# Classical Loop Gravity?

Gold Member
Can classical gravity be represented in the loop representation, analogously to LQG?
If yes, is it equivalent to the standard general relativity?
Corresponding references would be wellcome.

marcus
Gold Member
Dearly Missed
Demystifier said:
Can classical gravity be represented in the loop representation, analogously to LQG?
If yes, is it equivalent to the standard general relativity?
Corresponding references would be welcome.

Hi Demy., I'm forgetful but I think you are a physicist with some published articles---you gave some arxiv links to your work in another thread. So you are apt to be pretty knowledgeable with a sophisticated understanding of this business.

So I expect you already know about Ashtekar's reformulation of classical Gen Rel in terms of a connection. Still, I would appreciate if you would confirm that just so I can try to avoid saying stuff that is already completely obvious to you

Gold Member
Yes, I am familiar with Ashtekar reformulation of classical gravity, say at the level of Rovelli's book, although I am not an expert. However, Ashtekar variables are local (just as the usual formulation of GR), while the loop variables are nonlocal. Therefore, formulation of classical gravity in terms of loops seems nontrivial to me. I also suspect that this could be a part of the reason why it is difficult to get the correct classical limit of LQG.

What do you mean by local/nonlocal?

I guess some people define the observables in LQG as "nonlocal" because of diffeomorphism invariance, opposed to QFT, which is locally dependent on a spacetime background. But at the same time, in classical GR you have nonlocal observables (vectorial quantitites), ie, those which you cannot have a consistent local definition, like mass, energy and momentum, because the spacetime manifold can be curved in an arbitrary way, and then those quantities assume different values when transported along different paths in spacetime. You can only have a consistent definition of them in some kind of asymptotically flat region.

There is also the http://en.wikipedia.org/wiki/Principle_of_locality" [Broken] ("distant objects cannot have direct influence on one another: an object is influenced directly only by its immediate surroundings"), which I guess this is not what you mean here, since, as far as I understand, LQG is supposed to be consistent with that principle.

However, to make matters worse, there is also the problem of "local realism", which is rejected in the Copenhagen interpretation of quantum mechanics, but I do not know what LQG makes of this, as far as I understand it incorporates a usual canonical quantization scheme.

So I guess I am somewhat confused about what is the technical definition of locality and nonlocality. Is there any paper that treats these issues in a pedagogical way?

Christine

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Gold Member
Yes, you seem to be a little confused because you mix several different notions of locality.

Here I mean the following. Classical GR deals with metric or connections, which are locally defined quantities (not necessary observables), defined on spacetime points. On the other hand LQG deals with loops, which are essentially integrals of connections along a closed curve. They are nonlocal in the sense that they are not defined on points.

Right, thanks!

But it is not that I was completely mixing those concepts (somewhat, perhaps). I was mostly confused on what sense you were using them.

Best wishes,
Christine

Gold Member
By the way, I never completely understood why experts for quantum gravity insist on formulation in terms of diffeomorphism invariant quantities (that they call "observables"). As classical gravity works fine without such a formulation, I do not see an a priori reason why quantum gravity could not work that way as well.

The main arguments are summarized in Smolin's paper (http://arxiv.org/abs/hep-th/0507235) [Broken]. But, indeed, what you mention is a fundamental question and one should face the known difficulties in those approaches, and think over why one should go into that route.

Perhaps, somewhat related to this context, you can find interesting arguments in Torre http://arxiv.org/abs/gr-qc/9306030, a paper cited by Baratin and Freidel, in their last paper.

Best wishes,
Christine

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Sorry for the multiple posts. Something wrong with my connection. The last one is the correct one, the others were deleted.

Thanks

Christine

Gold Member
I have seen the paper of Smolin, but I do not think that he directly explained why diffeomorphism invariance is more important in quantum theory than that in classical theory. But perhaps I missed it.

I do not know how far his arguments on that paper were only of qualitative nature. He does mention that one can view GR as a partial relational theory. I do not know his exact arguments on why one should push forward to a purely relational quantum gravity theory. Maybe it is a question of personal strategy, but I could be wrong. It's a long paper with several arguments, and I am unabled to analyse it right now . I hope others can write about the question raised by Demystifier.

Christine

Gold Member
If anybody here knows the answer, then it is probably marcus.

turbo
Gold Member
ccdantas said:
The main arguments are summarized in Smolin's paper (http://arxiv.org/abs/hep-th/0507235) [Broken]. But, indeed, what you mention is a fundamental question and one should face the known difficulties in those approaches, and think over why one should go into that route.
As luck would have it, I was going back over some papers in your archived blog this afternoon that I wanted to re-read and that was one of them. The feeling that I took away from that paper was that independence from a fixed background would allow a relational theory of quantum gravity to exploit the concept of locality - something necessary to allow a dynamical theory of gravity.

This strikes a chord in me because I feel strongly that Einstein was right when he said that gravitation and inertia are entirely local effects arising from matter's interaction with the space-time in which it is embedded. It is this concept that makes me take exception with Padmanabhan's view of the vacuum. He models it as an elastic solid a la Sakharov (and has for a number of years), but he denies that the vacuum can be modified in its variable properties by embedded matter, relegating the vacuum to the role of backdrop instead of player in gravitation.

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For classical Loop Gravity, have a look at page 28 of D. Oritis thesis gr-qc/0311066 and the references he mentions.

For the neccessity of diffeoinvariance, well classical GR get's out of needing this by the intuitive way in which the gauge symmetry is broken by reference matter systems, however, it is quite reasonabl to assume that our classical intuition of this breaking will need to be reformulated.

Still it's by no means universally accepted by QG experts that things have to be formulated in diffeo invariant ways.

marcus
Gold Member
Dearly Missed
Demystifier said:
If anybody here knows the answer, then it is probably marcus.

thanks for the kind words Demy
let me defer to F-H here (see above post)

Background independence always sounded conceptually interersting to me, given that before I ever thought about quantum gravity, I was interested in the problem of inertia (as mentioned by turbo-1), that is, as arising from some relational point of view. So it would be natural to think on how inertia arises in the context of quantum gravity. (Such a theory must offer a fundamental formulation of inertia). It is clear, however, that the difficulties are very deep (actually the difficulties seem to be deep in any current approach to quantum gravity, so I guess anyone approaching the field should of course invest their time and energy on those ideas that sound conceptually more satisfying or logical, or better yet: create their own routes of investigation).

For the moment, I'm going way back to the 60's by thinking over the paper by http://adsabs.harvard.edu/cgi-bin/n...pe=HTML&amp;format=&amp;high=455a5a8c9621037".

Christine

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turbo
Gold Member
Yes, relationally-emergent inertia implies a contiguous field and denies any Machian action-at-a-distance. This treatment of GR demands a background-independent view of the vacuum and allows the development of a dynamical model of gravitation at the quantum level. This is going to be uphill sledding though. :grumpy:

Dmistifyer, it is not the firs time that I read some similar argumentation about the classical variables of LQG, and I don´t see the poin in them, so it would be good go to the actual equations and see where is the errata.

I am using the equations such as they appear in the classical report paper of Thieman from October of 2002 on Arxiv:gr-qc/0210094. I once tried to read the original Astekhar paper and I recognize that it seems to follow diferent (long and obscure) paths, but I assume that in the end the results would be equivalente

Well, lets go with eqs. I dont describee the ADM formalism as I gues you are familiar with it. Well, the two basic variables of LQG are (see page 22 of the article):

$$A_\alpha ^j = \Gamma_\alpha ^j + \betha K_a_be^b_j$$

Where, as expected, A is the "gauge" field, $$\Gamma$$ is the spin connection and e is the 3-d vielbein (the "spatial" part of the folliation in the ADM formalism). All of them are clearly local quantities.

The only "suspectious" quantity could be the K. What is it?

The article doesn´t preciselly says it in that point, but if we go to page 19 we find that he uses K to denote the extrinsic curvature of the ADM mechanism so I guess it is just that, and so, once again, a local quantitie.

The other basic variable is:

$$E^\alpha_j=\sqrt(det(q))e^\alpha_j/ \beta$$

These is the "field" (in analogy with the electromagnetic tensor field) part of the gravity. q is the thre dimensional metric and, once egain, e is the vielbein asociated with that metric. All of them are local quantities.

The betha factor in both equations is the famous inmirizi parameter, wich, beeing a number is of course a local quantitie.

So, where are ther nonlocalities in the classical LQG variables?

Also I have readed an slighly diferent argumentation. That you can´t make these change of vraibles form ADM ones to AStekhar ones globally. I don´t know why and i don´t see an obvious reason in these definition to think so, but of course it can easilly be my fault.

These is about the classical part. Later in the LQG formalism these fields must become operators. And following the usual way in constructive field theory in order to make them well defined they are smeared. But that is not especific of LQG and it would apply equally to the constructive ("rigurous") version of the self interacting klein gordon field. And i never have seem a claim that K-G is a nonlocal theory ;).

And if we go further in the LQG canonical formalism we find that teh hilber space is defined in terms of holonomies (wilson loops) of the fields. Well, holonomies are nonlocal quantities undoubtly. But on one hand we have that Willson loops are an standard tecnique in latice QCD and i never have seem a claim that QCD is non local. Ad anyway, that would be the quantum part of the theory, not the classical.

Sorry if I am focousing in too elementary aspects, but i guess that the claim of nonlocalities were preciselly against these basic facts.

About the need of background invariance i would post somtime later if I find time and nobody else do it.

Hope that the Latex code would be o.k. because i have not time to revise it just now if it wouldn´t. If it is wrong any moderator can feel totally free to correct it if he whises.

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