# This is what Baez was talking about earlier

1. Jan 24, 2005

### marcus

http://arxiv.org/abs/hep-th/0501191
Quantum gravity in terms of topological observables
Laurent Freidel, Artem Starodubtsev

"We recast the action principle of four dimensional General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance. The coupling constant becomes dimensionless (G_{Newton} \Lambda) and extremely small 10^{-120}. We give an expression for the generating functional of perturbation theory. We show that the partition function of quantum General Relativity can be expressed as an expectation value of a certain topologically invariant observable. This sets up a framework in which quantum gravity can be studied perturbatively using the techniques of topological quantum field theory."

Baez gave a report on the October 29-November 1 LQG conference at Perimeter (waterloo canada) and this was the main development he talked about.

a perturbation series in which the expansion is in powers of a very small number namely the cosmological constant Lambda.

2. Jan 25, 2005

### Chronos

I'm still dazed by the exponential Lamda connection between GR and QFT. The fit seems almost too perfect to be real.

3. Jan 25, 2005

Staff Emeritus
This is a major advance in the spin foam side of LQG; given Thiemann's apparent major advance in the Canonical side with his Master Constraint Program, we advance both pieces one square. Will one of them capture the other? Stay tuned for the next move!

(Added) Whatever happens with the spin foam path integrals, the deeper understanding of the Barbero-Immirzi parameter contained in this paper is already a major contribution to the field.

Last edited: Jan 25, 2005
4. Jan 25, 2005

### Kea

Agreed. One of those fantastic and simple ideas that makes you slap yourself a few times.

Laurent and Artem are both very bright guys, and they're nice too.

5. Jan 26, 2005

### marcus

I am not sure I would recognize them face to face, though I've seen there photos in the "people" section of the Perimeter website

the key idea here seems to be BF theory and the
MacDowell-Mansouri discovery of how to say General Realtivity in BF terms
Now since MacD/Mansouri's paper was back in 1977 and is apparently not online, please anybody who knows of a substitute introductory treatment of BF or a tutorial, please post the link!

John Baez has a 1995 paper discussing BF theory as applied to gravity.
http://arxiv.org/abs/q-alg/9507006

Lee Smolin and Artem Starodubtsev have this 2003 paper
http://arxiv.org/abs/hep-th/0311163

This refers to related earlier work by Smolin
A holographic formulation of quantum general relativity
hep-th/9808191,
Holographic Formulation of Quantum Supergravity
hep-th/0009018

I dont know which if any of these might be useful in understanding the present paper by Freidel and Starodubtsev

Last edited: Jan 26, 2005
6. Jan 26, 2005

Staff Emeritus
I tried looking up BF theory on google scholar. All I know about it is from brief descriptions in papers. But all the recommended sources seem to be on paper, and all 1995 or before. It's like it was discovered (as some kind of alternative to Chern-Simons, which I don't understand either). Then everyone scarfed it up and became an instant expert. After which nobody ever described it fully again. Professors probably assign developing it as an excercise for their students the way Peskin & Schroeder did sigma models. All I can tell you is that B is like a magnetic field, F is the curvature of a connection A, and the Lagrangian involves B wedge F.

7. Jan 27, 2005

### marcus

I'll take a look at those references I found in the Smolin/Starodubtsev paper to some earlier work by Smolin, and a further reference found in one of them
A holographic formulation of quantum general relativity
http://arxiv.org/hep-th/9808191 [Broken],
Holographic Formulation of Quantum Supergravity
http://arxiv.org/hep-th/0009018 [Broken]
Linking topological quantum field theory and nonperturbative quantum gravity
http://arxiv.org/gr-qc/9505028 [Broken]

what I want is just a grain of intuition about why taking the (wedge) product of B and F makes a good action

the first reference begins a section on page 4 which derives Gen Rel from a BF theory, it begins like this:
---quote Smolin---
2 General relativity as a constrained TQFT

In this section we introduce new way of writing general relativity as a constrained topological quantum field theory, which we call the ambidextrous formalism 3. For the non-supersymmetric case we study here, the theory is based on a connection valued in the Lie algebra G = Sp(4), (which double covers SO(3, 2) the anti-deSitter group.) Thus, this approach is similar to that of MacDowell-Mansouri, in which general relativity is found as a consequence of breaking the SO(3, 2) symmetry of a topological quantum field theory down to SO(3, 1)[29]. However it differs from that approach in that the beginning point is a B wedge F theory...

Footnote 3
We may note that there is more than one way to represent general relativity with a cosmological constant as a constrained topological quantum field theory. The earliest such approach to the authors knowledge is that of Plebanski [26], studied also in [27]. Alternatively, one can deform a topological field theory of the form of TrFwedgeF, as described in [28] (see also [8]).. What is new in the present presentation is the representation of general relativity as a constrained topological field theory for the DeSitter group SO(3, 2). For reasons that will be apparent soon, the present formulation is more suited both to the Lorentzian regime and to the theory with vanishing cosmological constant.
---end quote---

and this goes on until the middle of page 8 where he says he has now finished deriving Gen Rel from BF TQFT.

---quote Smolin--
Plugging (24) and its primed double into equation (25) we then find the Einstein equation.
.....[I wont copy this]..... (26)
Thus, we have shown how general relativity with a cosmological constant may be derived as a constrained Sp(4) BwedgeF theory.
---end quote---

Now I will check out the next reference. BTW doesnt it look as if Smolin is the originator of the BF approach to Gen Rel. Although there is that Baez paper of 1995 which also talks about BF theory I thought in the same sort of way but I must be mistaken.

Last edited by a moderator: May 1, 2017
8. Jan 27, 2005

Staff Emeritus
I seem to remember one of the recent papers of the Potsdam school of LQG had a good discussion of BF theory. I was reading it and it was the first good discussion I had seen. Polchinski doesn't have anything about it in the index Maybe last summer? Perhaps one of your handy dandy arxiv searches could find it?

9. Jan 27, 2005

### marcus

I can certainly try an arxiv search with "BF theory" and quantum gravity as keywords.

I am puzzled by the extra information that it was by someone at Albert Einstein Institute (MPI Potsdam). There are lots of people there in several disciplines but I can only think of Thomas Theimann, Martin Bojowald, and Bianca Dittrich right now, and of course Hermann Nicolai one of the AEI directors. There are visitors always coming through mostly for a few months. Oh, Hanno Sahlmann is connected there, but is currently at Perimeter. So right now my mind is drawing a blank about what Potsdammer it could be.

I will just try a simple arxiv search with BF theory.

Oh yes, Renate Loll was at AEI Potsdam a long time but now has moved to Utrecht. I wonder if it could have been her.

10. Jan 27, 2005

### marcus

http://arxiv.org/abs/gr-qc/0406063
maybe that is it?

I just noticed that John Baez has something about BF theory with the word "Introduction" in the title, maybe it could be helpful (at least to me)

http://arxiv.org/gr-qc/9905087 [Broken]

An Introduction to Spin Foam Models of Quantum Gravity and BF Theory
John C. Baez
55 pages LaTeX, 31 encapsulated Postscript figures
Lect.Notes Phys. 543 (2000) 25-94

In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory.

Last edited by a moderator: May 1, 2017
11. Jan 27, 2005

### marcus

this is the right introduction (for me and I hope others)
Page 3 of Baez "introduction" gr-qc/9905087

section 2: BF theory, classical field equations

I will just repeat what he says but in a more limited sloppy way (not so general)

M is a 4D manifold "spacetime"
G is a Lie group whose Lie algebra g has an invariant bilinear form
P is a principal G-bundle

A is a connection on P
ad(P) is the vector bundle associated to P via the adjoint action of G on its Lie algebra
E is a 2-form on M with values in ad(P)

The curvature of A, called F, is also an ad(P)-valued 2-form

(notice that for reasons best known to himself Baez is using E rather than B as his notation)

now we loosen our ties and take off our elbowpatch jackets and make a small admission. IF WE PICK A LOCAL TRIVIALIZATION WE CAN THINK OF
A as a g-valued 1-form on M,
and F as a g-valued 2-form on M,
and E as a g-valued 2-form on M

so we DONT REALLY NEED TO HAVE ad(P) the adjoint vectorbundle, after all. It is just nice so that we can converse elegantly. but we can think of these things as being one and two-forms valued in the Lie algebra of the gauge group.

And Baez says (still on page 3) that the Lagrangian for BF theory is
"trace"(E wedge F)

HERE IS WHERE WE USE THE BILINEAR FORM
because in wedging E with F we do the ordinary wedge of their
differential form parts and then we need to multiply the "coefficients" of their differential form parts which are in the Lie algebra so we need
a kind of multiplication (not the bracket but more like ordinary multiplication than that: a bilinear numerical-valued binary operation)
and that gets us a numerical-valued 4-form. IT ALL WORKS OUT!

And then a further confession. If G is semisimple then there is no mystery about the bilinear form and we can take the quotemarks off the "trace" because it can be just be the familiar trace we have of matrices representing elements of the Lie algebra.

and then instinctive-teacher and explainer that he is, Baez explains in 3 lines why this is a GOOD LAGRANGIAN (i was wondering about that for several days now)
you just set the variation of the Lagrangian equal to zero and use an identity
and you get field equations
F must = 0
derivative along A of E must equal zero

the identity is that the variation of F (the curvature of A) is equal to the derivative along A of the variation of A (it is the time-honored throwing away of little things which we learned from Brothers Leibniz and Newton).
Thank goodness for Baez when he says "introduction" in the title. Everything BF was a waste of time up to now (at least for me).

and I am just come to the bottom of page 3!
so it may be hoped that there will be more enlightenment in what follows

Last edited: Jan 27, 2005
12. Jan 28, 2005

### marcus

now why does M have to be 5-dimensional.
where does this extra dimension come from
and why does it have to be there

BTW that other paper I mentioned isnt bad either
it is by Rovelli, Oriti, and Speziale
http://arxiv.org/abs/gr-qc/0406063
"In 4 dimensions, general relativity can be formulated as a constrained BF theory; we show that the same is true in 2 dimensions. We describe a spinfoam quantization of this constrained BF-formulation of 2d riemannian general relativity, obtained using the Barrett-Crane technique of imposing the constraint as a restriction on the representations summed over. We obtain the expected partition function, thus providing support for the viability of the technique. The result requires the nontrivial topology of the bundle where the gravitational connection is defined, to be taken into account. For this purpose, we study the definition of a principal bundle over a simplicial base space. The model sheds light also on several other features of spinfoam quantum gravity: the reality of the partition function; the geometrical interpretation of the Newton constant, and the issue of possible finiteness of the partition function of quantum general relativity."

If it was last summer you were reading a BF thing, it could have been this one since the date is June 2004

13. Jan 28, 2005

Staff Emeritus
Sorry, my phrase "Potsdam School" just meant Thiemann, Sahlmann, and their coauthors; I didn't intend any deeper description. And then the actual paper, as you intuit, was of the Rovelli school! That's the one, though I had misremembered the authors.

14. Jan 28, 2005

### marcus

We found it, then!
and it seems more than average explanatory to me too
so we have the original paper that sparked our interest
(Freidel/Starodubtsev) plus two or three others that
might help us understand it

Baez "Introduction"
M and M
Rovelli/Oriti/Speziale
Smolin/Starodubtsev
and, as the saying goes, references therein

Last edited: Jan 28, 2005
15. Jan 28, 2005

### ohwilleke

One of the infinitely frustrating things about physics papers is that their authors routinely are sloppy about defining terms, identifying variables, and contextualizing their research in light of the preceding work that they use as a jumping off point. You get a whole paper on how one theory is related to BF theory, and nobody so much as bothers to say what "B" or "F" stand for (of even if they are vectors, tensors, scalars, determinants, or whatever), and they don't even always use the letter "B" for the "B" part!

I come from mathematics and law, and in both, you simply can't get away with not definining something of importance, and you routinely rehash the fundamental assumptions upon which you are relying before going forward.

Physicists seem to assume that you read every paper in their footnotes (and every paper in the footnotes in those papers) before you read their paper, which is not a good way to communicate.

16. Jan 29, 2005

### marcus

I agree. Baez is a mathematician by culture who has made significant contributions to quantum gravity physics, the difference in expository style is noticeable.

But I think in this case we are in good shape. (foolish of me to risk a wild guess but) I think this Freidel-Staro paper is major. sort of "decade-class" in the way that some Einstein papers are "century-class" if that makes sense. And we seem to have the stuff we need for reading it, like Baez "Introduction to BF for QG" gr-qc/9905087

I was thinking about what Freidel-Staro say they are coming out with next
(I think I noticed along about page 4 that their analysis seems to want DSR and so it is not surprising that they expect to co-author a paper with Kowalski-Glikman known for his work in DSR)

Willeke you might actually be interested in that! It has a MOND connection, Smolin when talking about MOND is always referring to this conjectured new fundamental (length) constant which is the inverse sqrt of Lambda and is sometimes called the "cosmological length"
It is on order of 10 billion LY and not to be confused with inverse Hubble parameter (which is not a constant). So smolin is flirting with the existence of a new universal constant---the "cosmological length"----which is really not so new because it is just reciprocal sqrt of Lambda a presumed constant curvature. But if these things really are fundamental constants, shouldnt they be the same for all observers? And that leads to thoughts of DSRs, deformations of special relativity.

That's vague on my part and may not matter here, i dont know. But the two planned papers they say are "to appear" are

Freidel Starodubtsev Perturbative Gravity via Spin Foam
Freidel Starodubtsev Kowalski-Glikman Background Independent Perturbation Theory for Gravity: Classical Analysis

Last edited: Jan 29, 2005
17. Jan 29, 2005

### marcus

that Freidel and Starodubtsev have on page 7 and 8

$$\gamma_I$$

and especially

$$\gamma_5$$

(this gamma is not to be confused with the Immirzi parameter also denoted gamma)

see equations 26, 27, 30, and 31

and also later on page 13, equations 71, 75,76

in a naive way I am thinking of these gamma matrices as so(5) analogs of pauli spin matrices

they seem to me to belong to the Lie algebra (not the Lie group) and to be defined by a simple Lie bracket and delta-function equation

$$\{\gamma_I, \gamma_J\} = 2 \delta_{IJ}$$

Oh, now I see there is something about these things in the brief appendix on page 18. But I could still use some help if anyone wants to talk us thru it.

Last edited: Jan 29, 2005
18. Jan 29, 2005

### marcus

PETER WOIT PICKED UP ON THE FREIDEL STARODUBTSEV PAPER
just this morning, at Not Even Wrong
http://www.math.columbia.edu/~woit/blog/archives/000145.html

he also has Lee Smolin reply to Nicolai's critique of LQG
(broad-based well-researched discussion by string theorist of LQG not narrowly focused on one particular set of papers but addressing LQG as a whole, hence quite constructive.) Glad Smolin replied and will be
very interested to see what he says.

19. Jan 29, 2005

Staff Emeritus

Those are the Dirac matrices: $$\gamma_1 - \gamma_4$$ are the basis of the transformation matrices for Dirac's 4-component spinors, and $$\gamma_5$$ is i times the product of all of them together. The Lie algebra is $$\mathfrak {su}(4)$$.

Last edited: Jan 29, 2005
20. Jan 29, 2005

### marcus

thanks!
I found the Dirac matrix page at mathworld
http://mathworld.wolfram.com/DiracMatrices.html

now what is puzzling me is the last sentence on page 18, where it says
"the so(5) can be represented in terms of gamma matrix.....
where
$$\gamma_I$$
are gamma matrices satisfying...."

it looks like some 4x4 matrices are providing a representation of a Lie algebra of 5x5 matrices, namely so(5).

Last edited: Jan 29, 2005
21. Jan 29, 2005

### marcus

well I should take the trouble to learn about gamma matrices because
(not only basic importance in quantum electrodynamics but) right here
in FreidelStarodubtsev this is how they reduce the symmetry down from SO(5) to SO(4)

First I recall the pauli matrices sigma_i i=1,2,3
Code (Text):

0  1
1  0

0 -i
i  0

1  0
0 -1

In mathworld a bunch of different Dirac matrices are defined
alphas, sigmas, deltas and also gammas: in equations
(25, 26, 27, 28)
http://mathworld.wolfram.com/DiracMatrices.html

the dirac matrices are 4x4 ones made by duplicating pauli 2x2 sigmas either along the main diagonal or along the crossdiagonal.
mathworld shows all the different ways this is done and i havent the energy to type it in right now.
Anybody know of a good link for Dirac matrices----I suspect I could do better than Mathworld but dont know of one at the moment myself.

22. Jan 30, 2005

Staff Emeritus
I'm glad you found that. I was going to say they broke the symmetry down from SO(5) to SO(4) because that was what all the commentators on the paper said. But I hadn't read how they did it in the paper itself . Got to get to work on it. Yeeks! And the Master Constraint Program still on my plate! Feels like grad school deja vu all over again.

Did you notice Smolin's comments on the MCP over at Not Even Wrong? He thinks it's a sideshow. Of course he would, being a spin foam partisan, but he seemd to have good arguments. This might be a response to Thiemann's comments on spin foams in the intro to his Phoenix paper, like it was just something people got into because they couldn't do the Hamiltonian constraint. Even the good guys can't keep from zinging each other.

23. Jan 30, 2005

### marcus

If i am not too much of a polyanna I would like to say that it could be just sincere belief with no spark of malice or thought of zinging.

It could be fraternal scuffle---trading put-downs----could well be. but also
I respect each viewpoint and can see how one could adopt either

TT: let's finish what we started and get a superhamiltonian that actually works

LS: what we want is a theory of gravity, not a quantization of one particular equation (even if it was Einstein's that doesnt make it sacred) so let's find a path integral version with the right largescale behavior and not get obsessed with the hamiltonian.

[edit: come to think of it, on further reflection, I think they were zinging.
but the both lines of investigation should clearly be pursued and might even interfere constructively down the road]

Last edited: Jan 30, 2005
24. Jan 30, 2005

Staff Emeritus
I have been working with an old Cambridge Monograph I bought years ago: Group structure of gauge theories by L. O'Raifeartaigh. From his description of the representations of SO(2n+1), I understand where the gammas come from, but now I don't see how gamma_5 breaks down the SO(5) symmetry to SO(4). Here's a paraphrase:

and he earlier defines the tilde symbol as the double cover of SO(n) so that

$$SO(n) = \tilde{SO}(n)/Z_2$$

He goes on
Put l = 2 and gamma = gamma_5 and you have the representation in the definition (27) of the Friedel-Starodubtsev paper. Now while 1/4 the brackets of the Dirac matrices with each other and with gamma_5 are a basis for the representation of SO(5), gamma_5 by itself is not obtainable within that algebra, so it has to be taken as an extranious factor. OK so far, but I don't see how that breaks down to SO(4), since the 2 representations for even numbered groupd SO(2l) are

$$\frac{1}{8}(1 \pm \gamma) [\gamma_{\mu},\gamma_{\nu}]$$,

so where do these come from? O'Raifeartaigh doesn't discuss this issue.

-

Last edited: Jan 30, 2005
25. Feb 1, 2005

### marcus

I should get some of the bibliography in order

John Baez (1995)
4-Dimensional BF Theory as a Topological Quantum Field Theory
15 pages
http://arxiv.org/q-alg/9507006 [Broken]

"Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G = GL(4,R) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4-manifolds."

Smolin (1995)
Linking topological quantum field theory and nonperturbative quantum gravity
http://arxiv.org/gr-qc/9505028 [Broken], (TQFT + QG)

Smolin (1998)
A holographic formulation of quantum general relativity
http://arxiv.org/hep-th/9808191 [Broken], (explicitly BF + QG)

"...Thus, this approach is similar to that of MacDowell-Mansouri, in which general relativity is found as a consequence of breaking the SO(3, 2) symmetry of a topological quantum field theory down to SO(3, 1)[29]. However it differs from that approach in that the beginning point is a BF theory... "

John Baez (1999)
An Introduction to Spin Foam Models of Quantum Gravity and BF Theory
55 pages, 31 figures
http://arxiv.org/gr-qc/9905087 [Broken]

"In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of 'spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a 'spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory."

Smolin (2000)
Holographic Formulation of Quantum Supergravity
http://arxiv.org/hep-th/0009018 [Broken]

Smolin, Starodubtsev(2003)
General relativity with a topological phase: an action principle
http://arxiv.org/hep-th/0311163 [Broken]

"An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced and depending on the value it takes the theory has solutions that reduce it to 1) general relativity in Palatini form, 2) general relativity in the Ashtekar form, 3) F wedge F theory for SO(5) and 4) BF theory for SO(5). This theory then makes it possible to describe explicitly the dynamics of phase transition between a topological phase and a gravitational phase where the theory has local degrees of freedom. We also find that a boundary between adymnamical and topological phase resembles an horizon."

Freidel, Starodubtsev (2005)
Quantum gravity in terms of topological observables
http://arxiv.org/abs/hep-th/0501191

Last edited by a moderator: May 1, 2017