Is the Lagrangian for LQG similar to that of Yang-Mills theories?

[jfy4]

Hi,

There is clearly an analogous relationship between Yang-Mills theories and LQG, for example the covariant derivative, the field strength tensor, the field and its conjugate momentum etc...

I am wondering how far this analogy extends. Specifically, is the Lagrangian for LQG the following?

$$\mathcal{L}=-\frac{1}{4}F^{\,i}_{ab}F^{\,ab}_{i}$$

where

$$F^{\,i}_{ab}=\partial_a A_{b}^{i}-\partial_b A_{a}^{i}+\epsilon^{i}{}_{jk}A_{a}^{j}A_{b}^{k}$$

 

[atyy]

The latest models are based on the Holst action.
http://arxiv.org/abs/0909.0939v4

A different take is to try to link the formalism to string theory.
http://arxiv.org/abs/0907.2994
http://arxiv.org/abs/0905.1317

 

[marcus]

Hi,

There is clearly an analogous relationship between Yang-Mills theories and LQG, for example the covariant derivative, the field strength tensor, the field and its conjugate momentum etc...

I am wondering how far this analogy extends. Specifically, is the Lagrangian for LQG the following?

$$\mathcal{L}=-\frac{1}{4}F^{\,i}_{ab}F^{\,ab}_{i}$$

where

$$F^{\,i}_{ab}=\partial_a A_{b}^{i}-\partial_b A_{a}^{i}+\epsilon^{i}{}_{jk}A_{a}^{j}A_{b}^{k}$$

If you are entering a PhD program at U. of Iowa (with an Astronomy undergrad background) then the kind of introduction you want is no more specialized or advanced than
http://arxiv.org/abs/1102.3660.

An expanded revised version of this was just posted today on arxiv. That link will get you the latest version.

It is a short up-to-date 34-page "textbook" covering the most widely researched form of LQG written by a lead researcher. (Most of the active young people in the community are his PhD students or have worked in his group as postdocs.)

Regarding your question, see Section V Derivations. See subsection A Dynamics. It starts on page 24.

This year there was a 2-week "winter school" for young researchers interested in getting into LQG. It was held at a ski-resort in south Poland, called Zakopane. This introduction+survey of current LQG status and open problems is called the Zakopane lectures.

Atyy recommended you look at a highly technical 2009 paper by Lewandowski and two of his students. That paper gives no overview and is focused on a highly technical paper. I would not recommend it to a person beginning a PhD program unless I did not like them and wanted to throw them off the track But obviously people's ideas differ. Atyy knows a lot and obviously wants to help you get an introduction to LQG--I just don't understand recommending that Lewandowski paper! It doesn't even have a list of the open problems in the field and what progress has been made on them in the past two years! Or historical overview, or a sketch of math prerequites, or some suggested homework exercises, or any of that introductory stuff.

There is also an undergrad LQG book by Gambini and Pullin coming out this year if anyone is interested. The book might be an easier slowerpaced intro, but the 34page Rovelli paper is free and immediately available.

Also google "Loops Madrid" and go to the 2011 conference, select the scientific program, scroll down to Rovelli's wednesday morning talk, and download it. I find I can play the video of the talk with VLC (I have a Mac). With windows you may find it easier. The video talk is a valuable adjunct to the new version of the 34-page paper called Zakopane lectures.

 

[jfy4]

If you are entering a PhD program at U. of Iowa (with an Astronomy undergrad background) then the kind of introduction you want is no more specialized or advanced than
http://arxiv.org/abs/1002.3660.

An expanded revised version of this was just posted today on arxiv. That link will get you the latest version.

It is a short up-to-date 34-page "textbook" covering the most widely researched form of LQG written by a lead researcher. (Most of the active young people in the community are his PhD students or have worked in his group as postdocs.)

Regarding your question, see Section V Derivations. See subsection A Dynamics. It starts on page 24.

This year there was a 2-week "winter school" for young researchers interested in getting into LQG. It was held at a ski-resort in south Poland, called Zakopane. This introduction+survey of current LQG status and open problems is called the Zakopane lectures.

Atyy recommended you look at a highly technical 2009 paper by Lewandowski and two of his students. That paper gives no overview and is focused on a highly technical paper. I would not recommend it to a person beginning a PhD program unless I did not like them and wanted to throw them off the track But obviously people's ideas differ. Atyy knows a lot and obviously wants to help you get an introduction to LQG--I just don't understand recommending that Lewandowski paper! It doesn't even have a list of the open problems in the field and what progress has been made on them in the past two years! Or historical overview, or a sketch of math prerequites, or some suggested homework exercises, or any of that introductory stuff.

There is also an undergrad LQG book by Gambini and Pullin coming out this year if anyone is interested. The book might be an easier slowerpaced intro, but the 34page Rovelli paper is free and immediately available.

Also google "Loops Madrid" and go to the 2011 conference, select the scientific program, scroll down to Rovelli's wednesday morning talk, and download it. I find I can play the video of the talk with VLC (I have a Mac). With windows you may find it easier. The video talk is a valuable adjunct to the new version of the 34-page paper called Zakopane lectures.

Thank you very much, but are you sure you linked the right paper?

Also I got a B.S. in Physics too, I need to put that in...

 

[marcus]

Thank you very much, but are you sure you linked the right paper?

Also I got a B.S. in Physics too, I need to put that in...

OOOPS! I wrote 1002 instead of 1102. the correct link is http://arxiv.org/abs/1102.3660

I think that paper could work out for you! I'm glad you got a B.S. in Physics. You mentioned Astro so I was fairly certain you had almost the equivalent. But someone else could be reading who wants something more basic so I will mention the forthcoming undergrad text.

For someone who wants a REAL entry-level basic introduction here is the first undergrad text specifically on LQG
http://ukcatalogue.oup.com/product/9780199590759.do

The amazon page has an estimate of November 1, 2011
https://www.amazon.com/dp/0199590753/?tag=pfamazon01-20
Here's the amazon blurb:
"This book provides an accessible introduction to loop quantum gravity and some of its applications, at a level suitable for undergraduate students and others with only a minimal knowledge of college level physics. In particular it is not assumed that the reader is familiar with general relativity and only minimally familiar with quantum mechanics and Hamiltonian mechanics. Most chapters end with problems that elaborate on the text, and aid learning. Applications such as loop quantum cosmology, black hole entropy and spin foams are briefly covered. The text is ideally suited for an undergraduate course in the senior year of a physics major. It can also be used to introduce undergraduates to general relativity and quantum field theory as part of a 'special topics' type of course."

But for now, the Rovelli "Zakopane lectures on LQG" is the best intro.

 

[jfy4]

Thanks a lot Marcus,

you have been a great help over the past year and a half I have been on here in getting me looking at LQG papers.

 

[marcus]

You are welcome Jfy4! Congratulations on entering gradschool at U Iowa! One of my favorite astronomy teachers is at University of Northern Iowa, but I think that is different.
I see that advanced review copies are out, of the Gambini Pullin *A First Course in Loop Quantum Gravity*

Here is the TOC:

1: Why quantize gravity?
2: Special relativity and electromagnetism
3: Some elements of general relativity
4: Hamiltonian mechanics including constraints and fields
5: Yang-Mills theories
6: Quantum mechanics and elements of quantum field theories
7: General relativity in terms of Ashtekar's variables
8: Loop representation for general relativity
9: An application: loop quantum cosmology
10: Further developments
11: Open issues and controversies

They have gotten good reviews from two leaders in the field of Quantum Gravity---Rovelli and Kiefer both praise the book. Kiefer is not specifically a Loop guy. His research perspective covers all approaches to QG. He especially likes that G&P do not gloss over the open problems---possible limitations---of this approach.

 

[Demystifier]

Y
I see that advanced review copies are out, of the Gambini Pullin *A First Course in Loop Quantum Gravity*

Here is the TOC:

1: Why quantize gravity?
2: Special relativity and electromagnetism
3: Some elements of general relativity
4: Hamiltonian mechanics including constraints and fields
5: Yang-Mills theories
6: Quantum mechanics and elements of quantum field theories
7: General relativity in terms of Ashtekar's variables
8: Loop representation for general relativity
9: An application: loop quantum cosmology
10: Further developments
11: Open issues and controversies

Can it be downloaded from somewhere?

 

[marcus]

Can it be downloaded from somewhere?

Demy, thanks for asking! I don't know of any place it can be downloaded. Besides Oxford press presumably. Your credentials as a reviewer or possibly prospective teacher might be sufficient.

They have a reviews by Carlo Rovelli and by Claus Kiefer. I infer that Oxford press must have gotten advance reviewer material out to them (perhaps simply a final draft or proof version) so they could supply the brief reviews OUP is using for promotion.

All I have is the link I gave to the Oxford press main page about *A First Course in LQG*:
http://ukcatalogue.oup.com/product/9780199590759.do

That page has a prominent button that you can press to get the Table of Contents.

 

[atyy]

Atyy recommended you look at a highly technical 2009 paper by Lewandowski and two of his students. That paper gives no overview and is focused on a highly technical paper. I would not recommend it to a person beginning a PhD program unless I did not like them and wanted to throw them off the track But obviously people's ideas differ. Atyy knows a lot and obviously wants to help you get an introduction to LQG--I just don't understand recommending that Lewandowski paper! It doesn't even have a list of the open problems in the field and what progress has been made on them in the past two years! Or historical overview, or a sketch of math prerequites, or some suggested homework exercises, or any of that introductory stuff.

I don't know a lot, and certainly am not intending to help grad students with research (I'd point them to strings if I were;) I'm a biologist, so what I suggest is really just what the general public might be interested in:)

 

[Demystifier]

I'm a biologist

I never met a better physicist among biologists.

 

[marcus]

I never met a better physicist among biologists.

Nor have I (theoretical). Atyy said (I think I recall) he did his bio degree at MIT where he was also interested in theoretical physics---and took a course with XG Wen. Atyy please correct me if I misremember some detail and am wrong about this.

BTW I think there may be a lot of extremely good physicists among the biologists now because of career crossover. People get degrees in physics and then move over into postdocs in biology (e.g. biolphysics, cell biology, etc.) because of the interesting activity, new problems, opportunities etc. I can think of a couple of examples.

To get back on topic, Atyy is saying that what he recommends in the way of papers is stuff that would itnerest the "general public". I don't exactly agree. Atyy I think of you as a "virtually" a physicist---i.e. that your taste is more technical than that. The Lewandowski paper you offered is, in my view, highly technical, effectively lacking overview, and specialized in one corner of LQG---one approach that is not representative of most of the current research activity in the field.

Lewandowski's contribution is surely valuable and someday it could be successful! in which case it would expand and become a larger fraction of LQG---take in a larger fraction of the Loop community. I admire him and am glad he is doing what he is doing. He is persisting with embedded spin networks, which can even be knotted. It is worth pointing out the difference I think. Kind of distinguishes what he is doing (with his Warsaw group) from the larger scene.