## June 25, 2006

### The LQG Landscape

Over at Cosmic Variance, in a long, and somewhat histrionic comment thread, Lee Smolin makes a new (to me, at least) claim about LQG

LQG easily incorporates most proposals for beyond the standard model unification including supersymmetry.

Since I’m afraid it will get buried over there, I thought I would drag the discussion of this rather important physics point over here. Hopefully, some LQG experts can chime in and explain Lee’s statement.

1. What classes of quantum field theories can be incorporated in LQG and what classes cannot?
2. In what sense do the former constitute “most”?
3. In light of the fact that “most” can be coupled to LQG, how are we to deal with Georgi’s objection (which is discussed at greater length here) ?

In the same comment, Lee also says

… someone might earn a Clay prize by rigorously constructing quantum Yang-Mills within LQG. It will certainly not be me, but there are people working on exactly that program. The conjecture is that background independent QFTs are more likely to exist rigorously in 3+1 dimensions than Poincare invariant QFTs.

It would also be interesting for someone to chime in with an explanation of the intuition for why coupling to quantum gravity should make the problem of constructing quantum Yang Mills theory easier, rather than harder.

Posted by distler at June 25, 2006 12:53 PM

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### Re: The LQG Landscape

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?

Posted by: nc on June 25, 2006 2:07 PM | Permalink | Reply to this

### Re: The LQG Landscape

This paper seems to indicate that the Immirzi parameter cannot be set arbitrarily once fermions are included.

Thus the question of “What classes of quantum field theories can be coupled to LQG and what classes cannot?” should include the sentence “and still get the correct value for the BH entropy”.

Posted by: wolfgang on June 25, 2006 2:33 PM | Permalink | Reply to this

### Immirzi

The Immirzi parameter is one problem. The cosmological constant is another (already vexing, even before coupling LQG to matter).

But I wasn’t hoping for an answer to all questions about LQG, here. Just a simple “foundational” one: what sorts of quantum field theories can be coupled to it?

P.S.: I hope you don’t mind that I turned your reference to that paper into a clickable hyperlink.

Posted by: Jacques Distler on June 25, 2006 3:07 PM | Permalink | PGP Sig | Reply to this

### Re: Immirzi

There is an even more foundational question, which no one has ever answered to my satisfaction: is LQG a well-defined theory? I always get the impression they effectively are choosing some action at a cutoff scale, then cloaking this in obscure mathematics. In other words, is there not possibly some infinite set of nj-symbols (or something along these lines) that reflects the infinitely many parameters of nonrenormalizable gravity? If so, it seems to me that it wouldn’t be at all surprising that one can couple any field theory to this, since I can always write down a nonrenormalizable action for gravity coupled to whatever I want.

Posted by: Anon. on June 25, 2006 8:31 PM | Permalink | Reply to this

### Re: Immirzi

This comment is a bit late, but I’ll post it anyway–the Immirzi parameter can still be set arbitrarily with fermions. The paper mentioned previously was more or less superceded (and subsequently rewritten) by hep-th/0507253. The original hope was that the Immirzi term would yield parity violating effects via torsion in the effective field theory. It was then shown that these effects come from the choice of what you might call a “shadow” term in the fermionic Lagrangian which is also parity violating but has no classical effect in the absence of torsion. With the approproate choice of the shadow term, one can reproduce Einstein-Cartan gravity coupled to fermions (with the correct torsion terms) for an arbitrary value of the Immirzi parameter. This was shown in hep-th/0510001 and again in more detail in gr-qc/0601013. With other choices for the shadow term you can get parity violating effects or even no torsion at all. This of course is all at the classical level–I’m not sure that it has been investigated at the quantum level.

Posted by: Andrew Randono on June 26, 2006 8:19 PM | Permalink | Reply to this

### Hodge-*

Dear Andrew,

Thanks for chiming in. Maybe you can explain something that puzzles me about the discussion in these papers, which purport to couple fermions to gravity in Hilbert-Palatini form.

They write expressions like $\int *e_a \wedge \overline{\psi} \gamma^a D \psi$ where “$*$” is the Hodge-* operator.

How does one define the Hodge-* operator, without assuming that the vierbein is invertible (which one presumably does not want to assume, if one is planning an Ashtekar-like quantization)?

Posted by: Jacques Distler on June 27, 2006 2:31 AM | Permalink | PGP Sig | Reply to this

### Hodge-*

Hmmm.

I suppose what you must really mean is $\int \epsilon^{a b c d}e_a\wedge e_b \wedge e_c \wedge \overline{\psi}\gamma_d D \psi$

Is that it?

Posted by: Jacques Distler on June 27, 2006 3:50 AM | Permalink | PGP Sig | Reply to this

### Re: Hodge-*

Yes, this is correct. The trick is to write everything in a way that all the metric information, in this case the dual, is in the SO(3,1) representation space prior to quantization. Then the e’s, which project the metric back down to the base manifold, become operators.

Posted by: Andrew Randono on June 28, 2006 11:42 AM | Permalink | Reply to this

### Re: The LQG Landscape

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.

Posted by: Haelfix on June 25, 2006 2:40 PM | Permalink | Reply to this

### Re: The LQG Landscape

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
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
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
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
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

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

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
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
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

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

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

Posted by: Lee Smolin on June 26, 2006 6:25 AM | Permalink | Reply to this

### Core Dump

Thanks for the long list of papers. They will, I’m sure, make for some interesting reading.

But, first to Georgi’s objection. You say:

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.

I have no idea why you think that Freidel-Livine addresses Geogi’s objection, let alone answers it.

First of all, Freidel and Livine integrate out gravity (not the matter) to obtain an effective theory of matter on a noncommutative spacetime (without gravity).

It’s nifty that one can do that in 2+1 dimensions, where there are no local degrees of freedom in the gravitational field. But it obviously doesn’t generalize to 3+1 dimensions (where the gravitational field has two local propagating degrees of freedom). Attempting to do the same thing in 3+1 dimensions would yield (if it were even possible) a disgusting nonlocal mess.

But that’s neither here nor there. Freidel-Livine has nothing to say vis-a-vis Georgi’s objection, and I don’t know why you brought it up.

Turning to the general question of coupling matter to LQG, am I to understand from your comments, and the lengthy list of references, that where you previously said “most”, you actually meant “all”?

Is there an example of a QFT that cannot be coupled to LQG?

And, if any QFT can be coupled to LQG, does that not make the “Landscape” problem infinitely worse (there being no “Swampland” in LQG)?

I have to say that, looking over the abstracts (I haven’t looked at more than a couple of the papers themselves, yet) gives one an interesting sense of the more cautious, methodical, “foundational” style of the LQG community, in contrast to the shoot-from-the-hip, over-hype and over-claim style of the string theorists.

Here’s Thomas Thiemann’s abstract

It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum. Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized.

Since QCD is part of the Standard Model, I suppose that it is only Thomas’s natural modesty that prevented him from submitting this paper for the Clay prize.

Posted by: Jacques Distler on June 26, 2006 1:11 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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

Posted by: Lubos Motl on June 26, 2006 11:47 AM | Permalink | Reply to this

### Re: The LQG Landscape

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. ;-)

Posted by: Lubos Motl on June 26, 2006 12:00 PM | Permalink | Reply to this

### Re: The LQG Landscape

An addition. The papers by Ling and Lee, and Ling and others, e.g. more recent SUGRA paper

http://arxiv.org/abs/hep-th/0310141

are always built on the assumption that you can view gravity as a deformed topological field theory. It is a purely classical argument that requires unusual quantization procedures and has no reasons to give the right physics at the quantum level.

The fermionic part of the supergroups they have is manifestly non-isomorphic to the actual spacetime supersymmetry (or superdiffeomorphisms), and this strongly indicates that you can never get such a supersymmetry inside them because it is not there to start with and there is no reason why it should appear. Similar criticism would however also apply to Petr Horava’s holographic field theory and other approaches that many of us have been trying.

In loop quantum gravity, it is their assumption that the only task is to find a different description of the classical system, a field redefinition, and then it works inevitable at the quantum level. In effect, all their excitement is based on counting the number of classical off-shell degrees of freedom. Of course, we now know dozens of reasons why such an assumption is naive.

In holography, the number of classical degrees of freedom in the two dual descriptions looks completely different - even the spacetime dimension is different - nevertheless the systems are fully equivalent. It’s because of dynamics in the quantum theory.

The notion of the “number of degrees of freedom” in a full quantum theory (at generic coupling where the classical limits are not applicable) only makes sense if you count the actual number of quantum states (and entropy) at some energy, which requires you to know the Hamiltonian.

Their alternative paradigm is that they assume that you can count the number of degrees of freedom before you know what the dynamics is, and then keep the number as you completely change dynamics from topological gauge theory to gravity or anything else. It’s just a completely flawed assumption that is enough to show why hundreds of their papers are wrong both morally as well as in details - and of course, there are very many additional reasons why these papers are incorrect.

Even the simple toy model, 3D gravity, does not really confirm the naive preconceptions that 3D gravity is isomorphic to Chern-Simons theory. It is just a classical coincidence that is invalidated by virtually every quantum effect you look at: different ranges and signs in the path integral etc. In higher dimensions, the differences become even more striking. Most of the problems in the LQG line of reasoning, much like Prof. Penrose’s reasoning etc., can be summarized by the fact that they don’t take quantum theory seriously and think that it is always just a straightforward addition to classical physics. They’re not right.

Posted by: Lubos Motl on June 26, 2006 12:33 PM | Permalink | Reply to this

### Re: The LQG Landscape

Most of the problems in the LQG line of reasoning, much like Prof. Penrose’s reasoning etc., can be summarized by the fact that they don’t take quantum theory seriously and think that it is always just a straightforward addition to classical physics.

I get the same vibe reading some of the articles. This is incorrect on so many levels, of course. The insights of QFT are too many(anomalies, instantons, SUSY, duality, effective field theories etc..) to be appreciated from merely thinking about the EPR paradox or similar such “deep” philosophical matters.

A couple of 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?

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? I realize that in some cases a clever trick can make a seemingly hard problem more tractable(like Fadeev-Popov ghosts); I see nothing like that here.

However, maybe I am missing something since work in such areas is being published by leading journals like Adv Theo Math Phys(home to many important papers and with distinguished members in the Editorial Board).

Posted by: ignoramus on June 26, 2006 1:50 PM | Permalink | Reply to this

### Re: The LQG Landscape

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

Posted by: Jacques Distler on June 26, 2006 12:54 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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,

Posted by: Michael on June 26, 2006 6:33 PM | Permalink | Reply to this

### Re: The LQG Landscape

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.

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.

Posted by: Lee Smolin on June 26, 2006 7:23 PM | Permalink | Reply to this

### Matter, not (super)gravity

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.

That’s about changing the (super)gravity theory. Michael (and I) were interested in fixing a particular (super)gravity theory and asking which matter QFTs could be coupled to it.

Obviously, the matter QFT must have at least as much supersymmetry as the supergravity theory you wish to couple it to (i.e., the matter theory coupled to $N=2$ supergravity must have, at least, $N=2$ supersymmetry).

But, beyond that, are there any restrictions?

Posted by: Jacques Distler on June 26, 2006 7:50 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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

Posted by: allan on June 26, 2006 8:12 PM | Permalink | Reply to this

### Re: The LQG Landscape

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.

Posted by: anonymous on June 27, 2006 12:59 AM | Permalink | Reply to this

### Any QFT?

Another way to state your question is:

1. Can one couple an $SU(N)$ gauge theory with a single Weyl fermion in the fundamental representation, $N$, to LQG?
2. If not, how about an $SU(N)$ gauge theory, with a Weyl fermion in the $\overline{N}\oplus \wedge^2(N)$ representation?
3. If LQG distinguishes between cases 1,2, how does it do so?

The statement that LQG can incorporate the Standard Model (let alonemost proposals for beyond the standard model unification”) hinges on this “foundational” question.

As far as I can tell, none of the long list of papers that Lee referred to even attempts to answer it.

To the contrary, people like Thomas Thiemann insist that LQG provides a manifestly gauge-invariant, nonperturbative UV regularization, and hence any theory coupled LQG (even case 1 above) is automatically free of anomalies.

Posted by: Jacques Distler on June 27, 2006 1:42 AM | Permalink | PGP Sig | Reply to this

### Re: Any QFT?

The reason I asked the question about which field theory is background independant (in lqg terms) is b/c coupling to a *background dependant* field theory (virtually all known examples other than tqfts, at least in the language I understand) will manifestly break their diffeomorphism invariance and afaics that is explicitly forbidden in their framework by internal theorems and so forth (pls correct me if im wrong).

Posted by: Haelfix on June 27, 2006 2:45 AM | Permalink | Reply to this

### Re: Any QFT?

I suppose that’s another way of stating the question I asked Andrew above.

Posted by: Jacques Distler on June 27, 2006 2:51 AM | Permalink | PGP Sig | Reply to this

### Re: Any QFT?

Based on what I read and heard about LQG, I feel that Jacques is right.

There are no papers that would even attempt to incorporate Standard-Model-like chiral gauge couplings into LQG. If you don’t have chiral couplings, there are no gauge anomalies.

LQG is a background-free, anomaly-free, chiral-coupling-free, scalar-free, graviton-free, Yang-Mills-free, spacetime-free, Lorentz-invariance-free, unitarity-free theory. And it’s for free.

On the other hand, there are various other anomalies that particle physicists are not familiar with that LQG practitioners are solving such as the unitarity anomaly and off-shell anomalies in the algebra of constraints discussed by Nicolai et al.

Posted by: Lubos Motl on June 27, 2006 8:08 AM | Permalink | Reply to this

### Re: The LQG Landscape

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

Posted by: anonymous on June 27, 2006 1:44 AM | Permalink | Reply to this

### Re: The LQG Landscape

Dear anonymous,

at least so far, I have not written any criticism of Varadarajan’s papers. What I wrote was that Nicolai et al. cite Varadarajan’s papers.

Of course, indirectly, the fact that a paper XY is cited by Nicolai et al. review of loop quantum gravity is indirectly a criticism :-), but I did not make this paradigm manifest.

I will try to write a comment about these papers if I successfully finish the trip.

Best
Lubos

Posted by: Lubos Motl on June 27, 2006 7:55 AM | Permalink | Reply to this

### Re: The LQG Landscape

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

Posted by: Lee Smolin on June 27, 2006 5:11 AM | Permalink | Reply to this

### Like a Lattice

Many a time, when I was first attempting to understand what y’all were doing in LQG, I was chided for assuming, even tacitly, that LQG was like a lattice theory.

Now, when faced with the obvious contradiction between the statement that LQG provided a gauge-invariant, nonperturbative regulator for field theories coupled to it and the strongly-held belief among particle theorists that chiral gauge theories with randomly-chosen fermion representations suffer from gauge anomalies, you assert that perhaps LQG suffers from fermion doubling.

(At least Thomas Thiemann has the strength of conviction to assert that the particle theorists are wrong.)

In Lattice Gauge Theory, it’s easy to see where the doublers come from: they are modes of the lattice fermions with momenta of order the inverse lattice spacing. Is there any way to see where the doubler come from in LQG?

In any case, regardless of where they come from, if LQG suffers from fermion doubling, then the correct answer to the question I posed in this blog post is

LQG is (at present) incapable of incorporating QFTs with chiral fermions.

That would include both the Standard Model and all “proposals for beyond the standard model unification”.

To say anything else would be somewhat misleading, no?

You write:

This is one reason for the interest in composite or emergent chiral states, in our recent work.

Well, as you know, I’ve expressed grave doubts about that paper. In that discussion, you never did explain how the Standard Model (without a Higgs, or a Higgs-replacement) managed to remain unitary at energy scales above a TeV.

Amusingly, this ties directly into Allan’s question. The “Higgless” Standard Model is, in this regard, no different from the toy field theory he asked you about.

Posted by: Jacques Distler on June 27, 2006 10:44 AM | Permalink | PGP Sig | Reply to this

### … and supersymmetry

LQG easily incorporates most proposals … including supersymmetry.

Just to complete a thought, that might not be obvious to the non-cognoscenti, fermion doubling is a key impediment to putting supersymmetry on the lattice ($N=1$ supersymmetric theories are inherently chiral).

It’s only when one gets to $N=4$ supersymmetry that Kogut-Susskind-type fermions (with the attendant replication of fermion flavours) prove to be exactly what one wants, in order to put extended supersymmetry on the lattice.

But, then, no one has attempted an LQG formulation of $N=4$ supergravity…

Posted by: Jacques Distler on June 27, 2006 12:24 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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!

Posted by: Lord Sidious on June 27, 2006 12:11 PM | Permalink | Reply to this

### Fixed points

I think that this comment was really intended for either this entry or this one. Unlike some people, all my old entries have comments open, and it’s better to add to the discussion there, rather than veering off-topic here.

But, anyway…

There are really two issues.

1. Does the erstwhile fixed point persist when you add new degrees of freedom to the theory? Sometimes, this is guaranteed by a symmetry. But, in the case at hand, no one knows of any symmetry that guarantees the existence of a fixed point in the first place.
2. Does the low-energy theory lie on an RG trajectory emanating from the fixed point?

Even if the modified theory still has a nontrivial UV fixed point, we will need to tune the couplings of the low-energy theory so that we recover the property that it lies on a trajectory emanating from the fixed point (recall that a UV fixed point has an infinite number of UV-repulsive directions).

When we don’t know what new field content lies just above accessible energies, we have no idea how the couplings of the low-energy theory are to be tuned.

Put differently, in the context of effective field theories, there’s no way to extract any useable information from the knowledge/belief/hope that there exists a UV fixed point, which is approached only at energy scales much much higher than accessible energies.

Posted by: Jacques Distler on June 27, 2006 12:49 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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”?

Posted by: urs on June 27, 2006 4:38 PM | Permalink | Reply to this

### Definitions

Since it was Lee’s statement

LQG easily incorporates most proposals for beyond the standard model unification including supersymmetry.

that I set out to try to understand, I will leave it entirely up to him to choose whatever definition of “LQG” makes the statement true.

Posted by: Jacques Distler on June 27, 2006 5:48 PM | Permalink | PGP Sig | Reply to this

### Re: Definitions

Unfortunately, it does not seem that we can expect a straight answer even to basic questions. An even simpler question is whether or not we should be surprised.

Posted by: Michael on June 27, 2006 8:24 PM | Permalink | Reply to this

### Cynicism

You can choose to be cynical about this, if you wish. For myself, I choose not to.

I’ve found this whole discussion quite informative, if a little lopsided. I was hoping that some other LQG experts besides Lee (and Andrew) would chime in with responses the questions raised here. After all several of them either have their own blogs, or are frequent commenters on other blogs.

Still, one takes what one can get. I think I got something out of this discussion; I hope others feel likewise.

Posted by: Jacques Distler on June 27, 2006 10:11 PM | Permalink | PGP Sig | Reply to this

### Re: Cynicism

Am I being cynical, or just honest? You and I both asked whether there is an example of a matter QFT that can’t be coupled to LQG. An acceptable answer would be “Yes, for example XY.” or “No such example is known.” Instead I get: “there is as yet no published work on supergravity with N greater than 2” Hmmm, and the question was?! His second attempt was even more nebulous: “Let us start with the claim is that there is a universal mechanism, as discussed by Thiemann […]” – No thanks, I asked a question!

I don’t know, Jacques, if the PC-police allows you to say so, but you *know* that the “different octopi”-paper is contentless, incosistent and childish. You have cherry picked an example of a question Lee refuses to answer straight up: What prevents unitarity violation above a TeV in their model? Am I being cynical because I know the unique correct answer to this question and can state it in a concise sentence?

If I ask string theorists such as yourself, I typically get sharp verifiable and correct answers. Do you mind if I notice the difference?

Oh, I just decided you don’t get that praise for free. ;) Please summarize for me what it is you learned. Thanks!

Posted by: Michael on June 27, 2006 11:18 PM | Permalink | Reply to this

### Re: The LQG Landscape

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

Posted by: anonymous on June 28, 2006 2:40 AM | Permalink | Reply to this

### Re: The LQG Landscape

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.

Posted by: boreds on June 28, 2006 1:17 PM | Permalink | Reply to this

Not stupid at all.

It’s been on my “TODO” list for a long time to write a javascript which would allow the reader to sort the comments into chronological order. I’ve seen similar things on other sites, and they work quite well.

In the meantime, you can always subscribe to my comment feed, which delivers the last 20 comments in chronological order.

Posted by: Jacques Distler on June 28, 2006 1:30 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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.

Posted by: Lee Smolin on June 28, 2006 3:57 PM | Permalink | Reply to this

### Once More on Anomalies

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.

No one ever claimed that there is anything inconsistent about coupling chiral fermions to 4D gravity.

The issue only arises when you have gauge fields and you want to couple chiral fermions in a complex representation of the gauge group.

And then the issue only arises in the quantum theory, not at the level of the choice of canonical variables in the classical theory.

The issue is not whether there is consistent coupling to chiral fermions at the Planck scale-there is.

This is either some sort of semantic quibble or a basic misunderstanding.

Lattice gauge theorists have no trouble writing down lattice Lagrangians that naïvely look like they contain chiral fermions. But appearances to the contrary, the spectrum is actually doubled, and the theory is non-chiral.

(Again, since you appeared to be confused about this, one can always choose to count Weyl fermions of a fixed chirality. The invariantly-defined distinction is between whether these Weyl fermions form a complex representation, $R$, versus a real representation, $R\oplus\overline{R}$, of the gauge group.)

The lattice gauge theorists don’t go around saying, “Sure, we can put chiral fermions on the lattice; there’s just this pesky little problem of whether there’s a suitable low energy limit.” They say, “We have trouble putting chiral fermions on the lattice.”

Above, I gave the example of an $SU(N)$ gauge theory coupled to a Weyl fermion in two different possible representations of $SU(N)$:

1. The $N$ representation.
2. The $\overline{N}\oplus \wedge^2(N)$ representation.

Is there any evidence that LQG treats these two cases in a qualitatively different way? What would that evidence be?

This is the central question of this subject. The reason why lattice gauge theory has trouble incorporating chiral fermions is that it doesn’t distinguish (despite the efforts of many smart people) between these two cases.

If LQG doesn’t distinguish between the two cases, either, then there is no way that it can possibly incorporate chiral gauge theories.

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,

That’s exactly what anomalies spoil.

The only reason why anomalies ever arise is the absence of a short-distance regulator that preserves the symmetries of the classical theory. If you have a short distance regulator which allows you to build a quantum theory in which the symmetries (in this case, SU(N) gauge-invariance) are preserved, then you are done. There are no anomalies.

If the gauge symmetry is unspoiled in the high energy theory, there’s no reason for it to be spoiled in the low energy limit.

Now, it could be that fermion-doubling renders would-be chiral gauge theories coupled to LQG non-anomalous (by rendering them non-chiral). But, if that’s the way gauge invariance is “saved” in the LQG approach, then no one in his right mind would say that you’ve solved the problem of coupling chiral gauge theories to quantum gravity.

A third is that the low energy effective theories live on non-commutative manifolds on which there are no anomalies.

How would non-commutativity change anything?

But as I emphasized, we don’t know the answer, this is an important unsolved problem.

It doesn’t trouble me that there are unsolved problems. “Unsolved problems” are how we physicists earn a living. But incorporating chiral gauge theories into LQG still every bit as much of an unsolved problem as it was in 1997, when Thiemann claimed to have solved it.

Maybe this whole chiral gauge theory business is too complicated. Maybe you need to tackle a simpler problem first.

I know that your colleagues don’t much care for thinking about quantum gravity in other dimensions, but you have.

How about LQG in 6 dimensions? Can one couple it to a single Weyl fermion (no gauge theory, or anything complicated like that; just pure gravity, coupled to a Weyl fermion)?

Posted by: Jacques Distler on June 28, 2006 9:23 PM | Permalink | PGP Sig | Reply to this

### dynamics?

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?

Posted by: confused on June 29, 2006 4:28 AM | Permalink | Reply to this

### Re: The LQG Landscape

Dear Jacques

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?

Posted by: boreds on June 29, 2006 12:32 PM | Permalink | Reply to this

### Re: The LQG Landscape

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

Posted by: Lee Smolin on June 29, 2006 6:38 PM | Permalink | Reply to this

### Too Naïve

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’ve explained to you why I think that’s wrong, and, long ago, I pointed you to a paper which explains, from the Hamiltonian perspective, where the (unremovable) Berry-phase in the ground state of the fermion system arises.

It’s certainly possible that a naïve; quantization of the fermions might miss this phase but I don’t think one should put much stock in a quantization more naïve than what Schwinger was able to do in the 1950s.

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.

I’d be a little surprised if you had somehow managed to shift the anomaly into the Hamiltonian constraint. But, since you haven’t been able to find it elsewhere …

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

I shall look forward to hearing about the 6D case. Stripped of all the gauge-theoretic complications, it should be a very good test of LQG quantization methods.

I have a lot of questions about your comments to “Boreds”. But I’ll leave those for later (or for others to pursue).

Posted by: Jacques Distler on June 29, 2006 7:41 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

I remarked here that there are quite different methods that are called “LQG”. One of them is CQL, “canonical quantization in terms of loop variables”. Another is SF, “spin foam models”.

As far as I am aware, there is nothing known about a direct relation between these. Or is there? Is there a way to construct a spin foam model which reproduces CQL? Or the other way around?

I am aware that the introduction of “spin foam” was motivated as an attempt to view “spin networks” evolving in time, somehow. But has this ever been made precise?

As Lee Smolin emphasized once again, most people have abandoned CQL more than ten years ago due to the problems that Nicolai et al have recently listed in two papers:

[…] 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.

This would imply that there is indeed no equivalence of CQL and certain spin foam models, since otherwise there would be no reason to abandon one in favor of the other.

From all this it seems one would have to conclude that when we talk about LQG, we should be talking about spin foam models, not about CQL.

But in the above reply, Lee Smolin says for instance

[…] 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.

This is a statement which concerns CQL and CQL only, as far as I can tell. It is in CQL that one has to solve a Hamiltonian and a Gauss law constraint. In spin foam models one computes state sums instead of solving constraints.

So this is the reason why I am having trouble figuring out which precise formalism it is we are discussing.

Given all the indications, it seems that there should be a certain (class of) spin foam models for 4d gravity wherein all these statements about coupling of gravity to fermions etc. are to be interpreted.

Which model is that?

Posted by: urs on June 30, 2006 6:17 AM | Permalink | Reply to this

### Re: The LQG Landscape

Dear Lee

“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.”

This issue is something I don’t fully understand. Particularly focussing on uv finite’ theories sounds unnecessarily restrictive from a particle theorist’s perspective. Renormalization is not a sin, and uv divergences in field theory are usually thought to be an artefact of perturbation theory, in some sense—and not necessarily an indicator of anything unpleasant.

Is the notion of uv finiteness in this context somehow more fundamental than uv finiteness in regular QFT? If not, why would we restrict to this set of theories? They might seem nicer, but is there a deeper reason?

“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. “

This sounds interesting, and maybe addresses my point above—but I’m not sure as I haven’t read the papers.

“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.”

That sounds like a fair point, but if the idea is that the dynamics of a given spin foam is related to a particular choice of Hamiltonian constraint, then it still seems legitimate to discuss the ambiguities in this choice.

“So the evidence is strongly against the analogy Nicolai et al suggest.”

Maybe so, but I’d certainly be interested in understanding what the correct analogy is! In general I think particle theorists would understand much better what LQG means if you could relate features of LQG to a features of a low energy effective field theory. Maybe this is just too hard a problem at the moment, but is it something that people work on in LQG?

James

Posted by: boreds on June 30, 2006 10:58 AM | Permalink | Reply to this

### Re: The LQG Landscape

Dear Lee (and myself)

I should probably clarify what I said above:

“Particularly focussing on uv finite’ theories sounds unnecessarily restrictive”

Obviously one would like theories to be finite in the sense that amplitudes calculated are (ultimately) finite.

You probably understood what I was getting at, but let me rephrase what I was asking: are the divergences that you allude to in the excluded spin-foam models analogous to the UV-divergences in regular field theory? Or are they actually much more dangerous?

Posted by: boreds on June 30, 2006 11:45 AM | Permalink | Reply to this

### Re: The LQG Landscape

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?

Posted by: Lee Smolin on June 30, 2006 8:17 AM | Permalink | Reply to this

### Fermions first

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.

Sorry, but that’s bull. The metric is only an operator if the quantum theory exists … which it doesn’t, in this case.

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.

And it’s at that stage that the trouble arises. You can try to gussy-up the problem by attempting to quantize ‘everything at once’. It won’t make the problem go away; at best it will hide it where your formalism is too crude to detect it.

It’s very easy to miss the existence of anomalies, even in the conventional Feynman-diagram approach. The integral you have to do is finite (cutoff-independent, not UV-divergent). And, combining terms in the integrand, one easily “shows” that the anomaly vanishes.

Similar naïve manipulations, in other contexts, can lead you to all sorts of similarly false conclusions, which is why seeing whether LQG captures the anomaly is a good test of the formalism.

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?

If you want to be able to claim that you can couple chiral gauge theories to LQG, it’s a good first step.

(There are some other challenges to surmount, before you can legitimately claim to have coupled LQG to chiral gauge theories, but we’ll leave those for another day.)

Posted by: Jacques Distler on June 30, 2006 12:05 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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

Posted by: Lee Smolin on June 30, 2006 11:14 PM | Permalink | Reply to this

### Have a good vacation

It depends on the use of a regularization procedure defined carefully so that the diffeo invariance and gauge invariances are preserved.

I realize that’s the claim. I’m sure you also realize why there very strong reasons to doubt that claim.

Diffeomorphism invariance was never at issue in $D=4$. In $D=4$, the entire issue revolves around gauge invariance.

If you had a convincing story to tell, which would reconcile the purported properties of this regulator with known topological facts about chiral fermions, I would be all too eager to listen.

But “All the rules are changed, 'cuz, instead of doing the Gaussian path integral over the fermions first, we’re going to quantize everything all at once.” is not exactly convincing.

But please stop insinuating that its features are due to our ignorance about ordinary Poincare covariant QFT.

Poincaré invariance has nothing whatsoever to do with the matter.

But to say something useful you have to at least grasp the scientific question that the work addresses.

I’m sorry if you find my harping on the subject of anomalies “not useful.”

I do it because the nontriviality of the Determinant Line Bundle (in the cases above) is a topological fact robust enough (that’s the virtue of topological facts) that any correct formalism ought to see it.

Perhaps your original guess — that fermion doubling renders all would-be chiral gauge theories coupled to LQG non-chiral — is correct. If so, then I’d stop going around claiming that one can couple chiral gauge theories to LQG.

Either way, tell me a convincing story about the realization of anomalies in LQG, and I’ll be happy to listen.

Posted by: Jacques Distler on July 1, 2006 12:54 AM | Permalink | PGP Sig | Reply to this

### anomalies

Jacques and Lee,

this thread was great because it forced me to brush up on the ABJ anomaly and all that.
I want to take the LQG = lattice comparison serious for another minute.
One way people deal with the fermion doubling problem on the lattice is the domain wall approach. In other words, they introduce an extra dimension. Perhaps LQG needs to be studied in 4+n dimensions as well 8-)

Posted by: wolfgang on July 1, 2006 10:53 AM | Permalink | Reply to this

### Re: anomalies

By the way, Georg von Hippel explains domain wall lattice fermions here.

Posted by: wolfgang on July 1, 2006 11:19 AM | Permalink | Reply to this

### Re: anomalies

Thanks, Wolfgang.

It’s great to hear that someone is getting something out of this thread. To the naïve observer, it didn’t appear to be going anywhere.

If it it turns out that that LQG quantization suffers from fermion doubling then perhaps the solution to obtaining chiral fermions will be to imitate these domain-wall contruction (in which there are chiral fermions on the wall, but the 5-dimensional physics only decouples for anomaly-free fermion representations).

The first order of business, however, is to understand how anomalies are realized in LQG. If this thread spurs some LQG theorist to sit down and figure that out, then it will all have been worthwhile.

Posted by: Jacques Distler on July 1, 2006 11:27 AM | Permalink | PGP Sig | Reply to this

### Re: first order of business

I have yet to understand how LQG escaped from the conclusions of the Helling - Policastro paper.

Posted by: wolfgang on July 1, 2006 12:06 PM | Permalink | Reply to this

### Re: anomalies

I’m a complete ignoramus on LQG but here’s 2c worth anyway.

“The term the chiral fermions appear in […] it is […]:
Psi E D Psi,
[…] You cannot simply integrate over the fermions […]”

I can and I will. It will give me det(ED), whatever that is. To make this well-defined, some regularisation is required. But we don’t need this to study chiral gauge anomalies; these can be uncovered without regularisation as follows.

Consider a disc in the space of gauge fields whose boundary points are all gauge-equivalent (so that the disc corresponds to a sphere in the gauge orbit space). There are winding numbers associated with each (generically isolated) gauge field in the disc for which ED has a zero-mode. These winding numbers are determined by how the lowest eigenvalues of ED vary with the gauge field (restricted to the disc) in neigbourhoods of fields in which ED has a zero-mode. The sum of these winding numbers coincides with the winding number of det(ED) as the gauge field is varied around the boundary of the disc. The latter is an obstruction to gauge invariance; if it is nonzero then a chiral gauge anomaly is implied. This is all standard stuff, described e.g. in the 1984 NPB paper by Alvarez-Gaume and Ginsparg, and presumably one could proceed to study the obstruction to gauge invariance as was done there, which is basically an application of families index theory for the Dirac operator (or, in the present case, ED) coupled to gauge fields.

But perhaps ED itself is not apriori well-defined as an operator on the space of fermion fields? (As mentioned, I know nothing about this.) In that case some regularisation of ED itself would first be required. Since people have mentioned lattice above, let’s assume a lattice regularisation is used. In that case there will be a problem to sort out before chiral anomalies can work out correctly: unless some modification of usual chiral symmetry is made, the anomalies will always vanish, no matter what representation of the gauge group the fermions live in.
Fermion doubling has been mentioned, but this isn’t the real origin of this problem (although it does have some connection to it). (Note that fermion doubling is a statement about free field behaviour, i.e. when the gauge field is turned off, whereas anomalies have to do with variation of gauge fields.) The right way to understand the problem is that it is a manifestation of the fact that index theory is automatically trivial for operators acting on finite-dimensional vectorspaces. As elaborated on in the following.

In the continuum setting, for the study of obstructions to gauge invariance mentioned above, the spacetime first needs to be compactified. (Otherwise the operator will generically have continuous spectrum and the winding numbers discussed above won’t be well-defined.) Compactifying to a 2n-torus would be a good choice if you later want regularise with a hypercubic lattice. But then when you implement the lattice regularisation the space of lattice fermion fields will be finite-dimensional. Now, it is an easy exercise to show that if you have a “Dirac” operator with the usual chirality property (i.e. anticommutes with some operator playing the role of gamma_5), and acting on a *finite-dimensional* vectorspace, then its index automatically vanishes. This is a special case of the general fact that index theory is trivial for operators on finite-dimensional vectorspaces. Also, families index theory can be seen to be automatically trivial in the finite-dimensional setting. Since the obstructions to gauge invariance can be expressed via families index theory for the Dirac operator coupled to gauge fields, a lattice Dirac operator with the usual chirality property will automatically lead to vanishing anomalies.

In the unlikely case that anyone is interested, the solution to this problem has been known in lattice gauge theory for some years. It goes under the names of “overlap formulation” and “Ginsparg-Wilson formulation” (these are mathematically equivalent, although each offers different perspectives). (Btw the overlap formulation is related to, but not the same as, the domain wall formulation mentioned by Urs.) Basically it involves modification of chirality on the lattice in a way that preserves its physical consequences while getting around the problems with usual chirality.
Lattice familes index theory has been set up in this setting and related to obstructions to gauge invariance and chiral anomalies in the same way as in the continuum. The continuum obstructions have been shown to be reproduced in the lattice setting. All that remains to be done is to show that there aren’t any new obstructions in the lattice setting (associated with spheres in the orbit space of lattice gauge fields which don’t correspond to continuum ones). If this can be done then a nonperturbative lattice formulation of chiral gauge theories will be achieved with anomalies vanishing precisely when the usual (continuum) anomaly cancellation conditions are satisfied. Some of us would like to work on this, but…(well, better that I don’t get into that…).

Btw, there is a general expectation (or at least hope) that anomaly cancellation in the Standard Model is not some fundamental feature of nature but rather a derivable property of the low energy effective theory for some more fundamental theory. So, ideally, a “theory of everything” should not involve putting in by hand an anomaly-free coupling to fermions; it should instead emerge automatically in a low energy effective description. But this doesn’t seem to be what is happening in LQG, where, as far as I can tell, the fermion fields are being stuck in by hand.
On the other hand, in string theory there’s the landscape situation so I guess anomaly cancellation is not something derivable but rather a criterium to be used when looking for suitable vacua…

Posted by: amused on July 3, 2006 11:24 AM | Permalink | Reply to this

### Re: anomalies

On the other hand, in string theory there’s the landscape situation so I guess anomaly cancellation is not something derivable but rather a criterium to be used when looking for suitable vacua…

Not really.

There are no anomalous gauge theories in the Landscape of vacua. These are all just different vacua of an underlying UV theory which is anomaly-free. Hence they, too, are anomaly-free.

Moreover, the “Swampland” conjecture is that there are large classes of — otherwise perfectly consistent and anomaly-free — quantum field theories that do not arise as low-energy limits of string theory.

The LQG theorists have a much bigger “Landscape” problem because, not only is there no “Swampland,” but they even claim to be able to couple arbitrary (anomalous) chiral gauge theories to quantum gravity. So there’s a complete lack of uniqueness of the UV theory (Georgi’s objection).

Posted by: Jacques Distler on July 3, 2006 11:46 AM | Permalink | PGP Sig | Reply to this

### Re: anomalies

>On the other hand, in string theory there’s the landscape situation so I guess anomaly cancellation is not something derivable but rather a criterium to be used when looking for suitable vacua…

This is a severe misunderstanding about the nature of string theory. String theory is a consistent theory, and therefore automatically anomaly free, without adjusting anomaly free spectra by hand. In particular there is a proof for (heterotic) strings that space-time anomalies always cancel, and this originates from modular invariance, which is one of the crucial ingredients guaranteeing consistency. It has no analog in standard particle QFT.

Posted by: MoveOn on July 4, 2006 8:56 AM | Permalink | Reply to this

### Re: anomalies

Well that provoked me into taking a look at the Green-Schwarz anomaly cancellation paper for D=10 superstring theory (something I had otherwise been postponing until string theory made a prediction that was experimentally verified ;-) ). If I understood rightly, the result there is that D=10 supergravity coupled to superYangMills with gauge group SO(32) becomes free of gauge and gravitational anomalies after additional terms from the low energy expansion of superstring theory are included in the lagrangian.

If you compactify the theory down to D=4, and consider a vacuum for which the compactified dimensions are very small, and if the low energy effective theory associated with this vacuum can be formulated in terms of gauge fields coupled to chiral fermions in D=4, then i guess it’s inevitable that the anomaly-free symmetries of the original theory will reduce to usual anomaly-free gauge and lorentz symmetry of the effective theory.

But I was under the impression (perhaps wrong?) that these days people don’t put in gauge fields by hand in string theories; instead, they are supposed to arise intrinsically from stuff to do with D-branes. (I vaguely recall some string theory talks where a bunch of D-branes were stacked together and claimed to somehow produce a U(N) gauge symmetry…) In that case the origin of anomaly-free gauge symmetry in low energy effective theories in D=4 seems quite murky.
It isn’t enough just to say that the degrees of freedom of these D-branes at low energy can somehow be identified with gauge fields, and that these get coupled in a gauge-invariant way to chiral fermions. What needs to be shown is that the quantum effective action for the low energy theory is gauge invariant (i.e. anomaly-free). Can this be done for each of the effective theories associated with all the zillions of vacua that apparently exist? It seems like quite a nontrivial task to me.

Posted by: amused on July 5, 2006 4:42 AM | Permalink | Reply to this

### Re: anomalies

But I was under the impression (perhaps wrong?) that these days people don’t put in gauge fields by hand in string theories;

People never put in gauge fields “by hand.” Because you can’t. In fact, you can’t couple any ‘external’ degrees of freedom (gauge fields, scalar fields, …) to string theory.

instead, they are supposed to arise intrinsically from stuff to do with D-branes. (I vaguely recall some string theory talks where a bunch of D-branes were stacked together and claimed to somehow produce a U(N) gauge symmetry…)

Open strings, ending on D-branes, is one place where gauge fields can arise. There are others.

What needs to be shown is that the quantum effective action for the low energy theory is gauge invariant (i.e. anomaly-free). Can this be done for each of the effective theories associated with all the zillions of vacua that apparently exist? It seems like quite a nontrivial task to me.

You could ask the same question about the low-energy degrees of freedom that arise in the effective theory stemming from an underlying strongly-coupled (confining) chiral gauge theory.

But there you know the answer: if the UV theory is free of anomalies, so is the low-energy effective theory.

It’s very comforting to see this happen in examples. But it follows from general priciples.

That’s not to say that one shouldn’t study anomaly-cancellation in string theory. Far from it: many important insights (the shifted quantization condition for the 4-form field strength in M-theory, to pick just one example) have followed from carefully studying how anomalies are cancelled.

But it’s not something that proceeds by case-by-case enumeration of examples.

Posted by: Jacques Distler on July 5, 2006 2:41 PM | Permalink | PGP Sig | Reply to this

### Effective actions

I should have responded to this:

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, …

Of course the effective action from integrating out the fermions is nonlocal!

That’s true of integrating out any massless field (which is why the Freidel-Livine trick of integrating out gravity in 2+1 dimensions cannot be repeated in 3+1).

That’s irrelevant for the anomaly, where we are interested in computing the gauge-variation of this effective action, which does have a local expression (for the students who, unlike Lee, are not well-versed in these matters: the anomaly is local, but it cannot be written as the gauge-variation of a local expression).

In fact, the anomaly 6-form (of Wess and Zumino) is completely metric-independent. So all your purported problems with background dependence (by which you mean metric-dependence) are absent from the anomaly.

Posted by: Jacques Distler on July 1, 2006 11:08 AM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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

Posted by: Lee Smolin on June 30, 2006 11:51 PM | Permalink | Reply to this

### Re: The LQG Landscape

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.

Posted by: fh on July 2, 2006 1:13 PM | Permalink | Reply to this

### Welcome.

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.

Physics, to me, is not synonymous with “representing interactions on Fock spaces.” Except in the case of fermions, which, since they enter quadratically in the action, yield – upon quantization – a Fock space bundle over the configuration space of the other fields.

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.

I think you misunderstand Georgi’s objection.

That is, we don’t try to work our way from an effective field theory and work our way towards an UV completion.

Do you rely on divine revelation for all of the particle physics degrees of freedom from 100 GeV up to $10^{19}$ GeV?

Or do you suppose that information will be revealed in a blinding flash of insight, once you properly understand “octopi”?

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.

I don’t see how that addresses, let alone is “the way around” Georgi’s objection.

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.

And coupling to gravity is supposed to make understanding that easier?

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.

Well, …

1. Nothing about Thiemann’s constructions relied on the peculiar $U(1)$ hypercharge assignments of the fields in the Standard Model. Nor would it have changed in the slightest, had one omitted the right-handed ($SU(2)$-singlet) up-quark from the theory. In other words, it would have applied equally well to an anomalous chiral gauge theory which, I continue to insist, cannot be turned into a well-defined quantum theory, with – or without – coupling to quantum gravity.
2. Whether in flat space, or coupled to quantum gravity, if one has indeed written down a complete quantization of Yang Mills, which is “entirely non-perturbatively defined and second quantized” then it contains all the relevant information (confinement and the existence of a mass gap) necessary to collect the Clay Prize.

But, of course, both you and Lee, and most everyone else in the LQG 'biz take refuge in the plaint that “We don’t know how semiclassical spacetimes emerge.” to avoid having to make any contact with actual physics (anything, even in principle, measurable).

The difficulties that LQG have in this regard are both a telling symptom that something deep may be wrong and largely irrelevant to the kind of questions I’m interested in.

First of all, in any theory, including quantum gravity, the asymptotics of the fields are not fluctuating degrees of freedom, but rather represent superselection sectors in the theory.

Thus, it is possible to speak of “asymptotically-flat spacetimes” (with fixed ADM mass, possibly zero), even if those spacetimes are in no sense semiclassical.

And many interesting questions about quantum gravity can be phrased in terms of the physics of such asymptotically-flat spacetimes, regardless of whether they are dominated by any semiclassical configurations.

[The same words can be said about asymptotically-AdS spacetimes. And there, string theory has very powerful, nonperturbative, and “background-independent” things to say.]

But this is a whole 'nother topic, deserving of a post in itself. Perhaps I will get around to doing that sometime.

Posted by: Jacques Distler on July 2, 2006 2:27 PM | Permalink | PGP Sig | Reply to this

### Re: Welcome.

Whether in flat space, or coupled to quantum gravity, if one has indeed written down a complete quantization of Yang Mills, which is “entirely non-perturbatively defined and second quantized” then it contains all the relevant information (confinement and the existence of a mass gap) necessary to collect the Clay Prize.

First, isn’t Thiemann’s construction on a spin-foam model with no clear connection to any kind of space-time? Second, if Thiemann qualifies for the conditions of the Clay prize, would not the lattice guys also qualify?

Posted by: Arun on July 4, 2006 2:12 AM | Permalink | Reply to this

### Re: Welcome.

First, isn’t Thiemann’s construction on a spin-foam model…

No, it’s in the canonical “CQL” framework.

Second, if Thiemann qualifies for the conditions of the Clay prize, would not the lattice guys also qualify?

The Lattice guys don’t claim to have proved anything. (Though, for pure Yang-Mills, they do have numerical results, believed to be accurate to within a few percent.)

Posted by: Jacques Distler on July 4, 2006 2:25 AM | Permalink | PGP Sig | Reply to this

### Re: Welcome.

Really, if you are honestly interessted in discussing LQG and it’s failures and successes (rather then winning a shouting match) it is indispensible that you a) take the time to carefully read at least the abstracts, b) don’t assume that anything is claimed beyond what the abstract says, c) stop assuming everyone else is stupid and d) accept that LQG is approaching Quantum Field Theory from a completely different perspective, and with (in parts) completely different goals. In particular it does not have the goal to single out a theory of everything, it is developing a new and more general class of QFTs.

Posted by: fh on July 4, 2006 7:15 AM | Permalink | Reply to this

### Re: Welcome.

> LQG is approaching Quantum Field Theory from a completely different perspective, and with (in parts) completely different goals.

But can we agree at least that it should reproduce well understood results of ‘conventional’ QFT?

If quantization of 2d gravity gives different results than everything else (including the lattice = dynamical triangulation) and if even the harmonic oscillator comes out wrong, how can you just go on?

Posted by: wolfgang on July 4, 2006 8:45 AM | Permalink | Reply to this

### Re: Welcome.

Yes, and everybody keeps saying that, it should make contact with ordinary QFT, Freidel showed that in 2+1 it does.

But at the same time it’s fundamentally different from ordinary QFT. So making this contact is a nontrivial task.

Soon we will hopefully see Freidels work extended to 4D, getting an ordinary field theory out of defects in an otherwise topological one.

So we are close to knowing that the types of QFTs LQG hqs produced are an honest non-trivial generalization of the framework of flat space QFT.

If this framework is powerfull enough to capture Quantum Gravity nobody knows.

Helling’s objections are a red herring, too. There is only one bit of quantization that differs drastically from ordinary quantization, the implementation of diffeomorphisms, and that is protected by the LOST theorem. (the rest is unusual, but clearly connects to standard methods)

Posted by: fh on July 4, 2006 10:35 AM | Permalink | Reply to this

### Re: Welcome.

Helling’s objections are a red herring, too. There is only one bit of quantization that differs drastically from ordinary quantization, the implementation of diffeomorphisms, and that is protected by the LOST theorem. (the rest is unusual, but clearly connects to standard methods)

I was going to get to this over at Christine’s place, but this isn’t really true. Besides the fact that you want to represent the diff symmetries on a Hilbert space, the actual representation so obtained is rather non-continuous. Given that Helling obtains physically incorrect results, it better be true that there’s some weirdness in there somewhere.

As for the F-L result, I’m not sure what it has to do with the canonical LQG. As best I can tell from a quick skim, it’s an application of a Regge-type quantization of gravity in 2+1, not a canonical quantization a la Thiemann. I do not believe that the LOST theorem guarantees an equivalence, but I’d be open to hearing about other results that might – 2+1D is special enough that some of this stuff might work there. Given the completely different quantizations obtained in 2D, I’m not particularly optimistic, though.

Posted by: Aaron Bergman on July 4, 2006 11:14 AM | Permalink | Reply to this

### Re: Welcome.

Perez showed that cannonical methods and Ponzano Regge/Spinfoams are equivalent in 2+1D. You are right, this has nothing to do with LOST.

But yes for 3+1 it would be a Spinfoam that reproduces the flat spacetime results, not a cannonical quantization.

Posted by: fh on July 5, 2006 3:53 AM | Permalink | Reply to this

### Re: Welcome.

But yes for 3+1 it would be a Spinfoam that reproduces the flat spacetime results, not a cannonical quantization.

I am still trying to make somebody tell me which spin foam model that is, precisely.

All that I am aware of is John Baez et al.’s work on soliton-like strings coupled to a gerbe, as in the followup of this.

I understand that it is hoped that in 4 dimensions this will give rise to a theory with string-like defects that would be describeable by a spin foam model.

Is that the spin foam model you have in mind? Has it already been constructed? Does it really describe gravity?

Posted by: urs on July 5, 2006 6:09 AM | Permalink | Reply to this

### Re: Welcome.

Hi Urs,
this is unpublished, it’s not the Perez+Baez Stringy Spinfoam, It’s a whole new Spinfoam model by Baratin+Freidel, apparently quite unlike the ones we considered for something gravity like. Baez talks about it in this thread:

This is powerfull to me because it suggests that Spinfoams are a much more powerfull formalism then anyone could have assumed a priori.

There has been a long standing idea in Gravity research going back all the way to Riemanns introduction of differential geometry that topology of space and matter are tightly interwoven, this actually makes the correspondence explicit.

(My own ramblings:)

Trying to quantize gravity forced people to come up with ideas like Spinnetworks and Spinfoams. I think these days a lot of people working on them would not neccessarily think that this is a direct quantization of Gravity but that these structures are powerfull enough to stand on their own and that, for various physical reasons, we might be well guided to mistrust the naive interpretation as a quantized geometry all the way down to the planck length.

There might be a geometry limit as well as a QFT limit and so on, but the geometric notions become just as meaningless as the flat spacetime QFT ones for the fundamental excitations. There is no way to regard E² as the physical area, and there is no S-Matrix at this level, Really just what you would expect from taking GR and QM serious.

Posted by: fh on July 5, 2006 7:38 AM | Permalink | Reply to this

### Re: Welcome.

[…] this is unpublished […]

Ok, thanks. So it seems to me that for the present discussion to lead anywhere, we would have to wait for that particular spin foam model to be published and then try to figure out the issue of coupling to chiral matter in that particular model.

Posted by: urs on July 5, 2006 9:26 AM | Permalink | Reply to this

### Re: Welcome.

It’s a whole new Spinfoam model by Baratin+Freidel, apparently quite unlike the ones we considered for something gravity like. Baez talks about it in this thread:

Great, thanks.

Let’s see. What John Baez announces is not a spin foam model that reproduces gravity, but that

This spin foam model is thus a candidate for the $G \to 0$ limit of any spin foam model of quantum gravity and matter!

He also conjectures that this spin foam model is equivalent to the one proposed by Louis Crane and Marni Sheppeard ($\to$), which - correct me if I am wrong - implies that the Crane-Sheppeard model can also only be a $G\to 0$-limit of quantum gravity? Is that right?

I have no problem with this being just a first step toward a hoped-for spin foam model for quantum gravity. In fact, I am pretty intrigued by these theories of $d$-brane-like objects coupled to higher gerbes, as you can imagine.

But I do wonder what we tried to talk about when we were apparently discussing the coupling of quantum gravity to chiral matter in the context of LQG.

Judging from what has been said, this seems to be a topic for the future. No?

Posted by: urs on July 5, 2006 9:39 AM | Permalink | Reply to this

### Re: Welcome.

(Disclaimer, I’m a beginner, too, just studying this stuff, far from expert!)

I think what you say is perfectly right. As for what we were talking about, well, there are different ways to look at this. Freidel doesn’t actually couple matter to Spin Foams, in some sense it arises from the possible topologies. Also their model IS Ponzano-Regge.

On the other hand what Lee Smollin was talking about was putting matter into the lagrangian and quantizing that. For gaugefields that’s straightforward, LQG is nothing but a way to quantize gaugefields on a manifold without a metric, nothing more. Yang Mills coupled to Gravity falls under this category, Yang Mills on flat spacetime does not.
The Standard Modell + GR Lagrangian defines a theory on a manifold without metric and can be done to. (Of course this classical theory doesn’t actually describe anything we see in nature, since the low energy limit of most of the standard modell does not at all look like it’s field content). Can be done means it defines an algebra of observables and a hamiltonian constraint that can be promoted in a well defined way to a quantum operator algebra.
This is rigorously defined but in a way those who chide Thiemann for speaking about the Standard Modell and QFT are right, since they use these terms to think of the physics these effective theories capture and not of some abstract algebras.
And we do not know if Thiemanns construction captures these physics (to the best of my knowledge).
Does this actually give matter in some sense? One might suspect so, but the quantization used is very different, as pointed out here. It allows for the inclusion of virtually arbitrary lagrangians,
One might conjecture that upon renormalization in the flat spacetime limit only renormalizable effective theories remain by the standard Wilson argument, and that only the non-anomalous parts of the algebra survive because what in flat spacetime is the anomaly is a lack of semi classical flat spacetime states in LQG or something.
That’s of course *wild* speculation. Emphasize on wild.

Posted by: fh on July 5, 2006 10:30 AM | Permalink | Reply to this

### Baratin-Freidel

This spin foam model is thus a candidate for the $G\to0$ limit of any spin foam model of quantum gravity and matter!

It’s not even clear to me that that’s what they are doing. Judging by their previous paper, what they are trying to do is show that one can generate the Feynman graphs of a flat-space field theory from a spin foam model.

However, as “fh” emphasized to me, even a flat space quantum field theory is not merely the sum of its Feynman graphs.

But, this discussion is getting pretty lame, if we are reduced to speculating about the results contained in not-yet-published papers.

Posted by: Jacques Distler on July 5, 2006 12:41 PM | Permalink | PGP Sig | Reply to this

### Re: Baratin-Freidel

Judging by their previous paper, what they are trying to do is show that one can generate the Feynman graphs of a flat-space field theory from a spin foam model.

I guess so. I haven’t tried to read it, but it seems like there is a nonperturbative theory whose observables produce the perturbative expansion of a perturbative theory.

While I am pretty much lacking the intuition for it, I understand that the hope is that there could be a deformation of this nonperturbative theory which analogously produces something like GR+SM Feynman diagrams.

(BTW, to those who work on this: would this be a nonperturbative model for perturbative quantum gravity, in a way, or would it really be non-perturbative quantum gravity itself?)

Posted by: urs on July 5, 2006 1:40 PM | Permalink | Reply to this

### Re: Welcome.

BTW to answer your specific question, a Spinfoam model with Stringlike defects has been constructed by Perez/Baez:

http://arxiv.org/abs/gr-qc/0605087

This is not Gravity but “only” BF theory.

Posted by: fh on July 5, 2006 8:39 AM | Permalink | Reply to this

### Re: Welcome.

I hope by “Helling obtains physically incorrect results” you don’t really mean that.

Rather, my reading of our results is that we (or rather the standard procedure) provide a “counter example” to this LOST theorem, at least on $S^1$ (where strictly speaking it does not apply): The usual Fock highes weight representation is a continous representation of the diffeomorphisms that does not share the pathologies of the polymer represenation.

Rather, the diffeos are spontaneously broken (only the positive frequency half is preserved by $|0\rangle$) for the rest you have to work more but in the end you get the unitary implementers.

This sponatneous breaking is what the LQG people do not allow in the assumptions of this LOST theorem however already any classical solution (i.e. metric, including Minkowski) provides it: Any metric breaks diffeos down to its isometry group. The remaining diffeos are non-linearly realised.

Posted by: Robert on July 11, 2006 7:11 AM | Permalink | Reply to this

### Re: Welcome.

the diffeos are spontaneously broken

But only weakly. Expectation values $\langle 0 | L_{\pm n} |0\rangle$ still vanish.

Posted by: urs on July 11, 2006 7:17 AM | Permalink | Reply to this

### Re: Welcome.

> my reading of our results is that we (or rather the standard procedure) provide a ‘counter example’ to this LOST theorem

‘anonymous’ told us that the LOST theorem does not apply to 2d gravity or the harmonic oscillator, because it requires (at least) 2d spatial slices.

Posted by: wolfgang on July 11, 2006 7:34 AM | Permalink | Reply to this

### Re: Welcome.

[…] the LOST theorem does not apply to 2d gravity […]

We need to be quite careful with quoting theorems, as long as we do not even have a well defined setup in which we could check that the assumptions of these theorems hold.

As I have stated several times before, I am (still) quite unsure what particular formalism people have in mind when talking about “LQG”.

In this example, Thiemann’s discussion of the quantization of the string is rather not an example of a CQL quantization of 2-dimensional gravity.

He does not consider a metric in 2d, does not replace it by a connection, does not pass to a space of Wilson network states for that connection, does not try to couple the scalar matter to these gravitational degrees of freedom, does not try to represent the diffeomorphism constraints on that space, does not try to solve the Hamiltonian constraint.

At least not explicitly.

Instead, he makes use of some simplifiying properties in 2d and proposes a shortcut.

If that shortcut is really still an example of CQL is at least unclear.

Posted by: urs on July 11, 2006 7:53 AM | Permalink | Reply to this

### SSB

This sponatneous breaking is what the LQG people do not allow in the assumptions of this LOST theorem

The LOST theorem does not allow for the possibility that the gauge symmetries are spontaneously broken??

If so, that would make it perfectly useless, both for gravity (where, as you mention, the Minkowski vacuum spontaneously breaks $Diff$ down to Poicaré) and for the Standard Model (where $SU(2)\times U(1)$ is spontaneously broken).

Posted by: Jacques Distler on July 11, 2006 10:09 AM | Permalink | PGP Sig | Reply to this

### Re: SSB

Could somebody with the reference at hand please confirm that this LOST (what does it actually stand for?) theorem assumes that the state is invariant?

Posted by: Robert on July 11, 2006 11:57 AM | Permalink | Reply to this

### LOST

[…] LOST (what does it actually stand for?) theorem […]

Jerzy Lewandowski, Andrzej Okolow, Hanno Sahlmann, Thomas Thiemann, Uniqueness of diffeomorphism invariant states on holonomy-flux algebras, gr-qc/0504147.

Posted by: urs on July 11, 2006 12:44 PM | Permalink | Reply to this

### Re: SSB

As I said, LOST does not apply to the circle but I would not be surprised if there were a similar statement. You would have to prove something like:

Take $S$ to be the set of test functions on the circle which have 0 integral. Find a functional $w:S\to C$ such that it is 1 for the zero test function for any diffeomorphism $\phi$ of the circle and $f\in S$ you have $w(f) = w(f\circ \phi)$ plus some positivity condition. Then it should not be too hard to show that your only choice would be to take $w(f)=1$ if $f=0$ and $w(f)=0$ in all other cases (currently taking the maximum of $|f|$ plus one sounds like another possibility but I am sure there is some condition that rules that out, similarly $w(f)=1$ for all $f$).

The non-trivial alternative (as done for example by Gupta and Bleuler) is to take a $w$ that is not invariant under diffeos, build the GNS Hilbert space on that and then find unitary operators for the diffeos that are broken by $w$.

Posted by: Robert on July 11, 2006 1:10 PM | Permalink | Reply to this

### Re: SSB

Find a functional […] such that […] for any diffeomorphism […]

This is, by the way, a procedure which does not strictly follow from Dirac quantization.

Dirac demands to first represent all constraints as operators, and then pass to the (weak, maybe) kernel of these.

In the CQL approach of LQG, this is indeed what is done for the Hamiltonian constraint. For the diffeo constraints, however, it seems that by fiat one declares that instead one wants to find functionals invariant under all reparameterizations. It is not a priori clear that this coincides with the correct Dirac presription.

Of course representing all ADM constraints (in more than 2+1 dimensions) as operators yields an ill-defined algebra (due to divergencies).

Posted by: urs on July 11, 2006 1:33 PM | Permalink | Reply to this

### Re: Welcome.

I hope by “Helling obtains physically incorrect results” you don’t really mean that.

I meant that the quantization of the harmonic oscillator doesn’t agree with the observed physics thereof. Or did I misinterpret your paper?

Posted by: Aaron Bergman on July 11, 2006 10:16 AM | Permalink | Reply to this

### Re: Welcome.

This is a slightly tricky issue. For the technical statements, see the paper. I would phrase it like this as well but there are ways to wiggle out.

The situation is as follows: Instead of a Hamiltonian, there is only the unitary time evolution operator $U(t)$. Unfortunately, the map $t\mapsto U(t)|\psi\rangle$ is not continious except for the ground state (which is invariant) and cannot be differentiated. Thus there is no Hamiltonian.

In the paper, we give a family of hermitean operators $H_\epsilon$ with the property that for a generic state they have expectation value 0 but their square has an expectation value $\sim \epsilon^{-2}$ and if that limit existed $\lim_{\epsilon\to 0}$ would be the Hamiltonian. If that limit existed, that Hamiltonian would have a very strange spectrum, but the limit does not exist. So?

Another remark is that the polymer state is formally the limit of a KMS (finite temperature) state in the limit of infinite temperature. Thus, with an infinitely hot temperature bath around you would not be surprised to have funny energetics.

The other question you should answer (and we had thought about that a lot) is “what exactly is the known observed physics of a harmonic oscillator?”. Naively, you would think “it’s the even spaced spectrum of the Hamiltonian”. But how exactly do you observe that? The standard procedure would for example be to look at the optical absorbtion spectrum of the oscillator (think: molecular excitation spectrum for example). However, to say that, you have to couple the photons to the oscillator and strictly speaking you now have an interacting system and all your reasoning about the oscillator is not exactly true anymore. So you might be worried that funny things are happening (especially if you want to avoid pertubation theory at all costs).

Another possible observation is that the only stationary state (a vector that only changes by a phase under time evolution and thus the state (projective vector) is invariant) in the polymer Hilbert space is the ground state.

This however is no longer the case if you do not allow all real $t$ as arguments of $U(t)$ but only discrete values (“multiples of the Planck time”). Basically, $U(t)$ rotates the complex plane (phase space) by an angle $t$, thus if you only allow $t\in 2\pi Z/N$ for some integer $N$ and $Z$ the integers then there are of course states that are invariant under this restricted time evolution.

Thus this discontinious physics is quite funny: If you believe you can make measurements at arbitrary times (like the generic real number being irrational) then you cannot see stationary states. However, if somehow time is a discrete lattice and the frequency of the oscillator happens to be rational in units of this lattice, things are different.

Note however, that in the latter case your Hilbert space is effectively finite dimensional, your time evolution is just a matrix and thus it is no surprise that you can find a vector such that the $N$ linearly independent measurements you can make in that world have prescribed outcomes (“known physics”).

In the end, you have a philosophical question: Given that in the end, we will make only a finite number (say $M$) of observations. Why don’t we give the final theory just as a list of $M$ numbers? That is as good as any theory of everything.

Posted by: Robert on July 11, 2006 12:35 PM | Permalink | Reply to this

### Re: Welcome.

If quantization of 2d gravity gives different results than everything else (including the lattice = dynamical triangulation) and if even the harmonic oscillator comes out wrong, how can you just go on?

Both examples you mention live in CQL. According to Lee Smolin, most practitioners of LQG have abandoned that and moved from CQL to spin foams.

It seems to be me that hence the question is if there is a spin foam model for $d$-dimensional gravity, in particular for $d \gt 3$, that we could throw all these questions at.

Posted by: urs on July 4, 2006 10:54 AM | Permalink | Reply to this

### Re: Welcome.

But can we agree at least that it should reproduce well understood results of ‘conventional’ QFT?

and atleast in the case of free field theories on flat spaces,it does. See my comment above on the work of varadarajan (posted on june 28th). Also regarding Helling-policastro paper and work on Thiemann on strings,it is important to note that his quantization of the embeddings is unitarily INEQUIVALENT to the loopy quantization of scalar fields.

Posted by: anonymous on July 4, 2006 11:54 AM | Permalink | Reply to this

### Re: Welcome.

sorry, I should have added. Thiemann in his LQG-string paper, manages to violet the LOST theorem as his exponentiated momenta are not weakly continuous ( one of the requirements of the LOST theorem.)
thanks

Posted by: anonymous on July 4, 2006 12:11 PM | Permalink | Reply to this

### stupidity

Infact Lost theorem only works for theory of connections-fluxes and that too only when spatial slice is atleast two dimensional. So
it is not even applicable to his string-paper. So although his quantization is unitarily inequivalent to loop-quantization of scalar fields, it has nothing to do with LOST theorem. I apologise for my completely wrong comment above.

Posted by: anonymous on July 4, 2006 12:17 PM | Permalink | Reply to this

### Re: stupidity

anonymous,

are you suggestingg that LQG may not work in 2d (at least the LOST theorem is not available) but it does work in 3d (F-L would support this) and in higher dimensions? Or do I understand this wrong?

Posted by: wolfgang on July 4, 2006 4:19 PM | Permalink | Reply to this

### Re: stupidity

No. not suggesting that at all. Only that lost theorem is only established when spatial slice is atleast 2 dimensional and when one has theory of connections and fluxes. Loop(More commonly known as polymer) quantizations (where spatial diffeomorphism group or even the Virasoro group) acts unitarily on some Hilbert space
can always be obtained in 2 dimensions.

Posted by: anonymous on July 4, 2006 4:58 PM | Permalink | Reply to this

### Re: stupidity

ok, then back to square one (at least for me). I think somebody noticed earlier that we are not really getting anywhere on this thread 8-)

Posted by: wolfgang on July 4, 2006 5:16 PM | Permalink | Reply to this

### Re: Welcome.

Sorry, this comment isn’t getting through…

Where does Thiemann claim that his theory reduces to Yang Mills theory on flat spacetime, has a mass gap, and has confinement?

Nothing of that sort is ever claimed!

It’s a quantisation of a different theory, there is certainly hope that, as you say, it contains all the neccessary information, but as Arun correctly points out, in principle, so does Lattice QFT.

Posted by: fh on July 4, 2006 7:22 AM | Permalink | Reply to this

### Re: Welcome.

“Nothing of that sort is ever claimed!”

Well, that’s a problem then, because if someone claims to have found a complete non-perturbative definition of QCD, it had better address these central issues.

“It’s a quantisation of a different theory”

Maybe you want to start reading the abstracts more carefully yourself now: “[…] we are dealing with a particular phase of the standard model coupled to gravity”.

Posted by: Michael on July 4, 2006 10:47 AM | Permalink | Reply to this

### Re: Welcome.

Yang Mills coupled to gravity is a different theory from Yang Mills in flat spacetime.

We quantize them differently, since the quantization methods used for the latter can not work for the former.

Even classically: EM in flat spacetime has divergent self energy and wavefronts propagate with c, in EM+GR neither of these is the case.

Posted by: fh on July 5, 2006 3:39 AM | Permalink | Reply to this

### New Kind of Science?

LQG is approaching Quantum Field Theory from a completely different perspective, and with (in parts) completely different goals.

Do you really want to assert that LQG is a “new kind of Science”, which cannot be discussed using conventional scientific criteria?

[I]t is developing a new and more general class of QFTs.

You can’t “generalize” something that you can’t reproduce.

Where does Thiemann claim that his theory reduces to Yang Mills theory on flat spacetime

Do you really want to assert that quantum gravity coupled to Yang Mills theory does not have a flat space limit? Or even that this is “an open question”?

The only “open question” is whether the theory discussed by Thiemann has anything to do with quantum gravity coupled to Yang Mills theory.

Where does Thiemann claim that his theory… has a mass gap, and has confinement?

Let me quote, again his abstract:

Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized.

He’s not claiming to discuss some random theory that has nothing to do with anything. He’s claiming to have a rigorous, nonperturbative quantization of the Standard Model. That implies (among other things) the above claims.

Posted by: Jacques Distler on July 4, 2006 10:52 AM | Permalink | PGP Sig | Reply to this

### Re: New Kind of Science?

“This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory”.

Nope - string theory does correctly provide running coupling constants. A theory that is completely finite is in gross contradiction with experimental observations.

Posted by: MoveOn on July 5, 2006 2:29 AM | Permalink | Reply to this

### Re: New Kind of Science?

“Do you really want to assert that LQG is a “new kind of Science”, which cannot be discussed using conventional scientific criteria?”

I want to assert that science (or more precisely fundamental physics) is not equal to standard QFT.

“You can’t “generalize” something that you can’t reproduce.”

See the work by Freidel (this is getting repetitive and tiresome) Spin Foams can reproduce ordinary flat spacetime QFT.

Also way to go to clip the context, the sentences preceding what you make bold are:

“It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum.”

So we have a theory that has the Gauge group and algebra of the Standard Modell but we do not know and nowhere claim to know if and how this theory reproduces the flat spacetime limit. Though we suspect that it does we can’t prove that. The situation is the same as Lattice QFT.

Anyways, what are you suggesting? That THiemanns results are wrong? That he’s lying about what he is doing?

Posted by: fh on July 5, 2006 3:49 AM | Permalink | Reply to this

### Re: New Kind of Science?

So we have a theory that has the Gauge group and algebra of the Standard Modell but we do not know and nowhere claim to know if and how this theory reproduces the flat spacetime limit. Though we suspect that it does we can’t prove that.

As I pointed out above, nothing of Thiemann’s analysis would have changed, had he attempted to couple an anomalous chiral gauge theory in place of the (nonanomalous) Standard Model.

That he would claim to have a consistent nonperturbative quantization of an anomalous chiral gauge theory coupled to quantum gravity ought to tell you something.

The situation is the same as Lattice QFT.

Not even close. First of all, the Lattice Gauge theorists know that they cannot currently study chiral gauge theories. Second, they have abundant numerical evidence that they can achieve the continuum limit.

Anyways, what are you suggesting? That THiemanns results are wrong?

Yes.

That he’s lying about what he is doing?

“Wild exaggeration” would be more accurate.

Posted by: Jacques Distler on July 5, 2006 10:47 AM | Permalink | PGP Sig | Reply to this

### Re: New Kind of Science?

You can tell that they are wrong from reading the abstract?

While demonstrating amply that you have not attempted to understand what he is claiming or trying to do?

While it is clear that there is a different use of language as well, resulting from the different perspectives, that you fail to acknowledge?

I’m sorry but this is a pure (and not the first in this thread) ad hominem. You have not pointed out ANYTHING about Thiemanns paper that is wrong.

You’re free to claim that the quantization of SM+GR THiemann is producing is unphysical for various reasons (including the fact that it apparently doesn’t know about anomalies), you can not claim thaat he doesn’t have one.

Posted by: fh on July 5, 2006 11:46 AM | Permalink | Reply to this

### Re: New Kind of Science?

The fact that he claims to be able to quantize an anomalous gauge theory, merely by coupling it to quantum gravity, is, indeed sufficient grounds to be confident that he is wrong.

Yes, I could carefully work through his paper to find the precise point where he goes astray. But what would that gain?

True-believers, like yourself, would dismiss it as a mean-spirited attack. And people like Lee would say, “It doesn’t matter, because we’ve all moved on to study spin foams, anyway.”

The US Patent Office does not accept applications for Perpetuum Mobile, without a working model. Similarly, I am not interested in entertaining any purported quantizations of chiral gauge theories, without a detailed explanation of how anomalies are realized.

Posted by: Jacques Distler on July 5, 2006 12:08 PM | Permalink | PGP Sig | Reply to this

### Re: New Kind of Science?

I think what we have here is, as they say, a failure to communicate.

The problem (at least from the point of view of this discussion) is that we have a fantastically successful theory called quantum field theory which has been incredible precisely tested in experiment. It states that certain field theories are inconsistent, and the standard model obeys this nontrivial constraint.

Now, the LQG guys come along and say, we have this new exciting framework for doing things without the presence of a nondynamical metric. They also say that their framework works even on theories that our old framework says are inconsistent. Given the reams of experimental evidence that the old framework has and the fact that nature satisfies to nontrivial consistency constraints that it imposes, I would think the very first question one would ask is, “is this really quantization?”.

Now, anomalies can be subtle things, and it’s possible that there’s a mistake somewhere in Thiemann’s work. It’s also possible, as Lee supposes, that the ‘anomalous theories’ somehow don’t engender old-style QFTs. In some sense that is begging the question, however, because one is still left with the question of why we don’t live in a universe with one of those theories (of which there are many more) rather than the ones that satisfy anomaly cancellation.

But, I hope you can understand that, when presented with this situation, a lot of people’s response is that it’s more likely that these theories have nothing to do with the real world. This is only reinforced by something like Helling and Policastro which shows explicitly that this sort of quantization gives experimentally incorrect answers for something as simply as the harmonic oscillator. At this point, I would say that the burden of proof is on the proponents of these theories to show that they at the very least encompass our prior succesful framework of QFT and hopefully give some a priori explanation why the standard model satisfies anomaly cancellation.

I don’t see how Freidel-Livine is relevant for this question as it seems to apply to a path integral method which is not obviously equivalent to LQG-canonical quantization. In fact, I keep hearing from Lee that most LQG people (the term seems to be a bit of a moving target, unfortunately) have abandoned canonical LQG and moved on to spin foams. So, perhaps this entire discussion is irrelevant and we should be asking about spin foam models and the real world?

Posted by: Aaron Bergman on July 5, 2006 1:00 PM | Permalink | Reply to this

### Re: New Kind of Science?

which connects polymer and Fock representation as an example of how old-qfts are related to the loopy quantization) I think Lee’s assessment is pretty inaccurate. Many people who were working in canonical LQG still do work in canonical LQG. Thiemann, Bojowald, Ashtekar, Varadarajan to name a few.

Posted by: anonymous on July 5, 2006 1:37 PM | Permalink | Reply to this

### Re: New Kind of Science?

Many people who were working in canonical LQG still do work in canonical LQG. Thiemann, Bojowald, Ashtekar, Varadarajan to name a few.

Originally the question was what the solution to the problems listed by Nicolai and others would be. Lee Smolin said, essentially, that the answer is to switch to different approaches.

So what is the perception in the community that still follows CQL. Do people ther think Nicolai’s objections can be addressed?

Posted by: urs on July 5, 2006 1:44 PM | Permalink | Reply to this

### Re: New Kind of Science?

Hi urs,
So in essense the upshot of nicolai et al.’s review is that there are many questions at the formulation level that remain to be answered before one can start thinking about the physical output. As far as i can see,the general perception seems to be to try to apply the LQG techniques to certain mini and midi superspace models (cosmology, black-holes, cylindrical waves , certain two-dimensional models to name a few) and try to understand how to tackle the various issues (problems with Thiemann’s hamiltonian constraint, problem of observables, semi-classical limit of the theory etc.) in these toy models.

Posted by: anonymous on July 5, 2006 4:07 PM | Permalink | Reply to this

### Re: New Kind of Science?

> try to understand how to tackle the various issues [..] in these toy models.

Why not stick to the one toy model (2d gravity) which is pretty well understood, but where CQL gives an unexpected (to avoid the word ‘wrong’) result? Or even simpler, the harmonic oscillator?
Are you aware that somebody is working on this?

Posted by: wolfgang on July 5, 2006 4:16 PM | Permalink | Reply to this

### Re: New Kind of Science?

Actually it is not completely clear to me that harmonic oscillator gives wrong answer when polymer quantized. Before the Helling et al. paper, the quantization was done in gr-qc/02071063. There the hamiltonian is treated in analogy with how the hamiltonian constraint is quantized in LQG, and the spectrum remains bounded from below. The hamiltonian in this case depends on an arbitrary length-scale \mu, which seems to be a generic feature of all the toy-models quantized via LQG methods.
(such an arbitrary parameter is also a generic feature of loop quantized cosmological models i think). This paper also illustrates how the Schrodinger physics can be recovered from the polymer physics by essentially mapping coherent states of Schrodinger Hilbert space to certain elements in the dual of polymer Hilbert space. Helling et al’s quantization as far as i can recall is unitarily inequivalent to the quantization given in the above paper. Also I think one of the ideas of studying these toy models is that, there are many inequivalent choices of hamiltonian which can be used in these loopy quantized theories, so one should use that hamiltonian (or hamiltonian constraint depending on the model) which gives the correct “low energy” limit. So infact studying these toy models can help reduce the ambiguities that plague the hamiltonian constraint of LQG.

Posted by: anonymous on July 5, 2006 4:57 PM | Permalink | Reply to this

### Re: New Kind of Science?

I assume you refer to this paper? (surplus 3 in your ref.)

Posted by: wolfgang on July 5, 2006 5:54 PM | Permalink | Reply to this

### Re: New Kind of Science?

yes, sorry about that. I was referring to
gr-qc/0207106.
thanks

Posted by: anonymous on July 6, 2006 1:41 AM | Permalink | Reply to this

### Re: New Kind of Science?

Let me use this opportunity to emphasise that our second paper, hep-th/0601129, does specifically address spin foam models as well.

Posted by: Kasper Peeters on July 6, 2006 3:31 AM | Permalink | Reply to this

### Re: New Kind of Science?

[…] our second paper, hep-th/0601129, […]

From section 9:

From this point of view, the finiteness properties established so far [about spin foam models, U.S.] say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging
procedure where one sums over all foams,
as discussed above.

Hm, so apparently the claim here is that people have not studied what happens when the integral over all spin foams is performed.

Posted by: urs on July 6, 2006 4:41 AM | Permalink | Reply to this

### “UV finite”

Indeed.

When Lee said, above, that spin foam models are UV-finite (here and there), he doesn’t mean what you think he means by that phrase.

When quantum field theorists use the phrase “UV-finite”, they mean “cutoff-independent”. In the case of a lattice cutoff, that means “invariant under block-spin transformations”, AKA the lattice Renormalization (Semi-)Group.

That’s not what Lee means. He means that the sum over spin labels is convergent, which is roughly what we denote by the phrase “has a cutoff.”

The funny thing is that this finiteness condition is a weird combination of a UV cutoff and an IR cutoff. Normally, one would expect a divergence, associated to the infinite volume of spacetime. We want that divergence to be present. And, though I’m no expert on spin foam models, it appears to me that they do their darndest to suppress it.

Posted by: Jacques Distler on July 6, 2006 8:07 AM | Permalink | PGP Sig | Reply to this

### Re: “UV finite”

Hi Jacques (and Lee),

I am also confused about the significance of UV finiteness’ here, hence my post to Lee, above.

I think the claim is that different choices of Hamiltonian constraint in LQG might correspond to different spinfoam models. Lee then said that:

“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.)…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.”

If nothing else, there is a language barrier here, because (using Lee’s analogy) the divergence of momentum integrals in QFT turns out not to be problematic when understood properly, and in a way leads to the richness of the RG.

So what is the significance of this UV-finiteness, and must we restrict to spin-foams which are UV finite in this sense?

Posted by: boreds on July 6, 2006 6:26 PM | Permalink | Reply to this

### Re: The LQG Landscape

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.

Posted by: Haelfix on July 4, 2006 11:00 PM | Permalink | Reply to this

### Re: The LQG Landscape

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

Posted by: Lee Smolin on July 5, 2006 10:43 AM | Permalink | Reply to this

### Sceptical, not hostile

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.

I am not hostile. I am sceptical. There is a difference.

I don’t believe the claim that an anomalous chiral gauge theory can be turned into a consistent quantum theory merely by coupling it to quantum gravity.

Thiemann’s paper may be nine years old; more’s the pity that understanding anomalies in LQG is still an “open question.”

Posted by: Jacques Distler on July 5, 2006 11:00 AM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

(I tried using blockquote here and failed for reasons I am unable to discern.)

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’m confused again (it seems to happen a lot). I thought that we had established that the connection of this new sort of quantization to the old sort we all know and experimentally verify was still unknown. In particular, I thought that your answer to the objection that new-quantization has no problems with anomalous gauge symmetries was that the corresponding old-quantized theory in the flat space limit and away from the Planck scale would not be nonanomalous or perhaps may not exist.

If these semiclassical states perform as you say, can you not see explicitly what the right answer is? Or by QFT in the above do you mean new-QFT rather than old-QFT?

Posted by: Aaron Bergman on July 5, 2006 11:10 AM | Permalink | Reply to this

### Re: The LQG Landscape

hi Aaron,
sorry for repeating again but my post(on june 28th) seems to have gone unnoticed. The relation between polymer and fock quantization is not totally unknown as shown in varadarajan 2000-2001 papers on U(1) theory. However i think in the above statement Lee is referring too Thiemann’s papers gr-qc/0207030 and its followup. These papers i must say seem rather adhoc to me. Not only does he assume that the matter field hamiltonian is some sort of true hamiltonian and not part of the hamiltonian constraint, his final results depend on extremely ad-hoc regularization schemes(choice of lattice) and so far i do not think there is any justification for it. Also he derives effective matter hamiltonian only for scalar and electro-magnetic fields so issue of fermionic fields and how anamoly might possibly arise(if they arise at all) even in his framework is completely open.

Posted by: anonymous on July 5, 2006 11:45 AM | Permalink | Reply to this

### markup

(I tried using blockquote here and failed for reasons I am unable to discern.)

I’m gonna guess it’s because you tried to write

<blockquote>fubar</blockquote>

<blockquote><p>fubar</p></blockquote>

If that’s too much trouble, you could use the Markdown filter and write

>fubar

Posted by: Jacques Distler on July 5, 2006 12:56 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

“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.”

I just took a look at Thiemann’s paper. The formalism is dense and will take a fair bit of time and effort to penetrate for people who are not familar with this. Before deciding whether to invest this time and effort, hopefully it is ok to ask questions to the LQG people who have already thoroughly checked and examined it.

When looking through the paper I didn’t see a specific statement on gauge anomalies. Perhaps it is implicit in some more general statement that he makes?
Specifically, I would like to know this: Did he implement the generators of gauge transformations as operators on his Hilbertspace and show that they commute with the Hamiltonian? (Presumably this is what gauge anomaly-free means in the Hamiltonian framework - I’m mostly familiar with the Lagrangian formulation.)
Does this commutivity hold with chiral fermions in any representation of the gauge group, or does it break down for representations which don’t satisfy the usual anomaly cancelation condition?

Posted by: amused on July 7, 2006 6:15 AM | Permalink | Reply to this

### Hamiltonian Anomalies

Very brave of you to launch in and attempt to digest Thiemann’s paper. But it seems to me you are going to have a rough time satifying your curiosity if you’re unfamiliar with how gauge anomalies are realized in a Hamiltonian framework.

Above, I linked to Alvarez-Gaumé and Nelson, the standard reference on the subject.

But, before we even get to anomalous theories, one should note a few facts about gauge theories in general.

The situation we’re interested in is one in which one quantizes first, and then imposes the gauge constraints as operator equations in the quantum theory. A crucial subtlety is that, on the classical configuration space, the gauge-orbits are noncompact.

If the gauge orbits were compact, we could take $L^2$-functions on the configuration space (the “kinematical Hilbert space” in their lingo) and average them over the gauge-orbits, to obtain gauge-invariant $L^2$ functions (the physical Hilbert space).

You can’t do that averaging when the gauge orbits are noncompact. In fact, instead of imposing the constraints strongly (the constraints annihilate physical states), you can only impose them weakly (matrix elements of the constraints between physical states vanish) in the context of this larger “kinematical Hilbert space.”

(You will run smack-into this problem fairly early on, in reading Thiemann.)

In the end, though, in a non-anomalous theory, the result you obtain is the same as you would have obtained if had quantized the quotient stack, {classical configurations}/gauge transformations.

It’s in the latter context that Alvarez-Gaumé and Nelson work. Quantizing the fermions first, one gets a Hilbert bundle of fermion Fock spaces over the “physical” bosonic configuration space ({classical configurations}/gauge transformations). The ground state of this Fock bundle has a Berry Phase if the theory is anomalous.

Posted by: Jacques Distler on July 7, 2006 8:47 AM | Permalink | PGP Sig | Reply to this

### Re: Hamiltonian Anomalies

Thanks for the background info. I’ll take a look at the Alvarez-Gaume and Nelson paper; in fact I’ve been meaning to read up on Hamiltonian anomalies for some time since they still haven’t been elucidated in the lattice framework. But for the moment I’ld just like to check if my general understanding of this stuff is right:

In going from the Lagrangian to Hamiltonian formulation via the transfer matrix formalism, you end up with constraints on the “kinematic” Hilbertspace: The generators of the symmetries of the theory get implemented as operators on the Hilbertspace, and only the subspace annihilated by these operators is physically relevant. (E.g. the partition function becomes the trace of the evolution operator over the physical subspace.) Now, to say that a theory is free of (gauge) anomalies just means that it is (gauge) invariant at the quantum level. In the Hamiltonian framework, this must surely mean that the Hamiltonian maps the physical subspace to itself. A sufficient condition for ensuring this is that the Hamiltonian commutes with the operators which implement the generators of the symmetries on the Hilbertspace, and I had envisaged that a breakdown of gauge invariance (i.e. gauge anomlies) would probably manifest itself through a breakdown in this commutivity. Is this correct? If so, it would seem that the presence of gauge anomlies in the Hamiltonian framework could be uncovered just by working out the commutator of the Hamilonian with the operators generating gauge transformations. It doesn’t seem necessary to actually construct the physical subspace for this, you just work out the commutator of the operators on the full kinematic Hilberspace. Since Thiemann has apparently constructed a well-defined kinematic Hilbertspace, and well-defined Hamiltonian acting on it, we just need to know the operators generating the gauge transformations, and then work out their commutator with the Hamiltonian. But it could be that there is a problem with actually implementing the gauge transformation generators as operators on this Hilberspace, as Aaron’s comment indicates. In that case it would seem that one can’t even start to ask about gauge anomalies in this formulation.

It would be good if the LQG experts who have studied Thiemann’s paper could chime in and say what the status of gauge anomalies is in this framework - do his results show that they are always absent, or has the question simply not been addressed? I’m not being hostile or “minimalising” but am genuinely interested to know this.

Posted by: amused on July 9, 2006 9:09 AM | Permalink | Reply to this

### Re: Hamiltonian Anomalies

I’m not being hostile or “minimalising” but am genuinely interested to know this.

I find it sad that anyone should feel it necessary to add that remark.

I think everyone asking questions here is trying to assiduously stick to the physics. And I think it proper to try to answer their questions, rather than impugn their motives.

I, for instance, did not take umbrage at your question about anomalies on the Landscape, or huffily tell you to go off and read the papers. If my answer was insufficient, I would be happy to expand upon it.

It would be good if the LQG experts who have studied Thiemann’s paper could chime in and say what the status of gauge anomalies is in this framework - do his results show that they are always absent, or has the question simply not been addressed?

In the standard Hamiltonian treatment of anomalies, it is the algebra of (Gauss Law) constraints that fails to be faithfully represented on the Hilbert space. Instead, it is, at best, only projectively-represented.

This is the origin of the Berry phase (found by Àlvarez-Gaumé and Nelson) of the fermionic Fock vacuum, viewed as a section of a Hilbert space bundle over $\mathcal{A}/\mathcal{G}$.

The LQG theorists, however, simply postulate that the algebra of constraints is faithfully represented. By definition, there is no anomaly.

Since the constraints contained a term quadratic in the fermions (and it was the commutator of these terms that was the source of the anomaly), this means that the fermions themselves are not represented on the Hilbert space. (If they were, then the algebra of the constraints would be something you would calculate, not something you could postulate.)

Under those circumstances, one might well wonder in what sense they are actually fermions (e.g. is there a Pauli Exclusion Principle?)

Posted by: Jacques Distler on July 9, 2006 5:27 PM | Permalink | PGP Sig | Reply to this

### Re: Hamiltonian Anomalies

“In the standard Hamiltonian treatment of anomalies, it is the algebra of (Gauss Law) constraints that fails to be faithfully represented on the Hilbert space.”

Thanks for pointing that out. Now that you mention it I remember hearing this previously; unfortunately these things don’t always stick in the memory.

“Since the constraints contained a term quadratic in the fermions (and it was the commutator of these terms that was the source of the anomaly), this means that the fermions themselves are not represented on the Hilbert space.”

But according to his abstract Thiemann achieves a represention of the gravity + SM Hamiltonian on his Hilbertspace. This contains fermion fields. How can there be a representation of the Hamiltonian if the fermions themselves are not represented on the Hilbertspace?

Posted by: amused on July 10, 2006 8:40 AM | Permalink | Reply to this

### Re: Hamiltonian Anomalies

How can there be a representation of the Hamiltonian if the fermions themselves are not represented on the Hilbertspace?

You must be new here.

For a mini-introduction to this debate, see these discussions (and the paper by Helling and Policastro) of Thiemann’s quantization of a much simpler system: the worldsheet theory of the free bosonic string. In the end of the day, the Fourier modes, $a^\mu_n$, of the spacetime coordinates (worldsheet free scalar fields), $X^\mu(\sigma,\tau)$, are not represented on the LQG kinematical Hilbert space.

The prevailing attitude among the LQG theorists is summed up above by Lee

Quantization of a QFT involves choices of which sub-algebra of observables is to be represented faithfully as well as, when allowed, choices of representations of that algebra. In the case of the bosonic string these choices lead to different unitarily inequivalent quantum theories. To my understanding only experiment can decide between such different versions of a quantum theory.

As discussed with “anonymous” above, they’re even willing to (radically) change the form of the classical configuration space (for Yang Mills Theory), if that leads to a more congenial setting for their quantization procedure.

Posted by: Jacques Distler on July 10, 2006 9:15 AM | Permalink | PGP Sig | Reply to this

### Re: Hamiltonian Anomalies

Thanks for the links - I was blissfully unaware of the previous extensive discussions of all this in the blogsphere.

From what I saw in the discussions, the situation seems to be that (i) in Thiemann’s quantisation the classical symmetries get implemented on the Hilbertspace without any anomalies, which doesn’t sound good, but then (ii) quantisation of gravity is required in this approach, so it can’t be applied to, say, chiral gauge theories in a fixed spacetime background. Still, it’s difficult to imagine that an anomaly-free chiral gauge theory with quantised gravity could ever reduce in a low energy limit to an anomalous gauge theory on a flat spacetime background, which is what you would expect if the fermion rep doesn’t satisfy the anomaly cancellation condition.

Btw, does the absence of anomalies statement also apply to the axial U(1) anomaly of QCD in Thiemann’s quantisation? (This is a physically crucial anomaly for explaining pion decay into photons.)

Posted by: amused on July 11, 2006 3:44 AM | Permalink | Reply to this

### Re: Hamiltonian Anomalies

Er, I meant of course the axial anomaly of QED for pion decay into photons. The one for QCD has to do with the eta’ mass…

Posted by: amused on July 11, 2006 3:53 AM | Permalink | Reply to this

### Re: The LQG Landscape

The impression I have is that they represent the finite gauge transformation, but, as the representations end up being very discontinuous, you can’t differentiate to represent the Lie algebra of generators.

Posted by: Aaron Bergman on July 7, 2006 9:35 AM | Permalink | Reply to this

### Re: The LQG Landscape

Hi Aaron,
That is certainly true for spatial diffeomorphisms but not true for, for example the internal SU(2) transformations. (I am talking about pure gravity case here).
The generators of the SU(2) gauge transformations (the Gauss constraint) is a well defined(essentially self adjoint) operator on the kinematical Hilbert space.

Posted by: anonymous on July 7, 2006 12:18 PM | Permalink | Reply to this

### Re: The LQG Landscape

The generators of the SU(2) gauge transformations (the Gauss constraint) is a well defined(essentially self adjoint) operator on the kinematical Hilbert space.

And, since the gauge-orbits are noncompact, there are no normalizable states in the kernel of constraint.

That’s why you need to impose the constraints weakly (taking its matrix elements between physical states to vanish), rather than strongly (taking it to annihilate physical states).

(Some would argue that the same considerations apply to spatial diffeomorphism as well. But, since that is irrelevant to the anomaly in 4D, “amused” can ignore that issue for the present.)

Posted by: Jacques Distler on July 7, 2006 12:38 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

Hi jaques,
In LQG one indeed looks for the solution to the Gauss’s constraint in the dual of a dense subspace of the kinematical Hilbert space.( So very vaguely speaking it is like requiring the matrix elements to vanish but not exactly). And the solutions then turn out to be normalizable in the kinematical Hilbert space (They are spanned by gauge-invariant spin networks.)
thanks

Posted by: anonymous on July 7, 2006 12:50 PM | Permalink | Reply to this

### Gauss’s Law

In LQG one indeed looks for the solution to the Gauss’s constraint in the dual of a dense subspace of the kinematical Hilbert space.

The dual of a dense subspace of Hilbert space is a huge space, much bigger than the original Hilbert space.

( So very vaguely speaking it is like requiring the matrix elements to vanish but not exactly).

Very vaguely, indeed. I don’t see the connection at all.

And the solutions then turn out to be normalizable in the kinematical Hilbert space

?? The solutions don’t lie in the kinematical Hilbert space. They can’t. That’s why you looked for solutions in the above, much larger space.

Posted by: Jacques Distler on July 7, 2006 1:02 PM | Permalink | PGP Sig | Reply to this

### Re: Gauss’s Law

Hi, yes, the dual is a much bigger space. But any solution(which does lie in the dual) can be written as a finite linear combination of gauge invariant spin-networks. Sorry, I do not know how to insert math symbols here, but one looks for solutions of the type $\Psi(\hat{C}f) = 0$. Where $\Psi$ lies in the dual and $f$ are in the dense subspace spanned by the so called cylindrical functions. It turns out that all such $\Psi$’s are also (gauge-invariant) cylindrical functions.

Posted by: anonymous on July 7, 2006 1:18 PM | Permalink | Reply to this

### Re: Gauss’s Law

(To get equations, you select one of the “... with itex to MathML” text filters, and type honest TeX equations, delimited by $...$, as usual. I “fixed” your previous comment for you. Hope that’s OK.)

$\Psi(\hat{C}f) = 0$. Where $\Psi$ lies in the dual and $f$ are in the dense subspace spanned by the so called cylindrical functions. It turns out that all such $\Psi$’s are also (gauge-invariant) cylindrical functions.

The last sentence is completely opaque to me.

There are no normalizable gauge-invariant state in the original kinematical Hilbert space. Since the gauge-orbits (on the classical configuration space) have infinite volume, gauge-invariant functions on the classical configuration space are not square-integrable and, conversely, square-integrable functions cannot be gauge-invariant.

This seems like such an elementary point; how can we be disagreeing about it?

Posted by: Jacques Distler on July 7, 2006 2:04 PM | Permalink | PGP Sig | Reply to this

### Re: Gauss’s Law

Dear jacques,
I sincerely apologise for not explaining this clearly. Probably some expert can do a better job then a confused graduate student but I will try my best here. The main point is, In the quantum theory the “so called quantum configuration space” which is a distributional extension of the classical configuration space(space of all morphisms from the space of path to SU(2))
is compact Hausdorff in a suitable topology. Also the space of gauge transformations on this space is an extension of the space of smooth gauge transformations. This space of “distributional” gauge transformations is also a compact Hausdorff space in a suitable(actually Tychonov) topology. These distributional gauge transformations
admit a continuous group action on the quantum configuration space whence the orbits are compact (in the topology in which the quantum configuration space is compact). Again I am sorry if I am not explaining anything properly. The canonical reference is Thiemann’s 2001 review.
Thanks for the patience

Posted by: anonymous on July 7, 2006 2:16 PM | Permalink | Reply to this

### Re: Gauss’s Law

space of all morphisms from the space of path to SU(2)

Huh?

How did paths in $SU(2)$ enter here? We were talking about connections on a principal $G$-bundle, $P\to M$, over a (topological) 3-manifold $M$.

That’s a contractible space. What topology do you want to put on it that makes it compact (and makes the Aut-$P$ orbits finite volume)?

Posted by: Jacques Distler on July 7, 2006 3:33 PM | Permalink | PGP Sig | Reply to this

### Re: Gauss’s Law

Hi jacques,
The (quantum) configuration space in LQG
is constructed as follows. In classical theory one starts with a sub-algebra of the full Poisson algebra generated by holonomies along (piecewise analytic) paths, and fluxes (obtained by integrating triads along analytic 2-surfaces). The holonomies themselves generate an abelian C* algebra. The spectrum of this C* algebra is the quantum configuration space of LQG(The square integrable functions on it with respect to a diffeo-invariant measure defines the kinematical Hilbert space of the theory.)In the so called Gelfand topology this space is compact Hausdorff.
This quantum configuration space can also be understood as the set of all
(groupoid) morphisms from the set of paths
(in the 3-manifold) into the gauge group(which is SU(2) for Ashtekar variables.) In this viewpoint, this space can be understood as a projective limit of a projective family, each member of which is a compact Hausdorff space. One can now put Tychonov topology on the projective limit such that it is compact and hausdorff as well.
In a similar way the space of smooth gauge (here by gauge i mean the internal SU(2) rotations generated by Gauss’s constraint) transformations is extended to a distributional space(basically all functions from 3-manifold to SU(2) without any continuity requirements) which can also be shown to be compact Hausdorff using projective limit techniques.
I have tried to condense too much stuff in here, but hopefully some of it makes some sense.
thanks again for these discussions

Posted by: anonymous on July 7, 2006 4:06 PM | Permalink | Reply to this

### Re: Gauss’s Law

Thanks for patiently trying to explain this stuff to me. I am, as you see, rather slow.

In classical theory one starts with a sub-algebra of the full Poisson algebra generated by holonomies along (piecewise analytic) paths, and fluxes (obtained by integrating triads along analytic 2-surfaces). The holonomies themselves generate an abelian C* algebra. The spectrum of this C* algebra is the quantum configuration space. … This quantum configuration space can also be understood as the set of all (groupoid) morphisms from the set of paths (in the 3-manifold) into the gauge group

Huh? Already there, you’ve lost me.

This configuration space — if you’re really going to take morphisms from the set of paths (in $M$) into $G$ — has all sorts of complicated topology (I think it’s not even simply-connected), whereas the space of $G$-connections on $M$ is an affine space (and hence, contractible).

Put differently, you switched from the $C^*$ algebra to its spectrum, without skipping a beat. What does this have to do with the configuration space (the space of connections on $P\to M$) we thought we were quantizing?

Posted by: Jacques Distler on July 7, 2006 5:26 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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?

Posted by: Arun on July 5, 2006 11:37 PM | Permalink | Reply to this

### Re: The LQG Landscape

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?

Posted by: boreds on July 6, 2006 6:32 PM | Permalink | Reply to this

### 2+1

The canonical reference (which has, in this long, long discussion, been credited with solving every outstanding problem, except for global hunger, as long as one is interested in 2+1 dimensions) is Freidel and Livine.

What they actually do was discussed above. They study free scalar field theory, coupled to 2+1 dimensional quantum gravity. Integrating out the gravitational degrees of freedom, they find that the theory is equivalent to free scalar field theory on a noncommutative spacetime without gravity.

Obviously, that’s not a trick that can be replicated (or even imitated) in 3+1 dimensions.

I don’t think that paper addresses any of your questions, and I’m not aware of a paper about LQG in 2+1 dimension which does. Perhaps one of the LQG experts can help us out here.

Posted by: Jacques Distler on July 6, 2006 7:30 PM | Permalink | PGP Sig | Reply to this

### Re: 2+1

Thanks Jacques.

You’re right, that’s not really an effective theory’ in the sense I meant.

I think it really would help a lot if someone working in LQG could explain how one might try to derive a low energy effective theory for LQG (in whatever number of dimensions), or what progress has been made….

Posted by: boreds on July 8, 2006 10:26 AM | Permalink | Reply to this

### Re: The LQG Landscape

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.

Posted by: Lee Smolin on July 8, 2006 10:56 AM | Permalink | Reply to this

### Re: The LQG Landscape

Lee,

> do you really believe that a lot of smart people have worked for 20 years on something and just fooled themselves?

I read that some people believe this sort of thing about string theorists and are publishing books about it this fall 8-)

Speaking of (bosonic) strings, how do you understand the LQG results for 2d gravity vs. the Helling-Policastro paper?
It was discussed here already, but I still cannot really comprehend it.

Posted by: wolfgang on July 8, 2006 12:08 PM | Permalink | Reply to this

### Re: The LQG Landscape

Wolfgang,

Re the bosonic string: here is my understanding. Quantization of a QFT involves choices of which sub-algebra of observables is to be represented faithfully as well as, when allowed, choices of representations of that algebra. In the case of the bosonic string these choices lead to different unitarily inequivalent quantum theories. To my understanding only experiment can decide between such different versions of a quantum theory. The fact that one kind of choice may be correct physically in one context is not an argument that it is correct in all contexts. Fock type quantizations are good for perturbative QFT and LQG type quantizations are good for diffeomorphism invariant gauge theories.

Its not surprising that the free string yields different quantum theories when approached from the two methods, this doesn’t invalidate either one. My understanding is that if you want L_0 to be a positive definite hermitian operator, which is the case in a gauge fixed quantization of the string in which it becomes, in that gauge, the Hamiltonian, you have to choose a rep with a central change. If you instead make a diffeo invariant quantization in which all the L_n for all integer n are generators of gauge transformations you should make a different choice.

Re the book: is precisely because I have taken great pains to be fair, accurate and respectful to all views, and credit the achievements while expressing my respect and admiration for those who made them, that I am sensitive to the lack of such considerations when they are not reciprocated.

Thanks,

Lee

Posted by: Lee Smolin on July 9, 2006 12:26 PM | Permalink | Reply to this

### Re: The LQG Landscape

Lee,

thank you for the answer and taking the time.

> Its not surprising that the free string yields different quantum theories when approached from the two methods, this does not invalidate either one.

I am still surprised and would have thought that there cannot be room for ambiguities; Either there is a central charge and critical dimension or there is not.

Posted by: wolfgang on July 9, 2006 4:12 PM | Permalink | Reply to this

### Re: The LQG Landscape

Fock type quantizations are good for perturbative QFT and LQG type quantizations are good for diffeomorphism invariant gauge theories.

But, as we agreed elsewhere (I thought), you don’t get to use one sort of quantization somewhere and another sort of quantization elsewhere. These diff-invariant quantizations have to encompass all the successes of the usual sort of quantization (I’m not sure Fock-type is really fair). The experiments have been done, and I don’t see how LQG-type quantization can explain them.

Posted by: Aaron Bergman on July 9, 2006 5:18 PM | Permalink | Reply to this

### Re: The LQG Landscape

Aaron,

> The experiments have been done

I assume you refer to the example of the harmonic oscillator. But in this case there is the additional complication that Ashtekar et al. (the paper mentioned by ‘anonymous’ above) get yet another result, different from the Thiemann/Helling-Policastro result, but also different from the conventional result. (But similar enough that experiments probably cannot distinguish between it.)

Posted by: wolfgang on July 9, 2006 5:57 PM | Permalink | Reply to this

### Re: The LQG Landscape

There’s the Helling-Policastro result, but there’s also issues like the beta function in QCD. To be honest (and I’m sure someone will correct me if I’m wrong), I’m not sure any scattering amplitudes have even been computed.

Posted by: Aaron Bergman on July 9, 2006 6:18 PM | Permalink | Reply to this

### Malice

[D]o you really believe that a lot of smart people have worked for 20 years on something and just fooled themselves?

I would rather not attempt to personalize (or psychologize) the matter, when directly addressing the physics issues would be more fruitful.

Since all the questions that been asked here are fairly obvious ones, I would be surprised if they had not occurred to the “lot of smart people [who] have worked for 20 years” on the subject.

Hence it would be helpful to the rest of us for y’all to share your insights on those questions.

If there’s a certain paper which addresses a particular question, it would not be unreasonable to direct us to it. But saying, ‘Go read all our papers, and then you won’t have these questions.’ is not particularly helpful. Or likely to achieve what you claim is the desired effect:

What I would love to do is to get you interested enough that you consider contributing to this.

Posted by: Jacques Distler on July 9, 2006 6:09 PM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

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

Posted by: Xiaofeng on July 11, 2006 12:43 AM | Permalink | Reply to this

### Re: The LQG Landscape

Dear Xiaofeng,

LQG is, among other things:

1) the result of a quantization procedure carried out on general relativity, supergravity and other classical gravitational theories.

2) An approach to the problem of quantum gravity, more specifically, the class of diffeomorphism invariant quantum gauge field theories.

It seems to me that anyone interested in the problem of quantum gravity would be interested in it, as well as in other approaches like causal dynamical triangulations, causal sets etc. that claim non-trivial results.

If you are not interested in quantum gravity then there is no reason for you to be interested in these theories.

People who don’t care about anything but ordinary non-relativistic quantum mechanics don’t have to do all the hard work of learning QFT. But if you are interested in the physics of QED or QCD you have no alternative but to learn a lot of new stuff.

Similarly, if you are a theoretical physicist sincerely interested in the problem of quantum gravity you should learn at a technical level how to derive the main results of all the well studied approaches. This is all I am saying, I don’t understand why it is controversial. I ask our LQG students to study the basics of string theory, CDT, causal sets etc and I also suggest to string theorists to learn the basics, at a technical level of the other leading approaches.

These other theories are not toy versions of conventional gauge theories. They differ, not on a wim from us to be different, but precisely because of, and in the ways that, a background independent, diffeomorphism invariant gauge theory must be different from a background dependent gauge theory.

You do not have to read all the papers. Different authors and groups use different methods and different levels of rigor, you can choose which you are more comfortable with. But you have to read some papers. And you have to be prepared to ask the question of what a diffeo invariant and background independent QFT might be and be interested in studying the answer.

Thanks,

Lee

ps: To Aaron,

“But, as we agreed elsewhere (I thought), you don’t get to use one sort of quantization somewhere and another sort of quantization elsewhere.”

Of course you can use different kinds of quantization in different application. And not only that, our claim is that the LOST uniqueness theorem says that you must use a different method to quantize a diffeo invariant theory than a free QFT in Minkowski spacetime.

What I hope we agree is that you must be able to recover experimentally relevent aspects of older theories in appropriate limits. As we’ve said many times, there are semiclassical states in LQG, expansions around which reproduce effective cut off versions of conventional QFTs, which reproduce known physics at long wavelengths. The fact,much discussed here, that we don’t know what happens to theories with chiral gauge anomalies does not negate that there are such results in other cases including gauge theories, scalar fields, gravitons etc.

Posted by: Lee Smolin on July 11, 2006 8:42 AM | Permalink | Reply to this

### Re: The LQG Landscape

As we’ve said many times, there are semiclassical states in LQG, expansions around which reproduce effective cut off versions of conventional QFTs, which reproduce known physics at long wavelengths. The fact,much discussed here, that we don’t know what happens to theories with chiral gauge anomalies does not negate that there are such results in other cases including gauge theories, scalar fields, gravitons etc.

Could you point to a paper where this is done for pure (nonabelian) gauge theory?

In the case of gravitons, could you point to a paper where the leading (4-derivative) corrections to the Einstein-Hilbert action are extracted?

And, for scalar fields, what value of the $\sqrt{-g}R\phi^2$ coupling do you obtain from LQG? Is this fixed, or is it an adjustable parameter in your approach?

Posted by: Jacques Distler on July 11, 2006 9:08 AM | Permalink | PGP Sig | Reply to this

### Re: The LQG Landscape

semiclassical states in LQG

Here “LQG” means CQL, I guess?

Does this refer to the Kodama state?

Posted by: urs on July 11, 2006 9:34 AM | Permalink | Reply to this

### Re: The LQG Landscape

Dear Lee

I’d also be interested in understanding exactly the kind of results you are hinting at. So I’ll second Jacques’ request.

I’m surprised and sorry if any of the earlier comments from me came across as hostile. They really were not intended as such, and I’m sorry if that’s the way it came across.

I’m genuinely interested in understanding what the low energy limit of LQG looks like, and in particular what the expansions look like around these `semi-classical states’. That’s all I was asking about above, really.

Posted by: boreds on July 11, 2006 9:35 AM | Permalink | Reply to this

### Re: The LQG Landscape

Hi Lee,

I was referring to this comment:

To Aaron, yes, when the standard model is coupled to gravity, the whole thing must be quantized using LQG technology. You cannot quantize a theory half by background independent methods and half by background dependent methods.

Also,

Of course you can use different kinds of quantization in different application. And not only that, our claim is that the LOST uniqueness theorem says that you must use a different method to quantize a diffeo invariant theory than a free QFT in Minkowski spacetime.

I’m not sure I understand what you mean. I thought the LOST theorem didn’t say anything about the physical Hilberrt space.

Thanx.

Posted by: Aaron Bergman on July 11, 2006 10:23 AM | Permalink | Reply to this

### Re: The LQG Landscape

The whole point of the LQG string is that it is both free and diffeo invariant at the same time so you can compare the two approaches.

Posted by: Robert on July 11, 2006 11:55 AM | Permalink | Reply to this

### BF theory

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

(1)$\hat A_1 \star \hat A_2$

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

(2)$\propto \int_X \hat A_1 \star \hat A_2 \,.$

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

(3)$\propto \int_X \, A_1\wedge \mathbf{d}\hat A_2 \,,$

(4)$\propto \int_X \, A_1\wedge F_{A_1} \,,$

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.)

Posted by: urs on July 11, 2006 11:30 AM | Permalink | Reply to this