The running of the spacetime dimension (AJL)

[marcus]

we are familiar with the "running" of coupling constants, as for instance in QCD or even in QED where alpha (the electrodynamic coupling) is about 1/137 at long distances and low energies but increases to about 1/128 as energies increase and distances diminish.

Who would have imagined that the spacetime dimension could be 4 at large scale (as we observe) but decrease to around 2 at very small scale.

Ambjorn Jurkiewicz Loll have been doing computer experiments with their universe model and have found a running of the spacetime dimension.

AJL tell us to expect a long article soon with a lot of detail about their CDT model (Causal Dynamical Triangulations----universe assembled from 4-simplices----no discreteness, no minimal length, "fractal" in character they say)

AJL say their long article is called Reconstructing the Universe and it is dated May 2005 from the University of Utrecht, the Netherlands. But it is not yet out on archiv.

What they posted today is just a short article which has this strange new result of the spacetime dimension going down gradually as you get into ultra small scale.

http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

I :!) Renate Loll.

Utrecht is a good place. They had Gerardus 't Hooft already. Then they got Renate Loll. Then, just this year, they have gotten Jan Ambjorn to come part time from Copenhagen and be part time at Utrecht. AJL is a kickass QG team.
CDT (causal dynamical triangulations) is one of or the most exciting thing going in QG. When you can model the emergence of the universe in your computer with like a quarter of a million simplexes and watch it evolve over and over again and you find that it has the right 4D dimensionality at large scale etc.

2005 is looking like another "annus mirabilis" (wonder year) like the other year of wonders that by coincidence occurred 100 years ago.

 

[marcus]

Here is a quote from

http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07

---quote---
...Secondly, after integrating out all dynamical variables but the scale factor a(t) in the full quantum theory, the correlation function between scale factors at different (proper) times t is described by the simplest minisuperspace model used 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. Instead, we have found evidence of a fractal structure (see [7], which also contains a detailed technical account of the numerical set-up). What we report on in this letter is a most remarkable finding concerning the universes’ spectral dimension, a diffeomorphism-invariant quantity obtained from studying diffusion on the quantum ensemble of geometries.

On large scales and within measuring accuracy, it is equal to four, in agreement with earlier measurements of the large-scale dimensionality based on the scale factor. Surprisingly, the spectral dimension turns out to be scale-dependent and decreases smoothly from four to a value of around two as the quantum spacetime is probed at ever smaller distances. This suggests a picture of physics at the Planck scale which is radically different from frequently invoked scenarios of fundamental discreteness: through the dynamical generation of a scale-dependent dimensionality, nonperturbative quantum gravity provides an effective ultraviolet cut-off through dynamical dimensional reduction.

---end quote---

this is from bottom page 2 and top page 3.

they have a point BTW about the FULL THEORY. when they say "Secondly, after integrating out all dynamical variables but the scale factor a(t) in the full quantum theory,..." they are pointing to a weakness of LQC which is a simplified version of the full LQG.

 

[selfAdjoint]

I just saw this paper! They suggest that this behavior could provide a natural UV regulator. That will be sure to put the cat among the pigeons in the QFT world. What a result!

 

[marcus]

yeah
it is so beautiful that this is happening when we can watch it



Hello nigthtcleaner in the next post! I will save space by editing this message rather than making a new one.
When I first encountered you I was trying to understand this CDT stuff
(an earlier causal dynam. triang. paper of Ambjorn Jurkiewicz and Loll)
We were talking about 4-simplices and I was trying to understand the different "moves" that AJL were using in their computer model----so called Monte Carlo moves that disassembled-reassembled a portion of the spacetime, locally changing the geometry.
I forget the name of the thread. You got immersed in visualizing 4-simplices.

[EDIT: I found the thread
https://www.physicsforums.com/showthread.php?t=57311
"Two World-theories"]

An AJL spacetime or universe is assembled out of 4-simplices and the way they pick a random geometry is analogous to shuffling a deck of cards. they keep doing "moves" on the universe where they pull a section of 4-simplices apart and recombine them. they have 7 or 8 different moves, the way a dealer might have 2 or 3 or 4 different styles of shuffle. he applies a series of these shuffles to the deck and eventually it is randomized and can be in any order. this is called ergodic (the primitive moves eventually get to all possible random arrangements, or from any ordering of the cards to any other)

so in this paper AJL randomize the geomtry of the universe this way, and then, inside a fixed universe chosen at random, they study PERCOLATION as a drop of ink dropped into a class of clear water gradually percolates and diffuses outwards.

and by studying the simulated diffusion process in that given random universe (this is elegant and maybe a bit unexpected) they learn its dimension at various different scales large and small.

then they record that and they move on to ANOTHER random universe and study it's dimensionalities by the same simulated diffusing spreading inkdrop method, and they record that new random universe's dimensionalities at a range of scales large and small

and finally they accumulate STATISTICS from a large SAMPLING of random universes, statistics about the typical dimensionalities at various scales. And then they plot a curve to show how the dimension depends on the scale you are looking at.

And at any reasonable scale it always looks like it is about D = 4, but at these teeny microscopic scales it somehow becomes feathery or something, can't quite picture it, what could spacetime D = 2 be like?

there are all different kinds of dimension definitions. Hausdorff dimension. how distances relate to volumes. the SPECTRAL dimension that they are using here has to do with HOW FAST THINGS DRIFT AWAY doing a random walk. the more dimensions, the slower things drift away. on a 1D straight line there is only one direction to go so drifting is really focused and efficient and you diffuse away fast, but in 2D or 3D you wander and forget where you were heading and kink and coil about and at the end of the day you are not very far from home as the crow flies. So that way you can tell what dimension you live in

 

[nightcleaner]

This is good. I suppose the fractal dimensions are what allows the transitions from 4 to 2 dimensions to be smooth. I'll have to have another go at understanding fractal math. Been wanting to do that anyway.

nc

 

[Spin_Network]

we are familiar with the "running" of coupling constants, as for instance in QCD or even in QED where alpha (the electrodynamic coupling) is about 1/137 at long distances and low energies but increases to about 1/128 as energies increase and distances diminish.

Who would have imagined that the spacetime dimension could be 4 at large scale (as we observe) but decrease to around 2 at very small scale.

Ambjorn Jurkiewicz Loll have been doing computer experiments with their universe model and have found a running of the spacetime dimension.

their simulation experiments also contradict what one might expect from LQG or LQC. However in their universe there is also a BOUNCE----they also find no big bang singularity.

AJL tell us to expect a long article soon with a lot of detail about their CDT model (Causal Dynamical Triangulations----universe assembled from 4-simplices----no discreteness, no minimal length, "fractal" in character they say)

AJL say their long article is called Reconstructing the Universe and it is dated May 2005 from the University of Utrecht, the Netherlands. But it is not yet out on archiv.

What they posted today is just a short article which has this strange new result of the spacetime dimension going down gradually as you get into ultra small scale.

http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

I :!) Renate Loll.

Utrecht is a good place. They had Geraldus 't Hooft already. Then they got Renate Loll. Then, just this year, they have gotten Jan Ambjorn to come part time from Copenhagen and be part time at Utrecht. AJL is a kickass QG team.
CDT (causal dynamical triangulations) is one of or the most exciting thing going in QG. When you can model the emergence of the universe in your computer with like a quarter of a million simplexes and watch it evolve over and over again and you find that it has the right 4D dimensionality at large scale etc.

2005 is looking like another "annus mirabilis" (wonder year) like the other year of wonders that by coincidence occurred aroung 100 years ago.

The consequence of this:Who would have imagined that the spacetime dimension could be 4 at large scale (as we observe) but decrease to around 2 at very small scale.
is that out in the large scale cosmos, there are area's and volumes that have concentric dynamic construction!

Example, according to an early paper of the above authors(circa 98' I believe)..the small scale discrete spacetime separates from 4-D volumes to 2-D area's, so spacetimes are not 'CO-JOINED', when one views the path integral from a fixed Dimension, ie..3+1..so spacetimes become spaceD1 +spaceD1..and timeD1 +timeD1.. off the top of my head I believe there was an argument which goes..Space Volume1..couples with Space Area1..and symetrically Time goes along the path..Time Volume1 couples with Time Area1

forgive the lack of links..but that was the gist of it.

Now interestingly the Fractal Path of time say, can only couple with space on certain scales, if for instance there is a certain compact 'time', this can only be contained within a Large Scale volume of Space, so the 'envelope/foldings' of Large scale structures, let's say a Galaxy for instance?..will have contained within its Dimensional make-up, area's that are bounded by Quantum Geometries?...the above paper is the first step to explaining why the Area's of Galactic Space, is surrounded by a 'SMALLER' scale dependant Volume!..or specifically the Vacuum of Space between Galaxies is made-up from discrete volumes and lengths!

For the 'un-initiated' Geometricians, this equates to the 3+1 spacetime of our Galaxy, being surrounded by an entangled fractal geometry that is
2-Dimensional space only!

And there is a farther consequence of the probing of Distances along certain world-lines..looking out into the cosmos from within a 3+1 domain, the spectrum will obviously contain some dimensional domains that cause interference, such as 2-D field scattering by E-M waves as it encounters 3-Dimensional spacetimes and 2-Dimensional Fields, the emmission line spectra allready shows this up within chemical elements for instance, but light coming to us from a far off cosmological distance wil 'not' be absorbed and re-emitted by matter, it would be absorbed and re-emitted by the Dimensional Volumes it encounters?

http://arxiv.org/abs/hep-th/9904012

P.S Marcus ...you do the maths?..take the Planck scale 2-d starting point..venture outwards to the first area/volume you encounter where the authors call spacetime = 4-D..now take the scale comparision of our Blackhole at the Galactic core, and venture outwards until you have the paramiter distance comparable to that of Planck scale --> 4-Dimension!

The fractal distance scale will no doubt have Cosmological Constraints?

 

[marcus]

...Example, according to an early paper of the above authors(circa 98' I believe)..the small scale discrete spacetime

...

forgive the lack of links..but that was the gist of it.

...

http://arxiv.org/abs/hep-th/9904012

You regretted the lack of links, and then you went and found the link you had in mind---or in any case a very interesting one from that circa 1999 period.

I just looked at the article you linked to. It is one of the first or even the very first where Renate Loll is working with Ambjorn and she has convinced him to try CAUSAL dynamical triangulations, or aka "Lorentzian" DTs.

He has been doing simplicial QG dynam. triang. for some years and it has been very frustrating. Renate is still at the AEI in Potsdam (near Berlin), it is before she goes to Utrecht. She urges to take seriously the idea of causality built into Lorentzian spacetime. they choose the simplest case of 1+1 D = 2D, one spatial and one time dimension. They do a MONTE CARLO study. In such a low dimension there is only one Monty move needed.

Look on page 16, you see the picture of the sole necessary Monty move (and its reverse) shown in Figure 5.

It is early days, Jan and Renate are working with Anagnostopoulos the Cretan. He does the computer work for them and generates motion pictures---computer animations of evolving spacetimes in attractive colors. IIRC they are online at Jan Ambjorn home page.

So this link you have supplied taps into some nice history. I will give the data on that article:
http://arxiv.org/abs/hep-th/9904012
A new perspective on matter coupling in 2d quantum gravity
J. Ambjorn, K.N. Anagnostopoulos (Niels Bohr Institute), R. Loll (Albert-Einstein-Institut)
24 pages, 7 figures
Phys.Rev. D60 (1999) 104035
"We provide compelling evidence that a previously introduced model of non-perturbative 2d Lorentzian quantum gravity exhibits (two-dimensional) flat-space behaviour when coupled to Ising spins. The evidence comes from both a high-temperature expansion and from Monte Carlo simulations of the combined gravity-matter system. This weak-coupling behaviour lends further support to the conclusion that the Lorentzian model is a genuine alternative to Liouville quantum gravity in two dimensions, with a different, and much `smoother' critical behaviour."

I guess very first CDT paper was the Ambjorn Loll one that preceded this one
http://arxiv.org/hep-th/9805108
J. Ambjørn and R. Loll
Non-perturbative Lorentzian quantum gravity, causality and topology change
Nucl. Phys. B 536 (1998) 407-434,

 

[marcus]

This is good. I suppose the fractal dimensions are what allows the transitions from 4 to 2 dimensions to be smooth. I'll have to have another go at understanding fractal math. Been wanting to do that anyway.

nc

I was just reviewing that other thread "Two World-theories" that we were both doing 4 simplexes for.

It seemed like the overall most helpful paper was
http://arxiv.org/hep-th/0105267
Dynamically Triangulating Lorentzian Quantum Gravity
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)
41 pages, 14 figures
Nucl.Phys. B610 (2001) 347-382
"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."

Here is a sample of that thread, from post #24. You can see I was using that paper to help understand the Monty moves:

in lower dimension versions of the theory the moves are few and comparatively easy to visualize. I want to get more familiarity with the Monty moves in 4D (where there are more moves and harder to visualize)

In 4D the authors (hep-th/0105267) give us a set of 10 moves
but some are just the REVERSE of others so, depending how you count there are really only 5 (or maybe 6) different moves

these can be given names which are just a couple of numbers like for example these:

(2,8)
(4,6)
(2,4)
(3,3)

The move called (4,6) takes a cluster of 4 chunks and redivides it so it becomes 6 chunks

the reverse of that could be named (6,4) and simply does the reverse.
the authors have a picture of how this redividing is done---Figure 7 on page 25---and they also spell it out by listing the vertices of the chunks---on page 24. It is actually pretty simple.

This redividing up of clusters of chunks RESPECTS the layering, or causality, or foliation, or Lorentzianity-----whatever you want to call it. If some tetrahedron starts out purely spatial and it gets divided up then the resulting things are purely spatial and so on.

I guess I would like to run down the list of these moves and try to say in words what they do.

Remember that Geom(M) is the bunch of all the 4D geometries of the universe from bang to crunch (so far in their simulations they have a finite life universe with a crunch)
We want to see how the geometry of the universe works---quantum style---and so we are exploring this huge Geom(M) set of possibilities by doing a random walk in it. We make little bitty steps from one 4D geometry to another by just changing some detail of some cluster of simplexes.

the marvel is if we make enough of these moves, take enough of these little bitty steps, we get a representative random sample of how the geometry of the universe works

check out some of their graphics, like Figure 5 of hep-th/0411152


On page 2 of hep-th/0105267 the authors refer to Geom(M) as
"the mother of all spaces" and also as the "space of geometries" which
seems reasonable since it contains all the possible spacetimes as its elements-----a spacetime geometry is a POINT in Geom(M)

this means that wandering around in Geom(M), the space of geometries, means visiting many different spacetimes geometries in turn.

these Monty moves may seem very weak and insignificant, each one changes just one cluster somewhere in the universe, and not by very much either. but the effect of the steps is cumulative

Anyway, I want to review these 5 or so Monty moves

That was post #24 of this other thread about AJL:
https://www.physicsforums.com/showthread.php?t=57311
from December of last year

 

[marcus]

here is a short reading list for AJL. so far there are just 4 essential papers, I would say (if you stick with the 4D case and don't pay attention to their work on 2D and 3D)

1.
http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

2.
http://arxiv.org/abs/hep-th/0411152
Semiclassical Universe from First Principles
J. Ambjorn, J. Jurkiewicz, R. Loll
15 pages, 4 figures
Phys.Lett. B607 (2005) 205-213
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."

3.
http://arxiv.org/abs/hep-th/0404156
Emergence of a 4D World from Causal Quantum Gravity
J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)
11 pages, 3 figures; final version to appear in Phys. Rev. Lett
Phys.Rev.Lett. 93 (2004) 131301
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."

4. Finally, this older paper gives the 4D Monty moves and explains stuff and is generally helpful. It may be replaced soon by a longer AJL paper called "Reconstructing the Universe" which is supposed to give details of how their computer runs work, but that one is not out yet.

http://arxiv.org/hep-th/0105267
Dynamically Triangulating Lorentzian Quantum Gravity
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)
41 pages, 14 figures
Nucl.Phys. B610 (2001) 347-382
"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."

 

[marcus]

How they measured the local dimension of the spacetime of one of their universes is kind of neat.

They used RETURN PROBABILITY in diffusion, essentially in random walks.

Say you have a 1D lattice world and you go for a walk where at each step you pick the direction randomly. So after 2 steps you are back home with probability 1/2

But in a 2D lattice world, after two steps you are back home with probability 1/4.

And in 3D you return after two step with probability only 1/6.

because in 1D you randomly chose either E or W and then on the second step you have prob 1/2 of chosing the opposite and getting back to where you started

but in 2D you randomly choose N, E, S, W and on second step you have only 1/4 chance of reversing and getting home

and in 3D you choose from N,E,S,W,Up,Down and you have only 1/6 chance.

So in more dimensions you have more chance of getting lost when you go wandering.

they did not do it for 2 steps they did it for 40 steps, and 41 steps, etc. all the way to 400 steps.

In any lattice you can FIND OUT THE DIMENSION EXPERIMENTALLY by doing a random walk in that lattice and seeing HOW OFTEN you happen to get back home after 40 steps, or 41 steps or 400 steps etc.

Now a subtle point is that to experimentally find out the "return probability" in a lattice, as a function of the number of random steps, YOU DONT NEED A METRIC. it is diffeomorphism invariant, the definition of the event does not depend on distance. you either arrived back home, or you didnt. if you REPARAMETRIZE and change all the distances the event that gets counted is still the same event.

So return probability is a good thing to measure because of this invariance, while the probability of getting "one meter away from home" is a bad thing to measure because you need some arbitrary prior choice of distance function in order to define it.

So their game is to crank out hundreds of universes at random (each one satisfying a simplex Regge form of the Einstein equation, so each one kind of like our universe, with a big bang etc, but closed so there is a final crunch which we don't know will happen to ours)

As an experiment they crank out a random sample of hundreds of universes. And then in EACH random universe they explore its dimensionality but at varying scale, letting the number of steps you take vary.

they found that if you only take 40 steps then the experimentally observed dimension is about 2.8,
but if you take 400 steps then it looks about like D = 3.8.

Look at Figure 1 on page 7
http://arxiv.org/hep-th/0505113
they plot a curve and it looks like for well over 400 steps it goes to 4D asymptotically
and for down near zero steps it looks like 2D, or even D = 1.8 (some kind of fractal)

Now 2D means 1+1, it means 1 space dimension and 1 time dimension. So it looks like in very small neighborhoods, whatever that means (since we do not have a fixed metric!), space is one dimensional. aaargh. that is very weird.

there is a lot that is confusing about this.

 

[Spin_Network]

How they measured the local dimension of the spacetime of one of their universes is kind of neat.

They used RETURN PROBABILITY in diffusion, essentially in random walks.

Say you have a 1D lattice world and you go for a walk where at each step you pick the direction randomly. So after 2 steps you are back home with probability 1/2

But in a 2D lattice world, after two steps you are back home with probability 1/4.

And in 3D you return after two step with probability only 1/6.

because in 1D you randomly chose either E or W and then on the second step you have prob 1/2 of chosing the opposite and getting back to where you started

but in 2D you randomly choose N, E, S, W and on second step you have only 1/4 chance of reversing and getting home

and in 3D you choose from N,E,S,W,Up,Down and you have only 1/6 chance.

So in more dimensions you have more chance of getting lost when you go wandering.

they did not do it for 2 steps they did it for 40 steps, and 41 steps, etc. all the way to 400 steps.

In any lattice you can FIND OUT THE DIMENSION EXPERIMENTALLY by doing a random walk in that lattice and seeing HOW OFTEN you happen to get back home after 40 steps, or 41 steps or 400 steps etc.

Now a subtle point is that to experimentally find out the "return probability" in a lattice, as a function of the number of random steps, YOU DONT NEED A METRIC. it is diffeomorphism invariant, the definition of the event does not depend on distance. you either arrived back home, or you didnt. if you REPARAMETRIZE and change all the distances the event that gets counted is still the same event.

So return probability is a good thing to measure because of this invariance, while the probability of getting "one meter away from home" is a bad thing to measure because you need some arbitrary prior choice of distance function in order to define it.

So their game is to crank out hundreds of universes at random (each one satisfying a simplex Regge form of the Einstein equation, so each one kind of like our universe, with a big bang etc, but closed so there is a final crunch which we don't know will happen to ours)

As an experiment they crank out a random sample of hundreds of universes. And then in EACH random universe they explore its dimensionality but at varying scale, letting the number of steps you take vary.

they found that if you only take 40 steps then the experimentally observed dimension is about 2.8,
but if you take 400 steps then it looks about like D = 3.8.

Look at Figure 1 on page 7
http://arxiv.org/hep-th/0505113
they plot a curve and it looks like for well over 400 steps it goes to 4D asymptotically
and for down near zero steps it looks like 2D, or even D = 1.8 (some kind of fractal)

Now 2D means 1+1, it means 1 space dimension and 1 time dimension. So it looks like in very small neighborhoods, whatever that means (since we do not have a fixed metric!), space is one dimensional. aaargh. that is very weird.

there is a lot that is confusing about this.

If you think that is weird !..then I can suggest that their forthcoming paper will be extreme?

Take a look at Baez week 215, for a similar perspective to 'choice' an random Chaotic movementsere I'm talking about the tiling of Klein's quartic curve by 56
equilateral triangles. We could equally well talk about its tiling
by 24 regular heptagons, which is the Poincare dual. Either way, the
puzzle is to fill in the question marks. I don't know the answer!

To conclude - at least for now - I want to give the driving directions
that define a "cube" or an "anticube" in Klein's quartic curve. Say
you're on some triangle and you want to get to a nearby triangle that
belongs to the same cube. Here's what you do:

hop across any edge,
turn left,
hop across the edge in front of you,
turn right,
then hop across the edge in front of you.

Or, suppose you're on some triangle and you want to get to another
that's in the same anticube. Here's what you do:

hop across any edge,
turn right,
hop across the edge in front of you,
turn left,
then hop across the edge in front of you.

(If you don't understand this stuff, look at the picture above and
see how to get from any circle or square to any other circle or
square of the same color.)

You'll notice that these instructions are mirror-image versions of
each other. They're also both 1/4 of the "driving directions from
hell" that I described last time. In other words, if you go
LRLRLRLR or RLRLRLRL, you wind up at the same triangle you started
from. You'll have circled around one face of a cube or anticube!

In fact, your path will be a closed geodesic on the Klein quartic
curve... like the long dashed line in Klein and Fricke's original
picture:

4) Klein and Fricke, Klein's quartic curve with geodesic,
http://math.ucr.edu/home/baez/Klein168.gif


there are some interesting correlations to the Baez posting and Klien Dimensional Transference, it may have its relevence in Stringtheory's, 'Vacuum-Choice'

But there is another problem, it's contained within the question I asked in my original reply..you do the maths

It seems that the answer, acccording to Hitchhikers and Stringtheorists..apppers to be close to, if not 42!

 

[marcus]

... generates motion pictures---computer animations of evolving spacetimes in attractive colors. IIRC they are online at Jan Ambjorn home page.

So this link you have supplied taps into some nice history. I will give the data on that article:
http://arxiv.org/abs/hep-th/9904012
A new perspective on matter coupling in 2d quantum gravity
...

since it connects with the link Spin_Network gave, I will see if I can find the link to Ambjorn's page that has the 1+1 = 2D spacetime animations

yeah, here it is
http://www.nbi.dk/~ambjorn/

http://www.nbi.dk/~ambjorn/lqg2/

these date back to 1999, the same time as the paper that Spin mentioned.
In the nomenclature AJL were using then, "LQG2" does not mean "loop quantum gravity". the letters stand for "Lorentzian quantum gravity" and the meaning is what we would now call CDT (causal dynamical triangulations). So for LQG2 read "2 dimensional CDT".


For a visual for 4D CDT look at figure 1 on page 5 of
http://arxiv.org/hep-th/0411152
it is much more like the balloon universe we are used to
(since they cannot draw the whole thing on a 2D piece of paper they use the circumference to plot the 3D spatial volume evolving along the vertical time axis)

 

[marcus]

the main topic of this thread is about how the small-scale DIMENSION REDUCTION that AJL found, in this recent paper, might help make gravity
RENORMALIZABLE

remember that AJL found the largescale macroscopic spacetime dimension turned out to be 4, as one would hope!, in their simulations but that at small scale it goes down to around 2.

It is a famous result that gravity isn't renormalizable above 2 but if you go down to 2D then it IS. so it seems like a piece of luck, and just what the doctor ordered, for AJL's microscopic dimension to go down to 2D

meanwhile we need to accumulate a bit of material about this off the web and get slightly familiar with it.

For starters here is something John Baez said in Week 139 sometime in 1999.
http://math.ucr.edu/home/baez/twf_ascii/week139

Scroll about halfway down and you will see things like this:

"...Okay, now for the really fun part. Perturbative quantum gravity
in 2 dimensions is not only renormalizable (because this is the
upper critical dimension), it's also asympotically free! ..."

And since he says that 2D is the upper critical dimension we want to know what that is, and why 2D is upper critical and like the threshold of bad perturbative gravity behavior. So scroll only about a third of the way down and you will see this:

"...This says G is proportional to p^{n-d}. There are 3 cases:

A) When n < d, our coupling constant gets *smaller* at higher momentum
scales, and we say our theory is "superrenormalizable". Roughly, this
means that at larger and larger momentum scales, our theory looks more
and more like a "free field theory" - one where particles don't interact
at all. This makes superrenormalizable theories easy to study by
treating them as a free field theory plus a small perturbation.

B) When n > d, our coupling constant gets *larger* at higher momentum
scales, and we say our theory is "nonrenormalizable". Such theories
are hard to study using perturbative calculations in free field theory.

C) When n = d, we are right on the brink between the two cases above.
We say our theory is "renormalizable", but we really have to work out
the next term in the beta function to see if the coupling constant
grows or shrinks with increasing momentum.

Consider the example of general relativity. We can figure out
the upper critical dimension using a bit of dimensional analysis
and handwaving. Let's work in units where Planck's constant and the
speed of light are 1. The Lagrangian is the Ricci scalar curvature
divided by 8 pi G, where G is Newton's gravitational constant. We
need to get something dimensionless when we integrate the Lagrangian
over spacetime to get the action, since we exponentiate the action
when doing path integrals in quantum field theory. Curvature has
dimensions of 1/length^2, so when spacetime has dimension n, G must
have dimensions of length^{n-2}.

This means that if you are a tiny little person with a ruler X
times smaller than mine, Newton's constant will seem X^{n-2} times
bigger to you. But measuring Newton's constant at a length scale
that's X times smaller is the same as measuring it at a momentum scale
that's X times bigger. We already solved the Callan-Symanzik equation
and saw that when we measure G at the momentum scale p, we get an
answer proportional to p^{n-d}. We thus conclude that d = 2.

(If you're a physicist, you might enjoy finding the holes in the
above argument, and then plugging them.)

This means that quantum gravity is nonrenormalizable in 4 dimensions.
Apparently gravity just keeps looking stronger and stronger at
shorter and shorter distance scales. That's why quantum gravity has
traditionally been regarded as hard - verging on hopeless. ... "

 

[Spin_Network]

I think there needs to be an inquiry into Dentritic Growth?
http://math.nist.gov/mcsd/savg/vis/dendrite/
as a means of lower to higher dimensional transformation?

There appears to be a genuine correlation to linear evolutions for B.E.C and that of Brane Lattice dimensional intersections?...the evidence is mounting that 'Branes' are evolutionally the Energy equivilent to 'States of Matter'.

Laughlin's emergent principle details the surface critical dimensional 'breakup' for evolving 'metal-like' condensates.

The recent paper by Ansari and Markopoulou, who have both worked with Smolin http://arxiv.org/abs/hep-th/0505165

shows that the landscape is being investigated a various levels, for instance the Ansari/Markopoulou Paper above, is the LQG equivilent of Cellular Orientations!

take a look at this:http://www.gps.jussieu.fr/engl/cell.htm

and one starts to see that Smolins 'Self-Creation' model is gently being molded by Smolin, the influence on those he works with appears to be one dimensional

 

[marcus]

I think there needs to be an inquiry into Dentritic Growth?
http://math.nist.gov/mcsd/savg/vis/dendrite/
as a means of lower to higher dimensional transformation?

There appears to be a genuine correlation to linear evolutions for B.E.C and that of Brane Lattice dimensional intersections?...the evidence is mounting that 'Branes' are evolutionally the Energy equivilent to 'States of Matter'.

Laughlin's emergent principle details the surface critical dimensional 'breakup' for evolving 'metal-like' condensates.

The recent paper by Ansari and Markopoulou, who have both worked with Smolin http://arxiv.org/abs/hep-th/0505165

shows that the landscape is being investigated a various levels, for instance the Ansari/Markopoulou Paper above, is the LQG equivilent of Cellular Orientations!

take a look at this:http://www.gps.jussieu.fr/engl/cell.htm

and one starts to see that Smolins 'Self-Creation' model is gently being molded by Smolin, the influence on those he works with appears to be one dimensional

you keep dragging in all this interesting stuff, Spin! thanks!

 

[arivero]

I will read the paper, but at a first glance I do not expect the "self-regularisation" to work. Path integrals for lambda phi^4 interaction have also a fractal dimension 2, and we keep renormalising it in the traditional way.

 

[marcus]

I will read the paper, but at a first glance I do not expect the "self-regularisation" to work. Path integrals for lambda phi^4 interaction have also a fractal dimension 2, and we keep renormalising it in the traditional way.

I cannot help you figure these things out (you are way ahead of me about conventional path integrals!) but I can say a couple of things:

1. I am very glad you plan to read the paper. Any informed comment you are able to make will be especially helpful.

2. I can simply REPEAT something that AJL said that is related to this analogy between conventional path integrals and CDT. This may or may not bear on the issue you raised.
They pointed out what I thought was a significant difference:

a particle track can be imagined as an unsmooth (nowhere differentiable) path which is approximated by piecewise straight paths. but the particle path is embedded in a surrounding space
or perhaps I should say its worldline is embedded in a surrounding spacetime.

the CDT spacetime can be imagined as an unsmooth continuum which is approximated by piecewise flat manifolds
but the spacetime is not embedded in any surroundings. the CDT spacetime is all there is.

when the CDT people start adding matter fields they will be drawing them not in ordinary spacetime but in CDT spacetime, which will mean drawing approximations in the simplicial manifolds (also called "PL" manifolds IIRC) and then taking the limit. the matter fields will be drawn in this unconventional spacetime, which seems to have dimension 2 at small scale though it looks normal 4D at large scale. You have to decide for yourself if this could make a difference----I cannot say if it is relevant or not.

 

[nightcleaner]

take the Planck scale 2-d starting point..venture outwards to the first area/volume you encounter where the authors call spacetime = 4-D..now take the scale comparision of our Blackhole at the Galactic core, and venture outwards until you have the paramiter distance comparable to that of Planck scale --> 4-Dimension!

The fractal distance scale will no doubt have Cosmological Constraints?

I thought this was an interesting idea, altho it may be difficult to do anything with it other than appreciate the mystery. We think we know where the "black hole" at the center of the galaxy is, in relation to say, our own position, at least if we don't try too hard to give it local coordinates other than 0,0,0. Or perhaps there should be four zeros. If we try to pin the zero's on anything else, then the singularity issue makes it hard to "measure" the "distance" from the singularity to any other object. So how do we determine scale if we can't detemine exact local positions? In fact, it seems to be scale itself that "runs" down to the singularity. A problem for Zeno.

Still, it is provocative to imagine that large structural members such as galaxies may have self-similar relationships to the geometry of the very small, the Planck scale. Where exactly then does the transition from four to lower dimensions become an issue?

The answer to that question has to depend on first fixing a value for the minimum discrete length...and AJL discard that option. If the minimum length is going to go running off when we try to approach it, we can't really say that four dimensionality breaks down at the Fermi scale, or the Planck scale, or, horrors unimaginable, at even smaller size. It breaks down. Where? Someplace over there, near the horizon. Now there is some hand waving for you.

However the AJL approach to this problem is still justified. We don't have to fix the transition zone as some part of an attometer or some number of parsecs. We only need to realize that there is a kind of scale of scales...in other words, we may be able to draw similarities between events on a large scale (galaxies) and events on a small scale (point particles) by order of magnitude difference between the two scales. Perhaps we will find that there is a spectrum of self-similarities displayed along a line defined by differences of orders of magnitude.

Maybe we can find a relationship between the shape of planetary systems and the shape of atoms after all. Perhaps the differences in planetary behavior and electron behavior can be shown to be a matter of dimensionality. When we look at planets, we view them in one set of dimensions, but when we look at atoms, we have to view them in a different set of dimensions.

Anyway that is what I have been thinking about, in between tending to the animals.

Be well,

Richard

 

[marcus]

I will read the paper, but at a first glance I do not expect the "self-regularisation" to work. Path integrals for lambda phi^4 interaction have also a fractal dimension 2, and we keep renormalising it in the traditional way.

Hi Alejandro, you mentioned reading the Ambjorn Jurkiewicz Loll paper

http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
10 pages

If you have gotten around to doing so, I am curious to know if you have any more ideas about the reduced dimensionality at small scale helping with the renormalization problem?

 

[marcus]

I will read the paper, but at a first glance I do not expect the "self-regularisation" to work. Path integrals for lambda phi^4 interaction have also a fractal dimension 2, and we keep renormalising it in the traditional way.

it may be our fate not to get a followup on this from arivero

I don't know enough to be sure but I suspect that what Alejandro is talking about takes place in the CONTEXT OF 4D spacetime. and there, whatever the path integral is, "we keep renormalising it in the traditional way"
as Alejandro says

however that is not the case with CDT, where the SPACETIME DIMENSION ITSELF is 2D, at short range, and opens to usual 4D at long range

so because the spacetime dimension is 2D, we maybe do not have to keep renormalizing in the traditional way-------gravity does not need that kind of kidglove treatment in 2D

would be great to get a followup on this from A. but absent that I shall not worry too much about the objection raised.

 

[arivero]

Yep I was referring to classical results about fractality of path integrals; particularly it is known that the lambda phi^4 theory is a sort of random walks of fractal dimension two. But now I look at it, this paper is not claiming fractality but the contrary, its absence in the case of quantum gravity. They say they are able to measure a kind of interpolation between two, at short distances, and four at long.

I still believe it could be just the way that quantum fields manifest themselves in gravity; after all, at short distances minkowski rules, and then the D=2 of field theories could appear. But for other view, visit 't Hooft:
http://citebase.eprints.org/cgi-bin/citations?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9310026

 

[marcus]

They say they are able to measure a kind of interpolation between two, at short distances, and four at long.
...

if that describes reality then no differentiable manifold can be a model of spacetime (I guess that is clear and goes without saying)

Alejandro I have a question for you

Fotini Markopoulou says that nonperturbative quantum gravity means
seeking a consistent quantum dynamics on
the set of all Lorentzian spacetime geometries

how would you describe that set?
has anyone provided a clear description for the set of all Lorentzian spacetime geometries?

 

[marcus]

some thoughts about the set of L. spacetime geom.

--------
in case anyone is curious, the quote comes from her recent article with M. Ansari
http://arxiv.org/hep-th/0505165
which begins by saying
<<The failure of perturbative approaches to quantum gravity has motivated theorists to study non-perturbative quantization of gravity. These seek a consistent quantum dynamics on the set of all Lorentzian spacetime geometries. One such approach which has led to very interesting results is the causal dynamical triangulation (CDT) approach[1, 2]. In the interest of understanding why this approach leads to non-trivial results, in this paper we study...>>

--------
describing the set of L. spacetime geometries is apt to be an interesting problem because by now it seems clear that it will NOT consist of a set of metrics on some diff. manifold!

or of anything related to a metric, like a connection

nor will it be a set of anything that lives on a diff. manifold at all, because such things cannot have dynamic dimensionality

but up til now spacetime geometries have been represented, in mathematics, by stuff living on diff. manifolds---including algebraic structures that come by considering functions defined on diff. manifolds.

so representing the set of all L. spacetime geometries could involve some innovative mathematics. some new concepts
-----------

to represent the set of all Lorentzian spacetime geometries it may turn out to be essential to include those L. spacetime geoms. where there is some (at least limited) topology change

that is why the CDT papers that are most interesting to me right now are
Loll-Westra 0306, and Loll-Westra 0309

http://arxiv.org/hep-th/0306183
http://arxiv.org/hep-th/0309012