# Covariant Loop Gravity and Livine's Thesis

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A spin foam is a "mousse de spin"
On Friday Rovelli is giving a symposium talk on spin foams and he was Etera Livine's thesis director.
My uninformed guess is that Rovelli will talk about Livine's thesis and in particular chapter 8 (Covariant loop gravity) which reflects potentially important work by Alexandrov, some of which Livine co-authored. This work extends the symmetry of loop gravity to SL(2,C) and seems to get rid of the Immirzi parameter (!). The thesis is at http://arxiv.org/gr-qc/0309028 [Broken]

Quite recent, another reason it would seem like a good thing to bring up at the Strings Meets Loops symposium this Friday. Here is a quote from the abstact, giving links to a paper of Alexandrov/Livine among other things:

------------quote from abstract--------

'Thu, 4 Sep 2003
Boucles et Mousses de Spin en Gravite Quantique
Etera R. Livine
165 pages, in French; PhD Thesis 2003, Centre de Physique Theorique CNRS-UPR 7061 (France)

I review the formalism of loop quantum gravity, in both its real and complex formulations, and spin foam theory which is its path integral counterpart. Spin networks for non-compact groups are introduced (following hep-th/0205268) to deal with gauge invariant structures based on the Lorentz group. The whole formalism is studied in details in three dimensions in both its canonical formulation (loop gravity) and its spin foam formulation. The main output (following gr-qc/0212077) is the discreteness of timelike intervals and the continuous character of spacelike distances even at the quantum level. Then it is explained how to extend these considerations to the 4-dimensional case. I review the Barrett-Crane model, its geometrical interpretation, its link with general relativity and the role of causality. It is shown to be the history formulation of a covariant canonical formulation of loop gravity (following gr-qc/0209105), whose link with standard loop quantum gravity is discussed. Similarly to the 3d case, spacelike areas turn out continuous. Finally, ways of extracting informations from the non-perturbative spin foam structures are discussed."
------------------end quote---------

BTW Livine seems to think it matters that time is discrete but space is continuous---a result from Freidel/Livine/Rovelli
http://arxiv.org/gr-qc/0212077 [Broken]
Why should this result be so important? It is admittedly odd.

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ranyart
Originally posted by marcus
A spin foam is a "mousse de spin"
On Friday Rovelli is giving a symposium talk on spin foams and he was Etera Livine's thesis director.
My uninformed guess is that Rovelli will talk about Livine's thesis and in particular chapter 8 (Covariant loop gravity) which reflects potentially important work by Alexandrov, some of which Livine co-authored. This work extends the symmetry of loop gravity to SL(2,C) and seems to get rid of the Immirzi parameter (!). The thesis is at http://arxiv.org/gr-qc/0309028 [Broken]

Quite recent, another reason it would seem like a good thing to bring up at the Strings Meets Loops symposium this Friday. Here is a quote from the abstact, giving links to a paper of Alexandrov/Livine among other things:

------------quote from abstract--------

'Thu, 4 Sep 2003
Boucles et Mousses de Spin en Gravite Quantique
Etera R. Livine
165 pages, in French; PhD Thesis 2003, Centre de Physique Theorique CNRS-UPR 7061 (France)

I review the formalism of loop quantum gravity, in both its real and complex formulations, and spin foam theory which is its path integral counterpart. Spin networks for non-compact groups are introduced (following hep-th/0205268) to deal with gauge invariant structures based on the Lorentz group. The whole formalism is studied in details in three dimensions in both its canonical formulation (loop gravity) and its spin foam formulation. The main output (following gr-qc/0212077) is the discreteness of timelike intervals and the continuous character of spacelike distances even at the quantum level. Then it is explained how to extend these considerations to the 4-dimensional case. I review the Barrett-Crane model, its geometrical interpretation, its link with general relativity and the role of causality. It is shown to be the history formulation of a covariant canonical formulation of loop gravity (following gr-qc/0209105), whose link with standard loop quantum gravity is discussed. Similarly to the 3d case, spacelike areas turn out continuous. Finally, ways of extracting informations from the non-perturbative spin foam structures are discussed."
------------------end quote---------

BTW Livine seems to think it matters that time is discrete but space is continuous---a result from Freidel/Livine/Rovelli
http://arxiv.org/gr-qc/0212077 [Broken]
Why should this result be so important? It is admittedly odd.

Marcus, if I may it is the New Physics and interpretation of 'Spacetime', spacetime is infact discrete, Galaxies are where spacetime resides. The MINKOWSKI SPACE VACUUM that intervines between Galaxies is continuous, spacetimes are coupled to matter in Galaxies, where as, Space-Vacuum has no observed matter, but is an Energy field.

My money is going to be on the re-classification of Space energy, and Spacetime energies, seperated into distinct Horizons, Time-depandant and NonTime-dependant.

E= Mc2...space energy(non-mass), converts to Spacetime energy (mass)

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Originally posted by ranyart

My money is going to be on the re-classification of Space energy, and Spacetime energies, seperated into distinct Horizons, Time-depandant and NonTime-dependant.

E= Mc2...space energy(non-mass), converts to Spacetime energy (mass)

I hope your bets and mine are in play money---monopoly money--because real mathematics is happening this time. There is an equation like eee-equals-em-cee-square that tells the dark energy that empty space contains but you and I do not know this equation and, what is more interesting, real mathematicians (who are characterized by uncanny ability to intuit and guess theoretical results---this is how they know what to try to prove logically) these people who are professionally selected for almost prophetic intuition about exactly this "space and time" kind of stuff---THEY ALSO do not know. We are in a dark room and there is something in the room with us that we cannot see and have never seen before and not even the people with very good eyes and ears can picture it. This is an amusing situation.

On usenet spr you can hear turmoil and cries of pain and frustration because the most articulate branches of physical theory seem unable to predict the positive cosmological constant, or even to ACCOMODATE it in any graceful way. the cosmological constant is the density of dark energy----observed to be 0.6 joules per cubic kilometer.

Instead of guessing, I propose we relax and enjoy a historical moment in science

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Livine generalized on Abhay Ashtekar

In his thesis Livine does a remarkable thing in 10 clearly written pages----he generalizes the Ashtekar-Lewandowski measure to the non-compact case. Yes this sounds like a mouthful of dry sand----something of purely technical interest.

But if the rest of the community checks this out and it is correct, and if this has not been done before (which I dont think it has but I could be mistaken) then it is a notable event. Like when somebody crossed the English Channel in a pedalpowered aircraft---they have you know. It goes in the edition of Guinness Book of Records that has a picture of Gauss on the cover.

Well maybe I'm mistaken but I will explain why anyway. So far Rovelli/Ashtekar/Lewandowski and all have only done loop gravity in the case where the group of motions or symmetries is limited in size----doesnt stretch off to infinity. Spin networks are labeled with integer numbers (or half integers) because the group of symmetries is so bounded and controllable, so-called "compact", that all the ways of physically representing it are like countable and labeled by "spin" numbers. Now for about 100 years if you were a mathematician and you proved something in the compact case you ALWAYS tried to extend the result to the more difficult less controllable non-compact case. this is a kneejerk professional reflex learned by generation after generation of graduate students. But Ashtekar and Lewandowski did not extend their construction to the non-compact case evidently because they COULD not----and that was where the group was the Lorentz group: wooooooo special relativity symmetries! important physically as well as mathematically.
But Livine does this in 10 pages somewhere (pages 47-57) in the middle of his PhD thesis at the University of Marseille.

So what is going on? What does this mean to us? Well to quantize anything you have to make a hilbertspace which means that you have to be able to integrate functions defined on the space of all possibilities.
In loop gravity the basic set of possibilities is all possible geometries on the particular manifold you're looking at.
this set of all possible geometries is denoted A.

You have to be able to define wavefunctions---just number-valued functions on this A. And you have to be able to INTEGRATE their squares because the square of the amplitude is the probability and you have to be able to sum it up under the integral sign. the whole thing depends on being able to write integrals and sum up functions defined on the configuration space or space of possibilities. So Ashtekar and Lewandowski defined the "measure" which is what the "means of integration" is called. You could call it the Ashtekar/Lewandowski "tool for integrating functions on A", or their technique for building the basic loop gravity hilbertspace of squareintegrable wavefunctions.

The hilbertspace of squareintegrable functions is fundamental and you cant get it unless you can integrate functions defined on the configs and you cant integrate unless you have a "measure" on the configs.

Its like having a surface measure or a volume element to integrate with in more familiar situations.

In a certain sense, progress was held up a while because there was no measure in a certain important version of the theory and Livine just constructed one-----at least it looks like it based on my imperfect knowledge of the situation.

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This morning I've been reading the Friedel-Livine paper, Spin Networks for Non-Compact Groups, which you cited above as hep-th/0205268. In this they do the math behind Livine's definition. It's a marvelous paper, written with limpid clarity, and shows how they apply familiar theorems from algebraic geometry, plus a good deal of ingenuity, to define the crucial measure that enables the whole structure. This is exciting and new, and has that feel of "why wasn't this discovered ages ago" that you associate with important results.

ranyart
Marcus, you say:There is an equation like eee-equals-em-cee-square that tells the dark energy that empty space contains but you and I do not know this equation and, what is more interesting, real mathematicians (who are characterized by uncanny ability to intuit and guess theoretical results---this is how they know what to try to prove logically) these people who are professionally selected for almost prophetic intuition about exactly this "space and time" kind of stuff---THEY ALSO do not know.

And not wanting to continue or hijack your posts, I have to beg to differ, for I believe I KNOW!

Just to ask a little one thing, are fields three dimensional?, and why do all 2-dimensional fields surround matter?

Huh mmmm...Contraction something to do with it maybe?

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This morning I've been reading the Friedel-Livine paper, Spin Networks for Non-Compact Groups, which you cited above as hep-th/0205268. In this they do the math behind Livine's definition. It's a marvelous paper, written with limpid clarity, and shows how they apply familiar theorems from algebraic geometry, plus a good deal of ingenuity, to define the crucial measure that enables the whole structure. This is exciting and new, and has that feel of "why wasn't this discovered ages ago" that you associate with important results.

Yes selfAdjoint I too have been reading the Freidel-Livine paper this morning! It perches on my knees as I lean over to type because there is no place to put it down on the cluttered surfaces around here. Great paper. also in ENGLISH so easier to quote

yes the columbus-egg feel of important results

it would be terrible not to have someone to share this sense of discovery with---a kind of lonliness---at this time of day and so I am very glad for your post on the subject

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the idea of taking a maximal tree in any graph and contracting
that tree to a point and having a new graph consisting of one
vertex from which extend a number of loops like the petals of a daisy, nice and visual

it may be a standard proceedure in some other mathematical venue that they just imported and adapted to loop gravity

and then the use of algebraic geometry

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Originally posted by ranyart
I have to beg to differ, for I believe I KNOW!

that must be a great feeling, which most people only get
the first time they fall in love

you must be feeling kind of wired, with extra energy and alertness

don't mention hijacking threads since your excursions so far are completely friendly

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Originally posted by marcus
the idea of taking a maximal tree in any graph and contracting
that tree to a point and having a new graph consisting of one
vertex from which extend a number of loops like the petals of a daisy, nice and visual

it may be a standard proceedure in some other mathematical venue that they just imported and adapted to loop gravity

and then the use of algebraic geometry

I really enjoyed the way they used the basic tree property (no loops) again and again in proving their theorems. All this time LQG people have been saying network-network, and never did anyone see the maximal tree in it before.

And then as you say, the old theorems on varieties applied to the reductive groups. BTW I found a little tutorial on them: http://www.stieltjes.org/archief/rep9899/node11.html which is a nice complement to the paper.

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bet you will visit some campuses while on this road trip to Illinois with yr offspring, good time of year to do this

the F/L article "spin networks for non-compact groups" is majorly fantastic

a fock space (a direct sum of L2 spaces) with creation/annihil
of spinnetworks corresponding to adding/subtracting a loop

Rovelli has signed on. There is a significant paragraph on middle page 4 of Freidel/Livine/Rovelli "Spectra of Length and Area in
(2+1) Lorentzian Loop Quantum Gravity"
http://arxiv.org/gr-qc/0212077 [Broken]
"....we call a connection on a graph &Gamma; the assignment of group elements..."
and the footnote on the same page "AS A CONSEQUENCE THE FULL SPACE OF QUANTUM STATES OF THE GEOMETRY...CAN NOT BE OBTAINED AS A PROJECTIVE LIMIT ANY MORE...similar to a Fock space."

This strikes me as Rovelli's assimilation of a fundamental change in the theory he founded (with Smolin and Ashtekar) some ten years ago. The quantumstates of the geometry is now to be obtained by gluing lots of L2 spaces together and will have creation annihilation operators. As far as I can see it just gets better :)

this morning did not even think to turn on the computer, just picked up F/L's and F/L/R's papers and resumed reading
after an hour it occurred to me you might be as well

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Livine's action on an edge

This formula for the basic action, which in occurs in all these Livine papers, puzzles me. k-1s(e)ge(A)kc(e). The part in the middle is the holonomy of connection A along edge e, resulting in group element g, as we have seen before. My problem is with the k's. Apparently from his notation, k denotes an operator from the Lie Algebra of G, and the subscripts indicate they are evaluated at the two vertices at the ends of e, but where did Lie(G) get mapped into points of &Sigma;? I haven't been able to find an explanation of this

Has anybody got any ideas?

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selfAdjoint, David Louapre may answer your question, if not I will try. Incidentally he says that Laurent Freidel will be giving the talk on spin foams this Friday, in Rovelli's place. See his PF post:

But Freidel is the person whose papers (with Livine, and Rovelli) we have been reading. Even if the talk is only very general for a broad audience it seems like it could be interesting to hear Freidel's point of view. I hope we can get a link to the talk as soon as one is available.

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This formula for the basic action, which in occurs in all these Livine papers, puzzles me. k-1s(e)ge(A)kc(e). The part in the middle is the holonomy of connection A along edge e, resulting in group element g, as we have seen before.

I'm not sure I understand your confusion. This just gives the behaviour of the holonomy ge(A) under gauge transformations A &rarr; Ak = k-1Ak + k-1dk:

ge(Ak) &equiv; k-1s(e)ge(A)kc(e).

See p41 of livine's thesis.

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Originally posted by jeff
I'm not sure I understand your confusion. This just gives the behaviour of the holonomy ge(A) under gauge transformations A &rarr; Ak = k-1Ak + k-1dk:

ge(Ak) &equiv; k-1s(e)ge(A)kc(e).

See p41 of livine's thesis.

Yes I was at that page when I posted. I understand the gauge transformation of the connection form, but how did the Lie elemnts get evaluated at the two ends of the edge? So far the only source of Lie algebra elements is the connection itself - through its form.

I have been all over google looking for a development of this - none of my books have it - and nothing useful has turned up. I tried to work it out with stoke's theorem, treating the two points as boundary of the edge, but that didn't come out in the right form either. So I'm stumped.

dlouapre

I'm not sure I understand your puzzle but let me try to answer it. The gauge transformation of the connection is parametrized by k which is a field over the manifold, with values in the group G. So k in

k^{-1}Ak + k^[-1} dk

is actually k(x) for x point of the manifold.

Now take a connection field A and g(A) the holonomy of A along an edge. Apply a gauge transformation (parametrized by a G-valued field k(x)) to A and look what happens to g(A). It appears that the way g(A) is modified depends only of the value of k(x) for x=start of the edge and x=end of the edge. So call k_{s(e)} and k_{t(e)} this values, write the transformation of g(A) and forget about the value of k(x) for every other x !

Yes I was at that page when I posted. I understand the gauge transformation of the connection form, but how did the Lie elemnts get evaluated at the two ends of the edge? So far the only source of Lie algebra elements is the connection itself - through its form.

I have been all over google looking for a development of this - none of my books have it - and nothing useful has turned up. I tried to work it out with stoke's theorem, treating the two points as boundary of the edge, but that didn't come out in the right form either. So I'm stumped.

...where did Lie(G) get mapped into points of &Sigma;?

The mapping is from &Sigma; to the lie algebra, as on p25.

I thought by "action" you'd meant one obtained from a lagrangian density.

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hi selfAdjoint, I will add a bit of detail to what David L. just said.
I've been busy elsewhere or would have replied earlier.
When you write down a holonomy along an edge it is an integral and I guess you can approximated it by a Riemmann sum
And all along the way the k and k-1 and k will cancel.
So only the first and the last k-1 and k will not be canceled and will appear. Something like that. I will try to find an online paper where this is spelled out.

the holonomy on an edge (gauge-transformed A) =
k-1(start) X holonomy on edge (original A) X k(end)
I am sure I have seen this proved maybe in some basic LQG
survey, somewhere anyway.

It looks like Louapre, alas, was just passing thru :(

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OK folks, thanks to all of you I get it now. I feel really dumb but what can you do. I do appreciate the answers, and they do completely enlighten me on the subject. Onward and upward.

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I looked in my basic beginner's textbook of LQG which is by Marcus Gaul and Carlo Rovelli and even THERE the thing is not proven. they just say on page 13

"despite the inhomogeneous transf. rule (19) of the connection
(which I think we already have written in this thread) under gauge...the holonomy TURNS OUT to transform homogeneously like"

and they write the same thing that you and I just wrote that just brackets the original holonomy by the endpoint k-inverse and k.

well that is Gaul/Rovelli
http://arxiv.org/gr-qc/9910079 [Broken]

its one of those times when the prof says IT IS EASY TO SHOW, I guess it is a Riemmann sum with a lot of cancelation, an integral on the interval [0,1], some exercise in bookkeeping
maybe it is just a fun thing to do at the blackboard so no one puts it in books, or they all do it in basic gauge theory for QFT years before

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Ambitwistor
A lot of details of gauge transformations are worked out in Baez and Muniain, although I don't about this particular case.

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Originally posted by Ambitwistor
A lot of details of gauge transformations are worked out in Baez and Muniain, although I don't about this particular case.

I just looked up Baez and Muniain at Amazon. Average proce used in any condition, \$78. Sorry, no can do. I'll get by with a little help from my friends.

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Typo on page 41 of Livine's thesis

Page 41 is a good place to start reading the thesis
but it has a potentially confusing typo that should be pointed out.

equation 2.43 should be

&Gamma;*f (A) = f(&Gamma;(A))

The reason its a good place to start (apart from the typo) is that before that he is developing the prior SU(2) theory.
It looks to me as if beginning on page 41, the first page of "Part 2" of the thesis, he takes his own track.

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On page 41 he strikes out in a new direction, tho it may start off seeming familiar

He says an oriented graph &Gamma; with E edges defines a mapping from the connections A to the E-fold cartesian product of the gauge group GE

&Gamma;: A --> GE

And given any Coo function f defined on GE he has the picture

A --> GE --> C

and that defines a cylinder function on A

and he refers to the "space of cylinder functions" as the collection of all those things

Then he defines A&Gamma; as the "discrete connections" on the graph &Gamma;
basically GE (an assignment of G-elements to edges) modulo an equivalence relation which involves assigning G-elements to each vertex and using the gauge-group action (2.44).

He writes A&Gamma; = GE/GV, as a reminder of how the discrete connections associated with the graph are built.

Then he constructs a measure on the space of discrete connections and obtains the hilbertspace.

L2(A&Gamma;, d&mu;&Gamma; )
...................

The fock space is made by merging all these indvidual hilbertspaces, one for each class of graph

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Originally posted by marcus

Then he defines A&Gamma; as the "discrete connections" on the graph &Gamma;
basically GE (an assignment of G-elements to edges) modulo an equivalence relation which involves assigning G-elements to each vertex and using the gauge-group action (2.44).

I think one point may be clarified. Why did he mod out the actions on the vertices? Correct me if I'm wrong but I believe this is a gauge fixing. He fixes the gauge by setting its actrion on the vertices to the identity. This leads him to mod out the vertices from his graph, using his tree constructions, and in turn that gets him into all the problems with quotient manifolds which he solves so cleverly.

So this euivalence relation is an important breakthrough; I don't recall any of the Ashtekar school who realized what you could do with that.

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I suspect your interpretation is right. As far as I can see, it is either a gauge fixing or else he at least wants to identify discrete (&Gamma; based) connections which are equivalent under a gauge transformation.

If there are two nominally different connections and a gauge transformation will turn one into another, then he wants to say they are essentially the same.

This was how I was looking at it, but you are looking a bit deeper at what is going on and have a more interesting take on it, so I will think about this some more.

dlouapre

I think one point may be clarified. Why did he mod out the actions on the vertices? Correct me if I'm wrong but I believe this is a gauge fixing. He fixes the gauge by setting its actrion on the vertices to the identity. This leads him to mod out the vertices from his graph, using his tree constructions, and in turn that gets him into all the problems with quotient manifolds which he solves so cleverly.

So this euivalence relation is an important breakthrough; I don't recall any of the Ashtekar school who realized what you could do with that.

That's correct. This gauge fixing along a maximal tree is new to the Freidel/Livine construction for two reasons : the gauge volume associated to the gauge invariance at the vertices is Vol(G)^V where V is the number of vertices and Vol(G) the volume of the group. Of course in the compact case it doesn't matter since the volume is finite (and even 1 if we work with the normalazied Haar measure). So the SU(2) spin networks don't require this gauge fixing.

The other reason that explains that this gauge fixing is not in the usual SU(2) spin network construction is that in some sense it creates some troubles : remember that to take the scalar product of 2 cylindrical functions (a priori based on different graphs) you embedded both graphs in a common graph and take the scalar product once you formulated the cylindrical functions both on the SAME graph.
If you impose the gauge fixing, ie that essentially each graph has to be a flower, it forbids you to use this scalar product.

So the gauge fixing is necessary from the point of view of finiteness but it leads to some problems for scalar product construction (this is discussed in the Freidel/Livine paper).

David

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To continue, Livine aims to define a measure d&mu;&Gamma; on the discrete connections A&Gamma;, living on a graph &Gamma;
this will allow him to define a hilbertspace L2(A&Gamma;, d&mu;&Gamma;) which by a slight stretch of notation he calls L2(&Gamma;, d&mu;&Gamma;)

And he observes that in the case that the gauge group G is compact the measure is just an extension of haar measure corresponding exactly to the celebrated Ashtekar-Lewandowski measure often encountered among us at PF but....

"In the non-compact case haar measure no longer suffices. In effect, since the group has an infinite volume, we must divide that volume out----that is to say: gauge-fix the group action. That's what this part of the thesis is about.

The construction of the measure is in two stages. First we will show that A&Gamma; is equivalent to Gh/Ad(G)..."

Where h = h&Gamma; is the GENUS of the fattened or inflated graph---determined as in topology by the number of handles.
And Ad(G) is just what we think it is namely the adjoint action of the group.

"...Subsequent to gauge fixing we will exhibit an isomorphism between
Gh/Ad(G) and Gh -1. The measure can be defined as the pullback of haar measure on Gh-1..."

it was at this point that I thought I would look at a concrete example of this construction of the measure---where the group is the lorentz group SL(2,C) instead of a more general group. What is done more abstractly and generally in the thesis is done in the "Projected..." paper IIRC for the specific case of SL(2,C), and also there is some useful explanation around page 25 of that other paper "Spin Networks for Non-Compact Groups"

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the key point in Livine's thesis, as I see it at the moment

It is on page 96 and 97 where he says there is a two-parameter family of (classical) connections depending on two real numbers l and m

this family includes the Ashtekar-Barbero as one case where

(l, m) = (- g, 1) and little gamma is, you guessed it, the Immirzi

and it also includes the covariant case Livine likes where

(l, m) = (0, 0)

He argues that these particular two cases of the connection are especially natural and/or advantageous for reasons he gives.
So he proceeds to quantize those two cases.

What grabs my attention is that the (0,0) one seems to have more going for it---it is the covariant one that makes an easy transition to Barrett-Crane Lorentzian spin foams. I'll try discussing this in a little more detail.

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Originally posted by marcus

this family includes the Ashtekar-Barbero as one case where

(l, m) = (- g, 1) and little gamma is, you guessed it, the Immirzi

and it also includes the covariant case Livine likes where

(l, m) = (0, 0)

He calculates the Dirac bracket of the connection A(l, m) with itself explicitly in terms of l and m, sets it equal zero, solves for the two parameters, and observes that

"the unique communtative connection" is
A(- g, 1), and this reduces to the Ashtekar-Barbero case

The other especially natural connection A(0, 0)is singled out for other reasons. I dont remember anybody else making the choice of classical GR variable so systematically. They could have done it, but in that case I just didnt notice. Anyway here he is laying out the choice rather comprehensively and it gets me thinking---what happens if you switch? What to make of the fact that the area operator's spectrum is different? (Not too surprising, different hilbertspace different basis---expectation values presumably still the same for any give surface measured) What happens to the hamiltonian constraint? Does it correspond to the classical one? Can one actually calculate with Livine's version of loop gravity?

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I think one point may be clarified. Why did he mod out the actions on the vertices? Correct me if I'm wrong but I believe this is a gauge fixing. He fixes the gauge by setting its actrion on the vertices to the identity. This leads him to mod out the vertices from his graph, using his tree constructions, and in turn that gets him into all the problems with quotient manifolds which he solves so cleverly.

So this euivalence relation is an important breakthrough; I don't recall any of the Ashtekar school who realized what you could do with that.

another possible breakthrough is in the new idea of what a spin network is

bottom of page 23 of Freidel/Livine "Spin Networks for Non-Compact Groups" definition 5: "We call these eigenvectors spin networks."

Instead of borrowing from Penrose a certain labeled graph construction, Freidel/Livine arrive at a set of eigenvectors by diagonalizing a set of commuting differential operators and they CALL these vectors spin networks. And it works, they get away with it. Is that what the hardware-version spin networks were all along?

I think I may be beginning to understand F/L pages 22 and 23 that lead up to that definition.

I think a "new" Loop Gravity is emerging with the graph-based connection, the F/L measure for non-compact G, this "natural" finessing of the spin-networks (all still on the basic graph &Gamma;) and finally the creation of a fock space by summing over all possible graphs &Gamma;

Maybe this is a "second" quantization of GR, if one thinks of the Ashtekar SU(2) quantization as the first. Bad analogy?

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next step: positive Lambda

It seems likely that the next step in developing Covariant Loop Gravity will be a continuation of Livine's thesis that introduces positive Lambda----positive dark energy, or cosmological constant---into the theory.

The key new idea we need to grasp is that the "quantum hyperboloid becomes a stack of fuzzy spheres"

see first paragraph page 2 of Girelli/Livine "Quantizing Speeds with the Cosmological Constant" gr-qc/0311032

The basis for "Quantizing Speeds..." is what now seems to be a very important paper by Karim Noui and Philippe Roche "Cosmological Deformation of Lorentzian Spin Foam Models" gr-qc/0211109
and in particular page 13 where the hyperboloid of possible speeds is put in terms of some 2x2 matrices---a right coset homog. space SU(2)\SL(2,C) and the Iwasawa decomposition is used to get it as A x N (diagonal and nilpotent) How bad can it be? It is just a few 2x2 matrices of especially simple form

lambda 0
0 lambda-1

plus

0 0
n 0

the first is the real number diagonal det=1 type
and the second is the complex number n for nilpotent
lower triangular type, the language here is heavier than the matrices

let's see if I can type the sum of those two matrices

Code:
[size=3][font=symbol]l[/font]  0
n  [font=symbol]l[/font][sup]-1[/sup][/size]

so the hyperboloid of possible speeds, or moving observers, or boosts or whatever, is happily pictured algebraically as some 2x2 matrices

NOW Noui/Roche will tell us how to q-deform them by introducing a cosmological constant.

See page 16 and 17
You will see elegant french style. first the quantum hyperboloid is presented in a fearsomely succinct and categorical way, then in equation (55) one sees that it is simply a stack of spheres made of essemtially the same matrices except the lower left entry, the complex number n, has been multiplied by something EXTREMELY NEAR ONE.

and then presto on the next page there is equation (58) that Girelli and Livine used to see the spectrum of quantized speeds.

The fearsome and succinct definition they give first is something else. They refer to the algebra of compactly supported functions on the quantum hyperboloid as
Func(H+q) = Func(ANq)

this is just the Iwasawa decomposition into diagonal (we saw before) and q-deformed nilpotent (here just means lower left nonzero entry)

And they say "therefore as an algebra it has the structure

Func(H+q)= +IMat2I+1(C)

And they say "this description is the deformation of the foliation of H+ by quantum fuzzy spheres. Quantum fuzzy spheres have been introduced and studied in hep-th/0005273 (Grosse, Madore, Steinacker "Field Theory on the q-deformed Fuzzy Sphere")

At this point my outrage knows no bounds. But what can one do. The speeds that things were traveling at the instant the universe began to expand has according to good authority somewhat to do with q-deformed Fuzzy Spheres. Speeds were quantized. Oh damn the matrix looks the same but the entries are "non-commutative numbers". Oh hell it is awful. It looks like

Code:
[size=3][font=symbol]l[/font]  0
n  [font=symbol]l[/font][sup]-1[/sup][/size]
except the n has been multiplied by something extremely close to one, namely
you can see that since the deformation parameter is very close to one namely like
q = exp(-10-123) as it is today, then
this square-root thingee is very close to &radic;(2/2) = 1

So the matrix Karim and Philippe (Noui/Roche) give us is
Code:
[size=3][font=symbol]l[/font]             0
&radic;((q[sup]2[/sup]+1)/2)n  [font=symbol]l[/font][sup]-1[/sup][/size]

Last edited:
Gold Member
Dearly Missed

Code:
[size=3][font=symbol]l[/font]             0
&radic;((q[sup]2[/sup]+1)/2)n  [font=symbol]l[/font][sup]-1[/sup][/size]

My outrage at the term "Fuzzy Sphere" has subsided and I can think more clearly. Actually this matrix is kind of intriguing. It is
a familiar sensible 2x2 matrix except EXCEPT the numbers in it, the lambda and the n, just barely DO NOT COMMUTE.

right under the matrix on page 16 of Noui/Roche it gives the equations of their non-commutativity and its rather nice. There is this parameter q very very near one. If q were exactly one then the arithmetic would not depend on the order of mults at all. But to the tiny tiny extent that q is not exactly one, the arithmetic depends on the order in which you multiply factors. I'm beginning to think its real cool.

Remember that q is the number e (the base of logs, 2.7...) raised to an unprecedentedly small number 10-123

one over (one followed by 123 zeros)

this is how we are incorporating dark energy into our local everyday business. the cosmo constant is 10-123 a number closer to zero than science has ever dealt with so far
and we take that number and we raise e = 2.7... to that power

well e raised to the zero is exactly one

so e raised to something extremely close to zero is extremely close to one-----unprecedentely close to one---I cant think of any number in science that differs from one by that little

and that number is by how much the numbers in the matrix do not commute. and that matrix affects local business---it tells how the speeds around us are quantized.

so suddenly the radius of the cosmological horizon----some 60 billion lightyears---which is basically the square root of that 10123 number---has been flipped over to be a very small 10-123---and Livine and Girelli are explaining to us that it enters into how speeds around us are quantized in little steps of speed.

and it hinges on the fact that some numbers we compute coordinate framechanges with, numbers which for all practical purposes are ordinary commutative numbers where AxB is the same as BxA, actually do not quite commute. What can I say. It is awesome.

true or not, it is awesome, and it could even be true.

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Ambitwistor