Tuesday, November 28, 2006

Non-metric gravity and renormalizability

Kirill Krasnov has asked me to debunk his paper
Now, I would really be happier if my reading could end up as an announcement of an important breakthrough ;-) but after having read his paper, it seems that he will get exactly what he asked for: a debunking.

Let me start with some general comments about the whole framework.

General strategy

First of all, there are many papers that attempt to rewrite gravity in different variables, combine them and recombine them, in order to get a better result. As we will discuss in detail, I think that this whole philosophy is fundamentally flawed. A change of variables is one thing but physics is an entirely different thing. In physics, we want to know what is the Hilbert space - the spectrum of particles - and what are the amplitudes (and consequently cross sections) of their interactions.

These amplitudes are completely physical and measurable in principle (up to an overall constant phase) and they cannot depend on your nationality, sex, or your choice of variables. If these people actually ever tried to compute at least one physical observable instead of this mumbo-jumbo with their formalism, they would have to see the same thing that everyone else does: the basic qualitative physical insights about quantum gravity are true and independent of any changes of your variables.

The coupling constant of gravity, Newton's constant, determines the length scale or the energy scale where quantum effects (such as loops in the Feynman diagrams) become as important as classical effects. The relative importance of the tree diagrams and multi-loop diagrams can be calculated by a simple dimensional analysis and the result doesn't depend on any field redefinitions. For example, the characteristic dimensionless coupling constant goes like "G.E^2" where "E" is the typical energy in your experiments and "G" is Newton's constant. Because it becomes of order one near the Planck scale, a better theory is needed as seen from the simple behavior of the "quantum foam" - the violent and topology-changing fluctuations of the metric near the Planck scale.

Kirill Krasnov seems to deny most of these facts, including the existence of the quantum foam. If there is an argument in that paper supporting such a denial, then I have missed it but it seems absolutely manifest to me that such a conclusion has to be wrong.

The self-dual split

OK, let's try to play Kirill's game anyway. He starts with a tetrad form of general relativity but the precise Lagrangian (unlike its variables) doesn't seem to play any role in the paper so we will ignore it. Then he recalls the first Plebanski self-dual Lagrangian for gravity. Its degrees of freedom are
  • B^a, an SU(2) Lie-algebra-valued two-form
  • Psi^{ab}, a two-form that contains the Weyl tensor components on-shell
  • A^a, an SU(2) gauge field.
Plebanski effectively worked in the four-dimensional Euclidean spacetime whose Lorentz group, SO(4), is isomorphic to "SU(2) x SU(2)": a special feature of a four-dimensional spacetime. The self-dual part and the anti-self-dual part are treated separately. Note that in the Minkowski space, these two sets become complex conjugates of each other so that they can't be separated as long as you consider Hermitean observables.

The metricity condition for "B^a" is that "B^a B^b" is proportional to the identity matrix "delta^{ab}" in the color space which implies that "B^a" can be related to the self-dual part of a two-form from the one-form formalism. The field "Psi^{ab}" acts as a Lagrange multiplier that imposes the metricity condition for "B^a". When the equations of motion are satisfied, "Psi^{ab}" itself contains components from the Riemann tensor.

The first intrinsically technical among Kirill's controversial statements is that the gravity action is effectively independent of Newton's constant because this constant can be reabsorbed to different fields. On physical grounds, this can be clearly seen to be wrong because the value of Newton's constant is quite physical. For example, its large enough value implies that we can't jump to the Moon.

Kirill says that Newton's constant only becomes physical when gravity is coupled to matter. That would be true in a classical theory. In a quantum theory, it's just wrong. Newton's constant determines the coefficient of the gravitational Einstein-Hilbert action, not the matter action. If any theory is called a theory of gravity, it should reduce to the classical Einstein's or Newton's laws at long distances. At some shorter distances, this classical approximation must break down. The scale where it breaks down is called the Planck scale and who believes that it never breaks down is a quantum mechanical denialist. Alternatively, you can define the Planck scale as the absolute value of the mass of the lightest massive black hole microstate that appears as a pole in the graviton-graviton scattering amplitude on the second sheet.

All these comments are completely physical - in principle, they're experimentally verifiable - and they don't depend on any field redefinitions as long as the field redefinitions are done properly.

Kirill argues that physics of gravity, when written in the Plebanski variables, is independent of the coupling constant. That seems obviously incorrect.

Witten's twistor approach to the N=4 gauge theory is also an expansion around a self-dual theory that is very analogous to the present case. We could make the same argument about the disappearance of the gauge coupling. Such a conclusion would, of course, be completely misguided. One must carefully remember the different powers of the gauge coupling that must be added to various diagrams. The power of the coupling constant may come from some unusual, topological terms in the action. But whatever their source is, it is clear that this dependence on the coupling constant can't disappear because it is a part of physics.

Whenever you split the variables to the self-dual and anti-self-dual ones, you must be careful because you are generally using a different scaling for these two parts. But their ratio still carries the information about the priviliged length scale - the Planck scale - in the gravity case or about the gauge coupling in the Yang-Mills case. In the Minkowski space, the relative scaling of the self-dual and anti-self-dual parts must really be one as long as they are complex conjugates to each other which seems necessary for you to be able to construct real actions whose reality seems to be necessary for unitarity.


Fine. So why does Kirill believe that this field redefinition magic makes the theory more convergent than the standard treatment? He argues that all the counterterms that are generated by UV divergences are of the form
  • function(Psi) B^a B^b: equation (4) in Kirill's paper
or, alternatively, you can redefine "Psi" to "Phi" so that all these counterterms are hidden in a different part of the Lagrangian. Later in the paper, he also speculates that one could also possibly generate a whole new pile of counterterms that have a different form, something that I am not able to confirm or deny.

Back to the "Psi" - "Phi" field redefinition. Whenever more than 10% of a paper is spent by one particular field redefinition, in this case the field redefinition in equation (5), you feel some compassion for the brave person who struggles with these vacuous operations so courageously even though you know in advance what the result will be.

Why does Kirill think that the counterterms he generates are less severe than the usual terms generated in the normal variables? Frankly, I have no idea. As we mentioned above, if you're on-shell, the components of "Psi^a" are nothing else than components of the Riemann tensor. So all the counterterms that are polynomials in "Psi^a" are nothing else than the usual polynomials in the Riemann tensor.

Kirill could have gotten confused by his unusual choice of dimensions in which "Psi" is dimensionless which could have led him to the idea that his counterterms are not higher-dimension operators. But they are. These choices don't change anything about the fact that "Psi" is really a curvature, something that we can actually measure as the tidal forces and other things. And even if you hypothetically did obtain an arbitrary counterterm-functional of dimensionless "Psi" variables, you would still be in trouble.

In fact, if one does the calculation properly, they should be exactly the same counterterms that one needs in the normal perturbative approach to general relativity, although they are written in different variables. He rewrites the action in terms of "Phi" instead of "Psi" in equation (6). Normally you would have no clue what he can ever gain by these rather straightforward operations. However, below equation (6), he summarizes all the counterterms that are at least cubic in "Phi" as "O(Phi^3)".

That almost looks like he wants to neglect them. Still, this nearly invisible "O(Phi^3)" contains all the infinitely many counterterms - the same counterterms - that are the technical reason why pure gravity can't be perturbatively quantized. Kirill says that the divergences in his theory are "clearly" much milder than in the normal treatment of GR which is why he calls the theory "quasi-renormalizable". As far as I understand, the adjective "quasi-renormalizable" is a politically correct word for "non-renormalizable": it's a theory with infinitely many types of divergences whose predictive power - besides the statements about the spectrum and symmetries that you put in - is equal to zero because you need to determine infinitely many parameters.

Kirill's counterterms could only depend on the self-dual part of the curvature (Psi) but it doesn't matter because his equation (7) shows that the metricity condition for B - something that is known exactly to all orders in normal GR - receives divergent corrections at all orders. So what he really does is to split the higher-derivative corrections to the effective action into the self-dual ones and others, and treats them separately. I don't know whether it is an interesting manipulation but what I know is that it can't change anything about the required counterterms in general relativity when it's done properly.

Later, Kirill only looks at the coefficient of the lowest new power of "Phi", namely "Phi^2", and claims that the coefficient of this new coupling is dimensionless (an overlined "g"). What the adjective "dimensionless" really means is somewhat obscured by Kirill's unusual absorption of Newton's constant into the fields. Had he used the conventional normalization of the action with the overall "1 / G" in front of it, his leading new term would have the same dimension as the familiar "R^2" terms. The coefficient contains a "length^2" besides the usual "1 / G" factor from the Einstein-Hilbert term. Every time you add a "Psi", you must add a "length^2" to the coefficient. In perturbative string theory, this "length^2" is of order alpha'. "Psi" or "Phi" is nothing else than some curvature.

Kirill then studies the running of this "overlined g" coupling constant which is, as we said, a coefficient of some "R^2" term, and argues it is asymptotically free. I forgot what the sign is but it is not too important anyway. This discussion is not physically crucial because there are three possible independent "R^2" terms in GR and all of them can be seen to be unimportant. One of them is the Euler density (the Gauss-Bonnet term) that is topological and doesn't influence perturbative physics. The other two depend on the Ricci tensor only and can be removed by field redefinitions: add a multiple of the Ricci tensor and of "R.g_{ab}" to the metric tensor "g_{ab}". Equivalently, you can say that the Ricci tensor is zero on-shell using the tree-level equations, so these two terms don't contribute.

Kirill speculates that all beta-functions in the theory are negative which would make GR as asymptotically free as QCD. This is essentially the old conjecture about the gravitational UV fixed point but I don't think that Kirill has any new evidence for this conjecture or a new identification of the Hilbert space that describes the UV physics if the conjecture is true. What are the excitations in this regime? Is the hypothetical UV fixed point a chiral theory including just the right-handed gravitons coming from the self-dual fields? This is just my conjecture that is not mentioned there but that at least offers an intriguing possibility. But I have no idea how you could make the left-handed gravitons disappear because they should appear at low energies and low energies are really equivalent to high energies for a single particle by Lorentz invariance. Removing some of the polarizations requires us to break Lorentz invariance.

Healthy one-loop graphs in general relativity

Kirill's test of the "Psi B B" terms was encouraging for him. Indeed, at the one loop level, there is no problem with general relativity, regardless of your choice of variables. We have recently discussed these issues in the context of finiteness of supergravity. The real loop problems of pure 4D gravity only start at the two-loop level. They resulting "Riemann^3" terms - in fact, "Weyl^3" terms - would get translated to "Psi^3.B.B" terms in Kirill's language and higher, and because Kirill apparently doesn't say anything about these terms, I think that his paper really says nothing about the divergences in quantum gravity. It's classical general relativity presented in such a way that it looks more complicated than it is.

There are calculations of a one-loop beta-function. I would like to emphasize that these beta-functions are determined by the logarithmic divergences which is why they're completely known just from the classical limit. They don't depend on any underlying theory. And we know what they are.

At the end, Kirill advocates a breakdown of GR and the replacement of dark matter by a MOND-like theory that could perhaps result from his formalism: the link is unclear to me except that the cosmological constant is incorporated into this guess in a way I don't follow.

After the dark matter was directly observed a few months ago, such alternative theories for dark matter have become really suspicious and I think that if the reality is described by a nice alternative without dark matter, its discoverer will have to complete her theory entirely before she is taken seriously by anyone else - simply because such a framework seems extremely unlikely right now.


I think that the hope that a change of variables will remove a physical inconsistency is logically inconsistent and impossible. The Feynman graphs written in the normal variables parameterize all the possible terms and most of the physics they imply is inevitable.

Throughout the history, the ultraviolet divergences only got cured by modifying the UV physics and adding new degrees of freedom and I am convinced it was no coincidence.

As the recent twistor uprising has demonstrated, the decomposition to self-dual variables may be a great tool to efficiently calculate some diagrams and to explain why some of them sum to zero. But it doesn't change anything about the results. The results can be obtained otherwise as long as the pictures are equivalent.

I am convinced that until someone in this "field redefinition" community has a result or a non-trivial condition that can be phrased in the normal variables and words that every high-energy physicist is familiar with, such as the metric tensor and the S-matrix, the serious physicists I know will continue to fully ignore this kind of papers because it just seems guaranteed at this moment that there can't be any interesting physics in them.

And that's the memo.

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