Musings

Page 190 of the book “Not Even Wrong” (Woit, Cape edition, 2006):

“A possibility consistent with everything known about superstring theory and loop quantum gravity is that, just as there are many consistent quantum field that don’t include gravity, there are many consistent quantum theories, some field theories, some not, that do include gravitational forces. If the loop quantum gravity programme is successful, it should construct a quantum theory of the gravitational field to which one can add just about any other consistent quantum field theory for other fields. If there is a consistent M-theory, it probably will depend on a choice of background spacetime and make sense for an infinity of such choices. Neither loop quantum gravity nor M-theory offers any evidence for the existence of a unique unified theory of gravity and other interactions. Even if these theories do achieve their goal of finding a consistent quantum theory of gravity, if they don’t have anything to say about the standard model such theories will be highly unsatisfactory since there is a serious question about whether they can ever be experimentally tested.”

Surely the standard model is entirely Yang-Mills exchange radiation based. So the loop transformation scheme has physical dynamics: force-causing gauge bosons flowing between masses. The there-and-back flow of gauge boson energy would constitute the loop.

I can’t believe that the widely held view of “rigor” in theoretical physics is such as to exclude the possibility of representing physical processes by any but the most intangible and sophisticated calculations which turn out to suffer landscape problems. Why is theoretical physics now stuck into a top-down abstract level methodology, instead of building representations of successful QFT based on experimental evidence? Is it entirely down to the fear of being submerged by crackpotism? Or just the fear that the subject might start moving?

Maybe someone can explain what it even *means* for LQG practioners when they talk about field theories being background independant.

AFAICs, the only candidates for such a loaded term are topological field theories, with the simplest nontrivial example being of the Witten type.

Dear Jacques,

There is a large literature on coupling matter to loop quantum gravity and spin foam models as well as extensions to supergravity, higher dimensional gravity, branes, etc. All forms of matter have been coupled including fermions, Maxwell and Yang-Mills, scalars, supersymmetric extensions, p-form gauge fields.

These are all constructed by hand by adding degrees of freedom by enlarging the gauge group that defines the labelings on spin networks. There is also a new point of view about matter, which is that it is emergent from spin foam and some other background independent models of quantum spacetime, because there exist automatically coherent excitations that can be interpreted as chiral matter fields. This is work of Markopoulou and collaborators: F.Markopoulou hep-th/0604120, D. Krebs and F. Markopoulou gr-qc/0510052,, S. Bilson-Thompson, F. Markopoulou, LS hep-th/0603022. This is of course to be preferred if it works as it is much more restrictive, but there is much here to be done here.

Some references to the old approach of adding matter by hand are in my review paper hep-th/0408048, where there is also a list of open problems (not every important question has been solved!). Below are a small set of references found quickly, sorry for the sloppiness of the listing.

Also, as to what we mean by LQG and spin foam models being background independent, look at Rovelli’s book and my hep-th/0507235.

As to Georgi’s objection, there is a test case, which is 2+1 gravity coupled to matter. There are no gravitons but for any Feynman diagram of the matter theory there are gravitational degrees of freedom. The theory seems consistent for all forms of matter it is coupled to. As shown by Freidel and Livine, hep-th/0512113, one can also in this case integrate out the matter degrees of freedom to find an effective field theory on kappa-Minkowski spacetime.

Below are the references, broken up into categories.

For the Hamiltonian version of LQG the coupling to all matter fields was worked out in detail in several early papers, see for example, the following and references cited:

Ashtekar et. Al. Phys.Rev. D40 (1989) 2572

gr-qc/9705019 [abs, ps, pdf, other] :
Title: QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories
Authors: Thomas Thiemann
Comments: 34p, LATEX
Journal-ref: Class.Quant.Grav. 15 (1998) 1281-1314

Thomas Thiemann gr-qc/0110034 [abs, ps, pdf, other] :
Title: Introduction to Modern Canonical Quantum General Relativity

hep-th/9210110 [abs, ps, pdf, other] :
Title: Quantum Einstein-Maxwell Fields: A Unified Viewpoint from the Loop Representation
Authors: R. Gambini, J. Pullin
Comments: 13pp. no figures, Revtex, UU-HEP-92/9, IFFI 92-11
Journal-ref: Phys.Rev. D47 (1993) 5214

For coupling of the path integral or spin foam formulation to Yang-Mills fields:

gr-qc/0210051 [abs, ps, pdf, other] :
Title: Spin Foam Models of Yang-Mills Theory Coupled to Gravity
Authors: A. Mikovic
Comments: 10 pages
Journal-ref: Class.Quant.Grav. 20 (2003) 239-246

3. gr-qc/0207041 [abs, ps, pdf, other] :
Title: A spin foam model for pure gauge theory coupled to quantum gravity
Authors: Daniele Oriti, Hendryk Pfeiffer
Comments: 18 pages, LaTeX, 1 figure, v2: details clarified, references added
Journal-ref: Phys.Rev. D66 (2002) 124010

Coupling to fermions (see the above general papers and:)

gr-qc/9705021 [abs, ps, pdf, other] :
Title: Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories
Authors: Thomas Thiemann
Comments: 26p, LATEX
Journal-ref: Class.Quant.Grav. 15 (1998) 1487-1512

gr-qc/9401011 [abs, ps, pdf, other] :
Title: Fermions in Quantum Gravity
Authors: H A Marales-Tecotl, C Rovelli
Comments: LaTeX file, 37 pages, no figures
Journal-ref: Phys.Rev.Lett. 72 (1994) 3642-3645

hep-th/9703112 [abs, ps, pdf, other] :
Title: Quantization of Diffeomorphism-Invariant Theories with Fermions
Authors: John C. Baez, Kirill V. Krasnov
Comments: 28 pages, latex, 7 ps-files (included) are needed to process the source file
Journal-ref: J.Math.Phys. 39 (1998) 1251-1271
SLAC-comments: Published in J.Math.Phys.39:1251-1271,1998

gr-qc/9506029 [abs, ps, pdf, other] :
Title: Quantum Loop Representation for Fermions coupled to Einstein-Maxwell field
Authors: Kirill V.Krasnov
Comments: 28 pages, REVTeX 3.0, 15 uuencoded ps-figures. The construction of the representation has been changed so that the representation space became irreducible. One part is removed because it developed into a separate paper; some corrections added
Journal-ref: Phys.Rev. D53 (1996) 1874-1888
SLAC-comments: Published in Phys.Rev.D53:1874-1888,1996

For general matter couplings to spin foams

gr-qc/0602010 [abs, ps, pdf, other] :
Title: Group field theory formulation of 3d quantum gravity coupled to matter fields
Authors: Daniele Oriti, James Ryan

For supersymmetry and supergravity in the Hamiltonian formulation:

hep-th/0009020 [abs, ps, pdf, other] :
Title: Introduction to supersymmetric spin networks
Authors: Yi Ling
Comments: 27 pages, 16 eps figures. Based on the talk given at Marcel Grossmann Meeting IX in Rome
Journal-ref: J.Math.Phys. 43 (2002) 154-169

16. hep-th/0009018 [abs, ps, pdf, other] :
Title: Holographic Formulation of Quantum Supergravity
Authors: Yi Ling, Lee Smolin
Comments: 30 pages, no figure
Journal-ref: Phys.Rev. D63 (2001) 064010

hep-th/9904016 [abs, ps, pdf, other] :
Title: Supersymmetric Spin Networks and Quantum Supergravity
Authors: Yi Ling, Lee Smolin
Comments: 21 pages, LaTex, 22 figures, typos corrected and references completed
Journal-ref: Phys.Rev. D61 (2000) 044008

For d=11 supergravity:

hep-th/0003285 [abs, ps, pdf, other] :
Title: Eleven dimensional supergravity as a constrained topological field theory
Authors: Yi Ling, Lee Smolin
Comments: 15 pages+7, Appendix added
Journal-ref: Nucl.Phys. B601 (2001) 191-208

hep-th/9703174 [abs, ps, pdf, other] :
Title: Chern-Simons theory in 11 dimensions as a non-perturbative phase of M theory
Authors: Lee Smolin

For supergravity in spin foam models:

hep-th/0307251 [abs, ps, pdf, other] :
Title: Three-dimensional Quantum Supergravity and Supersymmetric Spin Foam Models
Authors: Etera R. Livine, Robert Oeckl
Comments: 43 pages, 10 figures
Journal-ref: Adv.Theor.Math.Phys. 7 (2004) 951-1001
SLAC-comments: Published in Adv.Theor.Math.Phys.7:951-1001,2004

For higher dimensional gravity:

hep-th/9901069 [abs, ps, pdf, other] :
Title: BF Description of Higher-Dimensional Gravity Theories
Authors: L. Freidel, K. Krasnov, R. Puzio (Penn State)
Comments: 26 pages, Revtex; minor changes
Journal-ref: Adv.Theor.Math.Phys. 3 (1999) 1289-1324

For branes and p-form gauge fields:

gr-qc/9302011 [abs, ps, pdf, other] :
Title: Finite, diffeomorphism invariant observables in quantum gravity
Authors: Lee Smolin
Comments: Latex, no figures, 30 pages, SU-GP-93/1-1
Journal-ref: Phys.Rev. D49 (1994) 4028-4040

For attempts to use LQG methods to discover the background independent formulation of string and M theory:

hep-th/0002009 [abs, ps, pdf, other] :
Title: M theory as a matrix extension of Chern-Simons theory
Authors: Lee Smolin
Comments: Latex, 17 pages, no figures
Journal-ref: Nucl.Phys. B591 (2000) 227-242

hep-th/0104050 [abs, ps, pdf, other] :
Title: The exceptional Jordan algebra and the matrix string
Authors: Lee Smolin
Comments: LaTex 15 pages, no figures
Subj-class: High Energy Physics - Theory; Quantum Algebra

hep-th/0006137 [abs, ps, pdf, other] :
Title: The cubic matrix model and a duality between strings and loops
Authors: Lee Smolin
Comments: Latex, 32 pages, 7 figures
Subj-class: High Energy Physics - Theory; Quantum Algebra

hep-th/9712148 [abs, ps, pdf, other] :
Title: Nonperturbative dynamics for abstract (p,q) string networks
Authors: Fotini Markopoulou, Lee Smolin
Comments: Latex, 12 pages, epsfig, 7 figures, min

Dear Jacques,

I just wanted to list about 20 not-exactly-convincing (and mutually incompatible) papers that are clearly supposed to be the basis of the strange statements, but the commenter before me has already done it, so it’s OK. Thanks, Lee. How are you?

The most modern way to get particle physics in loop quantum gravity is described in two texts that you obtain if you search for “different octopi” (with quotation marks) on Google.com. ;-)

I have read many of these papers. As far as I can tell, none of the papers that claim to contain a supersymmetric model is actually supersymmetric; by a supersymmetry, I mean a fermionic symmetry where translations appear in the commutator. (A gauge supergroup is something different than SUSY, if you understand what I mean.) Most of the papers that claim to incorporate a gauge symmetry do not actually have any symmetry between the components of the multiplet at all - for example the trinion paper.

Some of them have a different strategy, and they just add extra fields in the normal way to the LQG treatment of the metric tensor.

None of the papers actually leads to low energy physics that can look like quantum field theory as we know it.

More generally, I think that you know very well that the question “what fields can you couple LQG to” cannot be invariantly answered. LQG does not work, even qualitatively, even before you couple it to anything. What feature of the LQG do you exactly want to preserve in order to be allowed to say that you can couple it to a field?

In Appendix A of the review by A. Ashtekar, J. Lewandowski in Class. Quantum Grav, 2004, they show what they imagine under the “Einstein-Maxwell” system in LQG. They add the gauge potential and hope for the best. There are papers explaining that chiral fermions and even scalars cannot be coupled to LQG, which also makes SUSY (and Higgs mechanism) impossible. I will write you more details if I re-find them.

All the best
Lubos

If you take hep-th/0501114 by Nicolai et al., you will see on page 12 that the Immirzi parameter being +-i was good for interpreting the SU(2) as the self-dual part of the Euclidean Lorentzian group SO(4). That would morally allow you to couple LQG to chiral fermions. However, the Immirzi parameter is taken to be something completely different, to adjust the black hole entropy etc.

On page 34, they tell you that the consensus LQG approach is to view LQG as lattice field theory, and try to attach scalars and fermions to the vertices of the spin networks. On this page 34, they explain how very awkward expressions you get if you need to obtain e.g. the Dirac kinetic term, by realigning things with viel-beins etc. They agree with me that it is completely uncertain whether the resulting lattice-like physics has anything to do with the Fock space of the matter fields anyway.

They cite Varadarajan 2000 and 2001 for some Fock space ideas in LQG, and Ashtekar Lewandowski gr-qc/0107043. Polymers and scalar fields in LQG are discussed by Ashtekar et al. in gr-qc/0211012, claiming to have filled holes that existed for scalar fields, and it’s up to you whether you still see some holes. ;-)

Thank you for your comments, Luboš.

I’ll now ask that you sit down, take a deep breath, and let someone else have a turn.

Dear Lee,

among the many questions here, one seems to stick out. Could you please tell us whether or not there exists an example of a quantum field theory that can *not* be coupled to LQG?

Thanks,

Dear Michael,

Sure, there is as yet no published work on supergravity with N greater than 2. This is an important problem that should be worked on.

To answer ignoramus’s questions:

1. I do not understand the big deal about gravity being “a kind of gauge theory” being used as the basis of LQG(as in review by LS 0408048). Was this not textbook knowledge before 80s(connection and curvature in GR analogous to gauge fields and field strength in Y-M)? What am I missing?

The missing thing is that it was not known before Ashtekar that GR and supergravity could be formulated so that the configuration space was a space of connections mod gauge transformations, while all the metric information was represented in an electric field conjugate to the gauge field. This is the first key result that everything depends on because it means that all components of the connection commute so you can define quantum states as functionals of connections.

2. It seems to me that LQG is based on the following: BF theory + constraints = Classical GR. It is not clear to me how it is quantized(at least from the LS review); simply do a path integral of the above? How can that solve ANY of the problems associated with quantum gravity, with or without matter?

The reason is that to define the path integral one has to choose a measure. What is new and powerful is that one can use the relation to BF theory to choose a path integral measure that is either a restriction or perturbation of a measure used to rigorously define BF theory. This is the second key result, it underlies the results on finiteness of sums over representation labels in spin foam models.

hi Lee,

I’m curious about some explicit examples. Let’s take a very simple theory which can’t be embedded in perturbative string theory: a single scalar field with canonical kinetic terms and leading interaction $c (\partial \phi)^4$ with $c$ negative. Can this theory be coupled to LQG? If not, why not?

best, allan

I’d like to know how the anomaly cancellation constraint (not in 10d with GS, but the ordinary 4d cancellation) arises in LQG framework. Thanks in advance.

This is regarding one of Lubos’s comments. I’d appreciate it if he can expand on his criticisms regarding varadarajan’s 2000-2001 papers.
Thanks

Dear Allen,

I don’t know the answer, it would be interesting to try. I would expect, though, that if the Hamilton of the matter theory is not bounded from below, this will also be true of the ADM energy when the theory is coupled to gravity. uv finiteness is not the only good property a theory must have, and a theory that is uv finite but whose energy is not bounded from below is probably not physical.

Dear Jacques,

In my first post in this series I mentioned that there are lots of open problems and to please see my review for details. One of the open problems, which you are raising, is what happens to anomalies in theories in which LQG is coupled to gauge fields and fundamental chiral fermions. One possibility, among several. is that the fermions double, as is common in lattice gauge theories. This is, as you note, not solved in the papers I mention and it would be very important to do so. This is one reason for the interest in composite or emergent chiral states, in our recent work.

Dear Michael and everyone,

To answer your questions I think it is important to emphasize what has and has not been shown in LQG and spin foam models (The open problems are not hidden and are emphasized in every review paper on the subject, but perhaps not everyone has read them.)

Let us start with the claim is that there is a universal mechanism, as discussed by Thiemann, which removes ultraviolet divergences when matter QFTs are coupled to gravity. What is meant is that, in the defining of the Hamiltonian and other constraints, a limit has been taken in which a regulator, imposed in the construction of the Hamiltonian and other operators, has been removed, in such a way that the theory remains diffeomorphsim invariant. In doing so operator products that would normally be divergent and require renormalization have been defined in such a way that the resulting operators are finite. In the analogous calculations in ordinary QFT renormalization is needed to remove divergences, here the operator products are defined in such a way that there are no divergences.

You might ask, what happens if you try to do a perturbation theory? Do you not get divergences again from something analogous to integrating over momentum?

Evolution ampltidues can be expressed as
an infinite series in spin foam models (they are in fact the Feynman diagram expansion of a certain dual QFT called group field theory.) The analogue of integrating over momentum is summing over labels on these diagrams which refer to representations and intertwiners of gauge groups. There are explicit calculations and in some cases proofs that for certain choices of these amplitudes, which can be derived from quantizing GR + matter, the sums over these labels also are uv finite.

The point is that diffeo invariant and background dependent QFT really are quite different, in ways that are understood in detail at a technical level.

To understand this you have to really go through the calculations, just as in any context. There are very non-trivial things happening in these calculations, and just as in QFT or string theory I don’t think you can really understand the claims unless you study them.

Now lets turn to what has not been shown.
-It has not been shown that all these theories have a good low energy or classical limit described by an effective field theory on a background spacetime.
-It has not been shown that the quantum Hamiltonian is, with appropriate choices of boundary conditions, bounded from below.

-It has not been shown whether or not the sums over spin foam diagrams, analogous to summing the Feynman diagrams up, is convergent, even if each diagraom is uv finite. (Apart from some results in 2+1 where the diagrams are Borel summable.)

-It has not been shown that fermions do not double so that, when the low energy limit exists, it is chiral.

I hope this makes clear what is meant and not meant by the claims I stated before.

Does this mean, as you imply, that there is a landscape issue because LQG does not restrict matter content? Possibly yes, so far as explicit couplings of matter. (Of course, if chiral fermions double then this is probably not the way to couple matter physically.)

But there is a very different situation with recent results of Markopoulou + others which show that many of these theories have emergent chiral matter. These are just beginning to be studied but there does not appear to be a lot of freedom-once the quantum gravity theory is defined the emergent matter content and dynamicds appears fixed.

A closing comment: the issues you are raising are important. As you imply, they are understudied. I have been urging people for years to study what happens to chirality and chiral symmetry breaking in LQG and spin foam models. Recently there are some results (our work with Bilson-Thompson and Markopoulou, recent work of Alexander, Soo and oa few others) but many questions remain open.

But please don’t assume that because there are important open issues it must mean that we don’t know what we are doing. In fact, there are relatively few of us, and many open problems, so the reason many problems remain open is that there has simply not been enough people working on the subject. But we are also careful, so if you read what we say, you will find that the precise claims-as stated-are backed up with detailed calculations and, in some cases, rigorous proofs. I would then hope that you would see the existence of open problems as opportunities for smart people like you, with good backgrounds in QFT, to get involved and contribute.

Finally, I do think that extending LQG and spin foam models to higher supersymmetry (N) is very important because it would make it possible to 1) understand the role of BPS states in this context and 2) make detailed comparisons possible between LQG and string theory in the AdS/CFT context and in the context of extremal black holes.

Thanks,

Lee

Jacques,

You say that, if the quantum gravity action has a non-trivial UV fixed point, adding matter might destroy this fixed point.

Your reasoning seems to be as follows: if we have a theory with a fixed point and then we embed this theory in another theory with more fields, the space of possible coupling constants is much larger than before and, it would be quite a surprise for the fixed point to survive. I completely agree with this, even if you didn’t spell it out like that.

Now, what bothers me is the this. We know that the existence of the the fixed points is an universal feature. For example, all $\mathbb{Z}_2$ invariant theories in $d=3$ have a non-gaussian fixed point. Even if I add some more fields, I think that as long as the $\mathbb{Z}_2$ symmetry is there, the fixed point survives. What I’m trying to emphasize here is that the existence of the fixed point is a robust feature, provided the original symmetry is unbroken.

Now, for gauge theories we don’t really have symmetries but we have gauge invariance which, I would say, is at least as constraining as a physical symmetry. Then, adding matter fields which also have this gauge invariance (as they have to because of the Equivalence Principle) might not kill the UV fixed point (assuming such fixed point exists).

Thanks!

What precisely do you all have in mind when you say “LQG”?

Originally, the term denoted attempts to quantize gravity by expressing the metric in terms of a connection and using Wilson loop observables for those connection degrees of freedom. This is the attempt to quantize gravity “canonically” by encoding the system in terms of Wilson loops.

I believe this is still what most people think of when hearing the term “LQG”. On the other hand, in a reaction to that Nicolai et. al paper, which listed technical difficulties with this approach, Lee Smolin wrote, on some blog, a long public reply where he said that practitioners of LQG have abandoned this canonical loop-basis quantization about ten years ago in favor of spin foam models.

As far as I understand, spin foam models are a rather general concept, which for instance also includes topological string theory (by constructiuon). So here it crucially depends on which particular spin foam model we are talking about. I am not aware of any canonical choice that people would be thinking of by default (but maybe there is one which I am ignorant of).

But, as far as I can tell, there is still more ambiguity. One or two years ago, there has been increasing activity in trying to understand if gravity can be understood as a perturbation about a topological BF-theory. While I have seen statements that such a setup would lend itself to “LQG-methods”, it appears to me to be a rather independent idea on how quantum gravity might work.

Even more recently, there has been increased activity in the LQG community in studying 3d gravity coupled to matter. I know that the idea is that what works for point particles in 3D should work for strings in 4D, and that this might lead to a description of 4D gravity in terms of spin foams. But at the moment it looks like yet another more or less independent line of attack to me.

Personally, I find some of these approaches more promising than others. So I would hesitate to make general statements about “LQG”. What is “LQG”?

I think before talking about which field theories can be coupled to LQG, there’s a (presumably) simpler question one can ask.
As “traditional canonical” LQG(or LQG type quantizations) essentially relies on choosing a particular Poisson subalgebra(the so called holonomy-flux algebra) and quantizing it using a peculiar GNS functional (one that is spatially diffeomorphism invariant) one can also apply this procedure to field theories on flat spacetime. So for example take U(1) theory on flat spacetime and quantize it this way. One will get a “loopy” Hilbert space with a spatially diffeomorphism invariant measure on which holonomies and smeared electric fields are well defined operators. Now one can ask the question, how is this representation
(with holonomies as well defined operators) related to the usual Fock representation on which holonomies(without smearing) are not even well defined. This question was answered by varadarajan in his 2000-2001 papers. The upshot is that,
there is a so called r-Fock representation which is very closely tied to the usual Fock representation and whose states are distributions over the loopy Hilbert space. One can play the same game for scalar fields (Ashtekar et al. 2001). But all this has only been done for free field theories. I think before addressing the question “what field theories can be coupled to lqg” one should probably study some interacting field theory (say \phi^{4} theory)on flat spacetime in the loopy(or r-Fock) Hilbert space, and see how do divergences arise, can be renormalized etc. As far as I am aware, this has never been done. So atleast from canonical LQG point of view, i think its too naive to say what field theories can or can not be coupled to LQG. Certainly saying that standard model can be easily incorportated in LQG seems like a bit of overselling. But then again, i am completely ignorant about spinfoams and so am unaware of progress happening on that front.
Thanks

Interesting thread.

Is it also possible to view entries in date order, rather than nested? I sometimes find it difficult to locate new comments but perhaps I am being stupid.

Dear Urs,

I agree, I use LQG to mean a collection of models and techniques which study the problem of quantum gravity incorporating the following elements:

-some version of background independence

-some version of a connection as configuration variable

-whose quantum theory is described in terms of the dual loop or spin network basis in which states and histories are expressed in terms of labeled graphs.

There are many things which have been investigated which fall under these characteristics.

Dear Jacques,

You suggest I should have answered, “LQG is (at present) incapable of incorporating QFTs with chiral fermions.” The problem is that in the self-dual representation based on the Ashtekar connection the fermions that are naturally coupled to gravity are chiral fermions. So your proposed statement is wrong.

The issue is not whether there is consistent coupling to chiral fermions at the Planck scale-there is. The issues are a) in which theories is there a good low energy limit described in terms of a low energy effective matter QFT and b) what happens to the fundamental chiral fermions in those cases when they are in reps that would be anomalous?

I speculated that in these cases the fermions might double, but I emphasized this is not the only possibility. Another one is that only those theories that could be represented as non-anomolous low energy theories have good low energy limits. A third is that the low energy effective theories live on non-commutative manifolds on which there are no anomalies. But as I emphasized, we don’t know the answer, this is an important unsolved problem.

The confusion seems to be that you take “incorporate” to mean incorporated in such a way that it has been shown there is a good low energy limit, where as I took it to mean, “can be incorporated so that the quantum constraints have a consistent algebra”, i.e. so there is a well defined quantum theory with a well defined Hilbert space, without regard to whether or not there is a good low energy limit.

Dear Michael,

I genuinely don’t understand your problem. Several questions were asked which I answered honestly. I was explicit about what is known and not known and mentioned various of the open problems. Because I understood that the context was unfamiliar I tried to state carefully what are the issues and questions involved.

The original question was, is there a landscape issue in LQG in that many different fields can be coupled to it, supersymmetry can be present or not etc? I answered yes and then I explained what was meant by saying that a field could be coupled to LQG on a technical level. When the issue was raised about chiral fermions I responded and explained what the issues were. I hope the above comments have further clarified the issues.

Your response above is meant to somehow criticize us but it just gives the impression you are not even trying to understand the issues.

After saying over and over again that there are open problems connected with defining the low energy limit you criticize me because we don’t have a quick answer to, “What prevents unitarity violation above a TeV in their model?” The reason we don’t have a quick answer is that we don’t have a low energy effective field theory in which it makes sense to ask your question. The paper and talks have been quite explicit about these and other open issues. The fact that you fail to get the point suggests to me that you have not spent any time trying to understand how one would even start to do QFT without a background metric.

Lee Smolin said… […]But there is a very different situation with recent results of Markopoulou + others which show that many of these theories have emergent chiral matter. These are just beginning to be studied but there does not appear to be a lot of freedom-once the quantum gravity theory is defined the emergent matter content and dynamicds appears fixed.

what exactly do you mean with dynamics?

Dear Jacques

Thanks for considering the chronological order. I might sign up to your feed, though checking here for new comments is good work-avoidance.

Dear Lee

I would like to understand better how people who work on LQG view the comments by Jacques or say p17 of Nicolai et al on LQG and effective field theory.

I.e. before one even considers adding matter d.o.f., are there already infinitely many LQGs with different Hamiltonians? If so, does this ambiguity correspond to choosing the infinite number of counterterms in perturbative gravity, as I think the intuition of Nicolai (and others) seems to indicate?

I’ve seen this question asked here and elsewhere, so apologies if I am going over old ground—but I’m not sure what the viewpoint is from people who work on LQG. I think it would help people who work on particle physics and string theory to understand what you are doing if you address questions posed in this kind of language.

Is your intuition is that some principles of LQG fix the effective action uniquely?

Dear Jacques,

So far as I know, at the kinematical level, there is no problem coupling to any gauge group and fermions in any representation, that is one can solve the quantum Gauss laws constraint and diffeomorphism constraints consistently in each case. I am not sure whether different choices of fermion reps affect the consistency of the Hamiltonian constraint, so why don’t I suggest to experts that they check this.

Similarly for the 6 dimensional case. It hasn’t been done, so the best thing is that someone look at it in detail.

Dear Boreds,

Different people working in LQG and related areas take different views of these questions. I believe I have already expressed mine many places. Ill try to be clear about them here.

For me LQG is nothing but the study of diffeomorphism invariant quantum gauge theories. At the kinematical level these theories are picked out by choice of a gauge group, manifold, differential structure and value of the cosmological constant. (There is no background metric or geometry.) The LOST uniqueness theorem tells us this there is a unique Hilbert space and observable algebra at the level of gauge and spatially diffeo invariant states, for each of these choices. This then gives a class of kinematical theories labeled by those choices.

These are then a class of theories, like gauge theories on background spacetimes with fixed metrics. As in those cases, the gauge invariances strongly restrict the forms of the quantum theories, kinematically.

I believe there is compelling reason to study this class of theories as possible quantum theories of gravity. These include the fact that all classical gravity theories we know can be expressed as diffeo invariant gauge theories, together with the uniqueness theorem. Furthermore, when one uses the unique representation given us by the LOST theorem, one can find explicit finite closed form expressions for the quantum dynamics derived from those gravitational theories, in both Hamiltonian and path integral forms. There is no other approach in which these things all work, down to rigorous details.

The many results that have been obtained from these theories, or from reduced models obtained from them, more and more supports the view that this class of theories captures a description of physics at the Planck scale which is consistent and plausibly true. The discreteness of quantum geometry is only the start of what is now a long list of results. It is as direct a consequence of the quantum kinematics in this setting as the relationship between energy and frequency is for ordinary quantum mechanics.

At the same time, there is as there is in any quantum theory, some freedom in the choice of dynamics. From the path integral point of view (which for reasons I’ve discussed a lot elsewhere is the more powerful one) a dynamical theory is picked out by 1) a choice of local moves and 2) specifications of the amplitudes for these local moves.

While this is a large class, there are preferred models which have been studied quite a lot. Some of these, such as the Barrett-Crane model, turn out to be uv finite (in the sense that the analogues of integrals over momenta are convergent.) These are also models for which the dynamics is derived through a well defined quantization procedure from classical GR (with or without matter.) These two facts make these special and worthy of a lot of study.

It is not known how large is the class of evolution amplitudes which lead to uv finite theories. I doubt it is an infinite set, but it is not known. This is an important open question.

At the Hamiltonian level there has been more work on this and it has turned out to be so far impossible to find more than a handful of regularizations of the Hamiltonian constraint that are consistent quantum mechanically. There is, so far as I know, no evidence that there is an infinite parameter family of consistent versions of the Hamiltonian constraint. Quite the opposite, most attempts I know about to find new consistent orderings and regularizations on top of the few that were known in the mid 90s have failed-and hence remained unpublished. So the evidence is strongly against the analogy Nicolai et al suggest.

Nevertheless, a few people, beginning with Markopoulou, have advocated a more general RG point of view with flow in some space of couplings. Unfortunately, so far this has not been easy to realize in detail, because of complications with applying the RG to spin foam models discussed in Markopoulou’s papers. But I think this is a fruitful direction and deserves more work. One doesn’t say the Ising model is not a good model of ferromagnetism because there are flows in the space of couplings, and for the same reason the existence of flows in the space of parameters governing evolution will not invalidate the use of models of quantum gravity in giving predictions for experiments that probe the Planck scale.

I hope this helps. One of the confusions in these discussions is that what I defend strongly is the study of this whole class of models, as a uniquely suitable setting for studying the problems of quantum gravity and unification. I do not up to now defend any particular model within it as a candidate for THE theory.

Nonetheless, in spite of the many open problems, and in spite of the freedom to vary the models, there are some generic features whose study suggests predictions which may be soon compared to experiment. While none of them has of yet been rigorously derived, there are to me strong reasons to suspect they are generic predictions of large classes of these theories. These have to do with DSR, emergent particles, and cosmological applications. These are where I am putting my energy.

As for my view of Nicolai et al, imagine someone criticizing string theory now, but not mentioning anything that has been learned since 1993. That is what Nicolai et al did. What they neglected to say is that the criticisms they raised about the Hamiltonian constraint have been well known since the mid 90s and they are the reason most (not all, but most) workers in the field shifted to spin foam models for the last ten years.

Finally, as to your question, is there a principle of LQG that fixes the effective action uniquely? It is not known. Much is fixed by the LOST theorem. I strongly suspect, but cannnot yet prove, that there are generic features of the low energy physics which are fixed, including whether DSR is the symmetry group of the ground state. If true, this is falsifiable experimentally. As I mentioned, attempts to widen the class of consistent quantum dyanmics in this setting have so far led to failure. And while there has been a lot of progress on the key issue of deriving the low energy physics, the problem is not solved in 3+1 (it was only solved in 2+1 last year). So we do not now know the answer to your last question.

I hope this helps clarify things,

Thanks,

Lee

Dear Jacques,

The problem with applying the usual arguments for the chiral gauge anomalies to LQG directly is that the methods used, such as calculating the phase of the fermion determinant in the path integral or calculating a triangle diagram are background dependent. When the metric is an operator there is not a linear operator on a fermionic fock space to compute its spectrum. Nor is there a Fock space of the fermions. So the problem, if it exists, has to be uncovered within the background independent framework. It seems to be not there because we construct explicitly the space of solutions to the Guass’s law constraints for any gauge group and fermion representations. So it either is hidden somewhere and has been missed and needs to be uncovered or the issue arises as a limitation on which theories have low energy limits expressible as conventional renormalizable QFTs. I would guess it’s the latter but I don’t think we can be sure which it is without more work. As I said at the beginning of this, I’ve been emphasizing for years that this was an important issue, thanks for the reminder.

Dear Urs,

At a general level, if a spin foam model is based on a group G and is embedded in a manifold S X R, it gives amplitudes for G-spin networks embedded in S to evolve to G- spin networks embedded in S and can thus be understood as giving a path integral form of dynamics for the diffeo invariant states embedded in S.

You can ask more specific questions: given a specific form of the Hamiltonian constraint operator, H, in CQL, is there a spin foam model which computes the projection onto its kernel whose amplitudes are given by the exponential of H? Or, given the spin foam amplitudes can one go backwards to a form of the Hamiltonian constraint operator?

In 2+1 this is shown to go through in explicit detail, see for example recent papers of Perez. In 3+1 there are steps towards this, for example, in work by Reisenberger and Reisenberger and Rovelli. Another approach to this in progress is the master constraint program of Thiemann, Dittrich et al. But it is true that the spin foam models most studied, such as the Barrett-Crane model, are not derived from the exponentiation of the Hamiltonian constraint, but by another method which involves constraining the measure of a BF theory.

So, as far as kinematics is concerned we understand the relationship between them, but whether there is an equivalence between the dynamics of a CQL theory and a SF model in 3+1 is still an open issue. Given this you are right that the issue of chiral gauge anomalies has to be discussed separately in each model and separately for the Hamiltonian and spin foam models.

At this stage in the discussion, is it fair to say that clarifying the issue of chiral gauge anomalies in LQG is a good research topic and that perhaps we should not speculate more but calculate?

Dear Jacques,

I’m sorry but, I’ll say this as gently as I can: you are completely missing the point in your remark below.

“The fermions appear quadratically in the classical action (and in the Hamiltonian). So, before you even attempt to quantize the gravitational degrees of freedom, you can deal with the fermions once and for all….Quantizing a quadratic action is something we all ought to be able to do.”

Can we please argue in good faith and not imply that somehow my colleagues and I are so stupid that we don’t know ordinary QFT? Yes, in case it is in doubt, I know QFT, I know what a gauge anomaly is and how to compute it.

One of the key challenges of background independent approaches to quantum gravity is to discover if there is a meaningful notion of QFT on a bare manifold with no metric, whose gauge invariances include the diffeo’s of that bare manifold. This means that you do not expand around a given classical metric, you consider the whole metric as a quantum degree of freedom

The term the chiral fermions appear in, in the Hamiltonian constraint, is not quadratic, it is, suppressing indices:

Psi E D Psi,

where E is the densitized one form of the three metric and D is the left handed derivative operator depending on the left handed spin connection plus, if present, the internal gauge field. When you don’t expand E around its flat value this is cubic and quartic. Furthermore, in a connection representation E is a functional derivative operator. You cannot simply integrate over the fermions as if E were a fixed classical field. If you try to do this you get infinite products of functional derivatives at the same point, which are not, to my knowledge, well defined.

There is in fact a successful approach to defining this quartic operator product in a background independent context, yielding a well defined and finite operator in a unique Hilbert space. It depends on the use of a regularization procedure defined carefully so that the diffeo invariance and gauge invariances are preserved. If you would like sometime to actually learn about it we are, as always, ready to explain it to you. But please stop insinuating that its features are due to our ignorance about ordinary Poincare covariant QFT. They are due to the fact that QFT has to be reinvented in order to be applied to a background independent and diffeo invariant context. This is a case where that reinvention succeeded. If the result is that the theory is finite even in cases in which there would be a perturbative gauge anomaly, so that in these cases the attempt to find a low energy effective field theory would fail because short distance modes don’t decouple, what is so hard to accept about that?

There are critical things you can say about LQG, and we, unlike you, are ready at every turn to admit to and discuss open problems. But to say something useful you have to at least grasp the scientific question that the work addresses. Without that, you certainly have no chance to understand the significance of the fact that LQG gives a unique and well defined answer to the question.

Have a good holiday; I’ll likely be off line till sometime next week.

Lee

Correction: sorry, I wrote too quickly. E is the densitized vector field formed from the frame field of the three metric, (equivalently, the * of the pull back of the self-dual 2-form of the metric).

Thanks,

Lee

A few comments from a newcommer.

Prof Distler, you keep asking what LQG has to say on the problems and strengths of effective field theoriy.

The simple answer is, little to nothing. If physics to you is synonymous to representing interactions on Fock spaces then LQG is not even wrong, it is silent and an utter failure to produce anything resembling this.

The idea of LQG is of course to take “Georgis objection” serious, there is no limit in which you’d look at a free gravitational theory and trying to describe it’s degrees of freedom as free field + interaction is doomed and non-renormalizable to boot.

The reaction of the LQG crowd is to point out that GR isn’t a theory on flat spacetime and that we can use the conceptional features of the theory to determine directly how it’s degrees of freedom should look, at least qualitatively.

That is, we don’t try to work our way from an effective field theory and work our way towards an UV completion. Therefore the emphasize on finiteness, renormalization get’s rid of infinities in effective field theories by shoving them into the irrelevant UV degrees of freedom. If we get them directly renormalization applies in reverse, surpressing non renormalizable terms in our effective action.

What imprints the fundamental degrees of freedom leave on these emergent effective theories is hard to tell and not yet understood except in some special cases. But it’s not a natural question to ask given what the theory attempts. It will eventually have to be addressed but it’s not the main focus.

Now on the degrees of freedom Prof Smollin and others developed the algebras of pretty much everything coupled to gravity can be implemented.

This implies conversely that contrary to what a lot of people keep saying LQG does not assume, in it’s key insights, that the metric degrees of freedom are good DoF at all scales. Change the labels and you have different DoF.
There are the area and volume operators but within LQG there are also people arguing that at thePlanckscale that are misnomers and these do not have a geometrical interpretation anymore.

Again, how to get effective field theory, and what the structure of these DoF proposed implies for these theories, and how all their language looks from that perspective is not known.
But just looking at classical gravity it is already clear that gravity has something to say on this, because, uniquely of all the unknown interaction above 1TeV it leaves a deep imprint absolutely everywhere. In a very very different way from the other three forces we know and understand. And this is the way around Gerogi’s objection of course, crucially we know that Gravity provides a background for the other forces, you can’t even write down these kinetic terms without assuming a certain form of the metric. Whatever the degrees of freedom of Quantum Gravity are, they are leaving a very universal imprint. Conversely then to answer your questions on effective fieldtheory, we would first need to understand how the full classical, low energy theory emerges from these DoFs. Something that isn’t even completely understood for QCD for example.

Basically LQG says that you are asking the wrong questions, eventually we will have to answer these, but in the meantime we have to develop a shitload of new tools to effectively work with this fantastic Spinfoam structures we stumbled upon.

As for the snipe against Thiemanns work, his work is on the level of mathematical rigour. If you have an objection to his claims voice it.
In the meantime you wouldn’t have needed to read the paper but just even the abstract to see that Thiemann is claiming a completely well defined theory of Yang Mills coupled to Gravity, not in flat spacetime (which is what the Clay prize is about), which would involve somehow miraculously replacing the quantum DoF in Tiemanns theory with a classical approximation of them.

Even then it is of course by no means guaranteed that this will look like flat spacetime YM or if this will generate other artefacts.

None of this makes any sense to me (from his abstract), im hopelessly confused. If Thiemann has a singularity free nonperturbative definition of say QED, well I for one would like to see it. Is he claiming things like the Landau pole dissappear?

Shouldn’t that be obvious upon inspection, if indeed his theory touches the standard model (a yang mills theory)?

Lattice QFT otoh makes perfect sense, there you see notable progress as you keep making spacing refinements and the problems that are associated are completely well defined and in fact expected. It most certainly touches the usual science that we have all grown up studying.

Dear MoveOn,

There are lots of finite theories that have running effective coupling constants: take any lattice QFT and choose the lattice spacing short enough to accomodate the present limits on uv breaking of Lorentz invariance.

Dear Jacques and others,

Yes, the results together amount to a demonstration of the existence of a new kind of QFT which has no background metric but is diffeomorphism invariant. Moreover, there are “semiclassical states” when gravity is coupled to matter fields, such that expansion around them reproduces at long wavelength a cutoff version of the matter QFT.

I understand your skepticism as I was also trained as a conventional QFT, but at some point you have to decide to take the chance that we are neither dishonest nor stupid and investigate whether these claims have merit. To do this there is no alternative but to study the books and papers. We do take great pains to be honest about open questions, but we also have to insist when questions are not open. These claims are not new (Thiemann’s paper in question is nine years old) and they have been thoroughly checked and examined by a community of smart, critically minded people. If your interest is, as I hope it is, in good faith-that is if this is science and not a debating club-you might consider taking the time to study the papers and understand exactly what the claims are and how they are demonstrated.

I would urge that this is only fair. I only expressed criticisms of string theory after having learned the basics, taught a graduate course in the subject from the standard textbooks, and published a dozen technical papers in the subject.

I promise you that if you do the same you will understand that everything we have claimed is true, as we have stated them. But I do not see how more repetition of those claims can help if you are implacably hostile to the possibility that they are right.

Thanks,

Lee

It seems to me that one of problems that the discussion here reveals is a great imprecision in language that is unfortunate.

The claims are being made in the form, e.g., “we can embed the Standard Model in LQG”, where more accurately one means “in the context of the XYZ model, by following such and such method of quantization, we can come up with a consistent algebra of observables (but have no semi-classical limit; we cannot compute a mass-gap or confinement or whatever)”. “Moreover the desirable features of MNOP model in LQG do not show up in the XYZ model”.

It costs a few more words to be precise, but I think it is worth it.

Second, perhaps one can ask, what would one have learned if the LQG program fails? Is it that quantization in the absence of a metric is not physically meaningful?

I realise this thread has gone on for an incredibly long time already, but…

quoting from Lee, many comments above:

“And while there has been a lot of progress on the key issue of deriving the low energy physics, the problem is not solved in 3+1 (it was only solved in 2+1 last year). So we do not now know the answer to your last question.”

I think from a particle physics perspective, understanding what is the low energy effective theory arising from (an) LQG helps immensely to understand what the theory means physically. So my question is, could someone summarise precisely what has been shown in 2+1?

Dear Jacques and Boreds,

If we are a bit slow in responding its because we would like to answer but we have the sense that anything we say is going to be minimized . At some point you guys just have to read the papers if you are going to appreciate what has been done.

Your tone continues to be one of suspicion that the whole thing is a dishonest trick that the right clever question will unravel. This is not very mature, do you really believe that a lot of smart people have worked for 20 years on something and just fooled themselves?

What I am hoping for but not seeing in these discussions is something as sophisticated and nuanced as we critics of string theory: Yes A, B, C and D have been accomplished and that is impressive. We understand why people are smitten with this approach. But, no, X and Y were never proven and there are no good ideas about how to do Z.

It seems to me a mature, scientific discussion is only possible if people take the point of view that we have in front of us a set of approaches, each partly successful, but so far only partly. The question is not the blue team vrs the red team, this was tiresome to begin with. The question is, what can we learn from the successes and failures of all the different approaches to quantum gravity that might help us to find the right theory?

Coming back to Freidel and Livine, the misimpression Jacques gives is that 2+1 gravity with matter is trivial because the gravitational degrees of freedom are only “topological”. But this is far from the whole story. When coupled to point particles, there are gravitational degrees of freedom associated with holonomies around each puncture, hence the more particles the more gravitational degrees of freedom.

If one then considers a diagramatic expansion of a coupled interacting QFT + gravity, in terms of spin foams with embedded Feynman diagrams, then the number of gravitational degrees of freedom is different for each spacelike slice of each diagram.

Integrating them out for a generic Feynman diagram of the matter, to re-express the theory as an effective field theory on a fixed background with no gravitational degrees of freedom is far from the trivial exercise Jacques implies it is. Go read the papers and then judge.

At the risk of repeating, yes, the problem of the low energy limit in 3+1 is not solved. There are however several promising directions in which there have recently been non-trivial results, among them are applying Freidel and Livine to 3+1 (and yes there is a strategy to do it-it depends on re-expressing spin foam models as an expansion around a TQFT), the work of Markopoulou and collaborators on identifying excitations as noiseless subsystems (which is more generL than our “preon paper”) and the work of Rovelli et al on the graviton propagator.

Yes, there is much to criticize as there remains much to do. What I would love to do is to get you interested enough that you consider contributing to this.

Perhaps you think these several different directions to the low energy limit are bound to fail. Fine, then invent your own approach to constructing a background independent theory and/or studying the low energy limit of such a theory. Perhaps you can do something better-I am sure there is something better because I believe the problem is solvable. Either way I suspect you will appreciate from struggling with it yourself that these are hard problems and that what has been done is substantial and useful.

Dear Lee,

I very much appreciate your efforts to help all of us understand the progress in LQG. However I can not understand your frustration about the lack of good faith on the side of some of the commenters here and I believe most (if not all) posters are guided by the true desire to grasp the exciting new results you are talking about.

However I think you should also understand that as long as there is no substantial amount of results from LQG that can be understood and appreciated *without* reading all LQG papers you will never be able to get people excited. That is just how things work for anyone working outside that field like myself. I am also not a string theorist but the results from AdS/CFT, twistor strings, etc, are all understandable to me without mentioning the word “string”. Thus string theorists have an easy time convincing me that their field is exciting and promising (at least as a toy model for studying gauge theories). I have read loads of posts from you as well as a couple of papers on LQG but I still couldn’t find analogous results that can be understood and appreciated *without* the LQG lingo and special LQG methods.

Once the amount of such results reaches a critical mass everybody will get excited about LQG but not before.

Best wishes,
Xiaofeng

By the way, since I am currently occupied with the recent work by Freed, Gomi, Moore and Segal (I, II, III), I feel like mentioning that what they study is also quite relevant for BF-theory, that class of topological field theories which underlies a few approaches used in the LQG community.

Given an abelian $(n_1-1)$-gerbe with connection $\hat A_1$ and an $(n_2-1)$-gerbe with connection $\hat A_2$, both on a space $X$ of dimension $\mathrm{dim}(X) = n_1 + n_2 + 1$, we can form an $(n_1 + n_2)$-gerbe with connection denoted by

on $X$ and take its $(n_1+n_2+1)$-dimensional holonomy over $X$ as an action functional

Locally, where $n$-connections $\hat A$ are given by $n$-forms $A$, the integrand is

which you can read as

where $F$ denotes curvature.

For the case that $n_2 = 1$, this is nothing but BF-theory.

Actually, as stated this is both more general and more restrictive than what is usually considered in BF-theory.

1) As stated, this is just abelian BF theory.

2) Usually, in BF-theory $\hat A_1$ is assumed to be given by a globally defined $n_1$-form. So in this respect the above is more general.

I think there should be ways to extend this to nonabelian BF theory and the result should be quite relevant to spin foam approached based for instance on gerbe-charged solitonic string-like objects in 4d ($\to$).

Interestingly, already the abelian case has plenty of highly interesting subtle effects, as Freed-Moore-Segal show in detail. One can even generalize the setup from the world of integral classes and $p$-forms to that of $K$-classes and differential K-classes. That would yield sort of a BF-theory for RR-charges, if you like. But that I haven’t learned yet.

(I know that greated minds than me have claimed before that BF-theory is a common link between some things done in the LQG community and things appearing in other approaches to quantum gravity.)