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NNTP-Posting-Host: feynman.harvard.edu Mime-Version: 1. Content-Type: TEXT/PLAIN; charset=US-ASCII X-X-Sender: In-Reply-To: Xref: solaris.cc.vt.edu sci.physics.strings:39 On Sun, 18 Apr 2004, Lubos Motl wrote: > On Sun, 18 Apr 2004, Urs Schreiber wrote: > > > This was supposed to be evidence that at least in the discussed limit > > the $E_{10}$ model is indeed equivalent to the BFSS model. > > Does this proof explain why the BFSS model is the discrete light-cone > quantization of the covariant $E_{10}$ model, and why "N" of BFSS is the > light-like momentum? (If it does not, it must be wrong.) Thanks for asking this question! (And thanks, in fact, for all of this rapid and very critical discussion.) I have thought about precisely this type of question a lot in the last couple of days. The point is: Can we, apart from noting that we have ensembles of the form $\exp(-Tr(M^2))$ on both sides in the given limit, can we identify the physical meaning of the matrices on both sides of the conjectured equivalence? In order to answer the question we would of course first of all have to figure out what the "meaning" of the matrices on the $E_{10}$ side of the conjectured equivalence actually is. Without any further work, all that we know is that the semiclassical limit of 11d sugra close to a spacelike sinmgularity is, due to chaoticity, equal to that of a theory of an ensemble of randomly chosen NxN matrices for large N. But what is the physical interpretation of N in this context? In fact this is a very old question. It has been known for a long time empirically (i.e. using numerics) that Random Matrix Theory (RMT) does universally describe the semiclassical limit of all chaotic quantum systems (i.e. the predictions obtained from the random ensemble of matrices, for instance concerning the statistic of the spectrum of the system's Hamiltonian, precisely coincide with the predictions obtained from the origina theory). But I am being told by specialists working on quantum chaos that the reason for this "unreasonable effectiveness" of RMT in quantum chaos has so far remained a mystery. In fact, there is quite some excitement at my institution about the recent results of one of our groups (S. Mueller, S. Heusler, P. Braun, F. Haake et al.) who have solved the long-standing problem of actually explicitly calculating the universal semiclassical spectrum of general chaotic quantum systems and checking equivalence with the resuts of RMT. So this finally improves the comparison with numerical results to a formal calculation. The prediction of RMT are now proven to be exactly those found from a full semiclassical analysis of the true systems. So this is reassuring, because it shows that the RMT-conjecture ("every chaotic quantum system is described by RMT") is correct. But unfortunately, at least as far as I can see, this still does not tell us WHY the conjecture is correct, i.e. why this ensemble of matrices describes single chaotic systems. It still does not give us a physical interpretation of the large matrices and of the parameter N, which is what we would need to answer your question above. For precisely this reason I have tried to figure out the answer myself, recently. You can find my proposed solution as well as some background information and discussion at the Coffee Table: http://golem.ph.utexas.edu/string/ar...2.html#c000927 I don't know if this works out as expected, but I think it looks promising. So my proposed solution is that we have to interpret the ensemble of systems used in RMT as associated with the ensemble of points in a "non-universal cell" of the config space of the chaotic system. More precisely, the classical chaoticity of the system implies that in the semiclassical limit (which is dominated by classical contributions to the path integral) we can introduce a coarse-graining of the confuguration space into cells, such that within each cell matrix elements of the Hamiltonian are correlated, while outside they are not. This reflects precisely the fact that on very small scales the system will have non-universal behaviour while on sufficiently large scales it will exhibt the universality of ergodicity and chaos. Please refer to the above links for some more details of this, admittedly simple but apparently original, idea. Let's assume this interpretation is approximately correct and try to understand what it implies for the interpretation of the matrices which would describe the rmt of 11d sugra close to a spacelike singularity. Now config space is the mini-superspace, and in particular the Weyl-chamber of E10, where every point corresponds to a specific set of values of the scale factors of the sugra universe. The matrices would now describe transition amplitudes between different "cells" of this config space and N would be the total number of such cells. Hence N would correspond roughly to a discretzation of the scale factors, I'd think. Ths does not look like it could have any relation to the interpretation of N in the BFSS model, does it? So maybe my conjecture is wrong. But maybe we should bea little more careful before making this conclusion, because of various reinterpretations which are possible. For instance in order for my conjecture to make sense we would really be looking at the canonical ensemble of BFSS theory and hence at BFSS at _finite temperature_. But we know that BFSS at finite temperature is nothing but the IKKT model! So it seems that we would rather have to identify the matrices with those of the IKKT model. There the above interpretation might make much more sense, since it is known that we can interpret in the IKKT model the integer N as the number of points in a discretization of spacetime. (i assume that that's what you were referring to when mentioning the relation of IKKT to "quantum foam".) So this would match nicely with the interpretation of N as characterizing discrete scale factors of the universe. Of course I am aware that all this is rather vague and very speculative. But I think it is a fun speculation and not uninteresting. Maybe it is wrong. Maybe not. In fact, if there is any truth to the E10 model then something along the lines sketched above must be true.