From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Spin foam continuum limits Date: 14 Sep 2000 21:59:07 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8pk4mf$2rn$1@crib.corepower.com>, Nathan Urban wrote: >On the off chance that any spin foam experts read this newsgroup, I >was wondering what the current state of progress is on the problem of >finding continuum limits of spin foam models. Though I try to avoid being pegged as an "expert" on anything (jack of all trades, expert on none), I guess this question is meant for me. First of all, I prefer the term "large-scale limit" instead of "continuum limit"". Mathematically, these are equivalent: they both amount to letting the size of the spacetime region under consideration go to infinity in Planck units. One can think of this either as making the region *big* or making the Planck length *small*. But I think one gets more insight into what's actually going on here if you say: we're treating the Planck length as fixed (since it is) and considering what happens in a region of spacetime much larger than the Planck length. See, this has a slightly different feel to it than lattice quantum field theory, where we let the lattice spacing go to zero and try to get a "continuum limit" where the lattice is gone. In spin foam models, spacetime is not a continuum, so the term "continuum limit" can lead to confusion about what we're trying to do. Second, I can answer your question pretty quickly since we know so little about this issue. (If I'm gonna be an expert, I prefer to be an expert on a topic about which little is known.) We understand the large scale limit very well for spin foam models that are TQFTs, like the G/G WZW model in 2d, the Turaev-Viro model in 3d, and the Crane-Yetter model in 4d. We could easily understand it if we wanted to for theories that are small perturbations of TQFTs. This works best for theories in 2d, which have an exactly solvable "dilute gas limit". In this limit, you have a big spin foam where only boring TQFT stuff happens at most vertices, and a few widely separated vertices where interesting interactions happen. I explained how to exactly solve these models in my paper "Spin foam perturbation theory", available on gr-qc or my website. (If there were a graduate student looking around for a manageable project to do as a kind of warmup for thesis work, they might find it interesting to study the "slight deviations from flatness" that you get in this kind of theory.) We are essentially clueless when it comes to physically interesting 4d spin foam models - e.g. the Lorentzian or Riemannian Barrett-Crane models. There is lots of good stuff to do here, but it will probably take about 10 years before we really understand the large-scale limit of these theories. One first step towards understanding the Barrett-Crane models is to understand the asymptotics of the 10j symbol, which is what you use to compute the amplitude for a 4-simplex in these models. It should be related to the exponential of the Regge action, but (as we've discussed already) not simply equal to it. These asymptotics are of limited value, because they concern what happens when the 4-simplices are large. What we really want to understand is what happens when spacetime is built from a lot of very small 4-simplices, not a few big ones! Nonetheless, physics consists of doing things we can actually do. If we study the asymptotics of the 10j symbol, we will better understand the relation between quantum geometry and ordinary geometry. That's bound to be a good thing. Once one knows the asymptotics of the 10j symbol one can do this: take a flat geometry built from largish 4-simplices and see if perturbations this geometry give gravitons. This is a doable project, but again, it's of limited value, because it doesn't take renormalization- group effects into account. In other words, besides the problem of of treating the 4-simplices as large (in Planck units), we have no good reason to think that near the Planck scale spacetime is almost flat with a few small wiggles thrown in; we could easily be in a phase where it's "foamy". For someone desperate to understand the large-scale limit of the Barrett-Crane model *right now*, its tempting to do some numerical simulations. If I were more of a hacker I might try this. But it's a highly nontrivial business, not an easy way out. The people who actually make a living doing simulations of this sort, like Loll and and Ambjorn, are wisely working on models where the calculations are easier. These models are too ugly to be "right" in a definitive and final sense, but they may shed a lot of light on what's going on. In short, the situation in quantum gravity is just as usual: desperate but not entirely hopeless. At least now we have some concrete models to study. When I was your age.... ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Spin foam continuum limits Date: 16 Sep 2000 00:02:28 GMT Newsgroups: sci.physics.research In article <8pu26p$itj$1@crib.corepower.com>, Nathan Urban wrote: >John Baez wrote: >> We understand the large scale limit very well for spin foam models >> that are TQFTs, like the G/G WZW model in 2d, the Turaev-Viro model >> in 3d, and the Crane-Yetter model in 4d. >> We could easily understand it if we wanted to for theories that are >> small perturbations of TQFTs. >Do we understand it (without resorting to perturbation theory) for *any* >spin foam model that isn't a TQFT, "interesting" or not, 4D or not? First of all, I don't see anything wrong with perturbation theory so long as you can prove that the perturbation series is convergent, or Borel summable, or something like that. It's not as if god hates functions that are defined by Taylor series. The trouble only starts when you have a power series and no clue as to what it means. In my paper "Spin Foam Perturbation Theory", I proved that perturbation about TQFTs gives a class of spin foam models where everything can be computed as a *convergent* power series in the coupling constant that governs the perturbation. This is about as understandable as something can possibly be. It's easy to cook up lots of theories in this class. All we need now is for some grad student to come along and play around with specific examples and see what interesting things can be said about them. There is a limit to how interesting they can be, because they're utterly trivial except in 2d spacetime. But anyway, to answer your question: there are non-TQFTs like 2d Yang-Mills theory which can be described via spin foam models and can be completely understood. (There's an essentially infinite literature on 2d Yang-Mills theory, because whenever someone learns about this theory they fall in love with it and write a paper about it.) You can describe 2d Yang-Mills theory as a spin foam model without resorting to perturbation theory: this is part of what Witten did in his famous paper on 2d Yang-Mills. But it's also possible - and enlightening - to describe it as a theory in the above class: namely, as a perturbation about 2d BF theory, which is a TQFT. Witten also wrote about this, though of course he didn't talk about "spin foams". I don't know any spin foam models which we understand, but can *only* be understood without use of perturbation theory. By now, in fact, I've almost listed all the spin foam models we understand. >> This works best for theories in 2d, >> which have an exactly solvable "dilute gas limit". In this limit, >> you have a big spin foam where only boring TQFT stuff happens at >> most vertices, and a few widely separated vertices where interesting >> interactions happen. >So what is there that one would want to know about "slight deviations >from flatness" in such models? Well, if I were gonna do this, first I'd compute a bunch of stuff, and then figure out which of these things someone might want to know. These models are very easy to do calculations with, so this "messing around" approach is probably the most efficient. But if I had to *guess* what *might* be nice knowing about these models, I'd say this: in any spin foam model we can try to compute the holonomy around a contractible loop. This will be a G-valued random variable if G is our gauge group. If we're doing 2d BF theory and our loop is contractible, this random variable is a delta function sitting at the identity of G. In other words: the geometry is perfectly flat. No quantum fluctuations from flatness! But if we add a small perturbation to the theory, we get quantum fluctuations. I.e., our random variable is no longer a delta function. These size of these fluctuations depends on the area enclosed by the loop. This has been worked out ad nauseam in the case of 2d Yang-Mills theory (which can be seen as a perturbation about 2d BF theory). See Witten's paper, which is cited in my big review article about spin foams. It would be easy and fun to do this for other examples too. It's too straightforward to be profound, but it'd be a good way to learn some stuff. >> [...] These asymptotics are >> of limited value, because they concern what happens when the 4-simplices >> are large. What we really want to understand is what happens when >> spacetime is built from a lot of very small 4-simplices, not a few big >> ones! >That would correspond to a large but still very quantum spacetime? Well, when the simplices get a lot bigger than the Planck scale, the quantum fluctuations in their geometry become small compared to their actual size, so they start looking classical. So a spacetime made of big 4-simplices is a spacetime made of nearly classical 4-simplices. On the other hand, now that I think of it, some of these asymptotic formulas get accurate really fast, so that they give good results even for simplices that have face areas equal to (say) 15 times the Planck area. So "a lot bigger than the Planck scale" can still be quite small compared to atoms. >> Once one knows the asymptotics of the 10j symbol one can do this: >> take a flat geometry built from largish 4-simplices and see if >> perturbations this geometry give gravitons. >How related (or not) is that likely to be to the existing weave state >constructions in the canonical formalism? Hmm, good question. Probably *very* related, but I haven't thought about this much, so I don't see precisely how. There is probably something very fun to think about here. Go for it!!! >> This is a doable project, >> but again, it's of limited value, because it doesn't take renormalization- >> group effects into account. In other words, besides the problem of >> of treating the 4-simplices as large (in Planck units), we have no >> good reason to think that near the Planck scale spacetime is almost >> flat with a few small wiggles thrown in; we could easily be in a phase >> where it's "foamy". >Couldn't it give a picture of an almost flat spacetime with gravitons in >it though? i.e. at least as good as linearized gravity? Yes. This sort of idea is very related to linearized gravity. >Would it be inherently incapable of doing anything that goes beyond current >graviton-based calculations? I'd first see if the plane flies, then try to fly it somewhere new. >> For someone desperate to understand the large-scale limit of the >> Barrett-Crane model *right now*, its tempting to do some numerical >> simulations. If I were more of a hacker I might try this. But it's >> a highly nontrivial business, not an easy way out. The people who >> actually make a living doing simulations of this sort, like Loll and >> and Ambjorn, are wisely working on models where the calculations are >> easier. >Well, Loll doesn't do the simulations, she just thinks about them. :) Even I *think* about them. >On a related note, I believe Fotini Markopoulou is planning to spend a >fair bit of time this year at the AEI trying to apply her new algebraic >coarse-graining techniques to the same models Loll and Ambjorn are >working on, to try to take renormalization-group effects into account in >quantum gravity (and hopefully lead to better applications to spin foams). >It's for the same reason, of course -- the calculations are easier -- and >also because the continuum limit of those models is already understood. That's good. I hope she gets somewhere with this - it's important stuff. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Spin foam continuum limits Date: 16 Sep 2000 23:13:32 GMT Newsgroups: sci.physics.research In article <8pulkm$rcj$1@crib.corepower.com>, Nathan Urban wrote: >In article <8pudak$qhm$1@mortar.ucr.edu>, baez@galaxy.ucr.edu (John >Baez) wrote: >> In my paper "Spin Foam Perturbation Theory", I proved that perturbation >> about TQFTs gives a class of spin foam models where everything can be >> computed as a *convergent* power series in the coupling constant that >> governs the perturbation. This is about as understandable as something >> can possibly be. It's easy to cook up lots of theories in this class. >Do any of them seem more "interesting" than others? Well, 2d Yang-Mills is the probably the most interesting, but everybody and his brother has studied that one already. If you study this stuff you *must certainly study that one*, but don't expect to come up with something people don't already know. Actually, the funny thing about most of these theories is that I defined them in a combinatorial/algebraic way, and have no idea how to obtain them by quantizing a classical field theory given by a Lagrangian! The one example where a Lagrangian is known is 2d Yang-Mills theory. I bet most or all of the other theories come from Lagrangians too, but one might have fun trying to figure out what these Lagrangians are. I'd start by understanding 2d Yang-Mills and then see how you can tweak that to get a wad of other theories in the above class. Some stuff about Lagrangians: Start with a 2d oriented manifold M as our spacetime. The basic fields in 2d BF theory are a connection A on a principal G-bundle P -> M and an Ad(P)-valued function which we call B. If we work locally and pick a local trivialization of P we can think of A as a g-valued 1-form, where g is the Lie algebra of G. Similarly, we can think of F as a g-valued 2-form and B as g-valued function. The Lagrangian for 2d BF theory is tr(B ^ F) where we assume our Lie algebra g consists of matrices and "tr" is the usual trace (to keep things simple). Exercise: work out the equations of motion and show they are F = 0, d_A B = 0, where d_A is the exterior covariant derivative as explained in (say) the book "Gauge Fields, Knots and Gravity". To get 2d Yang-Mills we need to introduce a metric, which lets us define a Hodge star operator *. Now the usual Lagrangian for Yang-Mills is (1/g) tr(F ^ *F) where g is the Yang-Mills coupling constant. Exercise: work out the equations of motion and show they are d_A *F = 0, i.e. the Yang-Mills equations. (Note that these classical equations do not depend on the coupling constant. However, the quantum version of Yang-Mills does depend on the coupling constant.) But we can also get Yang-Mills from BF as follows. Take as our Lagrangian tr(B ^ F + g B ^ *B) where g is a coupling constant. This is an obvious way to modify the original BF Lagrangian when we have a Hodge star operator at our disposal. What equations of motion does it give? Well, if we vary B we get the equation F + 2g *B = 0 and if we vary A we get the equation d_A B = 0. But the first equation lets us solve for B in terms of F, at least when g is nonzero. It says: B = (1/2g) *F so the second equation really says d_A *F = 0 Voila! Yang-Mills equation! Sometimes this trick is called "BF-YM theory" - it's a way of thinking of Yang-Mills as a perturbation of BF theory, which reduces back to BF theory as g -> 0. (It's hard to see this happening at the classical level, where the Yang-Mills equations don't depend on g. It's better to go to the quantum theory - see Witten's paper on 2d gauge theories for that.) Anyway, it might be fun to mess with the 2d BF Lagrangian in other ways and see if you can get other theories, and then see if the quantum versions of these theories are equivalent to the spin foam models that I get by perturbing the spin foam model for the quantized 2d BF theory. >> I don't know any spin foam models which we understand, but can *only* >> be understood without use of perturbation theory. By now, in fact, >> I've almost listed all the spin foam models we understand. >I notice the conspicuous absence of 3D Chern-Simons. Are we just as >bad off there as we are in, say, 4D Barrett-Crane? No! Chern-Simons theory is perfectly nice and well-understood. It's just not given by a spin foam model - at least not directly. If you want to think of it as a spin foam model, it's best to use the Crane-Yetter model, which is a 4d theory. Chern-Simons appears as a "boundary value theory" of this 4d theory. In other words: every compact oriented 3-manifold M is the boundary of some compact oriented 4-manifold M', and Chern-Simons theory on M can be thought of as the Crane-Yetter model on M'. I seem to recall explaining this to Aaron Bergman in a recent post on s.p.r.. Except I didn't say "the Crane-Yetter model", I said "BF theory with cosmological term". That's the theory with Lagrangian tr(B ^ F + g B ^ B) in 4 dimensions. When you quantize it, you (apparently) get the Crane-Yetter spin foam model. (You'll notice how all these Lagrangians I've been writing look very similar. That's why all the theories I've been talking about are all really just aspects of a single thing, which Crane calls "the dimensional ladder". This stuff may seem complicated but it's really just a single big fat juicy idea. In my quantum gravity seminar, which is just about to start up here at UCR, I hope to explain this idea.)