Diff invariance allows loop variables why?

In summary, Rovelli's explanation of why loop quantum gravity is UV-finite is based on the theory being diffeomorphism invariant. This can be seen in his 2008 LQG overview and survey talks, where he discusses the topic in a nontechnical style to make it more accessible to nonspecialists. The UV finiteness also extends to matter fields defined on the LQG spin network, as explained in his book and early draft versions. The proof for this can be found on page 281 of the book, with additional intuitive explanations provided by Rovelli.
  • #1
luxxio
44
0
I don't understand why a diffeomorphism invariance allows the extention of the loops variables in the continuum limit. Can someone give me some detailed reference?
 
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  • #2
luxxio said:
I don't understand why a diffeomorphism invariance allows the extention of the loops variables in the continuum limit. Can someone give me some detailed reference?

I wasn't sure what you were asking, but it could be you would like a reference for why loop quantum gravity is UV-finite. This is basically a consequence of the theory being diffeomorphism invariant.

Rovelli explained that to an audience of several hundred string theorists at the Strings 2008 meeting at CERN last summer. I will get the links to the PDF and the video. Actually the links are here
https://www.physicsforums.com/showpost.php?p=2040943&postcount=143
If you look down through that post you will come to this block of links

Rovelli's 2008 LQG overview:
http://relativity.livingreviews.org/Articles/lrr-2008-5/

2008 LQG survey talk to the Strings '08 conference, video and slides
http://cdsweb.cern.ch/record/1121957?ln=en
Slides (UV finiteness discussed starting at slide #22 or 23):
http://indico.cern.ch/getFile.py/access?contribId=30&resId=0&materialId=slides&confId=21917

2008 LQG survey talk to Loops '08, general, slides, audio
http://www.maths.nottingham.ac.uk/qg/wiki/index.php/QGsquared-slides
http://www.maths.nottingham.ac.uk/qg/wiki/images/3/3f/RovelliCarlo1214510081.pdf .

Others of these might be useful to you as well as the LQG survey talk given to the strings conference last August. In talking to the String experts, Rovelli used a somewhat intuitive nontechnical style, because talking to nonspecialists, this I think makes the talk especially useful. Also one can see what points the audience found hard to understand and what they needed to ask questions about.

I still am not sure about your question, so you could REPEAT the question and give a little more context. Are you talking about LQG spin networks?
Or do I misunderstand and you are asking about something else entirely! Are you talking about UV finiteness? Clarify, if you want to.
In the video (which is 43 minutes long) the discussion of UV finiteness starts around 17:00 minutes, when he comes to the section called Dynamics. Until today I could always get the video+audio. Today I could not get the full video right away, hope you have better luck. I had to pause it and wait for it to download a couple of minutes and then unpause it. It is an excellent talk.
 
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  • #3
i was referring exactly to that talk for string theorists. In presenting loop variables, rovelli said that loop variables borns in lattice theories and in the general cases you can't extend loop variables in the continuum space limit but for diff invariant theories.
why?
 
  • #4
luxxio said:
i was referring exactly to that talk for string theorists. In presenting loop variables, rovelli said that loop variables borns in lattice theories and in the general cases you can't extend loop variables in the continuum space limit but for diff invariant theories.
why?

There is a nice short formal proof of finiteness on page 281 of Rovelli's book.
(I have the hardcover first edition). He also gives some intuitive explanation to help understand why the mathematical proof works, down at the bottom of page 281.

As he mentions on the top of page 277, the UV finiteness extends to matter fields defined on the LQG spin network, and he discusses this formally starting on page 289, section 7.3 "Matter: dynamics and finiteness"

You asked for some references to things to read. I don't want you to have to go and buy Rovelli's book! :smile: What we need to do is find out what page 281 corresponds to in the free-online early draft version.

The section we have to look for is called 7.1.1 "Finiteness". I think if we go back to the early draft version the page number will be different but probably the section number, 7.1.1, will be the same. At least the chapter order will be the same, so look in chapter 7.

To find the free downloadable 2003 draft version, google "Rovelli"
which will give you
http://www.cpt.univ-mrs.fr/~rovelli/
and on that page you will find a link to the free copy.
Oh no! I just checked, and the link to the free copy has gone away! This presents a problem.
=====ADDED LATER=======
Whew! The book is still available free, here:
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
And the explanation you need is on page 202 (of the old draft version) in section 7.1.1
Check it out! He is a clear writer who puts in a lot of intuition.
 
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  • #5
marcus said:
There is a nice short formal proof of finiteness on page 281 of Rovelli's book.
(I have the hardcover first edition). He also gives some intuitive explanation to help understand why the mathematical proof works, down at the bottom of page 281.

As he mentions on the top of page 277, the UV finiteness extends to matter fields defined on the LQG spin network, and he discusses this formally starting on page 289, section 7.3 "Matter: dynamics and finiteness"

You asked for some references to things to read. I don't want you to have to go and buy Rovelli's book! :smile: What we need to do is find out what page 281 corresponds to in the free-online early draft version.

The section we have to look for is called 7.1.1 "Finiteness". I think if we go back to the early draft version the page number will be different but probably the section number, 7.1.1, will be the same. At least the chapter order will be the same, so look in chapter 7.

To find the free downloadable 2003 draft version, google "Rovelli"
which will give you
http://www.cpt.univ-mrs.fr/~rovelli/
and on that page you will find a link to the free copy.
Oh no! I just checked, and the link to the free copy has gone away! This presents a problem.
=====ADDED LATER=======
Whew! The book is still available free, here:
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
And the explanation you need is on page 202 (of the old draft version) in section 7.1.1
Check it out! He is a clear writer who puts in a lot of intuition.

thank you very much.
 

1. What is diff invariance and how does it relate to loop variables?

Diff invariance is a mathematical concept that refers to the property of an equation or function remaining unchanged under a change of variables. In the context of loops, this means that the loop will still perform the same actions and produce the same output regardless of the specific values of the variables involved. This is important because it allows for more flexibility and generality in the use of loops.

2. Why is diff invariance important in scientific research?

Diff invariance is important in scientific research because it allows for the creation of more robust and generalizable models and theories. By ensuring that the equations or functions used in a study are diff invariant, researchers can be confident that their results will hold true even if the variables are changed or tweaked in some way.

3. Can you provide an example of how diff invariance applies to loop variables?

One example of diff invariance in loop variables is in the calculation of the sum of a series of numbers. The specific values of the numbers may change, but as long as the loop equation remains unchanged, the result will always be the same. This makes the calculation more efficient and adaptable to different scenarios.

4. How does diff invariance relate to the concept of symmetry?

Diff invariance and symmetry are closely related concepts. Symmetry refers to the property of an object or system remaining unchanged under a transformation or change. Diff invariance is a specific type of symmetry, where the transformation is a change of variables. In both cases, the key is that the essential properties of the object or system remain the same despite the change.

5. Are there any limitations to diff invariance in loop variables?

While diff invariance is a useful concept, it does have some limitations. For example, it may not hold true for all types of functions or equations. Additionally, there may be cases where a specific variable transformation could significantly alter the results, making diff invariance less applicable. As with any mathematical concept, it is important to consider the specific context and potential limitations when applying diff invariance to loop variables.

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