A new look at loop quantum gravity

[marcus]

http://arxiv.org/abs/1004.1780
A new look at loop quantum gravity
Carlo Rovelli
15 pages, 5 figures
(Submitted on 11 Apr 2010)
"I describe a possible perspective on the current state of loop quantum gravity, at the light of the developments of the last years. I point out that a theory is now available, having a well-defined background-independent kinematics and a dynamics allowing transition amplitudes to be computed explicitly in different regimes. I underline the fact that the dynamics can be given in terms of a simple vertex function, largely determined by locality, diffeomorphism invariance and local Lorentz invariance. I emphasize the importance of approximations. I list open problems."

 

[marcus]

To a large extent the discussion revolves around a simple equation (45) defining the new look LQG vertex amplitude. You can see references to equation (45) in this sample excerpt. I will highlight them to help spot them.

==sample excerpt from pages 8 and 9==
...
The vertex amplitude (45) has been found independently by different research groups [13, 30, 36–39], following quite distinct research logics; the different vertices have only later been recognized as the same.

The presentation I have given here does not follow any of the original derivations, and is taken from [2]. Quite astonishingly, the simple and natural vertex amplitude (45) seems to yield the Einstein equations in the large distance classical limit, as I will argue below.

A natural group structure based on SU(2) ⊂ SL(2,C) appears to turn out to code the Einstein equations. The incredulity called by the surprise for this claim is perhaps tempered by two considerations.

The first is that the same happens in QED. The simple vertex amplitude...<snip>...yields the full complexity of the interacting Dirac-Maxwell equations. In other words, QED, with its fantastic phenomenology and its 12 decimal digits accurate predictions, is little more than momentum conservation plus the Dirac matrices γ^AB_µ, which, like f_γ , are essentially Clebsch-Gordan coefficients.

The second consideration is that general relativity is BF theory plus the simplicity constraints. BF theory means flat curvature. Hence in a sense GR is flat curvature plus simplicity conditions (34). The map f_γ implements the simplicity conditions, since it maps the states to the space where the simplicity conditions hold (weakly); while the evaluation on H_l = 1 codes (local) flatness.

The last observation does not imply that the theory describes flat geometries, for the same reason for which Regge calculus describes curved geometries using flat 4-simplices. In fact, there is a derivation of the vertex (45) which is precisely based on Regge calculus, and a single vertex is interpreted as a flat 4-simplex [30, 37]. In this derivation one only considers 4-valent nodes and 5-valent vertices. On the other hand, the resulting expression naturally generalized to an arbitrary number of nodes and vertices, and therefore defines the dynamics in full LQG. This fact was nicely emphasized in [39].

The vertex amplitude (45) gives the probability amplitude for a single spacetime process, where n grains of space are transformed into one another. It has the same crossing property as standard QFT vertices. That is, it describes different processes, obtained by splitting differently the boundary nodes into “in” and “out” ones. For instance if n = 5 (this is the case corresponding to a 4-simplex in the triangulation picture), the vertex (45) gives the amplitude for a single grain of space splitting into four grains of space; or for two grains scattering into three, and so on. A picture of the vertex of Figure 2 interpreted as a 1 to 3 transition, with the future upward, is in Figure 3.
...
==endquote==

Here's equation (45) defining the new Lqg vertex in a compact way that the QG researchers now apparently like to use.

⟨W_v|ψ⟩ = (f_γψ)(1)

f_γ is a map from SU(2) spin networks to SL(2,C) spin networks. The subscript shows the dependence of this map on the Immirzi parameter. For more explanation consult the paper directly.

 

[MTd2]

Why didn't Rovelli touched the issue of holography and LQG? Doesn't he like Smolin?

 

[tom.stoer]

anything regarding the canonical approach?

 

[marcus]

Why didn't Rovelli touched the issue of holography and LQG? Doesn't he like Smolin?

He's focused. It would be off-topic. If you look at past Rovelli papers you will see (IMHO) logical, coherent organization of thought. Not scattered.

I don't know that R. has anything new to say about gravity=entropy. Even if so, it would be off-topic. If he has anything new to say he would be more likely to write a separate paper.

This present paper gives the current status of core LQG. It shows how today's version is built from the ground up, and what the open questions are.

The paper is essentially a revised write-up of the talk he gave at Zakopane LQG workshop in early March 2010. I think. You can compare with the Zakopane slides, which are online.
That workshop was about the current status of LQG and the remaining open problems---internal problems defined within context of the theory itself. As I recall there was not anybody at the workshop who brought up Verlinde stuff, entropic force stuff.

 

[marcus]

Here are Gene Bianchi's January 2010 slides:
http://wwnpqft.inln.cnrs.fr/pdf/Bianchi.pdf
Rovelli gives substantial credit to Bianchi for the current formulation of LQG and cites this talk given at the University of Nice campus (at Sophia Antipolis research park)

Anyone who read Rovelli's preprint when it came out on 12 April will have realized that it is a landmark paper. Among other things it reformulates Lqg without reliance on earlier "quantizations" like the EPRL and FK.

There are some key passages from the Rovelli paper worth quoting, to help folks get caught up who may not have read the original. I hope to find time to post some excerpts here.

 

[marcus]

new Lqg

In making a key point right at the beginning, Rovelli cites Vincent Rivasseau's talk at the 2008 Nottingham Loops conference ("QG-squared")
I'm trying to get the V.R. talk. The first thing I try is I go here:
http://www.maths.nottingham.ac.uk/qg/wiki/index.php/QGsquared-slides

I find that from there I can get the audio:
http://www.maths.nottingham.ac.uk/r...ference/audible/01Monday/VincentRivasseau.mp3

And the slides:
http://www.maths.nottingham.ac.uk/qg/wiki/images/a/a4/RivasseauVincent1214731400.pdf

Scroll down till you see several slides each numbered 5 in the lower left corner. They say this:

==quote==
Enlarging Quantum Field Theory

From loop gravity we learn that we should develop a space-time
independent formulation of QFT. In ordinary QFT, Feynman graphs and
parametric representation are closest to that goal.
==endquote== Listening to the audio I find that Rivasseau indeed right at the beginning gives a nuanced philosophy about what a background independent quantum field theory must look like! Based on abstract combinatorial objects rather than (say) a differential manifold continuum. Like on Feynman graphs. The audio gives additional information beyond what is in the slides. The philosophy is essentially what Rovelli summarizes from V.R.'s talk in the first couple of paragraphs of his paper.

Therefore a BIQFT (background independent quantum field theory) could arise from a new mathematical basis instead of being "derived" from some known mathematics (like Gen Rel) by a "quantization procedure". Rivasseau is willing to contemplate this and he discusses one interesting possibility in his talk.

 

[marcus]

Last year several of us discussed the Corfu QG school, where Rovelli, John Baez, Vincent Rivasseau,
Abhay Ashtekar and John Barrett taught minicourses---each gave a series of five lectures on new QG results and active research areas. That was in September 2009.
http://www.physics.ntua.gr/corfu2009/qg.html
Rovelli's course was about "new look LQG" or words to that effect. The abstract summary he posted was intriguing, but we never got to see the slides. (The Corfu organizers seem to have suffered from technical problems or lack of resources, as most of their school's QG lectures were never made available on line.) So we've been waiting to learn the content. What is the "new LQG" that Rovelli was talking about in Fall 2009?

It now appears that the Corfu course description seems to correspond to some of the material in section 3 on pages 6 and 7 of the "new LQG" paper we just got. There may be other parts that match as well.

See what you think. I will recall Rovelli's September 2009 course description:
==quote==
Covariant loop quantum gravity and its low-energy limit

I present a new look on Loop Quantum Gravity, aimed at giving a better grasp on its dynamics and its low-energy limit. Following the highly succesfull model of QCD, general relativity is quantized by discretizing it on a finite lattice, quantizing, and then studying the continuous limit of expectation values. The quantization can be completed, and two remarkable theorems follow. The first gives the equivalence with the kinematics of canonical Loop Quantum Gravity. This amounts to an independent re-derivation of all well known Loop Quantum gravity kinematical results. The second the equivalence of the theory with the Feynman expansion of an auxiliary field theory. Observable quantities in the discretized theory can be identifies with general relativity n-point functions in appropriate regimes. The continuous limit turns out to be subtly different than that of QCD, for reasons that can be traced to the general covariance of the theory. I discuss this limit, the scaling properties of the theory, and I pose the problem of a renormalization group analysis of its large distance behavior.
==endquote==

 

[tom.stoer]

In the meantime I studied the paper in more detail (but not all references Rovelli cites :-)

It seems that he leaves - at some stage - the standard road from classical to quantum theory behind and claims that a certain mathematical model (spin-network Hilbert space, scalar product, ...) is the correct setting for QG. Rovelli leaves behind the idea that spin-foams are just quantum histories of spin networks; nor does he insist on the embedding of a graph in a manifold; he just uses these spin-networks as in- and out-states in rather a standard QFT setting to sandwich certain observables.

This view is supported by the fact that different research programs including the canonical approach seem to "converge" to this general spin-network framework, not necessarily a unique one, but rather closed to some "final theory". So all the old ideas of Ashtekar variables, loop space, cylinder fiunctions, ... are no strict derivation but only a motivation (it is a derivation of the kinematical framework, but not of the whole dynamics of the theory).

Doing calculations his focus is - as exact solutions are not available - on semi-classical or coherent states. Rovelli's aim is to study their physical consequences in certain regimes.

As usual Rovelli closes a review paper with a section regarding the main open question. I always appreciate these conclusions as they sometimes provide more insight than detailed technical calculations. You do not only learn about the current status - but you understand his objectives for the still ongoing research program. Unfortunately the big conceptual issues are hidden between rather technical statements:
- is it possible to construct a Hamiltonion in the canonical approach that matches to the new-look LQG approach? if not, why?
- how does one construct physical observables?

He mentiones a few problems - closely related to each other - which are not only "minor variations" of the framework:
- what is the nature of the cosmological constant? can it be derived from the theory or is it just an input parameter?
- what about q-deformations of SU(2)?
- what about the nature of the Immirzi-parameter?

So what do you think: what are the top-5 questions in LQG as of today?

 

[marcus]

Tom this is a perceptive and informative assessment of Rovelli's "new LQG" paper. The first I have seen. It helps me to see another's point of view. Somehow I missed this until just now, when I came here to fill in a detail.

Rovelli's 11 April http://arxiv.org/abs/1004.1780 paper says this on page 5:

==sample quote==
In summary, the Hilbert space H_Γ contains an (over-complete) basis of “wave packets” ψ_Hl=ψ⃗_{nl,⃗n′l ,ξl ,ηl},
with a nice interpretation as discrete classical geometries with intrinsic and extrinsic curvature.

These states define a natural holomorphic representation of H_Γ [8, 18]. In this representation, states are represented by holomorphic functions on SL(2,C)^L
ψ(H_l) = ⟨ψ_Hl| ψ⟩.

F. Derivation and relation with SL(2,C)

Above I have presented the kinematics of the theory without deriving it from known physics. Remarkably, there are a number of distinct derivations that converge...
==endquote==
The red highlight refers to something Tom said. Rovelli is shifting gears. The emphasis is no longer on "quantizing GR" and "deriving" the Feynman diagrammatics of spacetime geometry which new LQG represents. There are many derivations from different starting points using different proceedures which tend to agree on the main features. With this tendency of derivations to converge the game is no longer to find the perfect derivation from prior physics, but to develop the theory and explore its consequences.

Notice also the references [8, 18] with blue key phrase "natural holomorphic representation".
Reference [18] is a paper "to appear" by Bianchi Magliaro Perini.

Their paper appeared today:
http://arxiv.org/abs/1004.4550
Spinfoams in the holomorphic representation
Eugenio Bianchi, Elena Magliaro, Claudio Perini
9 pages
(Submitted on 26 Apr 2010)
"We study a holomorphic representation for spinfoams. The representation is obtained via the Ashtekar-Lewandowski-Marolf-Mourao-Thiemann coherent state transform. We derive the expression of the 4d spinfoam vertex for Euclidean and for Lorentzian gravity in the holomorphic representation. The advantage of this representation rests on the fact that the variables used have a clear interpretation in terms of a classical intrinsic and extrinsic geometry of space. We show how the peakedness on the extrinsic geometry selects a single exponential of the Regge action in the semiclassical large-scale asymptotics of the spinfoam vertex."

 

[Elliot.exe]

LQG and string theory have always seemed to clash. But as two un-proven theories each has merits. I think that for the purpose of research into the entropy of black holes string theory would be better to use because it is more conclusive in most respects. This is not to say that LQG is not a valid or possible idea as we see more evidence for it on a day to day basis (as long as you have the right tech).

What are everybody's thoughts on the matter?

(I should note I do not work in physics...yet. I am meerly a amateur physicist. But seeing how the physics community is divided on certain concepts interests me and I hope by observing debates I can hope to be a better physicist.)

 

[marcus]

Elliot, the situation in the two research communities is very different. LQG is just beginning to hit its stride. First Loops conference was 2005. There has been an increase in the number of faculty positions, LQG-related research has been springing up in new places. Still a small community by comparison with string, but the research output is growing: 40 papers in 2005 up to 140 in 2009, by my Spires search.
https://www.physicsforums.com/showthread.php?p=2708501#post2708501

Here's Francesca's world map of LQG, it lists permanent faculty at various locations.
http://maps.google.com/maps/ms?ie=U...985216139270436.0004843830d27f3e6c50e&t=h&z=0

By contrast the first String conference was back around 1990. String output has been level or declining recently, again by database search. The number of citations to string papers has dropped, according to Spires topcite data. The trend recently has been for researchers to stop trying to achieve unification and to find other applications of string math. You can see this by comparing the programs of the annual string conferences.

We have been hearing of disillusionment in string community over the past two years.
A number of prominent people have gotten out, or retain only part-time involvement.
The mood of rank and file is reflected in this recent anonymous blog comment:
==quote==
Anonymous says:
May 19, 2010 at 5:36 am
In my opinion it may be true that most string theorist are already disillusioned about the prospect of unification, as claimed by Trent and Jones. Hardly any string theory professors I met are fanatic about string theory. One of them teaching a course on string theory openly makes it very clear to students that string theory may have nothing to do with unification, but still it is worth studying for various other reasons. The PhD students doing string theory, as far as I can tell, don’t have much faith in string unification either, although when asked about what they are doing by some non-physicist, they always answer they are trying to unify nature’s forces.:-) Actually they choose this route simply because they want to get into formal particle theory, and given the current environment it does seem to be a natural choice for someone whose interest lies in this area.
==endquote==

So how to respond to your question? First off, they are not two theories that have been written down, they are two research programs. LQG has changed radically since 2006 and has entered a phase of rapid growth. The emphasis is increasingly on spin foam models, inclusion of matter and the area of quantum cosmology. The overall LQG program goal as stated by Rovelli in his talk at the Strings 2008 conference is to discover how to formulate QFT without any preselected background spacetime geometry. This is one strategy to unify physics---unite General Relativity (which is backgroundless) with Quantum Field Theory (which so far has been only backgrounded).
Some of the major players in the field are: Rovelli, Thiemann, Barrett, Freidel, Krasnov, and their younger co-workers. If you look at their output you will see that these people (and their collaborators) are publishing unusually rapidly these days. There is currently a lot of enthusiasm and innovation.

If you want to make a comparison, try this: Pick 5 names of string leaders and look up on arxiv to see what they have published say since 2006.

 

[czes]

For Black Hole fits better LQG because its independence of the background. It helps to solve the conservation of the Black Hole information. Recently developed holographic principle shows the emergent space-time what is natural in LQG. the AdS/CFT correspondence shows that String Theory also has to search the background independence.

 

[ensabah6]

inclusion of matter

Hi, What specific attempts do you refer to? Noncommutitive geometry and spectral triples, QFT in SF, Bilson-Thompson, or some other research program? thanks

 

[marcus]

There have been a number of signals recently.
If you go back to March 2009, there is this http://arxiv.org/abs/0903.3475

For example on page 2:
"As said, not only this is the first example of a derivation of a DSR model for matter from a more fundamental quantum gravity model, and one further example of the link between non-commutative geometry and quantum gravity formulated in terms of spin foam/loop quantum gravity ideas, but it is of great interest from the point of view of quantum gravity phenomenology. It is also interesting, more generally, as another possible way of bridging the gap between quantum gravity at the Planck scale and effective physics at low
energies and macroscopic distances."

And then for a kind of survey of one line of development you could go to minute
59:50, 1:01, and 1:06 of Livine's February 2010 talk
http://pirsa.org/10020079/

I think this may be what you were pointing to when you said "QFT in SF": efforts to realize quantum field theory in the context of spin foam and (closely related) group field theory.

Also you correctly suggested http://arxiv.org/abs/1005.1057 "Spin Foams and Noncommutative Geometry" by Marcolli et al.