Wednesday, August 6, 2014

Holographic principle

So just a while ago I received an email from World Scientific about my article "Easy Holography", which was published in their journal IJMPD. According to the email, my paper "...is a great contribution in understanding the relevant field of study, which is worth to be shared." Oo well gee, thanks for the recognition World Scientific (I'm sure only select few receive such encouragement)! Oh well, I guess I'll just do as I'm told and share the article!

Anyway, the idea is roughly as follows (it's been a while since I worked on this stuff, so my apologies for some possible inaccuracies): let's pretend we live in a two dimensional universe and that this universe is a two dimensional quantum gravity theory, which is "asymptotically $AdS_2$", which is very similar to a hyperbolic plane $\mathbb H_2$ but of minkowskian signature (google it or stop reading). Or just think of $\mathbb H_2$. The "asymptotic symmetries" of $\mathbb H_2$ are such diffeomorphisms which preserve the metric near the boundary. For $\mathbb H_2$ these symmetries are isomorphic to circle diffeomorphisms $Diff(S^1)$, which is an infinite dimensional group. The only thing we need to assume about the quantum gravity theory is that it is integrable w.r.t to the asymptotic symmetries, which in this context just means that every physical state can be mapped to another physical state by the asymptotic symmetries extended to all of the space (a la Conformal Field Theory).

The toy model I used in the "Easy Holography" paper was basically just a pixelated image (of a bunny, since it was submitted on Easter). A rough estimate of the entropy of that "model" is then a logarithm of the number of all distinct images that are mapped from an original image by the asymptotic symmetry group, which is isomorphic to $Diff(S^1)$.


So first I have a bunny. Now "coarse grain" it on an $N \times N$ grid, so that a grid is black if most of the (Euclidean) surface area under it is black and white otherwise. Then apply a map from the asymptotic symmetries on the original image and coarse grain that. If the coarse grained image is different from the original coarse grained image, count that as a distinct state/configuration. Repeat this for all as. symmetries (there's a noncountably infinite number of them) and write down the number of all distinct images, or configurations, as a function of the number of  $N$.

If the maps were just the usual diffeomorphisms, it's easy to see (see the image above or the paper) that the number of distinct configurations is just $2^{N^2}$ and thus the entropy is $S = N^2 \log(2)$. So the entopy is just proportional to the volume of the system, $N^2$. So that's the old physics. But if instead we use the asymptotic symmetry maps, it turns out that the entropy thus computed is now proportional to $N$, or the "area" of the system! This happens also for $AdS_3$, whose as. symmetries are the conformal group, isomorphic (if I recall correctly) to $Diff(S^1) \times Diff(S^1)$. In the $AdS_3$ the entropy would be proportional to $N^3$ (volume) for ordinary diffeomorphisms and to $N^2$ (area) for the asymptotic symmetries.

So the conclusion would seem to be that the asymtptotic symmetries and the asymptotic $AdS$ structure of the universe is somehow related to the Holographic Principle, which states that the entropy is indeed proportional to the surface are of the system, instead of the volume... but of course the universe is not three dimensional! So what happens in higher dimensions? The asymptotic symmetry group of $AdS_4$ is disappointingly not infinite dimensional, but instead the six dimensional $SO(2,2)$. That would actually just lead to a logarithmic entropy, so no luck there...

But what actually is an $AdS$ space and why would they be interesting? Let's look at (one) definition of an $AdS$ space. We can define $AdS_d$ as the homogeneous space $SO(2, d-1) / SO(2, d-2)$. For $d=2$, we have $AdS_2 \simeq SO(2, 1) / SO(2)$ and for $d=3$, $AdS_3 = SO(2, 2) / SO(2,1)$. For low dimensional Lie groups there exist so called "accidental isomorphies", which we can use to write $SO(2,1) \simeq SL(2)$ and $SO(2,2) \simeq SL(2) \times SL(2)$, where $SL(2)$ is the special linear group. So we can write the $AdS$ spaces in an equivalent way as $AdS_2 \simeq SL(2)/SO(2)$ and $AdS_3 \simeq SL(2)$ (Lie groups are manifolds too). For $d>2$ such accidental isomorphisms do not exist. More specifically, the higher dimensional $AdS$ spaces are not related at all to the special linear groups $SL(n)$.

A short recap: $AdS_2$ and $AdS_3$ are intimately related to the Lie group $SL(2)$, but higher dimensional $AdS$ spaces are not. $AdS_2$ and $AdS_3$ admit infinite dimensional as. symmetries, but higher dimensional $AdS$ spaces do not. Clearly the Lie group $SL(2)$ is the key here! Some natural questions are then

1) Do the higher dimensional Lie groups $SL(n)$ (or related homogeneous spaces) also admit infinite dimensional asymptotic symmetries?
2) If they do, will they also incorporate a "natural holography" as with $AdS_2$ and $AdS_3$?
3) Is it then possible come up with a higher dimensional quantum gravity theory, which is similarly exactly solvable as $AdS_3$ quantum gravity (well, it should be exactly solvable I guess...)?

The answer to part 1) above is "yes". I actually showed that for a homogeneous (symmetric) space $SL(3)/SO(3)$ in an earlier paper called "Infinite symmetry on the boundary of $SL(3)/SO(3)$". That paper is not very elegant, and I was working on an extended and more sophisticated version, but I didn't have time to finish writing it... the final equations were actually solved by Robert Bryant at mathoverflow. I couldn't get any research funding for that though and now I'm doing something completely different! Anyway, hopefully someone will some day find these results useful.