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Nonrenormalizability of GR

A reader has asked me what I thought about two preprints, namely

that are "beginning to understand that quantized Einstein's theory of gravity renormalizable using Wilsonian techniques". Well, that's surely an interesting statement because every student who has attended at least 10 lectures of Quantum Field Theory II of a good course knows how to prove that it is not renormalizable. ;-)

So what is going on? Let's discuss both science and sociology.


These papers and dozens of other papers that are found in the references are doing some sort of semiclassical treatment of general relativity. In the context of the renormalization group, they study the running of Newton's constant but throw away all other couplings of general relativity.

If you do so, you get some equations that have some solutions. In their particular formalism, they typically reveal an ultraviolet fixed point. But whatever the conclusion is, the conclusion can't be generalized to the full, untruncated theory of gravity. The truncated theory is not unitary i.e. it is not consistent. Also, it is not in the same universality class with the full theory.

How does the running work? In general relativity, it has been known for decades that the one-loop diagrams would keep it renormalizable. So various quantum corrections to Newton's force can be calculated at this precision - meaning terms proportional to the first power of Planck's constant.

It has also been known for nearly three decades that the two-loop diagrams are divergent and require us to add new kinds of counterterms that differ from the Einstein-Hilbert action. That proves that general relativity is not renormalizable. As we go to higher loops, the situation gets worse. Many more new terms are generated and the degree of divergence increases, too.

Can we just forget about all coefficients of these terms except for Newton's constant? The answer is that we can only forget them approximately and such an approximation will only be meaningful at energies much lower than the Planck energy. At energies comparable to the Planck energy, the higher-derivative terms are equally important as the Einstein-Hilbert term. You can even say that at energies above the Planck energy, the higher-derivative terms are actually more important than the Einstein-Hilbert term.

The existence of an ultraviolet fixed point - something that would almost uniquely determine what the theory is supposed to look like at extremely short distances - really depends on what happens at very high energies, namely energies above the Planck energy. It is very clear that if you throw away the higher-derivative terms, you are throwing away the most important thing that decides about the answer. What you get with this truncation is physically irrelevant: it has nothing to do with properties of quantum gravity.


I understand that in the eyes of a layman, most papers on the arXiv look the same. Some text with some equations, similar keywords, and so on. Can a layperson figure out that these papers probably don't include a proof of an unusual statement that Einstein's equations are renormalizable? I think that he can.

Assume that the number of people who actively study questions sufficiently related to quantum gravity and its nonrenormalizability is 2000. Now, let's be very critical and skeptical about the intellectual qualities of the science community. Assume that only 20% of this set, about 400 people, are evaluating the evidence and new papers really independently while the remaining 1600 tend to copy big opinions from others. (The active core would be much smaller than 20% in some other fields and higher in yet another group.)

In this arrangement, will the most famous physicists remain ignorant about a paper that has proven an important result that falsifies some lore?

Well, most of these 400 people look at the abstracts of preprints pretty much every day. So about 400 people who should know the stuff and who are thinking independently have looked at the papers above. By discussing them with others, the message about the new papers would spread to others. The papers are pretty simple so surely at least 200 people would understand the nontrivial essence of the proof. Because it would be a completely new approach to quantum gravity, 50 of them would start to think about related issues or write papers about this topic. That would surely include some well-known names.

Note that as you can see, I certainly don't claim that if there were a new key result in science, there would be an instant consensus of 2000 people that it is correct. Only pseudosciences can act in this way. But if you think about the events, you might agree that something comparable to 50 independent qualified physicists would start to be really interested in the topic. Let's estimate that about 20 of them would be considered to be "good physicists" by one of the leaders. How likely to do you think that these leaders would "miss" the new development?

Well, I think it is extremely unlikely.

What I want to say is that it is almost impossible to collectively miss an important paper. In other sciences, the places where papers are published are more chaotic and the percentage of independently thinking researchers may be smaller. But as long as these differences are "finite", my qualitative conclusions will hold.

The only realistic reasons why important papers like that could be missed is that almost all the people would have some unscientific or irrational reasons to reject a certain kind of results. Well, it may be true in other fields, especially if a certain kind of results means smaller funding. But it is hard to imagine that an important result is missed.

If you want to believe these papers make sense, it also means that you must believe that the community of 10 not-too-well-known physicists who work on these papers are systematically smarter than the people who are considered leaders of theoretical physics and who have made a lot of discoveries in closely related topics. I would find such a conjecture extremely unreasonable.

Of course if you are a postmodernist who believes that physics is a social construct and a random person who writes papers about "renormalizable GR" is intrinsically equally qualified as e.g. Edward Witten, you may end up with very different conclusions. But in that case, I think that you're not sane.

And that's the memo.

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