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:

 

[Locrian]

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

There is no zero choice? Because that is what it's predictive powers currently are. Maybe one day that will change, but by then you can be assured it will be a different animal than it is today, and will deserve a different name.

 

[marcus]

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)...

The CDT approach to quantizing gravity has no fixed link length 'a'. One particular triangulation will have a length 'a' which is the size of a spacelike tetrahedron. Then one let's 'a' go to zero.

the spacetime of CDT defined by taking the limit (as 'a' goes to zero) of finer and finer triangulated spaces using smaller and smaller simplexes.

the spacetime of CDT is not made of simplexes, the simplexes used in the approximations are, I guess I would say, a mathematical convenience

(as, in Freshman Calculus, "step functions" might be used in defining the integral, but ultimately the integral is not made of little skinny rectangles---the step functions are a convenience used along the way)

in some CDT papers, other simple geometrical objects are used besides simplexes.

the simplex is a very old mathematical object, it does not need NonCommutativeGeometry to define or validate it.

thanks for trying to show some fundamental overlap between NCG and CDT!
I still have hope that Kea will come up with an essential connection between the two----which would make NCG, in my view, considerably more promising as a possible way to describe gravity!

You too, Mike. Keep trying if you want. It was your notion that the two were connected (or so I interpret something you said) that I originally asked you, and later Kea when she appeared to concur in it, to substantiate.

 

[Kea]

From page 14 of Reconstructing:

"We will currently concentrate on the purely geometric observables, leaving the coupling to test particles and matter fields to a later investigation..."

Marcus, I'm afraid you are going to have to do some very smooth talking to convince the likes of kneemo and I that there is any such thing as gravity without matter.

Kea

 

[Kea]

kneemo,

I think our job here is to really convince Marcus that we're right, because if we can do that, if Marcus agrees with us, a whole lot more people will make an effort to understand NCG...and that's what counts.

 

[spicerack]

reagrdless of whether Marcus agrees i sure would like to know what it all means minus the geek speak and number crunching

Is plain fools english for plain english speaking fools like me too much to ask without making too much of an effort

 

[kneemo]

The CDT approach to quantizing gravity has no fixed link length 'a'. One particular triangulation will have a length 'a' which is the size of a spacelike tetrahedron. Then one let's 'a' go to zero.

Hi Marcus

By using NCG, one need not let the link length 'a' go to zero. Read pgs. 2-3 of J. Madore's gr-qc/9906059 for a simple example of how lattices become fuzzy in NCG.

 

[nightcleaner]

reagrdless of whether Marcus agrees i sure would like to know what it all means minus the geek speak and number crunching

Is plain fools english for plain english speaking fools like me too much to ask without making too much of an effort

Hi all

Kneemo, this has been my quest, too. However, I have tried to learn to speak geek and to crunch numbers because that is the language spoken here.

One problem I have encountered trying to translate geekspeak is that geeks are now trying to investigate spacetime relationships that are fundamental but not obveous to daily experience. Our language (English anyway) was developed to deal with daily experience. As a result, we have many enforced thought habits which do not serve us well when dealing with quantum spacetime.

Mathematics is descriptive of but not limited to our daily experience. So it is actually easier to talk about these things using math rather than English. But math is indeed another language, and the alphabet in that language is huge, the vocabulary immense. Even Chinese looks like wooden building blocks compared to the advanced architecture of math.

Don't give up. Keep trying to read the physics and the math. I keep reading even when the words become gibberish. Somehow things percolate in the subconsious, and even though you did not understand a word of it yesterday, today it seems to make a little sense, and tomorrow it may even appear reasonable.

Be well,

nc

 

[marcus]

Hi Marcus

By using NCG, one need not let the link length 'a' go to zero. Read pgs. 2-3 of J. Madore's gr-qc/9906059 for a simple example of how lattices become fuzzy in NCG.

In some versions of NCG (as far as I know, at least where applied to gravity), one is PREVENTED from making length parameters smaller than a certain amount by a minimal length barrier.

One of the interesting things about CDT, and something that makes it different from several other approaches, is that it HAS NO MINIMAL LENGTH.

at least until now, no minimal length has been found in CDT, here is a recent statement to that effect from hep-th/0505113, page 2

"in quantum cosmology. We have recently begun an analysis of the microscopic properties of these quantum spacetimes. As in previous work, their geometry can be probed in a rather direct manner through Monte Carlo simulations and measurements. At small scales, it exhibits neither fundamental discreteness nor indication of a minimal length scale."

this may point to a theoretical divide between CDT and NCG! For instance, as you can see from the first 5 pages of the Madore article you cited, the versions of NCG he discusses have minimal lengths

here is a sample from page 5 of the article you cited:

"... Such models necessarily have a minimal length associated to them and quantum field theory on them is necessarily finite [90, 92, 94, 24]. In general this minimal length is usually considered to be in some ..."

 

[selfAdjoint]

One of the interesting things about CDT, and something that makes it different from several other approaches, is that it HAS NO MINIMAL LENGTH.

That is, the simulations don't find any minimal length.

But the common a of all links, which then goes to zero (but only TOWARD zero in the simulations!) makes it all look more and more like what the lattice physicists do. Since the triangulation is only built to subsequently go away in the continuum limit, how is this fundamental?

 

[marcus]

Nightcleaner and Spicerack, this discussion of "geekspeak" has me chuckling.

I can imagine that to Spicerack ears it sounds pretty esoteric and technical to be saying that two pictures of spacetime are incompatible because one theory gives rise to a minimal length or a notion of fundamental spacetime discreteness (which I am not sure is quite the same thing although related)

and the other theory does NOT give birth to a minimal length---a barrier smaller than which length is meaningless----or to a discreteness idea.

We are not in some primitive discussion like "UGH, DIS IS GOOD! UGH DIS IS BAD!" We are trying, I hope, to sort out various models of spacetime and see whether and how they connect to each other.

So at this moment I am looking at two called CDT and NCG (which to me looks like a large family or tribe of theories really, not a single unique one like CDT).

And i am looking at CDT and the NCG tribe----both are interesting and show some promise----and trying to distinguish some significant details that can let me see objectively what possible overlap there is.

so of course it is going to sound technical.

If you are mainly interested in having your imagination INSPIRED by stimulating talk about different theories, or if you are looking for something to BELIEVE in, then almost certainly this kind of technical examination of details would not interest you one bit!

However it is the details about CDT that have made it suddenly change the map of QG.
CDT does not give rise to a minimal length, does not exhibit fundamental discreteness, and it appears to be MORE BACKGROUND INDEPENDENT than Loop Gravity. CDT is not built on a pre-established differentiable manifold continuum with a pre-established dimensionality and coordinate functions.
It changes the map because it makes radical departures. It is based on ROUGHER AND LESS PREDETERMINED objects or foundations.

this is not to make a value judgement like "UGH, DIS GOOD!" Indeed maybe it is bad. Who cares? What matters is not what you think is good or bad or what you want to believe in or not believe in or what makes appealing mental images in one's head. What matters right now is that suddently something new is on the table.

another thing with CDT is you can run computer simulations and generate universes "experimentally" and study them and find out things (like about the dimension, or the effects of the dark energy Lambda term or whatever). you can find out things that you didnt anticipate! The CDT authors have been experiencing this. It was something of a surprise to them when last year they got a spacetime with largescale 4D dimensionality for the first time. Must have been great to see that coming out of the computer, the first time.

anyway it is somewhat unusual that CDT has ample numerical opportunities, a lot of theories are so abstract that you cannot calculate with them. they are not very "hands on". CDT is very hands on and constructive. the computer builds spacetimes for you and you get to study them.

the objective sign of the "change in the map" that I am seeing is the change in the programme topics between May 2004 Loop conference (in Marseille) and October 2005 Loop conference (in Potsdam)

I sympathize with Spicerack puzzlement, but I am not sure "geekspeak" is the real problem. The real problem may be that there is no reason compelling for her to be learning about CDT because it may not offer the imagistic stimulation or the verbal excitement of something like Brian Greene-style String theory. It is kind of Plain Jane Spacetime, modeled with the most unpretentious possible tools, with the least prior assumptions, with little by way of grand shocking discoveries like "eleven dimensions with the extra dimensions rolled up" and "fundamental discreteness and minimal length" and "colliding brane-worlds" and such.

 

[marcus]

Since the triangulation is only built to subsequently go away in the continuum limit, how is this fundamental?

I don't know that the particular type of triangulation is fundamental. Did I say the triangulation was fundamental? As I pointed out several times, Renate Loll has used other shapes besides simplexes in some papers. Simplexes are simple tho, so there is probably no reason not to use the well-established theory of simplicial manifolds.

I remember in grad school in the late 1960s we got lectured about piecewise linear ( PL) manifolds. there was a guy who believed in studying PL manifolds rather than differentiable manifolds. At the time I did not see why, but maybe I see now. I did not guess that actual realworld spacetime might be better approximable using a quantum theory of PL manifolds instead of differentiable ones. CDT is based on PL geometry instead of Differential Geometry.

"Fundamental" something of a slippery term. I want to communicate what i think is fundamentally different about the CDT approach. The image is how a Feynman path is the limit of PIECEWISE STRAIGHT segments. And a CDT spacetime is the limit of piecewise flat, or PL, or piecewise minkowski, chunks.

Maybe Feynman would have been wrong if he had tried to approximate his path by smooth infinitely differentiable paths. Maybe we are wrong now if we try to approximate our spactime with smooth differentiable manifolds. maybe we should be approximating with PL manifolds, like they do in CDT.

But that is just a mental image. Let me try to list some ways CDT is DIFFERENT.

It is not based on a differentiable manifold (LQG and some others are)

It is not based on something using coordinates----curvature in CDT is found combinatorially, by counting

It does not automatically reflect a prior choice of dimension. the dimension emerges or arises from the model at run-time---it is dynamic and variable. again the dimension is something you find combinatorially, essentially by counting. (this feature is absent in some other quantum theories of gravity. one might hope that whatever is the final QG theory will explain why the universe looks 4D at large scale and this CDT feature is a step in that direction)

CDT has a hamiltonian, a transfer matrix, see e.g. the "Dynamically..." paper, one can calculate with it. The CDT path-integral is a rather close analog of the Feynman path-integral for a nonrelativistic particle using
piecewise straight paths. The simplexes are the analogs of the straight pieces. by contrast some other QG theories with which you cannot calculate much.

CDT is fundamentally different from some other simplicial QGs because of the causal layering. (the authors explain how this leads to a well-defined Wick rotation, which they say is essential to their computer simulations)
this layering actually has several important consequences, AJL say.

well, I can't give a complete list, only a tentative and partial one. maybe you will add or refine this

 

[selfAdjoint]

I don't know that the particular type of triangulation is fundamental. Did I say the triangulation was fundamental? As I pointed out several times, Renate Loll has used other shapes besides simplexes in some papers. Simplexes are simple tho, so there is probably no reason not to use the well-established theory of simplicial manifolds.

I wasn't talking about the detailed technology of the triangulation, but about the whole project of doing a triangulation, doing nonperturbative physics on it (if only via simulations), and then letting the scale go to zero to recover the continuum. That's the QCD lattice strategy, and it seems to be Ambjorn et al's strategy too.

I remember in grad school in the late 1960s we got lectured about piecewise linear ( PL) manifolds. there was a guy who believed in studying PL manifolds rather than differentiable manifolds. At the time I did not see why, but maybe I see now. I did not guess that actual realworld spacetime might be better approximable using a quantum theory of PL manifolds instead of differentiable ones. CDT is based on PL geometry instead of Differential Geometry.

Somebody mentioned finite element method in engineering. That's a valid refence too. To me PL manifolds seem a kludge - neither honest polyhedra nor honest manifolds. Do we have any important topological results from them that couldn't be obtained a step up or a step down the generality ladder?

"Fundamental" something of a slippery term. I want to communicate what i think is fundamentally different about the CDT approach. The image is how a Feynman path is the limit of PIECEWISE STRAIGHT segments. And a CDT spacetime is the limit of piecewise flat, or PL, or piecewise minkowski, chunks.

I am sure you know Feynmann's pretty little piecewise-limiting picture is problematic in the Minkowski context. Does the phrase Wick rotation ring a bell? How about paracompact?

Maybe Feynman would have been wrong if he had tried to approximate his path by smooth infinitely differentiable paths. Maybe we are wrong now if we try to approximate our spactime with smooth differentiable manifolds. maybe we should be approximating with PL manifolds, like they do in CDT.

Maybe so. Cerainly it's a valid research program. But you seem to be defending it the way Lubos used to defend string theory; as the One True Way. Neither LQG nor string theory, to name just two, is truly down for the count, and Kea's higher categories may come from behind to conquer all, or something entirely differnt may happen. Let us keep our options open.

But that is just a mental image. Let me try to list some ways CDT is DIFFERENT.

It is not based on a differentiable manifold (LQG and some others are)

It is not based on something using coordinates----curvature in CDT is found combinatorially, by counting

It does not automatically reflect a prior choice of dimension. the dimension emerges or arises from the model at run-time---it is dynamic and variable. again the dimension is something you find combinatorially, essentially by counting. (this feature is absent in some other quantum theories of gravity. one might hope that whatever is the final QG theory will explain why the universe looks 4D at large scale and this CDT feature is a step in that direction)

The dimension aspect was certainly the strongest aspect of it last year. It remains to be seen whether the running dimension of this year strengthens their case or weakens it.

CDT has a hamiltonian, a transfer matrix, see e.g. the "Dynamically..." paper, one can calculate with it. The CDT path-integral is a rather close analog of the Feynman path-integral for a nonrelativistic particle using
piecewise straight paths. The simplexes are the analogs of the straight pieces. by contrast some other QG theories with which you cannot calculate much.

Correct me if I'm wrong, but the Hamiltonian only subsists at the a > 0 level, it does not carry through in the limit. Or have they somehow discovered how to generate a non constant Hamiltonian in GR?

CDT is fundamentally different from some other simplicial QGs because of the causal layering. (the authors explain how this leads to a well-defined Wick rotation, which they say is essential to their computer simulations)
this layering actually has several important consequences, AJL say.

Some have expressed a suspicion that they built pseudo-Riemannian in with their "causal" specification. Lubos used to say their path integrations were unsound because they refused to include acausal paths, which must be done (he said) if you want to generate valid physics.

well, I can't give a complete list, only a tentative and partial one. maybe you will add or refine this

You have been a splendid defender of CDT. And I am not really a critic of it. But it disturbs me to see you so...evangelical.. about it.

 

[kneemo]

I remember in the 1960s or 1970s in grad school we got lectured to about PL manifolds. there was a guy who believed in studying PL manifolds rather than differentiable manifolds. At the time I did not see why, but maybe I see now. CDT is based on PL geometry instead of Differential Geometry.

Indeed there is power in the use of PL manifolds. Even more basic, however, is a zero-dimensional manifold. Zero-dimensional manifolds are naturally produced in noncommutative geometry, from the spectra of C^*-algebras. For a commutative, unital C^*-algebra \mathcal{A}, the Gel'fand-Naimark theorem ensures that we recover a compact topological space X=\textrm{spec}(\mathcal{A}), such that C(X)=\mathcal{A}. What Alain Connes did was extend the essentials of the Gel'fand-Naimark construction and apply it to noncommutative C^*-algebras.

In Matrix theory, higher dimensional branes are built using the spectrum of hermitian matrix scalar fields \Phi^{\mu}. Their spectrum alone, only yields a zero-dimensional manifold. What is important are the functions over the space, which are encoded as entries of the scalar fields \Phi^{\mu}. The hermitian scalar fields \Phi^{\mu} are elements of \mathfrak{h}_N(\mathbb{C})\subset M_N(\mathbb{C}). As M_N(\mathbb{C}) is a noncommutative C^*-algebra, a spectral triple is built, and noncommutative geometry ensues.

On further analysis, we see that only the hermitian scalar fields \Phi^{\mu}\in \mathfrak{h}_N(\mathbb{C}) are used for the spectral procedure. The spectrum of \mathfrak{h}_N(\mathbb{C}) is thus of more importance than the full C^*-algebra M_N(\mathbb{C}). \mathfrak{h}_N(\mathbb{C}) is a simple formally real Jordan *-algebra, thus is commutative, but nonassociative under the Jordan product. Hence, the spectral geometry is not a noncommutative geometry, but is rather a nonassociative geometry. I've been using the term 'NCG' to include these nonassociative geometries as well, as the spectral procedure is based on that of NCG.

The nonassociative geometry of \mathfrak{h}_N(\mathbb{C}) includes the projective space \mathbb{CP}^{N-1}, whose points are primitive idempotents of \mathfrak{h}_N(\mathbb{C}). The lines of the space are rank two projections of \mathfrak{h}_N(\mathbb{C}). By the Jordan GNS construction, \mathfrak{h}_N(\mathbb{C}) becomes an algebra of observables over \mathbb{CP}^{N-1}. The noncommutative algebra over \mathbb{CP}^{N-1} is the C^*-algebra M_N(\mathbb{C}). The gauge symmetry of this quantum mechanics arises from the isometry group of \mathbb{CP}^{N-1} which is \textrm{Isom}(\mathbb{CP}^{N-1})= U(N), with Lie algebra \mathfrak{isom}(\mathbb{CP}^{N-1})=\mathfrak{su}(N). This is how one properly formulates the N-dimensional complex extension of J. Madore's fuzzy sphere.

Now consider the N=3 case, which yields the projective space \mathbb{CP}^{2}, with \mathfrak{h}_3(\mathbb{C}) as an algebra of observables. The Jordan GNS eigenvalue problem provides three real eigenvalues over \mathbb{CP}^{2}, corresponding to three primitive idempotent eigenmatrices. This provides us with a three-point lattice approximation of \mathbb{CP}^{2}. We acquire a projective simplex by recalling the projective geometry axiom:

For any two distinct points p, q, there is a unique line pq on which they both lie.

This provides three unique rank two projections connecting our primitive idempotent eigenmatrices in \mathbb{CP}^{2}. The gauge symmetry of this projective simplex is U(3), arising from the isometry group of \mathbb{CP}^{2}.

The moral of the story is: a simplex is not just a simplex when points and lines are matrices. When we allow more general simplex structures, we see we can incorporate gauge symmetry. Now imagine the power of a full projective triangulation of this type with a richer isometry gauge group. :!)

 

[marcus]

You have been a splendid defender of CDT. And I am not really a critic of it. But it disturbs me to see you so...evangelical.. about it.

As for splendid, thanks! I am really just responding (partly as a mathematician but perhaps moreso) as a journalist. CDT is the hot story in quantum gravity at this time. The math is relatively fresh (more background independent and although the means are quite limited there seems opportunity for both computational experiment and new kinds of results)

If you have been reading my posts about this in various threads you can see that I am clearly not betting on any final outcome. It could turn out that LQG is right and NOT CDT, and it could turn out that NEITHER. Guesses about the final outcome are not so interesting to me as the story of CDT current developments.

An amusing side of it is that CDT has been achieving a series of firsts in the past couple of years (they point them out explicitly in their 4 recent papers so I probably don't have to list them for you if you have been keeping up) and yet----there are only 3 core workers!

String has on the order of 1000 active researchers and has been rather in the doldrums for past couple years. Not much to cheer about. Well maybe it is mathematically overweight or taking a pause to catch breath or something.

And Loop has on the order of 100 researchers and has made some notable progress in the past few years, I guess most notably in the cosmology department, getting back through the big bang, finding a generic mechanism for inflation, now beginning to understand the black hole.

and Loop output is growing sharply. Last time I looked it was posting around 170 papers per year on arxiv---a very rough measure, but I remember when the rate was more like 60 per year!

so from the journalist eye view Loop is showing outwards signs of success and robust health. But the hot story, for me, is what these THREE researchers have been achieving in a field where the basic output rate on arxiv is only around 4 papers per year!

The irony of this tickles me. The last shall be first and all that. So if you please you can consider my instincts not evangelical but news-houndish.

 

[marcus]

...Correct me if I'm wrong, but the Hamiltonian only subsists at the a > 0 level, it does not carry through in the limit...
...

Exactly, this is my reading too. remember the field is very very new. they only got 4D last year. but at least for now the limit is only a ghostly presence defined as a limit of concrete things. maybe it never will be any more than that (I am speculating here)

ANY calculating that you want to do, you can ONLY do in the approximations. all the features of the limiting spacetime are only accessible and calculable (as accurately as one pleases, in principle, but practically limited by computer size and power) in the approximating triangulated spacetime.