Going Beyond LQG: how to define QFT over Loops?

[MTd2]

Does anyone have any ideas? Marcus, do you have?

 

[marcus]

Does anyone have any ideas? Marcus, do you have?

There are some long-standing ideas for defining quantum fields on spin networks. As I recall, Rovelli reviewed them in his survey of LQG that he was invited to give at Strings 2008 at CERN.

You can get the slides PDF separately and check to see what he says, or watch the video.
I will find the links. I keep links like that in the "Introduction to LQG" thread.

The video and slides of Carlo Rovelli's talk at Strings 2008 provides a good introductory overview of LQG.
Here are the links:
Video:
http://cdsweb.cern.ch/record/1121957?ln=en
Slides:
http://indico.cern.ch/getFile.py/access?contribId=30&resId=0&materialId=slides&confId=21917

Let me know if the links don't work. I think this is the most recent authoritative survey. I don't know that much has been done with QFT defined on spin networks---you may not find what Rovelli has to say about it all that satisfying. It amounts to putting extra labels on the edges of the spin network graph, and there may also be isolated vertices (univalent, met by only one edge). I apologize if my memory is wrong and Rovelli does not talk about putting matter into the picture. Let me know also in that case and i will hunt up another review.

Oh, there is always Rovelli's book. If you don't have it, the 2003 draft is free to read on line. It has a section on putting matter into Lqg.

 

[MTd2]

On page 28 of the slides (14 of the pdf), he says matter fields can be achieved by braidings. I guess this is what Yidun Wan was trying to do, but where is he now Marcus ? Well, I thought he would be dead by now, but I found him on the participants' list:

http://www.loops09.org/asp/xtgl/part5.htm

 

[marcus]

On page 28 of the slides (14 of the pdf), he says matter fields can be achieved by braidings...

No, let us try to report accurately. He does not say that matter fields can be achieved by braidings. He describes the usual way (not braiding) by a label on the spin network.

Then afterwards, as a footnote really, he mentions braiding with a question mark. As far as I know, Rovelli has never indicated any interest in the possibility to represent particles by braiding.

 

[MTd2]

He describes the usual way (not braiding) by a label on the spin network.

But, wouldn't adding new lebels like adding new loops? Would't that be redundant since it would be topological modification and so would yield braidings anyway?

 

[marcus]

But, wouldn't adding new lebels like adding new loops? ...

I don't see the similarity.

 

[MTd2]

I don't see the similarity.

I am not sugesting anything. I am just clueless to what you mean by adding labels. I took it you were adding more loops/edges per vertex. What is the meaning adding new labels then?

 

[MTd2]

Please answer. I would really love to know.

 

[marcus]

... What is the meaning adding new labels then?

Maybe the thing for you to do is to actually listen to Rovelli's survey talk to the String 2008 audience.
He writes down explicitly what is adding new labels to the network. Gives an example.

BTW I think you know that the LQG theory is not about loops. So it could confuse people when you ask "how to define QFT over loops".

The theory began back around 1990 being about loops (so it got its name), but that changed very quickly and one never sees loops discussed in LQG papers. The quantum state of the spatial geometry is given instead by a labeled network. The evolution of the network is given by a labeled foam. The strategy for defining QFT on states of geometry has normally been to add labels to the networks.

A good way to become acquainted with the basics of LQG is to look at the free online version of Rovelli's book. More detailed and less elementary than his talk to the String audience.
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
He talks about how to put matter fields into the LQG picture. I can find a Chapter reference for you, if you need one. It should be clear from the Table of Contents, but if you need help please ask.

 

[MTd2]

Marcus, I take it loops are 1 dimensional topological manifolds, without internal space, which can be glued to others at its borders (the vertex). The intersection and the network link is where quantum states can be defined, because these are the unique topological quantities striclty preserved in so defined loops. What you are saying it is that Rovelli wants to give new labels to the vertexes? But this is weird, can you tell me any reasonable realization of that?

What I see more reasonable is to define is braidings by using 3 or more links between vertexes, that is, braids. This was succesfull enough to define feynmann diagrams: http://es.arxiv.org/abs/0809.4464

And not to forget, a few days ago, it was discovered that certain quantum states on a tetrahedrom had the quantum states similar to those of topological strings.

 

[marcus]

... What you are saying it is that Rovelli wants to give new labels to the vertexes? ...

No! If you would simply just read the PDF page that you already referred to you would see that is not the case! Please please learn something about LQG if you want to talk about it.
His talk to the String theorists at their annual convention is like a kindergarten basic intro. It is radically simplified for them to understand. Just the basics. It would be good to watch the whole video as a kind of minimum introduction. Have you got half an hour?

 

[MTd2]

If you would simply just read the PDF page that you already referred to you would see that is not the case!

Do not get mad at me. I just do not get the word "label". For instance, in the book you mentioned, there is not a single word "label" there. Try a PDF search.

Anyway, if you look at section 7.2.4, I don't understand why would that be any different from the definition I gave.

 

[marcus]

... For instance, in the book you mentioned, there is not a single word "label" there...

I opened my copy of Rovelli's book (Quantum Gravity) to page 19
In the half page following equation 1.11, I counted six occurrences of the word "label".
And an intuitive description of the significance of the labels on the network.

If you count the plural "labels" and the verb form "labeled" there must be hundreds of occurrences in the book, maybe thousands. Nothing is more basic than the labeled network, in describing LQG. So I am glad you had a look at Rovelli's book, I encourage you to read extensively and carefully! But I am astonished that you found no occurrence of that word in the book.

Anyway, if you look at section 7.2.4, I don't understand why would that be any different from the definition I gave.

I am very happy that you had a look at section 7.2.4! I hope you will read the whole section 7.2, about implementing matter! If you think that the description of gravity+matter state in 7.2.4 is what you had in mind when you were posting, that's fine. Wonderful even.

It doesn't matter that, to me, it didn't sound like it.

 

[MTd2]

I opened my copy of Rovelli's book (Quantum Gravity) to page 19
In the half page following equation 1.11, I counted six occurrences of the word "label".
And an intuitive description of the significance of the labels on the network.

That's why you are finding "label". In the pdf, there is no ocurance of that word anywhere! And I looked at the half page following eq. 1.11.

Look, my beef with such definitions of fields it is that at the scale of a link it is that you don't have the concept of scale to start with, because in this case, you should look at the contextuality, that is, how is the configuration of the network configure the probability of distribuition of the geometry and so of an emerging metric. Then, I think it is not possible to define an holonomy for fields, because it is not possible to say where something starts and ends at such scael.

 

[Coin]

MTd2: I think Marcus is resisting giving an explanation of this because it's too basic a question to really adequately explain in a comment thread? The word "label" does appear in the PDF, on the page labeled 14 that Marcus mentioned to begin with:

"Basis of H: abstract spin network states: graph labelled by spins and intertwiners."

The "labels" are simply part of the definition of a spin network. As I remember it the "label" is something like just some mathematical object that is "attached to" every edge of the graph. In Penrose's original proposal the objects are half-integers but I believe some LQG researchers attach other kind of data instead/also (?). Any document that explains the spin network definition will also explain the labels and what they're used for.

You seem to want to discuss these things in terms of loops rather than spinnets, I've never fully understood exactly how the spin network picture emerges from the wilson loop picture but I'm not sure you can just freely equivocate between the two pictures as you seem to in some of the comments here...

 

[MTd2]

I took a 3 hours rest, and suddenly I found lots of "labels"!

Coin, I guess it is because in a wilson loop you label its path in terms of position, in the case of a loop, there is intermediary position to speak of, just the relative position of nodes. I guess the distance between nodes is a quantized quantity dependent on the spins that bounds the loop, pretty much like the energy of an electron in a hydrogen atom depends on the quantum numbers associated with orbital energy and angular momentum.

 

[marcus]

Coin,... in the case of a loop, there is intermediary position to speak of, just the relative position of nodes. I guess the distance between nodes is a quantized quantity dependent on the spins that bounds the loop, pretty much like the energy of an electron in a hydrogen atom depends on the quantum numbers associated with orbital energy and angular momentum.

Excuse me if I jump in, your post was addressed to Coin, but I will add my response.

You show a sharp intuition here. The spin-network (which you can think of as composed of a possibly very large finite number of loops, if it helps you to picture it that way) is a minimal vehicle designed to carry the required quantum numbers of geometry and matter.

It is often said that the spin network is not located in space, rather the network of relationships is space. Or more accurately it is a geometry. The other fields "live" on the geometry.

So just at the moment (after your 3 hour nap) you seem by what you say to have a sharp intuition of what the basic LQG object, the spin network, is---its role as a minimal vehicle to carry the required information.

You are still calling networks "loops" which necessarily will confuse newcomers and give them the wrong picture, but desipite the eccentric terminology you personally seem to have in large part a correct picture (as far as I can tell, being myself an inexpert onlooker.)

 

[MTd2]

Marcus, what about the non trivial topologies created by bradings? That should at least create particles.

 

[marcus]

Marcus, what about the non trivial topologies created by bradings? That should at least create particles.

Well, I am an observer, not a researcher. Braids are not part of mainstream Loop/Spinfoam LQG.
Rovelli and the dozen or so active young researchers around him have never shown any interest in that.

That doesn't mean its a wrong path, it is just separate. As an observer I of course watch the braids developments, especially the work of Wan Yidun (or Yi-dun Wan). I like his work very much.

BTW braids is so primitive it does not even have labels on the links, or on the nodes. All the information is contained in the braids. There are enormous problems in getting braid models to work. The different braids must be able to move thru the larger graph, and they must be able to interact---braid with braid to make another braid. And they must be able to survive intact amidst random network fluctuations which might untie or unbraid them.
It is much more complex than the simple work of Sundance Bilson-Thompson back in 2005 who pointed out a correspondence between simple braids and particles.
I like Yidun Wan because he is very patient and persistent. It will be a sign that braids are a dead-end if he gives up and moves to another field. As long as he keeps working there is (in my view of the picture) a possibility of success.

 

[MTd2]

But the fact that braids doesn't have links it is not quite because it is primtive, but because one is interested that it can particle interactions through topological interactions. If you check the last paper of Yidun Wan, he labels the extremes of the borders of the braids so that he can define orientation in relation to a network.

Maybe what is missing in this picture it is that braidings should induce tree like patterns, besides emulating a feynmann diagram, so that would fit a huge scale. For example, braidings for a proton should have a network with ~(10^-18m/10^-34m)^3=~10^48 vertexes

BTW, is there a way to create and destroy links between vertex? Maybe you could superimpose links composing braids with those solely for the function of creating space. Or, that mean, a particle would just like be a hola-hola wave of extra links and delinks over the space.

 

[marcus]

...BTW, is there a way to create and destroy links between vertex?

I assumed you knew that when I wrote my previous post. There are standard network "local moves" by which a new node is created or an old node removed, and by which nearby vertices can be reconnected differently. These moves do naturally create and destroy links, in their normal course.

Local moves give a way for braids to propagate, and change, and interact---braids are affected by the local moves which have some amplitude/probability of occurring.

There are certain braids which the local moves will attack and destroy. These cannot be matter particles because they have no permanence. There are certain other braids which will always remain intact (or will until they interact with some other braid of the right kind). These have permanence and might be particles.

If you make the rules so that there are too many different local moves, then there will not be enough permanent particles, so you fail. If you make the rules so there are too few local moves then the particles will not be able to propagate thru the network. So again you fail.

This is why braids are an insanely difficult research topic, in my estimation, and may be doomed. Anyone who wants to know more can read the papers of Yidun Wan and the others. Jon Hackett, Bilson-Thompson, Louis Kauffman, and their co-authors. It would be inappropriate for me to comment any further.

 

[tom.stoer]

As far as I can remember the braids need framed graphs; is that correct?
Framing was to an quantum deformation of the SU(2) symmetry group; correct?
The deformation was something like q = 2 pi / G Lambda, so it works only with a cosmological constant; correct?

To formulate a theory with braids Smolin et al. had to put the cosmological constant in by hand; but in the meantime Smolin tries to derive a cosmological constant from LQG. So I think one approach must be wrong!

Personally I love the idea that all particles can emerge from topologically braided and twisted spacetime, but I am rather sceptical - see marcus' last post.

 

[MTd2]

I assumed you knew that when I wrote my previous post. There are standard network "local moves" by which a new node is created or an old node removed, and by which nearby vertices can be reconnected differently.

Do those moves naturaly exist within LQG? What I mean is, that in small networks with lots of loops per nodes, you are going to create physical dimensions higher than 4.

 

[marcus]

Do those moves naturaly exist within LQG? What I mean is, that in small networks with lots of loops per nodes, you are going to create physical dimensions higher than 4.

As I recall the moves Y-D Wan uses correspond to 3D Pachner moves, which preserve dimension.
I'm not especially sure about this. Just how I remember it. I thought you might like something on Pachner moves, so googled and got this:
http://sbseminar.wordpress.com/2008/12/05/tqfts-via-planar-algebras/
(this is too general, they talk about N-dimension moves, not just 3D)
Maybe Wikipedia has a more focused description.

Tom.Stoer reminds us that we are talking now about ribbon graphs, where the link can twist on its way from node A to node B.
This complicates the issue of moves. I would have to go back and consult articles and review this, if I was going to have a serious conversation about it.

Also Yidun Wan has specialized in looking at four-valent ribbon graphs.
For me, that seems to simplify everything because in the case with plain (non-ribbon) links a 4-valent graph corresponds to a triangulation of a compact 3D manifold by tetrahedrons.

Tom.Stoer or anybody please correct me if I am garbling this.

In the 4-valent case, what I said about Pachner moves makes sense, because Pachner's original idea was that the moves were transformations of trianguations of a compact 3D manifold.

In order to actually talk, I would have to review. This is just how it is coming back to me.

I don't want to get into braids right now.

At the moment I find what Smolin is doing with unimodular QG to be more interesting than braids, that's what I wish someone would talk about.

 

[marcus]

As far as I can remember the braids need framed graphs; is that correct?
Framing was to an quantum deformation of the SU(2) symmetry group; correct?
The deformation was something like q = 2 pi / G Lambda, so it works only with a cosmological constant; correct?

To formulate a theory with braids Smolin et al. had to put the cosmological constant in by hand; but in the meantime Smolin tries to derive a cosmological constant from LQG. So I think one approach must be wrong!
...

Yes. I remember now. This is correct. (About the ribbons and the quantum group and the CC.)
I had not thought of what you say---that one approach (at least!) must be wrong. That adds excitement somehow. Makes it a horserace. Increases the stakes.

 

[MTd2]

Marcus, when I said destroy and create links between vertexes, I didn't mean pachner moves, i really did mean just that: create and destroy links. Not only that, put as many links as it is desired between vertexes, one for every field, the fundamental one being the gravitational.

 

[marcus]

...put as many links as it is desired between vertexes, one for every field, the fundamental one being the gravitational.

If you didn't mean Pachner moves, then isn't the answer obviously "no"?
They do not run parallel links between the same pair of nodes, one link for every field. Instead they keep the object, the graph, as simple as possible.
Instead of running a new link, they simply add a label to the existing link, as necessary.

I think if you just read the half page which is section 7.2.4, and which you referred to, you will see how they are doing things.

If there is some alternative way of carrying matter on the quantum state of the geometry, then since I have not heard of it you will have to wait for someone else to come and tell you about it. Or go out into the LQG literature and find it yourself.

I guess one can understand approximately what is going on if one thinks, as I said earlier, of the spin network format as the minimal structure able to carry the quantum numbers of geometry and matter.

When matter is not being carried---when the network gives the quantum state of geometry alone---then the node labels tell us about volume and the link labels tell us about areas.

the volume surrounding the node. the area pierced by the link.

 

[MTd2]

They do not run parallel links between the same pair of nodes, one link for every field. Instead they keep the object, the graph, as simple as possible.
Instead of running a new link, they simply add a label to the existing link, as necessary.

But wouldn't be better to make a link infinitely complicated, if necessary, but in the simplest way possible? I mean, tying knots between several links between nodes. How would you specify topological obstructions? And considering that there is no intrinsic size to a network, propagating a topological obstruction is the correct wat to do.

Plus, by creating and destroying links you could strangle small parts of the space into small islands. For example, by deleting a few nodes of a 4D network, down to 3 dimensions, you could isolate a bud 4D network. As a lower dimensional example, but using 2D spheres, it is like strangling a part of the sphere into another small sphere and ataching a 1D line between the remaining parts. This happens in dynamical triangulations on small scale, doesnt't it? Just that, instead of a line, you have a linear or planar sequence of tetrahedrons.

 

[MTd2]

I guess it is just bettre to ask Yidun Wan in his own physics forums, isn't it?

 

[MTd2]

http://babelfish.yahoo.com/translate_url?doit=done&tt=url&intl=1&fr=bf-home&trurl=http%3A%2F%2Fwanyidun.com%2Fforum%2Fucp.php%3Fmode%3Dlogin&lp=zh_en&btnTrUrl=Translate

Damn, it doesn't accept any new people. I guess either I will die in ignorance or lqg (wan yidun's nick on this forum) or anyone else will have to help me! Oh damn, I feel awful. And Marcus hates me now.