June 16, 1995

This Week's Finds in Mathematical Physics (Week 56)

John Baez

I got a copy of the following paper when I showed up in Warsaw:

1) Lee Smolin, Linking topological quantum field theory and nonperturbative quantum gravity, available as gr-qc/9505028.

and then I spent a fair amount of time reading it and thinking about it throughout the conference. If the big hypothesis formulated in this paper is correct, I think we are on the verge of having a really beautiful theory of 4-dimensional quantum gravity, at least given certain boundary conditions. Mind you, I just mean a really beautiful theory, not necessarily a physically correct theory. But beautiful theories have a certain tendency to be right, or at least close, so let me explain this hypothesis.

First of all, we have to remember that Ashtekar reformulated Einstein's equation so that the configuration space for general relativity on the spacetime R x S, instead of being the space of metrics on a 3-manifold S, is a space of connections on S. A connection is just what a physicist often calls a vector potential, but for any old gauge theory, not just electromagnetism. Different gauge theories have different gauge groups, so I had better tell you the gauge group of Ashtekar's version of general relativity: it's SL(2,C), the group of 2x2 complex matrices with determinant equal to 1. And I should probably tell you which bundle over S we have an SL(2,C) connection on... but luckily, all SL(2,C) bundles over 3-manifolds are trivial, so I can cut corners by saying it's the trivial bundle. We can think of a connection A on the trivial SL(2,C) bundle over S as 1-forms taking values in the Lie algebra sl(2,C), consisting of 2x2 complex matrices with trace zero.

Okay, so naively you might think a state in the quantum version of general relativity a la Ashtekar is just a wavefunction ψ(A). That's not too far wrong and I won't bother about certain nitpicky technicalities here (again, for the full story try net.tex). But there's one very important catch I can't ignore: general relativity has constraint equations, meaning that ψ has to satisfy some equations. The first constraint, the Gauss law, just says that we must have

                      ψ(A) = ψ(A')

whenever A' is the result of doing a gauge transformation to A. Or at the very least, this should hold up to a phase; the point is that ψ is only supposed to record physically significant information about the state of the universe, and two connections are physically equivalent if they differ by a gauge transformation. The second constraint, the diffeomorphism constraint, says we need to have

                     ψ(A) = ψ(A')

when A' is the result of applying a diffeomorphism of space, S, to A. Again, the point is that two solutions of general relativity are physically the same if they differ only by a coordinate transformation, or - roughly the same thing - a diffeomorphism. The third constraint is the real killer. It's meaning is that ψ(A) doesn't change when we do a diffeomorphism of spaceTIME to the connection A, but it's usually formulated `infinitesimally' as the Wheeler-DeWitt equation

                        H ψ = 0

meaning roughly that the time derivative of ψ is zero. Think of it as a screwy quantum gravity version of Schrodinger's equation, where dψ/dt had better be zero!

It's hard to find explicit solutions of these equations. Indeed, it's hard to know what the heck these equations mean in a sufficiently precise way to recognize a solution if we found one! However, things were even worse back in the old days. Back in the old days when we thought of states as wavefunctions on the space of metrics, we didn't know ANY solutions of these equations. But nowadays we are very happy, because we know infinitely many times as many solutions! To be precise, we now know ONE solution. This is called the Chern- Simons state, and it was discovered by Kodama:

2) H. Kodama, Holomorphic wavefunction of the universe, Phys. Rev. D42 (1990), 2548-2565.

Now actually people have proposed other explicit solutions, but personally I have certain worries about all those other solutions, and I am not alone in this, whereas everyone seems to agree that, no matter how you slice it, the Chern-Simons state is a solution.

Now there is a slight catch: the Chern-Simons state is a solution of quantum gravity with cosmological constant. This is an extra term that Einstein threw into his equations so that they wouldn't make the obviously ridiculous prediction that the universe is expanding. When Hubble took a look and saw galactic redshifts all over, Einstein called this extra term the biggest blunder in his life. That kind of remark, coming from that kind of person, might make you a little bit reluctant to get too excited about having found a state of quantum gravity with this extra term thrown in! Luckily it turns out that you can take the limit as the cosmological constant goes to zero, and get a state of the theory where the cosmological constant is zero. I like to call this the `flat state', because it's zero except where the connection A is flat.

(In fact, if the space S is not simply connected, there are lots of different flat states, because there is what experts call a moduli space of flat connections, i.e., lots of different flat connections modulo gauge transformations. Not many people talk too much about these flat states, but I do so in my paper net.tex and also the harder one knot.tex.)

Now what is this Chern-Simons state? Well, there is a wonderful thing you can compute from a connection A on a (compact oriented) 3-manifold S, called the Chern-Simons action:

            CS(A) = integral_S tr(A ^ dA + (2/3)A ^ A ^ A)

which looks weird when you first see it, but gradually starts seeming very sensible and nice. The reason why folks like it is that it doesn't change when you do a small gauge transformation - i.e., one you can get to following a continuous path from the identity - and it changes only by an integral multiple of 8π^2 if you do a large gauge transformation. Plus, it's diffeomorphism-invariant. It's incredibly hard to write down many functions of A with these properties, so they are precious. There are deeper reasons why it's so nice, but let's leave it at that for now.

So, the Chern-Simons state is

                ψ(A) = exp(-6 CS(A)/Λ)

where Λ is the cosmological constant. Don't worry about the factor of 6 too much; depending on how you set up various things you might get different numbers, and I can never keep certain factors of 2 straight in this particular calculation. Note however that it looks as if things go completely haywire as Λ approaches zero, which is why my earlier remark about the `flat state' is a bit nontrivial.

Why does this satisfy the constraints? Well, I just said above that the Chern-Simons action was hand-tailored to have the gauge-invariance and diffeomorphism-invariance we want, so the only big surprise is that we also have a solution of the Wheeler-DeWitt equation. Well, we do: a two-line computation shows it.

But clearly nature, or at least the goddess of mathematics, is trying to tell us something if this Chern-Simons state, which has all sorts of wonderful properties relating to 3-dimensional geometry, is also a solution of the Wheeler-DeWitt equation, which is all about 4-dimensional geometry, since it expresses the invariance of ψ under evolution in TIME. I have been thinking about this for several years now and I think I finally really understand it. There are probably people out there to whom it's perfectly obvious, because it's not really all that complicated, but unfortunately none of these people has ever explained it. Let me briefly say, for those who know about such things, that it all comes down to the fact that the Chern-Simons action was born as a 3-dimensional spinoff of a 4-dimensional thing called the 2nd Chern class. (If you want more details, I explained this as well as I could at the time in knot.tex.)

What is the physical meaning of the Chern-Simons state? As far as I know Kodama's paper hasn't been vastly surpassed in explaining this. He shows that in the classical limit this state becomes something called the anti-deSitter universe, a solution of Einstein's equation describing a (roughly) exponentially expanding universe. If you are wondering what this has to do with Einstein's introduction of the constant to KEEP the universe from expanding, let me just say this. In our current big bang theory the universe is expanding, but the presence of matter, or any sort of positive energy density, tends to pull it back in, and if there is enough matter it'll collapse again. Einstein's stuck in a cosmological constant term to give the vacuum some negative energy density, exactly enough to counteract the positive energy density of matter, so things would neither collapse nor expand, but instead remain in an (unstable, alas) equilibrium. In the deSitter universe there's no matter, just a cosmological constant of the opposite sign, so that the vacuum has positive energy density. In the anti-deSitter universe (invented by deSitter's nemesis anti-deSitter) there's no matter either, but the cosmological constant has the sign giving the vacuum negative energy density, which pushes the universe to keep expanding faster and faster.

Now in addition to this physical interpretation, the Chern-Simons state also has some remarkable relationships to knot theory, which were discovered by Witten and, since then, studied intensively by lots of people. I have written a lot in This Week's Finds about this! But briefly, there should be an invariant of knots and links associated to any state of quantum gravity, and the one associated to the Chern-Simons state is the Kauffman bracket, a close relative of the Jones polynomial, which is distinguished by having a very simple, beautiful definition, and also lots of wonderful relationships to an algebraic structure, the quantum group SU_q(2). I should add that in addition to an invariant of knots and links, a state of quantum gravity should also give an invariant of spin networks, and indeed the Kauffman bracket extends to a wonderful invariant of spin networks. One can read about this in many places, but perhaps the most detailed account is Kauffman and Lins' book referred to in "week30".

So the question arises: are all these wonderful features of the Chern-Simons state of quantum gravity very special things that tell us very little about quantum gravity in general, or are they important clues that, if we understood them, would reveal a lot about the nature of all states of quantum gravity?

Of course, everyone who has fallen in love with the beauty of Chern-Simons theory would like the answer to be the latter. Louis Crane, for example, is deeply convinced that Chern-Simons theory is indeed the key to the whole business. He has written many papers on the subject, most of which I've gone over in earlier Finds, and also one brand new one, which is actually a very readable introduction to the grand scheme he has in mind:

3) Louis Crane: Clock and category: is quantum gravity algebraic?, to appear in the November 1995 special issue of Jour. Math. Phys. on diffeomorphism-invariant physics, preprint available as gr-qc/9504038.

This grand scheme involves a thorough refashioning of quantum gravity in terms of category theory, and uses some of the very beautiful mathematics of n-categories, but neglecting all these subtleties, let us simply say that he argues that if the universe is IN the Chern-Simons state, there is no need to understand any other states! He doesn't really think all there is in the universe is gravity, of course, so he envisages a souped-up theory containing other forces and particles, but he argues that a generalization of quantum gravity to include all these other forces and particles will still have a special state, and that that's the state of the universe.

Being a conservative fellow myself, I prefer to remain agnostic on this issue, but I did write a paper showing how, if you assumed that space, the manifold above I called S, is a 3-dimensional sphere - a so-called S^3 - then if quantum gravity was in the Chern-Simons state, there would be very nice Hilbert spaces of "relative states" on each "half" of space. The relative state notion is often associated with Everett, who made a big deal out of the fact that, even if a two-part system was in a single state (a "pure state" in the language of quantum theory), each part could be regarded as being in a probabilistic mixture of lots of states (a "mixed state"). Whether or not you like the "many-worlds interpretation" of quantum theory which Everett's thesis spawned, it is true that a single pure state on a two-part system specifies a whole space of states on each half. So my idea was to take S^3, arbitrarily split it into two balls (D^3's in math jargon), and work out the space of states on each half. If you want to wax rhapsodic of this you can call one half the "observer" and the other the "observed", though it's crucial that there is a symmetry interchanging the two - there's not any way to tell them apart.

On the more technical side, there is a lot of nice (though well- understood) knot theory involved. For example, a special property of the quantum group SU_q(2) - called the "positivity of the Markov trace", and discovered by Jones of Jones polynomial fame - equips each space of states with an inner product, even in this situation where we have no idea of an inner product to begin with. This paper is:

4) John Baez, Quantum gravity and the algebra of tangles, Jour. Class. Quant. Grav. 10 (1993), 673-694, also available (without the all-important pictures!) as tang.tex.

So what has Lee Smolin done? Actually I have spent so much time leading up to it that now I'm too tired to do it justice! So I'll explain it next time. But let me just say, in order to tantalize you into tuning in to the next episode, that he carefully analyzes quantum gravity on a ball, imposing boundary conditions that let you see why relative states of Chern-Simons theory give states of quantum gravity. And then he makes the hypothesis that I mentioned at the beginning of this article. This is that all states of quantum gravity with these boundary conditions come from relative states of Chern-Simons theory. He even gives some good evidence for this hypothesis, coming originally from Hawking's work on the thermal radiation produced by black holes! (To be continued.)

© 1995 John Baez