### Billiards, random matrices, M-theory and all that

#### Posted by Urs Schreiber

I am currently at a seminar on quantum chaos and related stuff. You cannot enjoy meetings like these without knowing and appreciating the *Gutzwiller trace formula* which tells you how to calculate semiclassical approximations to properties of the spectrum of chaotic quantum systems (like Billiards and particles on spaces of constant negative curvature) by summing over periodic classical paths.

One big puzzle was, and still is to a large extent, why *random matrix theory* reproduces the predictions obtained by using the Gutzwiller trace formula.

In random matrix theory you pick a Gaussian-like ensemble of matrices (orthogonal, symplectic or unitary ones) and regard each single such matrix as the Hamiltonian operator of some system. It is sort of obvious why this is what one needs for systems which are subject to certain kinds disorder. But apparently nobody has yet understood from a conceptual point of view why it works for single particle systems which are calculated using Gutzwiller’s formula. But there is quite some excitement here that one is at least getting very close to the proof that Gutzwiller does in fact agree with RMT, see

Stefan Heusler, Sebastian Müller, Petr Braun, Fritz Haake, Universal spectral form factor for chaotic dynamics (2004) .

One hasn’t yet understood *why* this agrees, only that it does so. My hunch is that it has to do with the fact that by a little coarse graining we can describe the classical chaotic paths as random jumps and that the random matrix Hamiltonians are just the amplitude matrices which describe these jumps.

But anyway. ‘Why all this at a string coffe table?’, you might ask.

Well, while hearing the talks I couldn’t help but notice the fact that I actually do know one apparently unrelated but very interesting example of a system which, too, is described both by chaotic billiards as well as by random matrices. This system is - 11 dimensional supergravity.

I had mentioned before the remarkable paper

T. Damour, M. Henneaux , H. Nicolai Cosmological Billiards (2002)

where it is discussed and reviewed how theories of gravity (and in particular of supergravity) close to a spacelike cosmological singularity decouple in the sense that nearby spacetime points become causally disconnected and how that leads to a mini-superspace like dynamics in the presence of effective ‘potential’ walls’ which is essentially nothing but a (chaotic) billiard on a hyperbolic space.

(This paper is actually a nice thing to read while attending a conference where everybody talks about billiards, chaos, coset spaces, symmetric spaces, Weyl chambers and that kind of stuff. )

So 11d supergravity in the limit where interactions become negligible is described by a chaotic billiard just like those people in quantum chaos are very fond of.

But here is the crux: 11d supergravity is *also* known to be approximated by the BFSS matrix model. Just for reference, this is a system with an ordinary quantum mechanical Hamiltonian

where the $X^i$ are large $N \times N$ matrices that describe D0-branes and their interconnection by strings or, from another point of view, blobs of supermembrane.

Hm, but now let’s again forget about the interaction terms. Then the canonical ensemble of this system is formally that used in random matrix theory!

Am I hallucinating ot does this look suggestive?

I think what I am getting at is the following: Take Damour&Henneaux&Nicolai’s billiard which describes 11d supergravity. Now look at its semiclassic behaviour. It is known that this is governed by random matrix theory (But we have to account for some details, like the fact that the mini-superspace billiard is relativistic. Maybe we have to go to its nonrelativistic limit.) We realize that the weight of the random matrix ensemble is the free kinetic term of the BFSS model. Therefore we might be tempted to speculate that the true ensemble of randowm matrices which is associated with 11d supergravity *away* from the cosmological singularity is obtained by including the $[X,X]^2$ interaction term of the BFSS Hamiltonian in the weight. With this RMT description in hand, try to find the corresponding billiard motion. Will it coincide with the speculation made by DHN about the higher-order corrections to their mini-superspace dynamics?

In any case, I see that apparently random matrix theory (‘like every good idea in physics’ ;-) has its place in string theory. I should try to learn more about it.

Posted at April 8, 2004 12:34 AM UTC
## Random matrices, M-theory and black holes

Several years ago I looked at the role of random matrices in Matrix theory black holes. Eventually, I wrote a paper, Eigenvalue Repulsion and Matrix Black Holes, in which I argued that the eigenvalue repulsion displayed by random matrices gives rise to the holographic behavior of matrix black holes. I did not

showthis, but I did argue it.The paper would probably be a pretty light read for you, but it might fit with your interest in the connection between Matrix theory and random matrices. It does not address any cosmological issues.

Cheers,

Gavin