## Tuesday, November 24, 2015 ... //

### Point-like QFTs in the bulk can't be a consistent theory of QG

Dixon's research is impressive applied science using deep insights by others, mainly string theorists

Lance Dixon is a prominent particle theorist at SLAC. A few days ago, he gave an interview about quantum gravity.

Q&A: SLAC Theorist Lance Dixon Explains Quantum Gravity
He's been most tightly associated with multiloop calculations in quantum field theory (including some calculations at four loops, for example) and various tricks to climb over the seemingly "insurmountably difficult" technical obstacles that proliferate as you are adding loops to the Feynman diagrams. However, as a Princeton graduate student in the 1980s, he's done important research in string theory as well. Most famously, he is one of the co-fathers of the technique of the "orbifolds".

Also, most of his claims in the interview are just fine. But some of his understanding of the big picture is so totally wrong that you could easily post it at one of the crackpots' forums on the Internet.

To answer the question "what is quantum gravity?", he begins as follows:
With the exception of gravity, we can describe nature’s fundamental forces using the concepts of quantum mechanics.
Well, one needs to be more specific about the meaning of "can". In this form, the sentence pretty much says that as we know it, gravity is inconsistent with quantum mechanics. But this isn't right. Frank Wilczek's view is the opposite extreme. Frank says that gravity is so compatible with the quantum field theory (The Standard Model) that he already clumps them into one theory he calls "The Core Theory".

The truth is somewhere in between. Gravity as a set of phenomena is demonstrably consistent with quantum mechanics – we observe both of them in Nature while the gravitational (and other) phenomena simply can't escape the quantum logic of the Universe. And in fact, even our two old-fashioned theories are "basically consistent" for all practical and many of the impractical purposes. We can derive the existence of the Hawking radiation, gravitons, and even their approximate cross sections at any reasonable accuracy from the quantized version of general relativity. Using a straightforward combination of GR and QFT, we may even calculate the primordial gravitational fluctuations that have grown into galaxies and patterns in the CMB.

The "only" problem is that those theories can't be fully compatible or the predictions can't be arbitrarily precise, at least if we want to avoid the complete loss of predictivity (the need to measure infinitely many continuous parameters before any calculation of a prediction may be completed).

OK, Dixon says lots of sane things about the similarities and differences between electromagnetism and gravity, the character of difficulties we encounter when we apply the QFT methods to gravity, and some new hard gravitational phenomena such as the Hawking radiation. But things become very strange when he is asked:
Why is it so difficult to find a quantum theory of gravity?

One version of quantum gravity is provided by string theory, but we’re looking for other possibilities.
You may also look for X-Men in New York. You may spend lots of time with this search which doesn't mean that you will have a reasonable chance to find them. There are no X-Men! The case of "other possible" theories of quantum gravity aside from string theory is fully analogous.

Moreover, I don't really think that any substantial part of Dixon's own work could be described as this meaningless "search for other theories of quantum gravity". Whenever gravity enters his papers at all, he is researching well-known i.e. old approximate theories of quantum gravity – such as various supergravity theories.

Dixon says that gravitons' spin is two, the force is weak, and universally attractive. But the next question is:
How does this affect the calculations?

It makes the mathematical treatment much more difficult.

We generally calculate quantum effects by starting with a dominant mathematical term to which we then add a number of increasingly smaller terms.
He's describing "perturbative calculations" – almost all of his work may be said to be about "perturbative calculations". However, it is simply not true that this is the right way to do research of quantum mechanics "in general". Perturbation theory is just an important method.

It is true that if we talk about "quantum effects", in the sense of corrections, we must start with a "non-quantum effect" i.e. the classical approximation and calculate the more accurate result by adding the "quantum corrections". But it is simply not always the case that a chosen "classical result" is the dominant contribution. Sometimes, physics is so intrinsically quantum that one must try to make the full-fledged quantum calculations right away.

Even more importantly, he tries to obscure the fact that the perturbative – power law – corrections are not the only effects of quantum mechanics. When he does these power-law perturbative calculations, and his papers arguably never do anything else, he is not getting the exact result. There almost always exist infinitely many non-perturbative corrections, instantons etc. The existence of the non-perturbative effects is actually related to the divergence of the perturbative series as a whole.

To summarize, he is just vastly overstating the importance of the perturbative – and especially multiloop – calculations, the kind of calculations his work has focused on. You know, these multiloop terms are only important relatively to the classical term if the quantum effects are sufficiently strong. But if they are strong enough to contribute $$O(100)$$ percent of the result, then the non-perturbative terms neglected by Dixon will contribute $$O(100)$$ percent, too. In other words, the multiloop terms are "in between" two other types of contributions, classical and nonperturbative, which is why they generally aren't the key terms.

In practice, Dixon's work has been about the question "up to how many loops do all the divergences cancel" in a given supersymmetric theory. Does $$d=11$$ supergravity cancel all divergences even at seven loops? True experts have to care about this question but ultimately, it is a technical detail. Supersymmetry allows the theory to "get rather far" but at the end, this theory and its toroidal compactifications can't be consistent and have to be completed to the full string/M-theory for consistency.

If you click the link in the previous sentence, you may remind yourself that nonperturbatively, the $$\NNN=8$$ $$d=4$$ SUGRA theory simply isn't OK. It wouldn't be OK even if the whole perturbative expansion of SUGRA were convergent (which I am not self-confidently excluding at all even though I do tend to believe those who say that there are divergences at 7 loops). This is why all the hard technical work in Dixon's multiloop papers consists of irrelevant technical details that simply don't affect the answers to the truly important questions. You don't need to know anything about Dixon's papers but you may still comprehend and verify the arguments in the following paragraph.

The theory has the noncompact continuous symmetry but the symmetry has to be broken because the spectum of charged black hole microstates has to be discrete thanks to the Dirac quantization rule (the minimum electric and the minimum magnetic charge are "inverse" to one another if the Dirac string is invisible). That's why the $$E_{k(k)}(\RR)$$ symmetry is unavoidably broken to a discrete subgroup of it, $$E_{k(k)}(\ZZ)$$, the subgroup that preserves the lattice of the charges, just like in string/M-theory, and all the other "purely stringy phenomena" that go beyond SUGRA (starting with the existence of low-tension/light strings in a weakly coupled, stringy limit of the moduli space we just identified) may then be proven to be there, too.

Also, the $$\NNN=8$$ SUGRA is too constrained because it's too supersymmetric. To get more realistic spectra, you need to reduce the SUSY and then the divergences unavoidably appear at a small number of loops. So effective gravitational QFTs are either realistic or relatively convergent at the multiloop level but not both. There is a trade-off here. Again, string/M-theory is the only way to make the theories realistic while preserving the convergence properties. In some sense, all the SUSY breaking in string theory may be said to be spontaneous (the compactification on a complicated manifold is a spontaneous symmetry breaking of symmetries that would be present for other manifolds).

SUGRA-like quantum field theories are wrong for other, perhaps more qualitative reasons. They can't really agree with holography or, more immediately, with the high-mass spectrum of the excitations. High mass excitations must be dominated by black hole microstates with the entropy scaling like the area. But the high energy density behavior of a QFT in a pre-existing background always sees the entropy scale like the volume. The real problem is that the background just can't be assumed to be fixed in any sense if we get to huge (Planckian) energy or entropy densities. It follows that the causal structure is heavily non-classical in the quantum gravity regime as well, and this is what makes the bulk QFT inapplicable as a framework.

This was an example but I want to stress a very general point that makes Dixon's argumentation totally weird:
Dixon uses all the self-evidently pro-string arguments as if they were arguments in favor of "another theory".
This paradox manifests itself almost in every aspect of Dixon's story. Let me be more specific. There are several paragraphs saying things like
We’ve succeeded in using this discovery to calculate quantum effects to increasingly higher order, which helps us better understand when divergences occur.
And these comments are implicitly supposed to substantiate Dixon's previous claim that "he is looking for other theories of quantum gravity". Except that virtually all the good surprises he has encountered exists thanks to insights discovered in string theory!

First of all, the cancellations of divergences in his SUGRA papers depend on supersymmetry – plus other structures but SUSY is really needed. (All known cancellations of divergences in $$\NNN=8$$ SUGRA may be fuly derived from SUSY and the non-compact $$E_{7(7)}(\RR)$$ symmetry!) In the West, SUSY was discovered when people were trying to find a better string theory than the old $$d=26$$ bosonic string theory. The world sheet supersymmetry was found to be necessary to incorporate fermions. And the spacetime supersymmetry emerged and seemed necessary to eliminate the tachyon in the spacetime. The ability of SUSY to cancel lots of (mostly divergent) terms was quickly revealed and became established. It was clear that SUSY is capable of cancelling the divergences; the only remaining questions were "which ones" and "how accurately".

You know, this kind of "silence" about the importance of SUSY for the cancellation of divergences; and about SUSY's role within string theory is unavoidably inviting some insane interpretations. In the past, the notorious "Not Even Wrong" crackpot forum has often promoted the ludicrous story – implicitly encouraged by Dixon's comments – that maybe we don't need string theory because field theories might cancel the divergences.

The following blog post on that website would attack supersymmetry.

The irony is that the good news in the first story are primarily thanks to supersymmetry which is trashed in the second story. So the two criticisms of string-theory-related physics directly contradict one another! You may either say that SUSY should be nearly banned in the search for better theories of Nature; or you may celebrate results that depend on SUSY. But you surely shouldn't do both at the same moment, should you?

But it's not just supersymmetry and the reasons behind the cancellation of divergences where Dixon's story sounds ludicrously self-contradictory. What about the relationship between gravitons and gluons?
What have you learned about quantum gravity so far?

Over the past decades, researchers in the field have made a lot of progress in better understanding how to do calculations in quantum gravity. For example, it was empirically found that in certain theories and to certain orders, we can replace the complicated mathematical expression for the interaction of gravitons with the square of the interaction of gluons – a simpler expression that we already know how to calculate.
So the insight that gravitons behave like squared gluons is also supposed to be an achievement of the "search for other, non-stringy theories of quantum gravity"? Surely you're joking, Mr Dixon. You know, this "gravitons are squared gluons" relationship is known as the KLT (Kawai-Lewellen-Tye) relationship. Their 1986 paper was called A Relation Between Tree Amplitudes of Closed and Open Strings. Do you see any strings in that paper? ;-) It is all about string theory – and the characteristic stringy properties of the particle spectrum and interactions (including the detailed analysis of the topologies of different strings).

The point is that an open string – a string with two endpoints – has the $$n$$-th standing wave and the corresponding modes to be excited, $$\alpha_n^\mu$$. A closed string – topologically a circle – has the $$n$$-th left-moving wave and $$n$$-th right-moving wave. The operators capable of exciting the closed string come from left-movers and right-movers, $$\alpha_n^\mu$$ and $$\tilde \alpha_n^\mu$$. So the closed string has twice as many operators that may excite it – it looks like a pair of open strings living together (its Hilbert space is close to a tensor product of two open string Hilbert spaces). Similarly, the amplitudes for closed strings look like (combinations of) products of analogous amplitudes from two copies of open strings. That's the basic reason behind all these KLT relationships. And now, in 2015, Dixon indirectly suggests that this relationship is an achievement of the search for non-stringy theories of quantum gravity?

This relationship was found purely within string theory and it only remains valid and non-vacuous to the extent to which you preserve a significant portion of the string dynamics. The relationship tells you lots about the dynamics of the massless states as well. But you can't really find any good quantitative explanation why the relationship works in so many detailed situations that would be non-stringy. It's only in string theory where the graviton $$\alpha^{\mu}_{-1}\tilde\alpha^\nu_{-1}\ket 0$$ is "made of" two gluons – because it has these two creation operators which are analogous to the one creation operator in the gluon open string state $$\alpha^{\mu}_{-1}\ket 0$$. The point-like graviton has the spin two, as two times the spin of a gluon, but you can't "see" the two gluons inside the graviton because all the particles are infinitely small.

And this kind of irony goes on and on and on. He has used SUSY and KLT relationships as evidence for a "non-stringy" theory of quantum gravity. Is there something else that he can use against strings? Sure, dualities! ;-)
We were also involved in a recent study in which we looked at the theory of two gravitons bouncing off each other. It was shown over 30 years ago that divergences occurring on the second order of these calculations can change under so-called duality transformations that replace one description of the gravitational field with a different but equivalent one. These changes were a surprise because they could mean that the descriptions are not equivalent on the quantum level. However, we’ve now demonstrated that these differences actually don’t change the underlying physics.
This is about equally amazing. You know, this whole way of "duality" reasoning – looking for and finding theories whose physics is the same although superficially, there seem to be serious technical differences between two theories or vacua – has spread purely because of the research done by string theorists in the early and mid 1990s. The first paper that Dixon et al. cite is a 1980 SUGRA paper by Duff and Nieuwenhuizen and the duality is meant to be "just" an electromagnetic duality for the $$p$$-forms. But before string theory, people indeed believed that such dualities weren't exact symmetries of the theories. Only within the string-theory-based research, many such dualities were shown to be surprisingly exact. They are just claiming a similar phenomenon in a simpler theory. They would probably never dare to propose such a conjecture if there were no stringy precedents for this remarkably exact relationship. The previous sentence may be a speculation but what is not a speculation is that they're far from the first ones who have brought evidence for the general phenomenon, a previously disbelieved exact equivalence (duality). Tons of examples of this phenomenon has previously been found by string theorists.

Most of the examples of dualities arose in the context of string theory but even the cases of dualities that apply to field theories, like insights about the Seiberg-Witten $$\NNN=2$$ gauge theories etc., were found when the authors were thinking about the full stringy understanding of the physical effects. They may have tried to hide their reasoning in their paper to make the paper more influential even among the non-stringy researchers but you can't hide the truth forever. Most experts doing this stuff today are thinking in terms of the embeddings to string/M-theory anyway because those embeddings are extremely natural if not paramount.

So what Dixon was doing was just trying to apply a powerful tool discovered in the string theory research to a situation that is less rich than the situations dealt with in string theory.

Near the end, Dixon joined the irrational people who don't like that string theory has many solutions:
However, over the years, researchers have found more and more ways of making string theories that look right. I began to be concerned that there may be actually too many options for string theory to ever be predictive, when I studied the subject as a graduate student at Princeton in the mid-1980s. About 10 years ago, the number of possible solutions was already on the order of $$10^{500}$$. For comparison, there are less than $$10^{10}$$ people on Earth and less than $$10^{12}$$ stars in the Milky Way. So how will we ever find the theory that accurately describes our universe?

For quantum gravity, the situation is somewhat the opposite, making the approach potentially more predictive than string theory, in principle. There are probably not too many theories that would allow us to properly handle divergences in quantum gravity – we haven’t actually found a single one yet.
I had to laugh out loud. So Dixon wants one particular theory. He has zero of them so he's equally far from a theory of everything as string theorists who have a theory with many solutions. Is that meant seriously? Zero is nothing! In Czech, when you have zero of something, we say that you have "a šit". Nothing is just not too much.

Moreover, Dixon's comment about "making string theories" has been known to be totally wrong since the mid 1990s. There is only one string theory which has many solutions – just like the equation for the hydrogen energy eigenstates has many solutions. There are not "many string theories". This fact wasn't clear before the mid 1990s but it became totally clear afterwards. When Dixon continues to talk about "many string theories", it's just like someone who talks about evolution but insists that someone created many species at the same moment. The whole point of evolution is that this isn't the case. Even though the species look different, they ultimately arose from the same ancestors.

To talk about very many ancestors of known species means to seriously misunderstand or distort the very basics of biology and Dixon is doing exactly the same thing with the different string vacua. What he's saying is as wrong as creationism. A professional theoretical physicist simply shouldn't embarrass himself in this brutal way in 2015.

Dixon wants to say that we want "one right theory" and he has "zero" why string theorists have "$$10^{500}$$" which is also far from the number he wants, one. But even if this "distance" were measuring the progress, the whole line of reasoning would be totally irrational because the number "one" is pure prejudice with zero empirical or rational support. You may fool yourself by saying that a theory of nuclei predicting 1 or 50 possible nuclei (or a theory of biology predicting that there should be 1 or 50 mammal species) is "more predictive" and therefore "better" but this rhetorical sleight-of-hand won't make the number 1 or 50 right. The right number of nuclei or mammal species or vacuum-like solutions to the underlying equations is much higher than 1 or 50. Emotional claims about a "better predictivity" can never beat or replace the truth! It's too bad that Dixon basically places himself among the dimwits who don't understand this simple point.

What we observe is that there exists at least one kind of a Universe, or one string vacuum if you describe physics by string theory. There is no empirical evidence whatever that the number isn't greater than one or much greater than one. Instead, there is a growing body of theoretical evidence that the right number almost certainly exceeds one dramatically.

At some moment, Lance Dixon decided to study the heavily technical multiloop questions and similar stuff. It's a totally serious subset of work in theoretical physics but it simply lacks the "wow" factor. Maybe he wants to fool others as well as himself into thinking that the "wow" factor is there. But it isn't there. A cancellation of 4-loop divergences in a process described by a theory is simply a technicality. A physicist who can calculate such things is surely impressively technically powerful and that is what will impress fellow physicists. But the result itself is unlikely to be a game-changer. Most of such results are entries in a long telephone directory of comparable technical results and non-renormalization theorems.

The true game-changers in the recent 40 years were concepts like supersymmetry, duality, KLT relations, holography and AdS/CFT, Matrix theory, ER-EPR or entanglement-glue correspondence, and perhaps things like the Yangian, recursive organization of amplitudes sorted by the helicities etc. Many of them have been use in Dixon's technically impressive research. But this research has been an application of conceptually profound discoveries made by others, not a real source of new universally important ideas, and it's just very bad if Dixon tries to pretend something else.

And that's the memo.