## Re: Supergravity Cosmological Billards and the BIG group

<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>On Thu, 15 Apr 2004, Urs Schreiber wrote:\n\n&gt; The most natural idea is to try to quantize the sigma model\n&gt; on E10/K(E10), or one of its consistent truncations. In\n&gt; principle this is straightforward, since quantizing 1+0 dim\n&gt; theories just involves quantum mechanics. But of course there\n&gt; are some thorny details. In any case, it hasn\'t been done yet.\n\n\nActually I now see that steps in this direction have been\ndone at least in\n\nJ. Brown, O. Ganor, C. Helfgott,\nM-Theory on E10: Billiards, Branes, and Imaginary Roots,\nhep-th/0401053 .\n\nIn section 7 the authors describe how to iteratively\nconstruct the E10 Laplacian Delta (i.e. the "wave operator"\non the group G10). And they also discuss how certain\nsolutions of the conjectured "Wheeler-deWitt equation\nof M-theory"\n\nDelta |psi&gt; = 0\n\nshould describe states containing brane/anti-brane pairs\nas a pure quantum effect of this framework.\n\nFurthermore they discuss a related "quantum test" of the\n\'Delta |psi&gt; = 0\'-conjecture related to the masses of the\nbranes.\n\nOne big open question is: What about fermions?\n\nSo far all these constructions seem to be purely bosonic,\neven though on p.50 of the above paper the possibility\nis mentioned that the fermions might be part of the E10\nvariables (in bosonized form).\n\nBut apparently one rather expects that a sort of\nsupersymmetrization of the E10 1+0d sigma model is\nnecessary.\n\nI still think that it should be interesting to study the\ndeRahm operators d and del on E10. We know that every 1+0d\nsigma model admits a straightforward N=2 susy extension\nby replacing Delta by {d,del}. It seems to me that with\nthe technology presented in hep-th/0401053 it should be\neasy to construct\n\nd = d_h + d_0 + d_1 + d_2 + ...\n\nand similarly for del in complete analogy to what is\ndone in section 7.2 of the above paper for\n\nDelta = Delta_h + Delta_0 + Delta_1 + ... .\n\nHopefully I\'ll find the time to look into that\nquestion. One should check if the fermions (differential\nforms) introduced this way match the dynamics of\nmodes of the 11d sugra fermions, in analoggy to the bosonic\nsector.\n\nOne important point at least seems to be immediate for\nthis kind of supersymmetrization of the E10 sigma model:\n\nAt the end of section 7.4 on p.53 of the above paper it\nis mentioned that in the bosnic theory there is a problem\nwith zero-point energies which should better vanish. Of\ncourse these do vanish automatically for the deRahm\nmodel for harmonic forms of the respective d_i.\n\n\nI see that Lubos is acknowledged by the above authors\nfor helpful discussion. Maybe he can comment on\nthe above issues.~\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 15 Apr 2004, Urs Schreiber wrote:

> The most natural idea is to try to quantize the $\sigma$ model
> on $E10/K(E10),$ or one of its consistent truncations. In
> principle this is straightforward, since quantizing 1+0 dim
> theories just involves quantum mechanics. But of course there
> are some thorny details. In any case, it hasn't been done yet.

Actually I now see that steps in this direction have been
done at least in

J. Brown, O. Ganor, C. Helfgott,
M-Theory on E10: Billiards, Branes, and Imaginary Roots,
hep-th/0401053 .

In section 7 the authors describe how to iteratively
construct the E10 Laplacian $\Delta (i.e.$ the "wave operator"
on the group G10). And they also discuss how certain
solutions of the conjectured "Wheeler-deWitt equation
of M-theory"

$$\Delta |\psi> =$$

should describe states containing brane/anti-brane pairs
as a pure quantum effect of this framework.

Furthermore they discuss a related "quantum test" of the
$'\Delta |\psi> =$ 0'-conjecture related to the masses of the
branes.

One big open question is: What about fermions?

So far all these constructions seem to be purely bosonic,
even though on p.50 of the above paper the possibility
is mentioned that the fermions might be part of the E10
variables (in bosonized form).

But apparently one rather expects that a sort of
supersymmetrization of the E10 $1+0d \sigma$ model is
necessary.

I still think that it should be interesting to study the
deRahm operators d and del on E10. We know that every $1+0d$
$\sigma$ model admits a straightforward N=2 susy extension
by replacing $\Delta$ by {d,del}. It seems to me that with
the technology presented in hep-th/0401053 it should be
easy to construct

$d = d_h + d_0 + d_1 + d_2 + .$..

and similarly for del in complete analogy to what is
done in section 7.2 of the above paper for

$\Delta = \Delta_h + \Delta_0 + \Delta_1 + .$.. .

Hopefully I'll find the time to look into that
question. One should check if the fermions (differential
forms) introduced this way match the dynamics of
modes of the 11d sugra fermions, in analoggy to the bosonic
sector.

One important point at least seems to be immediate for
this kind of supersymmetrization of the E10 $\sigma$ model:

At the end of section 7.4 on p.53 of the above paper it
is mentioned that in the bosnic theory there is a problem
with zero-point energies which should better vanish. Of
course these do vanish automatically for the deRahm
model for harmonic forms of the respective $d_i$.

I see that Lubos is acknowledged by the above authors
for helpful discussion. Maybe he can comment on
the above issues.~

 PhysOrg.com physics news on PhysOrg.com >> Promising doped zirconia>> New X-ray method shows how frog embryos could help thwart disease>> Bringing life into focus
 Dear Urs, I've been intrigued by Ori Ganor's proposals from the beginning. Several years ago, he conjectured that the whole M-theory is encoded in the Laplace equation defined on the $E_{10}$ group manifold. Partition sums of gravities, worldvolume theories and perhaps all other things should be encoded in this allegedly unique function that satisfies this Laplace equation on this huge manifold, see http://arxiv.org/abs/hep-th/9903110 http://arxiv.org/abs/hep-th/9910236 This would have the advantage of making all U-dualities etc. manifest. I think that the new paper by Ori et al. continues in this direction, and they would also like to find a one-to-one map between the basis vectors of the full M-theoretical Hilbert space on one side and the roots of $E_{10}$ on the other side, or something along these lines. We should perhaps ask Ori to write something interesting about it on SPS. Another purpose of this posting is to test LaTeX posted from physicsforums.com. Click the link below to see the math version of this post, including the nonsensical equation $$E=mc^2 + \frac{\sqrt{1-x^2}}{\log(3)} + \int {\mathrm{d}} x\, \bar f(x) g(x) + \dots$$ All the best Lubos


On Fri, 16 Apr 2004, lumidek wrote: > I've been intrigued by Ori Ganor's proposals from the beginning. Great to hear! :-) BTW, do you know if the approach by Ganor et al. started independently of the cosmological Billiard approach by Damour Henneaux and Nicolai and later merged, or what is the historical development? > Several years ago, he conjectured that the whole M-theory is encoded in > the Laplace equation defined on the $E_{10}$ group manifold. Yes. That would be remarkable, if true. It is entertainingly reminiscent of this famous saying, apparently due to Feynman, that most of physics is about "Laplace $\phi =$ " in some generalized sense. > We should perhaps ask Ori to write something interesting about it on > SPS. Are you going to contact him or shall I? By the way, while talking to Hermann Nicolai about these things in Golm I learned about the interesting fact that E10 can be regarded as sitting inside the vertex operator algebra of the string. Of course that's long and well known (e.g. http://www.arxiv.org/abs/hep-th/9411188) but I didn't fully appreciate before how already the symmetry algebra of the single string knows about "M-theory" in this sense. Right now I am visiting Ioannis Giannakis at Rockefeller University and he also emphasizes this fact. As you know, I am here in order to discuss deformations of superconformal worldsheet theories (as in http://www.arxiv.org/abs/hep-th/0401175 and references given there) which in particular include the symmetry transformations of the background (gauge transformations, dualities, etc.). Ten years ago Evans, Giannakis and Nanopoulos in Evans, Giannakis & Nonopoulos, An infinte dimensional symmetry algebra in string theory http://www.arxiv.org/abs/hep-th/9401075 have already discussed the immense symmetry algebra that is found by deforming the worldsheet CFT by similarity transformations of the form $$A -> \exp(-W) A \exp(W)$$ for W a general integrated vertex. I am not completely sure what currently the state of the art is with respect to our understanding of this algebra, though. I am being told that physicists are currently better prepared to discuss E10 than mathematicians are.

## Re: Supergravity Cosmological Billards and the BIG group

thanks for your response! Ori et al. is writing another paper, so he is
pretty busy - and at least one of his collaborator is a student that is
finishing $his/her$ PhD thesis, but I was told that eventually they will
write something about it on SPS.

> BTW, do you know if the approach by Ganor et al. started independently
> of the cosmological Billiard approach by Damour Henneaux and Nicolai
> and later merged, or what is the historical development?

My guess would be that it was independent, but they must know the answer
much better.

> Yes. That would be remarkable, if true.

Definitely. If true. ;-)

> It is entertainingly reminiscent of this famous saying, apparently due
> to Feynman, that most of physics is about "Laplace $\phi =$ " in some
> generalized sense.

Feynman also advertised the equation U=0 for the theory of everything
where U is Feynman's universal U function. Was the Laplacian comment more
serious?

> Are you going to contact him or shall I?

We have exchanged a couple of mails, but your mail might encourage him to
write something even more than one mail from me. ;-)

> By the way, while talking to Hermann Nicolai about these things
> in Golm I learned about the interesting fact that E10 can be
> regarded as sitting inside the vertex operator algebra of the
> string. Of course that's long and well known (e.g. http://www.arxiv.org/abs/hep-th/9411188)

This might be interesting, but let me ask a simple question. Are $you/they$
saying something beyond the simple claim that the $E_{10}$ current algebra
can be represented by a compactified bosonic CFT?

> have already discussed the immense symmetry algebra that is
> found by deforming the worldsheet CFT by similarity transformations
> of the form
>
> $A -> \exp(-W) A \exp(W)$
>
> for W a general integrated vertex.

By a general integrated vertex, do you mean an integral of a (1,1) primary
field? Or the integral of an arbitrary field? Don't you just get the
algebra of all operators in the CFT? Although I used to think about
exactly these ideas in the past, today I don't quite know how big object
should I imagine when you talk about these intriguing concepts.

Moreover, it reminds me of some of our discussions about Thiemann's stuff.
Do you realize that $\exp(W)$.$\exp(-W)$ is not equal to identity if W is a
general enough operator in a quantum field theory? For example,
$\exp(i.k.X(z))$ and $\exp(-i.k.X(z))$ have huge short-distance singularities in
their OPEs, and so on. I hope you don't want to ignore the subtleties of
QFT and build something along these lines of Thiemann's papers - such an
approach contradicts all properties of quantum field theory that
distninguish it from simple quantum mechanics.

> I am not completely sure what currently the state of the art is with
> respect to our understanding of this algebra, though. I am being told
> that physicists are currently better prepared to discuss E10 than
> mathematicians are.

You are obviously a fan of the idea that the deep underlying principle
behind M-theory is some huge symmetry/group. A favorite idea of many great
people many decades ago, as well as of Thomas Larsson and others today.
Well, let me admit that my belief that the things are that simple has been
reduced significantly (all symmetries in string theory tend to be local
symmetries, and local symmetries describe a redundancy of your description
which is not quite a physical and measurable concept, and can differ
between different descriptions), and moreover the $E_{10}$ group (or
hyperbolic algebra) just does not seem mysterious enough to me today. But
such exceptional groups must play *some* role.

Do you think that the existence of the $E_9$ and $E_{10}$ symmetries of
M-theory on 9-dimensional and 10-dimensional tori is equally well
established as in the case of $E_8$ and lower groups?

All the best
Lubos
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: $+1-617/496-0110$ Web: http://lumo.matfyz.cz/
eFax: $+1-801/454-1858$ work: $+1-617/496-8199$ home: $+1-617/868-4487$ (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^



On Sat, 17 Apr 2004, Lubos Motl wrote: > > Yes. That would be remarkable, if true. > > Definitely. If true. ;-) Heh. Well, so it should be checked! I guess the task is to proceed as in section 7 of http://www.arxiv.org/abs/hep-th/0401053 for higher orders and try to identify the degrees of freedom and their dynamical relations (e.g. as in section 7.4) described by the various orders $\Delta_i$ of the Laplace operator. At least for the pure supergravity degrees of freedom it has been checked in Damour, Henneaux, Nicolai E10 and a "samll tension expansion" of M Theory http://www.arxiv.org/abs/hep-th/0207267 that the E10 $\sigma-model$ reproduces these up to 30th order in some appropriate expansion. So I guess there can be no doubt that the E10 $\sigma-model$ does describe full 11d sugra plus lots of other stuff. Wouldn't it be pretty weird if this other stuff is _not_ that what we would hope it to be? > Feynman also advertised the equation U=0 for the theory of everything > where U is Feynman's universal U function. Was the Laplacian comment more > serious? I don't recall where I read it (must have been in the Feynman lectures, somewhere) but, yes, in the context that I read it it was supposed to be more serious than writing $$= U = (F-ma)^2 + .[/itex].. . Personally I think the more refined version of the statement that I am thinking about is the point of view expressed in Froehlich, Grandjean, Recknagel, Supersymmetric Quantum Theory, non-commutative geometry, and gravitation http://www.arxiv.org/abs/hep-th/9706132 where it is emphasized that any system of susy QM can be viewed as describing a particular (generalized) geometry. From this point of view it wouldn't be surprising if a TOE is solved by "harmonic forms" in some generalized sense (like e.g. harmonic functions on the group of E10 in the case which we are discussing). But I am digressing... > This might be interesting, but let me ask a simple question. Are $you/they$ > saying something beyond the simple claim that the $E_{10}$ current algebra > can be represented by a compactified bosonic CFT? I have barely begun learning about E10, so please bear with me. First of all I am confused why you refer to the "E10 current algebra". From http://www.arxiv.org/abs/hep-th/9411188 I seemed to learn that the whole point is that E10 is not a current algebra, because of its indefinite Cartan matrix. Only a Kac-Moody algebra with (semi)definite Cartan matrix is a current algebra. But what I was referring to was the fact that, according to equation (0.11) of the above mentioned paper E10 sits inside the Lie algebra of physical states of a completly compactified bosonic string. Is that a "simple" claim? If you feel that I am lacking some basic knowledge about E10, please go ahead and educate me! :-) > > have already discussed the immense symmetry algebra that is > > found by deforming the worldsheet CFT by similarity transformations > > of the form > > > > $A -> \exp(-W) A \exp(W)$ > > > > for W a general integrated vertex. > > By a general integrated vertex, do you mean an integral of a (1,1) primary > field? Or the integral of an arbitrary field? Don't you just get the > algebra of all operators in the CFT? Although I used to think about > exactly these ideas in the past, today I don't quite know how big object > should I imagine when you talk about these intriguing concepts. > > Moreover, it reminds me of some of our discussions about Thiemann's stuff. > Do you realize that $\exp(W)$.$\exp(-W)$ is not equal to identity if W is a > general enough operator in a quantum field theory? For example, > $\exp(i.k.X(z))$ and $\exp(-i.k.X(z))$ have huge short-distance singularities in > their OPEs, and so on. I hope you don't want to ignore the subtleties of > QFT and build something along these lines of Thiemann's papers - such an > approach contradicts all properties of quantum field theory that > distninguish it from simple quantum mechanics. Ok, good point. The exponents used in http://www.arxiv.org/abs/hep-th/9401075 are supposed to be normal orderd, but the full exponentials are not. So more explicitly I was referring to $A -> \exp(-:W:) A \exp(:W:) .$$ (cf. eq. (2.5) of http://www.arxiv.org/abs/hep-th/9401075). So this way [itex]\exp(-:W:)$ is indeed the inverse of $\exp(:W:), at$ least if one of them is well defined in the first place. The point of all this is that given any stress-energy tensor T of the worlsheet CFT you get a new worldsheet CFT by setting A=T in the above formula. The new CFT can be interpreted as coming from the old one by a generalized symmetry operation on the background. But, yes the W are supposed to be integrals over weight 1 fields, such that the $\sigma-reparameterization$ generator T- bar T is left unaffected by the operation, as it must be. The algebra of the :W: is huge and should have some close relation to that discussed on pp3 of http://www.arxiv.org/abs/hep-th/9411188. > > I am not completely sure what currently the state of the art is with > > respect to our understanding of this algebra, though. I am being told > > that physicists are currently better prepared to discuss E10 than > > mathematicians are. > > You are obviously a fan of the idea that the deep underlying principle > behind M-theory is some huge symmetry/group. A favorite idea of many great > people many decades ago, as well as of Thomas Larsson and others today. > Well, let me admit that my belief that the things are that simple has been > reduced significantly (all symmetries in string theory tend to be local > symmetries, and local symmetries describe a redundancy of your description > which is not quite a physical and measurable concept, and can differ > between different descriptions), and moreover the $E_{10}$ group (or > hyperbolic algebra) just does not seem mysterious enough to me today. But > such exceptional groups must play *some* role. I fully agree with what you are saying here. > Do you think that the existence of the $E_9$ and $E_{10}$ symmetries of > M-theory on 9-dimensional and 10-dimensional tori is equally well > established as in the case of $E_8$ and lower groups? I don't know. I still know too little about this stuff to make an educated guess. What I find impressive though is that E10 definitely knows all about 11d sugra (at least of the bosonic sector, that is) while still having way more variables. This alone seems to be very strong evidence for the fact that E10 must say something about M theory, I'd say.



On Sat, 17 Apr 2004, Urs Schreiber wrote: > Heh. Well, so it should be checked! Yes, it should be tried. > that the E10 $\sigma-model$ reproduces these up to 30th order in > some appropriate expansion. I would probably believe this particular technical claim even with a small number of orders than 30. ;-) I guess that you only talk about the bosonic sector of SUGRA? > So I guess there can be no doubt that > the E10 $\sigma-model$ does describe full 11d sugra plus lots of > other stuff. Wouldn't it be pretty weird if this other stuff is > _not_ that what we would hope it to be? If you ask me, my answer would be No, it would not be too weird. Constructing the full covariant formulation of M-theory is a big task, and it would seem more weird to me if someone suddenly found it using an obscure and apparently a slightly ill-defined $\sigma-model,$ especially because I don't expect M-theory to be *just* another string theory. :-) Nevertheless, I am gonna look at these things because what you say sounds highly nontrivial. > First of all I am confused why you refer to the "E10 current algebra". > From http://www.arxiv.org/abs/hep-th/9411188 I seemed to learn that the whole point is that > E10 is not a current algebra, because of its indefinite Cartan > matrix. Only a Kac-Moody algebra with (semi)definite Cartan > matrix is a current algebra. OK, so I wanted to say a more general word for the algebra that does not have any such constraints of (semi)definiteness. Does it make a big difference? > But what I was referring to was the fact that, according to > equation (0.11) of the above mentioned paper E10 sits inside the > Lie algebra of physical states of a completly compactified bosonic > string. Is that a "simple" claim? I may misunderstand something, but what you say sounds as a simple claim. If you read the paragraph below the equation (0.11) that you mentioned, you will see that the nontrivial statement is not that g(A) is included in $g_\Lambda,$ but rather the fact that the inclusion is "proper", i.e. that there are some missing states. I thought that just like the bosons compactified on an even self-dual lattice $\Gamma_8$ gives you an $E_8$ current algebra, the compactification on another even self-dual lattice $\Gamma_{9+1}$ gives you the $E_{10}$ hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra (indefinite one, in this case) has vertex operators $\partial{X^i},$ while the roots are represented by $\exp(ik_iX_i(z))$ where the momenta $k_i$ belong to the lattice, and they can be, in the $E_{10}$ case, both time-like as well as spacelike. Is there some huge difference that I missed? > ...So this way $\exp(-:W:)$ is indeed the inverse of $\exp(:W:), at$ least > if one of them is well defined in the first place. No, the colons don't change anything about my claim. Let me repeat my statements with the colons - which incidentally don't change the operators in my example at all: > > Do you realize that $\exp(:W:)$.$\exp(-:W:)$ is not equal to identity if W is a > > general enough operator in a quantum field theory? For example, > > $\exp(:i.k.X(z):)$ and $\exp(-:i.k.X(z):)$ have huge short-distance > > singularities in their OPEs, and so on. It is just not true that in quantum field theory you can write the inverse operator to $\exp(:W:)$ as $\exp(-:W:)$ for a nontrivial operator W. The subtlety is hidden in the OPE of W with itself. The exceptions where the naive formula works are the operators W that are constructed as the integrals of the (1,1) primary fields. The primary fields only have the simple singularities with the stress energy tensor, and they lead to "nonsingular" expressions after integration. Among these operators whose exponentials are invertible, you will find the total momentum, the generators of the Lorentz group (and its individual contributions, if properly defined), and some others. But it is not correct to assume that the exponential of a generic operator can be inverted in this classical way. > The point of all this is that given any stress-energy tensor T of the > worlsheet CFT you get a new worldsheet CFT by setting A=T in the above > formula. The new CFT can be interpreted as coming from the old one by > a generalized symmetry operation on the background. That's probably interesting, but I did not get it$. ;-/$ > The algebra of the :W: is huge and should have some close relation > to that discussed on pp3 of http://www.arxiv.org/abs/hep-th/9411188. Do you say that it is huge in a general case? The (1,1) primary fields are certainly rare. For example, the Ising model CFT contains 3 of them. > I fully agree with what you are saying here. Nice to hear. > I don't know. I still know too little about this stuff to make an > educated guess. What I find impressive though is that E10 definitely > knows all about 11d sugra (at least of the bosonic sector, that is) ... The absence of fermions is another reason why I feel that this $E_{10}$ is unlikely to be far-reaching and even more unlikely to contain the whole M-theory. The inclusion of the fermions would try to lead us to superalgebras such as OSp(1|32) or something similar, which is very different from $E_{10}$. Cheers Lubos __{____________________________________________________________________ ________} E-mail: lumo@matfyz.cz fax: $+1-617/496-0110$ Web: http://lumo.matfyz.cz/ eFax: $+1-801/454-1858$ work: $+1-617/496-8199$ home: $+1-617/868-4487$ (call) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^



Lubos Motl wrote (although Google was unable to retrieve it, so I responded to the next message instead): > You are obviously a fan of the idea that the deep underlying principle > behind M-theory is some huge symmetry/group. A favorite idea of many great > people many decades ago, as well as of Thomas Larsson and others today. > Well, let me admit that my belief that the things are that simple has been > reduced significantly (all symmetries in string theory tend to be local > symmetries, and local symmetries describe a redundancy of your description > which is not quite a physical and measurable concept, and can differ > between different descriptions), and moreover the $E_{10}$ group (or > hyperbolic algebra) just does not seem mysterious enough to me today. But > such exceptional groups must play *some* role. Hmrph. It is hardly some random symmetry such as $E_{10}$ that I suggest. The diffeomorphism group is *the* correct symmetry of GR (ask Rovelli!). [Moderator's note: I hope that Rovelli is not necessary to answer basic questions about classical general relativity. LM] As Urs Schreiber has pointed out, LQG is not really canonical quantization, which generically would give rise to lowest-energy reps of the symmetry group - this is true even for gauge symmetries such as worldsheet conformal symmetry. Thus, combining the symmetry principles underlying GR and $QM *and$ nothing $else* -$ experimentally confirmed physics - leads to LW reps of the diff group, and non-trivial such reps are anomalous. That's just the way it is. [Moderator's note: Sorry, I did not realize that you started to promote another group as the theory of everything, beyond your original mb(3|8) from http://arxiv.org/abs/http://www.arxi...hep-th/0208185 . I guess that $E_{10}$ is closer to the truth. LM] It seems that the basic idea in West's approach is that M-theory and 11D SUGRA should for some reason be described by non-linear realization of the Kac-Moody algebra E11. At least, this is what West writes in http://www.arxiv.org/abs/http://www....ep-th/0104081, which seems to be the seminal paper. A striking observation is that NO SUCH REALIZATION EXISTS! (if the underlying space is finite-dimensional or mildly infinite- dimensional). This follows immedately from the well-known fact that non-affine Kac-Moody algebras like E11 are of expontential growth. For the last 20 years at least, Victor Kac has emphasized that the lack of a natural realization, linear or nonlinear, is a major problem for non-affine Kac-Moody algebras; without a realization, these are somewhat useless. Kac states this very explicitly in http://www.arxiv.org/abs/q-alg/9709008: "It is a well kept secret that the theory of Kac-Moody algebras has been a disaster." [Moderator's note: this disaster has not influenced physics too much, and Kac-Moody algebras are very important and well functioning tool to study conformal field theories and perhaps other realizations. The users of physicsforums.com should probably click at "View the post in the original ASCII form" because otherwise they might have problems to see the equations and pictures in this post correctly.] Thus, if West and others would find a nice realization of $E_{11} it$ would be a major mathematical breakthrough, irrespective of its relevance to M-theory (and the relevance of M-theory to physics). However, I have glanced at quite a few of these $E_{11}$ papers, and I have never seen any mentioning of exponential growth. This suggests that string theorists do not realize that this is a major difficultly, which has stalled Victor and other mathematicians for decades. Nevertheless, West's original paper contains some beautiful math. First recall that nonlinear realizations are the same thing as gradings. E.g., the conformal realization of $so(n+1,1)$ is a 3-grading: $$so(n+1,1) = g_-1 + g_0 + g_1,$$ where $g_-1 =$ translations, $g_0 =$ rotations and dilatations, and $g_1 =$ conformal boosts. The grading is specified by defining the grading operator, which here is the dilatation - the degree is the dilatation eiginvalue. I had a nice discussion with John Baez and Tony Smith about 16 months ago. The spr thread "Structures preserved by e8" should be available from some archive. Wests starts from a 7-grading of E8, eqs $(4.1-11)E8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3$. If you formulate E8 in Cartan-Weil basis, $H^i, E^\alpha, a$ grading operator is of the form $Z = \alpha_i H^i$. There is thus $a 1-1$ correspondence between gradings and roots $\alpha.$ The relevant root here is * | $o-o-o-o-o-o-o$ which gives $$E8 = V* + A^6 V + A^3 V + (sl(8)+C) + A^3 V* + A^6 V* + V[/itex]. 248 $= 8 +$ 28 + 56 + 64 + 56 + 28 + 8 Here V is the $8-dim$ vector rep of sl(8) and V* its dual and $A^n V$ is the n:th anti-symmetric power. Note that $g_0 = sl(8)+C$ and you get the Dynkin diagram of sl(8) by removing the starred root - this is a general phenomenon. This grading corresponds to a realization on the dual of $g_- = g_-3 + g_-2 + g_-1,$ according to the following table: $g_-1 x_{abc} R^{abc} = d/dx_abc + .$.. $g_-2 y_{abcdef} R^{abcdef} = d/dy_abcdef + .$.... $g_-3 z^a S_a = d/dz^a$$ Aside: other E8 gradings have also occurred in the literature. The one corresponding to o | [itex]o-o-o-o-o-o-*(g_0 = E7)$ was discussed in a 2000 CMP paper by Gunaydin-Koepsell-Nicolai and o | $*-o-o-o-o-o-o(g_0 = so(14))$ is implicit in GSW, $(6.A.1-3).$ However, E8 cannot be right for M-theory, because the vector rep of $g_0= sl(8)$ is 8-dimensional and describes translations in 8D rather than 11D. Therefore, West seems to propose that the E11 grading corresponding to * | $o-o-o-o-o-o-o-o-o-o$ should be relevant to M-theory. Alas, this grading extends to infinity in both directions, E11 = ..$. + g_-4 + g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3 + .$.. . Moreover, dim $g_k ~ \exp(ck)$ for some constant c, i.e. E11 is of exponential growth. This means that E11 is HUGE, much bigger than any local symmetry. Local symmetries, e.g. diff and Yang-Mills in n dimensions, grow polynomially, dim $g_k ~ k^(n-1), i$.e. there are finitely many symmetry generators per spacetime point.



On Sun, 18 Apr 2004, Thomas Larsson wrote: > As Urs Schreiber has pointed out, LQG is not really canonical > quantization, which generically would give rise to lowest-energy reps of > the symmetry group - this is true even for gauge symmetries such as > worldsheet conformal symmetry. Thus, combining the symmetry principles > underlying GR and $QM *and$ nothing $else* -$ experimentally confirmed physics > - leads to LW reps of the diff group, and non-trivial such reps are > anomalous. That's just the way it is. BTW, do we now agree that this fact is not in contradiction to any known fact in string theory? You seemed to claim that the fact that gravitational anomalies for chiral fermions occur in $4k + 2$ dimensions is in contradiction with the above somehow. But clearly it is not, since what you write above has nothing to do with chiral fermions. > For the last 20 years at least, Victor Kac has emphasized that the lack > of a natural realization, linear or nonlinear, is a major problem for > non-affine Kac-Moody algebras; without a realization, these are somewhat > useless. Kac states this very explicitly in > http://www.arxiv.org/abs/q-alg/9709008: > > "It is a well kept secret that the theory of Kac-Moody algebras has > been a disaster." > > [Moderator's note: this disaster has not influenced physics too much, > and Kac-Moody algebras are very important and well functioning tool > to study conformal field theories and perhaps other realizations. Lubos, please note that Larsson (and Kac for that matter) referred to _non-affine_ Kac-Moody algebras, while it seems to me what you have in mind are current algebras, i.e. _affine_ KM algebras. KM algebras fall into 3 classes: 1) Cartan matrix positive definite --> finite algebra 2) Cartan matrix positive semi-definite --> affine algebra = current algebra in 2d 3) Cartan matrix indefinite --> exponential growth, desaster and all that (paraphrased from http://www.arxiv.org/abs/hep-th/9411188) > Thus, if West and others would find a nice realization of $E_{11} it$ would > be a major mathematical breakthrough, irrespective of its relevance to > M-theory (and the relevance of M-theory to physics). However, I have > glanced at quite a few of these $E_{11}$ papers, and I have never seen any > mentioning of exponential growth. This suggests that string theorists > do not realize that this is a major difficultly, which has stalled > Victor and other mathematicians for decades. This is at least not true for all physicist. In Gebert, Nicolai, $E_{10}$ for beginners, http://www.arxiv.org/abs/hep-th/9411188 it is emphasized quite strongly that there is exponential growth and that things are pretty problematic concerning $E_{10}$. But I am being told that the point is that one can study consistent finite truncations which can be handled and in terms of which we can do physics.



On Sat, 17 Apr 2004, Lubos Motl wrote: > > that the E10 $\sigma-model$ reproduces these up to 30th order in > > some appropriate expansion. > > I would probably believe this particular technical claim even with a small > number of orders than 30. ;-) I guess that you only talk about the bosonic > sector of SUGRA? Yes. But I was told that using some appropriate susy extension (I am not sure which one, though) you can similarly check that also the fermionic degrees of freedom of sugra are there. Aparently this is not published yet, though. > If you ask me, my answer would be No, it would not be too weird. > Constructing the full covariant formulation of M-theory is a big task, and > it would seem more weird to me if someone suddenly found it using an > obscure and apparently a slightly ill-defined $\sigma-model,$ especially > because I don't expect M-theory to be *just* another string theory. :-) Hm, the $\sigma$ model wouldn't be "another string theory", somehow, I think. It would rather be a vast extension of what in quantum cosmology is sometimes called "miDi superspace", i.e. the configuration space of all modes of all objects in the theory. I must say that I find the claim that such $a 1+0$ dim $\sigma$ model of all of M-theory exists (BTW, this is not $a 1+1d \sigma$ model describing string motion in some background!) plausible, because it is quite similar to the M=Matrix proposal, where the claim is, too, that all the degrees of freedom can be parameterized by $a 1+0$ dimensional quantum mechanics. Indeed, I have a little private speculation how one could maybe make a possible relation between the two $1+0d$ models which we know to reduce to 11d sugra in some limit, namely the $E_{10}$ model and BFSS. I have talked about that at the Coffee Table: http://golem.ph.utexas.edu/string/archives/000342.html . The rough idea is the following: We know that close to a spacelike singularity (super)gravity decouples in the sense that nearby points on spatial hyperslices no longer interact because they are outside each other's backward light cones and that in this limit the dynamics of each of the thus disconnected small causal patches of space are described in config space by the billiard with walls given by the Weyl chamber of $E_{10}$ (in the case of 11d sugra). This billiard motion is chaotic, a generalization of the old BKL "mixmaster" idea. Now the "first law" of quantum chaos is that the universal behaviour of any classically chaotic system (i.e. that behaviour which remains when you slightly coarse-grain the dynamics so as to get rid of system-dependent details of the dynamics on small scales) is described by Random Matrix Theory, i.e. by a theory where you pick a random ensemble of Hamiltonians and average over their properties (see the above link for more on this). In particular, the ensemble is in general Gaussian in the sense that the probability to find a particular u(N) matrix M in the ensemble is proportional to $$\exp(- tr(M^2)) .$$ Taking this well known fact in the context of the above cosmological Billiard tells us that 11d sugra in the limit where inteaction are negligible is well approximated by a canonical matrix ensemble $\exp(- tr(M^2))$. But isn't that suggestive? Take the BFSS canonical ensemble in the limit where the interaction terms are negected. It, too, looks like $\exp(- tr P^2)$. > Nevertheless, I am gonna look at these things because what you say sounds > highly nontrivial. Great. I hope you'll let us know what you come up with! :-) > > First of all I am confused why you refer to the "E10 current algebra". > > From http://www.arxiv.org/abs/hep-th/9411188 I seemed to learn that the whole point is that > > E10 is not a current algebra, because of its indefinite Cartan > > matrix. Only a Kac-Moody algebra with (semi)definite Cartan > > matrix is a current algebra. > > OK, so I wanted to say a more general word for the algebra that does not > have any such constraints of (semi)definiteness. Does it make a big > difference? Yes, apparently. I am no expert yet on this stuff, but according to the introductory article http://www.arxiv.org/abs/hep-th/9411188 it makes a huge difference. Non-affine KM algebras with indefinite Cartan matrices are apparently mind-bogglingly huge beasts which greatly exceed the complexity of the affine KM algebras. Hermann Nicolai has emphasized that to me over and over again. In order to emphasize this point he had shown me files of piles of pages of computer output where order-by-order the affine KM algebras contained in $E_{10},$ as well as their irreps and the multiplicity with which these appear, were listed. He said that mathematicians don't understand this huge algebra well yet, but that string theorists are slowly beginning to make progress in these matters. > I thought that just like the bosons compactified on an even self-dual > lattice $\Gamma_8$ gives you an $E_8$ current algebra, the compactification on > another even self-dual lattice $\Gamma_{9+1}$ gives you the $E_{10}$ > hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra > (indefinite one, in this case) has vertex operators $\partial{X^i},$ while > the roots are represented by $\exp(ik_iX_i(z))$ where the momenta $k_i$ belong > to the lattice, and they can be, in the $E_{10}$ case, both time-like as > well as spacelike. Is there some huge difference that I missed? Hm, no, probably not. It is just important that $E_{10}$ is not a current algebra, as far as I understand. > > ...So this way $\exp(-:W:)$ is indeed the inverse of $\exp(:W:), at$ least > > if one of them is well defined in the first place. > > No, the colons don't change anything about my claim. Let me repeat my The point of the colons was to distinguish $$:\exp(W):$$ from $$\exp(:W:) .$$ Due to OPEs we have $$:\exp(W): :\exp(-W): =[/itex] possibly lots of correction terms . But on the other hand $\exp(:W:)\exp(-:W:) = \exp(:W:-:W:) = 1 .$$ Ah, now I see what your are perhaps thinking: It is important that here W is the same object everywhere, i.e. what I wrote is _not_ supposed to be a shorthand for $$\exp(:W:(z)) \exp(-:W:(w))$$ with [itex]z \neq w .$ In fact, W is not supposed to depend on any coordinate anymore, because it is an integrated object. > > The point of all this is that given any stress-energy tensor T of the > > worlsheet CFT you get a new worldsheet CFT by setting A=T in the above > > formula. The new CFT can be interpreted as coming from the old one by > > a generalized symmetry operation on the background. > > That's probably interesting, but I did not get it$. ;-/$ There are really to statements here: 1) Take the Virasor algebra generated by $L_m$ satisfying $[L_m,L_n] = (m-n)L_{m+n} +$ anomaly . Now define $$L'_m := \exp(-W) L_m \exp(W)$$ with the exponentials defined as discussed above so that indeed $\exp(W)\exp(-W) = 1 .$ Then obviously we have defined an algebra isomorphism and the new generators still satisfy the Virasoro algebra: $$[L'_m, L'_n] = (m-n)L_{m+n} +[/itex] anomaly . 2) The second statement is that one can indeed interpret the new $L'_m$ as the generators corresponding to a different background of the string theory. Since the states annihilated by the $L'_m$ are just $\exp(-W)$ times the states annihilated by the original $L_m$ and the spectrum is the same and everything, the new background must be related to the original one by a symmetry of the background theory. > > The algebra of the :W: is huge and should have some close relation > > to that discussed on pp3 of http://www.arxiv.org/abs/hep-th/9411188. > > Do you say that it is huge in a general case? The (1,1) primary fields are > certainly rare. For example, the Ising model CFT contains 3 of them. I say that it is huge in genral. It is the full algebra of DDF invariants, i.e. of polynomials in the $A_n^\mu$ where $A_n^\mu \propto \int dz \partial X^\mu \exp(in k \cdot X)$$ (for the bosonic string). > The absence of fermions is another reason why I feel that this [itex]E_{10}$ is > unlikely to be far-reaching and even more unlikely to contain the whole > M-theory. The inclusion of the fermions would try to lead us to > superalgebras such as OSp(1|32) or something similar, which is very > different from $E_{10}$. As I said above, there is apparently a susy extension of $E_{10}$ which does correctly reproduce the fermionic degrees of freedom of 11d sugra (among other things). As far as I understood the corresponding publications are in preparation. Another (mayb equivalent) straighforward inclusion of fermions is the promotion of the $1+0d E_{10}$ model to the respective deRahm model, as I have mentioned before.



On Sun, 18 Apr 2004, Urs Schreiber wrote: > I say that it is huge in genral. It is the full algebra of DDF invariants, > i.e. of polynomials in the $A_n^\mu$ where > > $A_n^\mu \propto \int dz \partial X^\mu \exp(in k \cdot X)$ > > (for the bosonic string). Let me make that more precise: Pick any physical state by applying lots of the $A_n^\mu$ to the tachyon state. By the state/operator correspondence the resulting state comes from an integrated weight 1 vertex (the $V(\psi,z)$ in equation (0.10) of http://www.arxiv.org/abs/hep-th/9411188).



On Sun, 18 Apr 2004, Urs Schreiber wrote: > Yes. But I was told that using some appropriate susy extension (I am not > sure which one, though) you can similarly check that also the fermionic > degrees of freedom of sugra are there. Aparently this is not published > yet, though. I would like to see it first, to avoid building on someone's wishful thinking. Be sure that I will be among the first believers if such a theory really works, including the fermions. ;-) > I must say that I find the claim that such $a 1+0$ dim $\sigma$ model of all > of M-theory exists (BTW, this is not $a 1+1d \sigma$ model describing string > motion in some background!) plausible, because it is quite similar to > the M=Matrix proposal, where the claim is, too, that all the degrees of > freedom can be parameterized by $a 1+0$ dimensional quantum mechanics. Sure, why not. :-) But there is usually a long path from these big dreams to actual results. > ...dynamics so as to get rid of system-dependent details of the > dynamics on small scales) is described by Random Matrix Theory, Do you want to reproduce M-theory amplitudes approximately, or exactly? Is this connection with Random Matrix Theory a trick to get the right results? Or is it evidence that this description cannot work exactly? > But isn't that suggestive? Take the BFSS canonical ensemble > in the limit where the interaction terms are negected. It, > too, looks like $\exp(- tr P^2)$. There are also many systems that I don't like where you can obtain $\exp(-\tr P^2),$ too. :-) This exponential is not such a difficult concept for us to be shocked if we see it somewhere. > Yes, apparently. I am no expert yet on this stuff, but according > to the introductory article http://www.arxiv.org/abs/hep-th/9411188 it makes a huge > difference. Non-affine KM algebras with indefinite Cartan matrices > are apparently mind-bogglingly huge beasts which greatly... Well, I understand why $E_9$ is the affine $E_8,$ and why the hyperbolic algebra $E_{10}$ is even "bigger". There are many generators for a single root, and infinitely many roots organized in a Minkowski-like Cartan space, and so forth. Using the words "mind-boggling" for such things does not look quite appropriate to me. These objects are simply large, which is both a good as well as a bad message. I don't see anything amazing about an object only because it is "large"; and it makes such an object less ready for various operations. For example, we don't want to build Yang-Mills theories based on infinite-dimensional non-compact groups, at least so far. :-) What I still think is straightforward is that one can obtain such algebras, with the correct commutation relations, from a compactified bosonic CFT, and the maths justifying this fact is as trivial as in the $E_8$ case. Am I wrong? This is about the very definition of $E_{10}$. The compactified bosonic CFT simply reproduces (and allows you to check) the defining axioms of $E_{10}$ step by step. If there is something nontrivial about these things, you will have to be more specific to explain it. > The point of the colons was to distinguish > $:\exp(W):$ > from > $\exp(:W:) .$ OK, understood. But then one must note that the non-ordered $\exp(:W:)$ contains a lot of hard-to-interpret singular terms which make such an operator useless and, in fact, also ill-defined if W is a local operator. > Ah, now I see what your are perhaps thinking: It is important > that here W is the same object everywhere, i.e. what I wrote is > _not_ supposed to be a shorthand for > > $\exp(:W:(z)) \exp(-:W:(w))$ > > with $z \neq w .$ That's exactly one of the problems. If W(z) is a local operator, there is no working easy way to define $W(z)^2$. The only way to do so is to consider W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of singularities that you must deal with, subtract, normal-order, and so on, and these operations will destroy the naive identities that you are thinking about. > In fact, W is not supposed to depend on any coordinate > anymore, because it is an integrated object. I just don't see anything sharp about these objects. Of course, you can define the $U(\infty)$ group acting on the whole Hilbert space - which is the only thing I can imagine where you are led if you try to define an algebra that contains more than just a "couple" of operators. But every quantum mechanical theory has this $U(\infty) "$symmetry" group and we use it extensively all the time e.g. when we switch from one basis to another (it is only a symmetry of the norm, not a symmetry of dynamics). I don't precisely understand whether you want to define something more useful, constrained, and original than this universal and well-known $U(\infty),$ and if you want, what it is. Depending on your constraints what the "W" operators can be, you can obtain a couple of algebras, but because you have not described any clear constraints so far, it seems that you talk about the $U(\infty)$ algebra. > There are really to statements here: > > 1) Take the Virasor algebra generated by $L_m$ satisfying > $[L_m,L_n] = (m-n)L_{m+n} +$ anomaly . > Now define > > $L'_m := \exp(-W) L_m \exp(W)$ > > with the exponentials defined as discussed above so that indeed > $\exp(W)\exp(-W) = 1 .$ Then obviously we have defined an algebra > isomorphism and the new generators still satisfy the Virasoro > algebra: > > $[L'_m, L'_n] = (m-n)L_{m+n} +$ anomaly . Well, that's a basic lesson of undergraduate linear algebra that conjugation does not change the commutation relations. I guess that both of us realize that every physicist must know these things, and that we are using them all the time - and it is unlikely to make a revolutionary discovery just by noting that conjugation preserves commutators. > 2) The second statement is that one can indeed interpret the > new $L'_m$ as the generators corresponding to a different background > of the string theory. Since the states annihilated by the $L'_m$ > are just $\exp(-W)$ times the states annihilated by the original > $L_m$ and the spectrum is the same and everything, the new > background must be related to the original one by a symmetry > of the background theory. I find this statement physically vacuous. You have not constructed any new theory at all; you just renamed the basis vectors of its Hilbert space in a chaotic and arbitrary fashion. If you want to get a spacetime interpretation of the "transformed" theory, you will have to undo this unphysical $\exp(W)$ transformation, and return to the original operators such as $X^\mu(z)$ which encode the position of the string in spacetime - instead of $\exp(-W) X^\mu(z) \exp(W) -$ and you will finally obtain *identical* physical results. You have not even changed any moduli in the theory. It seems that what you did is nothing more than the $U(\infty)$ symmetry discussed above, and this whole "large" symmetry is not physically useful unless you restrict it to something that preserves some other structure of the theory beyond the norm on the Hilbert space. > I say that it is huge in genral. It is the full algebra of DDF invariants, OK, once again, it seems that you are talking about the full $U(\infty)$ acting on the Hilbert space which I find useless for physics. It is an algebra that does not commute with the spacetime Hamiltonian, for example. It does not even have any simple commutation relations with these spacetime Poincare generators. I think that what you are talking about is the algebra of everything - and everything is more or less isomorphic to nothing. It is the sort of group that allows you to "prove" that a harmonic oscillator is "equivalent" to the Standard Model: both of them have an infinite-dimensional Hilbert space, and they can be mapped to one another, and every operator in one theory can be identified with an operator in the other theory. But I say that such a conclusion is unphysical because the structures that physics requires are more than just the norm. You need some sort of dynamics and interpretable operators - for example the Hamiltonian - and the only interesting transformations are those under which the Hamiltonian (...) is invariant, or at least transforms in a controllable way. > As I said above, there is apparently a susy extension of $E_{10}$ which does > correctly reproduce the fermionic degrees of freedom of 11d sugra > (among other things). As far as I understood the corresponding > publications are in preparation. I thought that there is a theorem that no useful superalgebra extending $E_{10} ( or at$ least $E_8 )$ exists. > Another (mayb equivalent) straighforward inclusion of fermions > is the promotion of the $1+0d E_{10}$ model to the respective > deRahm model, as I have mentioned before. OK, but it is a random guessing, is not it? I sort of feel that you want to say that if you construct anything huge enough, it must be M-theory. Well, this is an assumption that I certainly don't share. But it is probably still more rational than the assumption of Lee Smolin that anything *simple* enough - e.g. the 3D Chern-Simons theory - must be equivalent to M-theory. ;-) Cheers, Lubos __{____________________________________________________________________ ________} E-mail: lumo@matfyz.cz fax: $+1-617/496-0110$ Web: http://lumo.matfyz.cz/ eFax: $+1-801/454-1858$ work: $+1-617/496-8199$ home: $+1-617/868-4487$ (call) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^



On Sun, 18 Apr 2004, Urs Schreiber wrote: > 1) Take the Virasor algebra generated by $L_m$ satisfying > $[L_m,L_n] = (m-n)L_{m+n} +$ anomaly . > Now define > > $L'_m := \exp(-W) L_m \exp(W)$ > > with the exponentials defined as discussed above so that indeed > $\exp(W)\exp(-W) = 1 .$ Then obviously we have defined an algebra > isomorphism and the new generators still satisfy the Virasoro > algebra: > > $[L'_m, L'_n] = (m-n)L_{m+n} +$ anomaly . In my discussions with Ioannis Giannakis I have come up with an alternative description in terms of string field theory, which might be instructive: Consider the equation of motion of open bosonic cubic string field theory: $$Q \phi + \phi \star \phi =$$ (setting the constant factor to unity). Here Q is the BRST operator for a string and $\star$ is the star product which merges two string states$. \phi$ is the string field. Now suppose we want to study single strings in non-trivial backgrounds. We could do that by trying to find a consistent worldsheet CFT. But the above equation should give us an alternative prescription: Pick any solution $\phi$ to the above string field equations of motion. Now perturb this solution by adding an infinitesimal "test field" $\psi,$ namely the field of a single "test string" probing the background described by $\phi:\phi \to \phi + \psi .$ By inserting this ansatz into the above equation of motion, using the fact that the original $\phi$ is a solution all by itself and that $\psi$ is infinitesimal, we get the equation $$Q \psi + \phi \star \psi + \psi \star \phi =[/itex] . If I introduce the notation $\{\phi, \cdot\} = \psi \star \cdot + \cdot \star \phi$$ this becomes $$(Q \psi + \{\phi, \cdpt\}) \psi =$$ which should be the quantum consraint for the single string [itex]\psi$ in the background described by the string field $\phi .$ Is this interpretation consistent? If it is, it should be possible ot interpret $$Q^\prime := Q + \{\phi , \cdot \}$$ as the BRST operator of the string in the background $\phi$. Can that be? Is $Q^\prime$ nilpotent? Hm, maybe I am confused. Comments are welcome. Anyway, what I wanted to say is that if the excitation $\phi$ is a symmetry of the theory in the sense that turning on $\phi$ does not really change the physics, then we would expect that $$Q^\prime = \exp(-W) Q \exp(W) .$$ This are symmetries of the type studied in Ioannis Giannkis, String in nontrivial gravitino and RR backgrounds http://www.arxiv.org/abs/hep-th/0205219



On Sun, 18 Apr 2004, Lubos Motl wrote: > Sure, why not. :-) But there is usually a long path from these big dreams > to actual results. Sure. > > ...dynamics so as to get rid of system-dependent details of the > > dynamics on small scales) is described by Random Matrix Theory, > > Do you want to reproduce M-theory amplitudes approximately, or exactly? > Is this connection with Random Matrix Theory a trick to get the right > results? Or is it evidence that this description cannot work exactly? This was supposed to be evidence that at least in the discussed limit the $E_{10}$ model is indeed equivalent to the BFSS model. Of course the limit is pretty radical and you may argue that it doesn't allow any conclusion about the equivalence of the two models away from this limit. > OK, understood. But then one must note that the non-ordered $\exp(:W:)$ > contains a lot of hard-to-interpret singular terms which make such an > operator useless and, in fact, also ill-defined if W is a local operator. > > > Ah, now I see what your are perhaps thinking: It is important > > that here W is the same object everywhere, i.e. what I wrote is > > _not_ supposed to be a shorthand for > > > > $\exp(:W:(z)) \exp(-:W:(w))$ > > > > with $z \neq w .$ > > That's exactly one of the problems. If W(z) is a local operator, there is > no working easy way to define $W(z)^2$. The only way to do so is to consider > W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of > singularities that you must deal with, subtract, normal-order, and so on, > and these operations will destroy the naive identities that you are > thinking about. We are still not talking about the same constructions, it seems to me. I am not considering things like $W(z)^2$. As I said, the W are integrated objects $W = \int dz$ something(z). For instance the W which induces T-duality is of the form $$W \propto \int d\sigma (\exp(i \sqrt{2}k\cdot X) - h.c.) .$$ Compare equation (3.18) of http://www.arxiv.org/abs/hep-th/9511061 . > I just don't see anything sharp about these objects. Of course, you can > define the $U(\infty)$ group acting on the whole Hilbert space - which is > the only thing I can imagine where you are led if you try to define an > algebra that contains more than just a "couple" of operators. That's too large and indeed without physical content. But as I said before, the restriction on the allowed W is that the similarity transformation preserves $T - \bar T$. Only under this condition can the resulting new algebra be interpreted as coming from a string in a certain background. > > $[L'_m, L'_n] = (m-n)L_{m+n} +$ anomaly . > > Well, that's a basic lesson of undergraduate linear algebra that Yes. :-) > conjugation does not change the commutation relations. I guess that both > of us realize that every physicist must know these things, and that we are > using them all the time - and it is unlikely to make a revolutionary > discovery just by noting that conjugation preserves commutators. It is interesting to examine how different admissable W give rise to certain symmetries/dualities of the string backgrounds. You can use this also to find deformations which are not symmetries but give new backgrounds, inequivalent to the one one started from. > I find this statement physically vacuous. You have not constructed any new > theory at all; you just renamed the basis vectors of its Hilbert space in True, but that's precisely what a symmetry/duality is supposed to do. As you know a string state with winding number n and momentum number m is sent to a state with winding m and momentum n and this is a symmetry. So T-duality, too, could be described as just a renaming of vectors in a Hilbert space, which it is. But of course that does not mean that it is not interesting. > I thought that there is a theorem that no useful superalgebra extending > $E_{10} ( or at$ least $E_8 )$ exists. Aha, don't know about that. Do you have a reference? > > is the promotion of the $1+0d E_{10}$ model to the respective > > deRahm model, as I have mentioned before. > > OK, but it is a random guessing, is not it? Not fully random, but, yes, it is guessing. :-)



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.) > $U(\infty)$ is too large and indeed without physical content. But as I said > before, the restriction on the allowed W is that the similarity > transformation preserves $T - \bar T$. $T-\bar T ?$ Do you mean the different between $T_{zz}$ and $T_{\bar z \bar z} ?$ That's very strange. You cannot just add different components of the same tensor. More precisely, this difference is not Virasoro invariant. The exact conformal symmetry is essential for string theory to work, at least at the loop level, and the condition $ {T-\bar T}$ is invariant under the elements of your group'' is not Virasoro invariant. So I guess that you now understand that I consider the group preserving ${T-\bar T}$ to be as (un)natural as the group that preserves $L_{2004}+2005 \tildeL_{2006}$ or anything else. What's exactly your reason to think that there is anything interesting about a group defined in this way, as opposed to any other random combination of words and symbols that respect the syntactic rules of our mathematical formalism? > > I find this statement physically vacuous. You have not constructed any new > > theory at all; you just renamed the basis vectors of its Hilbert space in > > True, but that's precisely what a symmetry/duality is supposed to do. No, it's not. A duality (and similarly for a self-duality, which is essentially the same thing as a symmetry) is a map redefining the variables of a theory, a redefinition that transforms the known observables in a first known theory AB exactly into some known observables of another known theory CD (which may be the same thing as AB). For example, if we work with the same theory AB and all the relevant operators such as the Hamiltonian are *invariant* under the map, then we talk about the symmetry of the theory AB. On the other hand, your conjugation by a random operator $\exp(W)$ transforms the quantities of a theory $AB -$ for example, the Virasoro worldsheet generators or the spacetime Lorentz generators - into some other, different randomly modified operators in a theory EF whose form has never been written before. The best way to solve the theory EF is to conjugate them back, and realize that EF is just an awkward way to write the theory AB; they are equivalent, and we don't earn anything by talking about a "new" theory EF. Such a conjugation by $\exp(W)$ is not a symmetry of AB, because the Hamiltonian and other operators are not invariant under it, and it is also not a duality because it does not relate two useful theories in a nontrivial way $- it$ relates one theory to the same theory written in awkward variables. A duality or a symmetry are very priviliged words, and a random redefinition of variables or a conjugation by a random operator certainly don't deserve to be called a "duality" or a "symmetry". In CFT and perturbative string theory, you should only use the word "symmetry" or "duality" for the transformations that preserve the form of all Virasoro generators, and be sure that the known spacetime symmetries and T-duality are the only solutions. > As you know a string state with winding number n and momentum number m > is sent to a state with winding m and momentum n and this is > a symmetry. So T-duality, too, could be described as just a > renaming of vectors in a Hilbert space, which it is. But of > course that does not mean that it is not interesting. But there is a huge difference between T-duality, which is a textbook example of a duality, on one side, and your generic conjugation on the other side. T-duality is a map, namely $X_L\to X_L, X_R\to -X_R,$ that preserves all Virasoro generators, for example, and at the self-dual radius, it even preserves the periodicities of all fields, and therefore is a nontrivial symmetry from the Hilbert space to the same Hilbert space (the same superselection sector). T-duality is a symmetry of string theory mapping two backgrounds that were thought to be independent (such as the Universe with radii R and 1/R of a circle) into one another. At the self-dual radius, it is even a symmetry of the background - in fact, it is an element of the $SU(2)^2$ enhanced gauge symmetry. On the other hand, your generic conjugation is a meaningless redefinition of fields that preserves neither the Virasoro algebra nor the spacetime or other operators. It is an uninteresting change of the conventions describing the same theory, not a map between two different theories or a symmetry of a single theory. You just can't use the word "symmetry" or "duality" for such contentless constructions. > > I thought that there is a theorem that no useful superalgebra extending > > $E_{10} ( or at$ least $E_8 )$ exists. > > Aha, don't know about that. Do you have a reference? Unfortunately I don't. I was asking the people who were telling me about this theorem, but no one has given me any reference either. > > > is the promotion of the $1+0d E_{10}$ model to the respective > > > deRahm model, as I have mentioned before. > > > > OK, but it is a random guessing, is not it? > > Not fully random, but, yes, it is guessing. :-) OK, so if it is not fully random :-), do you have some explanation why you said deRham model and not something else, for example the topological B-model on the $E_{10}$ group manifold? ;-) __{____________________________________________________________________ ________} E-mail: lumo@matfyz.cz fax: $+1-617/496-0110$ Web: http://lumo.matfyz.cz/ eFax: $+1-801/454-1858$ work: $+1-617/496-8199$ home: $+1-617/868-4487$ (call) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^



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:38 On Sun, 18 Apr 2004, Lubos Motl wrote: > $T-\bar T ?$ Do you mean the different between $T_{zz}$ and $T_{\bar z \bar z} ?$ > That's very strange. You cannot just add different components of the > same tensor. $$T - \bar T (in[/itex] modes $L_m - \bar L_{-m})$ is always the generator of spatial reparameterizations of the string at appropriately fixed worldsheet time. That it must be preserved under a change of background can be seen by either making an ADM-like analysis of the worldsheet gravity theory or else by considering deformations $T \to T + \delta T\bar T \to \bar T + \delta \bar T$$ and noting that consistency requires $$\delta T = \delta \bar T .$$ > [itex]L_{2006}$ or anything else. What's exactly your reason to think that there > is anything interesting about a group defined in this way, as opposed to It is precisely the group which gives you isomorphisms of the left+right Virasoro algebra. It is possible to identify the group elements which, for instance, correspond to background T-duality, background diffeo transformations, background gauge transformations. (see below) > > True, but that's precisely what a symmetry/duality is supposed to do. > > No, it's not. A duality (and similarly for a self-duality, which is > essentially the same thing as a symmetry) is a map redefining the > variables of a theory, a redefinition that transforms the known > observables in a first known theory AB exactly into some known observables > of another known theory CD (which may be the same thing as AB). Sure. And the conjugation which I mentioned in my last post does exactly that for T-duality. > On the other hand, your generic conjugation is a meaningless redefinition > of fields that preserves neither the Virasoro algebra I don't understand why you are now saying that it does not preserve the Virasoro algebra. In your last post you said that it is known to every undergraduate that conjugation preserves the algebra. > other operators. It is an uninteresting change of the conventions > describing the same theory, not a map between two different theories or a > symmetry of a single theory. You just can't use the word "symmetry" or > "duality" for such contentless constructions. Lubos, please have a look at F. Lizzi, R. Szabo, Duality symmetries and noncommutative geometry of string spacetimes, http://www.arxiv.org/abs/hep-th/9707202 where it is shown in great detail that and how the duality symmetries of the string are related to the conjugations in question. > > > OK, but it is a random guessing, is not it? > > > > Not fully random, but, yes, it is guessing. :-) > > OK, so if it is not fully random :-), do you have some explanation why you > said deRham model and not something else, for example the topological > B-model on the $E_{10}$ group manifold? ;-) Yes, I have a reason for this guess: I once checked that the supersymmetry generators of $N=1 D=3+1$ SUGRA (or rather appropriate linear combinations of them) are the exterior derivative d and its adjoint del on the configuration space of the theory (really the Dolbeault operators $\partial$ and $\bar \partial$ so that $d = \partial + \bar \partial)$. The susy algebra of the constraints of the theory are hence equivalent to the algebra which is schematically of the form {d,del} $= H + .$.. where H is the Hamiltonian constraint of the theory (and in particular something like the Laplace-Beltrami operator on config space) and the ellipsis indicates sa sum of patial diffeomorphism and Lorentz rotation constraints. It is to be expected that a similar statement applies to the constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have shown that the bosonic part of configuration space (which, to emphasize it again, is the space on which the above operators d and del act) is (a subspace of) the group manifold of $E10/K(E10)$ and that the (bosonic part of) the Hamiltonian constraint H is the Laplace operator on this manifold. Hence by the above it is a reasonable guess that the supersymmetric extension of this Hamiltonian constraint is the Laplace-Beltrami operator {d,del} on the exterior bundle over $E10/K(E10) (or$ rather of finite truncations of this object, since we don't know how to handle the full thing.)



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.



NNTP-Posting-Host: feynman.harvard.edu Mime-Version: 1. Content-Type: TEXT/PLAIN; charset=US-ASCII In-Reply-To: Xref: solaris.cc.vt.edu sci.physics.strings:40 On Sun, 18 Apr 2004, Urs Schreiber wrote: > $T - \bar T (in$ modes $L_m - \bar L_{-m})$ is always the generator of > spatial reparameterizations of the string at appropriately fixed > worldsheet time. Right, that's the very correct answer. So anything that you will obtain from such a special treatment of the "fixed" worldsheet time will be non-invariant under the general conformal transformations of the 2D worldsheet which can act nontrivially on the worldsheet time. These general conformal symmetries are necessary for decoupling of ghosts, and for calculations of the loop diagram, among many other things. If you take the comment about the loop diagrams seriously, your group of spatial reparameterizations does not even exist for loop diagrams. Such a group only looks meaningful if you stare at the cylindrical worldsheet of a single string, but it has no invariant meaning if you consider the interactions. The whole conformal symmetry (and the Virasoro algebra) must be treated as one group if you want to use its power - and believe me that you need its full power to calculate the scattering amplitudes, especially at the loop level. > It is precisely the group which gives you isomorphisms of the > left+right Virasoro algebra. It is possible to identify the group elements > which, for instance, correspond to background T-duality, background > diffeo transformations, background gauge transformations. (see below) These transformations (T-duality etc.; the spacetime diffeomorphisms are not the best example unless they are isometries, because the other transformations *do* change the background i.e. they *do* change the form of the worldsheet action and consequently of the Virasoro generators) preserve the full left as well as the full right Virasoro group, and therefore it is not surprising that they also preserve its subgroup or subalgebra of the "diagonal spatial" Virasoro generators. What I've been trying to say for some time is that the transformations that preserve the whole Virasoro algebra are interesting (they are symmetries of string theory, such as T-duality), while the remaining generic transformations that only preserve a part of it (such as the spatial diffeomorphisms) are not interesting (and are not physical) in string theory, and they are not symmetries, much like any other transformation or field redefinition that preserves nothing. There is a huge difference between the words "field redefinition" and a "symmetry" (or "duality"). Symmetries and dualities are much more special and rare concepts! > > No, it's not. A duality (and similarly for a self-duality, which is > > essentially the same thing as a symmetry) is a map redefining the > > variables of a theory, a redefinition that transforms the known > > observables in a first known theory AB exactly into some known observables > > of another known theory CD (which may be the same thing as AB). > > Sure. And the conjugation which I mentioned in my last post does > exactly that for T-duality. I don't quite understand. Is the conjugation that you talk about now the conjugation by a generic $\exp(W)$? T-duality is not a generic conjugation, but a very specific transformation that preserves - unlike the conjugation by a generic $\exp(W) -$ every single generator of the Virasoro algebra, and as far as I know, it was not you who discovered T-duality. ;-) > > On the other hand, your generic conjugation is a meaningless redefinition > > of fields that preserves neither the Virasoro algebra > > I don't understand why you are now saying that it does not preserve the > Virasoro algebra. In your last post you said that it is known to > every undergraduate that conjugation preserves the algebra. I guess that you only want to entertain us right now. What I wrote is that every kid knows that every conjugation preserves the form of the *commutation relations* between various generators, i.e. the form of the identities of the form $[A,B]=C$ which is transformed to $[A',B']=C'$. But that's not enough for an operation to be called a symmetry. For a transformation to be called "symmetry", the explicit form of every Virasoro generator, in terms of the fundamental field, must be unchanged, i.e$. A=A',B=B',C=C'$ in the example above, or for example $L_{m}$ must be still equal to $\frac{1}{2}\sum_{n} \alpha_{m+n}^\mu \alpha_{-n}_\mu$. While the generic conjugation by $\exp(W)$ preserves the form of the *commutation relations*, it certainly does not preserve the generators themselves (a generic transformation preserves neither $L_m$ nor $\alpha^\mu_m) -$ even though you seem to be saying exactly this (!), and therefore you cannot call it a "symmetry", and therefore it is also much less interesting than the real symmetries such as T-duality. > F. Lizzi, R. Szabo, > Duality symmetries and noncommutative geometry of string spacetimes, > http://www.arxiv.org/abs/hep-th/9707202 This paper offers a slightly non-standard approach, especially because of their usage of the Dirac operators, but nothing against it. I am not arguing against the fact that T-duality is important, interesting, and it is a very special example of a transformation i.e. conjugation. ;-) On the contrary, I was opposing your statement that you can extend the well-known T-duality to a much larger group, and I claim that there exists nothing beyond T-dualities (plus spacetime gauge symmetries and isometries) that is a symmetry of perturbative string theory. T-duality is very special. Some generic transformations that only preserve a subgroup of the Virasoro algebra are physically uninteresting. > Yes, I have a reason for this guess: > > I once checked that the supersymmetry generators of $N=1 D=3+1$ SUGRA > (or rather appropriate linear combinations of them) are > the exterior derivative d and its adjoint del on the configuration space > of the theory (really the Dolbeault operators $\partial$ and $\bar \partial$ > so that $d = \partial + \bar \partial)$. The susy algebra of the constraints > of the theory are hence equivalent to the algebra which is schematically > of the form > > {d,del} $= H + .$.. > > where H is the Hamiltonian constraint of the theory (and in particular > something like the Laplace-Beltrami operator on config space) > and the ellipsis indicates sa sum of patial diffeomorphism and > Lorentz rotation constraints. Lorentz rotation constraints? The RHS of the SUSY algebra in the flat space should contain the momenta, but no Lorentz generators or other diffeomorphisms. I also don't understand what you did with the spinor indices of the supercharges. Your schematic formulae look very Lorentz-non-invariant, also because of the special treatment of the Hamiltonian, and it just seems difficult at this moment to imagine a variation of these arguments that would be believable. Sorry, I am sure that you will convince me later. ;-) > It is to be expected that a similar statement applies to the > constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have > shown that the bosonic part of configuration space (which, to emphasize it > again, is the space on which the above operators d and del act) is > (a subspace of) the group manifold of $E10/K(E10)$ and that the > (bosonic part of) the Hamiltonian constraint H is the Laplace > operator on this manifold. Hence by the above it is a reasonable > guess that the supersymmetric extension of this Hamiltonian > constraint is the Laplace-Beltrami operator {d,del} on the > exterior bundle over $E10/K(E10) (or$ rather of finite truncations > of this object, since we don't know how to handle the full thing.) OK, I think that some work is needed before others will see what this proposal really means and why it is a reasonable guess. Good luck, LM __{____________________________________________________________________ ________} E-mail: lumo@matfyz.cz fax: $+1-617/496-0110$ Web: http://lumo.matfyz.cz/ eFax: $+1-801/454-1858$ work: $+1-617/496-8199$ home: $+1-617/868-4487$ (call) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^