I will be flying to California this weekend visiting the deep M-theory thinker John Baez , right after end of semester. Last preparations will probably keep me from chatting on the web too much today and tomorrow. But when I find the time I will get back to you, maybe next week.

One quick remark, though:

Nice that we managed to agree on what is going on. In summary it seems to me that the following has happened:

Somebody working on LQG/spin foams (namely Smolin and collaborators) has come up with some action S whose space of classical solutions (=extrema) contains (when suitably identified) those of 4-dimensional GR as well as those of some topological theory.

This is a curious formal observation, though I am not yet sure if it is really deep or useful. But let’s not argue about that. If something useful can be derived from this action (as suggested, but not yet demonstrated, by Freidel and Starodubtsev), fine.

However, it should be made rather clear that this new action S is just some formal construct of which nobody has any reason to expect that it directly describes our world. Rather, it plays a role of a calculational trick so far. And this trick also so far has been demonstrated to work only classically. I think the authors of the papers that I have seen would agree with this statement.

Even if any of these authors nourished hopes that this curious action S is actually phenomenologically important (in its full form, not just in its reduction to GR), one should make very clear that just writing down this action is not the same as doing LQG.

But it now seems to me that, since Smolin is a prominent representative of LQG, and since he wrote about this action S, Vafa got the impression that this action is part of the program called LQG. Since this action has a ’sector’ of solutions which are that of BF theory, it seems that Vafa concluded one can make the statement that
‘the topological sector of LQG is BF theory’.

Then, when he found BF theory in topological M-theory he commented that hence ‘the topological sector of LQG’ has appeared in topological M-theory.

(If that is not what happened I’d be grateful for corrections. It sure seems that this is what is going on.)

But really it is BF theory that has appeared in topological M-theory. BF-theory is not the ‘topological sector of LQG’, really, but merely the topological sector of the above discussed curious action – which was more or less designed to have BF theory solutions among its solutions.

The statement that ‘BF theory is the topological sector of that special action S, while GR is another ’sector” is true by construction of that strange action, really.

The motivation for the construction of S is that it might help (which has not been shown yet, though) to approach GR with tools of topological field theory. But that should not be confused with the stament that ‘LQG has a topological sector which is BF theory’.

Unless, of course, one would go ahead and redefine the term ‘LQG’. Maybe if something is investigated by Smolin people will tend to call it ‘LQG’. In that case however it should be made very clear in every discussion what precisely is to be meant by the term ‘LQG’.

So in conclusion I think what happens is that

Vafa found in topological M-theory a topological theory which, by construction is, classically, a certain sector of some action principle investigated by Smolin.

If this is true it is nothing to be axcited about at all, I think.

Urs said:
“Do you agree with this summary?
Again some comments:

1) What Vafa finds in topological M theory is BF theory, not any of its extensions that Smolin, Freidel and Starodubtsev discuss, right?

2) Having some action that involves BF theory is not yet the same as ‘doing LQG’, right?

3) I have more comments, but I gotta run. More later, if you like.”
Posted by: Urs at February 2, 2005 10:59 AM

To the extent that I can rely on my own inexpert judgment I agree fully with your comments:
1) absolutely! I do not hear Vafa connect top.M-theory to LQG (more like he creates a context or perspective in which both may be mentioned)
2) right! it is not the same as doing LQG
3) I am very glad you have more comments and hope you will share them with us.

One reservation is that I only heard the Vafa audio and did not see what he wrote at the board. I can only guess about some of what he said.

I also have several times read through your summary of the Smolin/Starodubtsev (2003) paper and find that I agree completely with your summary.

thanks for your efforts in replying and listing literature. It is appreciated.

I am short of time today so let me reply to your latest message first and postpone the previous discussion until later.

You wrote:

After listening to Vafa’s whole talk, I would say that what he said, including towards the end where he again discussed LQG, has significant overlap with Smolin, Starodubtsev(2003)
General relativity with a topological phase: an action principle

I have just had a look at this paper. Therein the following is discussed:

BF theory is a well known topological field theory. By adding a certain term to it that breaks both some gauge symmetry as well as the topological property one obtains an action that can be shown to be equivalent to the Einstein-Hilbert action.

Put the other way round: The Einstein-Hilbert action can be massaged into a form which is a topological term plus something else.

That ’something else’ involves a fixed background structure (that gamma5 vector) which is needed to reduces the 5-d theory to 4-d.

In their paper Smolin and Starodubtsev proceed by adding yet another term to the action which makes this background structure dynamical. By the above discussion the result is a modification of the Einstein Hilbert action.

Because that first extra term is now dynamical, there are solutions where it reproduces the first step and hence the EH action, but there are also solutions where it takes other values and yields topological theories.

Hence, in conclusion, the authors find that there is an extension of the EH action which has some solutions that reproduce those of the EH action and some that don’t.

Do you agree with this summary?

Again some comments:

1) What Vafa finds in topological M theory is BF theory, not any of its extensions that Smolin, Freidel and Starodubtsev discuss, right?

2) Having some action that involves BF theory is not yet the same as ‘doing LQG’, right?

3) I have more comments, but I gotta run. More later, if you like.

The first paper by Smolin dealing with QG in relation to BF theory explicitly, at least that I could find, is Smolin (1998), see link.

The history of the LQG/Spin Foam/BF connection has become interesting (for instance because of Cumrun Vafa’s discussion this month in Toronto, which was compensated by other remarks we saw tending to dismiss or downplay the relationship.
So I thought I would provide this sketchy bibliography to give some background perspective.

After listening to Vafa’s whole talk, I would say that what he said, including towards the end where he again discussed LQG, has significant overlap with
Smolin, Starodubtsev(2003)
General relativity with a topological phase: an action principle

It would be great to have a text copy of Vafa’s talk with some footnotes, because in the audio I couldnt catch what his sources were or if it was just general Vafa-knowledge

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

About Urs question #4, the answer is no, it would not imply that spinfoams have no dynamics. Indeed it has not been established that canonical LQG must have no dynamics. Nor has Smolin claimed this is necessarily the case. If anyone is interested in recent progress in LQG dynamics some relevant papers are
these five by Thomas Thiemann and Bianca Dittrich

http://arxiv.org/abs/gr-qc/0411138
Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
42 pages

“Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler-DeWitt constraint equations …. The models themselves will be studied in the remaining four papers. As a side result we develop the Direct Integral Decomposition (DID) for solving quantum constraints as an alternative to Refined Algebraic Quantization (RAQ).”

http://arxiv.org/abs/gr-qc/0411139
Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
23 pages

“This is the second paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity…”

http://arxiv.org/abs/gr-qc/0411140
Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
33 pages

“This is the third paper in our series of five…”

http://arxiv.org/abs/gr-qc/0411141
Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories
23 pages

“… We now move on to free field theories with constraints, namely Maxwell theory and linearized gravity…”

http://arxiv.org/abs/gr-qc/0411142
Testing the Master Constraint Programme for Loop Quantum Gravity V. Interacting Field Theories
20 pages

“… Here we consider interacting quantum field theories, specifically we consider the non-Abelian Gauss constraints of Einstein-Yang-Mills theory and 2+1 gravity. Interestingly, while Yang-Mills theory in 4D is not yet rigorously defined as an ordinary (Wightman) quantum field theory on Minkowski space, in background independent quantum field theories such as Loop Quantum Gravity (LQG) this might become possible by working in a new, background independent representation.”

Because of Urs’ ungrounded assumption in his point #4, I must re-emphasize that not only has it not been established that a proper LQG dynamics is unattainable, but also Smolin does not say this. He says that at a certain point in history (in the 1990s) some researchers, including himself, redirected effort because they saw serious problems with constructing the Hamiltonian constraint in canonical LQG. So they began exploring other paths, like spin foams with their connection to BF theory and related areas, which have turned out to be interesting and are still being pursued.

Urs said:
5) If they can not reproduce the canonical LQG kinematics, what does this imply for the experimental predictions that it seems Lee Smolin wants to derive from these kinematics?
Posted by: Urs at January 31, 2005 01:11 PM

I believe the answer to Urs’ question #5 is that it would not imply anything about Smolin’s predictions. The latest word on what experimental predictions Lee Smolin has gone on record with is in this recent paper:
http://arxiv.org/hep-th/0501091
Falsifiable predictions from semiclassical quantum gravity
Lee Smolin
9 pages

“Predictions are derived for the upcoming AUGER and GLAST experiments from a semiclassical approximation to quantum gravity. It is argued that to first order in the Planck length the effect of quantum gravity is to make the low energy effective spacetime metric energy dependent. The diffeomorphism invariance of the semiclassical theory forbids the appearance of a preferred frame of reference, consequently the local symmetry of this energy-dependent effective metric is a non-linear realization of the Lorentz transformations, which renders the Planck energy observer independent. This gives a form of deformed or doubly special relativity (DSR), previously explored with Magueijo, called the rainbow metric. The argument is general, and applies in all dimensions with and without supersymmetry, and is, at least to leading order, universal for all matter couplings. The argument is, illustrated in detail in a specific example in loop quantum gravity.
A consequence of DSR realized with an energy dependent effective metric is a helicity independent energy dependence in the speed of light to first order in the Planck length. However, thresholds for Tev photons and GZK protons are unchanged from special relativistic predictions. These predictions of quantum gravity are falsifiable by the upcoming AUGER and GLAST experiments.”

Urs gives the impression that he thinks Smolin’s predictions concerning the up-coming experiments are derived from details of LQG kinematics. But this is not the case, although they can be illustrated as Smolin says by considering the example of LQG.

LQG is falsifiable, Smolin argues, and risks being refuted by GLAST. And the prediction is not based on particular LQG detail but is in a sense “generic”. It applies to any of a broad class of approaches to Quantum Gravity which share the feature that they require an observer-independent energy scale but at the same time do not allow a preferred frame (they deform but do not break Lorentz invariance).

It will be interesting to see if GLAST falsifies LQG. This appears to me to be a valid prediction from the theory and a potentially solid experimental result. This has nothing to do with whether or not Spin Foams can or can not reproduce details of LQG kinematics (which was Urs question).

Both the super string camp and the LQG camp claimed their derivations of the Bekenstein-Hawking black hole entropy as their biggest success of their theories. In my judgement, claiming the derivation of Bekenstein Hawking entropy, such a trivial feat, as their biggest success, is completely “childish” and only shows the lack of “innate” ability on the part of each camp to comprehend what is the REAL physics behind the blackhole entropy!

I am going to show one very trivial derivation of the black hole entropy and how it is proportional to the event horizen surface area divided by Planck area. One that is different from Hawking’s but much simpler.

But first, one has to realize two things:
1.Hawking entropy is not an empirical experimental evidence, but merely the result of a gedanken “experiment”, e.g., mind exercise.
2. The entropy is a DIMENTIONLESS physical quantity.

See the complete message at
http://quantoken.blogspot.com/2005/02/blackhole-entropy-lqg-and-super-string.html

it seems to me that it will not take long before LQG is “unified” into the stringy framework.

I don’t think that the appearance of BF theory in ‘topological M-theory’ supports such an expectation.

Everybody interested in this question should pick up the recent paper

L. Freidel & A. Starodubtsev: Quantum gravity in terms of topological observables (2005) .

The crucial idea is expressed by formula (30).

It goes as follows:

Suppose we want to quantize some theory whose action can be written as a topological term plus a non-topological term.

Write down the generating functional for the action consisting of the topological terms alone.
Next consider the exponential of the non-topological part of the action as an observable. The expectation value of that observable can be computed by taking (infinitely many) functional derivates of the generating functional of the topological theory.

Since the topological theory is likely to be solvable exactly, this reduces the task of quantizing the full theory to that of computing that (highly nontrivial) expectation value of an exactly solvable theory.

Freidel and Starodubtsev demonstrate how this rewriting can be carried out for Einstein gravity in four dimensions with BF theory as the topological part.

The paper disucsses various path integral computations. It does not use any LQG techniques, though. At the end it says:

This suggests that the techniques of loop quantum gravity and spin foam model are adapted to describe our perturbative expansion and lead to a finite result.

The authors want to study that in a followup:

Our next paper is devoted to study in more details the perturbation theory in the context of spin foam.

So the relation of all this to LQG and spin foams is hypothetical at this point. Even if it can be made I don’t see how the appearance of BF theory in topological strings has any bearing on it.

After all, at least in this paper, BF theory serves the purpose of a calculational trick in a way. The message is that some very compliocated expectation values in some theories are equal to partition functions of other theories. The hope is that computing some expectation value in some auxiliary topological theory reproduces the partition function of some other theory T. Right now I cannot see how the appearance of the auxiliary field theory in any context allows to make a connection to that theory T.

If the connection top-M-theory -> BF-theory -> LQG were meaningful, it would imply that topological M-theory is about gravity. But it is instead ordinary M-theory, which is.

But as I have said above, apart from all these considerations there is as yet no demonstration that spin foams and/or LQG are helpful in performing the calculation described by Freidel and Starodubtsev. To me, their discussion rather suggests that instead ordinary perturbative path integral quantization of gravity might maybe make sense if we were able to find a suitable reformulation of the EH Lagrangian.

Whether it will be part of string theory proper, or only a “variation” on a stringy theme, is beside the point. I just don’t think that the two theories will remain “two different approaches”, the way that they are now, for too long.

Incidentally, it was Lee Smolin who predicted that this will be the case, many years ago, in an article in New Scientist.