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First of all I, it is my understanding that the problems one encounters with the non-renormalizability of gravity are very similar (if not the same) as one encounters in any non-renormalizable theory. As long as you are doing your calculations well-below the scale that suppresses irrelevant operators your calculations should make sense. As you approach that scale that suppresses irrelevant operators there is new physics that comes in at that scale - if you try to push your effective theory to the scale of new physics, perturbation theory will break down. This to my knowledge is the story that underlies the failure of Fermi-theory that necessitates introducing W bosons, and the failure of the standard model with out a higgs that necessitates the introduction of a higgs.

My questions is, is gravity, a priori, any different then the above 2 scenarios? That is, we need new physics to come in at the planck scale to UV complete the theory (or, put it another way, unitarize graviton scattering), but could it be as simple as one to a few new particles? Or for other reasons does it have to be something as drastic as String Theory or Loop Quantum Gravity? By drastic I mean paradigm shifts where its not some small modification to the current theory consisting of a few new particles, but more of an complete overhaul.

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I think you've just outlined the historical route to string theory :-) People tried for a long time with adding bits to the known particle physics model, but with each addition new problems cropped up, until they all get solved (maybe) with strings. LQG can be seen as a very conservative attempt to quantise pure gravity (one may chose ones poison with respect to the matter content of the theory) in line with the original qauntisation of QED --- find the appropriate symmetries, and deal with quantisation ambiguities as they come up, or just guess at a theory with the right semi-classical limit. –  genneth Apr 20 '12 at 14:44
@genneth: The historical route to string theory was not by adding particles, but by assuming that there would be a theory without fundamental particles at short distances, only linear Regge trajectories. This is a radical idea, not a conservative idea, and it is only called a conservative idea because those radical people who made the theory were politically removed from physics, and their work reappropriated by a new generation in 1984. The renormalization was secondary in S-matrix theory--- the main thing was to reject the over-counting of short-distance DOF's in QFT, using Regge theory. –  Ron Maimon Apr 20 '12 at 16:38
Related: physics.stackexchange.com/q/3636/2451 –  Qmechanic Apr 21 '12 at 8:54
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3 Answers

up vote 5 down vote accepted

First of all, loop quantum gravity is a model inconsistent with the existence of the spacetime, gravity, and Lorentz invariance. It doesn't solve any problem with the non-renormalizability of general relativity, either. Instead, the problem of infinitely many counterterms reappears in the infinite number of ambiguities in the "canonical Hamiltonian". See e.g. this most cited loop quantum gravity paper of 2005 and a related paper from 2006:


So we're only talking about string/M-theory here because it's the only known and probably the only mathematically possible consistent quantum theory that includes gravity.

This was a rudimentary correction of a misleading statement in the original question.

Now, to answer the question, general relativity differs from Fermi's theory or a Higgsless theory of massive gauge bosons because in those QFT examples, a consistent short-distance completion that is a local quantum field theory itself exists. In both examples, one finds massive particles at a high scale – the electroweak scale in both cases, in fact – which cure the divergences.

In particular, the seemingly contact (delta-function containing) four-fermion interaction isn't quite a contact one. It comes from the exchange of the W-bosons (and Z-bosons) that are massive. One may say that the original direct, contact, non-renormalizable four-fermion interaction doesn't exist at all; it is an effective description of some deeper, renormalizable interactions coupling the fermions to a gauge boson (cubic vertex).

For a different example, the addition of the Higgs boson cures the problems of the WW scattering but it doesn't change the fact that the W-bosons interact with one another; one only adds an additional term that makes the tree-level amplitudes unitary and that allows the theory to preserve unitarity at higher loops, too.

None of these two scenarios can work in naively quantized general relativity because the source of the non-renormalizability are gravitons, massless particles. They directly interact with each other because this is guaranteed even by the low-energy limit of Einstein's equations which are almost directly implied by the gauge symmetry underlying the metric tensor field (a gauge field of gravity), the diffeomorphism symmetry. Because the basic nonlinearities of the Einstein-Hilbert action induce long-range forces between the gravitons, these interactions can't be due to the exchange of a massive particle. That's why the gravitational interaction can't be generated as an exchange of a massive particle, to emulate the example of the W-boson that is being exchanged to produce the four-fermion interaction.

On the other hand, it's not enough to add some massive matter fields whose exchange cancels the bad behavior of gravity, something that would emulate the "added Higgs to regulate WW scattering" toy model. It's because quantized general relativity produces counterterms that depend on the metric only, like various $R^3$ terms at 2 loops (in the case with no SUSY), and by adding new matter fields, we're just increasing the number of possible types of counterterms depending both on the metric tensor and the matter fields, so these extra matter fields make the situation even worse, not better.

A possible and partial counterexample could be the $N=8$ supergravity which is believed by some people to be perturbatively finite. However, most of the best experts believe that the new counterterms start at the 7-loop level and even if the theory were perturbatively finite (all divergences cancel to all orders), it is clearly non-perturbatively inconsistent (because it doesn't contain the objects charged under the $U(1)$ groups with the correct Dirac quantization condition: one needs the stringy/M completion for that again) and it is phenomenologically unviable because $N=8$ SUSY is too much of a good thing. To consistently break SUSY in $N=8$ SUGRA, one has to apply "stringy" ways to break it, namely by more complicated compactifications of the maximally supersymmetric theories.

One may say the only way to make gravity consistent at high energies is to admit that the graviton is composite. However, quantum field theories don't really allow composite massless particles. The only loophole is string theory which makes graviton a "bound state of string bits", a closed string, and a theory of this kind is exactly the right compromise between the degree of "novelty" that is needed to deal with the harder problem, and the conservativism that is needed not to deviate from the safe consistency waters of quantum field theory. As David Gross would say, string theory is a radically conservative extension of the principles of physics. One may see that despite its not being a quantum field theory in the spacetime in the strict sense, it obeys many conditions and inequalities that may be derived for strictly local quantum field theories and other conditions.

Alternatively, one may consider stringy theories of gravity in various backgrounds, especially AdS-like, to be fully equivalent to a quantum field theory – but one on a spacetime with a different dimensionality. The AdS/CFT is yet another way to see that string theory is radically conservative. Not only string theory avoids radical departures from the rules of QFT that were apparently needed for consistency in the past decades; it is actually fully equivalent to a QFT.

There are many other ways in which string theory is "just a more clever way" to deal with quantum field theories. Matrix theories are QFTs that are also equivalent to sectors of string theory, for a large number of colors, much like AdS/CFT. One may talk about effective field theories and in the perturbative realm of open strings, one may extract all the amplitudes from string field theory, too. String field theory is a rather minor generalization of a quantum field theory although a string field is equivalent to infinitely many fields.

I haven't written the obvious point that gravity corresponds to the dynamics of the whole spacetime which makes things hard by itself. In particular, one cannot have any "universal cutoff scale" that removes excessively high-energy excitations of gravitons. It's because the diffeomorphisms change the proper distances between fixed points in the coordinate space (or spacetime). So many of the procedures to deal with the divergences don't work.

Also, because gravity contains the diffeomorphism group as the gauge group (one needed to remove the negative-norm polarizations of the graviton), general relativity admits no local gauge-invariant operators, see e.g.

Diff(M) as a gauge group and local observables in theories with gravity

That's why one can't talk about the local Green's functions in general relativity, at least not in a formalism that would preserve the Lorentz symmetry. Instead, only the scattering S-matrix may be computed in a manifestly Lorentz-invariant way. That's exactly what string theory does; even if we neglected that gravity shouldn't allow us to calculate the local Green's functions, string theory would force us to realize and learn this fact – not by the general wisdom and thoughts based on field theory but by the cold and indisputable formalism in which string theory just generates the answers.

There are several philosophical attitudes to learn general insights about quantum gravity – some of them build on arguments based on quantum field theory and its decent effects and consistency constraints; others start with very particular, well-defined vacua and calculational schemes in string theory that give us "some examples" what may happen in consistent theories of gravity. At the end, the lessons learned by both of these approaches agree. This agreement has been getting increasingly explicit in the recent decades.

I am afraid that it would make no sense to try to include loop quantum gravity into this discussion because the proponents of loop quantum gravity don't deny just string theory; they deny much of the key material learned in quantum field theory during the last 40 years, too, including the lessons about the renormalization group (especially that it's the infinite ambiguity, and not the "infinities" by themselves, which are the problem), various phases of gauge theory (which have been mapped to the behavior of string theory), the role played by SUSY, monodromies, emergence of extra dimensions in various ways, and so on.

It's really inevitable for the loop quantum gravity proponents to deny most of the modern insights and realizations (both technical and philosophical ones) about quantum field theory because these insights really do imply that approaches such as loop quantum gravity or anything based on "atoms of space" are inevitably inconsistent and have nothing to do with the "real problems". It's really the conservative wisdom about quantum field theory that implies that string theory is the only possible consistent quantum theory of gravity. Thirty years ago, people would be split into various camps (like SUGRA vs string theory) but this ain't the case anymore. People understand that they're investigating many aspects of the same structure from various directions and they understand how some people's findings match other people's findings obtained in a different way so that the whole structure makes sense. It's just a fact that loop quantum gravity or any other research of "atoms of space" fails to be a part of this structure of modern physics.

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+1 for accurate technical comments (as far as I can see), as usual, but I don't think you need to be so pessimistic--- perhaps LQG is only missing SUSY to ensure flat space--- it's almost as hard to get flat space is SUSY-less strings (you need a SO(16)xSO(16) projection of SUSY heterotic strings). All near-flat backgrounds have used SUSY to enforce this, and loops lack SUSY. Further, the large N limit in Matrix theory can be reasonably viewed as a "atoms of space" theory, because it reproduces an approximate continuum spacetime from a finite calculation. –  Ron Maimon Apr 20 '12 at 16:24
@RonMaimon: the comments on LQG are certainly not accurate. There now exists a completely finite theory which has every indication (some might even call it a proof) of having GR as the semi-classical regime. I always find it hard to vote one way or the other on Lubos' answers --- there's so much that is right, tarnished with casual, disingenuous and misleading nuggets. –  genneth Apr 20 '12 at 17:15
@genneth: You are not right. I have looked at the literature on LQG, and there is no full theory there--- there are only indications of a theory. There are problems of exactly the kind Lubos states, and there is no way you have an embedding of weak field GR today--- this is a major unsolved problem. But I am optimistic that it is possible that there is a theory along these lines, but the fine-tuning of the immirizzi parameter is a problem, as is the restriction to 4 dimensions. You need a principle that turns a spin-network predominantly flat--- and this principle is not present today. –  Ron Maimon Apr 20 '12 at 18:26
@RonMaimon: you've just outed yourself as being not up to date. There is no fine-tuning required of the Immirzi parameter (Asheketar et al, PRL 2011). There have been a boatload of papers in the last 12 months showing that the semi-classical spinfoam amplitude is the exponential of the Regge action. You say that the restriction to 4D is a problem --- aesthetically I think that's a bonus --- we only see 4D! It is the highest hubris to claim that because string theory would prefer more that implies it is our empirical observations which are incorrect. –  genneth Apr 20 '12 at 18:54
@genneth:I'll look at the recent papers--- you're right, I haven't been keeping up with this. But I think the skepticism is appropriate as a default position, since there is a lot of hype here--- the reason I want higher dimensions is because string theory is fully consistent, so if there is a gravitational representation in terms of spin-networks, it should work wherever there is a quantum gravity, whether it's the world we see or just some hypothetical world we don't see. –  Ron Maimon Apr 20 '12 at 19:03
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You can't unitarize gravity with a small modification, because the number of degrees of freedom is different at short distances--- it really has to become some sort of discrete structure, although that is not a naive discrete structure, and it must be holographically correct. The holographic principle is what prevents a naive completion of quantum gravity, and it is what all-but-requires strings.

I agree with Lubos's technical comments regarding LQG, but not with his pessimistic conclusion. The idea that one can build up a spacetime from spin-network loops in some limit is interesting, and it might only take a small addition from string theory (like incorporating supersymmetry) to make the theory work well. The major problem with LQG is that it is restricted to 4d the way I see it (I only read a little), and this makes it very unlikely that it will give something that can link up to the known quantum gravity in string theory in any obvious way. But I am optimistic that the mathematical ideas, which are so orthogonal to string ideas, can link up to something coherent.

The reason gravity is different was established by t'Hooft in the late 1980s. The path integral near a black hole has an infinite entropy, because of near-horizon memory. In a classical space-time around a black hole, you can shove infinite amount of entropy near any black hole horizon, no matter how small, because the time-dilation factor diverges along with the local Hawking temperature. If you have fluctuating thermal fields near a black hole, to have a sensible Hawking entropy, you need to cut off the space-time in some way, and to make a mathematically precise idea of how to cut-off spacetime, you need a holographic reconstructed space.

These ideas mean that the degrees of freedom at high energy in quantum gravity are much reduced as compared to a quantum field theory. String theory makes this reduction using Regge theory with its soft scattering, and it makes a consistent description--- the result describes the holography near a black hole. Loops try to produce a geometrical gravitational description of spacetime, but the resulting structure is also much smaller in size than a path integral over classical fluctuations, and ends up being a different kind of space-time atomic structure than Matrix theory or AdS/CFT.

But in order for loops to be consistent with holography, you have to adjust a free parameter, the Immirizzi parameter, to be some crazy unnatural value, and there is no fundamental reason it should take this value. This suggests that there is something missing still from the loop idea. Further, one needs to be able to describe all sorts of space-time with all sorts of matter, and the additional matter changes the entropy relation near black holes. In string theory, the central charge acts as a certain kind of sum-rule for all the matter fields in the theory. There is no such analog in LQG as far as I can see.

But the two approaches are pretty similar in terms of the gross number of degrees of freedom at small distances, and both are inconsistent with the ridiculous unphysical over-estimate derived by assuming a continuous spacetime and doing a naive path integral. So both approaches are worth studying, although string theory is further along in the development.

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This is a nice and fair answer, +1 –  Dilaton Apr 20 '12 at 23:08
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The non renormizability of gravity is of a tougher kind than what was seen with the standard model as you describe. The corresponding paragraph on this link is enlightening:

For a quantum field theory to be well-defined according to this understanding of the subject, it must be asymptotically free or asymptotically safe. The theory must be characterized by a choice of finitely many parameters, which could, in principle, be set by experiment. For example, in quantum electrodynamics, these parameters are the charge and mass of the electron, as measured at a particular energy scale.

On the other hand, in quantizing gravity, there are infinitely many independent parameters (counterterm coefficients) needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since we can never do infinitely many experiments to fix the values of every parameter, we do not have a meaningful physical theory:

At low energies, the logic of the renormalization group tells us that, despite the unknown choices of these infinitely many parameters, quantum gravity will reduce to the usual Einstein theory of general relativity.

On the other hand, if we could probe very high energies where quantum effects take over, then every one of the infinitely many unknown parameters would begin to matter, and we could make no predictions at all.

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