What are the best introductory resources for learning Loop Quantum Gravity?

[atyy]

Do you say it's false because string theory has calculated the entropy of Schwarzschild and Kerr black holes?

 

[atyy]

@atyy: It seems that you have some issue with LQG. May be I simply misunderstand you, but it seems that that you have a personal problem with it. And you are not a string theorist. :)

I would hope it's fair to say that lots of workers in LQG have personal problems with it. And I'm not an LQG theorist either:)

 

[martinbn]

@physicsmonkey: Which parts are the propaganda? I may be wrong but it seems that you are biased and searching for propaganda. For example figure 1.3 seems innocent to me. May be you are irritated because you assume that the last picture, the quantum gravity, is supposed to be LQG. But I don't think that's what they claim. It is meant to be the final and complete theory of quantum gravity, it may very well be string theory.

 

[martinbn]

I would hope it's fair to say that lots of workers in LQG have personal problems with it. And I'm not an LQG theorist either:)

That is why I don't understand you. They, the workers in LQG, may have problems, but you a side observer, why do you have a problem with it?

 

[atyy]

That is why I don't understand you. They, the workers in LQG, may have problems, but you a side observer, why do you have a problem with it?

I think many major claims of the Rovelli-Vidotto book are in doubt:

Does LQG really calculate from first principles any black hole entropy?

Is the EPRL model the consensus model, with the correct semiclassical limit?

Is the Rovelli-Vidotto approach to spin foam cosmology correct?

And an earlier major claim of LQG (but not, I think, in the Rovelli-Vidotto book) about the discreteness of the spectrum of geometric operators is also questioned.

I provided links above to show that all have been questioned by LQG insiders. I think the book should be more explicit about these problems and alternative viewpoints.

But dare I say, the real reason is I'm not a Jimi Hendrix fan!

 

[Physics Monkey]

Do you say it's false because string theory has calculated the entropy of Schwarzschild and Kerr black holes?

I think its fair to say that no one has really computed the entropy of a Schwarzschild black hole in flat space. To be sure there are many partial results, including semi-classical stories, hand-waving string calculations (described e.g. in Susskind's thin book on the holographic principle), and some LQG calculations. Regarding LQG stuff, I don't see that the issue of areas not being gauge invariant has been resolved, but I'd be happy to be corrected about this. For me, the conceptual bottom line is that we don't really know what this entropy even is, e.g. what is the hilbert space or whatever? We know S = A/4G but I think we don't really know the analog of S=tr(rho log(rho)).

Then there is Kerr/CFT. This does in principle describe Kerr black holes in flat space, but the CFT in question is associated with near horizon physics and is in general just part of a larger unknown structure. But at least one has a calculation of the entropy of the black hole and some better idea of the analog of S=tr(rho log(rho)).

One could also mention extremal black holes. Presumably Rovelli-Vidotto discount this case with their comment about "realistic", but I don't think we should dismiss these so quickly given that we understand the microstates.

Undoubtedly the nicest examples come from AdS/CFT. There the entropy of Schwarzschild black holes and Kerr black holes are precisely calculated and understood microscopically in the CFT. Fine, the asymptotics aren't right, but these results are clearly triumphs for quantum gravity.

Most of the successes above are string successes (or more generally holographic successes), and I don't see how LQG can claim anything like these successes. The state counting always requires tuning the IP and the other "dynamical" calculations are at best semi-classical statements that the whole community has been discussing in various forms for years.

I've been saying for years that the obvious thing to do is study LQG in AdS, but for some reason I can't fathom, almost no one works on this.

 

[Physics Monkey]

@physicsmonkey: Which parts are the propaganda? I may be wrong but it seems that you are biased and searching for propaganda. For example figure 1.3 seems innocent to me. May be you are irritated because you assume that the last picture, the quantum gravity, is supposed to be LQG. But I don't think that's what they claim. It is meant to be the final and complete theory of quantum gravity, it may very well be string theory.

I'm certainly biased, no doubt about that. However, I think you have to really bend over backwards to not see Fig. 1.3 as anti-string and anti-AdS/CFT propaganda. The crazy color picture even has little triangles in it and the surrounding discussion makes it clear, in my opinion, that Rovelli-Vidotto are claiming to be genuine quantum gravity theorists unlike the rest of us poor post-maldacenians :)

My claim is simply that such statements are unnecessary and seriously detract from my enjoyment of the book.

 

[atyy]

Undoubtedly the nicest examples come from AdS/CFT. There the entropy of Schwarzschild black holes and Kerr black holes are precisely calculated and understood microscopically in the CFT. Fine, the asymptotics aren't right, but these results are clearly triumphs for quantum gravity.

What are the references for these?

 

[martinbn]

You might be right. I may be too naive or uninformed to see the figure that way, but then I guess I am lucky because I am reading the book and so far no bad taste in my mouth. :)

 

[marcus]

Hi Atyy,
thanks for including the reference to Bianchi's 4-page paper. It's clear from the context, pages 219-220, that they are talking specifically about that paper and the derivation (from LQG first principles) there in that paper.

They are not talking about other attempts within Lqg to derive BH entropy.

I think that Rovelli Vidotto's statement is probably correct. That is, that Bianchi has given a valid derivation in that paper, of the S=A/4 formula. Now whether Hawking's formula is correct is another matter. It could be wrong. But Bianchi has derived this formula, and that is what they are saying. It is made explicitly clear from context that they are discussing Bianchi's derivation.

...
"The entropy can be computed from first principles, including the famous 1/4 Hawking factor, it is finite and the calculation can be done for realistic black holes, such as a Schwartzschild or Kerr hole. This is beautiful achievement of the theory which for the moment is not matched by any of the other tentative quantum theories of gravity." http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf (p220)

Here, for reference, is "Bianchi 2012b" http://arxiv.org/abs/1211.0522.

I hope we can all agree that this statement of Rovelli-Vidotto is just false.

No, I don't agree. I think Bianchi's derivation is OK. But it doesn't matter, you are very welcome to think it doesn't work, or whatever suits. Very glad to see all the interest in this new textbook! Thanks everybody for all the comments!

 

[Mordred]

I'm not positive if these articles have been posted, they are ones I've been studying.

http://arxiv.org/abs/1201.4598 "Introduction to Loop Quantum Cosmology by Abhay Ashtekar
http://arxiv.org/abs/1302.3833 Loop quantum Cosmology by Ivan Agullo, Alejandro Corichi
http://arxiv.org/abs/1108.0893 "Loop Quantum cosmology: a status report Abhay Ashtekar, Parampreet Singh.

by the way I like your article: Space_cadet

 

[Physics Monkey]

No, I don't agree. I think Bianchi's derivation is OK. But it doesn't matter, you are very welcome to think it doesn't work, or whatever suits. Very glad to see all the interest in this new textbook! Thanks everybody for all the comments!

I hate to get into this again, but fine:

There are a number of nice points in Bianchi's paper http://arxiv.org/abs/1204.5122, but I cannot agree that it provides a derivation of S=A/4.

For starters, there is no actual definition of S ever given in the paper until the end where a comment about spin foams is made. However, what the Euclidean spin form partition function he writes has to do with earlier manipulations is not clear.

A derivation should have a definition of S and then an argument to show that S=A/4.

The sudden appearance of Clausius from nowhere, the assumption that the horizon has a temperature because of an elementary transition rate calculation, and so on means that we cannot regard this paper as a derivation. The coupling V between the thermometer and the geometry is ad hoc. The fact that the spin foam area states are not gauge invariant is not obviously addressed. I could go on ...

Many of these points were raised by me and others years ago when we first discussed this paper on Physics Forums. And again, the paper has nice ideas and results, all I'm claiming is that it cannot be viewed as a full derivation of S=A/4.

We can compare all this with, say, the calculation of the BTZ black hole entropy in AdS/CFT. We say that the BH entropy is defined to be the thermal entropy of the quantum state of the CFT. One derives that the AdS radius is proportional to the central charge in Planck units. Then one evaluates the area of the black hole horizon and finds that it is precisely A=4S. One is also free to study the dynamics of matter in the usual ways to see that matter can experience thermal effects. And although this in the context of AdS/CFT, Bianchi says nothing about the asymptotics, hence we're free to transplant Bianchi to AdS and compare. All I can say is that I know which set of ideas looks more like a derivation to me.

 

[Demystifier]

Atyy, your examples demonstrate that the book should be read critically. There are claims in it which are controversial. But still, I find the book good because I find many illuminating explanations in it. Rovelli certainly has a talent to explain complicated technical stuff in a relatively simple and intuitve way, and that's what I like about this book.

 

[atyy]

I think one of the most interesting sections of the book is 2.4.2 on the boundary formalism and observables in quantum gravity, and its application on p241 & 242. I have never understood this properly, and would love to know whether others think it makes sense. What I don't understand about it is that it seems that the observer is outside the quantum region of spacetime, and so presumably spacetime is classical where he is?

 

[Demystifier]

I think one of the most interesting sections of the book is 2.4.2 on the boundary formalism and observables in quantum gravity, and its application on p241 & 242.

Yes, that's one of my favored parts too.

I have never understood this properly, and would love to know whether others think it makes sense.

I think it does.

What I don't understand about it is that it seems that the observer is outside the quantum region of spacetime, and so presumably spacetime is classical where he is?

I think one cannot say where and when spacetime becomes classical before studying measurement in terms of decoherence, which Rovelli doesn't. An important part of it are (almost classical) coherent states which he does study, but what is missing is an explanation how the wave function of the universe becomes decohered in the basis of coherent states.

 

[marcus]

I think one of the most interesting sections of the book is 2.4.2 on the boundary formalism and observables in quantum gravity, and its application on p241 & 242. I have never understood this properly, and would love to know whether others think it makes sense. What I don't understand about it is that it seems that the observer is outside the quantum region of spacetime, and so presumably spacetime is classical where he is?

Yes, that's one of my favored parts too.I think it does.I think one cannot say where and when spacetime becomes classical before studying measurement in terms of decoherence, which Rovelli doesn't. An important part of it are (almost classical) coherent states which he does study, but what is missing is an explanation how the wave function of the universe becomes decohered in the basis of coherent states.

I think it's similar to the fact that a map does not need to contain an accurate picture itself. And the idea there is no super observer or god's eye view. So the boundary formalism DOES NOT CONTAIN a "classical" rest-of-universe. There is the process being studied, which is defined as occurring in a bounded spacetime region. And there is the boundary. And nothing more. But if I learn that there is somebody else observing the process, I can redraw the boundary so that the region now includes THEM. Other observers are quantum and modeled like everything else. I'm the USER of the boundary formalism, not something formalized in it

So the boundary formalism does not need to contain any "classical" spacetime region.

I could be wrong, that is just how I interpret it. For more reliable clarification, people might google "relational cosmology rovelli" and get the talk Rovelli gave in June 2013 at Oxford conference on Cosmology and Quantum Foundations


If I remember right the first 15 or 16 minutes provides a kind of philosophical argument why there is no "wave function of the universe"

 

[atyy]

marcus, so if I am using this boundary formalism, am I inside the boundary or outside? I thought I'd be outside, since everything inside the boundary is quantum, but I the observer am not quantum.

 

[marcus]

marcus, so if I am using this boundary formalism, am I inside the boundary or outside? I thought I'd be outside, since everything inside the boundary is quantum, but I the observer am not quantum.

I would say that you, as user, are not in the spacetime region being studied, Atyy. If you consider yourself quantum, then you are quantum. But you are outside the process being studied and modeled.

Officially, "boundary formalism" is the work of Robert Oeckl. I think he is currently at Erlangen. Google him if you want and read one of his expositions of "boundary formalism". My impression is that the user is not represented in the formalism. So the user of the theory is neither "quantum" or "classical".

I could be wrong of course. Maybe Oeckl talks about a classical observer as part of his formalism.
Would you like to check? Or shall I?

 

[marcus]

I think that it is probably wrong to think of the world divided between "quantum" and "classical". this false dichotomy is a relic of the Copenhagen interpretation. There is no need to think that everything is this or that.
But I don't speak for these various experts.

My impression is that Rovelli conceives of the world composed of quantum processes separated by boundaries which are to some extent MOVABLE. And which interact (exchange info) across the boundaries.

there is nothing "classical" in such a world. However one does not try to study the whole shebang all at once. One defines a finite process in a bounded spacetime region. there are interaction "facts" or events on the boundary allowing one to study that process and compute enclosure amplitudes.

One COULD perhaps choose to study one's instruments, one's hands, one's eyes as themselves quantum processes. In principle there is some flexibility as to where one puts the boundaries. As I recall at the Oxford conference there was some talk about being faithful to ordinary scientific practice---what's customary lab practice.

There are probably some distinctions between the relational view and Oeckl's boundary formalism as he presents it, but they're certainly closely related.

The new introductory textbook that Demystifier linked to is a really good source on LQG, as he says! I'll repeat the link in case anyone is new to the thread.
http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf
But I also can't recommend too highly this video of the talk Rovelli gave at Oxford last June
on "Cosmology and Quantum Theory: the Relational View"

 

[atyy]

I think Rovelli in his earlier review http://relativity.livingreviews.org/Articles/lrr-2008-5/fulltext.html section 5.4 says to pick your favourite interpretation. As a Bohmian, my favourite interpretation is Copenhagen, so I should be able to use a classical observer, at least according to Rovelli in 2008.

Also, in Oeckl's http://arxiv.org/abs/gr-qc/0312081 (second bullet point at top of p3) he says it is not necessary to endorse a wave function of the universe, which is very much a Copenhagen point of view.

So presumably, if I am classical, the spacetime around me is classical, and a physically relevant boundary state should be one that is strongly peaked on a classical geometry?

 

[marcus]

Also, in Oeckl's http://arxiv.org/abs/gr-qc/0312081 (second bullet point at top of p3) he says it is not necessary to endorse a wave function of the universe, which is very much a Copenhagen point of view…

If anyone wants an up-to-date presentation of GBF by Oeckl, they should look at
http://arxiv.org/abs/1212.5571
This paper gives the axioms as of 2013 when the paper was published in Foundations of Physics. and also, in an appendix at the end, gives OLD axioms as of, say 2003, so that readers can compare the current version with the one of ten years back.

In neither set of axioms is there mention of a "classical observer". Neither the old or new formulation by Oeckl have much to do with "Copenhagen point of view" at least that I can see.

I think Atyy that you may just be determined to force your own interpretation on this stuff and there cannot be much to discuss for now

As for "endorsing a wave function of the universe", the first 15 minutes of Rovelli's Oxford talk explained his non-endorsement of such, but his POV was not Copenhagen. So what you say does not follow. It simply is not true that non-endorsement of "wave function of the universe" implies a "Copenhagen point of view".

 

[Demystifier]

As a Bohmian, my favourite interpretation is Copenhagen, ...

 

[Physics Monkey]

As a Bohmian, my favourite interpretation is Copenhagen,

Seconded! I'm quite curious what you mean, atyy.

 

[Demystifier]

Marcus and atyy, I don't think that the boundary formalism by itself prefers any particular interpretation of quantum mechanics (Copenhagen, relational, many-world, Bohmian, or whatever), even if the creators or users of this formalism do have some interpretational preferences. But whatever is the someone's preferred interpretation, I think the quantum measurement cannot be correctly understood without taking into account the phenomenon of decoherence.

 

[Demystifier]

Seconded! I'm quite curious what you mean, atyy.

Maybe he likes my solipsistic interpretation
http://lanl.arxiv.org/abs/1112.2034
the only interpretation I am aware that tries to reconcile Copenhagen with Bohm.

 

[marcus]

I don't think that the boundary formalism by itself prefers any particular interpretation of quantum mechanics (Copenhagen, relational, many-world, Bohmian, or whatever),...

I agree.
To say the obvious, GBF is a formalism and RQM is an interpretation. As far as I can see, GBF does not entail a preference for RQM.

It's a different question to ask if RQM entails a leaning towards some version of boundary formalism (not necessarily in the exact manner of Oeckl). I don't know that we are discussing that question, or have any need to in this thread, given that the topic is "good introductions to LQG".

But it interested me that the Oxford phil. of physics people invited Rovelli to present the Relational view at that Cosmology and Quantum Foundations conference last year, and in his first lecture he said he considers himself a Realist. I think Rovelli is in the course of defining or "staking out" a type of *Quantum Realism* (which is different from what was called strict Einstein realism in the 2007 Relational EPR paper with Smerlak).
I think when you put that type of Realism together with the Relational approach to quantum spacetime it becomes almost inevitable to focus on the basic entity which is the distinct process occurring in a definite region, interacting with other contiguous processes, similarly defined.

As I said, this is not necessarily germane to this thread and there's no special need to pursue it here, but I don't see how this particular nexus of ideas (Realism, Relationalism, quantum spacetime) can fail to have a "preference" (as you say) for separate interacting processes occurring in bounded regions.

 

[atyy]

Seconded! I'm quite curious what you mean, atyy.

I just mean that since Copenhagen is emergent from Bohmian mechanics, I can just use Copenhagen, since it is more user friendly.

I think it is analogous to someone believing that AdS/CFT can emerge from lattice models.

Maybe they are even related, since Bohmian mechanics is more natural in a non-relativistic environment.

So if: quantum equilibrium -> BM -> lattice models -> AdS/CFT -> quantum gravity in AdS.

Could it be: quantum non-equilibrium -> quantum gravity in non-AdS ?

 

[julian]

Rovelli and Vidotto have written a new introductory textbook on loop quantum gravity:
https://www.amazon.com/dp/1107069629/?tag=pfamazon01-20

It is not yet officially published, but a draft can be freely (and legally) downloaded from
http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf

I have started reading it, so I can tell its excellent.

A lot of introductory material on loop quantum gravity can also be found at the wikipedia page
http://en.wikipedia.org/wiki/Loop_quantum_gravity
where I have found the link to the draft above.

Thanks, I wrote up most of the wiki page on LQG - and most of it is still there - nobody has changed much - go figure. Not happy with it, incomplete, probably mistakes. Also related topics (to which I put links) incomplete. It covered mostly canonical LQG, not so much covariant LQG. Although have also started reading the Rovelli/Vidotto book and learning/reminding myself more about it (and reading papers about the connection between them - in particular a paper on the 3D curvature operator and references therein). Perhaps a new wiki page should be devoted to "Covariant Loop quantum gravity"?

Been looking at their book, excellent, very readable which they should be congratulated on - but also the feeling that this may be at the expense of presenting a picture of things that are maybe a bit too hunkydory. But then in the last exercise of the book they say "Show that the theory defined in this book is fully consistent; if it is not, correct it"...

There are certainly details left out...

Demystifier also said

"I would even go that far to propose that someone who never says anything silly, probably has nothing important to say at all." To which I say Thomas Thiemann.

People on this forum are capable of not just talking about this stuff but also writing articles for wiki for LQG related stuff...I know Marcus has much to contribute to "time" in quantum gravity and some other Rovelli type stuff I think.

I also had a go at wiki pages for

"Hamiltonian constraint" needs work.

"Hamiltonian constraint of LQG" - I'm missing stuff on matter coupling there and inclusion of their qunatization.

"Tetradic Palantini action"

"Self-dual Palatini action" - important proofs of stuff I could not find in any text and only came across in some quasi-paper, I have written up on wiki and checked it is correct.

Put up stuff on the Ashtekar variables page.

Spin connection.

Mean to include the derivation of Ashtekar's formalism from the Self-dual action.


THERE is a lack of any wiki pages on:

Dirac observables!

Problem of time.

Complete, partial observables.

Mandelstam identities to do with trace of SU(2)/SL(2,C) matrices generalized for LQG purposes (Mandelstam was quite big on identities in general and there are a few different identities with his name).

The loop representation.

Reduced phase space quantization.

Evolving constants of motion.

Consistent discretetizations.

momentum representation of QM! Erh? why not done yet?

Just to mention some.

 

[Physics Monkey]

I just mean that since Copenhagen is emergent from Bohmian mechanics, I can just use Copenhagen, since it is more user friendly.

I think it is analogous to someone believing that AdS/CFT can emerge from lattice models.

Maybe they are even related, since Bohmian mechanics is more natural in a non-relativistic environment.

So if: quantum equilibrium -> BM -> lattice models -> AdS/CFT -> quantum gravity in AdS.

Could it be: quantum non-equilibrium -> quantum gravity in non-AdS ?

I don't understand the first two implications between quantum equilibrium, BM, and lattice models.

(May I ask, why are you a Bohmian?)

However, I do think quantum non-equilibrium should have something to do with non-ads gravity, e.g. dynamical loss of entanglement could represent some kind of singularity.

 

[Demystifier]

So if: quantum equilibrium -> BM -> lattice models -> AdS/CFT -> quantum gravity in AdS.

Can you explain
BM -> lattice models
and
lattice models -> AdS/CFT?

 

[atyy]

I don't understand the first two implications between quantum equilibrium, BM, and lattice models.

(May I ask, why are you a Bohmian?)

However, I do think quantum non-equilibrium should have something to do with non-ads gravity, e.g. dynamical loss of entanglement could represent some kind of singularity.

I don't think we mean the same thing by quantum equilibrium. In BM, "quantum equilibrium" is a distribution of initial conditions. So BM is quantum equilibrium and deterministic dynamics of hidden variables. I was thinking that if AdS/CFT has a Bohmian version, then a generalization would be keeping the deterministic dynamics, but with initial conditions such that QM fails. I think QM still holds in what you mean by quantum nonequilibrium.

The relationship between BM and lattice models is that lattice models are nonrelativistic, with relativity being emergent. Since BM is more natural with nonrelativistic QM, it seems to me more natural from the BM point of view for relativity to be emergent.

I'm not really a Bohmian since I don't know if it's true. But since it allows a naive conception of reality, it accords with my intuition. Also it assuumes that QM fails at some level, so it shares with Copenhagen the possibility that QM is only an effective theory.

 

[atyy]

Can you explain
BM -> lattice models
and
lattice models -> AdS/CFT?

The idea is that with BM, it is more natural if relativity is not fundamental, but is instead emergent.

So I'm hoping there is a lattice model from which the relativistic supersymmetric CFT of AdS/CFT is emergent.

 

[Demystifier]

The idea is that with BM, it is more natural if relativity is not fundamental, but is instead emergent.

So I'm hoping there is a lattice model from which the relativistic supersymmetric CFT of AdS/CFT is emergent.

Makes sense to me.

 

[marcus]

Rovelli and Vidotto have written a new introductory textbook on loop quantum gravity:
https://www.amazon.com/dp/1107069629/?tag=pfamazon01-20

It is not yet officially published, but a draft can be freely (and legally) downloaded from
http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf

I have started reading it, so I can tell its excellent.

A lot of introductory material on loop quantum gravity can also be found at the wikipedia page
http://en.wikipedia.org/wiki/Loop_quantum_gravity
where I have found the link to the draft above.

Atyy, your examples demonstrate that the book should be read critically. There are claims in it which are controversial. But still, I find the book good because I find many illuminating explanations in it. Rovelli certainly has a talent to explain complicated technical stuff in a relatively simple and intuitve way, and that's what I like about this book.

Thanks again for the links! I didn't realize that a draft of the book "Covariant LQG: an Elementary Introduction" was online, free to download. I also think it is an excellent book.

Rovelli just gave a 5 lecture "minicourse" introduction with the same title "Covariant Loop Quantum Gravity" at the June 2014 SIGRAV school at a villa on Lago di Como. I understand the course went well! Students had the new introductory textbook available, and I think the course was based in part on portions of the text.
The other six lecturers at the June school were Abhay Ashtekar, John Barrett, Eugenio Bianchi, Alain Connes, Laurent Freidel, and Roberto Percacci. Typically each gave a 5 lecture mini course. Here are the topics and brief course outlines:
http://www.centrovolta.it/sigrav2014/Schedule.pdf

The brief outline for Rovelli's course at the villa was:
COVARIANT LOOP QUANTUM GRAVITY
1. Quanta of space
2. The quantum states of the geometry
3. Dynamics
4. Measurements and predictions in quantum gravity
5. Planck stars

I was interested to see that part of the course was devoted to the Planck star model of black holes, and that phenomenology (measurements and predictions) was covered. The subject of CLQG is not purely theoretical. Here is the website for the 2014 SIGRAV school
http://www.centrovolta.it/sigrav2014/

 

[marcus]

I've got a copy of the book and have been skipping around reading different chapters. It's great. It reminds me of a "Guide Bleu" travel guide to various lively research areas in quantum gravity, and cosmology. Written at the most elementary possible level, so as to make the different topics accessible.

Here's the link that Demystifier gave (see post #84), to the free online (earlier draft) version of the book, I just checked and it still works. People might want to download (save as PDF on your desktop for convenient reference).
http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf
I'll get the amazon link too:
https://www.amazon.com/dp/1107069629/?tag=pfamazon01-20
It is an updated version of the one Demystifier gave.

 

[marcus]

The book is loaded with pictures/diagrams which are really useful in understanding the material. At the moment my favorite sections of the book are Chapters 10 and 11. Two short chapters: one on Black Holes and one on Cosmology. They explain stuff that I recognize from recent research papers, but seen from a different angle and in a broader context making it easier to understand.

 

[julian]

Just ordered mine but got it from:

http://www.amazon.com/Covariant-Loop-Quantum-Gravity-Introduction/dp/1107069629/ref=sr_1_1?s=books&ie=UTF8&qid=1419776044&sr=1-1&keywords=loop quantum gravity

and bought it in pounds. :) "Very British".