Freidel does it: BeeF, it's what's for dinner

[selfAdjoint]

Aiyeee! Nakahara of course! Now where did my copy get itself to?

Actually http://store.doverpublications.com/0486661814.html" has a useful section on representations of orthogonal groups that is accessible (pp. 391-399) and it's cheap at Dover.

 

[marcus]

a BTW comment:
Back in January 2005 this article (I think it was this one) was listed as "To appear", but I think the title changed.

In the first Freidel Starodubtsev article there were two "to appear" references

[6] L. Freidel, J. Kowalski-Glikman, A. Starodubtsev, “Background independent perturbation theory for gravity coupled to particles: classical analysis”. To appear.

[13] L. Freidel and A. Starodubtsev, “Perturbative gravity via spin foam” To appear.

I think what we are reading now is [6], retitled to be

"Particles as Wilson lines of gravitational field"

Anyway that is how I interpret it, because it is by the same three authors, and also it IS the classical analysis.

But I could be wrong, there could still be two further papers of Freidel with Starodubtsev (not just one, reference [13]) in the works.
===============
John Baez said Freidel has several in progress with Starodubtsev, and one with Baratin. I will look at the most recent Freidel Baratin http://arxiv.org/abs/gr-qc/0604016 and see if there is some mention.
Yes! their April 2006 paper was the 3D case of "gravity hidden in Feynman diagrams" and they say in the abstract that "We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results." The companion paper is their reference [1]

[1] A. Baratin, L. Freidel Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams, To appear.

 

[marcus]

Aiyeee! Nakahara of course! Now where did my copy get itself to?

Nakahara Geometry, Topology, and Physics IoP Bristol 2005
is reference [12] of Freidel, Kowalski-Glikman, Starodubtsev
gr-qc/0607014

Hope you found your copy!

 

[selfAdjoint]

Nakahara Geometry, Topology, and Physics IoP Bristol 2005
is reference [12] of Freidel, Kowalski-Glikman, Starodubtsev
gr-qc/0607014

Hope you found your copy!

Yes, 'twas in the living room next to my favorite chair. Studying it (section 5).

 

[john baez]

SO(4,1) is deSitter - a close relative of Poincare

Hi! I'm having too much fun in http://math.ucr.edu/home/baez/diary/july_2006.html#july5.06" to post much, but I'm happy to hear that Freidel is finally getting more of his work out.

Yes! I agree that BF can be done with any of the usual groups and what makes this special is SO(4,1). I would like to know more about the deSitter group SO(4,1) and its algebra so(4,1). Just about anything you or Kea can say would be a help---either to me or to other readers of the thread.

The main thing to realize is that SO(4,1) is the deSitter group. In other words, it's the symmety group of a 4d spacetime called deSitter spacetime - an exponentially expanding universe not unlike our own. The curvature of this spacetime is proportional to the cosmological constant, which is positive.

In the limit where the cosmological constant goes to zero, deSitter spacetime reduces to good old flat Minkowski spacetime - and the deSitter boils down - or contracts, in the usual jargon - to the Poincare group.

So, you should think of SO(4,1) as a close relative of the Poincare group. To ants like us who can't see far enough to notice that spacetime is curved, there's no way to tell if the symmetry group of the universe is the Poincare group or SO(4,1).

The group SO(p,n-p) has dimension given by the triangle number n(n-1)/2. Since 4+1 = 5, SO(4,1) has dimension 5(5-1)/2 = 10:

o
oo
ooo
oooo

The Poincare group also has 10 dimensions. That should be reassuring.

Another close relative of the Poincare group is SO(3,2), which also has dimension 10. This is the symmetry group of anti-deSitter spacetime, where the cosmological constant is negative.

You may find it odd to describe a group like SO(4,1) in a way that depends on a number - the cosmological constant - and see what happens as we send this number to zero. But, it's common in physics. One of the first guys to study this limiting process was Wigner. It may have been he who invented the term contraction for this process. For example, he noticed that the Poincare group contracts to the Galilei group as the speed of light goes to infinity. The Galilei group is the symmetry group of Newtonian physics, generated by translations, rotations, and Galilei boosts.

 

[marcus]

Maybe it is time to recall the topic of this thread---and copy in the abstract, which I don't seem to have done yet.

The paper's arxiv number is easy to remember----just think of July 14 as Bastille Day and write this year's Quatorze Juillet holiday as 0607014

http://arxiv.org/gr-qc/0607014

...

Kea just referred to this paper at N.E.W.

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."

It's been 5 weeks since Freidel posted it and Kea and the rest of us have had time to reflect some about it so it is not just a first reaction. To me, Kea's intense comment on NEW had a ring of truth. She was saying it is time for some people at Peter's blog to look at this paper.
Here is what she said

http://www.math.columbia.edu/~woit/wordpress/?p=444#comment-14467

…and the lack of any definition of string theory when supersymmetry is broken by a positive CC, and thus the background is deSitter…

de Sitter…which is, er, cough, in agreement with Riofrio’s analysis of, for instance, the type IA supernovae data, and also with a simple interpretation of certain interesting background independent QG models for which we understand the derivation of classical gravity.

http://www.math.columbia.edu/~woit/wordpress/?p=444#comment-14468

How many nails need to go into this freaking coffin?

and then another commenter asked

Can you please provide a reference to a “background independent QG model” for which

A) we understand the derivation of classical gravity, by which I assume you mean the existence of a semiclassical limit

and

B) which leads to the appearance of de Sitter space?

I honestly don’t know what’s being referred to here, which is why I’m asking.

to which Kea replied
http://www.math.columbia.edu/~woit/wordpress/?p=444#comment-14471

This paper[/color] is a good gateway into the literature on BF theory and its variants, in the context of de Sitter gravity.

where paper[/color] was the link to Laurent, Jerzy, and Artem's paper that we're talking about.

 

[marcus]

"Duh, it's deSitter!"

maybe a way to paraphrase Kea's comments would be
"It's deSitter space, stupid!"

We tend to forget, or I do anyway, and slip back into the habit of expecting Lorentz or Poincaré invariance.

And she is reminding me forcefully of something that has been STARING US IN THE FACE ALL ALONG, or at least for quite a few years now since 1998, that the universe is after all deSitter and it is THAT symmetry, not some other symmetry, that it would be smart to concentrate on.

 

[marcus]

connection to "a" Kodama, not "the" Kodama

A commenter at N.E.W. thought Kea was talking about "the" Kodama state, and Kea said no she was NOT talking about "the" Kodama state. And note the emphasis, she added.

In trying to follow, I found that commenter's suggestion served mainly as irrelevant distraction---a red herring. The paper we are talking about doesn't have any reference to Kodama or any mention of his type of state(s) singular or plural.

Apparently there are a bunch of different Kodama-type states. Andy Randono (U Texas postdoc) gave a talk at Perimeter last week on generalized Kodama states.

But even if there is only a tentative connection between that and this BF+matter paper of Freidel et al, it could still be interesting to consider. Right now I can't see the contact, but Lee Smolin referred to one in a comment he contributed to
that same N.E.W. thread. Here is an exerpt:
http://www.math.columbia.edu/~woit/wordpress/?p=444#comment-14477

-The Kodama state also appears in a different, recently introduced approach, in which GR is constructed by an expanding around an SO(5) BF theory, in hep-th/0501191[/color]. That BF theory has a single bulk solution classically, which is (A)dS and a single bulk quantum state, which is related to Kodama. This may resolve the old issues about the Kodama state because it is a sensible state of BF (Indeed it’s the only bulk state of BF theory, work on this is in progress.

where hep-th/0501191[/color] http://arxiv.org/abs/hep-th/0501191 is the January 2005 paper by Freidel and Starodubtsev Quantum gravity in terms of topological observables
this was the paper that for many of us awakened interest in MacDowell-Mansouri (deSitter gauge) gravity and stimulated interest in the idea of PERTURBATION around beef, instead of around Minkowskispace.

 

[marcus]

BTW JB gave us some good clues about deSitter group a few posts back in this thread. At the time I did not understand and was uncertain enough that I didnt ask, but later these ideas were elaborated in another SO(4,1) thread.

Anyway here are exerpts from post #35, just a few back.
https://www.physicsforums.com/showthread.php?p=1035189#post1035189

...The main thing to realize is that SO(4,1) is the deSitter group. In other words, it's the symmety group of a 4d spacetime called deSitter spacetime - an exponentially expanding universe not unlike our own. The curvature of this spacetime is proportional to the cosmological constant, which is positive.

In the limit where the cosmological constant goes to zero, deSitter spacetime reduces to good old flat Minkowski spacetime - and the deSitter boils down - or contracts, in the usual jargon - to the Poincare group.

So, you should think of SO(4,1) as a close relative of the Poincare group. To ants like us who can't see far enough to notice that spacetime is curved, there's no way to tell if the symmetry group of the universe is the Poincare group or SO(4,1).[/color]

The group SO(p,n-p) has dimension given by the triangle number n(n-1)/2. Since 4+1 = 5, SO(4,1) has dimension 5(5-1)/2 = 10:

o
oo
ooo
oooo

The Poincare group also has 10 dimensions. That should be reassuring.

...You may find it odd to describe a group like SO(4,1) in a way that depends on a number - the cosmological constant - and see what happens as we send this number to zero. But, it's common in physics. One of the first guys to study this limiting process was Wigner. It may have been he who invented the term contraction for this process. For example, he noticed that the Poincare group contracts to the Galilei group as the speed of light goes to infinity. The Galilei group is the symmetry group of Newtonian physics, generated by translations, rotations, and Galilei boosts.

In that other thread JB discussed the Inonu-Wigner contraction
there is a Wikipedia about that
the deSitter group SO(4,1) contracts to the Poincaré group as the cc (cosmoconst.) goes to zero
and the Poincaré group contracts to the Galilei group of Newton physics as c ( speed of light) goes to infinity.

So IT SEEMS LIKE WE WERE JUST HERE ONLY 100 YEARS AGO when we backed off from the Galilei group into something that contracts into the Galilei group but which is TRUER, namely the Poinc.

And now there is this eerie feeling of the ground shifting in a dreamy tectonic way and we are backing off again this time from the Poincaré group into something which contracts into the Poincaré group but which is TRUER, namely deSitter.

they are all 10 dimensional groups which is why the Great Spinoza made us have ten toes-ah, so you could count the dimensions of His symmetry on your toes. Oh I mean the ALBERT, that's his name, isn't it heh heh
This is fun.

 

[marcus]

I loved what Kea said

http://www.math.columbia.edu/~woit/wordpress/?p=444#comment-14468

How many nails need to go into this freaking coffin?

in other words when will you guys catch on that the symmetry is deSitter?
===============

anyway, the SO(4,1) thread that we managed (with JB help) to put together is
https://www.physicsforums.com/showthread.php?t=126285

 

[Kea]

...she is reminding me forcefully of something that has been STARING US IN THE FACE ALL ALONG...

Oh dear, I hope I wasn't too rude. Personally, I haven't been thinking that much about de Sitter, but something holographically related, namely the conformal symmetry that gets put into twistor theory.

By the way, some quantum group people did work out how to deform de Sitter so that it gives deformed Poincare in the appropriate (flat) limit.