Steven Weinberg offers a way to explain inflation

[marcus]

A more serious question:
Weinberg starts with the most general generally covariant action. But Krasnov has an even more general one. What is the difference? I might guess Weinberg has the most general generally covariant *local* action, but is Krasnov's non-local?

Now you're talking! This paper of Weinberg's is awesome. I'm trying to focus on it.
There seems to be two parts or two "stages" to the paper.
In the first he considers the "completely general generally covariant action" (middle of page 3)
He uses the symbol Lambda for the scale (not for cosmological constant). And he has an infinite series of couplings which run with Lambda. And he assumes he knows the beta functions for all these couplings and that they make the couplings converge to the fixed point. To actually calculate he would need to truncate, but he doesn't want to calculate, he wants to set up the formalism.

Then he goes to the second stage where he assumes the usual uniformity (homog and isotropy) associated with the classic Friedman model. It isn't clear to me at a level of detail how he gets from stage one to stage two---from the full theory to the symmetry-reduced theory called FRW model that is normally used in cosmology. But he makes that transition, and then he can start talking about inflation. That part begins on page 5.

It also isn't clear how you define scale in a background independent manner. I think he
says at the top of page 5 that Lambda can be defined in either of two ways and it makes no difference which.

Either define the scale Lambda by limiting the loop diagrams at a certain level of complexity (akin to Rivasseau's idea of scale=complexity)
Or else define the scale as a momentum cutoff, which he notes is often denoted by letter k,
"a regulator term added to the action, or a sliding renormalization scale."

How exactly, if there is no background metric, does one define the scale?
I suspect this is just a minor problem, I may be the only one puzzled by it.

General covariance is a synonym for diffeomorphism invariance (as other parts of the community call it). Maybe someone can help us understand how the scale Lambda is defined in a diffeo invariant context.

 

[marcus]

If we assume an analogy to the case of the gases at critical point, ...

MTd2, I would urge you to contact Raoul Abramo about Weinberg's paper. Just ask questions. Don't interject your ideas. I think this paper of Weinberg is important and hot right now. The people at USP will want to be discussing its implications. Especially Abramo will want to discuss this. Or so I guess. If you can, get him to explain the significance, as he sees it.

They may have a journal club at the IFT that meets every week to discuss new papers. A discussion may be coming up about this paper. They would probably allow you as an interested outsider to sit in at the informal discussion meeting.

I could be mistaken, but I think this is the thing to focus one's attention on right now.

 

[Haelfix]

OK, I'm very confused. Is AS really incompatible with Asymptotic Darkness? AD means if you collide two things at high enough energy, you will form a big black hole, so the horizon will be pretty flat and semiclassical. I understand that AS seems to say that black holes will evaporate to a remnant (http://arxiv.org/abs/hep-th/0602159), whereas string theory seems to say black holes will evaporate completely (http://arxiv.org/abs/hep-th/0601001). But isn't that a different issue from AD?

The question of whether or not a black hole decays to a remnant or something like that is besides the point, why are we even talking about this? I just read the Bonnano-Reuter paper, and it says nothing about high energy quantum collisions, only that when the mass of the black hole is very low (after the Hawking radiation evaporates away most of the mass of the hole) that it turns off.. Quite on the contrary, it seems to agree with the relatively pedestrian notion that a black hole forms when the mass M is large enough (and you can arrange for collisions off arbitrarily high energy in this little thought experiment, makign the shock waves as big as you want, even making an astrophysical sized one if you want) and that indeed it remains more or less classical in that regime.

And there the scaling argument comes into play, b/c a local conformal quantum field theory cannot satisfy an area law.

If on the other hand, black holes do not form in the AS scenario at high energies (which I think none of the AS authors claim), then that indeed makes the point off the paper and you are back to trying to show which of the generic assumptions fail in the AD arguments. For instance, why the 1 graviton exchange eikonal regime ceases to be well described semiclassically and why it doesn't dominate the density of states.

 

[atyy]

The question of whether or not a black hole decays to a remnant or something like that is besides the point, why are we even talking about this? I just read the Bonnano-Reuter paper, and it says nothing about high energy quantum collisions, only that when the mass of the black hole is very low (after the Hawking radiation evaporates away most of the mass of the hole) that it turns off.. Quite on the contrary, it seems to agree with the relatively pedestrian notion that a black hole forms when the mass M is large enough (and you can arrange for collisions off arbitrarily high energy in this little thought experiment, makign the shock waves as big as you want, even making an astrophysical sized one if you want) and that indeed it remains more or less classical in that regime.

And there the scaling argument comes into play, b/c a local conformal quantum field theory cannot satisfy an area law.

If on the other hand, black holes do not form in the AS scenario at high energies (which I think none of the AS authors claim), then that indeed makes the point off the paper and you are back to trying to show which of the generic assumptions fail in the AD arguments. For instance, why the 1 graviton exchange eikonal regime ceases to be well described semiclassically and why it doesn't dominate the density of states.

Yes, I agree. Let me just paraphrase to see if I got what you are saying right: AD is a general argument goimg back to Bekenstein that suggests if AS works, then something interesting is happening maybe with the dimensionality or with asymptotically dS space. The Bonanno and Reuter papers don't address AD and are about something else.

 

[Finbar]

The question of whether or not a black hole decays to a remnant or something like that is besides the point, why are we even talking about this? I just read the Bonnano-Reuter paper, and it says nothing about high energy quantum collisions, only that when the mass of the black hole is very low (after the Hawking radiation evaporates away most of the mass of the hole) that it turns off.. Quite on the contrary, it seems to agree with the relatively pedestrian notion that a black hole forms when the mass M is large enough (and you can arrange for collisions off arbitrarily high energy in this little thought experiment, makign the shock waves as big as you want, even making an astrophysical sized one if you want) and that indeed it remains more or less classical in that regime.

And there the scaling argument comes into play, b/c a local conformal quantum field theory cannot satisfy an area law.

If on the other hand, black holes do not form in the AS scenario at high energies (which I think none of the AS authors claim), then that indeed makes the point off the paper and you are back to trying to show which of the generic assumptions fail in the AD arguments. For instance, why the 1 graviton exchange eikonal regime ceases to be well described semiclassically and why it doesn't dominate the density of states.

Let me try and explain the situation for high energy scattering and black holes in AS.

Classically when I have a energy E>>M_p located in a region of radius R<2GE a black hole will form. Where M_p is the Planck mass G is Newtons constant. But as E>>M_p we also have R_s>>l_p the Planck length where R_s is the radius of the black hole. So here we can neglect quantum gravity effects at the horizon and throughout most of the spacetime apart from at the singularity. So the semi-classical approximation is still valid.

The Black hole will then evaporate and the semi-classical approximation will break down once the energy E of the black hole falls to the Planck scale E~M_p. Here AS predicts that a remnant forms which stops the black hole from evaporating further.

On the other hand if we take if we begin with an energy E~M_p in a region R<2GE, where the curvature will be Planckian, we already cannot trust classical physics and AS predicts a black hole will not form.

I think a key point here is when we have to worry about QG effects. Note that it is not when E>>M_p but when the density~ E/R^3 is high this follows from the Einstein equations that relate the strength of the gravitational field with the energy density. If R~2GE then density ~ 1/E^2 so the smaller the black hole mass the more we need to worry about QG effects.

Another consequence of the density~1/E^2 is that it is indeed very "easy" to create black holes with a large energy who's formation can be described with classical physics.

 

[marcus]

The discussion has not been limited to black holes forming remnants. Bonanno's recent paper argues that BH simply do not form below a certain critical mass. This does not have to do with evaporation. But evaporation and remnants are also discussed in the same paper.

Right. Did you already cite Bonanno's recent paper? It's a good readable review and it mentions the 2000 result of Bonanno and Reuter to that effect.

http://arxiv.org/abs/0911.2727
Astrophysical implications of the Asymptotic Safety Scenario in Quantum Gravity
Alfio Bonanno
(Submitted on 13 Nov 2009)
"In recent years it has emerged that the high energy behavior of gravity could be governed by an ultraviolet non-Gaussian fixed point of the (dimensionless) Newton's constant, whose behavior at high energy is thus antiscreened. This phenomenon has several astrophysical implications. In particular in this article recent works on renormalization group improved cosmologies based upon a renormalization group trajectory of Quantum Einstein Gravity with realistic parameter values will be reviewed. It will be argued that quantum effects can account for the entire entropy of the present Universe in the massless sector and give rise to a phase of inflationary expansion. Moreover the prediction for the final state of the black hole evaporation is a Planck size remnant which is formed in an infinite time."
Comments: 28 pages, 6 figures. Invited talk at Workshop on Continuum and Lattice Approaches to Quantum Gravity. Sept. 2008, Brighton UK. To appear in the Proceedings

The point you were making is around the top of page 18. If the mass is below critical, no horizon exists.

I'm skeptical when I hear talk of imparting transplanckian energies to two particles and having them collide and form a black hole. It's speculative and has no clear connection with Weinberg's paper.

 

[atyy]

Classically when I have a energy E>>M_p located in a region of radius R<2GE a black hole will form. Where M_p is the Planck mass G is Newtons constant. But as E>>M_p we also have R_s>>l_p the Planck length where R_s is the radius of the black hole. So here we can neglect quantum gravity effects at the horizon and throughout most of the spacetime apart from at the singularity. So the semi-classical approximation is still valid.

The Black hole will then evaporate and the semi-classical approximation will break down once the energy E of the black hole falls to the Planck scale E~M_p. Here AS predicts that a remnant forms which stops the black hole from evaporating further.

On the other hand if we take if we begin with an energy E~M_p in a region R<2GE, where the curvature will be Planckian, we already cannot trust classical physics and AS predicts a black hole will not form.

Isn't AD limited to the case where E>>M_p? For example, Tong's notes say "Firstly, there is a key difference between Fermi’s theory of the weak interaction and gravity. Fermi’s theory was unable to provide predictions for any scattering process at energies above sqrt(1/GF). In contrast, if we scatter two objects at extremely high energies in gravity — say, at energies E ≫ Mpl — then we know exactly what will happen: we form a big black hole. We don’t need quantum gravity to tell us this. Classical general relativity is sufficient. If we restrict attention to scattering, the crisis of non-renormalizability is not problematic at ultra-high energies. It’s troublesome only within a window of energies around the Planck scale." http://www.damtp.cam.ac.uk/user/tong/string/string.pdf

So it's that case which leads to the information paradox and the suggestion that maybe gravity cannot be a local quantum field theory unless something interesting happens.

 

[Finbar]

Isn't AD limited to the case where E>>M_p? For example, Tong's notes say "Firstly, there is a key difference between Fermi’s theory of the weak interaction and gravity. Fermi’s theory was unable to provide predictions for any scattering process at energies above sqrt(1/GF). In contrast, if we scatter two objects at extremely high energies in gravity — say, at energies E ≫ Mpl — then we know exactly what will happen: we form a big black hole. We don’t need quantum gravity to tell us this. Classical general relativity is sufficient. If we restrict attention to scattering, the crisis of non-renormalizability is not problematic at ultra-high energies. It’s troublesome only within a window of energies around the Planck scale." http://www.damtp.cam.ac.uk/user/tong/string/string.pdf

So it's that case which leads to the information paradox and the suggestion that maybe gravity cannot be a local quantum field theory unless something interesting happens.

This is exactly my point "...the crisis of non-renormalizability is not problematic at ultra-energies" when E>>Mpl gravity the black holes are large and described by gravity in the IR. "It's troublesome only within a window of energies around the Planck scale".

AD is the assumption that gravity is not AS and hence gravity is not sufficiently strong to disallow black holes with a radius r<<lpl.


The information paradox is a different problem and AS still needs to deal with it. Personally I don't think the remnant picture is good enough if one assumes all the information is stored in the remnant and doesn't get out some how.

 

[marcus]

... It’s troublesome only within a window of energies around the Planck scale." http://www.damtp.cam.ac.uk/user/tong/string/string.pdf
...

I strongly agree. If there are any problems that are ready for us to confront they are on the way to Planck scale. This is the perspective that Nicolai adopted at the Planck scale conference. At Planck scale some new physics is expected to take over, his program is, if possible, to get all the way to Planck scale with minimal new machinery and have the theory testable.

And this range E < E_Planck is exactly where Bonanno's assertion applies. It is also where Roy Maartens and Martin Bojowald found, in 2005, that black holes could not form (given the Loop context).

We may in fact not have a problem. The sheer existence of black holes of less than Planck mass is questionable. There is no evidence that they exist, and there are analytical results to the contrary.

This is exactly my point "...the crisis of non-renormalizability is not problematic at ultra-energies" when E>>Mpl gravity the black holes are large and described by gravity in the IR. "It's troublesome only within a window of energies around the Planck scale".
...

I agree strongly again. I'm glad you made these points.

 

[atyy]

OK, looks like we all agree on the physics heuristics but maybe not the names of various hypotheses.

 

[atyy]

How exactly, if there is no background metric, does one define the scale?
I suspect this is just a minor problem, I may be the only one puzzled by it.

General covariance is a synonym for diffeomorphism invariance (as other parts of the community call it). Maybe someone can help us understand how the scale Lambda is defined in a diffeo invariant context.

They use a particle physicist thing called the "background field method". You pick a background, but the background is arbitrary. Take a look at http://arxiv.org/abs/0910.5167's discussion beginning before Eq 56 "We can write g=background+h. It is not implied that h is small." up to Eq 59 "Also the cutoff term is written in terms of the background metric ... where is some differential operator constructed with the background metric."

AS is basically not very rigourous (Rivasseau complained about this in a footnote in his GFT renormalization paper) and kinda hopeful, but my impression is that it's often that way in condensed matter. For example in Kardar's exposition at some point he says (I'm doing very free paraphrase) well, how do we know there's not non-perturbative fixed points - we don't, but luckily we can do experiments and they even more luckily match our perturbative calculations! He also says there are several different coarse -graining schemes which actually no one has proven are mathematically equivalent, but they all seem to match experiment, so we live in blissful ignorance! In condensed matter the predictions are "universal", so for example the critical temperature is different for all sorts of materials and the theory cannot predict the temperature - what it gets right is the critical exponent which seems to be independent of material and dependent only on symmetries and dimensionality. So I guess Weinberg and co are hoping for some such generic predictions.

 

[Finbar]

Just a note on possible confusion. When one says "high energy" in gravity it can be confused for "low energy" and vice versa. The reason is the following: Newton's constant is dimensionful. It has mass dimension [G]=-2 such that when I write GM this is a length or an inverse mass [GM]=[G]+[M] =-2+1=-1.

One consequence of this is the strange property of black holes that when I increase there mass their temperature drops T=1/(8 pi G M) i.e. they have a negative specific heat.

Other consequences of [G]=-2 are that the entropy of a black hole goes as the S=area/(4G) since G is the Planck area and the infamous power counting non-renormalizability of general relativity.

 

[atyy]

Does AS really need a fixed point? Could it live with, say, a limit cycle?

 

[Finbar]

They use a particle physicist thing called the "background field method". You pick a background, but the background is arbitrary. Take a look at http://arxiv.org/abs/0910.5167's discussion beginning before Eq 56 "We can write g=background+h. It is not implied that h is small." up to Eq 59 "Also the cutoff term is written in terms of the background metric ... where is some differential operator constructed with the background metric."

AS is basically not very rigourous (Rivasseau complained about this in a footnote in his GFT renormalization paper) and kinda hopeful, but my impression is that it's often that way in condensed matter. For example in Kardar's exposition at some point he says (I'm doing very free paraphrase) well, how do we know there's not non-perturbative fixed points - we don't, but luckily we can do experiments and they even more luckily match our perturbative calculations! He also says there are several different coarse -graining schemes which actually no one has proven are mathematically equivalent, but they all seem to match experiment, so we live in blissful ignorance! In condensed matter the predictions are "universal", so for example the critical temperature is different for all sorts of materials and the theory cannot predict the temperature - what it gets right is the critical exponent which seems to be independent of material and dependent only on symmetries and dimensionality. So I guess Weinberg and co are hoping for some such generic predictions.

If you use the back ground field method rigorously then (slightly paradoxically) you actually ensure background independence. In a sense you quantizing the fields on all backgrounds at the same time. Up until recently however it has not been done rigorously enough though.

The relevant paper is
http://arxiv.org/pdf/0907.2617

Also checkout

Frank Saueressig's talk at perimeter.

 

[Haelfix]

Let me try and explain the situation for high energy scattering and black holes in AS.

Classically when I have a energy E>>M_p located in a region of radius R<2GE a black hole will form. Where M_p is the Planck mass G is Newtons constant. But as E>>M_p we also have R_s>>l_p the Planck length where R_s is the radius of the black hole. So here we can neglect quantum gravity effects at the horizon and throughout most of the spacetime apart from at the singularity. So the semi-classical approximation is still valid.

The Black hole will then evaporate and the semi-classical approximation will break down once the energy E of the black hole falls to the Planck scale E~M_p. Here AS predicts that a remnant forms which stops the black hole from evaporating further.

On the other hand if we take if we begin with an energy E~M_p in a region R<2GE, where the curvature will be Planckian, we already cannot trust classical physics and AS predicts a black hole will not form.

I think a key point here is when we have to worry about QG effects. Note that it is not when E>>M_p but when the density~ E/R^3 is high this follows from the Einstein equations that relate the strength of the gravitational field with the energy density. If R~2GE then density ~ 1/E^2 so the smaller the black hole mass the more we need to worry about QG effects.

Another consequence of the density~1/E^2 is that it is indeed very "easy" to create black holes with a large energy who's formation can be described with classical physics.

I agree with most of what you just said (some technical quibbles aside), which is why I'm now very confused about what we are arguing about. B/c that's exactly what asymptotic darkness says. At transplanckian center of mass energy densities, as you go further and further into the UV you expect larger and larger black holes to form, which by the above argument implies that you are getting closer and closer to classical GR and QG becomes less and less relevant. Its immaterial what happens at the Planck scale (or say within an order or two thereof). No one knows exactly what goes on there, its only at much smaller energies, or conversely at much larger energies where we enter regimes that we can actually calculate in.

 

[marcus]

Agreement about "on the way" heuristics

I strongly agree. If there are any problems that are ready for us to confront they are on the way to Planck scale. This is the perspective that Nicolai adopted at the Planck scale conference. At Planck scale some new physics is expected to take over, his program is, if possible, to get all the way to Planck scale with minimal new machinery and have the theory testable.

And this range E < E_Planck is exactly where Bonanno's assertion applies. It is also where Roy Maartens and Martin Bojowald found, in 2005, that black holes could not form (given the Loop context).

We may in fact not have a problem. The sheer existence of black holes of less than Planck mass is questionable. There is no evidence that they exist, and there are analytical results to the contrary.
...

OK, looks like we all agree on the physics heuristics but maybe not the names of various hypotheses.

I think that's a good way to put it. IMO the reason for strong interest in the research community in what physics might be like in the range from say 10⁹ TeV up to 10¹⁶ TeV, is because of interest in high-energy astrophysics and the early universe.

The paradigm of colliding two particles at higher and higher energy, and equating that with physics, has become less interesting. It's a mental rut (almost an obsession) left over from the accelerator era. For example Weinberg was talking about inflation, which is a different business.

Different concepts, and different sources of data, come into play.

You could say that the range 10⁹ TeV up to 10¹⁶ TeV is the range from just over "cosmic ray" energy up to "early universe" energy.

A billion TeV is kind of approximate upper bound on cosmic ray energies. It's quite rare to detect cosmic rays above that level. And 10¹⁶ TeV is the Planck energy.

I would say this is a new erogenous zone for theoretical physics. The putative "GUT" scale, of a trillion-plus TeV, comes in there. But it impressed me that in Nicolai's new model there is no new physics at GUT scale. What Nicolai and Meissner have done is project a model which

*is falsifiable by LHC (once it gets going) and
*is conceptually economical, even minimalistic---based on existing standard model concepts,
*pushes the breakdown/blow-up points out past Planck scale, so it
*delays the need for fundamentally new physics until Planck scale is reached.

Whether Nicolai and Meissner's model is correct is not the issue here. What this example suggests is that this kind of conservative unflamboyant goal, this kind of unBaroque proposed solution, will IMO likely become fashionable among theorists. You could think of it as a reaction to past excesses, or a corrective swing of the pendulum.

This same economical or conservative spirit is the essence of what Weinberg is doing.
The new paper of his that we are discussing simply carries through on what he was talking about in his 6 July CERN lecture, where he said he didn't want to discourage anyone from continuing string research, but string theory might not be needed, might not be how the world is. How the world is, he said, might be described by (asymptotic safe) gravity and "good old" quantum field theory.

I assume that means describing the world pragmatically out to Planck scale (10¹⁶ TeV) so you cover the early universe. An important part of the world! And not worrying about whatever new physics might then kick in, if any does.
It's a modest and practical agenda, just getting that far, compared with worrying about putative seamonsters and dragons out beyond Planck energy. But of course that's fun and all to the good as well.

================================
In case anyone new is reading this thread, here is a link to video of Weinberg's 6 July CERN talk:
http://cdsweb.cern.ch/record/1188567/
It gives an intelligent overview of what this paper is about, where it fits into the big picture, and what motivates the Asymptotic Safe QG program (which he describes in the last 12 minutes of the video).

As a leading example of extending known and testable physics out to Planck scale, here is Nicolai's June 2009 talk:
http://www.ift.uni.wroc.pl/~rdurka/planckscale/index-video.php?plik=http://panoramix.ift.uni.wroc.pl/~planckscale/video/Day1/1-3.flv&tytul=1.3%20Nicolai
Here's the index to all the videos from the Planck Scale conference
http://www.ift.uni.wroc.pl/~rdurka/planckscale/index-video.php

 

[Finbar]

I agree with most of what you just said (some technical quibbles aside), which is why I'm now very confused about what we are arguing about. B/c that's exactly what asymptotic darkness says. At transplanckian center of mass energy densities, as you go further and further into the UV you expect larger and larger black holes to form, which by the above argument implies that you are getting closer and closer to classical GR and QG becomes less and less relevant. Its immaterial what happens at the Planck scale (or say within an order or two thereof). No one knows exactly what goes on there, its only at much smaller energies, or conversely at much larger energies where we enter regimes that we can actually calculate in.

Ok so we're getting somewhere. The problem is exactly the one I was pointing out in my post yesterday...

"Just a note on possible confusion. When one says "high energy" in gravity it can be confused for "low energy" and vice versa. The reason is the following: Newton's constant is dimensionful. It has mass dimension [G]=-2 such that when I write GM this is a length or an inverse mass [GM]=[G]+[M] =-2+1=-1. "

So for the argument about the non-renormalizability of gravity based on its scaling in the UV to be valid the "Asymptotic" in Asymptotic darkness and needs to be the same as the Asymptotic in Asymptotic safety. The reason it is false is because they are not for exactly the reason above.

If I have a large mass black hole M>>Mpl then r=2GM is large r>>lpl. This is what the "Asymptotic" in AD refers to and as you say you get closer and closer to classical GR. But the "Asymptotic" in AS refers to exactly the opposite limit that is when k>>Mpl where k=1/r this is where we are very far from classical GR and hence where we need a full theory of QG to answer any questions appropriately.

This is exactly the point David Tong is making

""Firstly, there is a key difference between Fermi’s theory of the weak interaction and gravity. Fermi’s theory was unable to provide predictions for any scattering process at energies above sqrt(1/GF). In contrast, if we scatter two objects at extremely high energies in gravity — say, at energies E ≫ Mpl — then we know exactly what will happen: we form a big black hole. We don’t need quantum gravity to tell us this. Classical general relativity is sufficient. If we restrict attention to scattering, the crisis of non-renormalizability is not problematic at ultra-high energies. It’s troublesome only within a window of energies around the Planck scale.""

So you see its not the AD scenario that I'm arguing about. Its that AD(an IR property of classical gravity) has any baring on AS/renormalizablity(which is a UV problem of quantum gravity).

 

[marcus]

...
So you see its not the AD scenario that I'm arguing about. Its that AD(an IR property of classical gravity) has any bearing on AS/renormalizablity(which is a UV problem of quantum gravity).

I was surprised anyone would bring up AD in this context. It seems like a red herring. Just distracts from considering the main burden of what Weinberg is doing.

Could it be that some people want to deny or dismiss the significance of AS suddenly coming to the forefront? It seems to me when something like this happens----greatly increased research, first ever AS conference, possible alliance with CDT and even Horava, connection with cosmology revealed---that the appropriate thing to do is to pay attention, and focus on it, not try to dismiss (especially not by handwaving about transplanckian black holes )

Haelfix, could you have been misled by someone with a vested interest that felt threatened by Weinberg's CERN talk, or recent paper, and is grasping at straws? or just blowing smoke? Be careful, maybe a bit more skeptical?

 

[atyy]

So you see its not the AD scenario that I'm arguing about. Its that AD(an IR property of classical gravity) has any baring on AS/renormalizablity(which is a UV problem of quantum gravity).

If AD suggests that gravity cannot be described by a "normal" local quantum field theory even at IR, then it suggests that AS may be wrong - only suggests, since Wilsonian renormalization indicates AS is a logical possibility - but in which case an interesting issue is in what way AS is not a "normal" local quantum field theory, even though the heuristic behind AS is that it is a "normal" local quantum field theory.

One thing I don't understand is that Weinberg's paper (the one being discussed in this thread) starts with the most general generally covariant Lagrangian (http://arxiv.org/abs/0911.3165) - but Krasnov has recently proposed an even more general generally covariant Lagrangian (http://arxiv.org/abs/0910.4028 ) - so presumably Weinberg's is less general - is that because Weinberg admits only local terms, while Krasnov's contains non-local terms? Usually renormalization flows don't generate non-local terms, I think, and naively I would expect the same for AS, but is that true?

Edit: Krasnov says his terms are all local - so what is the difference between his stuff and AS?

Litim's http://arxiv.org/abs/0810.3675 says "A Wilsonian effective action for gravity should contain ... possibly, non-local operators in the metric field." So I guess non-local terms can come about through coarse-graining, which is not intuitive to me - can someone explain? Also what are these terms, and did Weinberg include these?

Edit: As far as I can tell, Weinberg, as well as Codello et al, only included local (or quasilocal) terms. So what are these non-local terms Litim is talking about, and why would they arise?

 

[atyy]

http://relativity.livingreviews.org/Articles/lrr-2006-5/
"a canonical formulation is anyhow disfavored by the asymptotic safety scenario"

What!?

 

[Haelfix]

"So you see its not the AD scenario that I'm arguing about. Its that AD(an IR property of classical gravity) has any baring on AS/renormalizablity(which is a UV problem of quantum gravity). "

AD is a UV property of QUANTUM gravity by definition ... You are summing up ladder diagrams and things like that after all. THe peculiarity here is that it effectively looks semiclassical again. The quantum effects which may have been important at the Planck scale as well as the nonperturbative physics, at transplanckian energies, must drop out.

 

[Finbar]

"So you see its not the AD scenario that I'm arguing about. Its that AD(an IR property of classical gravity) has any baring on AS/renormalizablity(which is a UV problem of quantum gravity). "

AD is a UV property of QUANTUM gravity by definition ... You are summing up ladder diagrams and things like that after all. THe peculiarity here is that it effectively looks semiclassical again. The quantum effects which may have been important at the Planck scale as well as the nonperturbative physics, at transplanckian energies, must drop out.

Who is summing up ladder diagrams in quantum gravity?! you can't go beyond 2 loops in perturbation theory one has to go to effective field theory and work at energies below the Planck scale.

What your saying here is not the case and I assure you you have been mislead.

Please can you cite a paper where ladder diagrams in QG are being computed and the result is that gravity becomes semi-classical again?

 

[Haelfix]

"you can't go beyond 2 loops in perturbation theory one has to go to effective field theory and work at energies below the Planck scale. "

You can sum up however many orders of perturbation theory that you want in gravity, the thing is you may or may not get an underspecified answer (for instance, depending on constants arising from the counterterms of the next order) or alternatively a divergent answer (for E --> infinity). But for E finite, you will get some number. Incidentally that's what AS presuposes. Namely that as you sum up the perturbation theory, there are cancellations that take place within the divergence structure of the theory (so bad '2' loop terms like GS and the Rs coupling will presumably cancel out)

But anyway, here we are talking about a theory of 2 body scattering. The approximation under consideration is where you take the first exchange term with a graviton, and then 'exponentiate' it by summing up all the associated ladder diagrams. For large impact parameters, this approximation is valid and exact (this is the Eikonal regime).

 

[Finbar]

"you can't go beyond 2 loops in perturbation theory one has to go to effective field theory and work at energies below the Planck scale. "

You can sum up however many orders of perturbation theory that you want in gravity, the thing is you may or may not get an underspecified answer (for instance, depending on constants arising from the counterterms of the next order) or alternatively a divergent answer (for E --> infinity). But for E finite, you will get some number. Incidentally that's what AS presuposes. Namely that as you sum up the perturbation theory, there are cancellations that take place within the divergence structure of the theory (so bad '2' loop terms like GS and the Rs coupling will presumably cancel out)

But anyway, here we are talking about a theory of 2 body scattering. The approximation under consideration is where you take the first exchange term with a graviton, and then 'exponentiate' it by summing up all the associated ladder diagrams. For large impact parameters, this approximation is valid and exact (this is the Eikonal regime).

Papers?

I'm not sure that AS presupposes anything. AS is a possible scenario in which taking the cutoff to infinity will give you a finite theory.

I still insist that whatever approximation you are on about is certainly not valid at the UV fixed point of gravity. For sure it neglects non-perturbative effects if your just exponentiating the tree level graviton exchange.

My point has always been that all these arguments based on perturbation theory and the Einstein Hilbert action have nothing to say about AS. The Eikonal regime is sub-plackian
GE<lpl. It says nothing about graviton loops.

Look at fig 1. in http://arxiv.org/pdf/0908.0004v1
Its in the semi circle at the bottom that we need to know QG and can make comments on non-perturbative renormalisation. If AS is realized in nature this regime is controlled by a
UV fixed point and we don't expect black holes to be formed. AD is valid in the strong gravity regime where arguments can be made that we must see black holes here but these arguments have no bearing on the physics of a full non-pertubative theory of QG.

 

[Haelfix]

"The Eikonal regime is sub-plackian GE<lpl"

For the 10 th time.. Its transplanckian : E (CM) >>> Mpl! The papers I have already listed explain this in great detail, or see Veneziano's papers in the 80s (which are cited in Srednicki's paper)

Every point in that semicircle in figure 1 are at transplanckian energies!

 

[Finbar]

"The Eikonal regime is sub-plackian GE<lpl"

For the 10 th time.. Its transplanckian : E (CM) >>> Mpl! The papers I have already listed explain this in great detail, or see Veneziano's papers in the 80s (which are cited in Srednicki's paper)

Every point in that semicircle in figure 1 are at transplanckian energies!

I agree E>>Mpl but this means GE>>lpl because GE is a length not a mass. So we're in the IR physics of gravity 1/GE<<Mpl.

Sorry I meant GE>lpl in my last post.

I know its confusing that there's this UV/IR thing with gravity. But you need to think of the physics here. If I collide two tennis balls together then the energy E>>Mpl but I don't need QG to describe the physics. If I further take the mass of the tennis balls and compact them down such that when they collide there within a radius r<2EG then a black hole must form but the curvature at the horizon will be sub-plackian therefore I can still describe the physics without QG I only need semi-classical physics. Its only when I take a small amount of energy E~Mpl and confine it to a very very tiny space r=2GE~lpl that the curvature becomes Plackian and we need QG. In the Fig. 1 in Giddings paper this is the semi circle with the ? at the bottom left were both E and b are small i.e. a small energy confined to a small radius, its here and only here that the curvature is Plackian and we're in the UV.

 

[Haelfix]

Ok good, we are on the same page then. The regimes are paremetrized by the magnitude of the impact parameter relative to the Schwarzschild radius (well technically some sort of radius between two shockwaves, which is order magnitude the same as the schwarschild radius) and only for the case where b << R do you need to worry about strong coupling effects.. No one knows what goes on there exactly (although you can make the point that you need to smoothly match between regimes)

But the point is a generic field theory of gravity at transplanckian center of mass energies must be able to accommodate black hole states in their spectrum. It doesn't matter that those states are formed in regimes that are effectively classical or semiclassical (at large impact parameters). You still require that the entropy scales as the area, and there you run into a problem b/c at those ultra high energy scales the putative, apparently universal field theory under question has to be conformal and at no point can it have any states that satisfy this type of scaling.

If AS was an effective theory, there would be no problem, b/c you could just argue that you picked the wrong epsilon parameter to perturb around and you don't capture the correct physical regimes, but here this is supposedly *the* theory of all quantum gravity valid at all energy scales with arbitrary matter couplings. It has to be able to have a spectrum that contains high energy black hole states, since we know its low energy behavior is normal GR and will thus also have Eikonal and Coulomb regimes in high energy scattering experiments.

 

[Finbar]

Ok good, we are on the same page then. The regimes are paremetrized by the magnitude of the impact parameter relative to the Schwarzschild radius (well technically some sort of radius between two shockwaves, which is order magnitude the same as the schwarschild radius) and only for the case where b << R do you need to worry about strong coupling effects.. No one knows what goes on there exactly (although you can make the point that you need to smoothly match between regimes)

But the point is a generic field theory of gravity at transplanckian center of mass energies must be able to accommodate black hole states in their spectrum. It doesn't matter that those states are formed in regimes that are effectively classical or semiclassical (at large impact parameters). You still require that the entropy scales as the area, and there you run into a problem b/c at those ultra high energy scales the putative, apparently universal field theory under question has to be conformal and at no point can it have any states that satisfy this type of scaling.

If AS was an effective theory, there would be no problem, b/c you could just argue that you picked the wrong epsilon parameter to perturb around and you don't capture the correct physical regimes, but here this is supposedly *the* theory of all quantum gravity valid at all energy scales with arbitrary matter couplings. It has to be able to have a spectrum that contains high energy black hole states, since we know its low energy behavior is normal GR and will thus also have Eikonal and Coulomb regimes in high energy scattering experiments.

Ok I think we agree on the black hole scattering points now. But we still need to address the entropy scaling. What you said wasn't entirely correct.

A generic fundamental QFT has only to be conformal at the UV fixed point; by definition. So its only at this point that scaling arguments apply. So the question is when and where is physics at the UV fixed point. In gravity it is when the curvature becomes Plackian such that classical physics breaks down and we require UV completion. This happens only at very short distances r<lpl where the Weyl curvature C>Mpl^2. This is the case for the singularity of a generic black hole. But we can only say that all the physics of the black hole is at the UV fixed point when the curvature all the way the way up to the horizon is Plackian. This only happens when the radius of the BH is r~lp.

 

[marcus]

Ok I think we agree on the black hole scattering points now...

Then maybe you would explain something. Can you describe a scattering experiment which, if it were performed and came out as you imagine, would invalidate the AS approach?

 

[Finbar]

Then maybe you would explain something. Can you describe a scattering experiment which, if it were performed and came out as you imagine, would invalidate the AS approach?

There are researchers currently looking into this and I would expect papers to be published sometime in the near future.

This paper may be of some interest http://arxiv.org/pdf/0707.3983

 

[atyy]

Look at fig 1. in http://arxiv.org/pdf/0908.0004v1

A generic fundamental QFT has only to be conformal at the UV fixed point; by definition. So its only at this point that scaling arguments apply. So the question is when and where is physics at the UV fixed point. In gravity it is when the curvature becomes Plackian such that classical physics breaks down and we require UV completion. This happens only at very short distances r<lpl where the Weyl curvature C>Mpl^2. This is the case for the singularity of a generic black hole. But we can only say that all the physics of the black hole is at the UV fixed point when the curvature all the way the way up to the horizon is Plackian. This only happens when the radius of the BH is r~lp.

In fig 1 if I fix impact parameter and move to higher and higher energies, I'll move into the the strong gravity or classical black hole region. Won't I need the UV completion at this point - the classical theory won't work because we expect Hawking radiation from thermodynamics, and semi-classical theory won't work because of information loss?

 

[atyy]

I've asked this before, but still don't understand the answer, so here it is again. The UV fixed point should be scale invariant - under what assumptions is that equivalent to conformal invariance?

 

[Finbar]

In fig 1 if I fix impact parameter and move to higher and higher energies, I'll move into the the strong gravity or classical black hole region. Won't I need the UV completion at this point - the classical theory won't work because we expect Hawking radiation from thermodynamics, and semi-classical theory won't work because of information loss?

Ok so if we're in the semi-classical regime the black hole is radiating so the horizon shrinks. But as it shrinks and the horizon approaches the Planck length and the temperature approaches the Planck mass we then need a UV completion.

So far AS "solves" the information paradox by saying that the information is stored in a Planck size remnant. But I don't think this is a satisfactory solution.

 

[atyy]

Ok so if we're in the semi-classical regime the black hole is radiating so the horizon shrinks. But as it shrinks and the horizon approaches the Planck length and the temperature approaches the Planck mass we then need a UV completion.

So far AS "solves" the information paradox by saying that the information is stored in a Planck size remnant. But I don't think this is a satisfactory solution.

Hmmm, let me think about that. I was hoping that AS would prevent high energy scattering experiments by allowing only asymptotically dS spaces by enforcing a positive cosmological constant through the renormalization flow. (No, I don't really know what I'm talking about.)

What's wrong with the remnant solution?

 

[Finbar]

I've asked this before, but still don't understand the answer, so here it is again. The UV fixed point should be scale invariant - under what assumptions is that equivalent to conformal invariance?

I think that scale invariance is a sub group of conformal in variance. But I'm not sure. So if a theory is conformal in the UV it has to be scale invariant and hence at a fixed point.

 

[atyy]

I think that scale invariance is a sub group of conformal in variance. But I'm not sure. So if a theory is conformal in the UV it has to be scale invariant and hence at a fixed point.

Yes, I think that's true. What I don't understand is why the UV fixed point must be conformal, though I understand it has to be scale invariant. I think that under some additional assumptions like unitarity, Poincare invariance or something, then scale invariant theories are conformal - but I don't know what the exact conditions are.

Eg. http://arxiv.org/abs/hep-th/0504197 or footnote 3 of http://arxiv.org/abs/0909.0518

 

[Finbar]

Hmmm, let me think about that. I was hoping that AS would prevent high energy scattering experiments by allowing only asymptotically dS spaces by enforcing a positive cosmological constant through the renormalization flow. (No, I don't really know what I'm talking about.)

What's wrong with the remnant solution?

http://arxiv.org/pdf/hep-th/9508151

The Black Hole Information Paradox
Steven B. Giddings†
Department of Physics University of California Santa Barbara, CA 93106-9530
Abstract
A concise survey of the black hole information paradox and its current status is given. A summary is also given of recent arguments against remnants. The assumptions underly- ing remnants, namely unitarity and causality, would imply that Reissner Nordstrom black holes have infinite internal states. These can be argued to lead to an unacceptable infinite production rate of such black holes in background fields.
(To appear in the proceedings of the PASCOS symposium/Johns Hopkins Workshop, Baltimore, MD, March 22-25, 1995).


Theres also a another paper by Giddings but I can't find it right now

 

[marcus]

The paper by Weinberg which is our topic is
http://arxiv.org/abs/0911.3165
Asymptotically Safe Inflation
Steven Weinberg
13 pages
(Submitted on 16 Nov 2009)
"Inflation is studied in the context of asymptotically safe theories of gravitation. It is found to be possible under several circumstances to have a long period of nearly exponential expansion that eventually comes to an end."
================

The basic idea is to explain a self-terminating inflation episode, without making up some exotic "inflaton" matter field, as a natural consequence of the running of couplings such as Newton's G. The couplings can be assumed to be at or near their UV limit at the start of expansion. And this by itself, Weinberg shows, is sufficient to cause exponential expansion.

We can think of the scale as related to density. As the universe expands, the density falls off, and the couplings depart from their values at the UV-limit. After some 60 e-foldings of expansion the density is low enough that inflation ends.

Some readers may wish to question this statement of Weinberg:

"We will work with a completely general generally covariant theory of gravitation. (For simplicity matter will be ignored here, but its inclusion would make no important difference.)"

More about matter in the context of Asymptotic Safety is here:
https://www.physicsforums.com/showthread.php?t=349513
in the "Grav. + GUT" thread.

 

[uniquo87]

I would like to know the derivation of the field equations of quantum electrodynamics.

 

[marcus]

We should try to get this thread back on track as per the original Weinberg paper.

There is no physical reason to assume black holes are especially relevant or significant in this context, and the Weinberg paper is not about black holes. It is about inflation. Asymsafe QG provides a neat and economical way to explain inflation.

Since I started the thread a great paper by Shaposhnikov and Wetterich has come out. http://arxiv.org/abs/0912.0208 I'll quote some excerpts. Here's from page 2.

From the studies of the functional renormalization group for Γ_k one infers a characteristic scale dependence of the gravitational constant or Planck mass,

M_P² (k) = M_P² + 2ξ₀k²​

where M_P = (8πG_N )^−1/2 = 2.4 × 10¹⁸ GeV is the low energy Planck mass, and ξ₀ is a pure number, the exact value of which is not essential for our considerations.

From investigations of simple truncations of pure gravity one finds ξ₀ ≈ 0.024 from a numerical solution of FRGE [5, 11, 12]. For scattering with large momentum transfer q the effective infrared cutoff k² is replaced by q² . Thus for q² ≫ M_P² the effective gravitational constant G_N(q² ) scales as 1/(16πξ₀q²) , ensuring the regular behavior of high energy scattering amplitudes.

 

[marcus]

I made this point earlier in the thread. As the cutoff momentum k increases to infinity, the Planck mass goes to infinity. The Planck mass runs as k, and the S&W paper shows that it is asymptotically proportional to k.

All that M_P means is the low energy Planck mass. In the asymsafe picture, M_P(k) is the physically relevant Planck mass at scale k and it is scale dependent. At high energies, the low energy Planck mass is not relevant to black hole/particle physics. The physical Planck energy goes to infinity, so what does "transplanckian" particle collision mean? (The word "transplanckian gets thrown around not always thoughtfully or with clear significance.)

Newton constant is even more strongly scale dependent. It goes to zero as 1/k². I mentioned that in a post quite a few days back.
This is why I regard some of the old (say 1995-2003) discussion of "transplanckian" particle collisions forming black holes as unconvincing.
And even more dubious was the talk about "asymptotic darkness", but happily one hears very little about that nowadays. People were theorizing way beyond their base of solid understanding.

I see no indication that the obsolete discussion took the running of Newton's constant into account. What we have nowadays is a growing suspicion that gravity has an RG fixed point, and IF IT DOES, as many numerical studies now indicate it does, then G(k) falls off as 1/k².

The Shaposhnikov paper can even tell you the proportionality. So how is a black hole supposed to form? According to the asymsafe assumption gravity is essentially turned off at very high energy density, or at very high momentum transfer (q in the S&W paper) if we are discussing particle collisions.

So there is no indication that what the earlier authors had to say fits into OUR discussion which takes seriously the possibility that gravity is asymptotically safe and that the Renormalization Group plays an important role.

 

[marcus]

We can take a cue from the S&W paper http://arxiv.org/abs/0912.0208 .

Here's from the abstract:
"There are indications that gravity is asymptotically safe. The Standard Model (SM) plus gravity could be valid up to arbitrarily high energies. Supposing that this is indeed the case and assuming that there are no intermediate energy scales between the Fermi and Planck scales we address the question of whether the mass of the Higgs boson m_H can be predicted..."

Steven Weinberg's paper is in the same spirit. There are indications that gravity is asymsafe and that changes the picture. So let us see what we can do, assuming that it is.

These people show us what we ought to be going and what directions we ought to be looking. Assuming asymsafety we get
*a nice simple explanation of inflation.
*predictions that can be tested at accessible collider energies and probably also by astronomical observation.
*minimalistic approaches to unification.

And of course the assumption might be wrong! Predictions like Shaposhnikov and Wetterich derived might be falsified by LHC!
The point is that asymsafe unification is a good place to look for results and smart experienced people are focusing on it.
A defensive dismissal, at this point, based mostly on 1995-2003 papers or whatever, does not seem astute.

Here's from page 10 of Shaposhnikov and Wetterich:

"In conclusion, we discussed the possibility that the SM, supplemented by the asymptotically safe gravity plays the role of a fundamental, rather than effective field theory. We found that this may be the case if the gravity contributions to the running
of the Yukawa and Higgs coupling have appropriate signs. The mass of the Higgs..."

 

[oldman]

We should try to get this thread back on track as per the original Weinberg paper...

While this thread is temporarily dormant, there may be time for a naive question:

One of the reasons why inflation was invented was to explain the uniformity of the observed universe over regions too large to be causally connected in the early universe of the standard model.

If indeed gravity is asymptotically safe and is

...essentially turned off at very high energy density, or at very high momentum transfer

would this raison d'etrefor inflation be affected? Or does Weinberg's scenario merely resolve a puzzle with inflation?

It seems to me that the running down or switching off of gravity is such a drastic change in the physics of an expanding or inflating universe, ruled by gravity throughout its postulated history, that this is worth asking.

 

[marcus]

...

...If indeed gravity is asymptotically safe and is would this raison d'etrefor inflation be affected? Or does Weinberg's scenario merely resolve a puzzle with inflation?

It seems to me that the running down or switching off of gravity is such a drastic change in the physics of an expanding or inflating universe, ruled by gravity throughout its postulated history, that this is worth asking.

Personally I like the question a lot. My standards of naive may be different from yours, in any case I don't think of it as naive. I think this idea of doing barebones unification and barebones early-U cosmology based on renormalization group flow is a new initiative and just getting under way.

In Reuter's treatment not only does G(k) -> 0, but also cosmological constant Lambda(k) -> infinity. Because their dimensionless versions Gk² and Lambda/k² go to finite values.
With a huge cosmo constant you get tremendous inflation, just as nowadays with a small cosmo constant we get very gentle acceleration.

Earlier I mentioned only the running of G(k) because it was relevant to the sidetrack distraction topic of blackoles.
If you are curious about the distinction between the dimensionless and dimensionful versions of the two basic quantities, ask. I'll attempt more explanation or someone else will jump in.
===================

Maybe I shouldn't have used the expression "switched off".
It is simply that at very high collision energies, or very short distances, or very high densities, the repulsive term Lambda is very large and the attractive term G is very small.

But the laws do not change, nothing goes away or gets turned off for any appreciable duration. It is simply that the effective physical magnitudes of forces are different for an extremely brief inflationary episode.

I don't know how you picture the start of expansion. I imagine it as a bounce or a rebound from prior contraction. I don't ASSUME that since so far it hasn't been proven. It is just a possible conjecture. One of several options for visualizing.
====================

We live with running constants all the time. Quarks attract each other when comparatively widely separate. Nearby quarks have little interest in each other. I shouldn't say "switched off". The law is still there and operative, but its force varies with proximity.

=====================

NOW THE MEAT OF YOUR QUESTION is whether running constants might explain other things that inflation was earlier postulated to explain! Or which it later turned out to explain so well. Two main things come to mind, I think.
*Flatness
*The angular power spectrum of the CMB (scale invariance of temperature fluctuations).

That's an interesting idea. At first sight I don't see how to avoid inflation. My mind may be so locked into the inflation picture that I can't easily get out. It seems to me that as a geometrical event a bounce with extremely rapid initial expansion would be just the thing to achieve flatness and the observed main overall features of the CMB.

It would make predictions though. I imagine one would not see as much gravity wave imprint on the CMB---just a wild guess.

And the asymptotic safety vision of the early-U would probably have something to say about entropy. A brief episode with negligible Newton G would, I imagine, reset the apparent entropy clock of "curdling" (your word for condensed structure formation). Black holes and other blemishes in the prior contracting phase would be erased by a kind of renormalization group "botox". How could wrinkles persist in a high density phase with G(k) negligible? A deplorably wild guess.

It's a good line of questioning. I'll think about it some.

I think right now, at first sight, that the answer is that with the asymsafe early-U picture
*you can't avoid inflation
*and inflation is still useful in explaining flatness and scale invariance
*and asymsafe early-U will be shown to predict observable effects and be falsifiable.

Any reactions?

 

[oldman]

...(the) idea of doing barebones unification and barebones early-U cosmology based on renormalization group flow is a new initiative and just getting under way ...We live with running constants all the time...the answer is that with the asymsafe early-U picture
*you can't avoid inflation
*and inflation is still useful in explaining flatness and scale invariance
*and asymsafe early-U will be shown to predict observable effects and be falsifiable.

Your perspective is illuminating, especially when taken together with your thread on "QMG" and Reuter's no-frills QG, which clarified what is meant by asymptotic safety for me. The kinds of questions that seem to be floating around these days are fascinating. One that interests me is:

Are there any fundamental all-embracing theories in physics?

Or are there only "effective" theories, like electromagnetism (which is important far from an electron, where the charge doesn't "run') or superconductivity (which is important when electrons and phonons co-exist only in cold solids). The importance of gravity as we know it seems to stretch over the lifetime of the observed universe, but if it didn't always rule in its present form, with a small cosmological constant, could it be classed as an "effective" theory that has running constants?

Incidentally, has anyone yet devised a dimensionless version of c that could run?

 

[marcus]

Interesting questions!

...
Are there any fundamental all-embracing theories in physics?

Or are there only "effective" theories, like electromagnetism (which is important far from an electron, where the charge doesn't "run') or superconductivity (which is important when electrons and phonons co-exist only in cold solids). The importance of gravity as we know it seems to stretch over the lifetime of the observed universe, but if it didn't always rule in its present form, with a small cosmological constant, could it be classed as an "effective" theory that has running constants?

Incidentally, has anyone yet devised a dimensionless version of c that could run?

Some people might say it's all cut and dried---they might have a "correct" answer for each of your questions. On the contrary, I am not even sure that humans know what a "fundamental" theory is, or would recognize one if it stared them in the face.

About c: what occurs to me is that the scale parameter k is a momentum and there is no way you can combine c with a power of k to get something dimensionless.

So apparently, according to renormalization conventions, c cannot run. (Yet people construct frameworks in which they can talk about variable speed of light. I think there's a radical difference though.)

According to what I think is normal usage, we set c = hbar = 1 and then, since k is a momentum, Gk² is dimensionless. And Lambda is a reciprocal area, so it is the square of a momentum, and Lambda/k² is dimensionless.

In electromagnetism the operative running constant is alpha (approx. = 1/137) that relates charge to attraction and distance. Charge does not have to run, because alpha runs. As I recall it increases to more than 1/137 at very high energy and close proximity. Seem to recall alpha can get as big as 1/128
================

What I think is an intriguing question is what is meant by "fundamental".

It's not as simple an issue as some people may imagine. Percacci has a bit about this in his chapter in Oriti's book. And the new paper by Shaposhnikov and Wetterich has some bearing on the issue. For very high k, say with k being the momentum transfer in a collision, the Planck energy itself increases as k.
The Planck mass and the Planck energy go to infinity as k increases. So Shaposhnikov and Wetterich deal with this, and set out formulas for it, and build it into their equations.

Not everybody is so astute or careful. Others may for example assume that the Planck mass and energy are always equal to their low-energy values.

So the question arises what do you mean by saying a theory purports to be predictive out to arbitrarily high energy. Do we know enough about how nature behaves at Planck scale to distinguish between a "fundamental" theory and one which merely aspires to be applicable out to Planck scale?

And what is the appropriate "k" to use? People use various different handles on the scale, all supposed to give the same physical results. But why? what makes something a good handle? Energy density, collision energy, momentum transfer etc etc.
And why do coupling constants run? Can you always explain it by screening and antiscreening---by vacuum myths in other words---just so stories about the vacuum. And what is the vacuum. What is it when we throw out Minkowski space and declare that geometry is a dynamical something included in what we wish to explain? Why then do coupling constants run with scale? And what is scale? My basic feeling is that humans are wonderful animals but still rank beginners in this game.

So I can't answer your question about are there any really fundamental, not merely effective, physical theories. But glad you asked. Maybe someone else will put it into perspective for both of us.

 

[atyy]

A fundamental theory is one that does not predict its own breakdown. So QCD is a fundamental theory, as is Newtonian gravity, but both do breakdown.

 

[humanino]

A fundamental theory is one that does not predict its own breakdown. So QCD is a fundamental theory, as is Newtonian gravity, but both do breakdown.

It is an interesting point of view. Then I would say fundamental theories are in some sense inferior to effective theories, because we usually do not expect Nature to "breakdown", the prediction of new physics is the best we can get of any theory.

 

[DarMM]

It's not just that. It's the question whether we live in a universe that lies on the critical surface. Since the critical surface is most likely finite-dimensional and the space of all couplings is infinite-dimensional, the a priori probability that we actually live in such universe is zero. It would require either some not-as-of-yet-understood mechanism that puts the gravity in the UV fixed point, or the incredible amount of fine-tuning, to justify this scenario.

I don't fully understand this comment, maybe I'm missing something about the renormalization group.

I thought that the renormalized theory (i.e. one which has a continuum limit), is the renormalization group flow which emerges (roughly perpendicularly) from the critical surface at the fixed point. Provided the fixed point has only a finite number of rupulsive directions, then you have a theory. As long as the workers in this field can show that there is a critical point with a finite number of repulsive directions, then there will be finitely paramterised flows emerging from the fixed point. Which means a continuum theory with a finite number of parameters. I don't see why tuning would be necessary.

 

[DarMM]

Does AS really need a fixed point? Could it live with, say, a limit cycle?

Yes, the Wilson-Kandoff renormalization group takes place on a block lattice with everything in units of the lattice spacing a = 1. In lattice units the regularized integrals of perturbation theory have no divergences as a \rightarrow 0, because a has disappeared.

However the problem of renormalization has been replaced by the problem of taking a continuum limit, with no a where is the continuum limit a \rightarrow 0. This problem is solved by the lattice correlation length, which roughly tells you how big correlations are in lattice units. If you fix the correlation length in physical units, then the lattice correlation length has to diverge as you approach the continuum, as lattice units are smaller and smaller compared to physical units.

So the continuum limit is associated with points with infinite lattice correlation length, which are fixed/critical points.