## Thursday, February 19, 2009

Various media outlets recently informed about the progress in AdS/QCD and AdS/CMT, methods to use Maldacena's holographic correspondence (AdS/CFT) to transfer our knowledge and results between string theory and gravity on one side - and nuclear physics or condensed matter on the other side.

Because nuclear physics and condensed matter physics seem to be different disciplines than high-energy physics and quantum gravity, many people naturally ask whether we're talking about the "same string theory" in both cases. If you're impatient, the answer is Yes, of course. There is only one string theory.

But many readers are hopefully more patient, so let us look at some issues in more detail.

The disk on the picture above - see hyperbolic cows for a similar picture - represents the Poincaré disk which is the Euclidean edition of an anti de Sitter space (AdS). If you want to know, the disk is EAdS2 and if you add the vertical time coordinate and assume that it has time-like signature, the resulting cylinder may be called AdS3. Several general aspects of the picture are worth noticing.
1. All the bats seem to have the right shape; none of their aspect ratios ever gets extremely distorted; we say that the cylinder is "conformally equivalent" to the AdS space.
2. There are infinitely many bats: the AdS proper volume is infinite.
3. The disk (or cylinder) has a boundary; this boundary is surrounded by infinitely many small bats.
In general, Maldacena's holographic correspondence implies that a gravitational theory of some "bats" (degrees of freedom) inside the cylinder (a theory that includes black holes and other objects predicted by string theory, which usually includes strings and branes) is equivalent to a non-gravitational theory on the boundary. Note that we are talking about the exact equivalence of two theories; one of them is gravitating and the other is not; the spacetime dimension in the gravitating theory is higher by one.
Commercial break: Stanleyfest at KITP, Santa Barbara. Fascinating talks about Mandelstam and physics. Polchinski's talk is really inspiring and funny. Via Dmitry Podolsky.
In string/M-theory, the equivalence between type IIB string theory on AdS5 x S5 and N=4 gauge theory on R x S3 is the most popular and the most well-established example. This equivalence has profound implications for our understanding of the structure of string/M-theory.

However, there also exist more "practical" or "mundane" ways to extract some interesting information from the duality. If you require that the theory on the boundary describes nuclear physics, you deal with the AdS/QCD correspondence. (Yes, I am including even those AdS/CFT researchers who focus on physical interpretations of the exact N=4 theory and who dismiss the AdS/QCD people as a different, less rigorous community.)

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If the boundary theory describes some condensed-matter phenomena, we are talking about the AdS/CMT correspondence (CMT stands for condensed-matter theory). Let me include various recent applications of AdS/CFT to atomic physics into this category, too.

Let us return to our original questions: are these "mundane" methods relevant for unification? You should be able to guess the answer. There are two important facts you should have in mind:
1. The holographic duality is exact, at least in principle.
2. The equivalence is a relationship that symmetrically applies in both ways.
Because of these two reasons, it is incorrect to say that AdS/QCD and AdS/CMT are "just" approximate methods to study nuclear physics and condensed matter physics.

First of all, while the description of the complex, strongly coupled phenomena is approximate in practice, it is only approximate for the same reasons why everything that people ever compute about the real world is approximate: the exact calculations are so difficult that they are undoable in practice (and even the methods to do these things exactly are not fully known in most cases - but that's the case in all fields of science).

Second of all, it is not true that the AdS/CFT duality only allows ideas and information to flow in one direction. It is an equivalence, a symmetric relationship, so the insights can flow in both directions. And they do.

The correspondence may be used to study nuclear physics and condensed matter physics in new ways that turn out to be more intuitive and more accurate than the old methods in many cases. This direction of the equivalence is widely reported by the media because the effects may be seen in the labs and the journalists can only understand the existence of something when they see it (or when they hear someone else who has seen it).

However, logically speaking, the opposite direction of the equivalence is equally real and equally important. In all cases, the bulk description of physics may be viewed as the more "macroscopic" one while the boundary description is more "microscopic". We may use the gravitational bulk physics to efficiently deduce how the macroscopic phenomena look like.

But we may also choose the opposite direction and use the boundary theory to find out how the microscopic description of the basic degrees of freedom in gravity or string theory may look like. The basic result of this method is the fact that the information is preserved even in the presence of (evaporating) black holes: the boundary theory makes this conclusion manifest. However, there are many other, more quantitative insights coming from this line of reasoning.

The first strategy, namely the method to use gravity to learn about the complex systems, is overreported by the media but it is arguably overstudied by the scientists, too. The reason is the old, well-known barbarian, anti-theoretical bias of most people.

The second strategy, i.e. using the duality to learn about the basic degrees of freedom in the Universe, looks theoretical in character. But it is not true that pure high-energy theorists are the only ones who care about such things. Many ambitious condensed-matter physicists, such as Robert Laughlin or George Chapline, have been trying to construct black holes and spacetimes out of building blocks that are similar to those in condensed matter physics that they know pretty well.

Their dreams about the gravitational phenomena's being emergent are pretty much equivalent to this interpretation of the duality. The way how black holes emerge from
the boundary gluons in AdS/CFT is analogous to the emergent phenomena that the condensed matter physicists hoped to be relevant for the physics of gravity.

The two cultures use different languages - the condensed-matter physicists would like to talk about emergence all the time while the high-energy phenomena think of the relationship as a case of reductionism - but they're excited about pretty much the same thing in this case. If you call the bulk physics "emergent" and the boundary physics "fundamental", you deal with a situation where "emergent" and "fundamental" physics are indeed equally important: they're equivalent.

Now, as our knowledge about AdS/CMT and AdS/QCD expands, we can see the actual correct path towards this goal. It has become clear that all the viable attempts to construct gravitating systems out of condensed matter ingredients have to use a lower-dimensional condensed-matter theory as a microscopic description and many other conditions have to be satisfied, too.

A partial summary

You may hear various people claiming that the recent insights of string theory about the complex systems have nothing to do with unification. A notoriously deceitful crackpot harbored by Columbia University is the most likely person who will try to convince you about this assertion.

But the reality is that holography has everything to do with unification - and with string theory's status of a unifying theory. It is an exact equivalence between quantum gravitational systems on one side and quantum, non-gravitational, strongly coupled systems on the other side. Can you imagine a tighter form of unification of two theories or two classes of phenomena than their equivalence? The equivalence tells us that quantum gravity and strongly coupled systems are just two faces of the same coin. It also informs us about the angles from which the coin resembles one of the two old pictures or the other.

It tells us that everyone who wants to seriously study the elementary building blocks of quantum gravity has to learn these methods and - some insights of nuclear physics and condensed matter physics. In the same way, it tells us that everyone who wants to understand nuclear or condensed-matter phenomena well should also learn the correspondence and some basics of gravity and string theory.

The relationship is a mathematical equivalence. It doesn't really require any experiments for us to know that it's true. It can't ever be unlearned and it is damn important.

Planck scale vs other scales

Finally, I want to answer one more aspect of these considerations. A critic could say that AdS/CMT and AdS/QCD don't tell us anything about the physical phenomena near the four-dimensional Planck scale - which is just too far from the typical scales of nuclear and condensed matter physics.

That would be a true assertion but a somewhat vacuous one. Indeed, condensed matter physics is rarely (directly) interested in the four-dimensional Planck scale, 10^{-35} meters, and nuclear physics is very far from it, too.

But this separation of scales doesn't imply that "the string theory" relevant for nuclear physics is "different" from "the string theory" relevant for quantum gravity at the fundamental Planck scale. It's always the same theory. And in some major phenomenological scenarios, namely in the bottom-up approach to string phenomenology (the Vafa et al. F-theoretical phenomenology is the most important recent sub-project), this proposition becomes manifest and geometrical.

If you look at the Poincaré disk with the bats, i.e. the image at the beginning of this article, you may think of the large bats in the middle of the disk as being relevant for physics at relatively long distances. On the other hand, some very small bats near the boundary describe what's happening with your theory at very short distances.

They're different groups of bats but what's important here is that they're connected with each other - they're two parts of the same AdS-like manifold. There is no way to separate them completely. So it's of course true that different scientists focus their attention on phenomena at different characteristic scales. But that doesn't mean that the phenomena at different scales are completely disconnected from each other.

The phenomena at longer distances are always emerging from the phenomena at short distances. And the phenomena at short distances must be such that the phenomena at long distances have a chance to emerge from them. ;-) To put it more symmetrically, the segments of the geometry with small bats and large bats have to fit together.

In the "big desert" scenarios, there is virtually no new physics between the LHC scale and the fundamental Planck scale (or at least GUT scale). In these scenarios, the two "bat regions" have to be connected with one another pretty much directly. That's a non-trivial condition. If you consider other scenarios that could be classified a "big jungle", there will be many more things happening in between the large bats and the small bats (because of the medium bats) and the interactions between small bats and large bats will be much less direct. But these influences will still exist.

The AdS/CFT correspondence may have been slightly unexpected - as many big discoveries usually are - but it has clearly become a crucial insight about the unification of all forces and matter. And it has taught us about relationships between different types of physical phenomena occurring at very different scales. It has identified many previously unsuspected links in between them.

In this sense, the correspondence has brought the Planck scale as close to the verification in labs as mathematics allows. You know, 10^{-35} meters and even 10^{-18} meters will always be much shorter a distance than one millimeter and the phenomena at these scales will always be "largely" disconnected from each other (and 10^{-35} meters will never be "directly" tested). But they're not "quite disconnected" and the more we understand the fundamental laws of physics, the tighter links we can see and use.

Holography is an important link of this kind. And it is just manifestly wrong to say that the lab tests of the predictions of AdS/QCD or AdS/CMT have nothing to do with string theory's being the unifying theory of gravity and other forces and matter, or a theory of everything, if you wish. They have everything to do with it.

And that's the memo.