \in \mathbb{N} such that the only available choice is c k α=0c^\alpha_k = 0 for all knk \geq n. Otherwise one says that gS intg S_{int} is non-renromalizable.

(If only a finite set of Feynman diagrams contributes to the (“re”-)normalization choices to be made for a renormaliable interaction, then it is called super-renormalizable).

Interpretation

Beware that this terminology may be misleading: The main theorem of perturbative renormalization says, among other things, that for all perturbative QFTs there do exist constructions of time-ordered products/Feynman amplitudes on the diagonal, hence of ("re"-)normalizations, hence of choices of these c k αc_k^\alpha, namely by Epstein-Glaser renormalization (Epstein-Glaser 73); even if there are (countably) infinitely many. In Gomis-Weinberg 96 this is called “renormalizable in the modern sense” (at least if renormalization conditions such as Ward identities may be met).

Therefore the issue expressed with the term “non-renormalizable” at this point is not a mathematical obstruction, but a philosophical sentiment: The need to specify an infinite set of further physical constants to specify a physical theory is (or has been) felt to defeat the purpose or meaning of having specified a physical theory.

But in practice this is less of a problem than it might seem: By the Gell-Mann-Low renormalization group flow/Wilson-Polchinski RG flow the higher the order of the renormalization constants, the higher the energy/the smaller the wavelength at which their effect is visible in experiment. Hence in practice only a finite number of renormalization constants is observable anyway. As the energy/resolution of an experiment is increased by a finite amount, a finite number of further renormalization constants may become visible.

For famous “non-renormalizable” theories such as perturbative quantum gravity the first such corrections would become visible to experiment only at energies way beyond current reach, so that for all practical purpose the construction of “non-renormalizable” perturbative QFTs such as perturbative quantum gravity via ("re"-)normalization does exist (Scharf 01, chapter 5). (A wide-open problem is, instead, the construction of interacting non-perturbative quantum field theories for spacetime dimension 3+1\geq 3+1.)

It has become popular to think of such “non-renormalizable” perturbative QFTs as “effective field theories”. See there for more.

Examples

References

The renormalization of any interaction in perturbative QFT (by a possibly countably infinite number of renormalization constants) was established by Epstein-Glaser renormalization in

This point was later amplified as “renormalization in the modern sense” in

Discussion in the rigorous context of causal perturbation theory/pAQFT is in

See also

Last revised on February 6, 2018 at 16:02:21. See the history of this page for a list of all contributions to it.