<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>NNTP-Posting-Host: feynman.harvard.edu\nMime-Version: 1.0\nContent-Type: TEXT/PLAIN; charset=US-ASCII\nX-X-Sender: <sps@feynman.harvard.edu>\nIn-Reply-To: <Pine.LNX.4.31.0404181249060.10181-100000@feynman.harvard.edu>\nXref: solaris.cc.vt.edu sci.physics.strings:39\n\nOn Sun, 18 Apr 2004, Lubos Motl wrote:\n\n> On Sun, 18 Apr 2004, Urs Schreiber wrote:\n>\n> > This was supposed to be evidence that at least in the discussed limit\n> > the E_10 model is indeed equivalent to the BFSS model.\n>\n> Does this proof explain why the BFSS model is the discrete light-cone\n> quantization of the covariant E_{10} model, and why "N" of BFSS is the\n> light-like momentum? (If it does not, it must be wrong.)\n\nThanks for asking this question! (And thanks, in fact, for all of this\nrapid and very critical discussion.)\n\nI have thought about precisely this type of question a lot in the last\ncouple of days. The point is: Can we, apart from noting that we have\nensembles of the form exp(-Tr(M^2)) on both sides in the given limit,\ncan we identify the physical meaning of the matrices on both sides\nof the conjectured equivalence?\n\nIn order to answer the question we would of course first of all have\nto figure out what the "meaning" of the matrices on the E_10 side\nof the conjectured equivalence actually is.\n\nWithout any further work, all that we know is that the semiclassical\nlimit of 11d sugra close to a spacelike sinmgularity is, due to\nchaoticity, equal to that of a theory of an ensemble of randomly\nchosen NxN matrices for large N.\n\nBut what is the physical interpretation of N in this context?\n\nIn fact this is a very old question. It has been known for a long\ntime empirically (i.e. using numerics) that Random Matrix Theory (RMT)\ndoes universally describe the semiclassical limit of all chaotic\nquantum systems (i.e. the predictions obtained from the random\nensemble of matrices, for instance concerning the statistic of\nthe spectrum of the system\'s Hamiltonian, precisely coincide with\nthe predictions obtained from the origina theory).\n\nBut I am being told by specialists working on quantum chaos that\nthe reason for this "unreasonable effectiveness" of RMT in\nquantum chaos has so far remained a mystery.\n\nIn fact, there is quite some excitement at my institution about\nthe recent results of one of our groups (S. Mueller, S. Heusler,\nP. Braun, F. Haake et al.) who have solved the long-standing\nproblem of actually explicitly calculating the universal\nsemiclassical spectrum of general chaotic quantum systems and\nchecking equivalence with the resuts of RMT. So this finally\nimproves the comparison with numerical results to a formal\ncalculation. The prediction of RMT are now proven to be exactly\nthose found from a full semiclassical analysis of the true\nsystems.\n\nSo this is reassuring, because it shows that the RMT-conjecture\n("every chaotic quantum system is described by RMT") is correct.\n\nBut unfortunately, at least as far as I can see, this still does\nnot tell us WHY the conjecture is correct, i.e. why this\nensemble of matrices describes single chaotic systems. It still\ndoes not give us a physical interpretation of the large matrices\nand of the parameter N, which is what we would need to answer\nyour question above.\n\nFor precisely this reason I have tried to figure out the answer\nmyself, recently. You can find my proposed solution as well as\nsome background information and discussion at the Coffee Table:\n\nhttp://golem.ph.utexas.edu/string/archives/000342.html#c000927\n\nI don\'t know if this works out as expected, but I think it looks\npromising.\n\nSo my proposed solution is that we have to interpret the\nensemble of systems used in RMT as associated with the ensemble\nof points in a "non-universal cell" of the config space\nof the chaotic system.\n\nMore precisely, the classical chaoticity of the system\nimplies that in the semiclassical limit (which is dominated\nby classical contributions to the path integral) we can\nintroduce a coarse-graining of the confuguration space\ninto cells, such that within each cell matrix elements of\nthe Hamiltonian are correlated, while outside they are not.\nThis reflects precisely the fact that on very small scales\nthe system will have non-universal behaviour while on\nsufficiently large scales it will exhibt the universality of\nergodicity and chaos.\n\nPlease refer to the above links for some more details of\nthis, admittedly simple but apparently original, idea.\n\nLet\'s assume this interpretation is approximately correct and\ntry to understand what it implies for the interpretation of the\nmatrices which would describe the rmt of 11d sugra close to\na spacelike singularity.\n\nNow config space is the mini-superspace, and in particular the\nWeyl-chamber of E10, where every point corresponds to a\nspecific set of values of the scale factors of the\nsugra universe. The matrices would now describe transition\namplitudes between different "cells" of this config space\nand N would be the total number of such cells.\n\nHence N would correspond roughly to a discretzation of the\nscale factors, I\'d think.\n\nThs does not look like it could have any relation to the\ninterpretation of N in the BFSS model, does it?\n\nSo maybe my conjecture is wrong. But maybe we should bea little more\ncareful before making this conclusion, because of various\nreinterpretations which are possible.\n\nFor instance in order for my conjecture to make sense we would\nreally be looking at the canonical ensemble of BFSS theory\nand hence at BFSS at _finite temperature_.\nBut we know that BFSS at finite temperature is nothing but\nthe IKKT model!\n\nSo it seems that we would rather have to identify the matrices\nwith those of the IKKT model.\n\nThere the above interpretation might make much more sense, since\nit is known that we can interpret in the IKKT model the integer\nN as the number of points in a discretization of spacetime.\n(i assume that that\'s what you were referring to when mentioning\nthe relation of IKKT to "quantum foam".) So this would match\nnicely with the interpretation of N as characterizing discrete\nscale factors of the universe.\n\nOf course I am aware that all this is rather vague and very\nspeculative. But I think it is a fun speculation and not\nuninteresting. Maybe it is wrong. Maybe not. In fact, if there\nis any truth to the E10 model then something along the lines\nsketched above must be true.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>NNTP-Posting-Host: feynman.harvard.edu
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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 [itex]E_{10}[/itex] 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 [itex]E_{10}[/itex] 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 [itex]\exp(-Tr(M^2))[/itex] 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 [itex]E_{10}[/itex] 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.