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In these forums and elsewhere it is routinely agreed that "we do not have a theory of quantum gravity." My question is, how do we know that canonical quantum gravity is "wrong"? I understand that the theory is perturbatively nonrenormalizable, but doesn't that just mean that we can't apply perturbation theory to it? It seems the theory is nonperturbatively renormalizable. A theory being nonperturbative doesn't make it "wrong". My understanding is that in modern QFT nonrenormalizability is not anymore considered such a big deal, and that even the Standard Model is expected to be an effective theory modified by nonrenormalizable terms appearing at higher energies. So barring practical considerations, why is canonical quantum gravity considered "not a theory of quantum gravity?"

EDIT: I am not sure that the term "Canonical Quantum Gravity" is correct. I am referring to what is called "Quantum Einstein Gravity" in this paper. If someone knows better please help me correct my terminology. I was not meaning to imply anything about LQG because I thought that LQG "brought more to the table" than just a nonperturbative approach to the same QFT, but I could be wrong about that.

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The problem is that the perturbation series, even in the best behaved theories, is not a sufficient criteria for reconstructing the theory. In the case of QCD, you can reconstruct the non-perturbative theory by defining a path integral on a lattice, and taking the limit of a fine lattice with the coupling logarithmically going to zero as the lattice spacing gets smaller, and this makes a consistent continuum limit which defines the non-perturbative path integral. This definition is computational and absolute--- it gives you an algorithm to compute all correlation functions in the theory.

For quantum gravity, you can start with a flat metric and do a perturbation series, and get the graviton interactions. But there is no reason to believe that there is a non-perturbative theory you are approximating when you do this. The path integral for quantum gravity is not lattice regularized very well, because the lattice spacing is dynamical--- you have a metric that tells you what the actual distance between lattice points is. When you take the limit of small lattice distance, there is no guarantee that you have a well defined quantity.

Further, the path integral might include sums over non-equivalent topologies. You could imagine a handle popping out of space time and disappearing later. If this is so, and if the sum is over arbitrarily small space-time structure, then there is a serious problem, because high dimensional topologies are known to be non-classifiable, so that it is impossible to give an algorithm which will sum over each topology once and only once. You can given an algorithm on simplices which will sum over all topologies in a redundant way, by summing over all possible gluing of the simplicies. But if you think the continuum object is well defined, then it seems that the simplex sum should reproduce the sum over topologies, which is a non-computable thing. This suggested to Penrose that the full theory of quantum gravity is capable of hyper-computation (stronger than Turing computation), but I personally am sure the concept of hyper-computation of this type is with scientific certainty an incoherent concept in a logical sense, since the logical properties of hypercomputation cannot be described in any finite way using axioms, even allowing the axiom system to increase in complexity with time.

Even if you just look at the perturbation series, and try to make sense of this, there is a serious problem when the scattering of particles is Planckian or above. If you hit two particles at the Planck energies or more, you should produce an intermediate black hole state, and the sum over intermediate states should then be over the number of degrees of freedom of this intermediate black hole. But a black hole of radius R only has R^2 worth of degrees of freedom, while the volume degrees of freedom are R^3. So the scaling laws of the perturbation theory particles for the maximum amount of information in a given region, one which can contain a black hole, is not consistent with gravitational holography.

Transitioning to an S-matrix picture resolves all these problems, because it gives string theory. In string theory, the perturbation series is on S-matrix particle states, not on field states, so that the intermediate states are not localized at individual space-time points. The sum over intermediate states reproduces an extended object fluctuations, whose degree-of-freedom count is holographically consistent. The algebra of external operators is by insertions on the string world sheet (or on a brane world-volume theory), and the number of degrees of freedom on the classical limit of large branes or black holes has the correct holographic scaling. This is not a surprise for gravity, but it is not possible with a naive field theory, because the field theory has many more degrees of freedom at short distances.

t'Hooft's argument

The essence of the very wordy argument above can be explained in a short calculation by t'Hooft. He asked, given a Schwartschild horizon, what is the entropy that you can store in the fields just outside the horizon. You have a fixed energy, and you assume the black hole is enormous, and you ask, how many different microstates can you fit in the region R>2M.

The answer is easily seen to be divergent. At energy E, the redshift factor introduces a factor of $\sqrt{r-2M}$ (near horizon approximation), which shifts energies to the red region. If you fix the total energy, the number of modes of energy less than E in a volume V has a field theory scaling law determined by doing Fourier transforms in a box, and this scaling law gives VE^4 (it's the same as the vacuum energy divergence, since it is counting all the modes once and only once). Because the energy redshifts, you get a divergent integral when you look outside the black hole horizon, so that the number of states of energy less than E near a black hole horizon is divergent in any field theory.

The resolution for this paradox is to adopt an S-matrix picture for black holes, and renounce most of these degrees of freedom as unphysical. This means that the space-time around a black hole is only a reconstruction from the much smaller number of degrees of freedom of the black hole itself. This is the origin of the principle of holography, and the principle is correct for string perturbation theory.

Within loop quantum gravity, the regulator is completely different, and might not be consistent, I am not sure, because I do not understand it well enough. The regulated theory should reproduce an S-matrix type thing when it has an S-matrix, but such states are not known in the loop gravity. The knot representation, however, makes loops and cuts down the field theoretic degrees of freedom in a way that is reminiscent of holography, so it is isn't ruled out automatically.

But just doing a path integral over spacetime fields when the spacetime includes black holes is plain impossible. Not because of renormalizability (you are right that this is not an issue--- it would be fixed by an ultraviolet fixed point, or ultraviolet safety in Weinberg's terminology) but because the number of degrees of freedom in the exterior of a black hole is too large to be physical, leading to a divergent additive constant in the black hole entropy which is physically ridiculous. A quantum field theory of gravity would, if it were consistent, have to be a remnant theory, and this is physically preposterous.

I am sorry that the above sounds more hand-waving than it is, this is more a limitation of my exposition style than the content. The papers of t'Hooft are from 85-93 era in Nuclear Physics B, and the papers of Susskind on holography in string theory are well known classics.

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Forgive me if I misunderstood something, but it seemed like all of what you wrote except your second-to-last paragraph (what is a "remnant theory"?) applies to a perturbative approach to the problem. But the QFT is defined outside of perturbation theory, right? I guess you are saying that the integrals defining the QFT (which one might use perturbation theory to try to solve approximately) sum over states or degrees of freedom that we know are unphysical? How do we know those kinds of terms don't "cancel out" (like they usually do) in the end? –  user1247 Feb 20 '12 at 14:43
@user1247: It is a (very persuasive IMO) physical intuition, not a rigorous argument, but it is hard to formulate a rigorous argument regarding a theory that in all likelihood doesn't exist. The most direct way to define a QFT is by the small lattice spacing limit of a lattice path integral, and this is obviously fine for QCD, and any other asymptotically free theory. It is a little less clear for scale-invariant theories, but you can embed those in asymptotically free theories. So there is no problem of principle in defining QFT--- you can write a computer program to extract all predictions. –  Ron Maimon Feb 20 '12 at 15:06
@user1247: The problem in quantum gravity is that the small spacing limit is not clearly ok, because the spacing is dynamical, it is determined by the metric. So there is an annoying issue of small lattices that you have bad control over which part of the dynamics is in the lattice and which part is in the metric. Further, the number of degrees of freedom at energy E looks like it scales in the wrong way, which does not reproduce holography. So you need to cancel out degrees of freedom. Degrees of freedom don't normally cancel out--- there are positivity arguments against this. –  Ron Maimon Feb 20 '12 at 15:08
So you have to formulate the theory differently. This means either getting rid of most of the variables in the path integral, leaving only a discrete set of variables, and this is the loop approach, or starting from an S-matrix description where you reconstruct the space-time from the scattering, and this is the string theory approach. Of course, if both are correct, both have to be self-consistent with each other, but this is a major open problem. The loop approach integrates over geometries and has a hard time getting scattering, while strings have scattering and geometry is harder. –  Ron Maimon Feb 20 '12 at 15:09
@user1247: A "remnant theory" is an obsolete idea that black holes do not decay completely, but leave the entropy behind in a pointlike small black-hole remnant, which is the end point of black hole evaporation. This was never really a serious possibility, because it is so physically absurd to pack so much entropy into such a small thing, but it is what quantum fields on curved space suggested would happen, because there is a diverging degree of freedom count near any black hole horizon, leading to an additive divergent constant in the area/entropy law. –  Ron Maimon Feb 20 '12 at 15:12

Loop Quantum Gravity is an example of a non-perturbative approach to canonical quantum gravity. In fact it's mentioned in the Wiki article you linked to. Inevitably views about LQG vary, but no-one has proven it wrong in the sense that it is experimentally disproven or mathematically inconsistent.

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Perhaps I have the wrong terminology, which perhaps you can help me with. By 'canonical quantum gravity' I was hoping to imply the naive attempt to quantize the E-H action (as in the 1960's) the same way other Lagrangians were quantized (like QED/QCD), which I would have thought would be more or less unique. From the article on LQG: "The native mechanisms in quantum gravity (successfully applied to other fields) fail to quantize gravity." This is what I am referring to. How do they "fail" other than perturbatively? –  user1247 Feb 20 '12 at 10:55
@user1247 The term "Canonical quantum gravity" is sometimes specifically used to refer to the quantization of the Hamiltonian description of GR (GR with foliation by spacelike hypersurfaces, lapse/shift etc). Some of the problems with this quantization program are discussed here (of course Isham is concentrating on the role of time in the problems). –  twistor59 Feb 21 '12 at 8:20

There is indeed nothing wrong with it; people (e.g. 't Hooft) have done perturbative calculations with canoncial quantum gravity (or simply with a L=R^2 Lagrangian). In this sense it's seen as an effective field theory at lower energy scales. Google on 'Wilsonian effective field theory' for more information (I couldn't find a Wiki page).

However, we would like to know what happens at higher energy scales (regardless of the question if they ever can be reached with earth-based experiments). There is a strong feeling that there should be some more fundamental theory which has GR (and the standard model, for that matter) as it's lower effective descriptions.

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But we don't know what happens at higher energy scales in QED or QCD, and yet we don't call those theories "wrong." Is it just propaganda? It seems like it is incorrect to say that we haven't married QM with GR, and yet it is one of the most common things to hear theorists say. –  user1247 Feb 20 '12 at 11:00
QCD and Electroweak theories are grounded in experimental data at the energies we have studied. They can be viewed as a shorthand of what the data tells us, right and wrong have no value in this case.The same for GR as far as we have data checked. Any theory of gravity, and certainly a Theory Of Everything, should have as a limiting case the appearance/embedding of the Standard Model and GR, otherwise it is just a mathematical game. String models promise this, and in addition allow the extension to high energies that should have held at the Big Bang, and that is why they are popular. –  anna v Feb 20 '12 at 11:36
I understand why a TOE should have as a limiting case the appearance/embedding of the Standard Model and GR, but why any theory of gravity? If we have an action for classical GR and we can successfully quantize it the same way we do for classical electromagnetism, it seems strange to say "we don't have a theory of quantum gravity." –  user1247 Feb 20 '12 at 13:21
If all You`re up to is QG (leaving the other forces out of the game) would it then not suffice to successfully get back GR as the classical limit ? –  Dilaton Feb 20 '12 at 13:34
"it seems strange to say "we don't have a theory of quantum gravity."" The implication is in the continuation: a theory of quantum gravity for our space and time, i.e. incorporating the SM and GR at the same time. There could be many theories of science fiction (solid mathematically) gravity. –  anna v Feb 20 '12 at 13:52

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