nLab renormalizable interaction

Definition

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

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

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

still leaves, in general, a countable set of finite dimensional affine spaces of choices $\{c^\alpha_k \in \mathbb{C}\}_{k \in \mathbb{n}}$ 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 $g S_{int}$ is renormalizable if this potentially countably infinite dimensional space of choices happens to be a finite dimensional space, hence if there is $n \in \mathbb{N}$ such that the only available choice is $c^\alpha_k = 0$ for all $k \geq n$. Otherwise one says that $g S_{int}$ 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 $c_k^\alpha$, 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 $\geq 3+1$.)

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

Examples

• Yang-Mills theory (…condition on gauge group…) hence QCD is renormalizable (t’Hooft, …)

• Einstein gravity is non-renormalizable (Goroff-Sagnotti 85, van de Ven 92)

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

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

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

• Joaquim Gomis, Steven Weinberg, Are Nonrenormalizable Gauge Theories Renormalizable?, Nucl. Phys. B469 : 473-487, 1996 (arXiv:hep-th/9510087)

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

• Günter Scharf, p. 271 of Finite Quantum Electrodynamics – The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition

• Günter Scharf, Quantum Gauge Theories – A True Ghost Story, Wiley 2001

See also

• Wikipedia, Renormalizability