## Saturday, April 15, 2006 ... //

### Carlo Rovelli and graviton propagator

in loop quantum gravity, an attempt described as a groundbreaking paper by a fellow blogger and included in the unfinished revolution by another blogger. It would be far too dramatic to say that I am flabbergasted but one thing is clear. The work is so manifestly incorrect that I just can't fully comprehend how someone who has attended at least one quantum field theory course can fail to see it.

But of course, yes, I am happy that people are still trying different things and some of them don't get discouraged by decades of failure - and I always open such papers with an enthusiastic hope that a new breakthrough will appear in front of my eyes. ;-)

The paper linked above is supposed to be a more complete version of Rovelli's previous graviton propagator paper. Indeed, you can see that several pages in these two papers are identical. Most of these two papers' assumptions are misguided, nearly all the nontrivial steps are erroneous, and the results are incorrect, too.

Semiclassical GR

Let us start with semiclassical gravity. At this level, the graviton propagator is philosophically analogous to the propagators of all other quantum fields you can think of - for example the electromagnetic field. You must start with a background; the simplest background is the flat Minkowski space. This means that you write the full metric as
• g_{mn} = eta_{mn} + sqrt(Gnewton) h_{mn}
Here, eta_{mn} is a background, i.e. a classical vacuum expectation value of the quantum field while h_{mn} is the fluctuation around this background that remains a quantum field and is treated as a set of small numbers. The full gravitational action can be expanded in "h_{mn}", to get

• S = integral sqrt(g) R / 16 pi Gnewton = integral (d_{l} h_{mn})^2 + ...

You see that the absolute (h-independent) term is zero because the curvature "R" of flat space is zero. The terms linear in "h" and its derivatives vanish because the flat space is a stationary point of the action; in other words, it is a classical solution. The first nontrivial terms are inevitably bilinear in "h". If you do the calculation right, you will obtain the correct tensor structure and the correct numerical coefficients. Dimensional analysis guarantees that the leading bilinear terms in "h_{mn}" contain two derivatives. Indeed, this is what you get. The bilinear part of the action can formally be written as "h.L.h" where "L" is an operator, and its inverse "L^{-1}" is the propagator.

The theory has a diffeomorphism symmetry and this local symmetry of course survives even if we write the theory in terms of "h_{mn}". In these variables, the symmetry becomes very analogous to the gauge symmetry of electromagnetism. Much like in the case of QED, this gauge symmetry must be fixed in some way to obtain a well-defined propagator; without gauge-fixing, "L" is not invertible. The steps are analogous to QED - and one extra index is what summarizes the main complications. Up to various metric tensors, you obtain a propagator that is proportional to "1/p^2" in the momentum representation, much like the propagator for all other massless bosonic fields. Its Fourier transform into the position representation is proportional to "1/x^2" (for timelike separation "x").

It is not a big deal to derive the propagator if you have a working theory.

What you need to see a propagator

We needed several completely necessary assumptions and steps to be able to talk about a propagator at all, namely
• the choice of a completely serious and fixed background (classical solution) around which we expand
• the existence of a unique quantum state corresponding to this background (even if we do thermal physics, there exists a unique state but it is a mixed state)
• the existence of continuous (or at least effectively continuous) degrees of freedom in which the action can be Taylor expanded
• in theories with local symmetries - which certainly includes general relativity - one needs to gauge-fix the gauge symmetry to obtain a non-singular propagator
• in the path integral formalism, we need to sum over all configurations - in fact, the generic configurations that contribute to the path integral are non-differentiable almost everywhere and they look like a mess to a classical physicist

These rules are satisfied in any quantum field theory (or effective field theory) as well as general relativity expanded in the derivative expansion. They are also satisfied by string theory. On the other hand, at various points of the paper by Rovelli et al., you can see that the authors violate every single one from the list of these important principles. Let me discuss these principles one by one.

Breaking the principles

The first couple of pages as well as the conclusions are dedicated to bizarre statements (treated almost as religious dogmas by the authors) that the propagator should be derived from a "background-independent formalism". Such a statement is an oxymoron. A propagator is, by definition, encoded in the coefficients of the bilinear terms of the action expanded around a particular background. A propagator depends on the background. It is absolutely necessary to choose a background. If you think that you have derived a background-independent propagator, then you can be sure that you're being dumb even though you may simultaneously impress certain people by the pompous word "background-independent".

Talking about the propagator without having a classical background is nothing else than talking about the Taylor expansion of "f(x)" without having a point "x_0" around which we expand: it's just a silliness.

Second, there is no indication that there exists a unique state in loop quantum gravity - the counterpart of "x_0" - that could describe empty space. There exists no "canonical spin network" and there exists nothing analogous either. It is hard to expand the action to the second order around a point "x_0" if you don't even have the point "x_0" itself. What the loop quantum gravity people apparently believe is that the vacuum could be some linear superposition of spin networks that look "pretty good". But that can't be the case. By the ultralocality of the Hamiltonian, such a "vacuum state" would be highly degenerate. You could choose details of the vacuum state in every Planck volume. The same argument can also be given in the spin foam formalism: there are no unique or natural boundary conditions for the spin foam.

That implies that the loop quantum gravity vacuum, even if it exists, carries a huge, Planckian entropy density. This is incompatible with the existence of physics as we know it. Incidentally, a Planckian entropy density also breaks the Lorentz invariance maximally because this huge quantity is the time component of a four-vector (a current). The state of the spin network would never look like an empty space but more like a gas heated up to the gigantic Planck temperature.

A Planckian entropy density of vacuum is much more serious a problem than a Planckian vacuum energy (i.e. than the cosmological constant problem): the energy is a sum of positive and negative contributions and can be cancelled. Entropy is never negative and it can't be cancelled.

The authors of course don't care about the problem from the previous paragraph: they look for a propagator - which is encoded in the second-order terms - even though they don't even have the zeroth order background. This problem is also related to the following principle from the list above: the model they consider does not really have any continuous degrees of freedom, so the action can't really be Taylor-expanded. This forces them to compute derivatives with respect to discrete quantum numbers at many places even though they can't prove that there exists any continuum limit; in fact, the limit almost certainly does not exist.

The following rule they violate is the gauge-fixing that is needed to determine propagators of gauge bosons - in this case the graviton propagator. They seem to argue that it is possible to derive a unique bulk-to-bulk propagator without any bulk gauge-fixing. This approach is justified by citing several other wrong papers. Most of them argue that you don't need to make any gauge-fixing in GR as long as you define the boundary conditions for the metric at the boundary of a finite spacetime region. This statement is, of course, another error. We always assume that the metric is fixed at the boundary of whatever spacetime or region we consider; but this is far from being enough to gauge-fix the diffeomorphism symmetry in the bulk.

First sentence

There are problems with virtually every sentence of their text. For example, look at the first sentence:

• An open problem in quantum gravity is to compute particle scattering amplitudes from the full background-independent theory, and recover low-energy physics. [1]

What a deep misunderstanding. The main problem with these colleagues of ours is that they are never willing to accept that they have been asking a wrong question. First of all, the term "the full background-independent theory" is meaningless. A physics theory per se cannot be background-independent. Background independence is a property of a particular way how a theory is formulated and how its predictions are computed: for example, the calculation of a propagator is always background-dependent.

String field theory is "background-independent" while the light-cone gauge matrix theory is not. But they describe the same physics. They describe the same physics in two different but equivalent formalisms. One cannot make an experiment to determine whether the world is "background-independent" or not. Background independence is just an aesthetic property of a particular mathematical formalism - a property that is useful to understand some questions more easily but one that usually obscures many other questions. These physicists would like to promote "background independence", a term that they moreover misunderstand, to a new criterion that should decide whether science is good or bad.

Eliminating inconvenient terms

One of the methods that has become essential for loop quantum gravity is erasing inconvenient terms. Does you theory demonstrably predict black hole entropy that is far too large? No problem. Pick a random subset - a right number of the microstates, describe them as politically incorrect, and erase them from your sum. The same method is applied to the path integral of loop quantum gravity, the spin foam.

Imagine how easy life would be if Green and Schwarz could have erased the anomalies in this loop quantum gravity way, instead of doing their precious calculations.

The purpose of loop quantum gravity was probably never to find new physics. On the contrary, it has always been an attempt to show that we don't need any new physics and that there exists no new physics. Loop quantum gravity may be thought of as general relativity rewritten into "new variables". The believers believe that just by writing a theory in new variables, we can cure its physical problems. Of course that it's not a reasonable expectation. For example, quantized general relativity has lethal multi-loop divergences. What happens with them in loop quantum gravity - for example in the spin foam formalism? Do they disappear? No way. They are just translated to new variables.

Degenerate simplices

In the case of the spin foam, the UV divergences reappear in the simple fact that the path integral is dominated by histories constructed from crumpled, degenerated four-simplices; in some sense, this problem occurs at the classical level - much earlier than the multi-loop problems in general relativity quantized in the standard way. See, for example, the paper by Baez, Christensen, and Egan (BCE). It's a useful paper that actually shows how some things cannot work. I don't want to argue that it is too difficult to show that loop quantum gravity is inconsistent - but still, some people could have hoped that the spin foam by some miracles removes the UV divergences from gravity. BCE show that it's not the case.

Rovelli et al. keep on changing their guess how the BCE problem (also confirmed by Barrett and Steele and by Freidel and Louapre) could be circumvented. At the beginning, they say that some extra phases could make the BCE singular contributions cancel. In the middle of the paper they realize that this proposal is undefendable because BCE did take these things into account. So they argue that the partition sum is dominated by singular configurations but the propagator is perhaps not. Well, this might very well be the case: the volume-extensive divergent term in the partition sum is the quantum-generated cosmological constant and the ad hoc subtraction of the divergent quantity from the partition sum is called a counterterm. Indeed, you may need no counterterms for the propagator at one-loop but you will need counterterms for more complicated correlators that do get contributions from the divergent part of the partition sum.

Eventually Rovelli et al. apply the usual method of LQG of erasing inconvenient terms. After spending 10 pages or so with the harmonic oscillator and free quantum field theory (apparently, a goal is to make the harmonic oscillator itself controversial), they look at the spin foam. How do they "cure" the obvious UV problems - degenerated simplices - mentioned above? The answer involves the infamous eraser of unwanted terms and for me, it is the least plausible part of the bulk of the paper.

The so-called "physical boundary conditions"

Rovelli et al. indeed define politically incorrect configurations that are inconvenient when one wants to obtain the propagators. Instead of the correct description "politically incorrect", they strangely use the adjective "physical" even though their condition is actually physically inconsistent.

In the case of the harmonic oscillator, on page 5 (above equation 27), they say that "if a classical solution interpolating between (q1,p1) and (q2,p2) exists, we call (q1,p1,q2,p2) physical". In a free theory such as the harmonic oscillator, such an unusual criterion might be inconsequential (even though the correct phase space path integral of the harmonic oscillator definitely allows all values of (q1,q2) and it is summing over all values of momenta), but it is certainly devastating in interacting theories.

To assure you that the sentence about the "physical boundary conditions" was not a typo, they explicitly write down the extension of the strange statement to GR on page 10 (above equation 50): "physical" boundary conditions, they say, are those that can be interpolated by a classical solution - a Ricci-flat metric.

This is what I call a relatively basic misunderstanding of elements of quantum theory. The authors simply don't want to accept a simple fact that a classical limit can be derived from a quantum theory, but particular contributions in a path integral can't be defined as unphysical and eliminated when they don't give the right limit that you would like to obtain. The classical theory is a limit of the quantum theory, not the other way around.

It is absolutely essential for the consistency of every quantum theory that its path integral is summing over all conceivable configurations of all allowed degrees of freedom, as Feynman has figured out, and the only constraints on the allowed configurations are those that can be written as local conditions in some variables. (The moderately careful wording of the last sentence means that you are allowed to impose integrality of various fluxes.)

Eliminating initial conditions that don't allow you a Ricci-flat background in between is also arguably the most brutal violation of their holy principle of background independence that you can think of, but that's not the real physical problem: the violation of unitarity is the problem. For example, imagine that you compute the evolution operator between the moments A and C. The evolution operator must be the product of the evolution operator from A to B and the evolution operator from B to C.

• U(A,C) = U(B,C) U(A,B)

Imagine that you decide to eliminate configurations between A and B ending at intermediate states in B that cannot be connected to the initial conditions at A by a classical solution, and you do the same thing for the intervals B-C and A-C. Then you clearly violate the product identity above. It's because U(A,C) will contain many histories with intermediate states at B that are very far from the states at A and C. These configurations are omitted from U(B,C) U(A,B). The failure of the product identity is lethal because if two of the operators above are unitary, the third won't be unitary, and vice versa.

Consequently, the total probability won't be preserved.

Similar ad hoc restrictions of the path integral defined by global criteria are strictly prohibited in any quantum theory. In fact, a typical configuration that contributes to any path integral is a wildly fluctuating function resembling the Brownian motion that has nothing to do with the smooth classical solution. The typical trajectories of particles in quantum mechanics are non-differentiable almost everywhere. That also means that the typical trajectories have superluminal (infinite) velocities almost everywhere - even in relativity. The fact that the full theory (e.g. quantum field theory of a Dirac field) respects causality exactly is a derived fact based on various cancellations.

The classical solution emerges as a stationary point of the action - where the Feynman phase is almost constant and the contributions of nearby trajectories tend to drag the result in the same direction. But this fact is a derived observation, too. You cannot define a quantum theory by requiring that some solutions will be exact. Moreover, this heuristic explanation of the emergence of the classical limit is only correct in the semiclassical approximation.

Loop corrections to the effective action

In fact, beyond the tree - i.e. classical - approximation, we know very well that the classical solutions are not exact. It's because the loop effects (virtual pairs of particles etc.) add their contributions to the effective action. The statement of Rovelli et al. that the quantum theory should be defined by eliminating pairs of initial and final states that can't be interpolated by Ricci-flat solutions (solutions of the classical equations) shows that they are apparently convinced that quantum mechanics can be ignored completely and the classical results will be confirmed exactly.

But the reality is very different than what they seem to think. In the quantum theory, the correct low-energy equations are not given by Ricci-flatness. They are given by Ricci-flatness modified by quantum corrections to the effective action - you can think of the Wilsonian effective action or the 1PI (one-particle-irreducible) effective action; in the latter case, you will encounter infrared problems. Among the corrections, the one-loop corrections can be universally determined from the classical action - because they follow from logarithmic divergences that cannot be subtracted by local counterterms - and they are nonzero.

The coefficients of the higher-loop corrections depend on the details of the short-distance physics and can only be determined if you know the full theory (e.g. in string theory). But at any rate, all these corrections matter. These corrections are the very reason why we use the adjective "quantum" in "quantum gravity" - at least the main reason in the context of graviton scattering.

Our loop quantum gravity colleagues - not only Carlo Rovelli - clearly fail to appreciate the difference between gravity and quantum gravity, and they throw the whole "quantum" into the trash can. They throw it away by definition - because they define their quantum path integral to give you uncorrected, classical equations of motion. And they only "derive" the correct classical result because they are assuming that it comes out and because they are defining their theory by the classical result. What a convoluted example of circular reasoning.

However, such a reasoning is not only circular. It is manifestly flawed.

It can easily be shown that general relativity with vanishing (or "erased") one-loop corrections (of order hbar) violates unitarity. What Rovelli et al. are doing contradicts some of the basic principles of physics and every attempt that has at least 5% of the conceptual errors of the present paper is more or less guaranteed to give an inconsistent quantum theory or something even less encouraging.

Note that you cannot fix the problem by replacing the adjective "Ricci-flat" by a quantum-corrected condition because a priori, you don't know what the quantum-corrected condition is. In a consistent theory, the effective action can be derived from the quantum formalism. Attempts to define a quantum theory by using the effective action - a quantity that should be, on the contrary, derived from the quantum theory - is a flawed circular reasoning that cannot lead anywhere.

Despite all of these missteps, they also seem to fail to derive anything that would resemble the graviton propagator as we know it. The best you can see is a "1/x^2" scaling of some terms in the propagator. But they only obtain this scaling after "N" undefendable steps where "N" is the number of terms in their calculation that have an undesired scaling; the neglected terms moreover have no relation with the actual nonlinear and quantum corrections to gravity required by general covariance and unitarity. It's nice if kids count the number of different cars that they can build from their LEGO, or if the adult physicists count the number of arrangements of some simplices, but computing nice things is something slightly different than having something to say about quantum gravity.

And that's the memo. ;-)