A couple weeks ago, three string theorists, (Nicolai, Peeters and Zamaklar) posted on the arXiv a critical assessment of loop quantum gravity. Today I received from Lee Smolin something he wrote responding to them, and I’m posting it here with his permission. Lubos Motl also has put up Smolin’s text on his weblog this morning, but I thought it would be a good idea to provide a version that doesn’t include Lubos’s interspersed rantings. Smolin has some very interesting things to say, and his comments are well-worth reading by anyone who wants to understand what is going on in this field.

Somewhat off-topic, I’d also like to mention a paper by Freidel and Starodubtsev from earlier this week called Quantum gravity in terms of topological observables. The idea of trying to use topological quantum field theory to understand quantum gravity is one that I’ve always found appealing, and this paper is an interesting attempt to make this idea work. I don’t think I find it completely convincing, for one thing they seem to be breaking the topological invariance by hand. For another, TQFTs are very subtle QFTs, and the kind that might be relevant to gravity is still very far from well-understood.

Dear Friends,

Thanks very much for all the time and work you put into your review.

While I disagree with a number of your assertions, both in point of detail

and of attitude, what is certainly very much appreciated is your evident

willingness to “get your hands dirty,” learn the technicalities and attack

key problems. It is very good that you do this, as indeed too few of us

loop people have taken the time to try to learn the details and attack

problems in string theory.

Some points you raise have been underappreciated. The issue of what

happens to the chiral anomaly, and whether there is fermion doubling in

LQG is one I have suggested to many graduate students and postdocs over

the years, but so far no one takes it up. It would be good to know if LQG

forces us to believe in a vector model of weak interactions.

At the same time, the major difficulties you raise were underestood to be

there more than ten years ago. This is especially true with respect to

issues concerning the hamiltonian constraint such as the algebra and

ultralocality.

What is missing from your “review” is an appreciation of how the work done

over the last ten years addresses these difficulties.

Indeed the fact that much work in the field has been on spin foam models

is exactly because the problems you worry about do not arise in spin foam

models. I will explain this below. Other work, such as Thiemann’s master

constraint approach, also is motivated by a possible resolution of these

problems.

As you will appreciate, like any active community of 100+ people there is

a range of opinions about the key unsolved problems. I have the sense that

you are aware of only one out of several influencial points of view.

The view your concerns reflect is what one might call the “orthodox

hamiltonian” point of view towards LQG. According to this, the aim of

work in lqg is not so much to find the quantum theory of gravity as to

work through the excercise of quantizing a particular classical theory,

which is Einstein’s. From this point of view, the program would fail if

it turned out that there was not a consistent canonical quantization of

the Einstein’s equations.

While I will refer to my own views so as not to implicate anyone else, you

should beware that this is not necessarily the dominant view in the field.

It is a respectable view, and I have the greatest respect for my friends

who hold it. But, were it to fail, many of us would still believe that

loop quantum gravity is the most promising approach to quantum gravity.

This is not avoidence of hard problems, there are good physical reasons

for this assertion, which I’d like to explain.

What I and others have taken as most important about Ashtekar’s great

advance is the discovery that GR can be writen as a diffeomorphism

invariant gauge theory, where the configuration space is that of a

connection on a manifold Sigma, mod gauge transformations and Diff(Sigma).

This turns out to be true not only of Einstein’s theory in 4d but of all

the classical gravity theory we know, in all dimentions, including

supergravity, up to d=11, and coupled to a variety of matter fields.

This is a kinematical observation and it leads to a hypothesis at the

kinematical level, which is that the quantum theory of gravity, whatever

it is, is to be written in terms of states which come from the

quantization of this configuration space. This as you know, leads

directly to the diffeo classes of spin net states. Furthermore, given the

recent uniqueness theorems, that hilbert space is unique for spacetime

dimension 3 or greater. Thus, o long as the object is to construct a

theory based on diffeomorphism invariant states, it cannot be avoided.

The main physical hypothesis of LQG is not that the quantum Einstein

equations describe nature. It is that the hilbert space of diffeo classes

of spin nets, extended as needed for matter, p-form fields, supersymmetry

etc, is the correct arena for quantum gravitational physics. Given that

the theorems show that this hilbert space exists rigorously, this is a

well defined hypothesis about physics. It may hold whether or not the

Einstein equations quantized give the correct dynamics.

A lot already follows from this hypothesis. It gives us states,

discreteness of some geometric diffeo invariant observers, a physical

interpretation in terms of discrete quantum geometry etc.

But there is also a lot of freedom. We are free to pick the dimension,

topology, and algebra whose reps and intertwiners label the spinnetworks.

This then gives us a large class of diffeo invariant quantum gauge

theories, of which the choices that come from GR in d=4 are only one

example. These are possible kinematics for consistent background

independent quantum field theories.

Now let us come to dynamics. I believe the most important observation for

an understanding of quantum dyannics in this class of theories is that all

gravitational theories we know, in all dimensions, super or not, are

constrained topological field theories. (See my latest review,

hep-th/0408048, for details and references for all assertions here.) This

means they are related to BF theories by non-derivative constraints,

quadratic in the B fields.

A lot follows from this very general observation. It allows a direct

construction of spin foam models, by imposing the quadratic constraints in

the measure of the path integral for BF theory. This was the path

pioneered by Barrett and Crane. The construction of the Barrett Crane and

other spin foam models does not depend on the existence of a well defined

hamiltonian constraint. The properties that have been proven for it, such

as certain convergence results, also do not depend on any dynamical

results from the hamiltonian theory.

The relation to topological field theory is also sufficient to determine

the basic form of fields and states on boundaries. In 4d these give the

role of Chern-Simons theory in horizon and other boundary states. Thus,

it gives the basic quantum geometry of horizons.

Once we have the basic form of spin foam models, which follow from the

general relation to BF theories, we can consider the problem of dyanmics

in the following light. Given the choices made above, the spin foam

amplitudes are chosen from the invariants of the algebra which labels the

spin networks. There is then a large class of theories, differing by the

choice of the spin foam amplitudes. Each is a well defined spin foam

model, which gives amplitudes to propgate the spin network states based on

the chosen dimension and algebra.

The lack of uniqueness is unaviodable, because there is a general class of

theories, just like there is a general class of lattice gauge theories.

These theories exist, and the general program of LQG as some of us

understand it, is to study them.

>From a modern, renormalization group point of view, the first phsyical

question to be answered is which of these theories lead to evolution that

is sensible, i.e. which spin foam ampltidues are convergent in some

approrpiate sense. The second physical question is to classify the

universality classes of the spin foam models and, having done this, learn

which classes of theories have a good low energy behaivor that reproduces

classical GR and QFT.

It is of course of interest to ask whether some of these theories follow

from quantizing classical theories like GR and supergravity, by various

methods. But no one should mind if the most successful spin foam model, in

terms of both matheamtical elegance and physical results, was not the

quantization of a classical theory, but only reproduced the classical

theory in the low energy limit. How could one object from a physics point

of view, were this true?

This is the point of view from which many of us view the problems with the

hamiltonian constraint you describe.

The next thing to be emphasized is that there is no evidence that a

successful spin foam model must have a corresponding quantum hamiltonain

constraint. There are even arguments that it should not. These have not

pursuaded everyone in the community, and this is proper, for the

healthiest situation is to have differing views about open problems. But

it has persuaded many of us, which is why many people in the field turned

to the study of spin foam models after the difficulties you describe were

understood, more than ten years ago.

For example, Fotini Markopoulou argued that, as the generators of

infinitesimal spatial diffeos do not exist in the kinematnical hilbert

space, while generators of finte spatial diffeos do exist, the same should

be true for time evolution. This implies that there should only be

amplitudes for finite evolutions, from which she proposed one could

construct causal spin foam models.

This was partly motivated by the issue ultralocality. (Btw, you dont

emphasize the paper that first raised this worry, which was my

gr-qc/9609034). The worry arises because moves such as 2 to 2 moves

necessary for propagation do not occur in the forms of the hamiltonian

constraint constructed by Thiemann, Rovelli and myself, or Borissov.

This is because they involve two nodes connected by a finite edge.

However, the missing moves are there in spin foam models. This concretely

confirms Fotini’s argument. In fact, as Reisenberger and Rovelli argued,

invariance under boosts generated by spacetime diffeo requires that they

be there. For one can turn a 1-3 move into a 2->2 (0r 1->4 into 2-> 3)

move by slicing the spin foam differently into a sequance of spinnetworks

evolving in time.

So we have two arguments that suggest 1) that the problem of ultralocaity

comes from requiring infinitesimal timelike diffeos to exist in a theory

where infinitesimal spacelike diffeos do not exist and 2) the problem

is not present in a path integral approach where there are only

amplitudes for finite timelike diffeos.

One can further argue that if there were a regularization of the

hamiltonian constraint that produced the amplitudes necesary for

propagation and agreed with the spin foam ampltidues, it would have to be

derived from a point splitting in time as well as space. This suggests

that there is a physical inadquancy of defining dynamics through the

hamiltonian constraint, in a formalism where one can regulate only in

space and not in time.

Let me also add that there is good reason to think that the other issues

such as the algebra of constraints arise because of the issue of

ultralocality. Thiemann’s constraints have the right algebra for an

ultralocal theory.

It was for these and other reasons that some of us decided ten years ago

to put the problems of the hamiltonian constraint to one side and

concentrate on spin foam models. That is, we take the canonical methods

as having been good enough to give us a kinematical frameowrk for a large

class of diffeo invariant gauge theories, but unnecessary and perhaps

insufficient for studying dynamics.

At the very least, making a point splitting regularization in both space

and time seems a much more difficult problem and hence is less attractive

than spin foam methods where one can much more easily get to the physics.

Given that the relation to BF theory gives us an independent way to define

the dynamics, and path integral methods are more directly connected to

many physical questions we want to investigate, there seemed no reason to

hold back progress on the chance that the problems of the hamiltonian

constraint can be cleanly resolved.

Nothing I’ve said here means that I am not highly supportive of Thomas’s

and others efforts to resolve the problems of the hamiltonian dynamics-I

am. But it must be said that a “review” of LQG that focues on this issue

misses the significance of much of the work done the last ten years.

Let me make an analogy. No one has proved perturbative finiteneess of

superstring theory past genus two. I could, and have even been tempted to,

write a review of the problem, highlighting the heroic work of a few

people like d’Hoker and Phong to resolve it. I think it would be useful

if someone did that, as their work is underappreciated. But it would be

very unfair of me to call this a review of, or introduction to, the state

of string theory. Were I to do so, I would rightly be criticized as

focusing on a very hard problem that most people in the field have for

many years felt was not crucial for the development of the theory. This is

not a perfect analogy to what you have done in your “review”, but it is

pretty close.

There are other mis-statments in your review. For example, there are

certainly results at the semiclassical level. Otherwise there could not be

a lively literature and debate about predictions stemming from LQG for

real experiments. See my recent hep-th/0501091 for an introduction and

references. Of course semiclassical states do not necessarily fit into a

rigorous framework-after all, WKB states are typically not normalizable.

But I would suggest that it may be too much to require that results in QFT

that make experimental predictions be first discovered through rigorous

methods. At the standards of particle physics levels of rigor, there are

semiclassical results, and these do lead to nontrivial predictions for

near term experiments. It is possible that a more rigorouos treatment

will in time lead to a rigorous understanding of how classical dynamics

emerges-and that is a very important problem. But given that AUGER and

GLAST may report within two years, may I suggest that it is reasonable to

do what we can do now to draw predictions from the theory.

In closing let me emphasize again that your efforts are very well

appreciated. I hope this is the beginning of a dialogue, and that you will

be interested to explore other aspects of LQG not covered by or addressed

in your review.

Sincerely yours,

Lee Smolin

Urs, have a good time in California!

Maybe we can resume this discussion next week when you get back.

Hi Ludwig,

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

seemsthat 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

designedto 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.

Hi Urs,

Hearty thanks for your reply!

I am pleased that you approve my brief listing of some papers.

For now I can only respond briefly to your question and comments.

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.

Hi Ludwig,

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:

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 connection of TQFT (BF theory especially) with Loop-and-related Quantum Gravity seems to go back to 1995, see for instance the paper linked below

Smolin (1995)

Linking topological quantum field theory and nonperturbative quantum gravityThe 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 principleIt 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 Theory15 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 gravityhttp://arxiv.org/gr-qc/9505028 (TQFT + QG)

Smolin (1998)

A holographic formulation of quantum general relativityhttp://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 Theory55 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 Supergravityhttp://arxiv.org/hep-th/0009018

Smolin, Starodubtsev(2003)

General relativity with a topological phase: an action principlehttp://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 observableshttp://arxiv.org/abs/hep-th/0501191

Urs said:

4) What if spin foams could reproduce the kinematics of the canonical LQG approach. Would that imply that spin foams have no dynamics, either?

Posted by: Urs at January 31, 2005 01:11 PM

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 gravityLee 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

illustratedas 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).

I think my message below is not off topic. But to not waste too much bandwidth on Peter’s blog, please see the complete message at:

http://quantoken.blogspot.com/2005/02/blackhole-entropy-lqg-and-super-string.html

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

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:

The authors want to study that in a followup:

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.

From what I’ve learnt so far (from Vafa’s talk, and comments by various people), and given past experience, it seems to me that it will not take long before LQG is “unified” into the stringy framework.

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.

Interestingly enough Lee Smolin is giving a colloquim at

Lubos Motls ‘s university but in a different department:-)

later this semester.

http://cfa-www.harvard.edu/colloquia/latest.html

Vafa: “…The dimension 4 makes full contact with another approach to try to quantize gravity, in the context of Loop Quantum Gravity, which has also the same flavor of replacing a metric degree of freedom with, in that case, not just a form but a gauge field…”

Lubos, Any comments on Vafa’s statement?

PS Ludwig, Thank you very much for that. The audio is quite unclear on my machine.

O said:

I listened to Vafa’s talk at the topo strings meeting in Toronoto, a few weeks ago. The audio is not very clear, but my recollection is that he mentions LQG as an example of “a form theory of gravity” in 3+1 dimension, and therefore related to topo string theory…

O piqued my curioslity so I listened to the audio just now. Vafa discusses LQG briefly around minute 6 and again in more detail at minutes 16-18 into the talk. On my speakers the audio was quite clear. I transcribed what he said starting around minute 5:35

Vafa: “…The dimension 4 makes full contact with another approach to try to quantize gravity, in the context of Loop Quantum Gravity, which has also the same flavor

of replacing a metric degree of freedom

with, in that case, not just a form but a gauge field…”

It was at this point you can hear Vafa interrupted by someone asking if he had really said LQG and he confirms yes he said LQG.

At minute 15:57, or about 16, he returns to a discussion of LQG but does not mention that he is talking about LQG until minute 18, where he says “…this is one of the starting points of Loop Quantum Gravity…”

What he is doing in minutes 16-18 is outlining the Ashtekar “new variables” formulation of General Relativity. His discussion thereafter is enlightening and makes me wish the slides were also available as well as the audio.

higgs boson said:

“At least Brian Greene seems open to the idea that string theory could be wrong and if it does end up being wrong then perhaps he would be able to steer the other string theorists to better theories? If he has no sway with the physics community at large then you’ll just have to wait until they all die off from old age.”

In other words particle physics has become like a communist state: we must wait for the leaders to die out before we can hope for progressive change! Physics will have to spend 50 or so years behind an iron curtain, with L. Motl manning the machine gun nest shooting at anyone to tries to escape.

Here is Vafa’s talk at this month’s Topological String workshop in Toronto

http://www.fields.utoronto.ca/audio/04-05/topstrings/vafa/

Here is a recent LQG/BF paper

“Quantum gravity in terms of topological observables”

by Freidel and Starodubtsev

http://arxiv.org/hep-th/0501191

O said:

I listened to Vafa’s talk at the topo strings meeting in Toronoto, a few weeks ago. The audio is not very clear, but my recollection is that he mentions LQG as an example of “a form theory of gravity” in 3+1 dimension, and therefore related to topo string theory.

Urs said:

I think Vafa is talking about BF theory. This is related to LQG, but it is not the same as LQG. I am not sure if it is helpful to call this the ‘topological sector of LQG’ as Vafa does…

Posted by: Urs at January 31, 2005 12:40 PM

There is also another LQG/BF paper by Smolin and Starodubtsev that may be of interest:

“General Relativity with a Topological Phase: An Action Principle”

http://arxiv.org/hep-th/0311163

So from reading Smolin’s reply I get the impression that he is saying that

– there is little hope for the canonical quantization of the Einstein-Hilbert action

– one should instead study spin foams and see if one can come up with any amplitudes for these

– if so, one should check if the resulting spin foam theory has a limit in which it reproduces any known theory .

– The canonical approach is only good for studying kinematics, not dynamics.

– In conclusion, as Smolin writes:

So LQG = ‘use diffeo classes of spin networks somehow’ ??

Some comments:

1) The ordinary states of 2-dimensional conformal gravity coupled to matter are not diffeomorphism invariant and hence can hardly be expressed by diffeo classes of spin networks.

2) It is not known if spin foams reproduce any known theory and in particular not if any of them reproduces the kinematical results of the canonical LQG approach. So from the spin foam point of view what reason is there to keep the kinematical results of the canonical LQG approach?

3) What is a physical theory which has a kinematics but not a dynamics? Isn’t kinematics pretty much just my choice of symbols that I am going to use for defining dynamics?

4) What if spin foams

couldreproduce the kinematics of the canonical LQG approach. Would that imply that spin foams have no dynamics, either?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.

6) If one says that ‘spin networks should play a role’ shouldn’t one go all the way and say that ‘generalized Wilson lines should play a role’. (Which in particular generalizes the allowed groups from rotation groups to arbitrary Lie groups and their representations.)

7) If yes, then there is IKKT theory. 😉

“I think the reason Lubos (and other string theorists) is rabid on the topic of LQG is that it threatens the only remaining argument for string theory.”

Perhaps they are afraid of what will happen if string theory is not valid. Think about it. They will have to drop string theory and spend a lot of time and energy to get up to speed on loop quantum gravity or they would have to think about creating some other theory. I would imagine that it would not be a pleasant prospect for your colleagues.

At least Brian Greene seems open to the idea that string theory could be wrong and if it does end up being wrong then perhaps he would be able to steer the other string theorists to better theories? If he has no sway with the physics community at large then you’ll just have to wait until they all die off from old age.

I think Vafa is talking about BF theory. This is related to LQG, but it is not the same as LQG. I am not sure if it is helpful to call this the ‘topological sector of LQG’ as Vafa does.

See also the comment on that point made by Nicolai et al. in their recent paper.

I listened to Vafa’s talk at the topo strings meeting in Toronoto, a few weeks ago. The audio is not very clear, but my recollection is that he mentions LQG as an example of “a form theory of gravity” in 3+1 dimension, and therefore related to topo string theory.

I think that someone in the audience was surprised by Vafa’s statement and asked him to repeat it, and Vafa did, saying somthing along the lines of “in that case, there is not just a form, but the form is a gauge field”.

Would anyone care to verify that my understanding is correct?

Blank said:Can we ignore the foam at the mouth and the shrill rantings and merely evaluate the merits and demerits of the arguments, please?

I agree it would be interesting to look at what Nicolai et al said in their article and how Smolin replied. I thought the tone was basically friendly and cool-headed and the points were substantive.

Nicolai et al pointed to difficulties with LQG dynamics, when attempted along lines of a canonical quantization of GR, and particularly with Thiemann’s hamiltonian constraint (1997?).

Smolin replied that these difficulties were seen in the mid-1990s and that Nicolai and the others seem to have overlooked most of what has been happening in the past ten years in LQG.

Smolin could have cited many papers from the mid-to-late 90s backing up what he said: Loop people finding difficulties with the hamiltonian approach (which Nicolai was stressing) and beginning alternative lines of development. In fact Smolin only cited one paper, he 1996 gr-qc/9609034. I dont know if that was the most representative or the most obviously germane–he might have mentioned papers of about that time or a couple of years later by Lewandowski, Marolf, Gambini and others. I dont remember offhand but could get links if you wish them.

I thought Nicolai et al critique of Loop thoughtful and constructive but merely too narrow. It would be much appreciated, I think, if we could hear similar comment on what Loop people have been working on (for, I would say conservatively, the past 5 years, not as Smolin does, 10). These are approaches like Thiemann’s master constraint programme, Gambini-Pullin discrete quantum gravity (which has an evolution operator, instead of a contraint), spin foams (with their connection to BF theory). To substantiate what I’m talking about I will try to fetch some links when I have time later.

The main thing that strikes me in the Nicolai-Smolin exchange is the absence of heat. Both Nicolai and Smolin work at institutes where research is balanced (both string and loop research is done at AEI-Potsdam and at Perimeter-Waterloo) and neither of them come across as threatened by the other’s discipline. I was impressed by Nicolai’s lively appreciation of the LQG research that he was familiar with and the overall friendly tone. Would like to see more like that and less of Lubos stink-bombs.

Only if you identify yourself … sorry to point out the obvious, but there is as yet no experimental evidence to suggest that we even

needa quantum theory of gravity.Can we ignore the foam at the mouth and the shrill rantings and merely evaluate the merits and demerits of the arguments, please?

here’s the Weinberg talk

http://online.itp.ucsb.edu/online/kitp25/weinberg/

and also Peter’s October 8 blog commented on the talk and its

“only hope of extending SM to include gravity” or words to that effect utterance.

http://www.math.columbia.edu/~woit/blog/archives/000089.html

so the Only Hope notion goes beyond the rabidity and schadenfreude of a few and is at the foundation of string apologetics

Peter said:… string theorists) is rabid on the topic of LQG is that it threatens the only remaining argument for string theory. With the “Landscape”, all hope is gone for ever saying anything at all about particle physics, so all that is left is to keep repeating loudly “only string theory can quantize gravity”…

I am hearing this as a kind of party line: string is “our one best hope”, or sometimes “our only hope”.

It has to be repeated a lot and defended zealously because appearances are rather to the contrary, I’d say, and it is the main premise justifying continued focus of research effort on string.

Steven Weinberg delivered the “one best hope” line in his KITP 25th anniversary talk last year. I’ll get a link if anyone wants.

Still it’s a thought that MTW fell on Lubos head when he was a baby.

I think the reason Lubos (and other string theorists) is rabid on the topic of LQG is that it threatens the only remaining argument for string theory. With the “Landscape”, all hope is gone for ever saying anything at all about particle physics, so all that is left is to keep repeating loudly “only string theory can quantize gravity”. If the idea gets around that LQG is a more promising idea about quantum gravity than string theory, there won’t be any argument at all left for research on string theory as a TOE.

I must confess that I am totally fascinated by Lubos M’s hysterical attitude to LQG. It obviously goes way beyond anything involving physics. Most people I know don’t have much time for LQG, but I don’t know anyone who gets even remotely as rabid as LM. In fact, it’s pretty clear that LM has an intense fear of anything connected with general relativity; hence the various bizarre misunderstandings of the subject he has revealed over the years. What gives? Did a copy of Misner Thorne and Wheeler fall on his head when he was a baby? Or what?

Caracciolo and Pelissetto investigated SO(5) on a 4D

lattice as a quantum gravity model back in the 80s.

The model goes back to a proposal by L. Smolin. I

wonder if the two lines of research are (or can be)

related ? It would be cute to dig out the old

results or redo the old simulations …

There was a great comment on Lubos’ weblog:

Q: Why is the debate (between LQG and strings) so heated ?

A: The stakes are so low !

Peter, thanks for replying. I understand your point better now. I’m struggling to get a preliminary notion of BF theory and how they apply it. there seems to be a 4-manifold M and a principal SO(5) bundle P. I could be way wrong about this but I imagine it as a copy of SO(5) at every point of M, but each group is only a “torsor” or has lost track of its identity, so it is just something at each point of M that the real McCoy SO(5) can act on in a nice way. Sorry, have to go to lunch, back later.

Maybe I shouldn’t have called it “breaking topological invariance”, it was the breaking of SO(5) symmetry by hand that I was wondering about, which is more of a gauge symmetry (although since it is not an internal symmetry, but a symmetry involving local translations, I’m not sure whether “gauge symmetry” is the right way to refer to it either).

I haven’t read this paper carefully yet, but often the subtlety with TQFTs formulated this way is that you have to somehow break topological symmetry to make sense of them, but then everything depends on how you do this. It seems to me that the whole game here is to start with a topologically invariant QFT, and somehow end up with a valid approximation to it that looks like perturbation theory about flat spacetime with the scalar curvature as low energy effective action. For such an argument to be convincing, I’d have to be sure that the final result one wants hasn’t somehow been subtly smuggled into the derivation.

I agree with Wolfgang where he says

“I find it very interesting what Freidel and Starodubtsev have to say about perturbation theory.”

It seems they have found a way to do background independent perturbative analysis, and they suggest that the non-renormalizablity and failure of perturbation in the past was due to going about it wrong: with a fixed background.

It does seem to be a very interesting paper and I hope we hear more comment.

I did not yet find any place where they broke topological invariance, either by hand or any other way.

It looks to me that they broke gauge symmetry by hand, from SO(5) down to SO(4)

but as to topological invariance per se they say in the abstract “We show that the partition function of quantum General Relativity can be expressed as an expectation value of a certain topologically invariant observable.”

What am I missing?

I find it very interesting what Freidel and Starodubtsev have to say about perturbation theory.