Another Constraint on Quantum Gravity

[selfAdjoint]

This paper by Mohammed Ansari and Lee Smolin, http://arxiv.org/abs/hep-th/0412307, seems to have been overlooked here. The authors argue that for any quantum theory of gravity (they mention LQG and causal triangulations) to produce macroscopic physics - which would be necessary for it to have GR as a limit - it has to obey a certain criterion on its renormalization group flow that is of measure zero in that configuration space. That is, the odds against this criterion being met randomly are infinity to 1. Then Ansari and Smolin try to set up an evolutionary path within complex systems theory to bring the constraint about naturally. They aren't able to prove it happens, but they show a happy outcome in a very simple example.

 

[marcus]

This paper by Mohammed Ansari and Lee Smolin, http://arxiv.org/abs/hep-th/0412307, seems to have been overlooked here. The authors argue that for any quantum theory of gravity (they mention LQG and causal triangulations) to produce macroscopic physics...

let me check it out;

Self-organized criticality in quantum gravity
Mohammad H. Ansari, Lee Smolin
9 pages, 9 figures

"We study a simple model of spin network evolution motivated by the hypothesis that the emergence of classical space-time from a discrete microscopic dynamics may be a self-organized critical process. Self organized critical systems are statistical systems that naturally evolve without fine tuning to critical states in which correlation functions are scale invariant. We study several rules for evolution of frozen spin networks in which the spins labelling the edges evolve on a fixed graph. We find evidence for a set of rules which behaves analogously to sand pile models in which a critical state emerges without fine tuning, in which some correlation functions become scale invariant."

 

[marcus]

---exerpt from introduction arXiv:hep-th/0412307 v3---

Since the work of Wilson and others [1] it has been understood that the existence of a quantum field theory requires a critical phenomena, so that there are strong correlations on scales of the Compton wavelength of the lightest particle. If this scaleis to remain fixed as the ultraviolet cutof flength is taken to zero, the couplings must be tuned to a critical point, so that the ratio of the cutoff to the scale of the physical correlation length diverges. This requires asymptotic scale invariance of the kind found in second order phase transitions.

Similar considerations apply to quantum gravity in a background independent formulation such as loop quantum gravity, or causal set models. The problem is not alleviated if the theory is shown to be finite due to there being a physical ultraviolet cutoff, as in loop quantum gravity[2]. Instead, the need for a critical phenomena is even more serious as there is no background geometry.

This means that away from a critical point the system may not have any phenomena that can be characterized by scales much longer than the Planck length. That is to say, the volume, measured for example, by the number of events, may become large, but there may still be no pairs of events further than a few Planck times or lengths from each other.
[my comment: this is the crumpling infinite hausdorff dimension case often found by the DT people in the 90s]

This is seen in detail in models whose critical phenomena has been well studied, such as dynamical triangulation models[3] and Regge calculus[4]. Away from possible critical points, the average distance between two nodes or points need not grow as the number of events (or the total space-time volume) grows. Instead, one sees that for typical couplings, statistical measures of the dimension, such as the hausdorff dimension, can go to infinity or zero.

---end quote---

the problems described here are familiar from DT papers in the 90s. This description could almost be taken verbatim from an Ambjorn and Loll paper some years back

" the average distance between two nodes or points need not grow as the number of events (or the total space-time volume) grows. Instead, one sees that for typical couplings, statistical measures of the dimension, such as the hausdorff dimension, can go to infinity or zero."

Ambjorn and Loll in almost every paper for some years would always draw pictures of the two kinds of pathological behavior (this was before Lorentzian DT): either the thing would CRUMPLE and have hausdorff dimension infinity with some nodes that almost every other node was directly connected with
or it would get all FEATHERY fractally and have hausdorff dimension less than what it was supposed to----Smolin says zero but it could also be, like, two when it was supposed to be four.

It was only in 2004 that Ambjorn et al showed that they were now getting hausdorff dimension 4 in their Monte Carlo runs, that is, what Smolin talks about was NOT happening.

before 2004 they had gotten nice behavior in lower dimension toy models, where the dimensionality came out the way it was supposed to and this
generic crumply/feathery behavior didnt happen. but last year was the first time they succeeded with a full 4D model.

I believe this may be a sign that now LQG can successfully tackle a similar problem----so it is an opportune moment for Smolin and others to try. Unless they want to all hop the fence and work on DT instead.

He also mentions Fotini Markopoulou's "Causal Sets" approach in the same breath as LQG. he says:
"Similar considerations apply to quantum gravity in a background independent formulation such as loop quantum gravity, or causal set models."
Causal Sets is not to be confused with Ambjorn Loll "Causal Dynamical Triangulations". Causal Sets is not a simplicial QG model but a kind of graph theory where you have Hilbert spaces at the nodes and Hilbertspace mappings from node to node along the links-----very abstract algebra. Causal Sets seems at the opposite end of the spectrum from Ambjorn Loll CDT stuff. It is not a primitive simplicial model that one could simulate in a computer.
However Smolin may be right to include a mention of it here. There may be some way in which the recent progress in CDT opens up the possibility of some new results in Causal Sets as well.

Another outcome is that this paper by Smolin Ansari will help generate a move of people from LQG to Dynamical Triangulations. But as I have suggested I think it is just as likely that they will do some re-building in LQG in order to match CDT.

obviously too early to be speculating like this but how else are we going to voice our responses to the paper?

 

[marcus]

Uh oh!
it looks more and more (reading the paper) like what you said about the merger of Sears and Kmart
smolin is doing triangles (which would be tetrahedra in the next iteration)

 

[Rybo]

triangulated gravity

Uh oh!
it looks more and more (reading the paper) like what you said about the merger of Sears and Kmart
smolin is doing triangles (which would be tetrahedra in the next iteration)

Marcus, you cna have tringualre polyhedra that are not tetrahedrons.

Triangulated gravity is not news to me. My Rybonic Extrpolations of Synergetics 1 and 2 has propose this for a few years now.
http://home.usit.net/~rybo6/rybo/index.html

Fuller has suggested that gravity is icosahedral for much longer time than I have.

Whats is new is my overlapping of the5-fold icoshadra to define 4-fold tetrrahedra and octahedra.

The other news is two fold.
1) that of Jim Lehmans only documented discovery of the 3D graphical depiction of the overlapping hexagonal "flower of life pattern" is also the curved version of Fullers Euclidean-only Isotropic Vector Matrix.

2) and that the 4-fold triangulated hexahedrons(Pods) may be the correct goemtrical representation of either a graviton(spin-2 boson) or a Higgs boson.

I think the former #1 is the correct and that the Higgs particle can be reprasented as a clustering of these triangulated pods as 4-fold tetrahedra or octahedra. Maybe both. I dunno.

Fuller often stated taht it was teh artist-mathematicians hwo made discoverys before the phyiscla science.

Fuller stated that he was using the term "spin" with great circle creation years before the particle scientist did so for subatomic particles.

Rybo

 

[Wave's_Hand_Particle]

This paper by Mohammed Ansari and Lee Smolin, http://arxiv.org/abs/hep-th/0412307, seems to have been overlooked here. The authors argue that for any quantum theory of gravity (they mention LQG and causal triangulations) to produce macroscopic physics - which would be necessary for it to have GR as a limit - it has to obey a certain criterion on its renormalization group flow that is of measure zero in that configuration space. That is, the odds against this criterion being met randomly are infinity to 1. Then Ansari and Smolin try to set up an evolutionary path within complex systems theory to bring the constraint about naturally. They aren't able to prove it happens, but they show a happy outcome in a very simple example.

I have been reading the Dec 27TH print-paper, I have downloaded the present print, but cannot see any revision at all?

The spin network embedding has an interesting aspect for 'ROTATIONS' caused by the overlapping triangulated network!

Quote: Einstein called the warping of spacetime to be 2-D.

A third rotation (one of the line's) from the solid lines of trianglated triangle, produces a 'double spin' of whatever it encompasses?

Another dimensional aspect for the quantum field(2-D) to be the outer energy of all 3-Dimensional spacetimes?..maybe?

There is an error contained in the paper on the 'critical behavior' of self orginization, the authors neglect the fact that all equilibriated systems are discrete, and any dynamical evolution will destroy 'equilibrium'?

Any system in equilibrium is as Einstein stated, in total isolation.

 

[selfAdjoint]

There is an error contained in the paper on the 'critical behavior' of self orginization, the authors neglect the fact that all equilibriated systems are discrete, and any dynamical evolution will destroy 'equilibrium'?

Any system in equilibrium is as Einstein stated, in total isolation.

By "dynamical equilibrium state" they surely mean the kind of state they have been discussing, analogous to the sandpile. An initial equilibrium state, which changes by discrete dynamic events to a different equilibrium.

 

[Spin_Network]

By "dynamical equilibrium state" they surely mean the kind of state they have been discussing, analogous to the sandpile. An initial equilibrium state, which changes by discrete dynamic events to a different equilibrium.

I have seen a 'self-organized' experiment where a wheel containing sand is angled and slowly rotated. Over Time the grains organize themselves into distinct patterns, I wonder if anyone knows of a good site that has a similar experiment available?

I would be interested on the geometric make-up of specific 'mixed' structured states, ie not just one geometric componant such as sand, but one that can be compared to Triangular and Hexagonal structures?

Thanks.

 

[selfAdjoint]

From Ansari & Smolin

If this scaleis to remain fixed as the ultraviolet cutof flength is taken to zero, the couplings must be tuned to a critical point, so that the ratio of the cutoff to the scale of the physical correlation length diverges. This requires asymptotic scale invariance of the kind found in second order phase transitions.

Ambjorn and Loll in almost every paper for some years would always draw pictures of the two kinds of pathological behavior (this was before Lorentzian DT): either the thing would CRUMPLE and have hausdorff dimension infinity with some nodes that almost every other node was directly connected with
or it would get all FEATHERY fractally and have hausdorff dimension less than what it was supposed to----Smolin says zero but it could also be, like, two when it was supposed to be four.

It was only in 2004 that Ambjorn et al showed that they were now getting hausdorff dimension 4 in their Monte Carlo runs, that is, what Smolin talks about was NOT happening.

before 2004 they had gotten nice behavior in lower dimension toy models, where the dimensionality came out the way it was supposed to and this
generic crumply/feathery behavior didnt happen. but last year was the first time they succeeded with a full 4D model.

I believe this may be a sign that now LQG can successfully tackle a similar problem----so it is an opportune moment for Smolin and others to try. Unless they want to all hop the fence and work on DT instead

If I understand what you are saying Marcus, you are not claiming that Ambjorn et al. have solved or even directly addressed the renormalization group issues that Ansari & Smolin raise. Rather you assert that the Ambjorn et al. achievement of a stable Hausdorff dimension, suggesting a fractal structure (but not proving it, you alsoo need self-similarity for fractals), gives hope that their approach may satisfy the RG fixed point constraint. If so, I don't see your reasoniing. Could you elucidate?

 

[marcus]

If I understand what you are saying Marcus, you are not claiming that Ambjorn et al. have solved or even directly addressed the renormalization group issues that Ansari & Smolin raise. Rather you assert that the Ambjorn et al. achievement of a stable Hausdorff dimension, suggesting a fractal structure (but not proving it, you alsoo need self-similarity for fractals), gives hope that their approach may satisfy the RG fixed point constraint. If so, I don't see your reasoniing. Could you elucidate?

the earlier pathologies that plagued simplicial gravity models until around 1998 were that the LARGESCALE dimension, instead of being 4D as it should, would turn out to be either 2 or infinity (roughly speaking)

I have a concert and must run, but basically the spacetimes would turn out either MASSIVELY CRUMPLED or else SPREADOUT FEATHERY.

this has nothing to do with solving the renormalization problem by having dynamical dimension going down to around 2 at small scale

this was a pathology in the large scale dimension and a cure for it was found in 1998.

since then they have been gradually checking the model out starting at 2D and working up to 3D and then, last year, in 4D.

You have already discussed the renormalization business yourself. that dynamical dimension business is very new. the largescale dimension is 4D but as you noted when I first posted about the "spectral dimension" paper there is this microscopic 2D dimensionality

no it is obviously not a usual kind of "fractal" although it can have fractional dimension and it can be related to fractal-like behavior. but it cannot strictly be a fractal because it is not selfsimilar----not the same at all scales.

In a rush so will leave you with the job of straightening out details as needed. fun isn't it , great stuff!

 

[selfAdjoint]

no it is obviously not a usual kind of "fractal" although it can have fractional dimension and it can be related to fractal-like behavior. but it cannot strictly be a fractal because it is not selfsimilar----not the same at all scales.

Hausdorff dimension can belong to other spaces than fractal ones. Indeed it corresponds to the usual dimension where that is defined.

My oversimplified description of of an RG fixed point is that the physics (say the action and lagrangian) is scale invariant near a given scale. If a geometry were fractal (globally self-similar at ALL scales) and the geometry necessarily detemined the action, as in most GR actions, then it would seem there was no fixed point, no PARTICULAR scale near which scale-invariance holds. But Ansari & Smolin seem to assert (I am going to reread their paper) that a fixed point is a necessary feature of any QG theory?

As you say, the current CDT theory, with its running spectral dimension, is not fractal. They don't appear to use a cutoff, either.

 

[nightcleaner]

Hi selfAdjoint

Not fractal because not self-similar at all scales? Scales are properties imposed by the observer. Rounding errors, again imposed by the observer, lead to a "washing out" of the fractal pattern after a few hundred magnification steps. In the other direction, entire fractal sets receed to a single point when the scale taken is very large. These charachteristics of fractal patterns appear to me to be very much like observed behaviors of spacetime phenomenology approaching the Planck scale. The fractal pattern appears to me to be clearly defined as occupying one region of a scale continuum, with an upper bound set by the scale of the entire fractal set under observation, and the lower bound set by the computational power of the observer.

The feathering, layering, foliation, time-like volumes, bones, simplices, triangles, edges, vertices, tetrahedrons, whatever, is due to the movement of the observer and related reference frame through timespace, and it is the timespace which is patterned fractally. In a sense, the observer selects self-similar regions of the fractal pattern and experiences them sequentially.

 

[spicerack]

ummm...what causes triangulations to become dynamic ?

 

[Rybo]

ummm...what causes triangulations to become dynamic ?

Triangulation is just the minimal trajectoral encircling of and area due to mass-attraction.
---------
“It is the persuasion of action( motion ), to attract attention( observation ) of another action( motion ). This is gravity.” (Rybo)

Why does mass attract?

Obviously, to cohere all of physical Universe, ergo no ultimate entropic annihilation of radiation a.k.a “heat death” of Universe.

Universe, as the only perpetual motion machine, is perpeutated by, a oscillating between a circumferential, postive octagonal rippling of space, to a inter-mediate hexagonal reippling of space, to a negative octagonal rippling of space, over time, and back again.

This phemomena is incurred by, crossing points of energy --[ or quasi-energy ]-- having three differrentiations in mass that defines it geometry and is modeled using R.B.Fullers Jiiterbugging cubo-octahedron a.k.a. the vector equlibrium.

The hexagonal ripple, has four kinds of mass point/crossings.

1) 6 single-mass points
2) 1 double-mass point
3) 1 quadrubple-mass point

The hexagonal ripple has 5 single mass points plus the one double-mass point defineing its chordal circumference of the hexagon adn 1 set of quadruple mass points as the toroidal nuclear center of the hexagon.

The octagonal ripple hs 8 single mass points defining its chordal circumference and the 1 quadruple mass point at the toridal nuclear center. This octagonal ripple is aslo known as the “saddle shape” curvature.

http://www.rwgrayprojects.com/synergetics/s04/p6000.html
Scroll to 460.1 and 461.08 for Fullers minimal graphic depiction.

Unfortunatly Fuller did not explore at least he did not publish a more complete set of geometric configuration attainaabe with his jbug model.

E.g.
1) the sine-wave topology ( a.k.a triangualar wave form )

2) the saddle-shape topolgy( negative curvature )

3) the octagonally square ripple ( postive and negative )

4) the hexagon ( flatten torus with tetrahedral nucleus )

http://home.usit.net/~rybo6/rybo/id8.html

Rybo

 

[nightcleaner]

ummm...what causes triangulations to become dynamic ?

Hi Spicerack

Good question. I'll give you a philosophical answer, if you care to read it. Your question reduces down to asking what time is. It seems clear to me that this is a crucial question. Assuming that there are spatial triangulations which behave statistically within definable limits in a time frame, such that we may interpret the flux of time as a sequence of spatial foliations separated by chronic regions. The chronic parts of the idyll are not layers, exactly, because the time-like volumes of the bones are variables. There can be a lot of time between adjacent spatial surfaces or the spatial surfaces can actually be in instantaneous contact.

We might ask if there is a Casmir-like effect between foliations, requring a minimization of time volume, but this would have to occur over time, which involves a second time dimension. Time volume changes over time you see.

Richard

 

[nightcleaner]

Hi Rybo
I think that Fuller had the right idea with the isomatrix, a geometry of densly packed spheres. Others on this forum might have hoped I had forgotten that idea, or at least abandoned it. Nope, sorry.
Richard

 

[Rybo]

LQGeometry

Hi Rybo
I think that Fuller had the right idea with the isomatrix, a geometry of densly packed spheres. Others on this forum might have hoped I had forgotten that idea, or at least abandoned it. Nope, sorry.
Richard

Richard, The spherical clsoe-packing, isotropic vector matrix( IVM ) is a 4-fold nature( regular tets and octs ) a baryon and leptosn respectivly.

What I try to show in my Rybonic website is that there exists a 5-fold matrix of regular icosahedra( gravity ) that bond to each tet in such a way that that each tet is internal to 4 of the bonding icosahedra and external to 4 of the bonding icosahedra.

That is the simple explanation, however it is of corse much more complicated than that and more complex I have explantions for at this time.

Suffice it to say, if Loop Quantum Gravity is validated/confirmed in 2006-7 then I thnk Rybonics that is that much closer to also falling onto the quantum geometry bandwagon.

Rybo

 

[spicerack]

Hi guys

once again pardon my ignorance but it still sounds like we're still talking about bubbles. In an ideal situation they would be prefectly spherical but in a foam they can have a multitude of sides shared with others like connected soccerballs

Does the search for an ideal shape/fundamental object making up spacetime require it to be an ideally fixed shape or is the dynamically changing shape due to the passage of time and the uncertainty principle meaning it is what it appears to be only when we put a light on it and observe it ?

What is it that LQG, strings And CDT all have in common besides the mahtematics or is it too early to say ?

 

[nightcleaner]

Hi Spicerack

I donno.

It sounds like bubbles to me too. Marcus says they take a simplical structure and run Monte Carlo diffusion tests on it...average number of steps to return. Then they change the simplical structure and do it again. Eventually they have a representative sample, I guess, of the space of all possible geometries.

Seems to me if you take any structure, simplical or not, even a single bone, string, edge, whathaveyou, and run it through every possible invarient transform, or even a big enough sample of the possible invarient transforms, the limits on the set of transforms will surely be spherical. Instead of doing all the statistics and repeat tests and so on, why not just assume a spherical "simplex" to begin with?

Unless of course they find that the limits on the set of transforms turns out not to be spherical. I havn't done any calculations, it just seems to me that diffusion tests as described "ought" to vary pretty equally in every direction from some average middle.

We shall see. Marcus?

Richard

 

[marcus]

We shall see. Marcus?

hi Richard, it sounds like you are pointing to spicerack's post
so I could respond to it

What is it that LQG, strings And CDT all have in common besides the mathematics or is it too early to say ?

first of all, two extremely smart ladies name Renate Loll and Bianca Dittrich have written a CDT paper where they try DIFFERENT mix of building blox, not just simplexes. that shows you can do that in CDT. but it hasnt been tried much. the field is new. simplexes are the simplest blox.

it just happened that Loll and Dittrich found that using a certain mix including other shape building blocks facilitated what they were trying to do in that particular paper

they will soon post a new Loll-Dittrich CDT paper, this time about using CDT to model BLACK HOLES. that will a first (because the field is still very new and even the first black hole model has not been done yet) and so that will be very interesting. we will see if they try any changing of the mix of blox.

that part seems to interst you all but it does not me. for me it basically does not matter the shape of the basic cells as long as they are very simple and fit together. I actually am partial to simplexes

SPICERACK please asimilate thing one about CDT that the spacetime IS NOT MADE OF SIMPLEXES. it is a continuum. Only thing is this CDT continuum
is not describable except by a process of finer and finer approximations, by things assembled out of building blocks. you make the size of the basic block go to zero and the approximation gets better and better.

so what is made of simplex blox is is only an APPROXIMATE spacetime to what the theory says is the real spacetime

so in some sense it hardly matters what the basic block is, as long it is adequate to take care of business and accomplish the approximation.

This spicerack was not a bad question, i will repeat it:

What is it that LQG, strings And CDT all have in common besides the mathematics or is it too early to say?

Maybe someone else will jump in and answer that. I do not know. right now
I AM INTERESTED IN THE OPPOSITE QUESTION namely what is about CDT that is radically DIFFERENT from both string and LQG. Does that interest you at all? Can you understand why I would be looking at that, for any new approach? For any new approach I think it is a good idea to see what is radically different about it.

In other approaches to quantum gravity, they tend to have the dimension of spacetime put in by hand. it is either 4 (put in by hand) or some other fool number like 11 or 17 (this not the real number but some fool number established ahead of time to get things to work right for the theorybuilder)

I suspect that theories like this, where the dimension of spacetime has to be an integer chosen and put in by hand, are TOAST

The reason these other approaches have a prior chosen spacetime dimension is simply because they are built on a mathematical object invented in 1850 called a smooth manifold. Smooth manifolds have coordinates (like x,y,z) and the number of coordinates is the dimension of the manifold.

the reason CDT is different is that it is NOT built on such a manifold, it is built on a later invention called a simplicial manifold where you DONT NEED COORDINATES and you don't have a fixed dimensionality. there are different ways to define the dimension, by performing various experiements inside the spacetime, and the dimensionality DOESNT HAVE TO BE AN INTEGER (because there are no xyz coordinates that you have to count up and have that be the dimension) and the dimensionality can VARY depending on the scale you are looking. if you look with a magnifying glass it can be different. that is the wonderful thing about simplicial manifolds. to my mind.

however you can set up Einstein gravity in a simplicial manifold. A man called Tulio Regge saw how to do that in 1950. you do it by counting the blocks that come together around the bones of the manifold (a simp manif has a web of borders of borders of the blocks and that web is called the bones and it is like a scaffolding of the manif.

do you know who Cicero was? He was a good writer of like 100 to 50 BC IIRC. his name was TULIUS. Tulio Regge was named after Cicero. Not all romans were named Julius, some were named Tulius.

CDT is exactly as different from LQG and string as 1950 is from 1850. heh heh some people might not like my saying that

 

[spicerack]

thanks Marcus, Nightcleaner and Rybo

That's more than enough information to keep me amused, confused and enlightened for quite some time

I like how you say Marcus, it's not what's the same it's what's different that makes it interesting, so yes I can understand your excitement

CDT sounds like modelling the wire frame of an object before you skin it up in a 3d computer program ?

 

[selfAdjoint]

In other approaches to quantum gravity, they tend to have the dimension of spacetime put in by hand. it is either 4 (put in by hand) or some other fool number like 11 or 17 (this not the real number but some fool number established ahead of time to get things to work right for the theorybuilder)

I suspect that theories like this, where the dimension of spacetime has to be an integer chosen and put in by hand, are TOAST

Wouldn't it be fun if this were so? But two caveats:

1) As you have posted the dimension that varies in Ambjorn et al. is the Spectral Dimension of the flow. They identify this with spacetime dimension, but is this justified. I get the impression that a low Hausdorff dimension in the flow context is not surprisng or new per se.

2) See arivero's post on the low fractal (=Hausfdorff) dimension of path integrals of \lambda \phi^4 toy quantum field theory.

 

[marcus]

...

1) As you have posted the dimension that varies in Ambjorn et al. is the Spectral Dimension of the flow. They identify this with spacetime dimension, but is this justified. I get the impression that a low Hausdorff dimension in the flow context is not surprisng or new per se.

2) See arivero's post on the low fractal (=Hausfdorff) dimension of path integrals of \lambda \phi^4 toy quantum field theory.

I have replied to arivero's post in the other thread.

though far from expert here, i have the impression that indeed just as you say it is not at all new or surprising for convedntional path integral PATHS to be very kinky and perhaps to have fractional dimension. But those paths are embedded in conventional 4D surroundings----there could also be very kinky surfaces in conventional settings, also embedded in conventional 4D spacetime.

the possible difference here is that the SPACETIME ITSELF (not some embedded object) has low fractional dimension, at certain scales. so here the kinky thing is all there is. It has no surroundings. and any matterfields must be defined IN IT, not in some conventional surrounding minkowski.

BTW as you may have noticed, they do not use the Hausdorff dimension in this recent work that we were discussing, where the dimension of spacetime is less at small scale. The Hausdorff dimension is based on VOLUME and the diffusion dimension they use is based on RETURN PROBABILITY. I was interested to see them using various different measures of dimension, which in their latest paper they point out give various different answers.

I would be glad to know how you interpret the term "Spectral Dimension". In different branches of mathematics there are dozens or maybe scores of different spectra and spectral this and that. In this case I see a heat kernel and associated Laplace operator and i see eigenvalues and I see that what they call the spectral dimension is explicitly related to these eigenvalues.
what do you mean when you say "spectral dimension of the Flow"?
What Flow are you referring to? I have been unable to find a meaning for spectral dimension outside of the statistical physics of diffusion processes.

In this paper they are not doing renormalization of any sort, they only speculate about the possibility later on, and the only flow in sight AFAIK is a simulated diffusion or random walk in 4D using an artificial time parameter. Is this diffusion (according to the heat kernel they write down) the flow you mean? Sorry if this is an obvious question, but want to make very sure.

 

[Rybo]

14 sides/faces on average

Hi guys
once again pardon my ignorance but it still sounds like we're still talking about bubbles. In an ideal situation they would be prefectly spherical but in a foam they can have a multitude of sides shared with others like connected soccerballs
What is it that LQG, strings And CDT all have in common besides the mahtematics or is it too early to say ?

Spicerack, 14 sides/faces is the average number that occur in clso packing of biological cells or any maleable cell-like structure.

The cubo-octahedron/vector equilibrium/jitterbug has 14 faces/openings( 8 triangular 6 square ) and is Fullers Euclidean based Operating System of Universe that is central to the all-space filling Isotropic Vector Matrix combination of tets and octs.

Also the tetrakaidodecahedron has 14 faces.

Im not sure what CDT is but I think that what LQG and String theory have in common besides mathematics is a Quantum Gravity scenario. LQG being diffrrent in there are soon ot be instruments in space to test its hypothsis, whereas Strings cannot be teste current and is not foreseen to be testable any time in the future.

Rybo

 

[marcus]

Im not sure what CDT is but I think that what LQG and String theory have in common besides mathematics is a Quantum Gravity scenario. LQG being diffrrent in there are soon ot be instruments in space to test its hypothsis, whereas Strings cannot be teste current and is not foreseen to be testable any time in the future.
...

Yes Rybo, quantum gravity is what we are talking about in this thread. CDT is a newer approach to quantum gravity. the first CDT papers appeared in 1998.

LQG goes back to more like 1990, string even further back in history.

CDT means causal dynamical triangulations
it is a way of implementing a quantum version of Einstein 1915 classical theory of General Relativity. Other approaches tend to do this using a differentiable manifold to represent spacetime but CDT does not. It uses approximation by PL manifolds (also called simplicial manifolds)

the einstein action is implemented without using coordinates.

it is especially interesting because they are more able to compute, with CDT, than with other approaches, so they have been running thousands of computer simulations of spacetime generating random geometries and studying its geometry to find out statistical properties of space time, or what the different geometries have in common

if you are interested in geometry then you may have thought some about probability distributions on the space of all geometries, that is you may have thought about random geometries. geometric probability. it can be interesting to learn about----there are some classic books on the subject.
I guess CDT can in a very general way be said to be part of that, as well as being a quantum theory of spacetime and gravity

here is a short CDT paper that could serve as an introduction
http://arxiv.org/hep-th/0404156

I have other CDT links in my sig at the bottom of this post

 

[Rybo]

Causal Dynamic Triangulations

Yes Rybo, quantum gravity is what we are talking about in this thread. CDT is a newer approach to quantum gravity.
CDT means causal dynamical triangulations
it is a way of implementing a quantum version of Einstein 1915 classical theory of General Relativity. Other approaches tend to do this using a differentiable manifold to represent spacetime but CDT does not. It uses approximation by PL manifolds (also called simplicial manifolds)
the einstein action is implemented without using coordinates.
here is a short CDT paper that could serve as an introduction
http://arxiv.org/hep-th/0404156
t

Thanks Marcus, I have not been following this thread close enough
Quantum Graivity is the topic indeed ( Rybo-duhhh :).
I did go to the link you gave and read and saved the PDF document and now have good intro into Causal Dynamical Triangulations. (1) Why are triangulations more significant than squares or other polygons ergo the title CDT?

Most of the mathematics disscussed here are way above my head, HOWEVER, I do like to skim, pick and choose what goes well with my Synergetic interpretations and extrapolations thereof.

(2) Does CDT interpret( may be ) or require( is ) the physcial universe as being macro-finite? Here below is quote from PDF introduction to CDT.

..."This leads us to conclude that we have observed the emergence of a four-dimensional macroscopic world with three-dimensional spatial geometries. Judging from the computer simulations, this dynamically generated quantum geometry acts as a background geometry around which small quantum fluctuations take place. Further numerical studies will be needed to make this into a quantitative statement. The situation is really rather remarkable: we started out from an explicitly background-independent formalism and have rederived a particular stable background geometry."...

Rybo

 

[marcus]

... saved the PDF document and now have good intro into Causal Dynamical Triangulations. (1) Why are triangulations more significant than squares or other polygons ergo the title CDT?
...

(2) Does CDT interpret( may be ) or require( is ) the physcial universe as being macro-finite? Here below is quote from PDF introduction to CDT.

..."This leads us to conclude that we have observed the emergence of a four-dimensional macroscopic world with three-dimensional spatial geometries. Judging from the computer simulations, this dynamically generated quantum geometry acts as a background geometry around which small quantum fluctuations take place. Further numerical studies will be needed to make this into a quantitative statement. The situation is really rather remarkable: we started out from an explicitly background-independent formalism and have rederived a particular stable background geometry."...

Rybo, I'm glad you found the article interesting. You quoted a key paragraph right at the end, in the conclusions section.
Ooops, have to attend to something. be right back.

Now I am back. I will try to respond. first I think Buckminster Fuller probably found simplicial manifolds interesting if he studied them at all. Or else would have, if he had encountered them

Some CDT work involves cubes and pyramids as well as simplexes. I see them in Renate Loll papers. Personally I tend to stick to what is simplest and easiest. the straight simplex-based manifold is the easiest to construct and think about.

one reason for sticking to the mainstream PL-manifold or simplicial manifold is because there is tons of previous mathematics work done on it going back for decades. Its always nice when mathematicians for their own abstract reasons have gone in and explored and mapped out a subject and proven a bunch theorems which the Applied person can then use!

Ambjorn has an excellent 1996 online tutorial covering PL (piecewise linear) manifolds, of course, as he points out, it is a category, what isnt?
but the original sources go back to 1960s, 1970s. they just are not online whereas Ambjorn tutorial is online.

Intuitively, triangle is such a simple shape that if you have a bunch points you can always draw triangles with those points as vertices, and it is clear what the analogs are in higher dimension, less to worry about. the points of a simplex form a complete graph with the edges as links, don't have to stop and think about which points to connect and which not. Fraid I am being incoherent, but you get the message: simplex is simple.

Here is your other question

(2) Does CDT interpret( may be ) or require( is ) the physcial universe as being macro-finite?

I think it does not. Only the SIMULATION in a finite computer requires that the model be finite.

have fun.

 

[Rybo]

2D-3D-4D and 11D

thanks Marcus, Nightcleaner and Rybo
That's more than enough information to keep me amused, confused and enlightened for quite some time
I like how you say Marcus, it's not what's the same it's what's different that makes it interesting, so yes I can understand your excitement
CDT sounds like modelling the wire frame of an object before you skin it up in a 3d computer program ?

Spice, keep in mind that Jacob Bekenstien stated in his Scientific American article Feb 2002 or 03 that we humans are 2D beings only having an illusion of the 3Dness. I know it is one of teh hardest things for any human toever believe but that is what his black hole mathematics has led him to deduce.

This is also explained by Lee Smolin in his 3 roads to Quatum Gravity book in the holgraphic chapter. Holgraphy being the 3rd possibility along with String and LQG.

SO, we have
1) Holgraphy( 2D superfically appearing as 3D ). This is based in and around the event horizon being equal to the the entropy inside the black hole. See my home page for how I see the geomtrical versin of this scencario palying out.
http://home.usit.net/~rybo6/rybo/index.html

2) CDT's 3D superfically based on 4D manifolds?

3) Strings 4D( includes time ) superfically based on 11D.

Here is another graphic rendition( my interpetation ), that I espesically like, of what I see as the outer perimter/circumference of the gravitatinal Universe. Disreagard the stars in the background for more accurate represntation IMHO.
http://gregegan.customer.netspace.net.au/SCHILD/SCHILD.html

Rybo

 

[Spin_Network]

I like this:http://home.usit.net/~rybo6/rybo/id6.html

What are the current advances you have via the "25" turnaround?

It reminds me of this:http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm

I also found Ian Stewart a very interesting Mathematician, being that I was working on something I came across in his Royal Christmas Lecture, great Man.

The 'Einstein Wave' is what I called it:http://www.freewebs.com/moorglade/riemannshypothesesthetru.htm

will be found to be the 'eternal' running constant, but that is for another day!

 

[Rybo]

Primes and physics

I like this:http://home.usit.net/~rybo6/rybo/id6.html
What are the current advances you have via the "25" turnaround?
It reminds me of this:http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm
I also found Ian Stewart a very interesting Mathematician, being that I was working on something I came across in his Royal Christmas Lecture, great Man.
The 'Einstein Wave' is what I called it:http://www.freewebs.com/moorglade/riemannshypothesesthetru.htm
will be found to be the 'eternal' running constant, but that is for another day!

SpinN, Glad you like that page. I've nt gone much further in that specfic research but the #25 as a turnaround number I found in Synergtics. Here is link that chapter.
http://www.rwgrayprojects.com/synergetics/s12/p2200.html
Specifically you wil want to get to the 3 sets of graphs at 1223.12

I looked at Valletta's pattern of primes and entropic speculations. Interesting indeed and I want to study it more. Do we have better place to discuss primes than this thread?

Here is link to my yahoo geophysic web site. You can join and receive no mail or if you do it won't be much becasue it rare it recives any posts excetp from me.

http://groups.yahoo.com/group/Geophysiclink
I also have most of my graphic ideas listed in files there. Check it out.
I had MSN site with same but it expired. :--(

Rybo

 

[Rybo]

CDT Graviton

Rybo, I'm glad you found the article interesting. You quoted a key paragraph right at the end, in the conclusions section.
Ooops, have to attend to something. be right back.
I think it does not. Only the SIMULATION in a finite computer requires that the model be finite.
have fun.

Marcus, does CDT have a geometric hypothisis, or other, for a graviton?
Strings have a closed loop and string bits of many possible ones?
Holopgraphy( Bekenstien ) has spin-networks of which one, or more, trianlge is a graviton?
LQG has several traingles and squares equaling one graviton?( Jan Ambojorn )

Thanks if you have any leads. Fun right back at you.

Rybo

 

[spicerack]

Spice, keep in mind that Jacob Bekenstien stated in his Scientific American article Feb 2002 or 03 that we humans are 2D beings only having an illusion of the 3Dness. I know it is one of the hardest things for any human to ever believe but that is what his black hole mathematics has led him to deduce.

This is also explained by Lee Smolin in his 3 roads to Quatum Gravity book in the holgraphic chapter. Holgraphy being the 3rd possibility along with String and LQG.

SO, we have
1) Holgraphy( 2D superfically appearing as 3D ). This is based in and around the event horizon being equal to the the entropy inside the black hole. See my home page for how I see the geomtrical versin of this scencario palying out.
http://home.usit.net/~rybo6/rybo/index.html

2) CDT's 3D superfically based on 4D manifolds?

3) Strings 4D( includes time ) superfically based on 11D.

Here is another graphic rendition( my interpetation ), that I espesically like, of what I see as the outer perimter/circumference of the gravitatinal Universe. Disreagard the stars in the background for more accurate represntation IMHO.
http://gregegan.customer.netspace.net.au/SCHILD/SCHILD.html

Rybo

nifty grafix Rybo :!)

so by projecting 2 spatial dimensions onto <enter arbitrary dimensional number here> and add 1 for time we get perceptions of 3d including movement ?

meaning we exist in a 2d brane plane of bubble like foam

forgive me for even trying to get my head around this stuff but it interests the hell out of me and i would like to think that somewhere along the line it may have implications for the average person in the home like me

 

[Spin_Network]

SpinN, Glad you like that page. I've nt gone much further in that specfic research but the #25 as a turnaround number I found in Synergtics. Here is link that chapter.
http://www.rwgrayprojects.com/synergetics/s12/p2200.html
Specifically you wil want to get to the 3 sets of graphs at 1223.12

I looked at Valletta's pattern of primes and entropic speculations. Interesting indeed and I want to study it more. Do we have better place to discuss primes than this thread?

Here is link to my yahoo geophysic web site. You can join and receive no mail or if you do it won't be much becasue it rare it recives any posts excetp from me.

http://groups.yahoo.com/group/Geophysiclink
I also have most of my graphic ideas listed in files there. Check it out.
I had MSN site with same but it expired. :--(

Rybo

Thats quite amazing!..I mean I had no knowledge of Fuller's work, except for the 'Buckyball' of the mid eighties, I presume its the same man.

I am going to start a new thread later, but first I will explore the pages further, thanks again.

 

[nightcleaner]

Hi Rybo and Spin Network

I too, have been interested in the cubeoctahedron. IIRC Fuller called the dense packed sphere structure the 'isomatrix.' Johannes Kepler, the famous astronomer, studied the dense pack relations of spheres also. I noticed yesterday that Mathworld now has a graphic header showing this same arrangement. It seems to me that interest in this form is spreading.

It is a highly symmetrical form. Thirteen spheres, twelve dense packed around one, makes fourteen faces, has twelve vertices, twentyfour edges. Eight of the faces are triangular simplexes, six are squares. The faces are parallel to each other on opposite sides of the figure, so that the faces define parallel sets of planes of spheres. There are four sets of parallel planes which are made of triangular simplexes, and three sets of parallel planes made up of square simplexes. The figure can be subdivided in various ways into collections of tetrahedra.

It is interesting to me how the platonic solids are generated by this isomatrix structure by the simple matter of increasing scale. Kepler thought for a while that there was a relationship between the platonic solids and the orbits of the planets, but later abandoned that idea when it was discovered that the orbits were eliptical, not circular.

My thought has been that progress might be made by thinking of the isomatrix as a structure resulting from the expansion of the individual spheres. One might begin by imagining empty space, then adding the small perturbation of a single expanding sphere. This sphere would expand indefinitely in empty space. If there were other similar spheres in the space, they would eventually come into contact.

If we assume that the spheres are Planckian, then they would be impenetrable to each other. Therefore the meeting of two expanding spheres would result in the centers of the spheres undergoing gravitic acceleration. In gravitation we like to think of the spheres as attracting each other, but in this model the attraction is seen to be dual to the expansion of the sphere.

Further expansion (or infall) and contact would result in the cubeoctahedral shape being formed preferentially to any other shape. However there would be irregular features on the surface of the expanding isomatrix. These irregularities would be energy states, since the cubeoctahedral form is the lowest state, or alternately the state of highest symmetry, and any variation from it would tend eventually to be "forced" into the cubeoctahedral form.

In this model the irregularities actually are the physical manifestation of energy, and hence, of mass. The "perfect" or most highly symmetric form of any collection of Planck spheres would be in the shape of a faceted crystal, possessing vertices and edges as in the Platonic solids. But the tendency induced by infall or by the dual process of expansion would be like a force pushing outward on the faces and inward on the vertices and therefore possessing a form of curvature along the edges, in general promoting a spherical shape to the agglomeration rather than the faceted shape.

I would be interested to know if you are thinking along similar lines.

Thanks,

Richard

 

[Rybo]

2D, 3D, etc...

nifty grafix Rybo :!)
so by projecting 2 spatial dimensions onto <enter arbitrary dimensional number here> and add 1 for time we get perceptions of 3d including movement ?
meaning we exist in a 2d brane plane of bubble like foam
forgive me for even trying to get my head around this stuff but it interests the hell out of me and i would like to think that somewhere along the line it may have implications for the average person in the home like me

Glad you like them Spice. Your are probably referring to the ones icosahedral buble-like foams on my home page.

I found those on the net somewhere. In my scenarios there would be a 3D ness type relationships hapening inside them also, th to what extent is something I am still not clear on.

Keep in mind that I am no implying any reality in only 2D. It is Jacob Bekenstien and his resultant holographic theories proposing those truly unbelieveable 2D ideas i.e. the 2D bubble-like foam scenarios taht may indeed be liken to the icosahedral bubble on my home page.

Jacob did use a icosahedron in his Sciencitic American article to explain the event horizons equanimity to the internal entropy of a black hole but I do believe his choice of the icosahedron for that demostration was insignificant.

Her again is my yahoo egroup with all my older files leading me to my current website and thoughts.
http://groups.yahoo.com/group/Geophysiclink

You don't have to receive any mail to just look at graphic files.

Rybo