Can Gravity be Renormalized?

A big achievement of XX century Physics was renormalizing quantum electrodynamics (QED), by Feynman and others. That showed how infinities could be canceled out, so the theory can make predictions in all energy ranges.

QED is a gauge theory on the circle group. Then 'tHooft showed renormalization applied to gauge theories over other groups. Since the known particles were classified by group representations of other groups, that opened the way to the Standard Model. They just had to use the groups already linked to the the particles, and apply gauge theory renormalization.

Gauge theory was the only known renormalizable theory, so there was no choice.

Not everyone agrees that renormalizability is so important. The later invention of effective field theory seemed to bypass renormalization. String theory also provides another approach.

Attempts to quantize gravity have failed because general relativity is not renormalizable. This led people to say string theory is the only game in town, except for maybe loop quantum gravity. Neither approach has produced a quantum gravity theory.

I did not know that general relativity could be easily modified to a theory that is renormalizable.

Luca Buoninfante posts a new paper:

An important theoretical achievement of the last century was the realization that strict renormalizability can be a powerful criterion to select Lagrangians in the framework of perturbative quantum field theory. The Standard Model Lagrangian (without gravity) is strictly renormalizable from a perturbative point of view. On the other hand, the inclusion of gravity seems not to respect this criterion, since general relativity is perturbatively non-renormalizable. The aim of this work is to provide concrete evidence that strict renormalizability is still a valid criterion even when applied to gravity. First, we show that adding quadratic curvature terms to the Einstein-Hilbert action gives rise to a strictly renormalizable theory known as quadratic gravity. Second, we argue that this unique theory represents the most conservative approach to quantum gravity and, at the same time, is highly predictive, as it can explain new physics beyond general relativity already in the sub-Planckian regime.

The simplest way to define a physics theory is to specify the Lagrangian. If you do that for the Standard Model, you can fit it on a t-shirt.

General relativity is the theory derived from the scalar curvature R being the Lagrangian. Or subtract a constant, for general relativity with a cosmological constant. This paper says that you just have to add a quadratic term in the curvature, such as R² or other contractions of the squared Riemann tensor, and you get a renormalizable theory. News to me. This model is sometimes called Starobinsky inflation, and used to explain the early universe.

We do not have any way to test quantum gravity, so the best argument for this approach is that renormalizability has been such a crucially important criterion in the past. It is how we got the Standard Model.

Adding a quadratic term is a bit like Einstein adding the cosmological constant to general relativity. It could not be measured at the time, and was intended to improve the global properties of the theory. It was only measured 80 years later.

Maybe someday this quadratic gravity will be seen as the natural way to modify general relativity to handle extreme conditions.

Everybody always says that quantum mechanics and gravity are incompatible. There is no experiment that shows a problem, so there is only a theoretical incompatibility that might only apply at the center of a black hole or in the first nanosecond of the big bang.

Now I question this. As this paper explains, just add a couple of quadratic terms to the gravity Lagrangian, and there is no problem renormalizing quantum field theory predictions. The only problem is that we do not have experimental data to determine the coefficients of those extra terms. Presumably they are small enough not to affect the known celestial mechanics and cosmology.

So we have a perfectly good quantum gravity theory, with a couple of undetermined coefficients. Those coefficients are too small to affect any of our observations. Viewed that way, it is incorrect to say that there is any incompatibility between gravity and quantum theories.

A few years ago, you could have said that general relativity was incompatible with the concept of a quantum zero point energy. Now the cosmological constant is accepted, and that is believed to be the energy. Maybe we just need to add one or two more cosmological constants, and quantum gravity will cease to be a theoretical issue.