Likelihood of M-theory: 1-10 Scale

[Mwyn]

ok on a scale from one to ten exactly how likely is it for M-theorie to be true?

 

[Chronos]

Right now it's a mess with the 'landscape' fiasco. There is, however, hope. Some of the more gifted string people, like Lubos, are trying to sort it out.

 

[notevenwrong]

ok on a scale from one to ten exactly how likely is it for M-theorie to be true?

One. At this point the situation with M-theory is that one doesn't know what it is and whether it exists, but if it exists at all it is completely unpredictive. The last remaining hope, that somehow one could get predictions by a statistical analysis of the landscape, had just collapsed. All other hopes people put forward (e.g. that cosmology will save the situation), are pure wishful thinking, with nothing at all to back them up.

 

[selfAdjoint]

I'd give it a 5 or 6. I think it needs something additional to be able to get ino its non-perturbative area. The fact that it hasn't matured more than it has in the years since it was discovered is due primarily to the fact that its perturbative sector is more or less just pre-existing superstring theory, plus the mirror symmetries. In other words it's hard to get anything more than same old same old by treating M-theory perturbatively, but currently you might as well whistle for non-perturbative ones.

 

[marcus]

... The last remaining hope, that somehow one could get predictions by a statistical analysis of the landscape, has just collapsed...

for more on the new paper by Shamit Kachru et al
http://www.math.columbia.edu/~woit/blog/archives/000195.html
http://www.arxiv.org/abs/hep-th/0505160

 

[kneemo]

ok on a scale from one to ten exactly how likely is it for M-theorie to be true?

M-theory, as a nonperturbative theory, does not yet exist. The best string theorists have done is to propose matrix quantum mechanics as a nonperturbative definition of M-theory. There have been matrix models that reproduce dynamics of the various string theories, but the matrix definition is still a conjecture. From a global view, all matrix models are just formulations of string theory using noncommutative geometry.

Noncommutative geometry, however, has surfaced implicitly in other approaches to quantum gravity, such as loop quantum gravity and dynamical triangulations. Thus, if these other branches happen to find a powerful noncommutative theory before the string theorists, they may not call the theory "M-theory", even though the theory may reproduce stringy effects.

So if we wish to say M-theory is the theory of quantum gravity that eventually works, I'll say a 10. But if the finders of the working quantum gravity theory do not wish to pay homage to Witten's "M-theory", you'll have a 1.

 

[marcus]

I would hope that we can distinguish between commenting on the apparent collapse of the string theory enterprise (it has been crashing and burning since January 2003 and looks worse month by month) and being cheerful about major progress and hopeful developments in Quantum Gravity.

Both kinds of discussion are, I believe, reasonable and legitimate. And we should have room at Physicsforums for both, since open discussion can help people make rational decisions about what studies to pursue (whether in or out of school, professionally or for fun).

However the present string debacle and advances in Quantum Gravity are going on in different arenas and the failing String enterprise's most articulate critics are not even the same people as those interested in QG.
QG and String/M are not rivals in the sense of competing to achieve the same ends. QG aims at a quantum theory of what spacetime is and how its geometry works, and a QG theory can only be considered a success if it reproduces Gen Rel at large scale. At present it looks like the QG people are getting ready for a battle royale among themselves as to which is better: Loop Quantum Gravity or Causal Dynamical Triangulations (LQG vs. CDT). They are mostly too busy with their own business to comment on the state of affairs in string theory.

One can speculate that once QG researchers have arrived at a useful quantum theory of spacetime and its geometry, (if they do, and to me right now CDT looks like the most promising approach), then a new theory of the particle fields and forces of matter can be constructed over it. the idea is to get the underpinnings right and then do the overlay.

In CDT the dimension of the continuum can vary with scale----the spectral dimensionality can be 4D at large scale and get down around 2 at small scale---which may get rid of some renormalization difficulties. This is very new. I have some links in my signature.

Anyway there is a great deal of squabbling and rivalry in store among different research lines in QG, having nothing to do with the troubled state of affairs in Stringland. As well as some remarkable signs of progress, like the Freidel/Starodubtsev paper (Quantum Gravity in Terms of Topological Observables hep-th/0501191). We should try to cover both stories without mistaking one for the other!

 

[marcus]

...Noncommutative geometry, however, has surfaced implicitly in other approaches to quantum gravity, such as ... dynamical triangulations...

On what page, in what CDT paper?

 

[kneemo]

One can speculate that once QG researchers have arrived at a useful quantum theory of spacetime and its geometry, (if they do, and to me right now CDT looks like the most promising approach), then a new theory of the particle fields and forces of matter can be constructed over it. the idea is to get the underpinnings right and then do the overlay.

In searching for a theory of quantum gravity, we cannot assume that fields and matter are independent of the construction of spacetime. In the Matrix formulation of string theory, the picture has emerged of spacetime being built from D0-branes and fundamental strings. This is tantamount to saying that dynamical triangulations are built from D0-branes (vertices) and fundamental strings (edges). In CDT, the precise form of a triangulation is not derived, but is rather defined. Thus, using insights from Matrix theory, we can understand CDT at a deeper level, and even have fluctuating dynamical triangulations. Even more, through noncommutative geometry, the dynamical triangulations would be fuzzy, and we would have a natural UV cut-off.

 

[kneemo]

On what page, in what CDT paper?

If this was in the CDT papers, it would be explicit. Just notice that spectral techniques are what NCG is all about. Triangulations from spectra is natural in NCG, and NCG can tell you exactly how to make the triangulation fuzzy. So it's possible to have a discrete space, that at the same time has a nice quantum mechanical uncertainty.

 

[marcus]

...This is tantamount to saying that dynamical triangulations are built from D0-branes (vertices) and fundamental strings (edges)...

I believe you may have confused two things which are different mathematically, because on a naive level they "look" the same to you.

Strings live in a differentiable manifold. Mathematically they are different objects from edges in a piecewise flat continuum.
The space of CDT does not live in a smooth manifold---it is not embedded in anything with a differentiable structure.
Indeed the space of CDT IS NOT PIECEWISE FLAT AND IT IS NOT MADE OF SIMPLEXES! This is very important to understand. the space of CDT is the limit of (an ensemble of) piecewise flat continua.

for an analogy, thing of the nowhere differentiable paths in a Feynman path integral.
CDT gets rid of the smooth continuum altogether
so the theories are built on different mathematical foundations.

I have to go, will try to explain this a little more later on.

 

[ohwilleke]

Three (assuming that ten is high and one is low).

 

[kneemo]

I believe you may have confused two things which are different mathematically, because on a naive level they "look" the same to you.

Strings live in a differentiable manifold. Mathematically they are different objects from edges in a piecewise flat continuum.
The space of CDT does not live in a smooth manifold---it is not embedded in anything with a differentiable structure.
Indeed the space of CDT IS NOT PIECEWISE FLAT AND IT IS NOT MADE OF SIMPLEXES! This is very important to understand. the space of CDT is the limit of (an ensemble of) piecewise flat continua.

Marcus, the confusion lies with you. In Matrix theory, the spectral space is a zero-dimensional manifold M. The strings emerge as elements of C(M). You are making reference to perturbative string theory, where a background manifold is specified. In Matrix theory, there is no pre-existing spacetime background; it must be generated. The most basic ingredient is an algebra, and the algebra used will determine the properties of the D-brane arising from the spectral construction such as dimensionality, gauge symmetry, etc.

My point is that a CDT is a derived concept. I've read through the CDT papers and have nowhere seen how to acquire a triangulation from more basic principles. When the authors eventually figure out how to do this, instead of presupposing the existence of a triangulation, they will realize they are doing noncommutative geometry.

 

[marcus]

...Noncommutative geometry, however, has surfaced implicitly in other approaches to quantum gravity, such as ... dynamical triangulations...

this is what you said that interest me and I would like you to substantiate with some online article and page references.

It is fine with me if you reference a page from an article by Alain Connes on non-commutative geometry. I just want to see some connection established.

So far, all I can see is handwaving. And you have brought up the word "spectral" which occurs all over mathematics. Yes it occurs in the "spectral dimension" probed by diffusion processes. And it occurs in good old classical operator theory where the set of eigenvalues is the spectrum. the term must be on the order of 100 years old in mathematics if not more----50 years for sure. And yes the word "spectral" occurs in NonCommut. Geometry.

But what I need is text from you that shows a more substantial connection than the accidental use of the same word in different contexts.


BTW if you want more clarification about what is meant by "spectral dimension" in the context of diffusion processes and quantum gravity, try this:

http://arxiv.org/abs/hep-lat/9710024
The spectral dimension of the branched polymers phase of two-dimensional quantum gravity
Thordur Jonsson, John F. Wheater
29 pages 7 figures
Journal-ref: Nucl.Phys. B515 (1998) 549-574

they are talking about the SPECTRUM OF THE HEAT KERNEL in classical thermodynamics, or the associate Laplacian. This is the "spectral dimension" concept used in CDT. Plain old-fashioned random walks and diffusion process stuff. Nothing fancy.
I shall applaud you if you can find this concept of spectral dimension in an Alain Connes paper, and thus draw the connection you say is implicit.

 

[Kea]

My point is that a CDT is a derived concept. I've read through the CDT papers and have nowhere seen how to acquire a triangulation from more basic principles. When the authors eventually figure out how to do this, instead of presupposing the existence of a triangulation, they will realize they are doing noncommutative geometry.

Thank you, kneemo.

I was too polite to interrupt Marcus because I know how much he adores CDT. Marcus, listen carefully to what kneemo is trying to tell you (and what I have been trying to tell you for a long time).

Cheers
Kea

 

[marcus]

..and what I have been trying to tell you for a long time.
...

what have you been trying to tell me about the relation of CDT and noncommutative geometry? I don't remember your ever talking about CDT, at all, Kea. but please make some clear points. I am interested as you can see, from my questions.

Here, I will quote the post i just wrote, and redirect the question to you Kea. maybe you will give me some definite online article and page reference

...Noncommutative geometry, however, has surfaced implicitly in other approaches to quantum gravity, such as ... dynamical triangulations...
...

...this is what you said that interests me and I would like you to substantiate with some online article and page references.

It is fine with me if you reference a page from an article by Alain Connes on non-commutative geometry. I just want to see some connection established.

So let me redirect this to you Kea. I would be delighted if there could be demonstrated some real connection between CDT and NCG. But I want a real connection. Some object defined in common. So find me a page in some CDT article and a page of NCG that I can study and compare and see if they are talking about the same stuff. then I can evaluate for myself whether I think the connection is just vague handwaving or whether there is some substance to it.

Would you be willing to do that, Kea?

 

[Kea]

So find me a page in some CDT article and a page of NCG that I can study and compare and see if they are talking about the same stuff. then I can evaluate for myself whether I think the connection is just vague handwaving or whether there is some substance to it.

Hello Marcus

As far as I am aware, the words dynamical triangulations are not synonomous with CDT. In particular, in the paper

Construction of Non-critical String Field Theory by Transfer Matrix Formalism in Dynamical Triangulation
Yoshiyuki Watabiki
http://arxiv.org/abs/hep-th/9401096

which is referenced by

On the relation between Euclidean and Lorentzian 2D quantum gravity
J. Ambjorn, J. Correia, C. Kristjansen, R. Loll
http://arxiv.org/abs/hep-th/9912267

there is a background connection with the old Matrix theory. The point is that there is a long and complicated history to the CDT papers. Do you really want to ignore the evolution on the more mathematical side of things?

I admire the CDT papers, but they are not fundamental. At least, I don't see anything in them that is.

Cheers
Kea

 

[Kea]

There seems to be a reluctance to accept that more abstract modern mathematics might have a simplicity sublime enough to do physics. Of course the mathematics looks complicated. Goodness knows I find it complicated. But who is the judge of what is simple? Posterity more than you or I. I've always felt I was much too stupid to understand anything that wasn't simple, and yet I find the combinatorics of Descent Theory to be essential to QG. Maybe I'm wrong.

I am sorry, Marcus, if I have been too lazy to investigate the connection between CDT and its related papers. I can see that it would be of interest.

Kea

 

[marcus]

hello Kea, I asked you to explain the connection of CDT and NCG. I am not asking about Watabiki's work (I know of him as a collaborator of Ambjorn and Loll). I am not asking about Ambjorn's work in string theory. When I checked over a year ago I saw he had done quite a bit in string.

what I want to be told about is the overlap between two interesting fields: causal dynamical triangulations and Noncommutative Geometry.

I want you to show me a mathematical object common to both.

Hello Marcus

As far as I am aware, the words dynamical triangulations are not synonomous with CDT. In particular, in the paper

Construction of Non-critical String Field Theory by Transfer Matrix Formalism in Dynamical Triangulation
Yoshiyuki Watabiki
http://arxiv.org/abs/hep-th/9401096

which is referenced by

On the relation between Euclidean and Lorentzian 2D quantum gravity
J. Ambjorn, J. Correia, C. Kristjansen, R. Loll
http://arxiv.org/abs/hep-th/9912267

there is a background connection with the old Matrix theory.

you may be presuming in me more ignorance than is actually there
I have read fairly extensively in the the papers by Ambjorn and others in the 90s. And am not disinterested in the history.

The point is that there is a long and complicated history to the CDT papers.

yes I know (and was aware of Ambjorn doing string research and other crossover type stuff, and that the words "dynamical triangulation" can occur in other contexts besides CDT and have other meanings)

But that is sort of beside the point IMHO. I am not asking about string, I am asking about Noncommutative Geometry (which string is far from having a monopoly on!) and the NCG connection specifically to CDT. Please show me.
I would love to see it!

Do you really want to ignore the evolution on the more mathematical side of things?

that sounds like asking someone "when did you stop beating your wife"?
Imagine if people (not me, I never would) were to be asking YOU such rhetorical questions. My training, as you probably know, is primarily in mathematics, and I love history. As someone who thinks primarily as a mathematician interested in physics, I pay close attention to the historical evolution. NO I do not want to be ignorant of the evolution of mathematical ideas. Do you?

...I admire the CDT papers, but they are not fundamental. At least, I don't see anything in them that is.
...

AH HAH! HERE WE HAVE IT! You and I are two mathematicians, roughly at the same level of sophistication, i imagine, although we may know about different things. We both have looked at the CDT papers. YOU DO NOT SEE ANYTHING FUNDAMENTAL. And I do. I see a fundamentally new model of spacetime, and an historical breakthrough. I do not think CDT could have been or would have been derived from fashionable conventional math such as "M-theory".

If NCG was pregnant with CDT then I want to know rigorously and exactly how it was. If you don't happen to know, that's fine, just say

Cheers,
marcus.

Let's follow up on this interesting difference of opinion. you see nothing fundamentally new in CDT, and i do. Let us talk it over. It might help clarify the ideas!

 

[Kea]

Let's follow up on this interesting difference of opinion. You see nothing fundamentally new in CDT, and I do. Let us talk it over. It might help clarify the ideas!

All right, Marcus. I will go away and look at the Reconstructing the Universe paper. It might take me a bit of time.

By the way, I'm more of a physicist than a mathematician. I don't understand the concept of a 'wavefunction for the universe'. Could you clarify this for us?

Cheers
Kea

 

[marcus]

There seems to be a reluctance to accept that more abstract modern mathematics might have a simplicity sublime enough to do physics.

My dear that is a total fantasy on your part, as applies to me!


I was in love with elegant abstract modern mathematics presumably before you were born (you are a postdoc now right?) which is why i specialized in math.

but I have standards of concreteness which I apply in your case, and in the case of anyone claiming to be a mathematician. We don't want people waving their hands and just spouting words, we want to know exactly what the words mean.

I am telling you FAR from being reluctant I would simply LOVE it if you could give me a reliable set of page references that would show me that CDT (causal dynamical triangulation approach to quantum gravity, not something else applied to something else, but THAT) can be derived from Alain Connes NonCom Geom.

show me how CDT comes from something presumably more fundamental in NCG

(or tell me frankly that you can't, no harm done)

I hope you can, since it would add considerably to my delight in elegant modern mathematics.

Of course the mathematics looks complicated. Goodness knows I find it complicated. But who is the judge of what is simple? Posterity more than you or I. I've always felt I was much too stupid to understand anything that wasn't simple, and yet I find the combinatorics of Descent Theory to be essential to QG. Maybe I'm wrong.

I am sorry to hear you sound discouraged by the difficulty. Please do not think of these things as impossibly complicated! Have courage.
As it happens I do not know anything about Descent Theory----or do not know it by that name. One has to budget one's time and right now the combinatorics of CDT is taking all I can give it, so i am not about to start on Descent theory.

However whatever mathematicians give their attention too, eventually will become simple---- like a river running over a stone till it is smooth and oval, or an irritating grain of grit that becomes a pearl, even if it takes 100 years----mathematicians are the oysters or perhaps the rivers, whose job it is to love things until they become simple easy and beautiful.

 

[Kea]

I will go away and look at the Reconstructing the Universe paper.

I'm sorry, Marcus. I have a problem with the first sentence (but I will keep going). They say:

...at the shortest scales.

What does that mean?

Kea

 

[marcus]

I'm sorry, Marcus. I have a problem with the first sentence (but I will keep going). They say:

...at the shortest scales.

What does that mean?

Kea

you are wonderful Kea, I am delighted you are looking at that paper.

Of course there is no shortest scale in CDT as they say explicitly later.
what they mean is simple "at very short scales"
you are in the introduction and the language is relaxed and has some leeway. Just go with it and keep reading

 

[marcus]

I have to go and get materials for cucumber and watercress sandwiches for a nice teaparty we are giving for a German friend. I will be back soon

 

[Kea]

I was in love with elegant abstract modern mathematics presumably before you were born (you are a postdoc now right?) which is why i specialized in math.

Oh, no, I haven't finished my thesis yet. And I suspect that I am not quite as young as you think.

I know it's just the introduction, but I'm very confused by the second sentence: Because of the enormous quantum fluctuations predicted by the uncertainty relations... Are they assuming that the UP applies 'as is' to QG?

All the best
Kea

 

[marcus]

I'm very confused by the second sentence: Because of the enormous quantum fluctuations predicted by the uncertainty relations... Are they assuming that the UP applies 'as is' to QG?

Well, let's try to understand. here is what they say:
"...Because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical. Although different approaches to quantizing gravity do not agree on the precise nature of these fundamental excitations, or on how they can be determined, most of the popular formulations agree that they are neither the smooth metrics g_{mu, nu}(x) (or equivalent classical field variables) of general relativity nor straightforward quantum analogues thereof.

In such scenarios, one expects the metric to re-emerge as an appropriate description of spacetime geometry only at larger scales.

Giving up the spacetime metric at the Planck scale does not mean discarding geometry altogether, since geometric properties such as the presence of a distance function pertain to much more general structures than differential manifolds with smooth metric assignments."

the AJL introduction paragraph here is MOTIVATIONAL and starts off easy with (not their results but) accepted wisdom. Already back in 1970 the famous John Archibald Wheeler was saying to expect spacetime to be very rough and nonclassical at small scale.

now you are asking what AJL HAVE IN MIND. You say do they imagine applying the UP "as is"? Well if you think that is a possibility, please say how YOU would apply the UP "as is".

Basically you can assume that Ambjorn and Loll are two of the smartest people in the business and have been thinking intensively about quantum gravity since about 1990. they will have thought concretely about how the UP applies to spacetime geometry. they will also have noticed that what they concluded about rugged smallscale geometry has also been figured out by many other people. So they don't explain IN WHAT PARTICULAR FORM they apply the Uncertainty Principle. That is not what the paper is about!

they are just motivating what they want to do. So we cannot tell how exactly they apply it or what exactly they have in mind in this case. You would have to ask them what they had in mind.

To summarize what they say:

A. spacetime geometry will be rugged at small scale, it will not simply be given by a smooth metric as we are used to in differential geometry (thanks to Riemann 1850)

B. One expects the metric to re-emerge at larger scales. so the Riemann 1850 description will still be good for macroscopic spacetime geometry and we will have the nice smooth distance function we are used to (at least approximately at large enough scale)

C. And giving up a conventional smooth metric at small scale does not mean the end of the world (they say) because there are lots of rough rugged structures (that can have UNsmooth distance functions defined on them). so we will be able to continue doing geometry, of sorts, at small scale----we just will not use a conventional manifold but will use some other structure.


this seems just some motivation and some generalities, not something to scrutinize at length. and it seems quite unexceptionable to me at least.
so let us move on quickly so that we can come to the part about THEIR work, namely where they begin to discuss CDT.

Do you have any questions about paragraph 3 of the introduction, at the bottom of page 1, which begins

"In the method of Causal Dynamical Triangulations..."?

 

[Kea]

Do you have any questions about paragraph 3 of the introduction, at the bottom of page 1, which begins
"In the method of Causal Dynamical Triangulations..."?

Yes. To quote...

"In the method of Causal Dynamical Triangulations one tries to construct a theory of quantum gravity as a suitable continuum limit of a superposition of spacetime geometries..."

This says to me that AJL believe that conventional quantum theory is a good guide to a definition of a quantum gravitational path integral. In M-theory/Category theory we can show very clearly why this doesn't work. That's not to say that CDT isn't useful in understanding the classical limit. Maybe it is.

In other words, I don't believe at all that one can apply ordinary quantum intuition to quantum geometry.

Still reading...
Kea

 

[marcus]

Still reading...

this is very encouraging. thank you, Kea

 

[Mwyn]

One thing I don't understand about String or M-theory is that if point like particles are represented as fundamental strings that contain 11 different dimensions to them and give off waves as they move through space, then how does it explain protons and neutons. the protons and the neutrons are all bunched up in the nucleus and can't really move much so how does M-theorie explain how the strings representing the protons and the neutrons can have waves to them if their not moving. How does M-theorie also explain the exsistence of quarks? I could see basically how electrons can be waves but I don't understand how the other particles can be waves.

 

[kneemo]

but I have standards of concreteness which I apply in your case, and in the case of anyone claiming to be a mathematician. We don't want people waving their hands and just spouting words, we want to know exactly what the words mean.

I am telling you FAR from being reluctant I would simply LOVE it if you could give me a reliable set of page references that would show me that CDT (causal dynamical triangulation approach to quantum gravity, not something else applied to something else, but THAT) can be derived from Alain Connes NonCom Geom.

show me how CDT comes from something presumably more fundamental in NCG

Hi Marcus

Let us return to the path integral in eq. (1) of hep-th/0105267. The path integral is re-written as a discrete sum over inequivalent triangulations T. A basic question is: how does one acquire just one of the many inequivalent triangulations T? And given a specific triangulation T_0, what action is performed to acquire a new triangulation T_1?

Answer these questions and we'll discuss how NCG comes into the picture.

 

[marcus]

Hi Marcus

Let us return to the path integral in eq. (1) of hep-th/0105267. The path integral is re-written as a discrete sum over inequivalent triangulations T. A basic question is: how does one acquire just one of the many inequivalent triangulations T? And given a specific triangulation T_0, what action is performed to acquire a new triangulation T_1?
.

Hello Mike,
I believe you indicated you were working towards your Masters at Cal State LA? Do I have that right? You can either take an exam or write a thesis, then---don't have to do both, please correct me if I'm wrong. How are things going?

It sounds like you have read at least the first page of hep-th/0105267. This is wonderful! I am delighted and urge you seriously to read more.

The answer to your question is section 7, page 20. there are some interesting details on pages 23-25.

the presentation is very clear and concise
Frankly I could not hope to do better. So I suggest you read their pages 20, 23-25, rather than my trying to paraphrase.
You will be pleased to see that they describe rather concretely and explicitly what action is performed by the computer program---or, to put it in your words, they say
"given a specific triangulation T_0, what action is performed to acquire a new triangulation T_1".

 

[Kea]

Hi Marcus, hi kneemo

Well, I had guests also. Just back. Up to page 4 now:

They put causality in by hand? Why these type of simplices? Why the requirement that the resultant spacetime be a simplicial manifold? Is that because they are only considering the classical limit? If that's so, I'm OK with that point. By the way 'classical limit' in this context should mean (IMHO) standard quantum logic plus classical manifold spacetime. But this does not appear to be what AJL mean. They appear to be discussing what they believe to be an approximation of a full analytical approach to QG.

Weights from the Einstein action ... seems reasonable but, once again, I don't see their justification for this use of naive quantum principles.

Hope you can clarify some of these points for us, Marcus.
Kea

 

[Kea]

By the way...

Marcus,

Obviously I have not stressed the following enough in my ravings about category theoretic logic.

1. Small scale = high 'particle' number = omega-categorical, implies dual 2D structure (although admittedly the details are still being worked out)

2. Large scale = minimal interaction = 2:2 qubit tetracat logic (which we also expect to mean 4D)

No manifolds put in by hand. No fixed dimension.

All the best
Kea

 

[Kea]

Marcus, will you allow me to skip the numerics for now?

On page 37 they state "all the geometric properties of the spatial slices
measured so far can be modeled by a particular kind of branched polymers..."

By 'branched polymer' they mean what kneemo and I would call a 'rooted tree'. These beasts appear en mass in NCG. Recall that on PF we have discussed Connes, Marcolli, Kreimer and the new rigour behind the standard model - and its connection to NCG. A nice random reference:

H. Figueroa, J.M. Gracia-Bondia
On the antipode of Kreimer's Hopf algebra
http://arxiv.org/PS_cache/hep-th/pdf/9912/9912170.pdf

Still reading...

 

[kneemo]

Dynamical triangulations are a variant of Regge calculus in the sense that in this formulation the summation over the length of the links is replaced by a direct summation over abstract triangulations where the length of the links is fixed to a given value a. In this way the elementary simplices of the triangulation provide a Diff-invariant cut-off and each triangulation is a representative of a whole equivalence class of metrics

We see that dynamical triangulations have a fixed link length 'a'. Now, if this assumption is valid will depend on the method by which we generate an elementary simplex.

Using NCG, we can attempt to generate an elementary symplex as a quiver (or pseudograph). Quivers arise in categorical approaches to D-branes and deconstruction (hep-th/0110146, hep-th/0502105), and have been discussed by Aaron Bergman and Urs Schreiber in sci.physics.strings:

In article
<Pine.LNX.4.31.0503091443440.19481-...man.harvard.edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

> Lubos' blog entry on deconstruction made me have a closer look at this
> stuff, which I should have had long before.
>
> If I understand correctly a quiver can equivalently be defined as a
> functor from some graph category to Vect. Given some graph, it associates
> finite vector spaces to vertices and linear operators between these to
> directed edges.

That's not the definition of a quiver. A quiver is just a directed
graph. A quiver representation is a vector space at each node and a map
for each arrow. The graph should define a category and you could look at
the functors from this category to k-Vect.

Equivalently, you can form the quiver algebra. Let the arrows be denoted $$a_j$$ and, to each node j, have an idempotent

$$e^2_j=e_j$$

You have two functions source and target which take these to nodes. For
the idempotents, both the source and the target maps take the idempotent
to its respective node. The source and target map for the arrows is
obvious. Then, take the free algebra on this set subject to the relation
that a product ab is nonzero iff $$t(b)=s(a)$$ .

Aaron Bergman's quiver algebra description can be realized in projective space. This means our quiver simplex can eventually be represented on a fuzzy sphere (hep-th/0503039), which is an NCG construction.

There is more to say, but alas, I must sleep. :zzz: