The String Coffee Table

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 show this, 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

Stefan Heusler and Sebastian Müller told me (if I understood them correctly) that the importance of their result nlin.CD/0309022 is that it makes progress towards elucidating why Random Matrix Theory, which seems to be about ensembles of physical systems, makes such good predictions about the spectral statistics of single chaotic systems.

Indeed, in the introduction of the above paper it says:

Understanding the observed fidelity of individual dynamics to RMT has been an elusive goal, in spite of considerable efforts […]. We shall here take a non-trivial step towards that goal, on semiclassical ground.

As I said above, I am interested in understanding this relationship because I feel that it may help see a conceptual connection between the BFSS Matrix Model description of supergravity with the one in terms of chaotic cosmological billiards by Henneaux,Damour&Nicolai.

To me it seems that it should be possible to understand why (conceptually) RMT has something to say about, for instance, chaotic billiards, over and above checking that the trace formula over periodic orbits does indeed reproduce the RMT prediction for the spectral ‘form factor’.

While visiting my and my gilrfriend’s parents over Easter holidays and while sipping coffee and eating easter eggs ;-) I had the leisure to spend a couple of thoughts on this question. These are what I would like to discuss in the following.

The key idea is the following: Stefan Heusler had emphasized to me that the problem is that in a single system it is not clear what should correspond to the ensemble used in RMT. I want to discuss the possibility that if we consider a coarse-graining of the phase space of the system into cells large enough that all non-universal behaviour takes place only within a single such cell, while the coarse-grained large-scale behaviour is truly chaotic, that then the ensemble of the RMT corresponds simply to the set of phase space points within a single such cell.

I would like to demonstrate some circumstantial formal evidence for this claim, which, while being short of a being proof, even in the physicist’s sense, should make the conjecture quite plausible.

So pick some classically chaotic system $S$ with classical phase space $P_S$ and partition phase space into disjoint cells $C_i$ with $i\in \{1,2,3,\cdots ,N\}$ such that

Consider furthermore the set of all $N\times N$ matrices of pairs of phase space points to any given pair of such cells, e.g.

where $x_{\mathrm{ij}}\in C_i$ and $y_{\mathrm{ij}}\in C_j$.

These matrices give rise to approximations to the exact Hamiltonian $H$ of our system by the following procedure:

For a given point $x\in P_S$ let $|x⟩$ be a state in the Hilbert space of the system which is of minimal uncertainty (a generalized coherent state) and peaked at values of coordinates and momenta given by $x$. (Note that $x$ denotes a point in phase space.)

Now construct all $N\times N$ matrices $h_{\lambda }$ indexed by the above matrices $\lambda$ and defined by

These matrices are essentially the transition amplitudes from a coherent state peaked at $y_{\mathrm{ij}}\in C_j$ to a state peaked at $x_{\mathrm{ij}}\in C_i$.

The idea behind all these definitions is that I want to decompose the trace over the system’s Hilbert space into a sum over the cells $C_i$ and a sum over coherent states peaked within each $C_i$. The point is that the ergodicity of the system allows to say something about averaged transition amplitudes between different cells. This information then can be used to say something about the distribution of the transition amplitdues between single points in these cells and hence about the ensemble of the $h_{\lambda }$

More precisely, the averaged amplitude for a system to move from somewhere within cell $C_j$ to somewhere within $C_i$ is

where the sum is a symbolic shorthand for a suitably normalized double integral over all coherent states peaked at $x_i\in C_i$ and $y_j\in C_j$. (These states will of course overlap into neighbouring cells.)

At this point I use the first essential piece of assumption/classical information: By the very construction of the $C_i$ they are supposed to be larger than the scales traversed by the system in a time span so short that non-universal behaviour still plays a role. This means that on very short time spans there is no transition on average between two cells $C_i$ and $C_j$. Therefore the above expression should vanish to first order in $t$, i.e.

I like to think of this as saying that there is an ensemble of systems $C_j$ that evolves into an ensemble of systems $C_i$ such that the averaged transition amplitude cancels out.

In order to make this more precise note that the above condition can be reformulated in terms of the matrices $h_{\lambda }$ as

Here $\lambda$ runs over all $\lambda$-matrices, as defined above.

It seems correct to think of the $h_{\lambda }$ of an ensemble of effective Hamiltonians each describing the transition of a particular system with a denumerable phase space to evolve from $C_i$ to $C_j$. The different ‘systems’ are of course nothing but sub-trajectories of the single system under consideration. They appear as different systems due to the coarse-graining.

With this interpretation, the above condition says that the average of each off-diagonal matrix entry over the ensemble of all $h_{\lambda }$ vanishes.

This is nice, because it agrees with the assumption made in RMT. So let’s see if we can similarly say something about the variance of the $(h_{\lambda }{)}_{\mathrm{ik}}$:

To that end, consider the cell-averaged trace over the propagator:

By the above considerations the term of first order in $t$ vanishes. Let’s concentrate on the leading non-vanishing order $t^2$:

By inserting an (approximate) decomposition of unity and using the fact that the $|x_i⟩$ are approximately orthogonal (here is obviously the point in my discussion where all these considerations are currently still very vague and sketchy) this can be rewritten as something looking like

(Please recall that sums over $x_i$ are supposed to denote suitably normalized integrals. The idea should be clear. I don’t want to clutter its discussion with overly exact notation at this point.)

In words: The probability to go along a closed orbit starting in $C_i$ and visiting given $C_j$ in the process is, to lowest non-vanishing order, the sum of the squares of the ensemble of transition amplitudes between $C_i$ and $C_j$.

Now I use the second crucial assumption/classical observation: It is known that the classical chaotic dynamics is such that on the coarse-granined level of the $C_i$ the time evolution is a complytely random hopping between the different cells. This should mean that the contribution to the above from closed paths visiting any of the $C_j$ must be the same, i.e.

Note that there is no sum over $j$ and $k$ here.

(One should probably assume a compact phase space in order to make this a little less vague than it is in any case.)

But this is nice: The above says nothing else but that the variance of the non-diagonal matrix elements of the $h_{\lambda }$, when avaraged over the full ensemble (all possible $\lambda$) is the same for all matrix elements.

This way one finds that the ensemble of the $h_{\lambda }$ is such that mean and variance of the matrix elements of the $h_{\lambda }$ have exactly the properties that agree with those postulated in RMT. (The mean and variance of the diagonal elements follow now from unitary/orthogonal/symplectic invariance.)

If one could argue now that due to the chaoticity of the system $S$ the ensemble of the $h_{\lambda }$ is such that it maximizes the information entropy with the given mean and variance of the matrix elements fixed, then one would have established that and how Random Matrix Theory describes single chaotic systems:

If the ideas sketched here are correct, they would indicate that we have to identify the random matrices that appear in RMT with the objects $h_{\lambda }$ defined above, which are Hamiltonians/transition amplitudes obtained from the full Hamiltonain of the system by restricting to certain microscopic trajectories such that they seem to describe an ensemble of systems on the phase space of coarse-grained cells $C_i$, where the $C_i$ are chosen in such a way that they precisely hide all small-scale non-universal properties of the system.

I should stop at this point. All comments and pointers to the literature are very welcome.

What I would expect is similar to 2 separate cells colliding - first making contact at one point then at several other points in varying succession. There would also be an uneven exchange between the two cells contents - (Think of two undulating water balloons colliding in slow motion).

Lance