Asymptotic Safety paper thanks Jacques Distler

[marcus]

"We thank J. Distler and R. Percacci for animated discussions which triggered this investigation. Furthermore, we are grateful to J.P. Blaizot, A. Codello, R.
Loll, E. Manrique, and M. Reuter for useful conversations..."

It's interesting to contemplate the different ways that progress in research can come about. Those familiar with Jacques Distler can probably imagine what those "animated discussions" were like.

MTd2 has thoughtfully called our attention to an excellent and I believe important paper by young researchers based at Perimeter in Canada, Loll's group in Utrecht, and Gif-Sur-Yvette in France.

http://arxiv.org/abs/0902.4630
Taming perturbative divergences in asymptotically safe gravity
Dario Benedetti, Pedro F. Machado, Frank Saueressig
16 pages
(Submitted on 26 Feb 2009)
"We use functional renormalization group methods to study gravity minimally coupled to a free scalar field. This setup provides the prototype of a gravitational theory which is perturbatively non-renormalizable at one-loop level, but may possesses a non-trivial renormalization group fixed point controlling its UV behavior. We show that such a fixed point indeed exists within the truncations considered, lending strong support to the conjectured asymptotic safety of the theory. In particular, we demonstrate that the counterterms responsible for its perturbative non-renormalizability have no qualitative effect on this feature."

As you see they are beginning to include matter---a general trend now in several quantum gravity approaches. Another good extra feature of this article is that if you want a clear readable introduction and overview of the Asymptotic Safe gravity approach, they give it here in the first section of their article with selected references.

Those who read in this forum are probably aware of the remarkable coincidence that three actively pursued approaches seem to agree on a contraction of dimensionality at microscopic scale. In all three (Loop, Triangulations, and UV-Safety) the microscopic geometry seems to become fractal-like and the spacetime dimensionality goes from 4D down to around 2D at very small scale.
Saueressig earlier collaborated with Martin Reuter who showed this for UV-safety gravity. Loll and co-workers showed it for Triangulations gravity. Benedetti recently posted a paper showing it for a broader class of theories giving some hint as to why it happens. Modesto recently posted an argument explaining how it comes about in LQG.

In a full theory of quantum gravity, the dimensionality of space and spacetime are things one observes, they are not merely fixed by assumption. Dimensionality is something that can be measured experimentally and which does not necessarily always take on integer values. It can differ from place to place and be different at different scales. Of two measures often cited, one compares radii and volume, the other uses a diffusion process or random walk.

So the UV-Safe or Asymptotically Safe approach is currently of interest and it is especially nice to see this paper come out on it, provoked so to speak by the animated discourse of Jacques Distler.

 

[atyy]

Isn't the dimension in asymptotic safety hypothesized to be integer, not fractal?

 

[marcus]

Isn't the dimension in asymptotic safety hypothesized to be integer, not fractal?

I'll check. Fractals can have whole number dimension, can they not? What I've always heard is that asymptotic safe gravity (ASG) and CDT have the same or similar behavior.
Dimension continuously going down with scale, naturally it goes thru non-integer values.
But I have to check.

 

[atyy]

I'll check. Fractals can have whole number dimension, can they not? What I've always heard is that asymptotic safe gravity (ASG) and CDT have the same or similar behavior.
Dimension continuously going down with scale, naturally it goes thru non-integer values.
But I have to check.

Yes, you are right. The Niedermaier and Reuter living reviews article says it's "fractal-like".

 

[marcus]

Yes, you are right. The Niedermaier and Reuter living reviews article says it's "fractal-like".

Thanks for checking. Family time intervened and I didn't get around to it. Just got back.
There should be a first paper where they have this result.

Maybe this:
http://arxiv.org/abs/hep-th/0508202
Fractal Spacetime Structure in Asymptotically Safe Gravity
O. Lauscher, M. Reuter
20 pages
(Submitted on 26 Aug 2005)
"Four-dimensional Quantum Einstein Gravity (QEG) is likely to be an asymptotically safe theory which is applicable at arbitrarily small distance scales. On sub-Planckian distances it predicts that spacetime is a fractal with an effective dimensionality of 2. The original argument leading to this result was based upon the anomalous dimension of Newton's constant. In the present paper we demonstrate that also the spectral dimension equals 2 microscopically, while it is equal to 4 on macroscopic scales. This result is an exact consequence of asymptotic safety and does not rely on any truncation. Contact is made with recent Monte Carlo simulations."

Now although it doesn't say that, I would expect the dimensionality to decline more or less continuously with scale, so it would go thru lots of non-integer values. That is how it works with Loll's setup (referred-to here as "recent Monte Carlo simulations").

But I may be wrong. It has been a while since I looked at this Lauscher-Reuter paper.

Ah! here is something on page 8 about a smooth change from 4D down to 2D:

==quote==
As we discussed in ref. [3] the QEG spacetime has an effective dimensionality which is k-dependent and hence noninteger in general. Our discussion in [3] was based upon the anomalous dimension ηN of the operator... It is defined as ηN ≡ −k ∂k ln ZN k
where ZN k...
In a sense which we shall make more precise in a moment, the effective dimensionality of spacetime equals 4 + ηN . The RG trajectories of the Einstein-Hilbert truncation (within its domain of validity) have ηN ≈ 0 for k → 0
and ηN ≈ −2 for k → ∞, the smooth change by two units occurring near k ≈ mPl . As a consequence, the effective dimensionality is 4 for ℓ ≫ ℓPl and 2 for ℓ ≪ ℓPl .
==endquote==

k is an inverse scale parameter, so k going to infinity is the same as scale going down to very small. Somehow spacetime gets to looking more like soap-suds (2D) and less like a solid block of 4D cheese when you peer very close at it. But the transition is smooth. According to these people.
======================================

But now I see that was not the first time Reuter reported the result. An earlier paper was:
http://arXiv.org/abs/hep-th/0108040
Ultraviolet Fixed Point and Generalized Flow Equation of Quantum Gravity
O. Lauscher, M. Reuter
99 pages, latex, 11 figures
(Submitted on 7 Aug 2001)
"A new exact renormalization group equation for the effective average action of Euclidean quantum gravity is constructed. It is formulated in terms of the component fields appearing in the transverse-traceless decomposition of the metric. It facilitates both the construction of an appropriate infrared cutoff and the projection of the renormalization group flow onto a large class of truncated parameter spaces. The Einstein-Hilbert truncation is investigated in detail and the fixed point structure of the resulting flow is analyzed. Both a Gaussian and a non-Gaussian fixed point are found. If the non-Gaussian fixed point is present in the exact theory, quantum Einstein gravity is likely to be renormalizable at the nonperturbative level. In order to assess the reliability of the truncation a comprehensive analysis of the scheme dependence of universal quantities is performed. We find strong evidence supporting the hypothesis that 4-dimensional Einstein gravity is asymptotically safe, i.e. nonperturbatively renormalizable. The renormalization group improvement of the graviton propagator suggests a kind of dimensional reduction from 4 to 2 dimensions when spacetime is probed at sub-Planckian length scales."

That means he clearly came to this conclusion before Loll did. Loll's group only got the result in 2005. They were manufacturing little random universes in the computer and studying them by running random walks inside, and calculating radii and volumes etc. So they came across this experimentally, so to speak, this variation of dimensionality with scale. I doubt Loll would have been looking to make contact with Reuter's results (may not even have known about the 2001 paper).

Let's see, here is Loll's 2005. Let's see if she cites Reuter:
http://arXiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure, Phys.Rev.Lett. 95 (2005)
(Submitted on 12 May 2005)
"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.

YES! The first version of the paper, posted in May, does not mention Reuter and says "to our knowledge this is the first etc etc".
But apparently Reuter contacted them after that and so the revised version had a footnote and a reference to Reuter's 2001 paper!
I like it because it came to each of them as a surprise out of the blue.

 

[atyy]

I think it'd be very interesting if LQG was somehow related to AS, given the Modesto result. I'm actually not too surprised that they thanked Distler. Both him and Motl actually seem quite friendly towards AS, compared to LQG. I think this is because like string theory, AS is motivated by thinking of the Standard Model as an effective field theory, very much the "particle physics" viewpoint against which Rovelli contrasts the "relativists" viewpoint in his living reviews article.

 

[Chronos]

The fractal-like thing is very interesting and a good fit to observation. Given that spacetime, as we perceive, breaks down at quantum levels, sub-integer [fractal] spatial dimensions are logical.

 

[JustinLevy]

Does the GLAST (Fermi) results rule out such fractal structure? If not, could such experiments in principle (ie. if they could just go a couple more orders of magnitude) verify or rule out such structure?

Also, since LQG is supposed to be "just a quantized theory of GR" (ie. you still need to add mass and the standard model in by hand), then how could it not agree with the path integral quantization approach of CDT? Wouldn't that be like claiming the path integral approach gives a different result than the wavefunction/fields approach (which I thought was shown to be equivalent when Feynman presented the path integral approach)?

 

[MTd2]

In fact, Distler didn't "win" the discussion this time:

This is the original post:
http://golem.ph.utexas.edu/~distler/blog/archives/000648.html

Here probably lies the issues that led to the animated discussion that happened:

http://golem.ph.utexas.edu/~distler/blog/archives/001609.html
http://golem.ph.utexas.edu/~distler/blog/archives/000648.html

On a related note, this is the original article that gave birth to RGE, according to Distler:

Renormalization and Effective Lagrangians.
Joseph Polchinski (Harvard U.) . HUTP-83-A018, Apr 1983. 36pp.
Published in Nucl.Phys.B231:269-295,1984.

http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B231,269

There is a scan of the original preprint there.

 

[Haelfix]

Kinda cool that research was performed by looking at a blog discussion. Percacci mentioned this particular coupling as something worth trying, b/c it satisfied the criteria Jacques mentioned for a nontrivial term. All previous AS papers have used perturbatively safe terms to probe fixed point structures which of course is fishy.

Anyway the R^2 + matter term does not contain as much divergence structure as the G-S term, but at least its not a topological invariant or something that can be absorbed into a field redefinition.

We'll have to wait for the real paper to come out though to get the details. This one is short on specifics and just lays out the results like a PRL format.