June 4, 1995

This Week's Finds in Mathematical Physics (Week 55)

John Baez

I recently went to a workshop on canonical quantum gravity in Warsaw, organized by Jerzy Kijowski and Jerzy Lewandowski, and I learned some interesting things. I'll talk about some of them in this issue, and some in the next.

Conferences are a funny thing. On science newsgroups on the net, there is very little talk about conferences. This is probably because the people who really understand conferences are too busy flying from one conference to the next to post to newsgroups very often. Academic success is in part measured by the number of conference invitations one receives, the prestige of the conferences, and the type of invitation. For example, a big plenary lecture on an impressive stage, preceded by a little warmup where someone explains how great you are, counts for infinitely many talks in those parallel sessions where dozens of people get 10 minutes each to explain their work before the moderator begins to make little coughs indicating that it's time for the next one, while all the while people drift in and out in a feeble attempt to find the really interesting talks. Still, giving any sort of talk is regarded as better than giving none, so academics spend a lot of time doing this sort of thing.

One of the great dangers of being a successful academic, in fact, is that one may get invited to so many conferences that one never has time to think. Winning the Nobel prize is purported to be the kiss of death in this respect. Of course, it's a universal platitude that the real thinking at conferences gets done not during the talks, but informally in small groups. But the funny thing is that at most conferences people are so worn out after going to a day's worth of talks that they have limited energy for serious conversation afterwards: they usually seem more interested in finding the good local restaurants and scenic attractions. If people could have conferences with no lectures whatsoever, or maybe one a day, it would probably be more productive. But the idea that a bunch of people could figure something out just by sitting around and chatting informally is absolutely foreign to our conception of "work". People expect to receive money from bureaucrats to go to conferences, but to convince a bureaucrat that you are deserve the money, you need to give a lecture, so of course all conferences have too many lectures.

Turning back towards Warsaw, a city with a marvelous mathematical history, I am reminded of Gian-Carlo Rota's biographical sketch of Stanislaw Ulam, in which (as a master of irony) he talks about how lazy Ulam was: all he wanted to do was sit around in cafes and come up with interesting conjectures and research rograms, and leave it to others to work them out. And this in turn reminds me of the Scottish Cafe, where Polish mathematicians used to hang out and write on the tablecloths, until the owner provided them with a notebook, in which many famous conjectures were formulated, and I believe prizes like bottles of wine were offered for their solutions. Was the Scottish Cafe in Warsaw? [No, Lwow.] Does it still exist? I completely forgot to check while I was there. The Banach Center, in which the conference participants stayed, comes from a later stratum of Polish mathematical history; it was built after the war, and one room still contains a portrait of Lenin. I know that because a film crew used it to shoot a scene for a historical movie!

Anyway, I enjoyed this conference in Warsaw quite a bit, because a lot of people working on the loop representation of quantum gravity were there, and I managed to have a fair number of serious conversations. Before going into what I learned there, I should say that I just found a fun thing for people to read who are interested in quantum gravity, but are not necessarily specialists:

1) Gary Au, The quest for quantum gravity, available as gr/qc-9506001.

This consists mainly of interviews with Chris Isham, Abhay Ashtekar and Edward Witten. What's nice is that the interviews are conducted by someone who knows physics. The questions and answers are technical enough to convey some of the real substance of the subject, while still (I hope) non- technical enough so that you don't have to be an expert to get a lot out of them. Isham talks mainly about the "problem of time" in quantum gravity, Ashtekar talks mainly about the loop representation of quantum gravity, and Witten talks about string theory.

Anyway, Ashtekar and a bunch of other good people were at this Warsaw conference, which is why I went. The main topics of conversation were spin networks and their use in studying the area and volume operators in quantum gravity. As I explained earlier in "week43", one may very roughly think of a spin network as a graph whose edges are labelled with "spins" 0,1/2,1,3/2, and so on, and who vertices are labelled with certain gadgets called "intertwining operators" (which in the simplest case are just the Clebsch-Gordon coefficients you learn about when studying angular momentum in quantum mechanics). Penrose introduced these as abstract graphs (see "week22" and "week41"), as a kind of substitute for thinking of space as a manifold, but more recently Rovelli and Smolin started thinking of them as graphs embedded into 3d space, and saw that these were a really natural way to describe states of quantum gravity: even better than loops, because they form an orthonormal basis! Actually, it was mainly me who proved in a really rigorous way that they form an orthonormal basis, but Rovelli and Smolin had already been doing calculations using this idea for a while. One thing they computed was the eigenvalues of the observables in quantum gravity corresponding to the area of a surface in space, or the volume of a region.

Now there are all sorts of technical caveats and subtleties that I don't want to get into here, but in a really rough sort of sense, what their answers suggest is that IF the loop representation of quantum gravity is right, and we are on the right track about how it works, then the area of surfaces comes in certain (not integer, but discrete) multiples of the Planck length squared, and the volume of regions comes in multiples of the Planck length cubed. Note: that was a big "IF". This is especially interesting because it doesn't arise by assuming from the start that spacetime has a discrete structure. In fact, their computations assume spacetime is a continuous manifold. Nonetheless this discreteness pops out. It's not completely surprising: after all, Schrodinger's equation for the hydrogen atom is a perfectly "continuous" sort of thing, a partial differential equation, but the energy of the bound states winds up being a discrete sort of thing. Still, it's sort of exciting and new.

An interesting thing happened at the conference. Renate Loll, who works on the loop representation of gauge theories and also lattice gauge theory, has recently developed a lattice formulation of quantum gravity closely modelled after the loop representation:

2) Renate Loll, Nonperturbative solutions for lattice quantum gravity, preprint available as gr-qc/9502006.

This has the wonderful feature that it's perfectly rigorous and also one can start using computers to start calculating things with it. For example, the most subtle aspect of the loop representation of quantum gravity is the Wheeler-DeWitt equation

                                 H ψ = 0

where H is an operator called the "Hamiltonian constraint". More on this later; my point here is just that physical states of quantum gravity need to satisfy this equation. Getting H to be well-defined is tricky when space is a continuum, but in Loll's lattice version of theory (which is an approximation to the full continuum theory) she has already done this, so one can now start trying numerically to find solutions and see what they look like. She has also found some explicit solutions.

Also, she did some work on the volume operator in her lattice approach, and came up with a result in contradiction to Rovelli and Smolin's paper on the subject (cited in "week43"). They had said that states corresponding to trivalent spin networks ---- spin networks with only 3 edges at each vertex --- could have nonzero volume. But using her version of the theory she computed that trivalent states --- states with only 3 nonzero spins at the edges of the lattice incident to any vertex --- all had zero volume, and that she needed at least 4 nonzero spins to get volume! The volume operator, in case you're wondering, acts as a certain sum over vertices: each one winds up contributing a certain finite amount of volume, which the theory allows you to compute.

This led to a whole lot of discussion and scribbling on the blackboards of the Banach center. I found it truly delightful to see all these physicists drawing pictures of spin networks and doing graphical computations just the way a knot theorist like Kauffman does all the time. It was as if the universe had this spin network aspect to it, and everyone was finally starting to catch on. Either that or mass delusion! I hadn't quite gotten the hang of how to compute these volume operators before, so it was a great chance to learn: one person would do a computation, then someone else would do it a different way and get a different answer, then someone else would do it yet another way and get yet another answer, and so on, so you could ask lots of questions without seeming too dumb. Even I did a computation after a while, and got zero volume for at least a certain class of trivalent vertices. The votes in favor of trivalent vertices having zero volume kept piling up. Finally Smolin noticed that he and Rovelli had made a sign mistake. This is incredibly easy to do, since there are lots of tricky sign conventions in spin network theory. Fundamentally these are due to the fact that spin-1/2 particles are fermions... but I don't think people fully understand the physical implications of this. (There is also a marvelous category-theoretic explanation of it, but I fear that if I go into that all the physicists will stop reading. Maybe some other time.) Rovelli and Smolin got pretty depressed about this for a while, but I tried to reassure them that only people who write really interesting papers ever get anybody to find the mistakes.

So perhaps we know a little more about the meaning of volume in a quantum theory of spacetime.

Spin networks are very beautiful and simple things. To learn about them, in addition to the various papers listed in the "weeks" above, one can now turn to Rovelli and Smolin's paper:

3) C. Rovelli and L. Smolin, Spin networks in quantum gravity, preprint available in LaTeX form as gr/qc-9505006.

If you are more of a mathematician, or less of an expert on quantum gravity, you might also try a review article I wrote about them, which starts with a quick summary of what the heck canonical quantum gravity is about, why it's hard to do, and why the loop representation seems to help:

4) J. Baez, Spin networks in nonperturbative canonical quantum gravity, preprint available in LaTeX form as gr-qg/9504036, or via ftp from math.ucr.edu, as the file net.tex in the directory baez.

Now so far I have been trying to make things sound simple, but here I should point out that when one talks about "states of quantum gravity" there are at least three quite different things one might mean. This is because the loop representation follows Dirac's general philosophy of quantizing systems with constraints, with some extra twists here and there. As I've repeatedly explained (e.g. "week43"), Einstein's equation for general relativity has 10 components, and if you split spacetime up into space and time (more or less arbitrarily --- there's no "best" way) 4 of these can be seen as constraints that the metric on space and its first time derivative must satisfy (at any given time), while the remaining 6 describe how the metric on space evolves in time (which makes sense, because the metric has 6 components). When you follow Dirac's procedure for quantizing the equations what you do is this. First you forget about the constraint and get a big space of states, the "kinematical state space". There are lots of mathematical choices involved here, but Ashtekar and Lewandowski came up with a particular nice way of doing this rigorously, and one calls this space of states "L^2 of the space of SU(2) connections modulo gauge transformations with respect to the Ashtekar-Lewandowski generalized measure". Spin networks form an orthonormal basis of this Hilbert space. All the stuff about area and volume operators above refers to operators on this space.

Then, however, you need to deal with the constraints. Now 3 of the 4 constraints simply amount to requiring that your states be invariant under all diffeomorphisms of space, so these are usually dealt with first, and called the "diffeomorphism constraint". Imposing these constraints are a bit tricky; naively one would first guess that this "diffeomorphism- invariant state space" is just a subspace of the original kinematical state space, but actually it's not quite so simple. In any event, there are also spin network states at the diffeomorphism-invariant level, corresponding not to particular embeddings of graphs in space, but to diffeomorphism equivalence classes thereof. This again has been used by Rovelli, Smolin and others for a while now, but it was first rigorously shown in the following paper:

5) Abhay Ashtekar, Jerzy Lewandowski, Don Marolf, Jose Mourao, and Thomas Thiemann, Quantization of diffeomorphism invariant theories of connections with local degrees of freedom, to appear in the November 1995 Jour. Math. Phys. special issue on diffeomorphism-invariant field theory, preprint available as gr-qc/9504018.

This paper is nice in part because it doesn't assume you already have read every previous paper about this stuff; instead, it describes the general plan of the loop representation before constructing the diffeomorphism- invariant spin network states. Also, buried in an appendix somewhere, it gives nice conceptual formulas for the area and volume operators, which serve as a complement to Rovelli and Smolin's explicit computations of their matrix elements in terms of the spin network basis.

Anyway, after taking care of the diffeomorphism constraint, one finally needs to take care of the Hamiltonian constraint, meaning one needs to find states satisfying the Wheeler-DeWitt equation. This is the hardest thing to make rigorous, and the most exciting aspect of the whole subject, because it expresses the fact that "physical states" of quantum gravity are invariant under diffeomorphisms of space-TIME, not just space. There is much more to say about this, but I won't go into it here.

Now besides Loll and Rovelli and Smolin, all the authors of the above paper except Mourao were at the conference in Warsaw, so there was a large contingent of spin network fans around, not even counting some other folks whose work I will get to in a while. This is why I was so eager to go there, especially because my own talk was on a rather esoteric subject which only these experts could be expected to give a darn about. Namely....

The breakthrough of Ashtekar and Lewandowski, when it came to making the loop representation rigorous, involved working with piecewise real-analytic loops rather than piecewise smooth loops. (Actually Penrose suggested this idea.) This is because piecewise smooth loops can intersect in crazy ways, like in a Cantor set, which nobody could figure out how to handle. But the price of this breakthrough was that one had to assume the 3-manifold representing space was real-analytic, and things then only work nicely for real-analytic diffeomorphisms, as opposed to smooth ones. This always bugged me, so I have been working away for about a year trying to deal with smooth loops, and finally I got smart and teamed up with Steve Sawin, and we recently figured out how to get things to work with smooth loops (at least a bunch of things, like the Ashtekar-Lewandowski generalized measure). Our paper will be out pretty soon, but for now anyone who wants a taste of the mathematical technology involved should look at:

6) Steve Sawin, Path integration in two-dimensional topological quantum field theory, to appear in the October 1995 Jour. Math. Phys. issue on diffeomorphism-invariant field theory, preprint available as gr/qc-9505040.

Loop representation ideas are applicable not only to canonical quantum gravity but also to path integrals in gauge theory, because in both cases one is doing integrals over a space of connections mod gauge transformations. Here Sawin uses them to give a rigorous formulation of 2d TQFTs in terms of path integrals. There aren't many unitary 2d TQFTs, and all of them are isomorphic to 2-dimensional quantum gravity with the usual Einstein-Hilbert action, with different values of the coupling constant, or else direct sums of such theories.

Next "week" I'll talk about cool new idea Smolin has about TQFTs, quantum gravity, and Bekenstein's bound on the entropy of a physical system in terms of its surface area.

© 1995 John Baez