Baratin and Freidel: a spin foam model of ordinary particle physics

CarlB

Well the one thing that this sort of reminds me of is David Hestenes' comments on the Cambridge Geometric Algebra Group's gauge theory of gravity (GTG).

The comment was that it was significant that the theory could be put onto flat space, and Hestenes' reason for why this was something needed in the context of his "geometric algebra" seems to resonate with these ideas about vectors, particularly section IX, pages 21-23 of this link (which pages may be read without reading the rest of the paper):

Spacetime Geometry with Geometric Calculus
David Hestenes, To be published in the Preceedings of the Seventh International Conference on Clifford Algebra
http://modelingnts.la.asu.edu/pdf/Sp....w.GC.proc.pdf

I'd give a brief description of the argument, but I don't think I can do it justice. Hetsenes does it so well and so clearly that I wouldn't want to butcher it by reducing its length and two pages is too long. Okay, but basically, the idea has to do with how one connects up an algebra to a manifold in such a way that one can do calculus on it.

For more on the geometric gauge theory of gravity, see the Cambridge geometric algebra group:
http://www.mrao.cam.ac.uk/~clifford/index.html

Carl

Hurkyl

That, actually, is where I was getting stuck!

The problem is that something like R^n comes equipped with a mental image -- it's a space of points! But, my mental image of 2-automorphisms is very much like a homotopy of maps. While that generally works fine for natural transformations, I can't get it to mesh with my picture of R^n.

And to make things even more confusing... the 2-automorphisms of the vector space R^n look like the 1-automorphisms of the affine space R^n.

I think I'm okay if I pretend I don't know what R^n is, and just picture it as a dot with a bunch of loops hooked up to it... but I really don't think that's the right way to approach this problem.

marcus

JB began the thread with mention of a paper in the works by Baratin and Freidel-----extending to 4D what they have already done in 3D.

I think this other paper may be relevant. It just posted today and is also by Freidel, but with Starodubtsev and Kowalski-Glikman


http://arxiv.org/abs/gr-qc/0607014
Particles as Wilson lines of gravitational field
L. Freidel, J. Kowalski--Glikman, A. Starodubtsev
19 pages

"Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom."

garrett

I just read that new paper.

As far as I can tell, it works out exactly as you would expect point particles to behave in MacDowell-Mansouri BeeF gravity. Judging from their introduction, they seem oddly excited about it though, so maybe I'm missing something. It could be the excitement stems from their description of these particles as field monopoles, but I'm not sure why that's so different than putting the point particle actions in by hand. Anyway, it's a decent treatment and I like the approach.

marcus

I am glad you had a look at it.
In a recent post, Baez mentioned that Freidel has 3 papers in the works with Starodubtsev and one in the work with Baratin. even if all don't come to fruition I'm inclined to expect at least a couple more in this same line of investigation.

It seems to me that you are especially well prepared to understand and comment, not only on this one but on the others when they come out.

The conclusions section speaks of a "forthcoming paper" in which they do a perturbation expansion in alpha

and thru that, they say, address the question of the flat limit of gravity and particles. I will get the quote

==quote==
First, since the alpha parameter is small, we can consider a perturbation theory of gravity coupled to particle(s) being the perturbation theory in alpha. The distinguished feature of this theory would be that it is, contrary to earlier approaches, manifestly diffeomorphism-invariant, so its framework it is possible to talk about weak gravitational field in the conceptual framework of full general relativity. These investigations, both in the case of beta = 0 and beta not = 0 will be presented in the forthcoming paper. The fuller control over the small alpha sector will presumably make it possible to address the outstanding question of what is the flat space limit of the theory of gravity, coupled to point particles. It has been claimed that such a theory will be not the special relativity, but some form of doubly special relativity
==endquote==

If they can show that the flatspace limit is not usual Lorentz but is, instead, some DSR, this would probably open up some possibilities to TEST. It would seem to me like considerable progress just to get a good flatspace limit of one sort or another.

marcus

I am mulling over this parameter alpha that they want to do the perturbation expansion in.
I think it came up in the earlier (Jan 2005?) Freidel Staro paper.
You see it on page 3 of this paper, equation 2.3

if alpha and beta were both zero then S would be a usual BF action, but alpha perturbs it and makes it deviate from the usual BF action. Am I wrong?

the nice thing is that we are now looking at a perturbation theory where we DO NOT HAVE A FIXED BACKGROUND GEOMETRY around which we perturb. I don't claim to have much grasp of this, but we seem to be contemplating the opportunity to "perturb around pure BeeF itself"

so they hold out the attractive notion of a background independent perturbation theory or I guess what they said was a "manifestly diffeomorphism invariant" perturbation theory. that was what they said in conclusions on page 15.

right now it looks to me as if they are proceeding with exactly what they promised in http://arxiv.org/hep-th/0501191 that they would do. rather than us getting new signals this time we are getting confirmation of progress along lines they said in january last year. Am I missing something?

garrett

Yep, that all sounds right.

The [itex]\alpha[/itex] term is what makes BF into gravity. With [itex]\alpha[/itex] itself proportional to the gravitational constant. Rovelli wrote about this as well, in his propagator paper. And, urr, I do BF too -- although I came to it rather circuitously.

"BF, it's what's for dinner."

marcus

Yes! and I am looking for you to surf this BF wave!

there is much truth in the saying
"BF, it's what's for dinner." I may adopt it as a signature.

selfAdjoint

Although Prof. Baez once remarked that it should be "EF" which spoils the pun. He said the so-called B part of BF theory was not in fact like magnetism (which B traditionally expresses) but like electricity, E.

marcus

the equations look prettier with E and F instead of B and F
and the analogy is more correct, true, but still everybody says BF.
maybe we have to go with it.

selfAdjoint

Yeah, I agee. And who would want to give up that great Sig line? Even Baez seems to have bit the bullet.

marcus

Garrett can have it back anytime he wants

On Thursday, two days hence, John Baez student Derek Wise will give a talk at Perimeter.

It is along the general lines Baez has been talking about but expecially about the papers of Baez, Wise, Crans and of Baez Perez.

I hope they put a video at the streamer site. here is the abstract:

Derek Wise
Exotic statistics and particle types in 3- and 4d BF theory
Thursday July 13, 2006, 1:30 PM
"Gravity in 2+1 dimensions has the remarkable property that momenta live most naturally not in Minkowski vector space but in the 3d Lorentz group SO(2,1) itself. Having group-valued momentum has interesting consequences for particles, including exotic statistics and a modified classification of elementary particle types. These results generalize immediately to 3d BF theory with arbitrary gauge group. Better yet, they generalize to 4d BF theory, where matter shows up as string-like defects. These 'strings' exhibit exotic statistics governed not by the usual braid group, but by its higher dimensional cousin: the 'loop braid group'. We discuss these statistics as well as the classification of elementary 'string types' in 4d BF theory."

http://perimeterinstitute.com/activi...&SeminarID=759

marcus

I had the wrong post here earlier. Here is a question. if anyone wants to comment.

In the first Freidel Starodubtsev paper they cited two "in preparation" papers
one of them was something we know for sure has NOT appeared
[13]Freidel Starodubtsev "perturbation gravity via spin foams"
that would be the SPIN FOAM QUANTIZATION OF THE CLASSICAL WORK WE JUST SAW
so if and when that paper comes out it will be kind of major.

the other was
[6] Freidel Kowalski-Glikman Starodubtsev "Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis"

Now my feeling is that Freidel has gotten cagey about saying "background independent" because that term is defined differently by string theorists and others and tends to provoke controversy. people feel threatened and start protesting that maybe string theory really IS "background independent" even though it might not be "manifestly" background independent, and then they go on to say "LQG" is not really background independent, and so on. The term irritates people---and has become associated with semantic conflict
So my suspicion is that the paper that JUST CAME OUT REALLY IS THIS PAPER but RETITLED in a kind of inconspicuous ivy-league coat-and-tie way.

the paper that just came out is titled
"PARTICLES AS WILSON LINES OF GRAVITATIONAL FIELD"
which is shocking if you think of it, but innocuous enough on the surface.
the number is
http://arxiv.org/gr-qc/0607014 (remember by Quatorze Juillet Bastille day)

So I guess the question is, what do you think? Do you also think that the promised paper
"Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis"

is actually the new one we have in hand called "Particles as Wilson Lines of Gravitational Field" but renamed?

Notice if you look at "Particles as Wilson Lines" actually wilson lines is only a part of what they are doing and
very much of what they are doing could be accurately described as a classical analysis of background independent (in the LQG sense) gravity-and-matter perturbation theory.

and if so, any idea why they decided on the new name?

================
to repeat another point, that I think JB made, or various people have: to say "background independent perturbation theory" is a real kicker of a headline. Because perturbation theory is the customary predominant way to do fields and UP TILL THIS MOMENT all the perturbation field theory ever done has used a fixed BACKGROUND SPACETIME geometry. so when you hear that phrase you hear a slight breaking noise.
(which among other things could motive people to deny that the paper could possibly be on the right track, causing the author a lot of bother answering them). I can understand how one might want the breaking noise to be inaudible.

john baez

I don't know. I remember Laurent saying they weren't even sure how many papers they were writing on this subject: two or three. They've done a lot of work, obviously, and for a big project like this one needs to keep rethinking the best way to slice the work into papers.

The paper they wrote doesn't actually do any "perturbation theory", apart from writing the MacDowell-Mansouri Lagrangian as the BF Lagrangian plus two extra terms, and analysing what this means... which they'd already done in a previous paper. The big new thing is to introduce particle worldlines as "defects" - curves removed from spacetime - much as had already been done in 3d gravity. So, it makes sense for their title to emphasize this.

In fact, their title is a bit more dramatic than what I might have chosen, because they don't really study these particle worldlines in the context of MacDowell-Mansouri gravity, except for one equation right near the end. Mostly they study these particles in the context of plain old 4d BF theory.

This nicely complements my own study, with Crans, Wise and Perez, of strings coupled to 4d BF theory. In fact the particles and strings fit hand in glove: particles like the A field, while strings like the B field - because particles have 1-dimensional worldlines and the A field is a 1-form, and strings have 2-dimensional worldsheets and the B field is a 2-form. I explain this a bit more in the latest issue of This Week's Finds, week235.

Unfortunately, Crans, Wise, Perez and I studied strings coupled to 4d BF theory for a general gauge group but didn't work out the details for the gauge group Freidel uses, namely SO(4,1). We focused on SO(3,1). It should be easy to do the SO(4,1) case now, though since Freidel & Company have worked out a lot of the necessary stuff.

After a talk I gave, Freidel guessed that the strings may be related to gravitons... or replace them, somehow. It's a big mystery: a nice structure is emerging, but it's not clear what it means! This is what makes physics fun.

selfAdjoint

Are you now finding it as much fun as math again? I know you were fed up for a while.

It is surely a wonderful gift you have to be able to work back and forth in the two areas; not only are you able to spot unnobvious connections, but there always seems to be something in one field or the other that really floats your boat.

marcus

I think it's all one thing, basically.
just the formalities of which department and which journal
but if you see a glint in the eye of the universe or a little
smile on the face of nature it probably doesnt matter much
whether it is one or the other

marcus

Great news! Glad you found time!

UNFORTUNATELY? That's the way it's SUPPOSED to happen
general group case first, then specialize to SO(4,1)
couldnt be sweeter
certainly maximizes the pleasure and excitement for the sidelines observers like us anyway.
hotdog

"After a talk I gave, Freidel guessed that the strings may be related to gravitons... or replace them, somehow. It's a big mystery: a nice structure is emerging, but it's not clear what it means! This is what makes physics fun."

Freidel: let's invent how spacetimematter works. My stuff can be the geometry and your stuff can be the gravitons that connect changes in the geometry, OK?


"but it's not clear what it means! This is what makes physics fun" at some point, this begins to sound like a memorable understatement

thanks for posting here, enjoy Shanghai, and don't forget to figure spacetime out for us