The Holographic Way

Those of you who have been following me on Twitter (and had the patience to read what I am posting there) probably noticed the huge amount of twits with the tag #holography attached. The reason is, naturally, that I am trying to learn it. But before I enter in details, I need to explain what it is all about. If you already know what it is, I will hardly say anything new. 

The term "holography" has two meanings in modern physics, and they are obviously related. The first and most popular one is the technique used to create holograms, those three dimensional images embedded in a two dimensional sheet of paper or plastic. The second one is derived from an analogy with this property of storing the information for a three dimensional environment into a two dimensional one. The story starts with Jacob Bekenstein, a theoretical physicist that was thinking about thermodynamics and black holes. Although I will cut the story a lot, the main point is that he discovered that the entropy of black holes should be proportional to the area of their even horizon, the surface after which nothing can come back. That's what we call, in statistical mechanics language, non-extensive. We call a property extensive when it's proportional to the volume of the object.

The story actually mixes a lot of things. But I will try not to rush in. Back to the black holes, they are in fact the most entropic "objects" in the universe. The argument is simple enough and works by, as in many situations, invoking the Second Law of Thermodynamics. Suppose that in a region of space of radius R there is more entropy than a black hole the size of that region. Then, by adding matter to the region you can increase its mass. If you do that with no care at all, you can always increase the entropy by creating disorder, which is actually very easy as anyone know. It's easy to see where it ends. With enough matter, you can create a black hole the size of the original region. If the black hole has less entropy, than you decreased the TOTAL entropy of the universe and broke the Second Law.

Enters statistical mechanics. In the late 19th century, Boltzmann discovered that the entropy can be understood microscopically as the number of states accessible to some system. And it was by using this concept, that two other physicists, 't Hooft and Susskind, suggested which became known as the Holographic Principle. Consider a region in space. The entropy of that region is bounded by the area of the event horizon of a black hole the size of that region, which means, that the maximum entropy of that region is given by this area. Therefore, the number of possible states in which the entire region can be is proportional not to the volume of the region, but to its area!

Now it's easy to see why it is called the Holographic Principle. The possible configurations of the whole three dimensional region are in fact limited by the two dimensional area of its boundary. Like a hologram. Well, the Holographic Principle actually go one step further by suggesting that the boundary actually ENCODES the degrees of freedom (the equivalent of the possible configurations in some sense) inside the region. That's a bit more difficult to accept, but around 1997, a string theory guy named Juan Maldacena, based on his work on strings proposed something called the AdS/CFT conjecture. In a few words, the conjecture says that the degrees of freedom of a quantum gravity theory in Anti-de Sitter space are encoded in a strongly coupled conformal field theory that leaves on its boundary.

The importance of this is that in some limit, the quantum gravity theory becomes classical gravity, which means general relativity. In fact, it means a classical field theory with a dynamical metric, where metric is the mathematical way of encoding the distance between two points in any kind of space. I am not sure if I understood this point precisely, but I guess that this classical limit is the limit where the conformal field theory becomes strongly coupled. A conformal field theory is a special kind of field theory with an additional scaling symmetry. The good thing is that although we don't know how to deal with strong coupled field theories, we more or less can calculate things in the gravity sector of the AdS/CFT duality.

To finish, let me explain finally why I am interested in it. Recently, there has been some work where the CFT part of the duality display a phenomenology very similar to some strong coupled systems in condensed matter. Now, these systems are quite important and very difficult to deal with with traditional methods like statistical physics or perturbation theory. One of the most famous example is the high temperature superconductor. These superconductors were discovered in 1986 and we still do not have a good understanding of them. It seems that AdS/CFT can shed some light on this. Another problem is called non-Fermi liquids, which are also  strong coupled systems of fermions in condensed matter.

Well, this was just an introduction to the topic. I will try to write more about it as I read. It's a selfish endeavour as it's meant to help myself to think more clearly and understand better this subject. If anyone have comments, suggestions or want to correct the probably lots of mistakes I wrote, or the ones I will write, feel free. That's the aim after all. :) Oh, and by the way, the video has really nothing to do with the text. I just thought of it as a nice example of a hologram. :)