What is Rovelli's new LQG formulation trying to do?

[ensabah6]

What is Rovelli's "new" LQG formulation trying to do?

By way of illustration: contemporary LQG is not some sort of methodical quantization of Ashtekar General Relativity (the connection version of classical GR). The field did indeed start out in the 1990s based on Ashtekar GR. But convergence of various attempts crystalized in a de novo reformulation different from, but combining aspects of each. Something like a leap occurred, as is described in the survey/status report http://arxiv.org/abs/1004.1780.

" Heck the new version of LQG is not even about the Einstein Hilbert action, so again I have no idea what they're really trying to do.
).

I've wondered about this to

 

[marcus]

Let's step through it gradually. Look at the abstract and the first three paragraphs of the introduction. Is there anything you do not understand?

For your convenience, so we can focus just on this bit, I will copy them here:

==quote abstract==
A new look at loop quantum gravity

Carlo Rovelli

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.
==endquote==

If there is anything here you don't understand, please say! The introduction spells things out more explicitly. It seems remarkably clear to me, so I will just copy it in the next post. Again let us know if there is anything you don't understand.

 

[marcus]

I think the introduction is remarkably clear (and concise) about what the intention is. I don't think any paraphrase could make it any clearer, or briefer. So here we go.

==quote 1004.1780==
I. INTRODUCTION
Significant developments in the last years have modified the state of the art in quantum gravity. The merge of the canonical and the covariant frameworks has yielded a rather well-developed background-independent theory, with a reasonable kinematics and an intriguing dynamics, where physical transition amplitudes can be explicitly computed and compared with the classical theory. Here is an account of the state of this theory, as I understand
it today.

I present the theory without “deriving it from classical GR” or other “quantization procedures”.¹ As emphasized by Vincent Rivasseau [1], a formulation of quantum field theory that remains meaningful in the background-independent context, is as a generating function for amplitudes associated to a combinatorial structure, as in the definition of QED in terms of Feynman-graphs. The amplitudes define the dynamics by assigning probabilities to processes described in terms of a Hilbert space. I use this language here.

I emphasize in particular the fact-–pointed out by Eugenio Bianchi [2]–-that the dynamics of the theory has a very simple and natural definition, largely determined by general physical principles. It is given by a natural immersion of SU(2) representations into SL(2,C) ones.
A simple group theoretical construction (Eq. (45) below) appears to code the full Einstein equations.
==endquote==

You might also want to note that this is a presentation of past results achieved by many people including the author. It is a "new look" or way of summarizing and packaging what was already known and can be found spread out in a number of earlier papers.

==quote==
I take responsibility for the presentation, but the results reported below are due to a number of people, including: Emanuele Alesci, Abhay Ashtekar, John Barrett, Eugenio Bianchi, Florian Conrady, You Ding, Bianca Dittrich, Richard Dowdall, Jonathan Engle, Winston Fairbairn, Cecilia Flori, Laurent Freidel, Kristina Giesel, Henrique Gomes, Frank Hellmann, Wojciech Kaminski, Marcin Kisielowski, Kirill Krasnov, Etera Livine, Jurek Lewandowski, Elena Magliaro, Leonardo Modesto, Daniele Oriti, Roberto Pereira, Alejandro Perez, Claudio Perini, Lee Smolin, Simone Speziale, Thomas Thiemann, and Francesca Vidotto.
==endquote==

In partial answer to your question, part of what Rovelli's "new look" paper is "trying to do" (as you put it) is what any good survey/review paper is "trying to do". It is a status report, but it is also more concise than a lot of review articles---he is able to condense and boil down to a concentrated form.

But my main question to you at this point would be: is there anything in those first three paragraphs you don't understand? To me they seem to make extremely clear what the intent is and what is going on. But you may find something unclear, in which case please say!

If you are clear on those three paragraphs, we can move on.

 

[marcus]

If anyone has gotten through those first three short paragraphs and feels they understand what is being said, I would suggest a careful reading of the two footnotes on page 1. These are really important to understanding the intent of this way of presenting/packaging LQG--this "new look" at the subject as it has developed so far. These footnotes are important enough, in my view, that I want to highlight them.

Here are the footnotes on page 1:
==quote==
1
In my opinion, after many years of attempts to “quantize general relativity”, it is time to leave the ladder behind, and start taking seriously what the various “quantization procedures” have produced. It is especially so since large overlaps have appeared between the results of the different quantizations techniques (canonical, path integral and others; see Section II F, below). I expect that it is now going to be more productive to study the theory and its consequences, in order to asses its viability, rather than keep trying to “derive” the theory.
==endquote==

Footnote 2 goes with this sentence of main text
==quote main text on page 1==
I mention below some possible alternatives in the definition of the theory. These are written in smaller characters.²
...
...

2
I do not view alternatives as problems, I view them as opportunities. In quantum gravity we are not in the embarrassment of riches: we do not have numerous complete and consistent theories. In fact, we haven’t any. The theory described here, too, in spite of the various results it yields, is incomplete: a list of open problems is in Section V. At the present state of our knowledge, worries about under-determinacy of the theory are, in my opinion, ill-judged. Rather than worrying whether this theory might have alternatives, or continuing to sketch new very incomplete models, we better ask if we have at least one complete consistent theory. This is hard enough, and, in my opinion, is today the relevant scientific question, and the one likely to be fruitful.
==endquote==

 

[marcus]

Now an "unofficial" paraphrase by me can hardly make it any clearer. If you want to cling literally to something make sure it is from the actual paper, not someone's interpretation But here's my impression.

LQG has been developing fast, so you don't want to depend on anything before 2009, and especially the Lewandowski et al "Spinfoams for all LQG". But suppose you already understood LQG somewhat and were reasonably up to date.

Then Rovelli's April paper basically does not change anything. If you understood it before, then you understand it now. The intent is a nice concise mathematically convenient presentation. To a substantial degree it repeats what we already saw in the earlier 2010 Rovelli-Vidotto paper, but with more explanation and without the application to cosmology.

Theories can be given various equivalent formulations. The question is not which mathematical formulation is "right" but which mathematics is most convenient. That may be a quote from somebody famous like Henri Poincaré What I like about the new presentation is the convenience, neatness, elegance.

I also really like the fact that it presents the essential LQG without using a spacetime continuum manifold. Basically using only graphs, Lie groups, Hilbertspaces. (And the 2-complex which is the evolutionary path of a graph---a graph is a 1-complex and this is the analog of a graph but up one dimension.)

I think manifoldy QG is somehow "mouldy". The old habit of representing space and time by a manifold has become an outdated liability. It is at the root of landscape agony, sterility, and stalled programs. The manifold was a wonderful invention in 1850 when Riemann introduced it. But even then, Riemann himself wrote that a finite/combinatorial model might turn out preferable.

Anyway is there anything which you don't understand about the intention of the paper we are discussing? The new presentation of, or "new look" at LQG.

BTW had to laugh at someone saying "not even about the Einstein-Hilbert action" Look at equation (58) on page 12. The Regge action is the relevant (combinatorial) form of the E-H. It is certainly "about" that---and page 12 can give some idea of how close or far-off the goal of proving limit is at present.

=============
Out of curiosity I looked up the Poincaré proverb--one occurrence is in La Science et L'Hypothése (page 151 of the 1968 Flammarion reprint) where he says that which mathematical formulation we choose, in other words which convention, is not “the outcome of our caprice; we adopt it because certain experiences have shown us that it will be convenient”.

 

[tom.stoer]

To cut a long story short:

Old-fashioned LQG tried to quantize EH / EC gravity using ADM (3-dim. spatial slices of spacetime) and Ashtekar variables, stepping via loops and implementation of the G- and D-constraint to spin networks.

The major achievements were a well-defined Hilbert space with inner product. The major weakness and obstacles were ad-hoc quantization procedure and missing uniqueness of the Hamiltonian (after regularization), constraint implementation i.e. no off-shell closure of quantum constraints.

Fortunately a second SF approach was available which led after tuning of the vertex / intertwiner rules to a reasonable theory which has some overlap with LQG regarding intertwiners, HIlbert space and possible even the notoriously difficult Hamiltonian. In addition the Immirzi parameter can be understood both as a quantization ambiguity and as a topological quantity with similar concepts as for the QCD theta angle.

The SF approach starts w/o manifolds, i.e. throws away the very basic principles of GR but converges with the spin-network embedding-based LQG towards similar quantum structures like Hilbert space, inner product and Hamiltonian. In addition the low-energy sector seems to become addressable via the graviton propagator and coherent state methods.

There's still some work to be done, but it seems that two approaches can now be shown to converge (which has been guessed for more than one decade).

 

[Demystifier]

So, if I understood it correctly, the new theory postulates certain new "Feynman" rules (without deriving them from more fundamental principles) which satisfy certain consistency requirements. Is that correct?

 

[tom.stoer]

Don't confuse the LQG / SF vertices or intertwiners with Feynman vertices. The intertwiners appear in the action in the PI, whereas the Feynman vertices appear in his graphs derived from the PI.

SF can be seen as the PI version of LQG which therefore could produce Feynman rules. But the original work by Feynman was not to derive his famous rules (which are a calculational toolset for solving the PI), but instead it was the derivation of the path integral (PI) itself. In that sense it's a very broad framework, not only a perturbation expansion. In that sense SF is something like the PI.

Originally SF was motivated by structures known from ordinary LQG. Now it turned out that one can write down a class of theories based on spin networks and that one is able to fix some reasonable limitations that restricts this class of theories towards something one expects from ordinary LQG (the convergence is still not perfect).

So SF as of today is somehow top-down coming from the general SF idea and converging towards LQG, whereas old-fashioned LQG is bottom-up quantization of a classical theory. Both approaches are complementary.

 

[Demystifier]

Thanks, Tom. It is clearer now to me.

 

[tom.stoer]

You're welcome; nevertheless I recommend Rovelli's paper.

I am expecting a kind of review paper to appear within the next few month; the LQG community was rather open-minded regarding discussions about current status, achievements and obstacles in the past (!) Smolin and Ashtekar wrote some remarkable papers in 2000 - 2005. The last discussion were the twppapers exchanged between Nicolai and Thiemann (Inside and outside view), but some the conclusions are outdated, in the meantime. I haven't seen such papers over the last couple of years, but in the meantime they would be highly appreciated.

 

[sheaf]

My competence only extends to an understanding of the "New Look" paper at a fairly superficial level, but if what he says is true, then it may be time for some cautious excitement.

It may be the case that the calculational techniques are not so easily usable at the moment, but I would imagine that the same was true in the early days of quantum field theory.

 

[marcus]

My competence only extends to an understanding of the "New Look" paper at a fairly superficial level, but if what he says is true, then it may be time for some cautious excitement...

I think the only thing here worth getting cautiously excited about is that mankind keeps struggling to better understand what underlies the relation of geometry to matter. I've been watching QG since before 2003 --it's always time for caution, and never time to get one's hopes up.

My experience with Rovelli papers is that he is consistently honest and forthright.
He writes carefully. You have to read carefully. Look at the list of 17 outstanding problems at the end of the paper. They are formidable---and problem #17 about combining matter with geometry is the tip of an iceberg.

The Loop enterprise is high risk.

It does seem to me (a subjective judgment call) philosophically sound.

It gets away from dependence on the manifold. The labeled graph (spin network) is an economical representation of the experimenters' geometrical knowledge (a finite web of volume and area measurements which can also carry particle-detector readings and stuff like that).

The program does seem at least to define a clear and reasonable direction.

Rovelli says that recent results provide some indication that they might get the Einstein equation for the simple matterless case. He explains why he thinks they might. That's all, he doesn't say they got it yet.

I am mostly talking to myself. With a name like "sheaf" you probably have enough sophistication to provide yourself with adequate caution and reserve. You don't need my advice. I tell myself not to get hopes up.

 

[sheaf]

LOL ! The sheaves were from a long time ago - from my twistor days.

Rightly or wrongly I'm impressed by the convergence of the various approaches. Also, being able to pull the Regge action out of that purely combinatorial framework sounds like good news to me. Even if all this, for the moment, only relates to the vacuum equations, that is an enormous achievement.

So yes, I'm watching all this with a great deal of interest.

 

[tom.stoer]

Another general remark: we are rather good in guessing Lagrangians respecting certain symmetries (gauge, diffeo, ...) see GR, QFT, ST. But we are bad in guessing Hamiltonians.

One reason why we are bad in guessing Hamiltonians is that they have to respect Poincare invariance; but this symmetry has 2*3+4 = 10 generators so we have to guess H, 3*J, 3*K and 3*P. Everybody who has ever seen the gauge fixed QCD hamiltonian knows that nobody will ever be able to guess something like that! But in addition we would have to guess another 9 operatores with the correct algebra ...

But also with the Lagrangians there is a problem, namely that they contain unphysical degrees of freedom which must be eliminated, e.g. via constraints, gauge symmetry, BRST etc. That's why usually one starts with a Lagrangian, goes through all the quantization and symmetry-fixing procedures and arrives (hopefully) at a viable, anomaly-free, convergent / renormalizable quantum theory including Hilbert space, measure etc.

What I described here was the standard approach in QFT, ST and even LQG.

Now LQG tries to turn things round, namely guessing directly the Hilbert space of the quantum theory with its physical degrees of freedom. This is rather easy as one has learned yow to treat spin networks. As they are unconstraint = w/o unphysical derees of freedom one cannot violate any symmetry (all gauge constraints + diff.inv. have been fixed = eliminated, so all what remains is H).
Now one just takes a Lie group, writes down its invariant tensors / intertwiners / vertices and automatically has a viable spin network theory.

Of course there is a price to be paid: it is unclear how to arrive at GR in the low-energy limit. So this is what spin network / LQG is currently about: try to constrain the class of spin networks such that one arrives at GR or some suitable extension like Einstein-Cartan.

 

[ensabah6]

Another general remark: we are rather good in guessing Lagrangians respecting certain symmetries (gauge, diffeo, ...) see GR, QFT, ST. But we are bad in guessing Hamiltonians.

One reason why we are bad in guessing Hamiltonians is that they have to respect Poincare invariance; but this symmetry has 2*3+4 = 10 generators so we have to guess H, 3*J, 3*K and 3*P. Everybody who has ever seen the gauge fixed QCD hamiltonian knows that nobody will ever be able to guess something like that! But in addition we would have to guess another 9 operatores with the correct algebra ...

But also with the Lagrangians there is a problem, namely that they contain unphysical degrees of freedom which must be eliminated, e.g. via constraints, gauge symmetry, BRST etc. That's why usually one starts with a Lagrangian, goes through all the quantization and symmetry-fixing procedures and arrives (hopefully) at a viable, anomaly-free, convergent / renormalizable quantum theory including Hilbert space, measure etc.

What I described here was the standard approach in QFT, ST and even LQG.

Now LQG tries to turn things round, namely guessing directly the Hilbert space of the quantum theory with its physical degrees of freedom. This is rather easy as one has learned yow to treat spin networks. As they are unconstraint = w/o unphysical derees of freedom one cannot violate any symmetry (all gauge constraints + diff.inv. have been fixed = eliminated, so all what remains is H).
Now one just takes a Lie group, writes down its invariant tensors / intertwiners / vertices and automatically has a viable spin network theory.

Of course there is a price to be paid: it is unclear how to arrive at GR in the low-energy limit. So this is what spin network / LQG is currently about: try to constrain the class of spin networks such that one arrives at GR or some suitable extension like Einstein-Cartan.

So this new LQG is just a new SF model.

 

[marcus]

So this new LQG is just a new SF model.

If you would glance at the April paper with even one eye you would see it begins with a several-page development of spin-networks and the equivalent of the canonical LQG Hilbert space in a new way. These things are not part of a spinfoam (SF) model, but they are the focus of the first part of the paper.

I don't understand your method of "learning", 'Sabah. Instead of reading for yourself, you seem to ask other people to paraphrase and interpret for you, and then you frequently misunderstand what they say and draw false conclusions, like this.

One of the interesting things about this presentation is that here spin-networks and spinfoams are working like the left and right hands of a single judo move---canonical loop and foam are seamlessly joined here and acting cooperatively or reciprocally, like two hands. Or like walking with left foot and right. Each helps to define the dynamics of the other.

To understand the first four pages or so, where the LQG hilbertspace with spin-network basis is developed, I actually found it helpful to glance back at the very basic "LQG Primer" by Rovelli and Upadhya, a paper written in 1998 in which there are no spin foams--just straight canonical oldfashioned LQG. It helped me get an intuitive feel for what was being done, but in a new way.
Don't misunderstand. The first four pages are not the whole paper! As you read on you begin to see spinfoam and they play an essential role too.

 

[tom.stoer]

So this new LQG is just a new SF model.

No!

It's about convergence of canonical approach and spin foams; it's about mapping of or identities between certain entities in both frameworks; it's about making LQG accessable for calculations; it's about long-distance limit / semiclassical approximations; it's about consistency of quantization, implementation of constraints, regularization of the Hamiltonian (which is notoriously difficult in old-fashioned LQG) ...

... there more you read the more you will find.