- #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?
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?
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?
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! 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.
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.
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.
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.
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.
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.