Covariant Loop Gravity and Livine's Thesis

marcus

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 :(

selfAdjoint

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.

marcus

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

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

Ambitwistor

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

selfAdjoint

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.

marcus

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

Γ*f (A) = f(Γ(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.

marcus

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

He says an oriented graph Γ with E edges defines a mapping from the connections A to the E-fold cartesian product of the gauge group G^E

Γ: A --> G^E

And given any C^oo function f defined on G^E he has the picture

A --> G^E --> 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_Γ as the "discrete connections" on the graph Γ
basically G^E (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_Γ = G^E/G^V, 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_Γ, dμ_Γ )
...................

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

selfAdjoint

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.

marcus

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 (Γ 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

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

marcus

To continue, Livine aims to define a measure dμ_Γ on the discrete connections A_Γ, living on a graph Γ
this will allow him to define a hilbertspace L²(A_Γ, dμ_Γ) which by a slight stretch of notation he calls L²(Γ, dμ_Γ)

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_Γ is equivalent to G^h/Ad(G)..."


Where h = h_Γ 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
G^h/Ad(G) and G^{h -1}. The measure can be defined as the pullback of haar measure on G^h-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"

marcus

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.

marcus

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?

marcus

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 Γ) and finally the creation of a fock space by summing over all possible graphs Γ

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

marcus

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
see also their equation (1)

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


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
Fun_c(H_+q) = Fun_c(AN_q)

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

Fun_c(H_+q)= +_IMat_2I+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

except the n has been multiplied by something extremely close to one, namely
√((q²+1)/2)
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 √(2/2) = 1

So the matrix Karim and Philippe (Noui/Roche) give us is

marcus

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.

Ambitwistor

Good, now you can survive the giant fuzzy moose.

http://arXiv.org/abs/hep-th/0111079