Likelihood of M-theory: 1-10 Scale

[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.

 

[marcus]

kneemo, thanks for the thoughts about NonComGeometry!
I am having a bit of difficulty reading some of the LaTex right now, hope it clears up.

 

[Kea]

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

Hi Everybody

One can't get a good night's sleep around here without missing a hot discussion!

Marcus, we all agree that any decent theory of QG can't use spacetime coordinates as fundamental entities. I was hoping you might address some of my questions from yesterday, but they seem to have been forgotten.

Another question: the cosmological constant appears to play an important role in the AJL simulations; what if we had good reason, observationally, to think it was zero?

Cheers
Kea

 

[marcus]

...neither honest polyhedra nor honest manifolds...

Neither of the two you mention seems likely to me exactly right for spacetime. "honest manifolds" means differential manifolds. IMO they are overdetermined, not background independent enough.
you have a single dimension (the number of smooth coordinate functions) which is good all over the manifold and at every scale even the smallest.

if you try to relax that by additional superstructure you get mathematically topheavy

on the other hand the usual polyhedron idea is a SIMPLICIAL COMPLEX and that can be a hodgepodge of differerent dimension simplices joined by toothpicks. It can be totally crazy and ugly. Not at all like spacetime ought to be. So going to simplicial complexes is relaxing too much.

the PL manifold (aka simplicial manifold) is intuitively (IMHO) relaxing the restrictions just enough. It is a simplicial complex which satisfies an additional condition which gives it a degree of uniformity.

they did not seem too interesting to me in the late 1960s when I was exposed to them, but I did not have foresight clairvoyance either. Now it seems just the ticket. we should probably have a tutorial thread on simplicial manifolds. Ambjorn has some online lecture notes aimed at the grad student level.

 

[marcus]

... the cosmological constant appears to play an important role in the AJL simulations; what if we had good reason, observationally, to think it was zero?

that is an excellent question. I believe that CDT is falsifiable on several counts.

I think this is one. If one could show that Lambda was exactly zero then I THEENK that would shoot down CDT.

In other words CDT predicts, and bets its life, on a positive cosmological constant. At least in its present rather adolescent form. this is only my inexpert opinion.

I happen to find theories interesting which risk prediction and bet the ranch on various things, the more the better because it gives experimentalists more to do.

I kind of think that finding evidence of spatial discreteness or a minimal length would ALSO shoot down CDT. well there is enough here for several conversations. I have another chorus concert tonight and must leave soon

 

[Kea]

Nice to see you here too, selfAdjoint.

Marcus, if you will allow me, I can give you a rough idea why the 'classical spacetime' limit produces causality from a more fundamental concept of observable:

F.W. Lawvere pointed out some time ago (1973) that the non-negative reals (plus infinity) form a nice symmetric monoidal category. A metric space may be thought of as a construction based on this category. $\mathbb{R}^{+}$ is used here in the same way that the category $\mathbf{2}$ of one non-identity arrow is used to construct posets. In other words, the two objects of $\mathbf{2}$ somehow represent the two values, true and false, of classical logic. Standard quantum logic, as we all know, relies on a principle of superposition and the replacement of a 2 element set by a number field. That is, we must introduce negative quantities, which forces the possibility of pseudo-Riemannian metrics.

More on zero $\Lambda$ later.

Cheers
Kea

 

[marcus]

...That is, we must introduce negative quantities, which forces the possibility of pseudo-Riemannian metrics.
...

it is difficult to apply this to what I am interested in Kea, because in CDT there is no pseudo-Riemannian metric in sight and I don't know of anyone, certainly not the CDT authors, who wants there to be.

there is no differentiable manifold in sight for such a metric to be defined on. so what use? maybe you have some non-standard construction in mind.

so something that "forces the possibility" of such a metric does not appear relevant to CDT, even if it was, as you say, discovered in 1973.

I want to park my old sig. get back to it later.
CDT http://arxiv.org/hep-th/0105267 , http://arxiv.org/hep-th/0505154
GP http://arxiv.org/gr-qc/0505052
Loops05 http://loops05.aei.mpg.de/index_files/Programme.html
CNS http://arxiv.org/gr-qc/9404011 , http://arxiv.org/gr-qc/0205119

concert went well, lot of fun. unfortunately it is now summer break

 

[selfAdjoint]

it is difficult to apply this to what I am interested in Kea, because in CDT there is no pseudo-Riemannian metric in sight and I don't know of anyone, certainly not the CDT authors, who wants there to be.

there is no differentiable manifold in sight for such a metric to be defined on. so what use? maybe you have some non-standard construction in

At the triangulation level they don't have pseudo-Riemannian, but that is what their "causality" does; it leads to pseudo-Riemannian in the continuum limit. No?

Kea pehaps you should start a new thread about these ideas? They are much worth looking at, but not so much under the rubric of CDT.

 

[marcus]

At the triangulation level they don't have pseudo-Riemannian, but that is what their "causality" does; it leads to pseudo-Riemannian in the continuum limit. No?
...

Obviously it does lead to pseudo-Riemannian if you have a differential manifold to put the metric on that is what pseudo-Riemannian is all about!

but we do not know that the continuum limit is a differentiable manifold

I thought I made that clear. the continuum limit of quantum theories of simplicial geometry may be a new type of continuum

it may not be just some old differential manifold like we have been playing physics with since 1850. in fact this is what the CDT authors work INDICATES, because they get things happening with the dimension, in the continuum limit, which do not happen with diff. manif.

in other words the CDT technique is a doorway to a new model of continuum which gives us some more basic freedom in modeling spacetime

and a pseudoRiemannian metric is a specialized gizmo that works on vintage 1850 continuums and not on the new kind---that is how it is defined---so it is irrelevant

however it should certainly be fun to study and learn about the new kind of continuum, and there is a lot of new mathematics for PhD grad students to do here

 

[Kea]

...but we do not know that the continuum limit is a differentiable manifold

Marcus,

In our approach we don't assume differentiable manifolds either. I was just trying to make the point that by putting causality in by hand you cannot possibly be doing something as fundamental as is required, IMHO. Actually, Lawvere is discussing generalised metric spaces. Forget the manifolds. In CDT they talk about lengths. What kind of a mathematical beast is that?

selfAdjoint, at some point I'll update the "Third Road" with these causality issues.

Kea

 

[kneemo]

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.

Excellent point selfAdjoint. The analogy to lattice QCD is accurate, and it is well known these lattice techniques are problematic. As an alternative to the lattice, Snyder proposed choosing a sphere instead with noncommuting position operators. The supersymmetric extension then becomes de Sitter superspace (hep-th/0311002). Mathematically, Snyder's sphere (and its generalizations) amount to higher-dimensional versions of the fuzzy sphere of Madore, so are inherently NCG.

 

[marcus]

I am still waiting for Kea or kneemo to carefully show a connection between NonComGeom and CDT. I find both interesting and I would like to be shown a rigorous connection, with page references in online articles.

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
...

Kea, you apparently have been telling me for a long time that there is a rigourous connection between CDT and NCG. I don't remember our ever discussing CDT at all, certainly not over the course of "a long time".

Please find me some links to your earlier posts that connect NCG and CDT.

For my part, I have initiated some NonComGeom threads, in part because the subject is interesting to me, but have not talked about Causal Dynam. Triang. in those threads.

If there is indeed a REAL connection (not just superficial verbal slop-over) between the two fields, that might be interesting. So please show it if you can. and make the connection simple step by step, as in a proof by Euclid, avoiding vagueness like the plague.

 

[marcus]

Kea post #55

...
F.W. Lawvere pointed out some time ago (1973) that the non-negative reals (plus infinity) form a nice symmetric monoidal category...That is, we must introduce negative quantities, which forces the possibility of pseudo-Riemannian metrics.
...

later Kea post

In our approach we don't assume differentiable manifolds either. I was just trying to make the point that by putting causality in by hand you cannot possibly be doing something as fundamental as is required, IMHO. Actually, Lawvere is discussing generalised metric spaces. Forget the manifolds.

pseudo-Riemannian metrics live on manifolds

"metric" on a metric-space is entirely different from "metric" on a manifold as all the mathematicians here know---just an unfortunate superficial verbal similarity----the metric on a manifold is defined on pairs of tangent vectors, not on pairs of points in the continuum

Kea please try to be less vague, do not jump around so much, and give online sources. If Lawvere work seems so important to you find some contemporary online exposition you can show us.

So, Kea, you were at first talking pseudo-Riemannian, and manifold, and then whoops you were not talking about manifolds, so forget manifolds.

You say:
"In our approach we don't assume differentiable manifolds either."

I am glad for you that you don't assume differentiable manifolds. Also I am happy that you (plural) have an approach. What is it an approach to? Who is "we"?

Are you collaborating with kneemo on an approach to quantizing general relativity, perchance?

That would be very nice.

 

[nightcleaner]

Excellent point selfAdjoint. The analogy to lattice QCD is accurate, and it is well known these lattice techniques are problematic. As an alternative to the lattice, Snyder proposed choosing a sphere instead with noncommuting position operators. The supersymmetric extension then becomes de Sitter superspace (hep-th/0311002). Mathematically, Snyder's sphere (and its generalizations) amount to higher-dimensional versions of the fuzzy sphere of Madore, so are inherently NCG.

Hi all

Given a simplex of any sort, be it a line segment or a triangle or a square or polyhedra of any order or polygon or (I presume) a simplex of any dimension,

rotate it around any chosen point to every degree of every possible dimension,

pick another point and repeat the universal rotation,

repeat this process until a representative sample of points in the simplex has been the chosen center of rotation,

call the space of all rotations possible to the simplex the universal rotation space of the simplex,

plot the density of each point in the universal rotation space of the simplex as a function of how many times that point is occupied by a structural member of the simplex in one rotation,

I postulate that the universal space of the simplex under the described rotations will always be a spherical analog in any dimension,

and that the universal space of any simplex other than a zero dimensional point will exhibit a unique spectrum of discontinuous densities in cross section or as a function on any radial line.

I propose that the universal rotation space of simplexes be cataloged and that their spectra be analyzed for dual relationships with quantum observables.

I predict that 1)The universal rotation space of simplices will be easier to use in calculations and as an element in timespace models than CDT; 2)The universal rotation space of simplices will be found to correspond to observable states of the many-body problem (Elementary nuclei down to the Planck scale) and 3)The sum of all universally rotated simplices will be a smooth continuum corresponding to a flat spacetime which can be approximated down to a few Planck lengths by a dense pack isomatrix composed of Planck radius spheres. At smaller scales the isomatrix breaks down into the tetrahedrons, triangles, and squares of the individual n-dimensional simplexes where n = (1,2,3).

(Of course I also predict that someone will step forward who wants to pay me and my research team to do this work!)

Any comments welcome. Marcus, is this the sort of thing you meant when you suggested that grad students would find new mathematical toys to play with behind CDT? If only I were a grad student. Oh well.

Thanks, and be well,

Richard

 

[marcus]

selfAdjoint, you replied to what I said here

... 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

your reply went in part

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.

a lot of mathematics including basic calculus uses the technique of setting something up with a parameter 'a', or 'h' or epsilon, and then letting it go to zero.

a lot of calculation all over physics and engineering uses lattices and let's the scale go to zero.

Are you saying that Ambjorn and Loll have not been innovative because they also have some parameter go to zero?

As mathematicians we realize the need to be definite and avoid handwaving and passing bad checks, at least some of the time. So how about being definite with me about what you see as the similarity between QCD and CDT.

Is it not the case that all kinds of quantum field theories are defined on Minkowski flat, or on a manifold? And is it not the case that one sets up a lattice that approximates that (say) manifold, with a finite cutoff, and calculates? And the limit, making the grid fine, is supposed to represent what you would get from the field on the manifold.

what I see here has little (except for superficial) resemblance to CDT. what I see is basically calculating with some function defined on some fixed static manifold----and approximating by looking at a grid of dots.

if, in your picture, you want the manifold itself to change shape, then you have immediately to appeal to its coordinate patches and the machinery of differential geometry.

What I see in CDT is that there are no coordinate functions and the shape of the manifold (not something defined on a fixed manif) is what is important, and the shape depends on HOW THE IDENTICAL BLOCKS ARE GLUED TOGETHER and is meansured not by diff.geom machinery by by combinatorics----by counting.

This much is DT, and it got started (according to a history by Loll) around 1985. That is already pretty revolutionary-----it is a new kind of continuum which breaks with the 1850 tradition of differential geometry.

Revolutions proceed by fits and starts, or by stages. DT became CDT in 1998 and that may not look so big to you, we will see who has the right perspective.
But in my view you cannot dismiss any of this by waving it off as just more latticework
It is a new kind of dynamic continuum, it is not just more lattice-QCD, it gives a new model of spacetime. The limit is not a differentiable manifold. As far as we know the limit does not have coordinate functions. The reason I can see that we are dealing with something new is because I CAN SEE THERE ARE A LOT OF THEOREMS TO BE PROVED HERE. that is a measure of how fundamental or new some territory looks. the rest is self-deluding glibness "oh that is just this, oh that is just derived from that".
If people who really know what they are talking about say that kind of thing then it is wonderful, and it means they can prove something. But otherwise just glib empty chatter. And if you CANNOT see that there are basic theorems to be proved in CDT territory, then of course it looks small to you. We are talking about subjective impressions based on our differences mathematical intuition.

 

[selfAdjoint]

Comments on the simplex rotation idea

Given a simplex of any sort, be it a line segment or a triangle or a square or polyhedra of any order or polygon or (I presume) a simplex of any dimension,

Simplices (or simplexes) are ONLY the triangle-kind of things. No squares or other polyhedra. To make an n-simplex, take an (n-1)-simplex and a point not on it in the new direction and draw all the lines from the point to the (n-1)-simplex; the result is your n-simplex. Alternatively take the set spanned by the unit vectors along the n axes of some basis in n-space, and that's an n-simplex. No tesseracts need apply.

rotate it around any chosen point to every degree of every possible dimension,

The number of possible dimensions is infinite. Every time you think you've reached the last n, you realize you can make another; n+1. If you rotate about every point (inside it?), that's a continuum of centers, and -> infinite dimensions gives you a continumm cross the integers different ways to turn, so I don't know what you wind up with, but it sure ain't surveyable.

pick another point and repeat the universal rotation,

repeat this process until a representative sample of points in the simplex has been the chosen center of rotation,

What do you mean, "representative sample"?

call the space of all rotations possible to the simplex the universal rotation space of the simplex,

So far i think it's still of cardinality c. I could be wrong, though.

plot the density of each point in the universal rotation space of the simplex as a function of how many times that point is occupied by a structural member of the simplex in one rotation,

By "structural member" you men the (n-1)-skeleton of (n-1)-hyperfaces, (n-2)-hyperfaces,...,faces, edges, and vertices? For every point inside the n-simplex, this count will be infinite. Because some rotation can be factored into two, the first of which brings the point into some cell of the skeleton and the second of which is such as to make the point describe a small circle within the cell, so every point on that small circle is a count by your definition.

I postulate that the universal space of the simplex under the described rotations will always be a spherical analog in any dimension,

I don't think I can pin down exactly what you mean here. What is a "spherical analogue"?

and that the universal space of any simplex other than a zero dimensional point will exhibit a unique spectrum of discontinuous densities in cross section or as a function on any radial line.

No, as I showed above, there will be a continuum of such points.

I propose that the universal rotation space of simplexes be cataloged and that their spectra be analyzed for dual relationships with quantum observables.

In so far is this is a well-defined idea, I propose to you that the "catalog" is in fact the group SO(n), as n -> infinity. This is a very interesting object, or class of objects, and for example the Yang-Mills theories for such groups have been intensively studied.

I predict that 1)The universal rotation space of simplices will be easier to use in calculations and as an element in timespace models than CDT; 2)The universal rotation space of simplices will be found to correspond to observable states of the many-body problem (Elementary nuclei down to the Planck scale) and 3)The sum of all universally rotated simplices will be a smooth continuum corresponding to a flat spacetime which can be approximated down to a few Planck lengths by a dense pack isomatrix composed of Planck radius spheres. At smaller scales the isomatrix breaks down into the tetrahedrons, triangles, and squares of the individual n-dimensional simplexes where n = (1,2,3).

I am totally unable to evauate these speculations. For Richard, you know that's what they are.

(Of course I also predict that someone will step forward who wants to pay me and my research team to do this work!)

Best of luck to you on funding!

Any comments welcome. Marcus, is this the sort of thing you meant when you suggested that grad students would find new mathematical toys to play with behind CDT? If only I were a grad student. Oh well.

Thanks, and be well,

Richard

 

[selfAdjoint]

Are you saying that Ambjorn and Loll have not been innovative because they also have some parameter go to zero?

Did I say one thing about innovative? The question I had was about BASIC. But since you bring up innovative the major innovation of AJL was the causality; everything else had been done previously by people working on the Regge triangulation project. Oh yes, and the Monte Carlo simulations too.

 

[Kea]

pseudo-Riemannian metrics live on manifolds;
"metric" on a metric-space is entirely different from "metric" on a manifold as all the mathematicians here know...
Kea please try to be less vague, do not jump around so much, and give online sources. If Lawvere work seems so important to you find some contemporary online exposition you can show us...

Hi Marcus

OK. I apologise for using the word pseudo-Riemannian when I shouldn't have. Unfortunately, I don't know of ANY easy expository notes, online or otherwise, on Lawvere's ideas - although Baez has mentioned him a few times on his website. The article I refer to is available at

http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html

...still planning to look into this branched polymer connection...
Cheers
Kea

 

[marcus]

Did I say one thing about innovative? The question I had was about BASIC. But since you bring up innovative the major innovation of AJL was the causality; everything else had been done previously by people working on the Regge triangulation project. Oh yes, and the Monte Carlo simulations too.

What I heard you say was

... 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.

there is a significant difference between the QCD lattice strategy and the whole Regge program, and the Dynamical Triangulation program after about 1985 (Ambjorn was a major figure in that and wrote the book, Cambridge UP, on it). And then CDT in 1998. this whole development is basic and revolutionary. It is not all due to those 3, certainly! and it did not happen all at once in 1998, for sure!

but it is not to be confused with ordinary lattice field theory, or other types of calculation on a fixed lattice (essentially a discrete approximation of a manifold).

You seem to wish to minimize the change that this represents, getting off of differentiable manifolds. I would be pleased if you would read and comment on the rest of my previous post.

you were comparing dynamical triangulation modeling of spacetime with lattice field theory:

...

what I see here [that is, lattice QCD] has little (except for superficial) resemblance to CDT. what I see is basically calculating with some function defined on some fixed static manifold----and approximating by looking at a grid of dots.

if, in your picture, you want the manifold itself to change shape, then you have immediately to appeal to its coordinate patches and the machinery of differential geometry.

What I see in CDT is that there are no coordinate functions and the shape of the manifold (not something defined on a fixed manif) is what is important, and the shape depends on HOW THE IDENTICAL BLOCKS ARE GLUED TOGETHER and is meansured not by diff.geom machinery by by combinatorics----by counting.

This much is DT, and it got started (according to a history by Loll) around 1985. That is already pretty revolutionary-----it is a new kind of continuum which breaks with the 1850 tradition of differential geometry.

Revolutions proceed by fits and starts, or by stages. DT became CDT in 1998 and that may not look so big to you, we will see who has the right perspective.
But in my view you cannot dismiss any of this by waving it off as just more latticework
It is a new kind of dynamic continuum, it is not just more lattice-QCD, it gives a new model of spacetime. The limit is not a differentiable manifold. As far as we know the limit does not have coordinate functions. The reason I can see that we are dealing with something new is because I CAN SEE THERE ARE A LOT OF THEOREMS TO BE PROVED HERE. that is a measure of how fundamental or new some territory looks. the rest is self-deluding glibness "oh that is just this, oh that is just derived from that".
If people who really know what they are talking about say that kind of thing then it is wonderful, and it means they can prove something. But otherwise just glib empty chatter. And if you CANNOT see that there are basic theorems to be proved in CDT territory, then of course it looks small to you. We are talking about subjective impressions based on our differences mathematical intuition.

 

[marcus]

Hi Marcus

OK. I apologise for using the word pseudo-Riemannian when I shouldn't have...

Thanks Kea, I am downloading the Lawvere, and will see if there is any conceivable connection with Causal Dynamical
Triangulations, even a very remote one.

I am still waiting for you to substantiate the claim about a connection between NonComGeom and CDT. I find both interesting and I would like to be shown a careful derivation of CDT from NCG, with page references in online articles, as per your and kneemo posts:

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
...

Kea, you apparently have been telling me for a long time that there is a rigorous connection between CDT and NCG. I don't remember our ever discussing CDT with you at all, before this, certainly not over the course of "a long time".

Perhaps you would like to move this discussion to selfAdjoint's new thread, which seems ideally suited for it!

cheers

 

[Kea]

Kea, you apparently have been telling me for a long time that there is a rigorous connection between CDT and NCG.

No, Marcus, I never said that. But I believe that NCG easily consumes CDT, and I'm hoping one of us will eventually convince you of this. What I have been trying to tell you is about some of the features that a decent approach to QG ought to have, and that CDT is clearly lacking.

Cheers
Kea