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.

Hi Dave,

There is a simple argument for holography that goes as follows: assume you increase the amount of energy is some closed volume till it collapses to form a black hole (which is generic once you reach the Schwartzschild radius), that black hole has entropy that grows like area. But, and that is the crucial point, the process of collapse could not have increased the entropy, by the second law. So the matter before collapse has entropy bounded by the Bekestein entropy of the resulting black hole.

Incidentally, AdS/CFT holography, as an argument for area scaling, may be less convincing than one may think as areas and volumes scale the same in AdS space.

PS: disocvered your blog only recently, and enjoy all the QIS posts I can understand…

replace above “could not have increased” by “could not have decreased” of course,preview may be a good idea sometimes…

Dave – very nice summary of our paper!

Moshe – the construction you mention was popularized by Susskind in ’95. The question is: what is the “entropy” to which the bound applies? We show that if it is entanglement or von Neumann entropy the bound is *automatically* satisfied by the condition that the stuff you collapsed in the black hole has not *already collapsed* into an even bigger black hole!

It isn’t necessary to interpret the area bound as a limit on the number of degrees of freedom in the volume (or equivalently on the dimension of Hilbert space). You can keep all the degrees of freedom, but gravitational collapse keeps you from building any state or density matrix which results in entropy > A.

Cheers!

Steve,

I see what you mean, thanks. That is an interesting point of view about the scaling of entropy of the final black hole. However, the holographic bound refers to the entropy of the initial non-gravitational matter before collpase. I still don’t see what is wrong with Lenny’s (and I think ‘thooft’s) argument that the number of DOF of any matter system will grow as area. There may be two things you mean:

1. The matter system before collapse indeed satisfies the holographic bound, but that bound should not be interpreted as “number of DOF”. Note that this is ordinary statistical machanical system, so presumably we know exactly what that entropy means.

2. The matter system before collpase has entropy scaling like the volume, it then seems inevitable that gravitational collapse will decrease entropy, for large enough systems.

But, I better go and read the details of the paper to unconfuse myself…

best,

Moshe