Why CDT changes the map of quantum gravity

[marcus]

the "why" is basically Loll's defintion of what a QG theory should be:

page 1 of "Discrete History" http://arxiv.org/hep-th/0212340
<< By quantum gravity I will mean a consistent fundamental quantum description of space-time geometry (with or without matter) whose classical limit is general relativity...>>

... a path integral because that is a common type of quantum description (essentially counting the paths except the paths are complete spactimes ...

...from page 3 and 4 of "Discrete History"
<< In this approach, “computing the path integral” amounts to a conceptually simple and geometrically transparent “counting of geometries”, with additional weight factors which are determined by the Einstein action. >>

For the purposes of this thread (why CDT changes the map of quantum gravity) we can take Lolls definition of QG. For our purposes here, what a theory should be, to be considered a theory of Quantum Gravity, is what she says in the above.

the use of standard physics tools, such as the real and complex numbers, does not require justification. path integral is one of a couple of traditional methods of building a quantum description and does not require justification

People have been trying to get a theory of quantum spacetime dynamics for a long time, and one of the ways they have been trying for a long time is PATH INTEGRAL quantum spacetime dynamics. And it has not worked so far. But now it seems to be starting to work.

It would be natural, if one was so inclined, to be in denial about this and to be thinking of a million reasons why this cannot possibly be right and why what Renate Loll means by "quantum gravity" cannot REALLY be the REAL quantum gravity, and so on. But I think we might as well listen to her, on her own terms, and try to understand and not be threatened by it.

what it appears to me is like this. Since the 1990s the Loop people have been trying a particular Path Integral approach called SPIN FOAM which was aimed at doing just this what Loll-type Triangulations does and being just this kind of Quantum Gravity theory, and it was not working very well, and no body got to feeling threatened and started redefining the rules so that Spin Foam could not be a REAL QG. People just kept speaking normal professional physics English language-----spin foams, like LQG and many other approaches, was a candidate QG theory.

Now what we are seeing is a particularly clear straight-forward case of a path integral formulation of Quantum Gravity. It has been making very rapid progress for the past two years, and it may come to be, for a while, our PARADIGM of what people mean by a quantum gravity theory.

And it seems to me that a natural reaction to this would be to put one's hands over ones ears, shut one's eyes, and start redefining what people OUGHT to mean when they say "quantum gravity" to be be something completely different from what they have mostly so far. So let's watch out for signs of that happening----not to say it HAS ALREADY been noticeable, but it might happen so let's watch for it.

Cheers

 

[marcus]

Just visited Peter Woit's blog, Not Even Wrong (often interesting posts about string theory there) and was struck by the relevance of this
http://www.math.columbia.edu/~woit/blog/archives/000200.html

it links to an article by Scott Berkun
http://www.scottberkun.com/essays/essay40.htm

that is illustrated, entertainingly enough, by Monty Python episodes

 

[marcus]

parts of the summary section of "Reconstructing the Universe" are worth quoting

---quote from hep-th/0505154---
8 Summary and outlook

This paper describes the currently known geometric properties of the quantum universe generated by the method of causal dynamical triangulations, as well as the general phase structure of the underlying statistical model of four-dimensional random geometries. The main results are as follows. An extended quantum universe exists in one of the three observed phases of the model, which occurs for sufficiently large values of the bare Newton’s constant G and of the asymmetry $$\Delta$$, which quantifies the finite relative length scale between the time and spatial directions. In the two other observed phases, the universe disintegrates into a rapid succession of spatial slices of vanishing and nonvanishing spatial volume (small G), or collapses in the time direction to a universe that only exists for an infinitesimal moment in time (large G, vanishing or small $$\Delta$$). In either of these two cases, no macroscopically extended spacetime geometry is obtained. By measuring the (Euclidean) geometry of the dynamically generated quantum spacetime *17* in the remaining phase, in which the universe appears to be extended in space and time, we collected strong evidence that it behaves as a four-dimensional quantity on large scales...

...

The most local measurement of quantum geometry so far is that of the spectral dimension of spacetime at short distances, which provides another quantitative measure of the nonclassicality of geometry. As we have seen, the spectral dimension changes smoothly from about 4 on large scales to about 2 on small scales. Not only does this (to our knowledge) constitute the first dynamical derivation of a scale-dependent dimension in quantum gravity, but it may also provide a natural short-distance cut-off by which the nonperturbative formulation evades the ultra-violet infinities of perturbative quantum gravity.

In summary, what emerges from our formulation of nonperturbative quantum gravity as a continuum limit of causal dynamical triangulations is a compelling and rather concrete geometric picture of quantum spacetime. Quantum spacetime possesses a number of large-scale properties expected of a four-dimensional classical universe, but at the same time exhibits a nonclassical and nonsmooth behaviour microscopically, due to large quantum fluctuations of the geometry at small scales. These fluctuations “conspire” to create a quantum geometry that is effectively two-dimensional at short distances...


FOOTNOTE *17*: Since our universe is a weighted quantum superposition of geometries, all “measurements” refer to expectation values of geometric operators in the quantum ground state.
-----end quote---

I think we are looking at a fundmental advance. A concrete picture of quantum spacetime has appeared. It is likely, I suspect, to become the paradigm for what the words "quantum spacetime" come to mean in our discussions.

 

[marcus]

We are talking about Reconstructing the Universe hep-th/0505154, a landmark CDT paper.

the phase diagram relevant to the discussion in foregoing post is Figure 3 on page 10,

the kappa here is the reciprocal of G, and region C of the phase diagram is the good phase where the universe has its familiar spatial and temporal extension.

what we see immediately is that the Newtonian gravitational constant G, the coupling constant, has to be sufficiently strong for spacetime to exist.

I am interested in understanding the other parameter, $$\Delta$$
which they call the "asymmetry parameter"

Delta = 0 corresponds to the simplexes being equilateral
You can see that in Figure 3 if Delta is near zero we get out of the good phase C and again the familiar spacetime fails to materialize. I'm wondering what this tells us---what the significance of Delta is. I can see that it affects the angles of simplexes and thus how one compares purely spatial curvature with curvature involving timelike lengths.

Visually the asymmetry parameter tells us "squatness" of the simplex in standard position. (by simplex I mean 4D simplex, the 5 pointed thing a little 4D pyramid-like thing sitting on a tetrahedron base--should I call it "pentamid"?)

each simplex has equal spacelike lengths and equal timelike lengths, but the spacelike and timelike lengths don't have to be equal, there is this positive number alpha with

alpha =- timelength²/spacelength²

if alpha were equal to one then we have the "equilateral" case, but the CDT people want to make alpha smaller than one, like 7/12, say. around one half but maybe a little over one half. they find what works good. it is not so critical, the Figure 3 shows as wide range works. But alpha should not be ONE, it should be somewhat less

this "asymmetry parameter" Delta is a measure of how much it is less than one. it is not simply the straight difference but it is an indicator of that. IIRC Delta occurs in angle formulas and in measuring curvature. IIRC its meaning has to do with how purely spatial curvature relates to other kinds.

 

[Kea]

It would be natural, if one was so inclined, to be in denial about this and to be thinking of a million reasons why this cannot possibly be right and why what Renate Loll means by "quantum gravity" cannot REALLY be the REAL quantum gravity...

Sigh.

Is that it? Are we supposed to accept path integrals because that seems to be the best thing around? The third road isn't hand waving - it has alternatives. These alternatives can explain, for example, quark confinement in a mathematically rigorous description of elements of the Standard Model. There are alternative cosmologies that suggest that there may be no cosmological constant, and which agree with the supernovae data. What evidence does CDT have to support it?

Oh well. I'm quite used to people thinking I'm crazy. Sigh.
Kea

 

[Kea]

the use of standard physics tools, such as the real and complex numbers, does not require justification

Oh. Do AJL use real numbers? I was under the impression they were doing p-adic geometry.

Kea

 

[marcus]

---quote from http://arxiv.org/hep-th/0505154 ---

...In summary, what emerges from our formulation of nonperturbative quantum gravity as a continuum limit of causal dynamical triangulations is a compelling and rather concrete geometric picture of quantum spacetime. Quantum spacetime possesses a number of large-scale properties expected of a four-dimensional classical universe, but at the same time exhibits a nonclassical and nonsmooth behaviour microscopically,...

the topic of the thread here is, of course, "Why CDT changes the map of quantum gravity". Also I want to do some introductory explaining about what CDT is----how the approach works and how spacetime is modeled as the quantum spacetime just mentioned.

for my working definition of what QUANTUM GRAVITY theories are, i am using Renate Loll's own concise description of a QG theory:

page 1 of "Discrete History" http://arxiv.org/hep-th/0212340
<< By quantum gravity I will mean a consistent fundamental quantum description of space-time geometry (with or without matter) whose classical limit is general relativity...>>

THE CLEAREST OBJECTIVE SIGNAL that CDT is changing the map of QG research is what we already know about the programme of this year's main QG conference.

this is the October 2005 "Loops 05" conference.

http://loops05.aei.mpg.de/index_files/Home.html
http://loops05.aei.mpg.de/index_files/Programme.html

the prominence of CDT in the October programme is greater than at any similar conference in the past. CDT is not only listed on the Loop 05 Homepage as a topic in its own right but it exemplifies another focus of the conference that was listed there, namely
Non-perturbative Path Integrals

CDT is one of the leading path integral approaches to QG. So there are clear signs that people at this year's main QG conference will be hearing a lot about CDT.

And my guess is that people in other branches of QG research will be devoting considerable effort to MAKE CONTACT with CDT. There will probably be some convergence of the approaches, with people in other departments of QG deriving analogous results to those found in CDT, or inspired by what has been discovered in CDT computer simulations of quantum spacetime.

 

[marcus]

IMHO Loll-type triangulations is the most important new development in quantum gravity for some years

and partly for that reason I think it is essential that a good description and explanation of the CDT authors' methods be devised that can work for laypeople. it is a landmark new development and interested non-technical people should be able to follow what's happening

another reason for working up some clear non-technical explanation is that it is so interesting! the CDT quantum spacetime is simple enough to grasp, very concrete (not floating ectoplasmically in some abstract algebra-land), and well-furnished with attractive novelty. so there are other good reasons for developing good explanations of it for laypeople.

also IMHO this PF forum can serve as an "explanation-lab" or "explanation-workshop" to try out ways of presenting new theoretical physics----quantum gravity physics----and see what works.

I am going to take some risks here, and i think I'll try calling the simplest 4D object (analog of triangle, conventionally called "4-simplex") by some made-up names. this may not work and I may have to go back to calling it by the conventional name of 4-simplex.

in CDT method, spacetime is assembled out of TWO TYPES of 4D building blocks. and all those of a given type are IDENTICAL
we (any possible readers and I) need to get to the point of visualizing the layered assemblage of spacetime made from some halfmillion building blocks of this type (roughly half of each type)

 

[marcus]

working with building blocks is good for several reasons. when you approximate quantum spacetime by assemblages of these 4-simplex blocks then

1. you can totally get rid of coordinate functions, the blocks are identical (by type) and you can gauge curvature by COUNTING because the geometry of spacetime is COMPLETELY EXPRESSED BY HOW THEY ARE GLUED TOGETHER. You don't have to futz around with drivatives and bundles and connections and diffeomorphisms and all that hardware that is based on coordinate charts. you have a directory in the computer memory that keeps track of how the blocks are glued together, and the different numbers in each type, and that directory is the geometry.

2. these tiny little blocklets are all cut out of Minkowski space, the space of special relativity, which is kind of sweet, isn't it? So they have two kinds of edges: spacelike edges and timelike edges. BTW a block has 10 edges in all. Think of a tetrahedron with 6 edges, and use that as the base of a pyramid, so there is an apex above the tetrahedron connected to its 4 corners by 4 additional edges

3. I was going to think of reasons it is good to approximate quantum spacetime by random assemblages of blocks. Well it makes everything finite so you can COUNT GEOMETRIES. essentially different geometries are equivalent to essentially different ways of assembling and gluing the blocks! and you can count those combinatorially.

4. you can also make formulas for curvature and volume and the Einstein action (measuring how "busy" a spacetime is) which ONLY INVOLVE COUNTING different kinds of simplexes. so it makes setting up a the usual physicist-gear kind of straightforward.

instead of doing a lot of calculus with drivatives and integrals most of which are purely theoretical because no one ever gets down in the mud and solves them. instead of all that, you just tell the computer to COUNT stuff

5. it is kind of a readymade situation for quantum mechanics---one of the standard tools is path integral!

have to go back later

 

[Mike2]

By now the TOE dream is a wild goose chase and a bunch of hype. the generation of guys that once seriously thought they could do that in the context of differential geometry, without first re-inventing spacetime, are now old men. IMHO

Just a second. How close are we, and what needs to be done? It seems to me that since everything is described by coordinates, etc, that a TOE would have to explain the emergence of a manifold from a singularity. But isn't this pretty close to what CDT is trying to do? Then it would have to explain how particles emerge from spacetime. So why does that seem such a fanciful endeavor? We seem to be touching on it already. I suppose one issue would be that even if we came up with a situation where particles emerged from spacetime through some symmetry breaking process, how would we know if it were the particles we are familiar with in the standard model or if they were some more fundamental particle that we will never connect with the standard model? Isn't there some reason to suspect that there can be no more fundamental particle beside quarks?

Also, if everything emerges from a singularity in a reasonable and traceable way, then it seems that it must have emerged in a continuous fashion. Otherwise, instantaneous changes would by nature defy any attempt at explanation. So I have my doubts that non-commutative geometry is truly fundamental, since it allows for spontaneous, unexplanable changes in the spacetime itself, IIRC. Comments welcome, thanks.

 

[setAI]

CDT is a very interesting computational formalism [as is LQG/ M-theory/ etc]- but I'm more interested in the "computer" itself (^_-)

that is why computationalism/ causal sets/ categories/ and the like are more promising IMO-

 

[marcus]

How close are we, and what needs to be done?...

Mike, I think we are very close (to a model of quantum spacetime)

and when we have a good concrete model of spacetime to build the fields of matter, and its forces, on top of, then the whole picture will change
how the fields are built will depend on what the spacetime foundations are like

so it seems illogical and unproductive to pursue the matter business now when you are still working with outdated models of the continuum---building fields on classical Minkowski space and the like. that has gone about as far as it can without remodeling the spacetime foundations

I gather that SetAI has a rather similar view, since he has also focused his attention on progress being made towards a model of quantum spacetime

 

[marcus]

...
that is why computationalism/ causal sets/ categories/ and the like are more promising IMO-

Look at the focus topics for the October "Loops 05" conference.
Rafael Sorkin (father or at least godfather of causal sets) is on the
invited speakers list and causal sets is one of the topics

Fay Dowker is also an invited speaker, if you know causal sets then you may know of her too.

quote: "The topics of this conference will include:

Background Independent Algebraic QFT
Causal Sets
Dynamical Triangulations
Loop Quantum Gravity
Non-perturbative Path Integrals
String Theory"

for more information see
http://loops05.aei.mpg.de/index_files/Home.html

for the list of speakers for the October conference see
http://loops05.aei.mpg.de/index_files/Programme.html

 

[marcus]

CDT is a very interesting computational formalism ...

setAI I am glad you find it interesting! that is a good sign.
but i think it is more (or perhaps you might say less) than a computational formalism

it is a concrete model of the continuum which may turn out to actually be the RIGHT model----that is become the new paradigm (replacing the differentiable manifold) for how we think of spacetime

the CDT continuum is NOT discrete
however the 4D simplex, cut out of Minkowski space (the space of special relativity) is a kind of "atom of relationship"

and these simplexes are assembled, and the arrangement of them randomly shuffled, in accordance with General Relativity

(so you have elements of both SR and GR)

spacetime becomes a PATH in the "path integral" quantization approach.

but in CDT one does not stop there, with a random assemblage made of a finite number of simplex building blocks, one goes to the limit of small blocks-----so in the end there is no discreteness, no minimal length like Planck length, no atoms of spacetime----there is a continuum

 

[marcus]

I will quote from the conclusion of the latest CDT paper because it is relevant to this response to setAI:


http://arxiv.org/hep-th/0505154
Reconstructing the Universe
J. Ambjorn, J. Jurkiewicz, R. Loll

<<...This paper describes the currently known geometric properties of the quantum universe generated by the method of causal dynamical triangulations, as well as the general phase structure of the underlying statistical model of four-dimensional random geometries. The main results are as follows. An extended quantum universe exists in one of the three observed phases of the model,...
...In summary, what emerges from our formulation of nonperturbative quantum gravity as a continuum limit of causal dynamical triangulations is a compelling and rather concrete geometric picture of quantum spacetime. Quantum spacetime possesses a number of large-scale properties expected of a four-dimensional classical universe, but at the same time exhibits a nonclassical and nonsmooth behaviour microscopically, due to large quantum fluctuations of the geometry at small scales. These fluctuations “conspire” to create a quantum geometry that is effectively two-dimensional at short distances...>>

 

[Kea]

the CDT continuum is NOT discrete...

Marcus

Why are you writing the same things over and over again? Are you even going to try and answer my questions?

Cheers
Kea

 

[Mike2]

Mike, I think we are very close (to a model of quantum spacetime)

and when we have a good concrete model of spacetime to build the fields of matter, and its forces, on top of, then the whole picture will change
how the fields are built will depend on what the spacetime foundations are like

so it seems illogical and unproductive to pursue the matter business now when you are still working with outdated models of the continuum---building fields on classical Minkowski space and the like. that has gone about as far as it can without remodeling the spacetime foundations

I agree with you here. However, it has not been proven yet that you can calculate quantume spacetime without matter. It may be that one cannot exist without the other. I tend to think that some form of very tightly curled up quantum spacetime must have existed, and then matter emerged from that some time later due to expansion. But it may be wrong to think that you can get a nearly flat, almost infinite, spacetime without plugging matter into the equations. Large spacetimes may actually require matter in order to solve for it. It may be that there is no alternative but that spacetime without matter can only be derived for very small universes in the very early moments of its expansion.

 

[marcus]

I agree with you here. However, it has not been proven yet that you can calculate quantume spacetime without matter...

Several of the CDT papers include matter. I would agree in turn with you, Mike, and feel that it is a very interesting direction for CDT research to be taking. Really essential, since matter and spacetime may be existentially linked so that they cannot be satisfactorily modeled separately. In CDT, so far just a beginning has been made! the inclusion of matter in the spacetime model is still very tentative---and has so far been investigated mostly in lower dimensions (simplex D < 4) where the demands for computer time are not so great.

 

[marcus]

has anyone besides me noticed the graphics at the Loops 05 website?

"Loops 05" is probably 2005's most important QG conference----at the AEI in Potsdam Germany 10-14 October

Hermann Nicolai, a director at AEI, is a "swing voter" in Quantum Gravity research, so plays a key role.

this page has one of the graphics as a large still:
http://loops05.aei.mpg.de/index_files/Home.html

this page has several graphics in sequence, including
logos for the AEI and Max Planck Institutes
http://loops05.aei.mpg.de/index_files/Contents.html

hints at how some visual artist imagines quantum gravity (quantum geometry) at the micro-scale and also emerging as a large-scale limit in the distance, or how the conference organizers picture it

BTW here's a dutch article about Renate Loll, for wide audience, if anyone reads dutch. It says she was born in Aachen, Germany.
www.phys.uu.nl/~loll/Web/press/knutselen.pdf
I think it says she just got a sizeable grant of funding for her quantum geometry group at Utrecht, but i can't read dutch.

Oh great, I just found a German translation of the dutch magazine article
http://www.phys.uu.nl/~loll/Web/press/NRCdeutsch.htm

 

[marcus]

...
"Loops 05" is probably 2005's most important QG conference----at the AEI in Potsdam Germany 10-14 October

this page has one of the graphics as a large still:
http://loops05.aei.mpg.de/index_files/Home.html

... artist imagines quantum gravity (quantum geometry) at the micro-scale and also emerging as a large-scale limit in the distance, or how the conference organizers picture it

the topic of this thread is how Loll-style Triangulations changes QG.
It changes the picture in a profound way. We need to explore how it does.
New picture or "paradigm" (see the Loops 05 conference website graphics), emphasis on computer experiments, probing a different quantum spacetime---which is not a smooth manifold like what LQG was based on---different measures of dimensionality from what we are used to with smooth manifolds

the quantum spacetime we get from Triangulations is not discretized, not broken up into little bits, or atoms of relationship, or little computers, or nodes of some imagined cosmic circuitry. Nor is it some fancy abstract algebra construction. It has no minimal length. It is just a new idea of continuum, one that breaks with 1850-style differentiable manifolds and Euclidean txyz space and Minkowski txyz----with all the old kinds of continuum that physics has used for 150 years.
somehow the new model continuum manages to be 4D in the large, but to be 2D at short range, and have spatial slices that are branchy.

 

[marcus]

So this is a basic new departure, and from the looks of the topic list and speaker list of the October Loops 05 conference what will result in the near term is something like a "Causal Coalition" of approaches to QG which will include core-Loop and spinfoams and causal sets (as preached by Rafael Sorkin, Fay Dowker and others). Because the new thing in the Triangulations approach is that spacetime is layered by causality, like the grain in wood that is layered on year by year, giving a direction to time and ideas of cause preceding effect. this was the contribution Loll and Ambjorn made in 1998, that paid off last year.

the hope would be that other approaches to QG can duplicate the gains made in Triangulations, and perhaps converge with it, or diverge in interesting ways.

and maybe just focusing on causality won't be enough, but the way Loops 05 program looks it seems like somebody thinks it might bring things together. the Loop-and-allied QG people need a common reseach program---some common theme that links them together. and they need bridges connecting their different lines of investigation.

Fraid I'm just thinking out loud.

Let's listen to what Lee Smolin and Des Johnston said in September 2004 about CDT Triangulations. They were talking about "Emergence of a 4D World" which we discussed here at PF starting April 2004, but which had just been published in the Americal Physical Society journal "Physical "Review Letters"

<<"It's exceedingly important" work, says Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, Canada. "Now at least we know one way to do this." Des Johnston of Heriot-Watt University in Edinburgh, Scotland, agrees the work is "very exciting" and says it underlines the importance of causality. "The other neat thing about this work is that you're essentially reducing general relativity to a counting problem," Johnston says. "It's a very minimalist approach to looking at gravity.">>

This was part of Adran Cho's article in another APS publication, Physical Review Focus.
http://focus.aps.org/story/v14/st13

 

[Mike2]

somehow the new model continuum manages to be 4D in the large, but to be 2D at short range, and have spatial slices that are branchy.

"Somehow"... I fail to see how 4D volumes can change to 2D volumes as some parameter approaches zero. This goes against my previous understanding of the calculus process. Perhaps it has something to do with the weight given to each volume coupled to how spacetime curves during monte carlo moves. Can you give a reference and page number to how this particular problem is addressed? For me, I have no proof that this change from 4D to 2D is anything more than an anomaly of the algorithm used. Not only that, but they have not explained the use of 4D to begin with. This leaves the background unexplained. Thanks.

 

[marcus]

the Adrian Cho article, with the quotes from Lee Smolin and Des Johnston, point to a few themes:

1. the idea that CDT Triangulations gives you something to duplicate with other Loop-and-allied approaches

2. the idea that introducing a built in causality direction or layering into spacetime has something to do with it

3. the idea that, as Renate Loll put it in a talk she gave in 2002,
"quantum gravity IS counting geometries"

quantizing general relativity boils down to counting geometries

it is the state sum strategy (as in the Feynman path integral) where you add up all the ways something can happen------combinatorial geometry---the probability/counting approach to shape and space---random geometry. there is an interesting literature of random geometry that goes back a long ways.

BTW the Adrian Cho article can give one a false impression, which I will try to correct here. You may get the idea that the CDT quantum spacetime continuum is MINKOWSKI txyz at very small scale, merely because in one of the approximations by flat building blocks it is!

And then the weird non-classical dimensionality only happens at LARGER scale. That is backwards. here is Adrian Cho, which is mostly good, but in this case gives a wrong impression

http://focus.aps.org/story/v14/st13


<<The researchers added up all the possible spacetimes to see if something like a large-scale four-dimensional spacetime would emerge from the sum. That was not guaranteed, even though the tiny bits of spacetime were four-dimensional. On larger scales the spacetime could curve in ways that would effectively change its dimension, just as a two-dimensional sheet of paper can be wadded into a three-dimensional ball or rolled into a nearly one-dimensional tube...>>

because you take a limit, all Minkowski familiar txyz flatness goes away at small scale. the small scale of the continuum is where the weirdness is, and in the very thin slices

it is the LARGER scale and the THICKER slices where things look normal.

so you can see that Adrian Cho has it backwards in one of his nuances, like as a journalist he put his undershirt or his socks on the wrong way, but basically he is very good, the best American reporter i have seen so far on this.

 

[marcus]

"Somehow"... I fail to see how 4D volumes can change to 2D volumes as some parameter approaches zero...

I can believe you have trouble picturing it! It is hard to visualize.
I can sort of dimly picture it but at least at this point I cannot put my mental pictures into your head. the piece of cloth analogy is something, but not very good, for a big creature walking on the cloth it is 2D but for a little mite crawling along a thread it is 1D. Intuitive notions of dimension are imprecise and don't include fractional dimension very well, like 1.4 D. or 1.46 D.

Fortunately the CDT people are able to determine the dimension objectively by running diffusion processes, and by other methods like comparing spatial separation and volume. these things give rigorous and practical or operational meaning to dimension. So we do not have to rely on possibly deceptive or inadequate intuitive mental images!

IF YOU GO OUT TWICE AS FAR, DOES THE VOLUME INCREASE 4-fold, or 8-fold, or 16-fold? (as with dimension 2D, or 3D, or 4D)

or maybe when you go out twice as far does it increase 5.66-fold, as when the dimension of the surrounding space is 2.5?

That is the wonderful thing I think---that one is able to measure the dimension objectively
Indeed the concept of dimension has no meaning, operationally speaking, except if you say how you are going to measure it. different ways to measure, different dimension numbers.

 

[marcus]

There is a telling quote on page 31 of "Reconstructing the Universe"
that comes right at the start of section 6.2 on Spectral dimension.

<<Given the results of the previous sections, one might be tempted to conclude that the geometry of our dynamically generated ground state simply is that of a smooth four-dimensional classical spacetime, up to Gaussian fluctuations. A more detailed analysis of the geometry of spatial slices makes explicit that this is not so... >>

what this says to me is that the picture of spacetime continuum coming out of this is not a differentiable manifold.

most spacetimes modeled whichever way---string, LQG---are basically differentiable manifolds or some fixed even more traditional txyz space.
this isn't.

they take a microscopic dynamic principle operating at Planck scale and even at scale arbitrarily smaller-----in the limit the size of simplex can go below Planck since it is not a physcial thing, just a mathematical tool to represent spacetime dynamics---and from that GROWS the largescale phenomenon of spacetime.

which they did not start by assuming was any dimension or had this or that properties

to see the contrast, in stringy theories you start with some readymade oldfashioned differentiable manifold for to be the "target space" where the strings live, and it is very elaborate with many dimensions and all specified which are curled up and which are extended etc etc. Like you already have this complicated graph paper world to work in even before you do anything.

they don't start with some elaborate pre-constructed graph paper "house" for their stuff to live in, they GROW it from dynamics operating at unrestrictedly small scale. If you have the computer time you can make the scale that the dynamic runs at smaller than Planck, there is no restriction on how small you can make the simplexes or how fine you can approximate. the seething turbulent thing you are studying is a CONTINUUM, it does not have a smallest.

it is a continuum
it is not readymade, it grows.
it is not like any differentiable manifold
what else can we say?

presumably it isn't coordinatizable either because at small scale you would just need to give two numbers to specify your location in space to some other person, and at a larger scale you would need to give the other person three numbers so they could find you.

 

[Chronos]

I don't think the 2D picture, or even fractional D, is far fetched. Look at all the 2D black hole models that started emerging like 10 years ago. They took different approaches yet somehow arrived at very similar versions of spacetime in the Planckian realm as does CDT. That is what caught my attention when CDT hit the pavement. I see different approaches converging with the same outcome - a smoke and fire thing. CDT may not be dead on, but I think it is scary close.

 

[marcus]

... - a smoke and fire thing. CDT may not be dead on, but I think it is scary close.

my intuitive feeling is in line with yours. I think other people as well (QG folks at AEI who organized the conference, AEI director Hermann Nicolai) must have gotten similar signals because of the way the Loops 05 conference has been set up. the topic list and choice of invited speakers give it a different direction from past Loops conferences. Some influential people must have had a similar impression back when the conference started taking shape and direction

maybe not dead on but scary close is a good way to put it. the other QG people may want to see if they can copy CDT, and what, if any, similar results they can get. in any case some serious openminded consideration will do no harm

MIKE2 I SEE YOUR NEXT POST and to save space I will edit in, and reply here. IMO one CAN think of dimension as the result of measuring and in terms of operators on a hilbert space. With one type of dimension one can think of operators corresponding to measuring distance and to measuring volume. One does a bunch of measurments of radius and volume and compares, to see if the volume is proportional to R^2.5 or R^2.9 or R^3.0 or R^3.1.

the word dimension only has meaning if you say what definition of it you are using and therefore what measurements it is going to depend on, and in this example I mean the concept of dimension that depends on measuring distances and volumes.

in their computer simulations they plot CURVES fitted to DATA about distances and volumes. that is rather like what you were talking about I think-----making measurements (which would correspond in theory to operators on a hilbertspace)

I hope you check out the curves they plot in various figures in their recent paper. it gives a concrete idea of the various meanings of dimension.

 

[Mike2]

my intuitive feeling is in line with yours. I think other people as well (QG folks at AEI who organized the conference, AEI director Hermann Nicolai) must have gotten similar signals because of the way the Loops 05 conference has been set up. the topic list and choice of invited speakers give it a different direction from past Loops conferences. Some influential people must have had a similar impression back when the conference started taking shape and direction

maybe not dead on but scary close is a good way to put it. the other QG people may want to see if they can copy CDT, and what, if any, similar results they can get. in any case some serious openminded consideration will do no harm

So is the dimensionality the result of an operator on the Hilbert space of various geometries in 4D? I can accept changing dimensionality on that basis, maybe. Is there a more common analogy with simple QM that could help visualize what's going on mathematically? Thanks.

 

[marcus]

So is the dimensionality the result of an operator on the Hilbert space of various geometries in 4D?...

Hi Mike, I edited my reply to your post yesterday morning into preceding one (#62) and that may have led to your missing it. the general idea is, I think, right.

in their approach one can make various measurements (which would convenionally correspond to operators), and from these measurements one determines various dimension numbers----for thin spatial slices, for thick slices, for the whole spacetime, for shortrange, for longrange...

there is no one right definition of dimension and no one correct dimension number (because spacetime is not a differentiable manifold with coordinates, where there would be)

I have not seen the construction of a hilbert space of "various geometries in 4D" as you say. With the path integral approach the focus is on the path inegral and not on the hilbert space. But I expect one COULD be constructed for the various spacetime geometries.

these geometries would not be "in 4D" though, I think. they would not be IN any larger space, they would not be embedded in anything, and some of the spacetimes would have dimension greater than 4

for the first 10 years or so that people did dynamical triangulations approach, one of the problems that dogged them was that when you tried building a space of low dimension, like 2 or 3 or 4, it might turn out to have unboundedly high hausdorff dimension. essentially the dimension would go infinite

(even if you were building the space out of 2-simplices and wanted it to be 2D, or when you were building it out of 3-simplices and wanted it to come out 3D--------a kind of crumpling occurred in the computer simulation that led to results of very high dimension)

these possibilities are presumably still there, they just have very small PROBABILITY. so now we have results where the EXPECTATION VALUE, or average value, of the dimension comes out 4, or 3.99 or 4.01
(look at the plots of their data in their paper, it does not come out exactly 4D)

so these geometries are not quite exactly "various geometries in 4D", as you said. But there would be some hilbert space of various geometries that you could construct and define operators on

=====================
Hi Mike just saw your post #65 (which follows) will reply here for compactness. Yes I agree it should be straightforward, but I cannot picture the explicit construction of the Hilbert space for the continuum limit as the simplexes shrink down to nothing. for the path integral corresponding to one fixed size of simplex, I can roughly form an idea of how the Hilbert space could be constructed, maybe also for spacetimes of a fixed volume.
A basis could be made from the discrete set of all possible gluings, which one could try to write down and enumerate combinatorially. I can see the advantage for people who are more familiar with the canonical formulation than with path integrals. But I have not noticed this construction having been done by any of the Triangulations people. Here is your post #65 I am responding to

If there is a path integral, then shouldn't it be an easy matter to convert it to a canonical version with operators on a wave function type of equation? It would probably be easier to understand things in this context, right? Thanks.

 

[Mike2]

so these geometries are not quite exactly "various geometries in 4D", as you said. But there would be some hilbert space of various geometries that you could construct and define operators on

If there is a path integral, then shouldn't it be an easy matter to convert it to a canonical version with operators on a wave function type of equation? It would probably be easier to understand things in this context, right? Thanks.

 

[marcus]

hi Mike, I edited post #64 to include a response to yours.

To get back to the main question----of how CDT changes the QG map:

I think TOPOLOGY CHANGE is a significant issue. the usual version of LQG does not deal with any change in spatial topology. there is a spatial manifold Sigma and the spacetime manifold is simply the cartesian product Sigma x R, that is Sigma plus a time axis. And the Sigma is usually just the 3D sphere S³. So topologically the usual LQG spacetime is S³ x R. LQG is exciting for other reasons than topology.

Loll and Westra have two interesting papers in CDT "Triangulations" gravity about extending the path integral so the sum is over various topologies.

I suspect there is another paper in the works which will come out in time for the October Loops 05 confence.

FAY DOWKER is one of the invited speakers at Loops 05, and she has written at least 5 papers bearing on spacetime topology in QG. Loll and Westra
http://arxiv.org/hep-th/0309012
cite these 5 papers of Dowker.

I have no opinion on Dowker papers cause I haven't looked yet, but one that they cite is
http://arxiv.org/gr-qc/0206020
Topology change in quantum gravity
Fay Dowker
18 pages. Contribution to the proceedings of the Stephen Hawking 60th birthday conference, Cambridge, January 2002

"A particular approach to topology change in quantum gravity is reviewed, showing that several aspects of Stephen's work are intertwined with it in an essential way. Speculations are made on possible implications for the causal set approach to quantum gravity."

Independent of how I like Fay Dowker's reseach, when I have a look later today at it, I can see the theme of topology change emerging in Renate Loll's CDT research and at the Loops 05 conference.

So how is CDT changing the quantum gravity map?

1. putting topology on the table (where Loop has a S³xR spacetime with constant spatial topology)

2. getting away from idea that spacetime is a differentiable manifold (very old idea going back to Riemann 1850)

3. dynamical dimension, able to be different at close range and to change continuously

4. really making the "state sum" or Feynmanian "path integral" work finally.

5. idea of quantum spacetime dynamics---that a microscopic dynamic principle operating down at Planck scale can GENERATE macroscopic spacetime with its wellknown properties.

 

[Mike2]

is a differentiable manifold (very old idea going back to Riemann 1850)

I'm not sure I believe that no manifolds are involved in CDT. For as you shrink the length scale to zero, then you are talking about a continuosly changing metric, a metric field, on what else... but a manifold, right?

 

[Mike2]

=====================
Hi Mike just saw your post #65 (which follows) will reply here for compactness. Yes I agree it should be straightforward, but I cannot picture the explicit construction of the Hilbert space for the continuum limit as the simplexes shrink down to nothing. for the path integral corresponding to one fixed size of simplex, I can roughly form an idea of how the Hilbert space could be constructed, maybe also for spacetimes of a fixed volume.
A basis could be made from the discrete set of all possible gluings, which one could try to write down and enumerate combinatorially. I can see the advantage for people who are more familiar with the canonical formulation than with path integrals. But I have not noticed this construction having been done by any of the Triangulations people. Here is your post #65 I am responding to

It seems to me that you simply replace the measure, x, in the traditional path integral with the metric, g, and convert to canonical form as usual. But I suppose that the detailed nature of the Action integral and the Lagrangian if the CDT path integral prevents knowledge of the Hamiltonian that we would then use in the canonical version. Is that your take on the subject?

 

[marcus]

I'm not sure I believe that no manifolds are involved in CDT. For as you shrink the length scale to zero, then you are talking about a continuosly changing metric, a metric field, on what else... but a manifold, right?

there is something called a "simplicial" manifold which is glued together out of simplexes. it can be a bit craggy and jaggy compared with a "differentiable" manifold. I think you know about this.

when the CDT people take the length scale down to zero they don't necessarily get a differentible manifold. In the first place, altho this is not the main reason let's make explicit that the length scale does not have to go to zero in a smooth way
it could go to zero in spastic jumps
1/17, 3.14x10^-6, (39 billion)^-1,...

A way to picture is that they have a sequence of BLURS made of many simplicial manifolds where in each blur the simplexes are all the same size 'a'. and they take 'a' down. they could, for example, divide 'a' in half each step. so the component simplexes get smaller and smaller

at each step you don't have just one particular collage of simplexes making one particular simplicial manifold, you have a blurry quantum cloud of possible collages

and the 'a' is jumping, maybe spastically, down in size, so the cloud consists of things getting finer and finer. But it is not clear that this process converges to a differentiable manifold, or to any kind of manifold that anyone has yet defined or studied.

BTW Mike you know historically the Greeks resisted the idea of the real numbers for a time, and could only believe in fractions.
nowadays mathematicians define the reals by various equivalent means of which a very common is as LIMITS OF SEQUENCES OF RATIONAL NUMBERS.

that is you only ever get your hands on rational numbers, but you fantasize having an infinite sequence of rationals with larger and larger denominator and the LIMIT of that sequence (which is an abstract thing you never actually get hold of) is the real number

and computers use rational number arithmetic, as an approximation of abstract real number arithmetic which they cannot do because no actual concrete data sequence is ever infinite.

And nevertheless we think of real numbers as REAL, even tho abstractly defined as limits of sequences of rationals.

Well it may be that differentiable manifolds are like the rational numbers were for the Greeks. We can't think of a continuum any other way. But, like, MOST continuums are probably not differentiable manifs!
just like most of the numbers on the real line are not rational---not expressible as fractions----only as limits of sequences.

BTW I believe you are right in saying that manifolds are (at least potentially) INVOLVED in CDT because they always can, for any particular simplicial manifold, coordinatize it and find some way to make a smooth, or mostly smooth, thing out of it. But that is kind of tangential because we are not talking about taking limits of individual manifolds. It is an option, but not part of the main business

 

[wolram]

http://citebase.eprints.org/cgi-bin/fulltext?format=application/pdf&identifier=oai%3AarXiv.org%3Agr-qc%2F0210061

a 2002 paper by Fay Dowker etc.

http://www.imperial.ac.uk/research/theory/research/quantum.htm

A link that may be of interest.