Going Beyond LQG: how to define QFT over Loops?

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.

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.

 

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.

 

I don't see the similarity.

 

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.

 

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?

 

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.

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 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...

 

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":rofl: 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.)

 

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 doesnt 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.

 

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.

 

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.

 

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.

 

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.