Spin Foam Models: A Perspective on Quantum Gravity

[marcus]

Now might be a good time to get some perspective on spin foam, if there are knowledgeable people around willing to help. Baez in some TWF mentioned a paper by Freidel/Louapre with "asymptotic 10j" in the title. It suggests a way to dispell the surprise over unexpected 10j numbers discovered by Baez/Christianson/Egan in mid-2002 IIRC. Rovelli is giving a symposium survey of spin foam in a week, 31 October yes I realize that is halloween, and he might talk about what significance this 10j business has.

But I stand no chance of understanding any of that without some basic perspective, so I will try to sketch out what could be basic perspective on spin foam and hope other people will correct or fill in parts I miss.

It seems that a spin foam is just a path getting you from one spin-net or spin-knot state to another. the original deeply confusing idea is by Feynmann: in a quantum picture trajectories don't exist and a system gets from A to B by following all possible paths---a spinfoam is just one of millions of possible paths for getting from spin-net quantum state of geometry A to spin-net quantum state of geometry B. As insane laughter rises, you AVERAGE all the possible paths with a whole lot of phasecancelation, you ADD UP all these millions of possible paths, and you get the amplitude of evolving from state A to state B. This actually seems rather nice.

I notice that there is a 2003 paper by Livine and Oriti called "Causality in spin foam models for quantum gravity" and I wonder if Rovelli will say anything related to it----there is something attractive about it: a Green function or a propagator of some kind that seems to be comprised of a going forwards piece and a going backwards piece, as if one of the problems that is always coming up is how do you select the right piece. I have a vague suspicion that the problems with spin foam and the problems with hamiltonian are neither of them *prohibitive* problems but are clues to a connection between the two. That is, the spin foam approach is in a fundamental way not all that different from a hamiltonian approach.

In some other thread I mentioned this strangely easy-to-read article "A simple background-independent hamiltonian quantum model" by Colosi and Rovelli. It is a simple toy model of a pendulum or something. I don't have the ability to judge if that article is in any way significant---it seems suggestive to me but I don't know enough to judge. there is a propagator in the toy model that gets you from one situation to another. Is this paper simply a "hamiltonian" toy model or is it a sort of hybrid toy model.
Does this paper, simple as it is, have any bearing on spin foams. Sorry about all the dumb questions. In case anyone wants to take a look the Colosi/Rovelli "simple background-independent quantum model" paper is

http://arxiv.org/gr-qc/0306059

I'll try to steer this back more to the main topic of spin foams proper if I post a follow-up

 

[marcus]

Oh yes the 10j problem, the Freidel/Louapre paper "Asymptotics of 6j and 10j symbols" is

http://arxiv.org/hep-th/0209134

It is dated December 2002, it came out not very long after the Baez/Christensen/Egan paper that revealed the surprising celebrated 10j misbehavior and in their abstract they say,
"We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease."

Rather a strong word, disease. And John Barrett, author of the particular sort of spin foam model which was discovered to have the disease, was not to be left out either. In January 2003 he posted Barrett/Steele

"Asymptotics of Relativistic Spin Networks"

http://arxiv.org/gr-qc/0209023

(the number looks like september 02 but its dated jan 03)

Which brings up the issue of just how Lorentzian spin foams are. The last sentence of Barrett/Steele is "Finally we discuss the asymptotics of the SO(3,1) 10j symbol."

Then finally, something that seems very promising appears. Freidel/Louapre post "Diffeomorphisms and spin foam models" dated 29 January 2003

http://arxiv.org/gr-qc/0212001

"We study the action of diffeomorphisms on spin foam models. We prove that in 3 dimensions there is a residual action of the diffeomorphisms that explains the naive divergences of the state sum models..."

that sounds almost too good to be true. It is how things are SUPPOSED to work. Baez et al say hey there is a divergence and
Freidel et al are compelled to think and find out something. But maybe that is not what happened.

Perhaps I will try to read Diffeomorphisms and spin foam models and report further, unless someone else here has looked at the paper already.

So there is all this stuff about spin foams. Which, this Halloween, Rovelli will talk about.

[BTW Baez posted all or most of these links in TWF some time back but I see no harm in repeating.]

And (I would say "finally" but it probably doesn't stop here) there is this paper dated 30 July 2003 by 5 people CDORT of which R stands for Rovelli. The paper is in the spinfoam department and it is called

"Minkowski vacuum in background independent quantum gravity"

http://arxiv.org/gr-qc/0307118

I would tell you about it but my wife wants a fresh seedy baguette this morning so I have to go out.

 

[eigenguy]

Originally posted by Ambitwistor
Spin foams originally became popular because they were viewed as a way of getting around problems perceived with the canonical Hamiltonian constraint. Nowadays, it's still kind of up in the air whether the canonical "problems" are problems, and spin foam progress has bogged down. It is commonly thought that there should be a spin foam model that corresponds exactly to the canonical theory, but nobody knows what it is

yeah this is what jeff says except for the part about

"it's still kind of up in the air whether the canonical "problems" are problems".

What are you referring to specifically?

 

[selfAdjoint]

Go check out Baez's This week's finds #85. It tells the story of Thiemann's original try at the Hamiltonian constraint. His paper was announced at a meeting of QG people. Baez was there and judged it a "blockbuster'. At one stroke, it seemed, the whole problem of QG seemed to be on the way to solution.

Then came the morning after.

You can see why Thiemann is very,very cautious with this new announcment. And maybe why he sort of dwells on the dark side of what has happened since.

I emailed him last week, and made so bold as to ask "will the Phoenix fly?". He was kind enough to respond. He is still optimistic, he says, but it's a big project, and we don't have final answers yet. That's good enough for me.

 

[marcus]

Originally posted by Ambitwistor
Yes, a spin foam is a "history", describing how one state evolves into another...


...should be a spin foam model that corresponds exactly to the canonical theory, but nobody knows what it is, though there has been a little work relating the approaches, by Arnsdorf, Livine, Alexandrov, etc. (gr-qc/0110026, gr-qc/0207084, gr-qc/0209105). Just today Perez gave a talk on trying to construct the physical Hilbert space inner product from spin foams in 2+1 gravity, which is certainly one thing needed to understand the relation between the two approaches.

...

Ambitwistor, could you give an intuitive explanation for why one has to calculate all these 10j symbols in the first place. I know there are good papers by Baez and others online about spin foams, but. ...I want something more basic.

a state of gravity is a wavefunction over 3D geometries of the manifold being studied
therefore it's a wavefunction on the space of 3D connections
an efficient way to define such functions on the connections is with a network
so the states of gravity are networks

a foam is the obvious way to connect two networks by a history
(that is the easy part because it's visual, you just drag the network out in another dimension and presto it's a foam)

the place I get stuck is when I want to understand why, when you want to associate an amplitude with one of these transitional histories (so you can sum up all the amplitudes), why do you then suddenly find yourself calculating 10j symbols for simplices in the foam.

I have an idea about this, ignore it if it doesn't make sense: simplices in the foam eventually after canceling might correspond to changes in the topology of the network----if you want to add or subtract a vertex in the network this might introduce a simplex or a series of simplices. so you want a number that you can calculate from any simplex in the foam that will accumulate a measure of the topological change going on as you evolve from one network (quantum 3D geometry state) to another network

 

[marcus]

You mentioned a talk just given by Perez (I think he is a Penn State) do you happen to know if the talk is online, or what the title was?

You gave some arxiv links which I am copying here in full for convenience

http://arxiv.org/gr-qc/0110026
http://arxiv.org/gr-qc/0207084
http://arxiv.org/gr-qc/0209105

these are to explorations of how canonical gravity relates to foam gravity. I'll have a look, with my second cup of coffee, and see if there is something there for bears of modest brain.

Whoah! the Arnsdorf seems pictorial and helpful!

 

[eigenguy]

Originally posted by marcus
a state of gravity is a wavefunction over 3D geometries of the manifold being studied
therefore it's a wavefunction on the space of 3D connections

My understanding is that the wavefunction is defined not on the connection but on it's holonomies along the edges of spin networks.

 

[marcus]

Originally posted by marcus
...
http://arxiv.org/gr-qc/0207084
...

Oh THAT Livine! Mousse and Boucles Livine! He says here that the Immirzi parameter is likely to go away! This is music to my ears.

I was just looking at a paper by Livine and Oriti about foam and diffeomorphisms, but had never seen this one

Hello eigenguy! just give me a moment to collect my wits.

 

[selfAdjoint]

Marcus is using here the one connection-one geometry satz, which I have questions about.

In canonical QG you have a space A of connections A, which all take values in the Lie Algebra of the gauge group. For each connection in A and each edge in each possible network on M, you have the holonomy, which is thus a "motion" of the group on the manifold (I avoid the term action to avoid misunderstanding). So the general kinematics ranges over the set of connections A[/i} and the set of networks on M and produces group motions on M.

 

[marcus]

Originally posted by eigenguy
My understanding is that the wavefunction is defined not on the connection but on it's holonomies along the edges of spin networks.

language has all these sources of misunderstanding so I will just go over what I MEANT to say and see if your understanding agrees or not

there is this set of connections A that is basic
and we have to define complex valued functions on it and
make a hilbertspace of those functions

so, pick a connection A out of that set

how, with what kind of machine, are we going to cook up a number from this connection?

a ("Wilson") loop would do it, we could define a loop in the manifold and say that our recipe for getting numbers is to go around the loop and get a matrix and take the trace of the matrix.

so that loop is, itself, a machine for getting a number from any connection A, so it is a "wavefunction"---a complex valued function defined on A

So we could have our hilbertspace just be all the loop functions and linear combinations and limits of those loop functions.

But the loop functions don't provide a clean efficient orthonormal basis for the hilbert space. there are too many. so Mssrs Smolin and Rovelli futz around and find a good basis which consists of slightly more complicated machines called networks, instead of loops

this sounds like an horrendous oversimplification and probably is but let us start there and see what the problems with that are

 

[eigenguy]

Originally posted by marcus
why, when you want to associate an amplitude with one of these transitional histories (so you can sum up all the amplitudes), why do you then suddenly find yourself calculating 10j symbols for simplices in the foam.

The relevant excerpt in "this weeks finds in mathematical physics (week 170)"

http://math.ucr.edu/home/baez/week170.html

is,

"compute the partition function as follows. First you take your 4-dimensional manifold representing spacetime and triangulate it. Then you label all the triangles by spins j = 0, 1/2, 1, 3/2, etcetera. Following certain specific formulas you then calculate a number for each 4-simplex, a number for each tetrahedron, and a number for each triangle, using the spin labellings. Then you multiply all these together. Finally you sum over all labellings to get the partition function."

My take on this is the following:

The spins assigned to each triangle represent their respective areas, these reflecting the number of spin network edges puncturing them. Eventually one sums over all possible labellings corresponding to different puncturings which collectively give different possible spacetime geometries. What we need to calculate is the probabliity amplitudes associated with these geometries. This is done by calculating amplitudes for each 4-simplex, 3-simplex (tetrahedron) and 2-simplex (triangle). The 10j symbols in particular are used to calculate the different amplitudes for the spins of the 10 faces of a 4-simplex to couple analogous to the use of clebsch-gordon coefficients (which may be expressed as 3j symbols) for adding two momenta in QM.

 

[marcus]

Originally posted by eigenguy
The relevant excerpt in "this weeks finds in mathematical physics (week 170)"

http://math.ucr.edu/home/baez/week170.html

...
My take on this is the following:
...

That helps some. Thanks for both the Baez excerpt and your take on it!

 

[marcus]

overdue replies to selfAdjoint's posts

selfAdjoint,

it is great you wrote Thiemann (and got a reply!) also thanks for sharing because then your contact puts us all more in touch with people doing actual research---guess that's obvious but will say it anyway

also did you follow what Ambitwistor said about the meaning of "geometry" having broadened in recent years until you could actually say that a (non-LeviCivita) connection describes a "geometry"

I don't know if we should go as far as that. But I want to say that equating the connections with the geometries is not altogether *my* satz (it is something Ambitwistor suggests others do too) and also that I am a bit leery of it.

I was just looking at a paper by a Dutchman named Arnsdorf who said what he meant by a "geometry" was an equivalence class of metrics under diffeomorphism! get that. woooo. pretty general.

We still have a lot of time to try things out and decide on what words to use how, it's ongoing.

the Arnsdorf paper was in a set of 3 links that Ambit. just gave:

...

http://arxiv.org/gr-qc/0110026
http://arxiv.org/gr-qc/0207084
http://arxiv.org/gr-qc/0209105

...

The first two seemed quite interesting, I'm going to check the third now. Why do people say foam gravity research is "bogged down"?

 

[marcus]

selfAdjoint you asked about getting copies of the symposium talks

I believe you mentioned having to be out of town on the weekend of the 31 thru 1 and were wondering if there was some way of getting copies of the talks to be given at Strings Meets Loops symposium at Berlin

I guess you would be particularly interested in what Lewandowski is going to say about the Hamiltonian constraint

also Thiemann probably is interested and may have an advance copy

have you or would you think it OK to ask?

Also maybe some other poster here knows of how we can get
copies of the talks after they are given. Will they be at the Albert Einstein Institute website? Does anyone know?

BTW I am interested in getting a copy of Rovelli's symposium talk on spin foams when it is available (this is the one I am most curious about)

 

[marcus]

Originally posted by Ambitwistor
Well, as you say, a spin foam is supposed to be a history of how spin networks evolve. Your idea about the role of vertices is also correct, but I will explain anyway:

So, imagine one spin network just staying the same over the history. Each edge of the network in "space" sweeps out a face of a spin foam in "spacetime". Similarly, each vertex of the network sweeps out an edge of a spin foam (a vertex is a place where network edges meet, and an edge is a place where foam faces meet).

We end up with a spin foam that has faces that meet in edges, but no vertices. Vertices appear in a spin foam when the spin network changes. So vertices describe the evolution of space -- without them, space couldn't change (the spin network would always be the same).

Spin foam vertices are labeled by 10j symbols. They're "10j" because 10 spin foam faces meet at a vertex, which in turn is because a 4-simplex has 10 edges, and when you compute the dual 2-complex of a simplicial "triangulation" to get a foam, simplex edges are replaced by foam faces (the simplex edges poke through them). "Nj symbols" are called that because they combine N spins, and in this case the 4-simplex edges (spin foam faces, or spin network edges) carry spins.

Thus, "10j symbols" are essential to define transition amplitudes in spin foams.

this is an example of exceptionally clear explanatory style, the word that comes to mind is "epiphanous" (dont know if that is actually an English word)

you just now gave a link to 3 spinfoam preprints of which the third was by Sergei Alexandrov and Etera Livine.

I like this explanation of why a vertex in the foam represents change in the state of the geometry so much that I would like to start a separate thread where I simply paraphrase this in my own words. Also the "dual" is a nice thing to visualize happening to a foam and could do with more discussion. But in a separate thread so as not to overload this one. Maybe.

I was impressed by the Alexandrov/Livine paper. It sounds the way a quantum gravity paper ought to sound and sets about doing what it ought to be doing (just a very subjective impression) also Livine is probably special he has co-authored recent papers with Oeckl, Oriti, Freidel, Rovelli, Smolin---not all in one of course but in a lot of separate recent papers. Also there is something impressive about Sergei Alexandrov, besides the Fock Institute Sankt Peterburg address, underneath his French one. The paper has a kind of old-school magisterial tone that makes one (me anyway) sit up and take notice. I better quote from the introduction.

 

[marcus]

Alexandrov and Livine, from the introduction

"Loop Quantum Gravity as developed today seems to be a promising approach for quantizing general relativity (for reviews see [1, 2]). Although it gives some interesting results like discrete quanta of area and volume [3, 4] and a derivation of the black hole entropy [5], there appear several problems. First of all, it is based on the use of a space triad and an SU(2) connection where SU(2) is the gauge group for the three dimensional space.

This particular choice of variables loses the explicit covariance of the theory and a space-time geometrical interpretation [6]. Moreover, there exists an additional puzzle: a free parameter in the theory, the so-called Immirzi parameter [7]. This parameter comes out of a canonical transformation but creates a full one-parameter family of quantizations which are not unitarily equivalent [8]. It was an open problem to understand the physical relevance of the Immirzi parameter and how it effectively influences the dynamics of the quantum theory. It turned out that this problem can be studied from a new point of view in the
framework of an explicitly covariant formalism [9]. The obtained results suggest that the Immirzi parameter should disappear from the physical output of a path integral formulation of quantum gravity [9] as well as of its canonical quantization based on this covariant formulation [10, 11]. The goal of the present paper is to explain how one can derive the SU(2) Loop Quantum Gravity (LQG) from the covariant canonical quantization. This will allow us to tackle the issues of LQG from this different point of view, and discuss the drawbacks of LQG."

from the introduction of a September 2002 paper of Sergei Alexandrov and Etera Livine

 

[eigenguy]

Originally posted by Ambitwistor
Meanwhile, the canonical program is making slow but steady progress with its Fock space / shadow state / coherent state approach

Again, I think the best place to start is with this paper

http://arxiv.org/abs/gr-qc/0207106

entitled

"Quantum gravity, shadow states, and quantum mechanics"

 

[marcus]

Originally posted by eigenguy
Again, I think the best place to start is with this paper

http://arxiv.org/abs/gr-qc/0207106

entitled

"Quantum gravity, shadow states, and quantum mechanics"

I hear it, eigenguy! I have it printed out. Looks like a busy day.

 

[selfAdjoint]

Originally posted by marcus
I believe you mentioned having to be out of town on the weekend of the 31 thru 1 and were wondering if there was some way of getting copies of the talks to be given at Strings Meets Loops symposium at Berlin

I guess you would be particularly interested in what Lewandowski is going to say about the Hamiltonian constraint

also Thiemann probably is interested and may have an advance copy

have you or would you think it OK to ask?

Also maybe some other poster here knows of how we can get
copies of the talks after they are given. Will they be at the Albert Einstein Institute website? Does anyone know?

BTW I am interested in getting a copy of Rovelli's symposium talk on spin foams when it is available (this is the one I am most curious about)

Let me think it over for a while, I may ask him, but I doubt if it's online so it's unsure when we could get it. I can ask where and when they would be posted and if we could have access to them. I think the issue of just what, in the fill modern context, a geoemetry on a manifold is is very much to our purose here.

BTW, give me some advice. I was working on an elementary presentation of the connection one-form using the torus as a principle bundle over the circle with group U(1). The advantage beig that everything can be visualized, T(P), V(P), H(P) and so on. But the problem is that the Lie Algebra of U(1) collapses to just the one dx, and the curvature coefficients become trivial. But the next compact Lie Group up from U(1) is SU(2) and that has a nice familiar (to some) Lie algebra, generated by the Pauli matrices, but it can't be visualized. So which would you recommend me to use?

 

[marcus]

Originally posted by Ambitwistor on 24 October 12:57
... It is commonly thought that there should be a spin foam model that corresponds exactly to the canonical theory, but nobody knows what it is, though there has been a little work relating the approaches, by Arnsdorf, Livine, Alexandrov, etc. (gr-qc/0110026, gr-qc/0207084, gr-qc/0209105)...

this last link, to gr-qc/0209105, proved a really good lead
it's the Alexandrov/Livine paper that Livine cites in the abstract to his thesis:

"...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..."

this, to me, seems to have some shock value. can this be right?
the Barrett-Crane (that Baez and everybody have been investigating) can this truly correspond (not to more familiar loop gravity versions but) to the Alexandrov version called "covariant loop gravity", that seems to be different in some significant ways.

the area formula, depends now on sums of square roots of Casimir operators of the lorentz group. SelfAdjoint or Ambitwistor would either of you care to give a brief explanation of Casimirs? Please. We are being deprived of our p's and q's and given Casimirs instead.

Oh, the link to Livine's thesis is
http://arxiv.org/gr-qc/0309028
I gave it in another thread but should put it here as well
in case anyone wants to look. It strikes me as remarkable,
new things dispersed widely throughout---for instance the
Lorentzian Barrett-Crane spin foam discussion is pages 135-140,
but there are other goodies too.

 

[selfAdjoint]

the area formula, depends now on sums of square roots of Casimir operators of the lorentz group. SelfAdjoint or Ambitwistor would either of you care to give a brief explanation of Casimirs? Please. We are being deprived of our p's and q's and given Casimirs instead.

Kaku: A Casimir of a group is an element of the group that commutes with every generator of the group.

Baez: A Casimir of a group is a generator of the center of the enveloping algebra of the group.

O'Raifeartaigh ("Group Structure of Gauge Theories", Cambridge University Press, 1986) "A concept useful for Lie Algebras...is that of the enveloping algebra...the linear span of the... symmetric products X_aX_b, X_aX_bX_c... An important property of the enveloping algebra is that it admits non-trivial central elements, i.e. elements that commute with all the elements of the Lie Algebra. According to Schurs lemma, the central elements are multiples of the identity in any irreducible representation.

The most usefuk of the central elements is the second degree Casimir element C₂ = - g^abX_aX_b where g^ab is the Cartan metric.


For SU(2) the second order Casimir element determines the representations. For higher groups there are families of Casimir elements and other elements derived from them that do the same.

 

[lethe]

Originally posted by Ambitwistor
Casimirs are just invariants you can construct from a Lie algebra. For instance, the J^2 = Jx^2 + Jy^2 + Jz^2 operator in SU(2); [/url]

I believe this quadratic fellow is a casimir invariant in any Lie algebra, that is sum over all vectors squared (in the right basis) commutes with anything.

 

[marcus]

Thanks all, Ambitwistor's second link corroborates what selfAdjoint says Baez says and is perhaps worth quoting:

"Since most mathematicians don't know what physicists mean by
"Casimirs", let me translate your question into math jargon:
Q: Given a finite-dimensional Lie algebra, is the dimension of a
maximal abelian subalgebra equal to the minimal number of generators of the center of its universal enveloping algebras?
(A physicist's "Casimir" is a mathematician's "element of the
center of the universal enveloping algebra".)"

------the rest is just concerned with loop gravity area formula---

Livine doesn't give a general definition of casimirs, but uses short simple formulas for them in the SL(2,C) case of interest.
On page 96 of his thesis, for SU(2) and SL(2,C), in case anyone should want to know them without going through generality.

And also includes a short appendix at the end where he describes the generators of sl(2,C) and describes the two casimirs of sl(2,C) as
C₁ = J² - K²
C₂ = JK

The irreducible representations are cataloged by pairs of numbers (n, m) and the Casimirs take on numerical values

C₁ = n² + m² - 1
C₂ = inm

Given a surface and a spin network state of gravity, the area of the surface is the sum of all the piercings by links the the network
where each link that pierces the surface contributes
planck area times the square root of a numerical combination of the casimirs. So the area formula in the Lorentzian case he is developing is reminiscent of the area formula in the SU(2) case that we have had since, I guess, Rovelli and Smolin work of mid 1990s. But the spin network quantum states are Lorentzian, that is the "colors" on the legs of the network are pairs of numbers indicating irreps of SL(2,C) rather than just half-integers indicating irreps of SU(2). then there is some theorem that the expected value of the area is the same whether you use one kind of quantum state or another---at least I hope so! I've seen indications of some such theorem deriving SU(2) loop gravity from covariant loop gravity but haven't had time to look at it in any detail. Livine is the person to have lunch with if you want to find out about quantum gravity, in my humble estimation. (Or his thesis director)

 

[selfAdjoint]

Thanks for digging into that 165 page thesis! Very interesting work, but having seen what happened to Thiemann, I'll wait till after the meeting before I cheer. I did email him to ask for the papers BTW, but he hasn't responded yet.

My daughter and I will be leaving for Illinois probably the afternoon of Hallowe'en, and returning most likely the afternoon of All Souls' Day (Those are October 31 and November 2 respectively for the calendrically impaired). I just might be able to post on Saturday, but I won't have access to email.

About that spectrum, everybody assumed after it was shown that length and area were quantized that they would come in discrete chunks. But of course without the eigenvalue spectrum they couldn't be sure of that. Now Livine comes along and claims to have computed the spectrum, and gotten a continuous one for the geometry operators. So there's the fox among the pigeons.

 

[marcus]

the calendrically impaired may actually be historically impaired
since the year's cycle of feastdays is a kind of Fourier transform
of human history----or the individual in history

Moses, Judas Maccabeus, Esther, Columbus, Martin Luther King Jr, Lincoln, one could do better and one could do worse

Webster's big gives three adjective formations of calendar:

calendrical
calendric
calendrial

so one could also be calendrially competent
or calendricly aware

 

[eigenguy]

I can understand surprise at the result that spacelike and timelike intervals in 3D LQG are continuous and discrete respectively, but so what? It's only the 4D case that matters, right? Maybe I'm missing something? What is the physical significance of this?

 

[selfAdjoint]

Thanks for that clarification.

 

[marcus]

Originally posted by Ambitwistor
After the work on 3D LQG, in his thesis (gr-qc/0309028) he went on to consider 4D models that have this property as well.

another remarkable thing, it seems to me anyway, is Asht/Lewand. used the compactness of SU(2) in an essential way to construct their measure on A----they took a limit of cartesian powers of Haar measure to make the celebrated AL measure. Now Livine is working with a non-compact group SL(2,C) and in what I've read it has always been described as a challenge to get a measure in the non-compact case, one of the outstanding problems.
He does this in clear, spare, elegant style. It takes 10 pages (Chapter 4, pages 47-57) and a theorem out of algebraic geometry---Rosenlich's theorem.
In my opinion those 10 pages are a thesis in themselves. I could easily be mistaken of course. He doesn't just wave hands, he let's you see just how he gets the measure---generalizing Ashtekar/Lewandowski to the non-compact case---and it has the columbus-egg simple-once-you-see-how feel. If there is some flaw in the proof, we can be sure that Jerzy Lewandowski will find it. Oh boy. (If Ashtekar and Lewandowski had seen how to extend to non-compact case you bet your booties they would have.) We must check out Rosenlich's theorem at PF. chortle chortle

 

[eigenguy]

Originally posted by Ambitwistor
After the work on 3D LQG, in his thesis (gr-qc/0309028) he went on to consider 4D models that have this property as well.

Are these 4d models as "tenable" as the conventional ones?

 

[marcus]

Originally posted by Ambitwistor
They are not yet well understood, as far as I know. I haven't read Livine's thesis, though.

there is a thought-provoking discussion in Livine's thesis of what remains incomplete or fuzzy----starting on page 106 with section 8.4 "The alternative: a covariant Loop Gravity"

He is talking about constructing a more general framework from which both canonical LQG and the Barrett-Crane model can be derived
and he describes this as an "explicit linkage" (lien explicite) and later on page 109 as a "bridge" (pont) between the canonical formalism and covariant spin foams.

And he describes the preliminary or partial character of the result:

"I am only going to give a representation of the Dirac brackets at a finite number of points of Σ. Nevertheless in this partial quantization exactly the same kinematic structures are found as in the Barrett-Crane spin foam model which I described in Part IV. In this way, the quantization provides an explicit linkage between the canonical formalism and the covariant spin foam theories..."

On page 108: "...In effect, these are the only points where we know the normal to [the submanifold] Σ explicitly. The other points are rendered fuzzy in space time." (fuzzy = "flous")

On page 109: "So the theory is quantified at a finite number of points. I do not know how to explicitly quantify the whole, although it seems possible to do that using the algebra of cylinder functions described earlier, or the limit of cylinder functions under refinement. Nevertheless we obtain simple spin networks which diagonalize the area operator of a surface and the operator's spectrum is continuous. These same spin networks are equally the kinematic states of spin foam models and form, therefore, a bridge
between the canonical formalism and the covariant formalism of spin foams. Finally, one obtains a quantum theory (and a spectrum of the the area operator) which is independent of the Immirzi parameter g. This result is compatible with the implementation of the path integral realized in [70], which shows that it is independent of the Immirzi parameter..."

this stirs things up some, I'd say

the reference [70] was to a paper by Sergei Alexandrov and D Vassilevich
http://arxiv.org/gr-qc/9806001
and also Phys. Rev. D, called
"Path integral for the Hilbert-Palatini and Ashtekar gravity"