This is what Baez was talking about earlier

[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.

 

[Chronos]

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

 

[selfAdjoint]

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.

 

[Kea]

This is a major advance in the spin foam side of LQG

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.

 

[marcus]

Glad you know them, Kea.
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 don't know which if any of these might be useful in understanding the present paper by Freidel and Starodubtsev

 

[selfAdjoint]

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.

 

[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 ,
Holographic Formulation of Quantum Supergravity
http://arxiv.org/hep-th/0009018
Linking topological quantum field theory and nonperturbative quantum gravity
http://arxiv.org/gr-qc/9505028

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 won't 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 doesn't 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.

 

[selfAdjoint]

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?

 

[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.

 

[marcus]

selfAdjoint, this was last summer and is about BF theory

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

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.

 

[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

 

[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 isn't 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

 

[selfAdjoint]

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.

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.

 

[marcus]

... That's the one,...

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

 

[ohwilleke]

Thanks for your efforts Marcus.

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.

 

[marcus]

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,...

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, shouldn't 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 don't 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

 

[marcus]

can anyone say something generally helpful about the gamma matrices
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.

 

[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.

 

[selfAdjoint]

can anyone say something generally helpful about the gamma matrices
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.

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 $$\mathfrak {su}(4)$$.

 

[marcus]

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 $$\mathfrak {su}(4)$$.

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).
I hope everybody else knows about this, I sure didnt.

 

[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

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 haven't 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 don't know of one at the moment myself.

 

[selfAdjoint]

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.

 

[marcus]

... 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.

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 doesn't 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]

 

[selfAdjoint]

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

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 haven't 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 don't know of one at the moment myself.

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:

...primitive tensor representation do not form a complete set for the orthogonal groups SO(n), because they do not include the spinor representations (faithful representations of $$\tilde{SO}(n)$$ which are two-valued representations of SO(n) )...

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

...there is one fundamental spinor representation for SO(2l+1) ... The representation is two valuedand constitutes the only primitive spinor representation (i.e. for spinor representations, primitive = fundamental).

The construction (Boerner 1972) of the fundamental spinor representations is a sraightforward generalization of the well-known Dirac construction for the Lorentz group. Let
$$[\gamma_{\mu}\gamma_{\nu}] = 2\delta_{\mu\nu}, \gamma = (i)^l\gamma_1\gamma_2...\gamma_{2l}$$
be the (2^l-dimensional) representation of the 2l+1 hermitian Dirac matrices $$\gamma_{\mu}, \gamma$$. Then the generators of (the fundamental spinor representation of SO(2l+1)) are
$$\frac{1}{4}[\gamma_{\mu},\gamma_{\nu}], \frac{1}{4}[\gamma_{\mu},\gamma]$$.

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.

















-

 

[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

"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 , (TQFT + QG)

Smolin (1998)
A holographic formulation of quantum general relativity
http://arxiv.org/hep-th/9808191 , (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

"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

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

"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

 

[marcus]

Another relevant paper I should have included in the short list is
Smolin
An Invitation to Loop Quantum Gravity
http://arxiv.org/hep-th/0408048

this gives an overview of LQG in the context of topological field theories
which begins on page 12, section 3.2 "Dynamics of Constrained Topological Field Theories"

Smolin's survey provides what I think is the most up-to-date and workable description of what "LQG" as a field of research consists of. He bases his account on 4 main premises which he calls "observations"

see page 4:
Section 2
The four basic observations
"While the principles assumed are only those of general relativity and quantummechanics, there are four key observations that make the success of loop quantum gravity possible. These are..."

and then on page 12

"...Observation IV means in details that all gravitational theories of interest, including general relativity and supergravity in 4 dimensions, can be described by an action, which is generically of the form,

S = S^topological + S^constraints + S^matter (17)

To describe the detailed form, its simplest first to fix the dimension to be four, in which case G = SU(2). The first term describes a topological theory called BF theory. It depends on a 2 form Bⁱ and the field strength Fⁱ of a connection, Aⁱ, all valued in a Lie algebra of SU(2). Thus, i = 1, 2, 3. The action is,..."

 

[marcus]

Another relevant paper I should have included in the short list is
Smolin
An Invitation to Loop Quantum Gravity
http://arxiv.org/hep-th/0408048

this gives an overview of LQG in the context of topological field theories
...
"...Observation IV means in details that all gravitational theories of interest, including general relativity and supergravity in 4 dimensions, can be described by an action, which is generically of the form,

S = S^topological + S^constraints + S^matter
..."

At the present time Vafa and others have suggested or hinted at some relation between the field of LQG and a subfield of String theory ("topological M-theory") so it is advisable to look and see what LQG consists of. It is obviously not limited to the "canonical" LQG program pursued by Thiemann because that is not what most LQG people do. So how does one describe the field of LQG into which Smolin is inviting researchers? (And to which Vafa is hinting at connections.)

Well, what does Smolin exclude?
see page 7
"...There are several different approaches to a background independent quantum theory of gravity. Besides loop quantum gravity, others include dynamical triangulations[37, 38], causal sets[36] and the Gambini-Pullin discrete quantization approach to quantum gravity[39]. Although these are independently motivated research programs, some of their results are relevant for loop quantum gravity, because they concern models which can be understood as arising from spin foam models by simplifications which eliminate certain structures..."

this is very interesting, I find. He also distinguishes LQG from String lines of research by contrasting them at the bottom of page 6.

And how does he describe what he includes in LQG? He bases it on his four premises or "observations" and he presents topological field theory as fundamental to LQG. In his discussion he develops the spin foam approach out of this TFT basis

He also also develops the canonical or hamiltonian formulation of LQG out of this TFT foundation. See page 13 and 14

"...For d = 2 + 1, the BF theory is equivalent to general relativity[50]. For spatial dimension d > 3 the extension has been given in [51]. It is also known how to express supergravity in d = 3 + 1 [32] and d = 10 + 1 [34] as constrained topological field theories. Now we can return to the canonical quantization, and discover the structure we assumed above...

...We can easily see how the restriction from the topological BF theory to general relativity by a quadratic equation works in the hamiltonian formulation. As the field equations of the BF theory, (19,20) are expressed as spacetime forms, they pull back to equations in the three surface ?. These must hold the canonical theory..."

In the second part of this quote he is talking about the 4D case of ordinary canonical LQG.

But Smolin's view of LQG is obviously based on topological field theory and not restricted to the canonical approach. Indeed to insist that LQG is strictly confined to a narrow hamiltonian programme would exclude most of those who go to LQG conferences and think of themselves as working in that general area of research.

I must say that i like Smolin's comparatively unrestrictive definition of LQG.
It is less confusing than the narrow one, because more in line with the way people actually use the word.
I believe it is also well justified logically and historically to some extent, see Smolin's footnote 12 on page 12:

Footnote 12
"This kind of formulation of general relativity was first discovered by Plebanksi[44] and later independently by [45, 46]. The corresponding simplification of the Hamiltonian theory was independently discovered by Sen[1] and formalized by Ashtekar[2]. By now several different connections are used in loop quantum gravity. These include the self-dual part of the spacetime connection[1, 2], and a real SU(2) connection introduced by Barbero[47] and exploited by Thiemann[48]. There are also alternate formulations that use both the left and right handed parts of the spacetime connection, [31, 49]."

what this suggests to me is maybe one could argue that what gave rise to LQG (including the Sen-Ashtekar "new variables) grew out of or was motivated by the Plebanski action.

[44] J.F. Plebanski. On the separation of einsteinian substructures. J. Math. Phys., 18:2511, 1977.

[45] T. Jacobson and L. Smolin, Phys. Lett. B 196 (1987) 39; Class. and Quant. Grav. 5 (1988) 583; J. Samuel, Pramana-J Phys. 28 (1987) L429.

I include the second reference for completeness. Perhaps I am misinterpreting, or drawing too tenuous a conclusion. This is a new view of things for me. Arguably, in other words, the broadly defined field of LQG including Spin Foams as well as the Hamiltonian formulation (and presumably Thiemann's current Master Constraint effort) may have been motivated by Topological Field Theory and a 1977 paper by Plebanski!
Or by independent rediscovery of similar ideas by others later.

Well, maybe that is wrong. But in any case I like the way Smolin presents LQG, basing the exposition on Topological Field Theory (his premise IV and the section beginning on page 12). I think some confusion could be avoided if people knew how Smolin defines LQG in his "Invitation" and that it is a broad-gauge definition inclusive of Spin Foams.

 

[marcus]

I see from discussion at Peter Woit's blog that some folks may be confused about what LQG means in the 0411 paper of Vafa et al.

At top of page 11, Vafa says:
In dimension four, there are several versions of “topological gravity”. Here we review a theory known as 2-form gravity [26,27,28,29,30,31,32,33], which also describes the topological sector of loop quantum gravity [34].

well that is certainly true in the sense intended by Vafa, because the reference [34] is precisely to Smolin's paper "An Invitation to LQG" where BF theory is the topological sector in equation (12) on page 12.
This is fundamental to LQG as outlined in that paper.

But some object and say this cannot be true, I think because of using a too narrow or restrictive idea of LQG ( to be more like what Thomas Thiemann does----the canonical approach, the hamiltonian formulation, and possible new variants of that.)

That kind of restrictive definition would exclude a lot that Smolin discusses in his survey (or that Rovelli discusses in his book)
it would include only a small part of the current LQG research papers and of what is discussed at conferences.

But still some do not like the Vafa et al. phrase "topological sector of loop quantum gravity" and say Vafa must be mistaken, apparently because of not conforming to the narrow definition of LQG.

So this is an unfortunate confusion of a verbal sort. We really should have a general name for the broad Loop-and-allied research area, that includes spinfoam and formulations using B and F-forms, and the recent papers by Smolin, Freidel, Starodubtsev, including the perturbation series one.
Regretfully, we do not have a general classifier word, except LQG, as used in broad sense of Smolin's title "Invitation to LQG".

 

[selfAdjoint]

Regretfully, we do not have a general classifier word, except LQG, as used in broad sense of Smolin's title "Invitation to LQG".

In my opinion the term Loop Quantum Gravity and its abbreviation LQG should be resticted to those who actually use Wilson loops in their derivations. This includes spin foams and Thiemann's Master Constraint Program, and pretty generally anything that starts out with the Ashtekar variables. But it does not include, I think, dynamical triangulations or stuff derived from Regge calculus. Maybe the blanket term Quantized gravity or Quantized Spacetime (QS?) could be used to cover them all.

 

[marcus]

...actually use Wilson loops in their derivations. This includes spin foams and Thiemann's Master Constraint Program, and pretty generally anything that starts out with the Ashtekar variables...

I confess I am a bit unclear as to how Spin Foams depends in its formulation on wilsonloops or ashtekar variables. Could it be that it doesnt?

Indeed I believe if you insist on wilsonloops you kind of exclude SpinFoam and that is why, when one is being longwinded-correct one calls a conference a
"LQG and Spin Foam Conference"

as e.g. rovelli did, in May 2004, but also is pretty typical.

And there are people, i have discovered, who insist that LQG cannot include Spin Foam because the "LQG programme" is essentially the canonical or hamiltonian approach----e.g. Thiemann is the main representative, out of 100+ researchers ordinarily said to be Loop.

So there is a real potential for confusion which I fear you and I cannot legislate away.

We ALSO need a general classifier for what you said (Loop in the broad sense of BF approach, Spin Foam approach, canonical, Master constraint, PLUS things that Smolin excludes, like dynamical triangulation).

but first of all, I think we need to be aware of the lack of a universally accepted term for what most people call LQG which is Loop in the broad sense topological FT-based, like BF, Spin Foam AS WELL AS stuff that is based on Ashtekar variables and has that hilbertspace built on connections.

 

[selfAdjoint]

Spin foams and MCP are from the same stable, just moving by different roads. From the Baez introduction paper you cited above;

Since loop quantum gravity is based on canonical quantization, states in this formalism describe the geometry of space at a fixed time. Dynamics
enters the theory only in the form of a constraint called the Hamiltonian constraint. Unfortunately this constraint is still poorly understood. Thus until recently, we had almost no idea what loop quantum gravity might say about the geometry of spacetime.

And recall that Thiemann, in his Phoenix paper, noted that some had gone off to study path integral methods because the Hamiltonian constaint had not been solved. But the networks come from the same source and they both started from holonomies along an edge (I should have said that instead of Wilson loops).

 

[marcus]

Spin foams and MCP are from the same stable, just moving by different roads. From the Baez introduction paper you cited above;

And recall that Thiemann, in his Phoenix paper, noted that some had gone off to study path integral methods because the Hamiltonian constaint had not been solved. But the networks come from the same source and they both started from holonomies along an edge (I should have said that instead of Wilson loops).

I don't think that Spin Foams is about "holonomies along an edge". I don't see a space of connections and spinfoams being a Hilbertspace of fuctionals on connections.
My understanding is that Hamiltonian LQG is not rigorously connected to SpinFoam (that is a problem to work on, that people have tried to resolve)

On the other hand, as I see it, Spin Foam is clearly linked to BF theory. See e.g. some mid 1990s papers of Baez and of Smolin. ( bibliography). It seems clear to me that the move into spin foam (even the invention of that approach) was inspired by BF. A paper of Witten. the Barrett-Crane model.

I would say it this way: the contact between BF and SpinFoam is logical and mathematical and evident in papers from the mid 1990s. And it is historical. Spinfoam approach was developed by people thinking TQFT and BF.

On the other hand the link between SpinFoam and Ashtekar variable and holonomies is somewhat vague and "sociological". It involves some imperfect analogies and things brought over when people moved. To some extent we can say SpinFoam is a part of LQG because at a certain point Loop PEOPLE moved over and did Spin Foam.

that reasoning would apply also to any future surge of LQG papers in constrained BF theory if we didnt already have the connection of LQG and BF going back to the 1990s

what I mean is this, you reason that LQG includes SpinFoam because Loop PEOPLE moved over, as you say: "...noted that some had gone off to study path integral methods because the Hamiltonian constaint had not been solved..." Well, if you allow that kind of argument, then definitions become community based. Loop is what Loop people do. String is what String people do, and so on.

It would be a lot clearer if everybody would just take verbatim what Smolin says in "Invitation". See how he defines terms and try to be consistent with that.

I will get a bit from page 12 that I already posted, but some may have missed.

 

[marcus]

Smolin
An Invitation to Loop Quantum Gravity
http://arxiv.org/hep-th/0408048

this gives an overview of LQG in the context of topological field theories
which begins on page 12, section 3.2 "Dynamics of Constrained Topological Field Theories"

BTW selfAdjoint, by LQG here I mean primarily SpinFoam (very different from hamiltonian or masterconstraint narrowlydefined canonical QG). when he says "invitation to LQG" I read it as largely an invitation to spinfoam and BF and stuff like that.

on page 12

"...Observation IV means in details that all gravitational theories of interest, including general relativity and supergravity in 4 dimensions, can be described by an action, which is generically of the form,

S = S^topological + S^constraints + S^matter

To describe the detailed form, its simplest first to fix the dimension to be four, in which case G = SU(2). The first term describes a topological theory called BF theory. It depends on a 2 form Bⁱ and the field strength Fⁱ of a connection, Aⁱ, all valued in a Lie algebra of SU(2). Thus, i = 1, 2, 3. The action is,..."

this is how Smolin does the exposition of what he calls "LQG"
Is there something here that is unclear, or that you disagree with?

My inclination is since he developed the field (with Ashtekar and Rovelli) that we should let him define the field the way he wants (and not insist on defining it the way we want)

 

[Chronos]

I am not comfortable with the S-constraints. The evidence is not very compelling.

 

[marcus]

I am not comfortable with the S-constraints...

I think that equation is a way of writing a generic form that various QG theories (the ones he is interested in) take and to judge the degree of empirical support we need to look at some particular case, like the Plebanski action (which is of this form and reproduces Gen Rel)

In that case i would say the observational evidence is as supportive as it is of Gen Rel, no more no less, because equivalent.