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 dont 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 dont have the ability to judge if that article is in any way significant---it seems suggestive to me but I dont 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
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 doesnt 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.
Yes, a spin foam is a "history", describing how one state evolves into another. I don't think path integrals are that confusing, though... they're quite elegant. 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, 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. I don't know what you mean by "Hamiltonian" vs. "hybrid" toy model. The paper you mention is concerned with obtaining a propagator from a canonical theory; this bears some relation to spin foams, since spin foam transition amplitudes are propagators. But the main idea is just to examine how to define the observbles and their evolution in a generally covariant theory, and to see how the problem of time plays out in a toy model.
Baez himself isn't so sure that it is a disease, because it's far from clear whether the continuum limit of the theory should be dominated by the asymptotic behavior of the 10j symbols in the first place.
Re: Re: spin foam models 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?
Re: Re: Re: spin foam models Take a look at Thiemann's "Phoenix Project" paper: http://arXiv.org/abs/gr-qc/0305080 He has some references to the debates. The cited problems raised with the Hamiltonian constraint are discouraging, but it's not conclusive whether they're really fatal problems. (Still, he is trying to remedy them.)
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 happend 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.
Re: Re: spin foam models 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 doesnt 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
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!
Re: Re: Re: spin foam models My understanding is that the wavefunction is defined not on the connection but on it's holonomies along the edges of spin networks.
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.
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.
Re: Re: Re: Re: spin foam models 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 dont 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
Re: Re: Re: spin foam models 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.
Re: Re: Re: Re: spin foam models That helps some. Thanks for both the Baez excerpt and your take on it!
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 dont 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: The first two seemed quite interesting, I'm going to check the third now. Why do people say foam gravity research is "bogged down"?
The talk was "2+1 Gravity and the Physical Scalar Product from Spin Foams" http://www.phys.psu.edu/events/index.html?event_id=777&event_type=17 The CGPG used to put all their talks online, but they appear not have done that this semester...
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)
Re: Re: Re: spin foam models 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.
Re: overdue replies to selfAdjoint's posts That would probably make gauge theorists, mathematicians, and mathematical physicists like Ashtekar and Baez very sad. The whole point of the original Sen connection variables was to describe the geometry of left-handed neutrino parallel transport, after all ... and the whole reason why Yang-Mills gauge theories work, like Maxwellian electromagnetism and the whole Standard Model, is because of the gauge symmetry which in turn leads to a picture of physical fields (like the electromagnetic field) being nothing but curvature. Well, you don't have a concept of distance like you do in metric geometries. But you do have parallel transport, geodesics, curvature, etc. No, that's just the usual notion of metric geometry. You want to mod out by diffeomorphisms since there are many metrics that represent the same geometry (i.e., many different coordinate systems give many metrics for the same geometry -- e.g., Schwarzschild, Eddington-Finkelstein, Kruskal-Szekeres, ... more or less.) I don't know if "people" say it ... I say it, and I know at least one spin foam researcher who found the results of Baez, Christensen, and Egan discouraging. That paper doesn't kill spin foams by any means, but it means that if spin foams are to work, it's either back to model-building (without a clear guide to models), or the 10j asymptotics aren't meaningful to the continuum limit (quite possible, but it means that people have to do a lot more work on what it means to take a continuum limit). The spin foam program has lots of different models floating around, and while it's nice to know that some of them are rigorously defined, there is no clear winner, and still nobody knows how to extract much in the way of physics out of them. Meanwhile, the canonical program is making slow but steady progress with its Fock space / shadow state / coherent state approach, whereas spin foams just have a mishmash of formally defined but largely unimplemented renormalization group ideas. Spin foams are still my favorite from a mathematical and and physical elegance standpoint, but it must be admitted, they have not yet progressed as far as the canonical approach in terms of useful physics. (Area and volume spectra, black hole entropy, quantum cosmology, etc.)