Warning: this post is about a subject I know a tiny tiny bit about. I suspect I will have to update it once I get irrate emails pointing out my horrible misunderstandings.
Roman Buniy and Stephen Hsu (both from the University of Oregon…quack, quack…the mascot of UofO is the Duck!) cross listed an interesting paper to quant-ph today: hep-th/0510021: “Entanglement entropy, black holes and holography.” (Steve posted about it on his blog) As many of you know, the idea of holography is that the number of degrees of freedom of a region of our universe scales proportional to the area of surface of the region. This strange conjecture is extremely interesting, and bizarre, because it raises all sorts of questions about how such theories work (I especially have problems thinking about locality in such theories, but hey that’s just me.) One line of evidence for the holographic principle comes from black hole physics. One can formulate a thermodynamics for black holes, and this thermodynamics gives an entropy for a black hole which is proportional to its area. Another interesting fact is the AdS/CFT correspondence which shows an equivalence between a certain quantum gravity theory in an anti-deSitter universe and a conformal field theory on the boundary of this space: i.e. quantum gravity in this space can be described by a theory on the surface of the space, a holographic theory, so to speak. Indeed, the fact that certain string theories have black holes which have a holographic number of degrees of freedom is taken as evidence that string theory might be consistent with our universe.
What Buniy and Hsu suggest in their paper is that the holographic bound is not a bound on the degrees of freedom of our theory of the universe, but that instead, the holographic bound should be thought about as a bound on entropy of a region in the presence of gravity. They point out that if you take gravity away, then the scaling of the degrees of freedom scales like the volume (although, if you take the ground state of a local quantum field theory, then this particular state has an entropy which scales like the area: such states that Buniy and Hsu consider are therefore necessarily not ground states of such theories. But this doesn’t mean that they don’t exist or that we can’t construct such states.) They then argue that if, on the other hand, you want to avoid gravitational collapse, then this requirement precludes such states, and indeed gives you states whose entropy scales like the area. What Buniy and Hsu seem to be arguing is that while one does obtain entropies which scale like the area using these arguments about black holes, this doesn’t imply that the degrees of freedom of the underlying theory must scale as the area.
One might wonder whether there is a difference between having an entropy scaling like the area and the degrees of freedom scaling like the area. Well certainly there would be for an underlying theory of quantum gravity: presumably different degrees of freedom can be accessed which give the same area scaling, but which represent fundamentally different physical settings. So, for example, I can access some of these degrees of freedom, and as long as I don’t create a black hole, these degrees will be as real for me as they can be. But if I try to access them in such a manner that I create a black hole, I will only see the effective degrees of freedom proportional to the area of the black hole.
Which is all very interesting. Just think, maybe one of the greatest achievements of string theory, deriving holographic bounds, actually ends up being a step in the wrong direction. And, no I’m not wishing this fate upon string theory. I wish no fate among any theories: I just want to understand what nature’s solution is.