# Gravity Renormalizable?

1. Jan 22, 2008

### BenTheMan

I remember marcus claiming somewhere that one of the QG researchers had claimed that Gravity was renormalizable. He had some graph of renormalization group flow or something, and it was some big result at the last loops conference.

The thing that has bugged me (and perhaps this has been asked already here) is that there are lots of effective field theories that have dimensionful coupling constants (eg Fermi theory), that are clearly effective field theories. What happens if the same treatment is applied to them? Why should gravity be special, in that respect?

Again, apologies if this has been asked before, but I know there are many here who can answer this question.

2. Jan 22, 2008

### marcus

Reuter's results have been discussed some here, but not enough. I'm glad you started a new Reuter thread.

I think Reuter's stuff (NONperturbative renormalizable) is extremely interesting and (in terms used in Percacci's review paper) could either lead to a theory that is effective or fundamental. At this point it is not clear which.

Percacci's review, an invited chapter for a book, is called Asymptotic Safety and would definitely be something to look at if you are interested. I'll get some links later.

Another thing to look at is the most recent paper by Frank Saueressig, which CHALLENGES some of Reuter's conclusions. This is especially important because Saueressig has been a junior co-author of several Reuter papers since 2001, and he now has this paper where he is the senior author where finds something different. Saueressig publishes both in string and non-string quantum gravity----that is, he does string papers and also asymptotic safety Reuter-type papers.

You should be very cautious about terminology. NONperturbative renormalizable may confuse you because you may be used to thinking only in the perturbation series context.

Reuter is NOT talking about taking the first few terms of a perturbation series.

Basically the concept was discovered by Steven Weinberg in around 1976-1979, if I remember right, and Weinberg gave up on it after he and others tried unsuccessfully, and it has almost not been heard of until Reuter's paper from around 1998.

You may also misunderstand what Percacci means by an EFFECTIVE theory, if you think of it in the perturbative context. You have to read his paper to understand. He doesnt mean what you might immediately think of, having heard the term "effective" in other QFT contexts.

Donoghue also treats gravity as an effective theory and gave an invited paper at Loops 07 which generated some interest. I am not familiar with his work. I recall either Reuter referred to Donoghue's work positively in his talk IIRC, or some of the blog commentary did. But I didn't follow up on that.

3. Jan 22, 2008

### marcus

First I will get a link to Saueressig's paper because it challenges Reuter's findings (making it of special interest). My impression is Saueressig is young and extremely smart. he is at Utrecht ('t Hooft's ITP same place as Renate Loll)
=====
Oh wait, I just ran across Percacci's review, so I will list that.
http://arxiv.org/abs/0709.3851
Asymptotic Safety
R. Percacci
To appear in "Approaches to Quantum Gravity: Towards a New Understanding of Space, Time and Matter", ed. D. Oriti, Cambridge University Press
(Submitted on 24 Sep 2007)

"Asymptotic safety is a set of conditions, based on the existence of a nontrivial fixed point for the renormalization group flow, which would make a quantum field theory consistent up to arbitrarily high energies. After introducing the basic ideas of this approach, I review the present evidence in favor of an asymptotically safe quantum field theory of gravity".
=========

I also found a recent Reuter-Saueressig collaboration

http://arxiv.org/abs/0708.1317
Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity
Martin Reuter, Frank Saueressig
Based on lectures given by M.R. at the 'First Quantum Geometry and Quantum Gravity School', Zakopane, Poland, March 2007, and the 'Summer School on Geometric and Topological Methods for Quantum Field Theory', Villa de Leyva, Colombia, July 2007, and by F.S. at NIKHEF, Amsterdam, The Netherlands, June 2006
(Submitted on 9 Aug 2007)

"These lecture notes provide a pedagogical introduction to a specific continuum implementation of the Wilsonian renormalization group, the effective average action. Its general properties and, in particular, its functional renormalization group equation are explained in a simple scalar setting. The approach is then applied to Quantum Einstein Gravity (QEG). The possibility of constructing a fundamental theory of quantum gravity in the framework of Asymptotic Safety is discussed and the supporting evidence is summarized."

As a sample, here are Saueressig papers from the past year
http://arxiv.org/find/grp_physics/1/au:+Saueressig/0/1/0/past/0/1
1. arXiv:0712.0445 [ps, pdf, other]
Title: On the renormalization group flow of f(R)-gravity
Authors: Pedro F. Machado, Frank Saueressig

2. arXiv:0710.4931 [ps, pdf, other]
Title: Recent results in four-dimensional non-perturbative string theory
Authors: Frank Saueressig
Comments: 7 pages, to appear in the proceedings of the European Physical Society HEP 2007 Conference, 19-25 July 2007, Manchester, England

3. arXiv:0708.1317 [ps, pdf, other]
Title: Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity
Authors: Martin Reuter, Frank Saueressig
Comments: Based on lectures given by M.R. at the First Quantum Geometry and Quantum Gravity School'', Zakopane, Poland, March 2007, and the Summer School on Geometric and Topological Methods for Quantum Field Theory'', Villa de Leyva, Colombia, July 2007, and by F.S. at NIKHEF, Amsterdam, The Netherlands, June 2006

4. arXiv:0707.0838 [ps, pdf, other]
Title: Membrane instantons from mirror symmetry
Authors: Daniel Robles-Llana, Frank Saueressig, Ulrich Theis, Stefan Vandoren

5. arXiv:0704.2229 [ps, pdf, other]
Title: Conifold singularities, resumming instantons and non-perturbative mirror symmetry
Authors: Frank Saueressig, Stefan Vandoren

You can see he has chosen to do both string and nonstring QG research. Interesting guy.

===========
HERE'S the paper that I think calls some finding of Reuter into question (but the issue is not yet resolved)
http://arxiv.org/abs/0712.0445
On the renormalization group flow of f(R)-gravity
55 pages, 7 figures
(Submitted on 4 Dec 2007)

"We use the functional renormalization group equation for quantum gravity to construct a non-perturbative flow equation for modified gravity theories of the form $$S = \int d^dx \sqrt{g} f(R)$$. Based on this equation we show that certain gravitational interactions monomials can be consistently decoupled from the renormalization group (RG) flow and reproduce recent results on the asymptotic safety conjecture. The non-perturbative RG flow of non-local extensions of the Einstein-Hilbert truncation including $$\int d^dx \sqrt{g} \ln(R)$$ and $$\int d^dx \sqrt{g} R^{-n}$$ interactions is investigated in detail. The inclusion of such interactions resolves the infrared singularities plaguing the RG trajectories with positive cosmological constant in previous truncations. In particular, in some $$R^{-n}$$-truncations all physical trajectories emanate from a Non-Gaussian (UV) fixed point and are well-defined on all RG scales. The RG flow of the $$\ln(R)$$-truncation contains an infrared attractor which drives a positive cosmological constant to zero, thereby providing a dynamical explanation of the tiny value of Lambda observed today."

Last edited: Jan 22, 2008
4. Jan 22, 2008

### marcus

This is an important paper of Percacci, which Reuter cites in his talks. It is probably the strongest evidence yet found for the existence of the UV fixed point of the RG flow

http://arxiv.org/abs/0705.1769
Ultraviolet properties of f(R)-Gravity
Alessandro Codello, Roberto Percacci, Christoph Rahmede
4 pages
(Submitted on 12 May 2007)

"We discuss the existence and properties of a nontrivial fixed point in f(R)-gravity, where f is a polynomial of order up to six. Within this seven-parameter class of theories, the fixed point has three ultraviolet-attractive and four ultraviolet-repulsive directions; this brings further support to the hypothesis that gravity is nonperturbatively renormalizable."

You can see why this is in a certain way a new field, even though Reuter has been working on it since 1998. It is only last year that this Percacci paper came out, and I expect quite a few people would only begin to notice based on this result.

5. Jan 22, 2008

### BenTheMan

Right, but none of this answers any questions as to why you can do this with gravity but not with, say, Fermi theory. Both have a coupling constant with the same mass dimensions, and both are (were) considered effective field theories. Except in the case of Fermi theory, the completion is clear (the SM), whereas in gravity, Reuter is telling me that there is no completion, and it is non-perturbatively renormalizable (whatever that means).

6. Jan 22, 2008

### marcus

You've put your finger on what you need to understand in order to grasp what is being talked about.

If your mind stays stuck in a perturbation theory context then you will not understand what any of these people (Saueressig, Percacci, Reuter etc) are saying, and you will necessarily be confused.

the other theories you are talking about are normally handled perturbatively.
I may be wrong of course---I may not understand what you are trying to say about these other theories. But it sounds to me like you may be trying to compare apples and oranges.

Maybe someone else can clarify for you.

7. Jan 22, 2008

### BenTheMan

I don't quite understand the difference. In one case, the Fermi constant goes like one over the W mass squared, and the Newton's constant goes like one over the Planck mass squared. Based on the effective field theory approach, I don't quite understand how you can claim the two are different---new physics comes in at the W mass and provides a natural UV completion for the theory.

8. Jan 22, 2008

### marcus

Have you looked at any of the papers I linked to?

9. Jan 22, 2008

### BenTheMan

I don't have time to read these things...I was hoping that someone could offer me a (semi-)technical explanation instead of linking me to a bunch of papers that I could have found on the arXiv myself.

10. Jan 23, 2008

### Haelfix

What Reuter originally did was to truncate gravity to first order (not surprisingly as divergences already appear at two loop), and then apply the 'exact renormalization group flow' to this quantity (see eg papers by Polchinksi which pioneer this method). Eg cosmological constant + (Einstein Hilbert terms + 1 loop contributions).

Distler explains the scheme here: http://golem.ph.utexas.edu/~distler/blog/archives/000648.html

Its a bit of a clever trick that you can indeed apply to say Fermi theory (though you won't find anything interesting), and it does give some control on the perturbative beta function, as this is now encoded at 1 loop, which he then uses to surmise a fixed point.

But... This is not a *proof* of a fixed point, but more of a hypothesis (that Reuter then goes on to test in a variety of circumstances, including strong coupling regimes). Indeed the effective action is not in the same universality class as the full shebang a priori, and there is absolutely no guarentee that it works at strong coupling (see Distlers objection) as you still need to consider the beta functions for the infinite set of couplings in the effective lagrangian and these are by no means guarenteed to vanish in the UV.

Last edited: Jan 23, 2008
11. Jan 23, 2008

### Demystifier

I think that the correct answer to your question is slightly philosophical. A priori, there is no reason to treat gravity in a different way than Fermi theory. However, many theoretical physicists find the classical Einstein theory of gravity so beautiful and elegant that they do not like the idea that such a beautiful theory is nothing but an approximate effective theory. For that reason, they desperately want to save that theory as a basis for quantum gravity as well. On the other hand, the Fermi theory is not that beautiful, so no one is too unhappy when it turns out that this theory is merely an effective one.
Is there a way to accept that the Einstein theory is merely an effective theory without sacrificing the elegance? Only if we replace it with something that seems even more elegant. And that is why string theory is believed by many physicists to be the right stuff, even without any observational evidence for it.

Of course, should the criteria of elegance and beauty be taken as valid scientific criteria, it's up to you. If you ask me, I think it should, but only as a guiding principle when no other scientific evidence is available. Quantum gravity is certainly one of such branches of physics.

Last edited: Jan 23, 2008
12. Jan 23, 2008

### Haelfix

The whole discussion hinges on the use of Polchinksi's form of the exact renormalization equation. Its primary purpose was to remove an approximation in the standard Wilson derivation (eg it takes into account irrelevant couplings), and it has a bit of a better bound on suppressing fluctuations above the cutoff scale (which is now smooth as opposed to sharp) as opposed to other exact renormalization group techniques. The issue is whether or not it sees nonperturbative UV effects and not just the IR ones since we have a regulator that is only perturbative in nature (eg its not on a lattice).

In the comment section of Distlers blog there was an interesting discussion arguing that it does not, and that make good sense, especially when we do not have any a priori knowledge about how much control we have in the UV. In general we expect (and this can be shown explicitly in exactly solvable models) quite a bit of extra nonperturbative corrections that are non analytic at the locus and that perturbation series will miss order by order.

13. Jan 23, 2008

### marcus

that was in 1998
as of 2007, the truncation is now at SIXTH order

the fixed point lies on a 3D hypersurface called the critical hypersurface
roughly speaking the fixed point is an attractor from three directions, and repells in four directions

Last edited: Jan 23, 2008
14. Jan 23, 2008

### marcus

Many people seem slow in registering this paper. I can understand this because it took me a while. My interest in the work of Reuter, Percacci, Saueressig etc was slight until I began to realize there was considerable suggestive, if not conclusive, evidence for the existence of the fixed point.
It could still prove NOT to exist, but there's reason to be interested in the approach. Another feature is that it looks like there might be only THREE attractive dimensions which means that there are only three parameters which need to be determined experimentally. If you get those numbers right then you are on track and get drawn in towards the fixed point. The physical significance of those three parameters is interesting to consider---one might be newton's G, one might be the cosmological constant. what is the third parameter?

15. Jan 23, 2008

### shalayka

If one keeps in mind the repulsive nature of the strong force at very close distances (http://arxiv.org/abs/nucl-th/0611096), then something like this may be a starting point:

Attractive:
1) Gravitation of spherically symmetric sources (stars)
2) Gravitation of axially symmetric sources (galaxies, dark matter)
3) Gravitation of minimally symmetric sources (filaments, dark matter)

Repulsive:
1) Weak nuclear force
2) Electromagnetic force
3) Strong force
4) Cosmological expansion

Last edited: Jan 23, 2008
16. Jan 23, 2008

### Coin

So... I am trying to understand this. As far as I get what you and Distler are saying, the idea is that this one-loop renormalization method is physically accurate in SOME models, but not ALL, and the models in which it is accurate are not models which we expect to find describing quantum gravity. The complaint however seems to be not that the model for quantum gravity CANNOT be one where the one-loop renormalization trick works, but that such a model would be contrived and there is no justification for adopting one.

Although this seems like a very good reason to expect the one-loop normalization approach wrong, it doesn't seem to rule it out completely since we don't really seem to know what the proper underlying model for quantum gravity is. So I'm curious-- what if, just for the sake of argument, we ASSUME the one-loop normalization approach is correct, and therefore spacetime and/or gravity must be described by some model to which the one-loop normalization hack can be applied? If this were true, what would it tell us? What if, for example, spacetime were "on a lattice"-- or rather, is there any choice of model in which the sentence "the regulator is on a lattice" can be made sensical in the qg arena? Alternately, what if we just assume some mechanism must exist which allows us to truncate to a fixed number of terms in the effective lagrangian, what would postulating the existence of that mechanism tell us about physics?

Please excuse me if this question is a little garbled :)

Last edited: Jan 23, 2008
17. Jan 23, 2008

### marcus

It sounds as if Haelfix and Distler are saying the same thing and that neither understands what has been happening.

In fact there IS reason to expect that only three constants are non-zero.
The conjecture is that at the UV fixed point you simply don't have an infinite list of coupling constants.

The conjecture is, once you get those three numbers right, the law is CORRECT.

There are various sorts of evidence for this, but the one that I've already pointed to here is the 2007 paper by Percacci Codello Rahmede where they tried out varying 7 parameters (a 6 degree polynomial in the curvature) and found that only the three lowest were nonzero. Reuter and the other already knew about those first three.

So you get the first three right and then you look at the next four and you find they vanish. OK what is your guess about the next? Very likely it is zero too. and it is just a matter of doing some more calculation to determine that.

It would seem there are people who are so used to looking at perturbation series that they have the illusion they are confronted with an infinite series of coupling constants even when they aren't!
What they've been doing is expanding around the wrong fixed point, so they get an endless bunch of coupling constants.

Start at the right fixed point and you may well avoid all that mess.

Last edited: Jan 23, 2008
18. Jan 23, 2008

### Coin

Well to the extent this is true I think Haelfix was basically paraphrasing Distler? The Distler quote is from the post Haelfix linked.

Hm, but that's only true if there is a UV fixed point, correct? I think Haelfix/Distler understand the argument here but are doubtful the UV fixed point exists.

I mean-- I don't know if I am understanding what you're saying right: Is the conjecture that given a fixed point, exactly those 3 terms will be non-negligable? Or is it known that if there is such a fixed point there will be exactly 3 relevant terms, and the conjecture is that the fixed point exists?

So are you saying there's a potentially soluble calculation that could show that terms > 3 in this effective lagrangian are zero?

If they did finish this calculation and proved, oh look at that, terms seven through infinity come out to zero-- would this prove the existence of the UV fixed point? Or would it prove that, if we assume the existence of the UV fixed point, gravity is renormalizable via this method?

19. Jan 23, 2008

### marcus

No Coin, what i am saying is something you may not recognize as kosher physics at all!
This is not an analytical calculation! I am not at all expert in these matters so I could be wrong, but what I think is that they did a NUMERICAL calculation. They ran the renormalization group flow using a computer.

Admittedly there are pitfalls when you use numerical calcuation, because you do not have infinite precision. But also it opens up new opportunities. I am talking about the Percacci Codello Rahmede paper. I will go back and check---perhaps I'm wrong and they were working analytically (as opposed to numerically).

Numerically they can never prove this! If they used a giant supercomputer and found that terms #4 thru #20 went to zero, it would still not PROVE anything. Because nature could be tricking us and term #21 might be significant!

I think what they are doing is, however, bold and exciting. It is using a computer to explore an infinite dimensional theory-space.

they explore the infinite dimensional space by projecting it down to----say, 6-degree polynomicals----a seven dimensional space and cruising around in that space.

and then there they find a 3D hypersurface on which the UV fix point is an attractor, so if you get on, it homes in on the fixed point.

It is fascinating to think that these structures, which one discovers empirically using the computer, might actually exist in nature. that this fixed point which for over 6 years keeps re-appearing, keeps coming up in study after study, might actually exist for real, and be the real theory. And always it is the computer sniffing the thing out for them, operating in effect on its own by an iterative process outside their direct control.

The aim is to have a predictive theory. To get the fixed point identified precisely enough so that one can use that action to calculate with and make predictions. We'll see.

To me the most worrying thing is not what Distler says (which I doubt matters much because of where he's coming from) but what Frank Saueressig says. His latest paper, with Machado, for me puts the UV fixed point in question. I am somewhat on pins and needles until the questions raised in that December 2007 paper are resolved.

20. Jan 25, 2008

### Haelfix

Coin. There are basically two seperate issues with what Reuter is doing.

One is that he truncates the series at some order (1, 7, 1000, it doesnt matter). This is *fine*, people do this all the time, however the theory that remains is an effective one, and not *a priori* in the same universality class as the full theory.

Now, the way that it can be *sometimes* in the same class, is if all those infinite counterterms that are being discarded have a certain pattern within them such that they cancel or at least are bounded to within some finite range. This happens when there is a fixed point within the theory. Basically, for a given truncated theory (gravity or other), there can be roughly three things that can happen. One is there is no fixed point (either in the effective truncated lagrangian or in the full renormalized lagrangian), two, there can be a fixed point in the effective truncated lagrangian but not the full renormalized lagrangian, and vice versa or both. The point is those last two options need not be the same theory, and absent some other way of solving the full path integral, its very hard to *know* the difference since we have no knowledge about what quantum gravity is. Reuters *hypothesis* is that they are coincident.

The other seperate issue that Distler mentions is I admit, a bit technical. It has to do with Polchinski's exact renormalization group method. Basically (very roughly) the technique starts by imposing a regularization cutoff that is a smooth function as opposed to the standard hardcutoff, the idea being eventually you send it to infinity after you have done the calculations and filtered out the high frequency modes. The whole point of using renormalization group flow is that it tends to give you some information about the nonperturbative behaviour of the theory in question. Now the issue is that this particular choice of regularization obviously lives in the perturbative sector, so consequently it will miss nonperturbative effects in the UV (but not the IR since its sensitive to these). You can contrast that with using a lattice regulator, which clearly is UV nonperturbative as you take infinite volume and zero lattice spacing limits. The point being that whatever fixed point you may find, could be spoiled by nonperturbative effects (even instanton like ones if you go to Euclidean space) and absent some degree of control from somewhere else, that remains a big assumption. (im glancing over a whole host of issues, like renormalon behaviour and so forth).

Anyway, you can see that its a hypothesis. And thats fine, but it should be clear there is good reason to be skeptical ipso facto and the burden of proof is high (we are after all talking about a full solution to QG if he were correct). I mean you can use the same technique for other nonrenormalizable field theories, as the OP originally asked. For instance certain quenched QCD models are probable with exact renormalization group flow methods.

Last edited: Jan 25, 2008