## Wednesday, May 12, 2010 ... //

### So is the (AdS/QCD) black hole real?

Clifford Johnson recently gave a colloquium about the AdS/CFT-inspired description of the quark-gluon plasma in terms of 4+1-dimensional black holes and someone asked him an obvious question:

But is it real? (Asymptotia.COM)
Clifford gave a thoughtful answer and I only agree with 2/3 of it. Moshe Rozali joined Clifford in the comments and it's clear that my degree of agreement with Moshe would be much closer to 100%.

In the AdS/QCD approach, one looks at a chunk of quark-gluon plasma or a similar material and he realizes that the quarks and gluons are no longer too useful to predict what's going on. Instead, one wants to deal with the material macroscopically, in a way. One wants to more directly capture the "emergent" degrees of freedom and their relationships.

But what are the properties of such a material? It turns out that the most accurate answer involves a black hole in a curved spacetime with one additional dimension! This is not just some random coincidence. It is a consequence of the most important mathematical equivalence found in physics during the last 15 years.

A large number of gluons and quarks can also be equivalently described in terms of a higher-dimensional theory of gravity. The most typical and universal "localized" object in any theory of gravity is a black hole. It's the "final phase" of matter because it maximizes the entropy among all localized or bound objects of the same mass and charges. So not too surprisingly, it describes the quark-gluon plasma, too.

This technology allowed the people to calculate the viscosity-to-entropy-density ratio and many other quantities much more accurately than by other, older, less creative and less imaginative approaches. It simply works. It is beautiful and annoyingly technical at the same moment. It shows that the gravity is also unified with other forces such as the nuclear ones if there are many quarks or gluons and the gravity is allowed to operate in a higher-dimensional curved space. The ideological network irreversibly connecting gravity with other forces is denser than people once thought.

Fine. So to describe a piece of a quark-gluon plasma, we need a black hole in a higher-dimensional curved spacetime. Let's start with the questions and answers:
Is the black hole real?
Clearly, I agree with Clifford that the notion of "real" is ill-defined. Moshe emphasizes that people often interpret the word "real" as "intuitive", something "they are familiar with". That's unfortunate because such a definition of the adjective depends on their psychology. With this definition, the meaning of the word "real" ceases to be objective in character.

I guess that almost no lay person can imagine higher-dimensional AdS black holes interacting in the same way as a quark-gluon plasma, so if the mental abilities of a layperson are implicitly included in the word "real", then, of course, almost nothing in advanced theoretical physics is real!

If you allow me to talk about somewhat more objective questions, and use the definition of "real" as understood by a physicist who knows what's going on, I would probably choose the answer, "Yes, the black hole is real." All of its physical properties are given by a five-dimensional theory of quantum gravity. It is a real black hole.

The only problem is that we can't take a spaceship and fly around this black hole because we are composed of the type of matter that can't be localized in the five-dimensional space; after all, we have a lower dimensionality. Our electrons "really" live on the boundary of the AdS space. So the black hole will never be a "real black hole in our neighborhood". It will be a "black hole in some space where we can't quite get".

However, this is, in a sense, our problem, not a problem of the black hole. The black hole is real while we are impotent when someone asks us to run around such a black hole. ;-)
Is there a black hole living in the lab somewhere?
Of course, I completely agree with Clifford that if the lab is defined as a region of a 3-dimensional space, the black hole is not "inside it" because higher-dimensional objects can't be "inside" lower-dimensional ones, at least assuming the conventional geometric interpretation of the word "inside".

Well, you could also imagine that there is a "bump" in the lab where the dimension increases but I would consider such a mixture of geometries inconsistent. The geometry on the boundary, where the CFT lives, is inherited from the bulk. More precisely, its conformal structure i.e. angles are inherited. But the overall scaling is independent (and needs to be rescaled by an infinite factor).

To summarize, when you talk about everything about the geometry, you must consider the geometries of the 5D spacetime and the 4D spacetime to be independent of one another and you shouldn't combine them. They offer the arenas for two descriptions of the same physical phenomena but you must consistently work only with one of them at a given moment. It follows that the black hole is not "inside the 3D lab".
Ah, ok. But do you believe that this higher dimensional spacetime you’re doing the computation in, along with the black hole you put into it to model the phenomena properly, “exist” somewhere? Are they “real”?
Well, the higher-dimensional spacetime is real or unreal in the same sense as the black hole itself. It's not intuitive for the laymen and we're not used to it. But it does exist. The physical phenomena in this higher-dimensional spacetime coincide with the phenomena we would once call "behavior of quark-gluon plasma".

The quark-gluon plasma description in 3+1 dimensions was found before the black hole description but it is just a historical coincidence and both of these descriptions are equally fundamental. For large chunks of quark-gluon plasma, the black hole description is more directly associated with the observable quantities - even though this fact may sound surprising to the newbies.
Gauge symmetries are unreal
Clifford also correctly says that the gauge symmetries are "unreal" in the sense that they're only properties of a particular description - and there can be (and there usually are) other descriptions that only include other types of gauge symmetries or no gauge symmetries at all.

Note that the conventional Lagrangians for (low-energy) AdS gravity include the diffeomorphisms as a gauge symmetry, but no SU(N) Yang-Mills symmetry, while this picture is equivalent to a boundary CFT without diffeomorphisms but with an SU(N) Yang-Mills symmetry. There are many equivalences of theories with very different gauge-theoretical principles. Moreover, any of the gauge symmetries may be gauge-fixed, and so on.

That's a reason why (all) gauge symmetries deserve to be called a "part of a mathematical formalism" that is otherwise unphysical. They never correspond to directly observable phenomena. All physical objects are required to be invariant under gauge symmetries - hadrons must be color singlets (color neutral, red+green+blue etc.) in QCD; field configurations have to satisfy the "div D = rho" Gauss' law in electromagnetism; and so on.

The whole representation theory of the groups that coincide with the gauge symmetries is "unphysical" because only the trivial, one-dimensional representations that don't transform under the group at all are physically allowed.

Of course, this doesn't mean that there can't be any "traces" of the gauge symmetries in physics. Of course, there are traces. The existence of the transverse (physical) polarizations of photons, gluons, and other gauge bosons (and, with diffeomorphisms as another gauge symmetry, also gravitons), may be viewed as a pretty direct consequence of the gauge symmetries.

The physical polarizations of these bosons are real and observables - but these polarizations are pretty directly associated with the whole gauge fields, including their unphysical degrees of freedom. That sounds natural but it's still true that we can fully describe the behavior of the physical polarizations without any unphysical degrees of freedom or gauge symmetries!

Also, the "global" subgroups of the gauge-symmetry groups - e.g. the U(1) electromagnetic transformations acting on the whole space - i.e. some transformations that do affect the fields at infinity are usually not required to annihilate the physical states (you can have non-singlets or charged fields). So there often exist "global symmetries" that arise as "remnants" of the local ones (the conserved energy-momentum and the angular-momentum can also be represented as "remnants" of the local symmetries called diffeomorphisms).

So if Clifford meant that we must carefully distinguish physical states (and degrees of freedom) from the auxiliary ones (because only the former are observable, while the latter are a part of an auxiliary toolkit that can be replaced by another one), I agree with his statement:
We should not mix up our computational tools with the thing we are trying to describe (Nature).
However, if he were only talking about the physical (gauge-invariant and satisfying other conditions) states and degrees of freedom, I would disagree with the sentence above. When we focus on the physical Hilbert space of the AdS theory, and the physical Hilbert space of the CFT theory, they're just isomorphic.

Because they're isomorphic, with unitarily equivalent Hamiltonians, so to say, they lead to identical predictions, so we can't ever say that one of them is more physically true than the other one. We have no right to say that one of them is "Nature" while the other is just a "tool". In this sense, we are obliged to mix the "tools" and "Nature" because they're the same thing!

Saying that one description is "more real" than the other one would be a fallacy. Of course, I am not preventing you from distinguishing "any description" on one side and the "reality" on the other side - but I am afraid that such a distinction is a pure philosophy because whenever you talk about the "reality", you have to talk about a "description" of it, anyway. ;-)

Elsewhere, Clifford wrote the following:
It's perfectly fine that the math invented from string theory is applied to other areas. That's great. But that establishes nothing about the hypothesis that strings are the fundamental constituents of matter.
Well, I would agree if he said that according to conventional, "mechanistic" methods to derive results in science, the "usefulness" of the mathematical structures from string theory in other fields doesn't establish that string theory is the right fundamental theory. There obviously exists no completely robust, water-proof argument that would allow us to prove such an implication.

But if Clifford also wanted to say that it is unreasonable to think that the probability that string theory is the right theory of the fundamental interactions does increase because string theory's maths has successfully explained some other classes of phenomena, then I completely disagree.

Because the previous sentence is convoluted, let me say what I believe: Nature inevitably recycles various mathematical structures at many places and if a mathematical structure appears in a higher number of contexts, it inevitably means that it's more likely for this structure to be universally important. The proper logical inference meant to estimate the relevance of string theory for the fundamental interactions simply has to take the success of the ideas in other contexts into account.

So the appearance of black holes - and their stringy collaborators, including open strings with quarks at the end points - inside the quark gluon plasma surely does mean that the string-theoretical objects are more important in Nature, even at the fundamental level. The quark-gluon plasma is just one example that shows that the right connection between the microscopic objects such as gluons and the macroscopic ones such as chunks of quark-gluon plasma or black holes is provided by the detailed rules of string theory.

Because this hypothesis is viable in the context of quantum gravity (and other elementary interactions) and because it's been established in nuclear physics, it makes the validity in the case of quantum gravity even more likely, especially if no other consistent theory of quantum gravity is known as of today.

Be sure that what I write is an opinion shared by most top minds in theoretical physics. In his 2005 popular article called Unravelling string theory (in the good sense), Edward Witten wrote:
And finally, string theory has proved to be remarkably rich, more so than even the enthusiasts tend to realize. It has led to penetrating insights on topics from quark confinement to quantum mechanics of black holes, to numerous problems in pure geometry. All this suggests that string theory is on the right track; otherwise, why would it generate so many unexpected ideas? And where critics have had good ideas, they have tended to be absorbed as part of string theory, whether it was black-hole entropy, the holographic principle of quantum gravity, non-commutative geometry, or twistor theory.
Well, I completely agree with that. While we have no "complete" proof that string theory is the right theory of all fundamental interactions and matter species, this circumstantial evidence speaks a pretty clear language.

The logic is the following: if a human being invents a random idea that is proposed to be relevant for the right understanding of a question in quantum gravity, it will almost certainly 1) be found inconsistent with some directly or indirectly observed facts or principles about Nature, 2) remain intellectually localized to the original place where the author suggested the idea to be relevant.

After all, this is not just a hypothetical situation: hundreds of ideas were proposed to be relevant for "a theory of everything". Nearly all of them were either found to be inconsistent with some basic and vital characteristics of the real world, and nearly all of them remained speculations whose only ambition remained to solve a particular small problem. Their importance never turned out to be broader than the original proposal suggested.

That's the typical behavior of wrong ideas.

String theory has been different. It remains consistent with all the features of the world, within the accuracy we can test today (which includes almost everything except for the detailed parameters of the Standard Model that can only be extracted from the right, so far unknown vacuum) - and while the set of these required features that are reproduced by string theory is not complete, it's already very constraining. And string theory has shown to be more important for a proper description of wider and more numerous classes of physical situations. As statistics shows, this usually doesn't occur by accident.

In the absence of accurate or unique predictions and/or direct experiments, scientists must still be affected by their findings, by some results of the work. It seems obvious to me that the continuing consistency with the observations - at the level we can check (which, of course, includes self-consistency) - and the universal importance of the ideas are the key criteria. And when these criteria are taken into account, it follows that every serious high-energy theoretical physicist should take the findings of string theory into account.

Three paragraphs ago, when I talked about the fate of the wrong theories, I didn't acknowledge one more group of exceptions: alternative theories that didn't turn out to be inconsistent with the reality. For example, people would find some interesting things about the black hole thermodynamics (and "string theory critic" Gerard 't Hooft would co-author the holographic principle), supersymmetry, supergravity, or twistors (due to "string theory critic" Roger Penrose), among other topics, that are still believed to be valid - at least as approximations describing some localized aspects of physics - today.

But these theories have been swallowed by string theory. They turned out to be just particular aspects of the same theory - string theory. It's the best thing that has happened to any "competing" theories in the last 30 years. Any other kind of fate has been much worse than that.

Just by statistical arguments (or other arguments), it's pretty reasonable to expect that the same thing will happen to any other successful "competing idea" in the near future, too. If you have a good idea that will turn out to be operational or deep, it's damn likely that it is a part of string theory. I think that to disagree with this expectation means to deny the qualitative insights we learned in dozens of important physics developments during the recent decades.

You may dislike the historical statistics and deliberately "work" on a program that will (hopefully) circumvent string theory completely. Fine. Scientists can start with any assumptions they want. But the success is not guaranteed and it's obvious that if most of the community were making this anti-string bet, the community would be acting irrationally simply because the assumption is unlikely to be true.

So yes, I think it is true that the main reason why I feel pretty much certain that string theory has to be the right theory of fundamental interactions is the argument that Clifford Johnson wants to declare a taboo. I think that he is wrong: it is not plausible for a wrong theory to preserve its consistency with all the new qualitative data for such a long time, to expand the range of its impact, to produce many more great ideas than those that were inserted (which shows that we're discovering something, rather than inventing/constructing it), and to steadily absorb other ideas in the same way as string theory has done.

That's the real "big picture" reason why string theory is almost certainly right as "a theory of everything". Not to pay attention to these "signals of light" would mean to "randomly walk" through the dark alleys of ignorance. Science simply couldn't proceed if we couldn't learn a lesson here.

And that's the memo.

#### snail feedback (2) :

What IS string theory? How many dense pages does it take to describe it? (A curious lay person)

Dear Harlow,

string theory is a system of ideas and equations meant to describe all elementary particles in Nature, and the forces between them including gravity, as different modes of vibrating strings that split and join, and therefore reproduce all phenomena in the world.

To describe string theory, you need space depending on your level of detail. Someone may find the sentence above enough. Undergrads learn from 500-page textbooks, grad students from 1000-page textbooks, usually in 2 volumes.

There are tens of thousands of papers on string theory. With 20 pages per paper or so, that can be half a million pages. Because much of the stuff is redundant, insufficiently concise, or wrong, I suppose that all ever published knowledge about string theory can be squeezed to 50,000 pages.

But it's not being invented or constructed by the humans. They just discover something that exists objectively and independently of us. In this sense, it's similar as the thousands of pages needed to classify all finite groups. But the task to understand string theory hasn't yet been completed.

Best wishes
Lubos