“Infinite Grid” by Georg Koch. |

To begin with quantizing gravity isn’t all that difficult. Gravity can and has been quantized much like the other interactions by promoting gravitational waves to quantum waves. That is called perturbative quantization and technically somewhat annoying but perfectly doable. The problem starts only after this because the so quantized theory does not make sense anymore when gravity becomes strong. It delivers infinities as results, no matter what, and that makes it not only useless but also meaningless as a fundamental theory.

You might shrug shoulders on the infinites because you have also probably heard that quantum field theory has a problem with infinities anyways. But that too is not true...

Yes, the occurrence of infinite results was historically a big issue. But quantum field theory development hasn’t stopped in the 1930s and we know today very well how to do these calculations. Whenever you get an infinite result, you need to use a measurement to fix a physical parameter. We call it renormalization and it’s a mathematically clean procedure.

However, if you need an infinite amount of measurements to fix parameters,

*then*you have a real problem because now your theory is no longer predictive: You need infinitely many measurements before you can make a prediction about the next measurement. A theory with that problem is said to be perturbatively non-renormalizable and on that diagnosis most physicists will not resuscitate the patient. Perturbatively quantized gravity has exactly this disease.

As long as gravity is weak, for all practical purposes you need only a finite amount of parameters to get to a desired precision. You can use the theory there. But once gravity gets strong, the theory becomes useless. This means it is not a candidate for a fundamental theory.

The use of quantum field theories and their properties thus depend on the energy scale. Interactions can for example become weaker or stronger depending on the energy of the interaction. Quantum Chromodynamics famously becomes weaker at high energies; it is “asymptotically free” and that insight was worth a Nobel Prize in 2004.

But how theories in general depend on the energy scale has only been really understood within in the last two decades or so. It has been a silent development that almost entirely passed by the popular science press and goes under the name renormalization group flow. The renormalization group flow encodes how a theory depends on the energy scale, and it is at the basis of the idea of effective field theory.

The dependence of a quantum field theory on the energy scale that is used to probe structures is much like Ted Nelson’s idea of the stretch-text, a text in which you can zoom or click into layers of more detail. The closer you look, the more new features, new insights, new information you get to see. It’s the same with quantum field theory. The closer you look, the more new layers you get to see.

Based on this, Weinberg realized in 1976 that perturbative renormalizability is not the only way for a theory to remain meaningful at high energies. It is sufficient if, at high energies (technically: infinitely high), you need to fix only a finite number of parameters. And none of these parameters should become infinite itself in that limit. These two requirements: A finite number of finite parameters that determine the theory at high energies are what make a theory asymptotically safe.

This then raises the question of whether quantum gravity, though perturbatively nonrenormalizable, might be asymptotically safe and meaningful after all. That this might be so is the idea behind “asymptotically safe gravity”. While the general idea has been around for almost four decades, it has only been in the late 1990s, following works by Wetterich and Reuter, that asymptocially safe gravity has caught on.

As of today, it has not been proved that gravity is asymptotically safe, though there are several arguments that support this idea. The problem is that doing calculations in an infinite dimensional theory space is not possible, so this space has to be reduced. But then the result can only deliver a limited level of knowledge. The other problem is that even if the theory is asymptotically safe, it might be physically nonsensical at high energies for other reasons.Another criticism on asymptotically safe gravity has been that it does not seem to take into account that space-time fundamentally might be described by degrees of freedom different from those used in general relativity. While that arguably is so in existing approaches, the idea of renormalization group flow is in principle perfectly compatible with changing to different – more ‘fundamental’ – degrees of freedom at high energies, as Percacci and Vacca have pointed out.

That is to say, this approach towards quantum gravity has its problems, its friends and its foes, as has every other approach towards quantum gravity. But it is a strong competitor. What makes this approach so appealing is its minimalism: Maybe quantum gravity makes sense as a quantum field theory after all! Depending on your attitude though you might find exactly this minimalism unappealing. It’s like at the end of a crime novel the murder victim comes back from vacation and everybody feels stupid for their conspiracy theories.

Whatever your attitude, asymptotically safe gravity has made some contact to phenomenology, mostly in the area of cosmology, though for all I know these studies haven’t yet resulted in a good observable. Most interestingly, asymptotically safe gravity has been shown to also lead to dimensional reduction and it has recently been argued that it might be related to Causal Dynamical Triangulation. It seems to me that whatever quantum gravity ultimately looks like, asymptotically safe gravity will almost certainly be part of the story.

So the next time somebody tells you that we don’t know how to quantize gravity, keep in mind the many layers of details underneath that statement.

## 50 comments:

Do you believe like Max Tegmark we have to get rid of infinity?

No, but I think it's an interesting question whether infinity is physical. I've tried some years ago to figure out if there's a way to draw an observational distinction between the two cases, but failed to come up with anything.

It is often claimed that the 126 GeV Higgs was predicted by asymptotic safety. OTOH, m_H = 126 GeV also seems to be the border where the SM ceases to be consistent. Do you know if there is a relation?

About the notion of infinity in Physics or mathematics, I strongly believe that if atomic theory is right, infinities are not possible. But if they are not possible, fields, systems with infinite degrees of freedom, can NOT be fundamental, and it seems it is not the case. Of course, what do you mean by infinity? Do we allow infinities in any type or only some of them? I mean, we can have "infinite quantities" (e.g., vacuum energy, speed of light, or the number of particles -thermodinamic limit), but yet we allow for fields. We also have atoms with infinite levels (e.g., the hydrogen atom or the particle in a box). So, I guess, that the question is more what kind of infinities are physical and what are, secretly, limits of "large quantities". Are fields in that category? I dislike the idea of "infinite theories" more and more accurate and precise, I dislike "infinite" quantities, but sometimes we get meaningful results from "infinite" values of some parameters:

1. Galilean relativity is the "large c" limit of special relativity.

2. Classical thermodynamics is the large N limit of "atomic and molecular physics", somehow.

3. Field theory is the large N limit of relativistic mechanics of the particle (more or less, I have always felt weird with this idea, even if natural).

And I guess we could write more of this about paramaters being infinite. So, it seems, paramaters taking a "formal infinite value" are "useful" and physically meaningful when we take approximations or "thermodynamic limits".

About renormalization of infinite quantities, I am more and more confident that NO ONE has explained yet WHY renormalization works. It is a trickery than works, it is self-consistent and it provides finite answers to formally infinite quantities (indeed, in QFT, most of the series are not even convergent or finite, and they are only making sense in the asymptotic limit, a puzzle that, to my knowledge, no one has tried to prove or improve).

A harder question, what types of infinities are possible? For me:

1) To fit something with an infinite number of parameters is nonsense. So, this case (in parallel with your article) should be avoided (experimentally, you can fit anything with a large amount of quantities; however, better theories use to have LESS fundamental parameters AND/OR they relate old constants). I don't like the MSSM only due to the enlargement of the parameter space. I like supersymmetry, but I don't like "complex models" if a single theory (like the SM) does it better.

2) The number of fundamental degrees of freedom should be FINITE (even if large). Of course, a finite number of fields or a finite number of "field configurations"?

3) The number of fundamental constants should be kept minimal (maybe "zero" and/or dimensionless?).

Tropical mathematics make sense of "infinite" number addition, for instance. So, something "infinite" can be meaningful. Therefore, values of "variables" equal to infinity are "allowed". And values of the variables "formally infinite" as well.

I disagree with your sentence: infinite dimensional spaces are meaningful, e.g., atoms have infinite levels, ...And mathematically, Hilbert spaces and other things...Even classical fields have a meaning.

I didn’t quite catch your explanation of asymptotic safety. "It is sufficient if, at high energies (technically: infinitely high), you need to fix only a finite number of parameters." Why is this not the same as renormalizability? And how does it avoid the earlier argument that an infinite number of parameters are needed?

In defense of the popular press, I've alluded to RG flow in my book and at least one Sci Am article.…

George,

Well, it is renormalizable - that's the whole point. It's just not perturbatively renormalizabile. Which is to say, it looks like it's not renormalizable if you quantize it perturbatively. The relevant part of the sentence is 'at high energies' (read: in the UV) you get a theory that has only finitely many finite parameters and is all well and fine.

Thomas,

Sorry, never heard of this prediction.

http://arxiv.org/abs/0912.0208

"...the idea of renormalization group flow is in principle perfectly compatible with changing to different – more ‘fundamental’ – degrees of freedom at high energies"

But it should still be a Conformal *field* theory. Correct? In that case the Bekenstein-Hawking entropy of a black hole cannot be reproduced...

If I remember the paper with the detailed argument I will tell you.

What are the implications of AS to both string theory, AdS/CFT and LQG?

Matters of Gravity

https://files.oakland.edu/users/garfinkl/web/mog/

http://files.oakland.edu/users/garfinkl/MOG/mog43/mog43.pdf

Winter 2014, top of page 8, "Prof. Piotr Bizon presented his striking result that the anti-de Sitter space-time, is unstable in full, non-linear general relativity although it was known to be perturbatively stable."

A "

perturbative" procedure, by definition, cannot pull emergent properties out of its hat. If new symmetries appear or disappear when more or fewer dimensions are considered, peturbative techniques will be mathematically rigorous but empirically wrong. Chirality is emergent. Physical theories of matter choke on matter chirality as parity violations, symmetry breakings, chiral anomalies, and Chern-Simons repair of Einstein-Hilbert action.A lucid and informative exposition. ("Amount" is measurable (cup of water), number is countable (infinite

numberof measurements. The fun arrives with a cup of ball bearings and, I suppose, more than an aleph-null of parameters).We all live in a 'Yellow King's' Submarine..

http://en.wikipedia.org/wiki/The_King_in_Yellow

If there were no absolute or potential infinity as part of physical reality in the most general conception of space we would not be able to move through it on the human scale. (Roughly a light nanosecond in length.)

Hints of dark fluid issues should be a hint as to how we should redefine mathematics as the most general space as well concepts of physics where in moving through space we encounter boundaries real as matter or imagined. Given certain consistent models the finite is introduced but without better definition of dimensions the dimensionless is ill defined. So are ideas like degrees of freedom asymptopic or otherwise.Or renormalization or safety, stability, excluded middles, mathematical induction, Pauli Exclusion, perturbation especially now that we can entangle indepent photons of different frequencies and color. This makes it hard to distinguish where a deeper level of quantization from intelligible ideas of quantum mysticism (which essentially makes a rigid distinction between the starry sky above and the moral law "within' (Kant) if there is such a concept of safety or some islands of atomic element magic and unexplained numbers is it so because there is still a deeper level of foundations by which in the chaos laws of the cosmos can persist as determined and measured predicting new total phase states thru concepts of quasi - singularity as well as Natures own dynamic quantiization as part of its spacious unity. But I suspect this post will ring most empty as does so much speculation evidenced by words alone or inadequate formulas, experiments, or particles trapped in complexity (of analysis or complicated and dismissed in an idea of non - linearity.) to which Leibniz called imaginary numbers a salamander between Being and Non Being. What is so hard to see that we can pass through the rim of a four dimensional volcano and like a neutron have to slow down to fall into it to harness the fission?

Gravitational waves have never been observed.

Never!

Hi Bee,

Do you know if there are any quantum gravity models out there that attempt to describe spacetime curvature in terms accelerating virtual particles near mass/energy?

(Not relying on gravitons at all)

Cheers, Paul.

Robert

Check out today's newscientist com discussing ways to measure gravity waves and detect them by phonons and other QM in curved spaces involving Casmir effect. Sabine has long been talking about such phenomenal physics.

BTW your statement on scale macro Meso and micro the same I too agree with as part of this more advanced physics where dark fluids may be intermittent in a sense not directly detectsble as such. But this is not simply problems of multiverse models in the cosmic model zoo.

Hi Captain Paul,

I don't even know what you mean with 'accelerating virtual particles near mass/energy'. There are approaches to describe space-time as emergent from other particle/fields via condensed matter analogies and such. There are also relational approaches that try to convince you that either space or time or both don't exist. (As you can tell I'm not a huge fan of this idea.) That's all that comes to my mind. Best,

B.

Thomas,

Thanks. Now that I see it, I believe I've seen it before. Don't have much to say about it though. Best,

B.

L. Edgar Otto, Robert,

Well, Robert is right in that there hasn't yet been any *direct* detection of gravitational waves. We do however have excellent indirect evidence. And hopefully soon direct one as well. Nobody in their right mind doubts the existence of gravitational waves at this point. Besides this, PLEASE don't call them gravity waves. A gravity wave is a cloud phenomenon. Best,

B.

"Gravitational waves have never been observed.

Never! "

As Sabine says, "tional waves. We do however have excellent indirect evidence. And hopefully soon direct one as well. Nobody in their right mind doubts the existence of gravitational waves at this point."

Readers can draw their own conclusions.

Several new species of animals and plants are discovered every year. Is this evidence that biology is fundamentally flawed and in need of a new paradigm? No.

I took a couple of decades to observe neutrinos after they had been predicted. Did this mean that Fermi was making speculative, ad-hoc predictions? NO.

Hi Bee,

perhaps you may find this a bit interesting.

http://arxiv.org/abs/1311.4979

Hie Bee

In your opinion what are the criteria for a self consistent quantum theory of gravity?

"Nobody in their right mind" doubted Newtonian gravitation until Einstein came along with his "wrong mind".

On first hearing about Einstein's new research on gravitation Planck told Einstein that he was almost certainly wrong, and even if he were right nobody would believe him.

Also, "nobody in their right mind" questioned simultaneity or the "divine creation of life".

Get the point?

Hello Bee,

the picture from Georg Koch reminds me of

ads from Wissoll chocolate in the 50ties.

http://www.visualitaeten.de/79901.html

This pictures influenced my vision of

infinity when I was a child.

Georg

There seems to be a tradition of spit minds on these issues. QM vs GR ; Stringy vs Loopy ; Asymtopic freedom vs Asymptotic safety in the conceptual

growth of details of inquiry to this day

Sabine in her blogs have touched on all the issues including this idea of something as a more general background with questions of infinity as phenomenal physicality in the illustration above of s Koch grid.

But the grid is a deeper level of concepts beyond existing math or physics as foundational. Beyond if a scale may concern concepts quasi finite or rejected to but one side that appears as a total theory to the exclusion of others. In perturbative uncertainty as the free will or determinism issue where all stands on QM the stringers reject such tablecloth fractal infinite regressions outright along with any ideas involving spin foams and wider ideas of ultimate dimensional group and energy steps of chain rule reduction

In effect there is not enough infinity as a physical unity to rotate Feynman diagrams and no intelligible linking of any spinning complex spaces beyond their limited although asymptotically infinite dimensions.

The use of the third thermodynamic law to refute QMG as the foundational law is an arbritary perhaps coincidence of choices applied that denies indirect and unified evidence of force and particle nature that excludes that beyond the standard theory which does not allow the full meaning of uncertainty as a pure QM physics.Do particles actually contain heat or is this s metaphor invoking eternal mystery, a measure of ignorance, or acceptance of illusions.

Clearly Lorentz can be extended into a wider unified theory. GR and QM with it.

My childhood idea of infinity was also inspired by a product label. A Pet Milk can with a cow holding a can with a cow holding and so on... then to walk down a hall with facing mirrors and in the blurring light trying to decide if the reflections went on forever. I tend to choose the finite when I have to choose. (but my quasic grid being more general has shown me better.)

Geometric models are not the foundational grounding either. We could interpret the unified model a either how Sabine describes or as a quasifinite grid. But it takes both concepts to begin a physics of reality, the Omnium if we can keep our ideas of dimensions staeight in symmetries and inquirey.

Yes, Robert, thank you, I understand you just fine. I don't think you understand what I say though.

Stuart,

Self-consistency is a mathematical requirement, it's not a matter of opinion. A theory that leads to internal contradictions is not self-consistent. Combining classical gravity with quantum field theory for example is for all we know not self-consistent. Best,

B.

The main idea of the asymptotic safety for gravity was very nicely explained by Weinberg in arxiv:0908.1964. Essentially, it may happen that infinitely many couplings

one have in gravity may belong to a finite-dimensional surface, so that one can describe the theory with a finite number of parameters.

"Nobody in their right mind" doubted Newtonian gravitation until Einstein came along with his "wrong mind".Well, there was the unexplained precession of the perihelion of Mercury, which indicated that something might be wrong with Newtonian gravity, and there was. Einstein's theory also made testable predictions which were different from Newton's, they were tested, the theory was confirmed and the rest is history. Also, Einstein's theory has Newtonian gravity as a limit. So, the comparison between yourself and Einstein is not a good one.

10 points for each favorable comparison of yourself to Einstein, or claim that special or general relativity are fundamentally misguided (without good evidence).

10 points for claiming that your work is on the cutting edge of a "paradigm shift".

I know very well what you said, which was "Nobody in their right mind doubts the existence of gravitational waves at this point."

Nice try, but no soap.

---------------------------

Mr. Helbig! Only a moron would conclude that I was comparing myself to Einstein.

You are a font of shabby disinformation. Get some integrity.

Perhaps it is a minor point but a strictly surface to volume holographic principle on meso scales at least, an idea for instance time is an emergent or illusionary property is not fundamental. Thus an intelligible but limited constent theory that may not account for hidden (or fine tuned) mechanisms while a part of a more general picture.

Differences in phase sine waves can add parameter value as measure which could be evidence directly but it needs not be interpreted as inflation.

This higher physics of time suggests a theoretician takes time to develop theory yet already given the total vision at birth to which as a measure of genius it integrates somewhat at any instance after and over a lifetime.

So I used a lot of soap exploring soap bubbles taking long hours in my uncles bathtub until he finally asked me what the hell I was doing. I told him I was trying to blow doughnut shaped bubbles. 1968. He replied, "Let me know if you do, every kid on the bloc will want one. "

OK, we all know the formal mysteries and issues involved. But it still brings out childhood wonder that the equivalent is two spheres with a flat disc between them.

There is no reason we cannot bring a little more order to Karnough maps or topological methods for the logic of it all. Nor in our digital notations with or without valid loopy circular reasoning despite the thermodynamic differences in laying down memories and retrieving them, subsets of encoding as in BCD as a 4 base code for decimal digits or that from any sequence we can eventually find any other (Shakespeare or Bacon? Newton and his Bible code parallels? ) Yet if we so quantize sequences and actually impliment it in DNA do we not get a lot in such a small space?

I would like to think there is s little Einstein in all of you, but hoe that can be measured is still beyond the modest scope of our inquirey save whom we imitate or emulate.

On the other hand it occurs to me tonight that there is no known reason to consider any theory of waves involving gravity any more consistent than any belief. Thus I have to consider I am not among those in my right mind at the moment anyway for I do find another path to explore as a possible way out of where the information goes if not another dead end and the universe and society ultimately makes sense.

Why the standard particle model or standard cosmology if we find no direct evidence. Why are we expecting some cosmic evidence in the background as the "B" mode?

New Scientist has an article today where an obscure paper from hundreds of years ago predicts by the earth centered shell models and translated into formal computer terms suggest this model predicted inflation and although the author did not explicitly say so raised questions of a multiverse said to plague today's models. Do we design such language and not expect it may be consistent by design, running out of anomalies or direct observations never to be seen? Is magnetism the swan song left to explain black hole transfers of momentum?

This is an important question as to how the next generation will treat science, why fund the last frontiers of inquirey for imagined new paradigms.?

Gravitational waves would then reasonably be in the realm of ESP ideas as interpretations.

In looking back will a Nobel prize for extreme star systems seem laughable save for the spirit of it; what if these were not physical objects could such waves independent if matter arise?

If no signal exceeds the velocity of light then how can we have wiggle room on a wave some say faster than light?

Where is that place theory corresponds that we can see things backward for decades, the Accretion of matter into planets the "Creaction" as well but not necessarily connected in mere 4 space. Einsteinian or 10 space Newtonian as equivalent descriptions?

Markus,

The scientist admits the possibility of gravitational waves, but certainly does not say they must exist.

Einstein told everyone many times that General Relativity was not the final word on gravitation and that there would be a further "deepening" of the theory.

The great failing of our era in theoretical physics has been the largely faith-based insistence on poorly, and completely untested, assumptions.

If we want progress we must question all dogma.

MarkusM, I think this blog and links going years back including commenter ideas and links to even alternative models cover the issues well enough that I am surprised at the disconnect of your depth of the question. Is there and intuitive picture here or do you assert there is no such picture?

I will give it a vague intuitive answer, try anyway although our model makers seem to miss each others point at times or what shared in vision. It seems true also that there is justification as science sometimes that a more sensible unified vision is beyond the grasp of lesser understanding as wisdom grows. The paradox is that your asking this question should always be asked to see such model differences but it is not clear this the conclusion or summation of all possible models.

So I vaguely intuitively assert the following reply:

Short of the supernatural is there something perhaps hidden but intelligible we imagine as many-world and multiverse?

There are three ways to view this two of which both may apply or together as geometric models. Leave aside what is h or c or universe as E like null (a simple parabola in Descartes quadratic symmetry plots, rather Euclidean, absolute zero and positive.

One B like model at or near singularity as if parallel continua of a universe of parallel universes is erected on a region flat at each point the region itself in hyperbolic symmetry invariant a particle or force mediator.

The other is a mediator as if spherical in symmetry but not clearly independent of scale or sign in a hierarchy question of our local range of physical dimensions h as a center scale an asymptopic point of uncertainty or perturbation. That and internal reflections of maximum reduction of maximum internal freedoms and symmetry.

As a poet I call them Hossions and Quasons as at least s wider Phenomenology needing both for a better discription of our Ominum (Reality) .

But as s poet quite happily independent to a culture of art and science where some are marginalized by our gifts of model theory and rocket makers from the world war Europe with computer diagrams and an exhausting winters going back thru my diagrams as not just art it takes awhile to recall what I was doing

Yet as always in such accretion of research looking back with better vision I find remarkable new things or paths I missed along the way by intuition alone.

I should add that where the idea of small changes can lead to greater asymotopic or exponentially of tidal effects unto chaos, these concepts in mathematics applied to spin ideas and frame dragging without details in and between formalism may remain isolated models to a higher physics. Thus the value of initial observations of experimental tests as a basis for awarding Nobel Prizes remains controversial in this realm of the frontiers of inquirey where hard established physical principles meet wider backgrounds as phenomenal.

Robert Oldershaw,

When you come up with your own replacement for General Relativity and Newtonian Theory that accounts for the Big Bang, the precession of Mercury's orbit, a whole bunch of other observations that GR explains so very well, AND that also accounts for the famous binary Pulsar observations from Taylor et al., the same observations that gravitational waves accounts for so very well, then please come back and spam us with your theory.

Robert Oldershaw,

When you come up with your own replacement for General Relativity and Newtonian Theory that accounts for the Big Bang, the precession of Mercury's orbit, a whole bunch of other observations that GR explains so very well, AND that also accounts for the famous binary Pulsar observations from Taylor et al., the same observations that gravitational waves accounts for so very well, then please come back and spam us with your theory.

I do not advocate replacing General Relativity but I do share Einstein's view that the theory should be further deepened with additional symmetries.

I have suggested that the gravitational coupling constant is not an absolute constant but has discrete self-similar scaling. This adds discrete relativity of scale to General Relativity, and is a form of discrete conformal symmetry.

The paper can be found here:

http://arxiv.org/abs/physics/0701132

and it has been published in a peer-reviewed astrophysics journal.

It's totally free so chow-down Vince baby!

Vincelovesfreefood, it is not necessary to explicitly confirm every prediction of GR. It suffices to prove that your replacement reduces to GR in an appropriate limit, because then it inherits all the successes of GR. Same goes for QFT.

String theory is such a replacement, since it allegedly reduces to GR (in too many dimensions) in some limit. You would not try to calculate Mercury's perihelion precession directly from string theory, would you?

Thomas,

I don't see why the string theory could not do it and wonder just what models cyclic or otherwise will be ruled out by this announcement. We should call "gravity waves " aether waves after all. That is why Einstein caught his mistake early on in Eddington 's data for the eclipse lensing on the force side of the equations. Buy on the energy side without scaling is the universe measurable or a free lunch of. Scaleless bootstrapping?

This is a common mistake which tonight Dyson on Cosmos suggests why (our eyes evolved in water and have not adjusted to land.) I see this as a compensation that directed our evolving multidimensional mind's eye.

Do we not know of Newton's relativity? That too the scale square root if the time.

Robert's paper is right on. On the energy side of the equation I suppose it is how many turtles up and down to count with mirror info complements.Einstein knew better.

Robert,

Well, thanks for confirming that you are in fact not able to parse my English. Nobody in their right mind doubts that gravitational waves exist means that in the face of presently existing evidence any theory which says otherwise is strongly disfavored and almost certainly a waste of time. You're welcome of course to waste your time. I hope that explains what I mean. This is, please note, not the same as saying that scientists claim the possibility has been excluded, which is what you seem to have read. Best,

B.

Markus,

Good question about the physical interpretation about the CC running. I don't know one, will watch out for it. Best,

B.

Markus,

Yes, I know... There is actually a longer discussion around that, but I can't recall the details and I don't think it's been settled. As I said, ASG has its friends and foes, see also eg this paper. I'm not familiar enough with the details to have a strong opinion on that. Best,

B.

@ Thomas "String theory is such a replacement, since it allegedly reduces to GR (in too many dimensions) in some limit. "

why do you say "allegedly "

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