Likelihood of M-theory: 1-10 Scale

[marcus]

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

My point is that a CDT is a derived concept. I've read through the CDT papers and have nowhere seen how to acquire a triangulation from more basic principles. When the authors eventually figure out how to do this, instead of presupposing the existence of a triangulation, they will realize they are doing noncommutative geometry.

Thank you, kneemo.
I was too polite to interrupt Marcus because I know how much he adores CDT. Marcus, listen carefully to what kneemo is trying to tell you (and what I have been trying to tell you for a long time).
Cheers
Kea
...

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

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

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

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

 

[marcus]

Kea post #55

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

later Kea post

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

pseudo-Riemannian metrics live on manifolds

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

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

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

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

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

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

That would be very nice.

 

[nightcleaner]

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

Hi all

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

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

pick another point and repeat the universal rotation,

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

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

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

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

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

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

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

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

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

Thanks, and be well,

Richard

 

[marcus]

selfAdjoint, you replied to what I said here

... Let me try to list some ways CDT is DIFFERENT.

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

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

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

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

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

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

your reply went in part

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

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

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

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

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

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

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

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

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

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

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

 

[selfAdjoint]

Comments on the simplex rotation idea

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

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

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

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

pick another point and repeat the universal rotation,

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

What do you mean, "representative sample"?

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

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

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

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

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

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

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

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

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

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

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

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

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

Best of luck to you on funding!

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

Thanks, and be well,

Richard

 

[selfAdjoint]

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

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

 

[Kea]

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

Hi Marcus

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

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

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

 

[marcus]

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

What I heard you say was

... and then letting the scale go to zero to recover the continuum. That's the QCD lattice strategy, and it seems to be Ambjorn et al's strategy too.

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

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

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

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

...

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

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

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

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

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

 

[marcus]

Hi Marcus

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

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

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

My point is that a CDT is a derived concept. I've read through the CDT papers and have nowhere seen how to acquire a triangulation from more basic principles. When the authors eventually figure out how to do this, instead of presupposing the existence of a triangulation, they will realize they are doing noncommutative geometry.

Thank you, kneemo.
I was too polite to interrupt Marcus because I know how much he adores CDT. Marcus, listen carefully to what kneemo is trying to tell you (and what I have been trying to tell you for a long time).
Cheers
Kea
...

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

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

cheers

 

[Kea]

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

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

Cheers
Kea

 

[marcus]

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

that sounds very interesting if you can give it substance!
BTW I am not interested in primitive arguments like UGH DIS GOOD!
MINE IS BIGGER THAN YOURS! UGH DIS BETTER THAN DAT!
So I hope that you mean some logical connection when you say
NCG "consumes" CDT.

I never heard "consumes" in a mathematical discussion. It sounds more like poetry. what I hope you mean, and can show, is that CDT can be LOGICALLY DERIVED FROM NonComGeom.

I am still waiting for some logical connection to be established. It would, as I've said several times here, delight me. Unfortunately I cannot simply take your word for there being any connection at all. I need hard online evidence.

As to what you think you have been telling me, your memory of what you have said is doubtless different from mine.

You claim to have told me features which a theory of quantum gravity should have.
I have no idea what you are tallking about. Would you list them please?

cheers, Kea , list them and please do not be vague or use esoteric terminology. Say clearly and simply what features a satisfactory QG should have.
I want it in simple terms so that as many PF posters as possible will understand.
Go for it kiwibird!

 

[Kea]

You claim to have told me features which a theory of quantum gravity should have. I have no idea what you are tallking about. Would you list them please?

Hi Marcus

All right. Let's begin with the short list below. I have left out all technical jargon, but that means it might appear a bit vague, for which I apologise.

Features that a theory of QG should have
----------------------------------------

1. Unprejudiced geometry All path integral type approaches that I am aware of, including CDT, make a selection of contributing geometries with no backup physical arguments. Without going into fancy maths, there is a way to generalise the notion of a space such that there is more than one option for the real numbers. Any restriction to more ordinary spaces should be backed up with good physical grounds.

2. Geometric observables A rigorous notion of observable needs to be defined whilst respecting point 1.

3. Quantum general covariance Discussed a little in the "Third Road", QGC is a sort of Machian equivalence principle. GR began with a consideration of processes between separated matter domains. Separation between observers is not to be mistaken for basic discreteness of an objective reality at small scales. The only objective reality (in the sense that there is a universal observer) is a classical one. QGC must define the interaction of basic observables.

4. Solve the measurement problem

5. Recover Einstein's equations This means more than a concrete recovery of the equations in the limit of universal observation. Newton's equations describe the orbit of Mercury perfectly well, but the answer happens to be wrong to the limit of early 20th century observation. GR provides the correct answer, but more to the point: GR shows us when Newtonian mechanics breaks down. QG must be very clear about when GR breaks down. For example, it might say that we will not directly observe gravitational waves.

6. Calculation of Standard Model parameters QG should eventually be able to calculate numbers such as the fine structure constant and (rest) mass ratios. Masses are like quantum numbers. QG isn't QG if it can't tell us something about them.

7. Make new quantitative predictions Obviously.

8. Explain the 4 dimensionality of local classical spacetime Some people think CDT does this, but the physics is far from clear. Some people think quantum computation explains this. This really comes under point 5.

Cheers
Kea

 

[marcus]

Now we seem to be making rapid progress. I am glad to see your list of QG desiderata. I have no reason to quarrel with any because it is your list. Maybe i will try to formulate my own list, or borrow a definition of QG from Renate Loll lecture notes hep-th/0212340

...

Features that a theory of QG should have
----------------------------------------

1. Unprejudiced geometry
2. Geometric observables
3. Quantum general covariance
4. Solve the measurement problem
5. Recover Einstein's equations
6. Calculation of Standard Model parameters
7. Make new quantitative predictions
8. Explain the 4 dimensionality of local classical spacetime

this is actually a pretty good list. it may be helpful in this or other threads.
congratulations on boiling it down like this.
oh, please be specific about number 4. the measurement problem

 

[Kea]

please be specific about number 4: the measurement problem

Hi Marcus

The measurement problem is about the 'logic of measurement' and the context of an observer in the universe at large. Since logical issues are already raised by the generalised geometries that I referred to, and certainly in the question of what causality means, a proper definition of QG observables should also solve the so-called measurement problem.

Must go again.
Cheers
Kea

 

[marcus]

The measurement problem is about the 'logic of measurement' and the context of an observer in the universe at large. Since logical issues are already raised by the generalised geometries that I referred to, and certainly in the question of what causality means, a proper definition of QG observables should also solve the so-called measurement problem.

I think maybe I can put it in more concrete terms than that...in usual quantum mechanics the observer who measures is distinct from the experiment---but with the universe we can't stand outside it and measure.

In usual QM there are two separate systems the cat-in-the-box or whatever, and the guy in the white coat standing outside. QM is about what the guy in the white coat can learn by preparing the experiment in a certain way and then making measurements. the observables are the measurements he is allowed to make.

the trouble with the universe is that the man in the white coat cannot stand outside it. so it breaks the sacred two-system model that is basic to QM.
now what can QM be a theory of?

maybe that is a concrete statement of the measurment problem

 

[Chronos]

The 'measurement problem' rubs me wrong. I'm not yet willing to concede it is physically meaningful. I try to keep an open mind, but the screen door remains shut - I'm trying to keep the flies out.

 

[marcus]

What is the absolute minimum list of requirements?

Oh, you re there Chronos, good. How would you pare kea list down to the bare bones?

 

[marcus]

Kea has a list of things that it would be NICE if a QG would eventually do for us. But we are just evolved fish on a small planet and we take what we can get.

...

Features that a theory of QG should have
----------------------------------------

1. Unprejudiced geometry
2. Geometric observables
3. Quantum general covariance
4. Solve the measurement problem
5. Recover Einstein's equations
6. Calculation of Standard Model parameters
7. Make new quantitative predictions
8. Explain the 4 dimensionality of local classical spacetime

Chronos let us chuck out #6. Once we get a QG, that is just a theory of spacetime and its geometry, then we can reconstruct the Std Mddle ON TOP OF IT and maybe things will improve, but that is a later chapter

the three I like very very much, of Kea list are these three, what about you?
5. Recover Einstein's equations
7. Make new quantitative predictions
8. Explain the 4 dimensionality of local classical spacetime

 

[Kea]

on Reconstructing...

The \gamma = \frac{1}{3} from p39 of Reconstructing comes from AJL's reference [60]

http://arxiv.org/abs/hep-th/9401137
http://arxiv.org/abs/hep-th/9208030

on 2D gravity models and Ising spin systems. To quote the conclusion of the first paper: "The model is closely related to the matrix models studied in [-] and to the c \rightarrow \infty limit of multiple Ising models studied in [-]. However, our approach has the virtue of being simple and avoids any use of matrix models..."

 

[marcus]

The \gamma = \frac{1}{3} from Reconstructing comes from AJL's reference [60]
...

Kea thanks for helping with the detective work. it can be like herding cats
to try to track down all the notation in a major paper like this.

since Ambjorn is or has been a string theorist and sees lots of connections there
one can get concepts and notation crossing over from string theory

any other things to point out

 

[Kea]

The \gamma = \frac{1}{3} from p39 of Reconstructing comes from AJL's reference [60]

http://arxiv.org/abs/hep-th/9401137
http://arxiv.org/abs/hep-th/9208030

The first paper's reference number [3] comes from the hey-day of String theory (sorry - no web version; library visits required)

Conformal field theory and 2D quantum gravity
J. Distler, H. Kawai; Nucl. Phys. B321 (1989) 509-527

On page 525 they have the formula for \gamma for any genus g, not just g = 0 as considered by AJL and others. This is explained in more detail in the supertitled paper

Super-Liouville theory as a two-dimensional, superconformal supergravity theory
J. Distler, Z. Hlousek, H. Kawai; Intl. J. Mod. Phys. A5 (1990) 391-414

The formula for g = 0, on page 400, is

\gamma = 2 + \frac{1}{4} (D - 9 - \sqrt{(9 - D)(1 - D)})

in terms of a String theoretic dimension parameter. Below the formula the authors state: "Note here that the expression makes sense only for D < 1". Clearly this isn't true, but they are motivated by the connection between D and a coefficient in their action, which they require to be real since they're afraid of ghosts. Since we don't care about String theory
we can happily ignore this statement and plug \gamma = \frac{1}{3} into the formula, to find

D = \frac{544}{60} = 9.066666666

How cool is that! I'm not sure what it means yet. By the way, the partition function, on the same page 400, may be written

Z = \int_{0}^{\infty} \textrm{d}V S(V)

where

S(V) = \nu e^{- \frac{\lambda}{G} V} V^{-3 + \gamma}

roughly in the terminology of Reconstructing. Compare this to page 33 of Reconstructing where AJL discuss critical exponents in the Baby Universe papers. In summary, it appears that the cool things in Reconstructing have quite a lot to do with early Strings and superconformal theories, or, on the other hand, perhaps some of the later topological Strings stuff. We need to rope in Distler to look at CDT.

 

[selfAdjoint]

Chronos let us chuck out #6. Once we get a QG, that is just a theory of spacetime and its geometry, then we can reconstruct the Std Mddle ON TOP OF IT and maybe things will improve, but that is a later chapter

I don't see how you can say this. M-theory, for all its problems, proposes to do better than that, to get both background-free spacetime and the standard model (maybe supersized) out of the ONE theory. And that's the goal of Thiemann's Phoenix Program too. Why should we give up on this goal just because AJL have made a breakthrough with the Regge Calculus?

 

[marcus]

The first paper's reference number [3] comes from the hey-day of String theory (sorry - no web version; library visits required)

Conformal field theory and 2D quantum gravity
J. Distler, H. Kawai; Nucl. Phys. B321 (1989) 509-527
...

Back in the heyday of String theory, Jan Ambjorn was doing String research himself----I just checked his papers on arxiv.org (which only goes back to 1991) and he has over a 100 papers going back to when arxiv opened, and quite a lot of them are String.

I'd guess we can expect to continue seeing stringy references and notation in CDT, if for no other reason because of Ambjorn's earlier research activities.

 

[Kea]

Super-Liouville theory as a two-dimensional, superconformal supergravity theory
J. Distler, Z. Hlousek, H. Kawai; Intl. J. Mod. Phys. A5 (1990) 391-414

The formula for g = 0, on page 400, is

\gamma = 2 + \frac{1}{4} (D - 9 - \sqrt{(9 - D)(1 - D)})

From page 406: "From the point of view of the random surface theories, the particularly interesting quantity is the susceptibility exponent because of its relation to the Hausdorff dimension. Actually we only need \gamma
[for genus zero] since d_{H} is proportional to \gamma^{-1}."

Of course this agrees with the d_{H} = 3 of section 6 of the Reconstructing paper.

 

[Mwyn]

how does m-theory explain quarks?

 

[marcus]

... to get both background-free spacetime and the standard model (maybe supersized) out of the ONE theory...

I see that as a longterm goal. (It seems obvious. I can't imagine anyone not looking forward to theorists putting quantum spacetime and matter into ONE theory.)

I prefer a minimalist definition of the "quantum theory of gravity" goal which can be inclusive of incremental efforts modestly aimed quantizing relativity, and I think overreaching efforts may prove a colossal waste of time.

I do not see the all-encompassing "ONE theory" criterion as a helpful way of deciding which models of quantum spacetime are interesting.

I object to defining "quantum theory of gravity" in a way that EXCLUDES those efforts which make no attempt at explaining the various particles and forces at this time.

I think that is rhetorically stacking the deck against the modest, one-step-at-a-time, approaches and in favor of the grandly ambitious (possibly premature) ones.

Getting a new quantum spacetime continuum is a hard problem. Indeed quantizing Gen Rel has been an historical roadblock. The CDT authors are focussing on that (not on incorporating the Std Mddle of Matter at the same time in one grand fell swoop) and I suspect that will prove the more efficient path for making progress towards the ultimate goal.

 

[Chronos]

Pardon me for being a little slow catching up with this thread [my regular computer commited suicide]. I would settle for the odd numbers on Kea's list. I think if you can accomplish that much, the rest should fall into place rather naturally. Of course I would also expect it to closely match all known observations supporting predictions of both GR and QFT. I also think it should be renormalizable at some scale. I'm not sure you could otherwise legitimately call it a quantum theory.

 

[marcus]

I would settle for the odd numbers on Kea's list...

Hi Chronos, you might wish to use a little caution, or get some clarification about the implications, before you buy #1. Here is it in full:

1. Unprejudiced geometry All path integral type approaches that I am aware of, including CDT, make a selection of contributing geometries with no backup physical arguments. Without going into fancy maths, there is a way to generalise the notion of a space such that there is more than one option for the real numbers. Any restriction to more ordinary spaces should be backed up with good physical grounds...

So, Chronos, certainly "unprejudiced" SOUNDS great and even PC and all, but am I "prejudiced" if i decide to use real and complex numbers, instead of fancier stuff like, say, quaternions, octonians, some noncommutative matrix algebra? what's the rhetorical slant here? shall physicists be saddled with the obligation to give solid physical reasons for using the real numbers? should they be called "prejudiced" if they don't justify NOT using quaternions? How much esoteric math will the thought police force me to eat, if I refuse to give them "physical grounds" for just using ordinary math.

my feeling is that IT IS UP TO THOSE PEOPLE USING ESOTERIC GEOMETRIES AND NUMBER SYSTEMS, to physically justify their choices if they want to. But those math tools which physicists usually suppose don't require justification, and which are traditional with physicists, they should keep on using without having to justify it. Especially, as recently with CDT, they work brilliantly in practice and lead to breakthroughs!

my thought is that working physicist like Renate Loll is too busy getting new results---using familiar, and rather modest, means like little chunks of Minkowski txyz space, and the real numbers. She does not have time to
justify, to some philosopher, her NOT using more esoteric math. She should not have to give "good physical reasons" for using the simple traditional mathematical materials that she find work.

 

[Kea]

I would settle for the odd numbers on Kea's list.

Hi Chronos

I always appreciate your point of view. I'm curious: what is your objection to point 2 (geometric observables)?

Cheers
Kea

 

[Chronos]

Hi Kea! No objection. I just think an unprejudiced geometry, which I interpreted as being diffeomorphism invariant and background independent, would naturally produce the correct geometric observables.