Nicolai digs back into string theory-supergravity, E8, octonions

[marcus]

Nicolai digs back into string theory--supergravity, E8, octonions

Hermann Nicolai is an interesting researcher to watch for several reasons. He is a leader in theoretical particle physics. Has been on the main advisory committee for the annual Strings conference for the last 10 years or so. A leader in the String community. Knows a lot about non-string QG and various approaches to unification. And when necessary can develop a new, testable, minimalist approach to unification that doesn't require string, or low energy supersymmetry. The kind of person who is recognized good at things you already heard about, but also will sometimes start cutting a path in a completely new direction you hadn't already heard about. So an instructive person to watch.

And for several years he has been working on a non-string program that he worked out with Kris Meissner, the extreme minimalist approach that makes the Standard Model extend out to Planck scale with almost no new concepts. That makes LHC testable predictions and if it were to turn out right it would make a lot of more elaborate theorizing unnecessary. (In that respect it resembles the asymptotic safety approach of Reuter Percacci Weinberg and others).

But just recently Nicolai came out with a string theory paper. And I'm interested to get a sense of what kind of string theory paper it is, and where it is going. So if anyone wants to interpret or comment or explicate for us that would be fine. Here is the paper:

http://arxiv.org/abs/0912.0854
Cosmological quantum billiards
Axel Kleinschmidt, Hermann Nicolai
18 pages
(Submitted on 4 Dec 2009)
"The mini-superspace quantization of D=11 supergravity is equivalent to the quantization of a E₁₀/K(E₁₀) coset space sigma model, when the latter is restricted to the E₁₀ Cartan subalgebra. As a consequence, the wavefunctions solving the relevant mini-superspace Wheeler-DeWitt equation involve automorphic (Maass wave) forms under the modular group W⁺(E₁₀)=PSL₂(Oct).* Using Dirichlet boundary conditions on the billiard domain a general inequality for the Laplace eigenvalues of these automorphic forms is derived, entailing a wave function of the universe that is generically complex and always tends to zero when approaching the initial singularity. The significance of these properties for the nature of singularities in quantum cosmology in comparison with other approaches is discussed. The present approach also offers interesting new perspectives on some long standing issues in canonical quantum gravity"

*The Octonions are used here. The authors use the letter O in a special font to designate the octonions. Here I have substituted the abbreviation "Oct" to make their abstract more immediately understandable.

E₁₀ is related to the exceptional Lie group E₈ used by Garrett Lisi and many other people. The official name of E₁₀ is the hyperbolic Kac-Moody group. It is described as an infinite dimensional extension of E₈.

I gather that SL(2, Oct) or alternatively written SL₂(Oct) would be the octonion analog of SL(2, C) which is the 2x2 matrices of complex numbers, with determinant equal one. In effect you just take those 2x2 matrices and substitute in octonions instead of complex numbers.

And SL(2, C) is familiar to many people as a stand-in for the Lorentz group, the transformations used in special relativity. So SL(2, Oct) could be a weirded-up version of the Lorentz group.

What I am hoping, or part of what I'm hoping, is that Garrett will comment.

 

[atyy]

Hey, this is very cute - quantum chaos! Maybe he wants to solve the Riemann hypothesis :tongue2:

I feel one motivation behind his work with Meissner is actually string theory or some sort of unification, quite unlike Asymptotic Safety. For example, in http://arxiv.org/abs/0907.3298 they state "known ansaetze at unification in general do not give rise to effectively Weyl invariant low energy theories, despite the ubiquity of dilaton-like fields in supergravity and superstring theory. For this reason we here suggest a different route ..."

The other motivation is his feeling that symmetries make theories finite, eg. like modular invariance in string theory, or some yet unknown symmetry for N=8 supergravity. The Cremmer and Julia paper about E7 he mentions as one of the motivations for the paper of this thread is also cited in this opinion http://physics.aps.org/articles/v2/70

 

[atyy]

Wow, marcus, this stuff is absolutely fascinating! Asymptotic Silence?

http://arxiv.org/abs/0710.1818
Spacelike Singularities and Hidden Symmetries of Gravity
Marc Henneaux, Daniel Persson, Philippe Spindel

http://arxiv.org/abs/0909.3329
Spontaneous Dimensional Reduction in Short-Distance Quantum Gravity?
Steven Carlip

 

[MTd2]

Marcus, have you seen the threads where I tried to discuss that supersymmetry is a low energy regime of a 12d M-Theory?

 

[tom.stoer]

... SL(2, Oct) ...

What is SL(2, Oct) ?

Another problem:

I think I asked this question some time ago: all Lie groups I am familiar with can be interpreted as a symmetry group of an scalar product of a certain vector space. In that sense the group is defined via the vector space.

Now let's look at E₈. Let t^a be a members of the algebra e₈ and define g = exp ix^at^a as a member of E₈. Assume there is a vector |x> with |x'> = g|x> and consider a scalar product which is invariant under g, namely <y'|x'> = <y|x>. Now find the simplest vector space for which such a representation is possible (facefully). This is somehow the definition of the fundamental representation of the group E₈. My question is: what is the vector space to which |x> belongs? how does it look like?

My feeling is that (as the fundamental and the adjoint reprsentation of E₈ are identical) the vector space is nothing else but the e₈ algebra. Is this correct?

 

[MathematicalPhysicist]

Sorry to but in (so to speak) into this discussion, but I have seen some of these topics around combining quantum chaos and quantum gravity, which seem uncorrelated.
Now next year (hopefully) I'll start my Msc study, so I think I found one supervisor from the mathematics department who is researching in QC, now I wish somehow also to combine my interest in QG with QC, so I guess I need also a supervisor who is well versed in this intermarraige between the two subjects.

I found a few names, like Hemann Nicolai, also Thibualt Darmour, and a belgium theoretical physicist (whose name I forgot), can you please provide other names of researchers in this marriage of two seemingly unrelated subjects?

 

[marcus]

MathPhys,
I don't know about "quantum chaos". When write the abbreviation QC it means "quantum cosmology" as in, for example, LQC which is the application of LQG to cosmology. Unfortunately, I can't suggest names of people for you to contact! Maybe others can.

What is SL(2, Oct) ?
...

Tom, I think you are way ahead of me on this, but I will go over some basics in case others are curious. My computer can't read the symbol which I call "Oct", it just gives a square box, which is in effect is a question (?) mark. From the context I deduce it is some ring, probably related to the octonions. Now I infer further and please correct me if you see that I'm off track.

What we are interested in is actually the projective version, PSL(2, Oct).

So let's back up, GL(V) is the general linear group on some vectorspace.
GL(n, F) is the general linear group on the n-tuples of field elements, and this extends to rings. (I just read that it extends to rings, I didn't check that it makes sense. Sounds like getting into modules over a ring.)

Once you have GL there is some rigamarole for saying what SL is (like "determinant = 1" or analogous modding out a subgroup, let's not worry, we know SL(V) in the usual vectorspace context.)

OK so now we need to understand PGL(V) and the less familiar but related idea of PGL(n, Ring). PGL(V) is the group defined on the LINES of the vectorspace, where you identify all the vectors which are the equivalent mod scalar multiplication. I think that PGL is essentially equal to GL modulo scalar mult. GL induces an action on the projectified version of the vectorspace V.

Now I didn't check to make sure all the definitions make sense and this really generalizes to the octonians. But I can sort of see how PSL(n, Ring) could be defined and we could have the special case where n=2 and Ring=Octonions.

And in this paper, or in the previous Kleinschmidt Nicolai paper, they introduce some variant of the octonions which they call the "octavians". I never heard of that before but presumably it's OK.

Since you are more familiar with stringy math, you can probably give a more rigorous and knowledgeable account of this than I can. But that's a brief starter on the basics regarding this unfamilar notation.

If anybody's computer is able to read what that symbol is, which I see as a box and which I surmise is Octonions, please tell us.

 

[marcus]

MTd2,
I didn't see the posts you refer to. You could give links, but my guess is that they are outside my scope. Thanks for suggesting to me that I start a thread on Kleinschmidt Nicolai. I posted partly because I thought other people might be interested. It is actually somewhat outside my usual range of interest, so I don't expect to be especially active in the discussion. I'm certainly a fan of Nicolai however. He does non-string quantum gravity and minimalist unification research, as well as string, and he supports mixed conferences where people from from different communities can share and compare ideas. So in a way I'm motivated to pay attentiuon to whatever paper by H.N.

Wow, marcus, this stuff is absolutely fascinating! Asymptotic Silence?...

http://arxiv.org/abs/0909.3329
Spontaneous Dimensional Reduction in Short-Distance Quantum Gravity?
Steven Carlip

Atyy, I am so pleased to hear that! I'm very glad when something turns up that you find especially interesting. I also remember Carlip describing "asymptotic silence" in his paper (and video talk at the Planck Scale conference). For anyone not familiar with Carlip's talk and paper "silence" happens classically at a type of singularity that one might have at classical big bang where the curvature gets so chaotic that it essentially fragments the geometry and space turns into tiny isolated regions, that are causally isolated so they can't hear each other or talk to each other, and which keep dividing into more and more causally isolated specks.
So silence prevails, nobody's lightcones can reach out to anybody else, existence is atomized, a honeycomb of constantly increasing complexity.

Unless I'm mistaken about what Carlip was saying, this happens at certain singularities based simply on classical 1915 general relativity. And he was presenting this in part as a way of giving some classical background to the fact that in some quantum versions of GR, some approaches to QG, the familiar 4D geometry we see at our scale breaks down into approximately 2D spacetime geometry at very small scale.

That was my take on the Carlip paper. You are putting it together with other things like this Kleinschmidt Nicolai paper and you may be getting some additional significance out of it.

 

[atyy]

That was my take on the Carlip paper. You are putting it together with other things like this Kleinschmidt Nicolai paper and you may be getting some additional significance out of it.

Asymptotic Silence is mentioned in both papers.

Yes, it somehow seems related to quantum chaos. Basically, quantum chaos is refers to quantum systems whose classical counterparts are chaotic. There is a conjecture, "half-proved" (I think) that classically chaotic systems have energy level statistics different from regular classical systems. The link to the Riemann hypothesis (not relevant here specifically, but in quantum chaos more broadly, is the Berry conjecture that the zeros of teh Riemann zeta are the eigenvalues of some chaotic Hamiltonian).

 

[atyy]

Asymptotic Silence is mentioned in both papers.

Ooops, I got mixed up. Asymptotic Silence is mentioned not in Kleinschmidt et al, but in Henneaux et al, which is the background to Kleinschmidt et al, and also in Carlip's paper, which is why I brought it up.

 

[atyy]

and a belgium theoretical physicist (whose name I forgot), can you please provide other names of researchers in this marriage of two seemingly unrelated subjects?

Henneaux? I linked a paper of his in #2.

 

[MathematicalPhysicist]

Henneaux? I linked a paper of his in #2.

Yes, that's the name I forgot.

To Marcus I know the abbreviations, but it seems to me that in this article and in other articles, they incorporate quantum chaos along with QG or Quantum Cosmology.

Btw, is there any difference between Quantum Cosmology which was initiated by Hartle and Hawking and LQC?

 

[atyy]

http://arxiv.org/abs/0710.1818
Spacelike Singularities and Hidden Symmetries of Gravity
Marc Henneaux, Daniel Persson, Philippe Spindel
"... in the BKL-limit, not only can the equations of motion be reformulated as dynamical equations for billiard motion in a region of hyperbolic space, but also this region possesses unique features: It is the fundamental Weyl chamber of some Kac-Moody algebra. ... The most celebrated case is eleven-dimensional supergravity, for which the billiard region is the fundamental region of E10=E8++"

Thanks to Lubos Motl's blog, here is a helpful exegesis of the subject by Nicolai's doctoral student:
http://arxiv.org/abs/0912.1612
Exceptional Lie algebras and M-theory
Jakob Palmkvist

 

[MTd2]

MTd2,
I didn't see the posts you refer to. You could give links, but my guess is that they are outside my scope. Th

Hey Marcus, here it is:

https://www.physicsforums.com/showthread.php?t=331294

There was a really good source about this, on a geocities website (gone now...). I guess you remember him. You said I couldn't give links from his website. But I saved his paper on the subject...

 

[apeiron]

There is a conjecture, "half-proved" (I think) that classically chaotic systems have energy level statistics different from regular classical systems.

Do you mean that chaos has powerlaw or fractal statistics and ordered systems have gaussian?

Anyway, I think the general picture of spacetime becoming a chaotic foam at small scale/high heat is a deep one. The whole problem of self-interactions surely disappears when spacetime itself breaks down and there is no possibility of interactions smeared across some spacetime surrounding some locale?

I view it in terms of hyperbolic geometry. Classical spacetime is flat and connected. A continuum. But at small scale/high heat, this flatness breaks down. Every point starts to show fluctuating curvature. Eventually the curvature is hyperbolic at every point so no two points of spacetime are in physical contact. A summing over histories could not happen over such a realm.

Carlip, as I suggested in another thread, finds dimensional reduction for this sort of reason. To be a 3D action, you need a stable context. You need a stable background of two orthogonal axes that then make an action in a third direction meaningful. A sum over histories would "see" both the action and the background of non-actions.

But as spacetime breaks down at the limit, each action would be taking place against a vagueness. The other dimensions would no longer be visible as concrete non-directions of that action. You would just have a 2D vector floating in a fog of undefined dimensionality.

Most QG thinking seems to be based on the presumption that self-interactions will keep getting stronger asymptotically to the limit of a singularity. But asymptotic silence would appear to be about the very ground of interaction giving way, crumbling to dust under your feet.

You always need something larger to make sense of something smaller. A container for the contents. But when the container - flat spacetime - is shrunk to the size of its Planckscale contents, you have a symmetry state where action becomes meaningless. The landmarks are lacking to flag the directions. A sum over histories over such a symmetry would have to give you a different kind of result?

 

[atyy]

Do you mean that chaos has powerlaw or fractal statistics and ordered systems have gaussian?

No. The conjecture is that generic regular classical systems have quantum counterparts whose energy level statistics are Poisson, while chaotic classical systems have quantum counterparts whose statistics are Wigner-like.

 

[apeiron]

No. The conjecture is that generic regular classical systems have quantum counterparts whose energy level statistics are Poisson, while chaotic classical systems have quantum counterparts whose statistics are Wigner-like.

Thanks atyy. Could you possible explain briefly what Wigner statistics are? Or point to a simple explanation? I googled and found only very technical references.

 

[atyy]

Thanks atyy. Could you possible explain briefly what Wigner statistics are? Or point to a simple explanation? I googled and found only very technical references.

Basically, the statistics are the histogram of the spacings between neighbouring energy levels. The Poisson distribution has a peak near zero, meaning that neighbouring energy levels can be very, very close. In contrast, the Wigner distribution has a peak away from zero. The Wigner distribution is only an approximation. You can find more precise statements about Gaussian Unitary Ensembles at http://terrytao.wordpress.com/2009/...e-riemann-zeta-function-on-the-critical-line/ and http://arxiv.org/abs/0708.4223