# Loop quantum gravity

1. Aug 12, 2004

### marlon

hello,

can anyone give me some introductory notions of what loop quantum gravity (i hope it is in the right order) and what are the differences with string theory. I know my QFT and i have some introductory notions on strings, so i don't mind a deeply elaborated physical answer.

Feel free to indulge yourselves

thanks
marlon

2. Aug 12, 2004

### marcus

take a quick look at a recent survey by Lee Smolin called
"Invitation to Loop Quantum Gravity"
http://arxiv.org/hep-th/0408048 [Broken]

LQG and who might be considering the field
It has FAQ----questions physicists from other fields frequently ask

It has a list of open problems to work on. It has a listing of results proven so far.

It has a summary of experimental testing planned or in progress (still pretty tentative but some good possibilities coming up near term)

In acknowl. section Smolin thanks string-theorist John Schwarz for giving him the idea to write this type of survey article. And it is meant for publication in the annual reference series "Reviews of Modern Physics".

So this paper seems like a good match with your needs and interests.

I will post this as a start and if I think of other stuff will add it later.

Last edited by a moderator: May 1, 2017
3. Aug 12, 2004

### marlon

regards
marlon

4. Aug 13, 2004

### marcus

I'd appreciate having someone to discuss the paper with, if you feel like doing going over parts of it later after having a look.

regards from here too

(btw there are parts I dont get, but lets try to overlook my deficiencies and see if there are some we can tackle )

Last edited: Aug 13, 2004
5. Aug 13, 2004

### Mike2

I certainly like the four key observations that LQG is based on.

1) background independence,
2) duality and diffeomorphism invariance
3) gauge field theory,
4) topological field theory.

It give a sense of deriving it from first principles, or physics from pure mathematics, as opposed to string theory which seems to be a contrivance (more curve fitting fudge factors).
I don't like the concept of quantized space. Or perhaps I don't understand it. I suppose I could except that quantized space is a quantum mechanical superposition of non-quantized space. But I'm not sure that this is a correct interpretation.

6. Aug 14, 2004

### marlon

ok, ok, sounds very nice marcus

Give me a few days to read it thourougly...

regards
marlon

7. Aug 16, 2004

### marlon

ok,marcus, let's get started on the Smolin-article.

Just for the record and the good mutual understanding. I wish to discuss the content of page 8 : The connections in LQG are the gauge fields. I think this has to mean that them gauge fields are covariant right??? But isn't that always the case, just like in QFT ???

All possible metrics and connections form the state space. All possible configurations of the gauge fields form the configuration space. But what do we mean by the configuration of a gauge field? ??

In order to describe the dynamics of the gauge fields we perform parallel transport of these gauge fields along some Wilson loop and we study how these gauge fields change during the transport. Now we have a system to study fields independently of the metric, like in GTR. Is this vision right.

Finally I have some difficulty with imagining how excitations (these are the particles) are defined by Wilson loops acting on the vacuum. Is this an analogy to QFT where particles arise as excitations of the QCD-vacuum-state??

regards
marlon

8. Aug 16, 2004

### marlon

And if the gauge field is viewed as a connection over a principal G-bundle, we do the parallel transport of the identity around the loop which would give an element of the Lie group G.

Let's suppose the used identity is a vector. This G-bundle projects this vector onto the tangent-vector which is to be parallell transported. Doing the holonomy means doing the actual transport and looking how the "connection" between the original vektor and the tangent vektor changes. The result gives an element of the Lie group.

WHAT IS THIS ELEMENT ? WHAT DO WE KNOW FROM IT ???

regards
marlon

9. Aug 17, 2004

### marcus

hi marlon, I am eager to continue the discussion of LQG started by your question.

today I found a good page of Quantum Gravity links assembled by Seth Major, a physicist at Hamilton College (somewhere in NY state, I think)

It looks like he has assembled this page for his students and has included
some good explanatory and introductory material, as well as surveys.
Also the page is seemingly up-to-date, which can be a big advantage.

You asked also about the physical meaning of a holonomy---the path integral of a connection. I believe it depends very much on the path
and that holonomy on a path going around a loop intuitively tells information about the curvature inside the loop. Also my feeling is that
you are probably better than me about describing these things! Or anyway I am not the best person at PF to try to do it! So I would be happy if you would do some of the exposition. (Or some other knowledgeable person if they wish to step in)

I will make an attempt in the next post but please anyone feel free to augment or correct this preliminary effort

10. Aug 17, 2004

### marlon

Hi, marcus

just have been looking at this site of Seth A Major. It is the best.

There is a nice reference to an article that explains the origin of spin networks, some crazy stuff. Assigning lines and so on to the eigenvalues of the angular momentum operator is freeky yet geniously work out in there.

I can only say : thanx, thanx, thanx
ps : there is still a lot of work to be done in this research. Maybe we will earn our Nobel prize here (lol)

regards
marlon

ps : what do you do marcus, what are you working on, are you a student ???

11. Aug 17, 2004

### marcus

Smolin's "Invitation" http://arxiv.org/hep-th/0408048 [Broken]
is a truly wonderful paper---a landmark in being finally a really good
QG survey boiled down into few pages----FAQ, list of solved problems, list of unsolved problems, thumbnail sketch of experimental situation etc.

But to read the paper one needs to get over an initial barrier, which Marlon has pointed to. What is a connection?

why does this arise? Because at first in 1970s and even earlier when Wheeler and DeWitt (and others) did Quantum Gravity they naturally considered the configuration set to be all possible metrics and then
around 1986 people switched to using the set of all possible connections.

One must have a configuration set representing all possible geometries of the manifold. At first it was clear to everybody that it is the metric or distance function defined on the manifold which tells its geometry (its angles, areas, volumes, distances etc). so the set of all geometries must obvously be the set of all metrics. Then the quantum states were functions defined on the set of all metrics.

Quantum states are somewhat like probability densities defined on a set of possible configurations of the world. Quantum states must be defined on something. Shall it be the set of all possible metrics on our manifold?

they tried this in the 1970s and encountered serious difficulty. Around 1986
Ashtekar and (I think) Sen suggested using the set of connections instead.
(this account is dangerously oversimplified but I need to have some overall perspective)

Using as configurations the set of all connections seems to have worked better. So one must ask "What is a connection and how does it represent essential features of a geometry?"

One should, I believe, picture the manifold and imagine all the possible frames at every point. If one could interconnect a choice of frame at every point then one could say "If I take an infinitesimal step in some direction, how does the frame begin to twist around?"

An infinitesimal step in some direction corresponds to an element of the tangent (or the cotangent) space at that point. An infinitesimal rotation corresponds to an element of the Lie Algebra (one makes some arbitrary choices to equip oneself with a Lie Algebra, and basis elements, at every point and one must pay later by having to factor the arbitrariness out, but at least one gets some good out of it)

So a connection needs to be, at every point, simply a linear map from the tangent (or cotangent) space to the Lie Algebra.
Because of arbitrarily chosing basis elements one can even present this with actual numbers in matrix form.
To remember: it tells how the frame will infinitesimally rotate as one takes an infinitesimal step in some direction.

With my bad memory, I cannot remember if one defines this map on the tangent space or the cotangent space at the point! Perhaps either will work.
Perhaps it is best described as a Lie-algebra-valued differential form.

Well, I am now expected to go to the market to get food for company coming, and must continue this later. But first I just want to mention that
one of the nice things one can do with information about infinitesimal steps at the microscopic level is integrate over a macroscopic journey.

so once we have the connection we can consider traveling along any path
and seeing how the frames twist and turn around! We can break the path down into lots of microscopic steps and do the integral of the corresponding Lie algebra elements and get a Lie group element corresponding to the path from one point to some other.

In particular if we take a path around in a loop, then the frames will twist around and perhaps will be something different when we get back! So the loop tastes something about the geometry inherent in the connection.
Well now i have to go buy meat and vegetables and when I get back I hope my head will still be pointed in the same direction as now. this marketing is also something of a holonomy, but it must be done. til later.

Last edited by a moderator: May 1, 2017
12. Aug 17, 2004

Staff Emeritus
Nice description Marcus! Here is a little addendum. The Lie Algebra as you said provides the differentials of the group actions; if you can stretch your mind this far, think of the group (gauge group) as itself a curved manifold, and somewhere on it is a point representing its group identity, the group element that doesn't change anything. Then the group manifold has its own tangent space, in particular there's a tangent "fiber" at the identity and that tangent fiber is a vector space of those differentials, spanned by some basis vectors. The vector space is made into an algebra, the Lie Algebra of the group, by defining a bracket product, but we don't need to go there. What we do need is a theorem that if you exponentiate the Lie Algebra vectors you get the group elements, and this provides a mapping from the Lie Algebra to the Lie Group.

So now look at Smolin's expression (4):
$$T[\gamma,A] = Tr Pe^{\int_{\gamma}A}$$

where $$\gamma$$ is a loop and the P operator enforces pathwise ordering. We take the frame around the loop, and we INTEGRATE all the differential motions to come up with a total motion, and we EXPONENTIATE to get the group element that equals the final result. The group element is a matrix, and we take its trace. And that map from loops to numbers is the Wilson loop.

13. Aug 17, 2004

### marlon

So these connections are indeed some map from the tangent space to the Lie Algebra in which we have the rotations that the chosen frame undergoes while we perform parallel transport.

Once we went around the loop we see for axample that the original frame has turned 90 degrees. We look for the matrix-representation of a 90-degrees-rotation and we take its trace. Then we have a number that tells us that the frame (which in this case is the actual gauge-field, right) rotates 90 degrees after the loop is completed.

Still one question though : is it ok to say that the Lie-group contains the infinitesimal paths of which we construct the loop by integrating. And given some gauge-field, how do we construct the associated Lie-group ??? Isn't the Lie-group the group of transformations that leave the gauge-field invariant, like in QFT ??? I guess no because this has something to do with how the frame (here the gauge field) changes when following the loop.

regards
marlon

14. Aug 17, 2004

### marlon

selfAdjoint, do you mean by this that in the Lie Algebra , we find all the possible transformations in order to go from the original frame to the "rotated" frame after completing the loop? So in like in my previous post, this would be the 90-degree-rotation ???

A differential of the group action means : "how does a frame change when we do something with it? In this case, taking it along the loop " Is this correct ???

regards
Marlon

15. Aug 17, 2004

### sol2

Last edited: Aug 17, 2004
16. Aug 17, 2004

### marcus

Yes sol, IMHO, and most thoughtful of you to ask.
Personally I would like it very much if we could keep this one thread as a sort of "working LQG thread"----keep other topics to a minimum and focus on whatever is recent in that particular approach, and hopefully some nuts-and-bolts.

17. Aug 17, 2004

Staff Emeritus
From one physics textbook, I remember this explanation:
The transformations (elements) of the original group are fine for math, but in physics we need a form that goes with the differential equations and integrals of mechanics, hence the differentials of the group elements. So yes, if we transport the frame around the loop, adding up (integrating) the differential transformations we wind up with a group element; instead of all the business with the loop you could have just rotated the frame from its original position to its final one with a single 90o turn. That single element is what we recover by exponentiating.

One thing in your other post bothered me; you ask if the differentials form pieces of the loop. No, the loop is there in the manifold; the differentials are pieces ("seeds" maybe) of the group transformations that act on the manifold. As you move the frame along some "d(loop)" it experiences some "d(rotation)."

Last edited: Aug 17, 2004
18. Aug 17, 2004

### jeff

What, no more poetry?!

19. Aug 17, 2004