# Some confusion about asymptotic safety

1. Jan 2, 2010

### Physics Monkey

I'm trying to understand why asymptotic safety is a reasonable approach to quantum gravity.

Here is my present understanding:

1. Asymptotic safety is roughly the statement that perturbative gravity (by which i mean a spin 2 field) has a UV fixed point where it becomes like any other conformal field theory.

2. As a conformal field theory, the UV fixed point has a particular scaling for the density of states as a function of energy which follows from conformal invariance.

3. In quantum gravity one expects (essentially because of large black holes) that the density of states at very high energy is different from that of a conformal field theory in the same dimension. For example, in anti de Sitter space the density of states is that of a conformal field theory in one lower dimension (holography).

I don't see how 1 and 2 can possibly be compatible with 3. Nevertheless, I assume people in asymptotic safety have an answer to this kind of concern. Perhaps I have misunderstood what asymptotic safety is, for example, maybe the theory is not an ordinary quantum field theory as I have assumed. Does anyone know?

2. Jan 2, 2010

### marcus

Asymptotic safety is often referred to as nonperturbative renormalizability.
In that sense it doesn't deal with perturbative gravity.

So your interpretation of what AS is seems to drag us out of the familiar context of AS.

If perturbative gravity is a spin 2 field, what geometry is field defined on?

If you are perturbing the geometry, what is the chosen initial geometry that you are perturbing around?

3. Jan 2, 2010

### marcus

I guess the standard mainstream expert on AS that one would turn to, for an up-to-date review and FAQ is Robert Percacci.

He has a website and FAQ and wrote the chapter on AS in the latest book on quantum gravity approaches.
He organized the first conference on AS---it was held November 2009 at Perimeter Institute.

Steven Weinberg only recently came back into AS research. And he says he did so primarily because of a couple of papers by Percacci, which put things on a more solid footing. So although Weinberg had the idea (back in 1976) it has gone through changes and your best bet, if you want to learn about it, is to read Percacci's review.

Here is the Asymptotic Safety FAQ:
http://www.percacci.it/roberto/physics/as/faq.html

Here is the main Asymptotic Safety website index:
http://www.percacci.it/roberto/physics/as/

Here is the AS bibliography:
http://www.percacci.it/roberto/physics/as/biblio.html

Here is Percacci's 2007 review:
http://arxiv.org/abs/0709.3851

Last edited: Jan 2, 2010
4. Jan 2, 2010

### Finbar

Hi. So I was having this discussion on a different thread.

Large black hole is the prediction of general relativity which is the IR limit of asymptotically safe gravity. When the centre of mass energy of a scattering experiment is E>>M_pl and all the energy is contained within a radius r_s~2GE we expect a black hole to form. But note that r_s>>l_pl therefore the relevant physical scale of the black hole defined at the horizon is IR. Therefore we do not expect that the black hole to be described by a conformal field theory. It is only at small radial distances r<<l_p close to r=0 that the UV fixed point controls the physics.

5. Jan 2, 2010

### marcus

Finbar, this is a good opportunity for me to ask you for some explanation and to get educated on how you think of this.

In AsymSafe gravity, newton's G goes to zero as the characteristic momentum scale k appropriate to the experiment gets large.

Shaposhnikov and Wetterich have an estimate of how G(k) behaves. Your 2GE may be incredibly small. Don't we need to actually do some numbers?

6. Jan 2, 2010

### marcus

I don't see how you get this. You mentioned a scattering experiment. I see that as, for instance, a collider experiment. In AsymSafe gravity we have to choose a scale k. Our hunch is it doesn't matter there are various possible choices, Weinberg mentions several including defining k = the momentum transfered in the collision.

So as the energy E goes up, k goes up. G(k) goes to zero.
I would expect the schwarzschild radius Rs to be much less than the planck length. But that is just an intuitive guess---we should put in Shaposhnikov's numbers and actually see what happens.

But you say that Rs is actually quite large! Much larger than the infrared planck length that we are used to. This is surprising.

It seems to me that you have gone around in a circle. You ramp the collider energy up above Planck energy so you are way way UV. But then you turn around and say the real physical scale is now IR. This is paradoxical. It seems to me that it does not work in the framework of AsymSafe, because it fails to take account of the running of newton's constant.

7. Jan 2, 2010

### Physics Monkey

marcus,

I appreciate you attempting to help with the links, but I don't think they really address the point. For example, the FAQ uses all the ordinary language of quantum field theory and renormalization both of which seem somewhat inappropriate in a quantum theory of gravity.

More specifically, you mention issues about "what geometry", etc, but doesn't the language of renormalization i.e. "high energy modes", "UV fixed point", etc depend strongly on some kind of background notion of energy and time? Its not so clear to me that asymptotic safety isn't already using a several of background notions. Perhaps these fall away in the end, but I'm not so sure.

Finbar,

Of course I agree that the large black hole is an infrared phenomenon. This is presumably why we think we understand it pretty well. Nevertheless, I think the high energy scattering experiment you mention should tell us something about the high energy density of states of the theory (by definition, almost?). Don't you think the result of a high energy scattering experiment would be different between an otherwise ordinary conformal field theory and quantum gravity (no matter what the conformal field theory is)?

But perhaps the scattering experiment (or my analysis of it) is too naive. Phrased more abstractly, doesn't the conformal field theory just have too many degrees of freedom?

8. Jan 2, 2010

### marcus

Fall away is a good image. Asymptotic Safety is work in progress and the last few talks and papers by Reuter have tended to focus on arguing why AS is background independent.

Naively this contradicts one's common sense for the reason you mentioned. And it can actually get a little tedious when he butts his head against the problem again and again. But at least he and some collaborators have made some progress on it.

Actually he raised the issue already at Loops 2007 and offered an argument for AS background independence.

What he tries to show, and various collaborators, is that sure you have to start with a crutch metric to get things going, but it doesn't matter which metric you start with. In the end after running the ERG flow, he can throw the initial crutch metric away. This is somehow "morally" tantamount to background independence.

My feeling is we don't know what Asymptotic Safety applied to gravity is, as yet. We don't know what the scale k really signifies. We don't know why couplings run. I assume that in order to get a valid theory of gravity the crutch metric will indeed "fall away" as you say.
Your feeling is you are "not so sure".

I'm hopeful, but also not sure.

Last edited: Jan 2, 2010
9. Jan 3, 2010

### atyy

The AS indications are that the fixed point is at positive cosmological constant, so we'd be in de Sitter space.

10. Jan 4, 2010

### Physics Monkey

atyy,

Thanks for the information. I have a few comments.

1. Presumably even if the UV fixed point has positive cosmological constant, the cosmological constant could flow as one moves to lower energies. It is the low energy effective action which determines the classical solutions, at least in the usual renormalization story. Very naively this suggests one could still have anti de Sitter or flat space backgrounds.

2. If 1 is not correct and asymptotic safety only describes de Sitter quantum gravity (whatever that means), then I would regard that as a defect of the theory. For example, we know quantum gravity exists in anti de Sitter space via holography.

11. Jan 4, 2010

### atyy

But then we'd be away from the presumed conformal fixed point, so there wouldn't be the worry of the wrong entropy scaling of a CFT.

Well, at least incomplete from the point of view of making a list of all possible theories of quantum gravity. OTH, isn't AdS/CFT not yet known to produce any cosmology consistent with our own?

BTW, have you seen the interesting discussions at http://golem.ph.utexas.edu/~distler/blog/archives/001585.html and http://golem.ph.utexas.edu/~distler/blog/archives/002140.html ?

Last edited by a moderator: Apr 24, 2017
12. Jan 4, 2010

### Physics Monkey

atyy,

I believe that regardless of the background, the high energy scattering amplitudes are controlled by the high energy fixed point. So even if its true that the low energy cosmological constant can potentially being negative, high energy scattering should still be controlled by the high energy conformal field theory. This argument seems to suggest that no matter the low energy background, the high energy density of states should be "universal" in the asymptotic safety setup. I feel this still leaves me with a contradiction.

You're right as far as I know that gauge/gravity duality has yet to produce any realistic cosmology. Despite this, I find it interesting partially because it appears to be a totally well defined quantum theory of gravity (with certain special asymptotic behavior).

Thanks also for the links to those dicussions. They were interesting, although I was disappointed to find that the black hole issue did not seem to be addressed after it was mentioned.

13. Jan 4, 2010

### Finbar

To get r_s>>l_p I'm just using the classical Schwarzschild relation r_s = 2GE where I have equated the centre of mass energy E with the mass. G is something like G= (l_pl)^2 the planck length squared. So if E>>m_pl=1/l_pl therefore r_s/l_pl = 2E/m_pl>>1.

Clearly it doesn't make sense to associate E with k. E is only the centre of mass energy it doesn't say anything about impact parameter b i.e. how "close" the particles come. Something more like k~ E^c b^-a would make more sense where a and c are positive constants say c=1 b=2(with some approiate power of G to give it the right dimensions).

14. Jan 4, 2010

### Physics Monkey

One possibility that occured to me is the following. When studying the perturbative physics of a spin 2 field $$h$$ on Minkowski space $$g_0$$, one finds that the effective metric is really something like $$g = g_0 + h$$. Physically, the spin 2 field couples universally (a la Weinberg) and hence can be interpreted geometrically.

I don't want to push this too far, except to say that perhaps something similar could be imagined to happen in the non-perturbative setup of asymptotic safety. The details are very negotiable, but I feel like something like this has to happen in order to reconcile what we know about quantum gravity with the statement that gravity is described by a conformal field theory at high energy. In particular, the scaling of degrees must become "area-like" rather than "volume-like" whenever an effective geometry description is appropriate.

15. Jan 4, 2010

### marcus

Yes I could see you were using the classical formula for the Schwarzschild radius, but you were using the IR value of G.

This is where I have doubts about the analysis. When you say "l_pl" you clearly mean the standard Planck length as we know it. We have to be clear about this. G(k) runs. But the Planck length does not run.

The Planck length is $$\sqrt{G \hbar/c^3}$$
where G, in this expression is the usual low-energy value of Newton's constant.

But in the formation of a black hole, what matters is the strength of the coupling at that scale. In other words the relevant value of G, in your imagined scattering experiment, is G(k).

In that case, isn't the equation you wrote wrong? It seems that

the relevant physical quantity G(k) is not equal to (l_pl)^2.

Last edited: Jan 4, 2010
16. Jan 4, 2010

### atyy

Agreed. I do think AdS/CFT is the most fascinating thing in quantum gravity! And that the black hole entropy issue and Asymptotic Safety is unresolved, and that both suggest that if Asymptotic Safety is to work, something very interesting must happen.

In the Asymptotic Safety literature it is often said that the anomalous dimension of gravity must be 2 if there is a UV fixed point, and that this means that gravity is somehow 2 dimensional near the fixed point - but I have no idea if this is related to the points you brought up.

17. Jan 4, 2010

### marcus

The renormalization flow, projected onto the G Lambda plane, has been studied quite a lot. Realistically speaking, there is no indication in all this work that the cosmological constant goes NEGATIVE as one one moves to lower energies.

In the plots (generated numerically with the beta functions) one can see trajectories where Lambda(k) does go negative---but on these trajectories G(k) does not behave right.

Have you looked at this plots? They usually have the dimensionless version of Lambda on the horizontal axis, and the dimensionless G(k) on the vertical.

A good introduction to the subject would be to watch Saueressig's talk at the Perimeter conference. For instance around slide 47, if I remember, there is one of these flow plots.

You seem to have the fixed opinion that a theory of gravity can not be right unless it fits into the AdS/CFT preconception. Isn't this a rather narrow view? Couldn't it be the case that Nature likes deSitter quantum gravity, and knows what it means, even if we don't?

In case you want to watch Saueressig's talk, here is the link:
http://pirsa.org/09110047/
The plot I was thinking of is actually slide #48. The link here lets you get the slides PDF separately, or watch the video, whichever you choose.

Last edited: Jan 4, 2010
18. Jan 4, 2010

### Finbar

Yes this is all true but I expect k~1/r_s so when r_s is large k is small thus I expect G(k=0)= (l_pl)^2. My point being that its only when r_s is small that I have to take account of the running of G(k).

19. Jan 4, 2010

### Physics Monkey

marcus,

I would actually be quite happy to see a consistent theory of quantum gravity in de Sitter. To the extent that asymptotic safety purports to provide such a theory, I am interested in it. And I definitely agree with the general sentiment that nature is pretty creative!

I only consistently mention holography in this context because it is a precise version of quantum gravity which makes sense and from which we are learning a great deal. For example, my comment about density of states at high energy makes complete sense in the holographic setup and has a satisfactory resolution.

My potential complaint about asymptotic safety could be phrased this way. If the starting point is basically only Einstein's equations, and the conclusion is that all we get is de Sitter, then I would be worried. I would be worried because we know that we can have Einstein's equations (approximately), anti de Sitter, and quantum gravity because of holography. So if "quantizing Einstein's equations" seemed to exclude anti de Sitter, I would feel something wasn't working right.

20. Jan 4, 2010

### marcus

Exactly! and in very high momentum particle collisions, the Schwarzschild radius can be expected to be extremely small.

So one would, in those situations be taking account of the running of G(k).

(And indeed that running is the reason that r_s is small, since r_s depends on G(k), which is going to zero.)

21. Jan 4, 2010

### Finbar

One expects the Schwarzschild radius to be small if E is small r_s=2GE. So actually E needs to be small but the energy density, something like ~E/b^3, needs to be large. When E~M_pl and b~l_pl then we're in the real UV QG regime and we can no longer use the reasoning of classical gravity because G(k) is running.

22. Jan 4, 2010

### marcus

I think we are beginning to reach an understanding on this.
At any given scale k, the physically meaningful Schwarzschild radius associated with a given energy E is r_s = 2G(k)E.

This is the radius such that if you get the energy E concentrated within that radius, a black hole will form. It naturally depends on the G(k) that is appropriate to that scale.

In what you have written so far, you have omitted to show the k dependence and you have written simply G instead of G(k), but I assume you know that when you write G it is not the low-energy version of Newton's constant GN that one would look up in a handbook.

We can put in some numbers. Shaposhnikov and Wetterich say that
G(k) ~ 1/k^2
and that is something any of us can figure out by ourselves, we don't need S&W to tell us.

Now as a rough back-of-envelope, suppose in a collider experiment we have
E ~ k
The momentum scale k is going up more or less proportionally with the collision energy. Say.

OK, then the Schwarzschild radius 2G(k)E, which determines the physical issue of whether a black hole can form, is proportional to 2 times 1/k^2 times k.

The Scharzschild radius is therefore proportional to 1/k.

It goes as 1/k while k is going to infinity, so it goes to zero.