February 24, 2002

This Week's Finds in Mathematical Physics (Week 177)

John Baez

This week I want to talk about some new developments in quantum gravity. But first, here's a little taste of Greg Egan's new novel:

1) Greg Egan, Schild's Ladder, Eos, May 2002. Synopsis available at http://www.netspace.net.au/~gregegan/SCHILD/SCHILD.html

Kusnanto Sarumpaet had lived on Earth at the turn of the third millennium, when a group of physicists and mathematicians scattered across the planet - now known universally as the Sultans of Spin - had produced the first viable offspring of general relativity and quantum mechanics. To merge the two descriptions of nature, you needed to replace the precise, unequivocal geometry of classical spacetime with a quantum state that assigned amplitudes to a whole range of possible geometries. One way to do this was to imagine carrying a particle such as an electron around a loop, and computing the amplitude for its direction of spin being the same at the end of the journey as when it first set out. In flat space, the spins would always agree, but in curved space the result would depend on the detailed geometry of the region through which the particle had travelled. Generalising this idea, criss-crossing space with a whole network of paths taken by particles of various spins, and comparing them all at the junctions where they met, led to the notion of spin network. Like the harmonics of a wave, these networks comprised a set of building blocks from which all quantum states of geometry could be constructed.

But this is just the start. The tale goes on to sketch how Sarumpaet found a "theory of everything" which goes by the name of Quantum Graph Theory:

"It's simple, it's elegant, and it's consistent with all observations to date." That handful of words sounded glib, but other people had quantified all of these criteria long ago: QGT as a descriptionof the dynamics of the universe with the minimum possible algorithmic complexity; QGT as a topological re-description of some basic results in category theory - a mathematical setting in which the Sarumpaet rules appeared as natural and inevitable as the rules of arithmetic; QGT as the most probable underlying system of physical laws, given any substantial database of experimental results that spanned both nuclear physics and cosmology.

There's more in this novel than just physics. Fans of Egan's "Diaspora" (see "week115") will recognize this world, but there are many new twists, too. Technophiles will enjoy the depiction of a time when virtual reality is commonplace, death and gender are things of the past, and everybody except a few "anachronauts" is fully backed up and fitted with a device that prevents their mind from splitting into different Everett branches when making decisions. Thoughtful readers will be interested to see what people worry about in a world like this - though I wish Egan had given more of a sense of everyday life. Finally, those of us who like math will enjoy reading of a world where people give theorems as presents - since everything else is too easy.

Would you like to see SO(4) as a principal SO(3)-bundle over S³, dear?

I should admit this is a wholly biased review of Egan's novel, since we're collaborating on a physics paper, and he cites my work at the end of his book. So you may want to take it with a grain of salt when I say: Read this!

If "Schild's Ladder" gets you hungry to learn more about what loop quantum gravity was like at the beginning of the third millennium, try these review articles by Ashtekar:

2) Abhay Ashtekar, Quantum geometry and gravity: recent advances, available as gr-qc/0112038.

Abhay Ashtekar, Quantum geometry in action: big bang and black holes, available as math-ph/0202008.

They're not too technical, and they show how the theory is going beyond "kinematical" results like the quantization of area and a tentative explanation of black hole entropy in terms of microstates of the horizon, towards "dynamical" results like a theory of the big bang.

In case you're wondering, "kinematics" in loop quantum gravity means the description of the geometry of space at ultra-short distance scales, taking quantum effects into account, while "dynamics" means a description of the geometry of spacetime. Loop quantum gravity has a much easier time with the former than the latter, basically because it's a form of "canonical quantization" - more jargon, which means that at the very start of the day one chops spacetime into space and time. Only recently have people like Martin Bojowald made progress on using loop quantum gravity to answer interesting dynamical questions - and even now, this work is fraught with difficulties.

To understand dynamics better in loop quantum gravity, people have tried to develop a form of "covariant quantization" that goes hand-in-hand with the canonical approach. Covariant quantization doesn't chop spacetime into space and time; it's very popular in particle physics, where it gives rise to the much-beloved Feynman diagrams. A Feynman diagram describes a "quantum history" in which particles zip hither and thither along edges, interacting at vertices. A theory of particle physics gives a rule for computing the probability - or more precisely, the amplitude - of any such history, and to figure out the probability that something happens, you need to sum over these histories, weighted by their amplitudes.

In loop quantum gravity the analogue of a Feynman diagram is called a "spin foam", because it looks a bit like a foam of soap bubbles. A spin foam has 2-dimensional faces in addition to the 1-dimensional edges and 0-dimensional vertices of a Feynman diagram. Again, these spin foams describe "quantum histories", and again we want to compute their amplitudes - but now these are histories for spacetime itself, rather than particles moving around in spacetime. Of course we eventually want to describe particles as well as spacetime, and it's unlikely that we'll get very far until we combine both aspects into a unified description, but work on that is just beginning:

3) Aleksandar Mikovic, Spin foam models of matter coupled to gravity, hep-th/0108099.

Aleksandar Mikovic, Quantum field theory of open spin networks and new spin foam models, available as gr-qc/0202026.

As usual, the hard part is deciding how much effort to put in going after the big enchilada, versus straightening out all sorts of details - details that could prove fatal if not handled properly. For example, it would be a real pity if the work on canonical quantum gravity weren't firmly bolted to the new work on spin foams. The best paper so far on this is by Arnsdorf, building on work by Rovelli and others:

4) Matthias Arnsdorf, Relating covariant and canonical approaches to triangulated models of quantum gravity, available as gr-qc/0110026.

This shows how, in principle, one can go back and forth between a recipe for computing spin foam amplitudes and a formula for the "Hamiltonian constraint" - the basic description of dynamics in canonical quantum gravity. Unfortunately, attempts to relate the most popular spin foam models to the most popular formulas for the Hamiltonian constrant are going slower. In particular, it's hard to see how Thiemann's formulas for the Hamiltonian constraint (see "week85") could arise naturally from a spin foam model. A tentative step in this direction has been made by Gambini and Pullin:

5)Rodolfo Gambini and Jorge Pullin, A finite spin-foam-based theory of three and four dimensional quantum gravity, gr-qc/0111089.

However, their theory is admittedly just a toy model, since it only handles a certain restricted class of solutions of Thiemann's constraint.

In a different direction, there has also been some good work lately on clarifying the inner workings of the spin foam formalism:

6) Robert Oeckl, Generalized lattice gauge theory, spin foams and state sum invariants, available as hep-th/0110259.

Florian Girelli, Robert Oeckl and Alejandro Perez, Spin foam diagrammatics and topological invariance, available as gr-qc/0111022.

Mathematicians will especially like the second paper, since it gives a slick way to prove the triangulation-independence of the Turaev-Viro model which avoids complicated calculations involving 6j symbols. Physicists, however, need to understand triangulation-dependent models; there is more about these in the first paper.

As for me, I've been spending the last half year working with Greg Egan, Dan Christensen, and two students of his trying to compare various spin foam models by means of computer simulations:

8) John C. Baez, J. Daniel Christensen, Thomas R. Halford and David C. Tsang, Spin foam models of Riemannian quantum gravity, gr-qc/0202017.

Riemannian quantum gravity is a toy model where the gauge group is the 4d rotation group rather than the Lorentz group. The reason we've been studying this instead of the real thing is that we don't yet have efficient ways of computing spin foam amplitudes in the Lorentzian theory. In the Riemannian case, Egan and Christensen developed an efficient algorithm that serves as the workhorse for the above paper - see "week172" for more on that. Without this algorithm, we'd be dead in the water!

In essence, what we did is take the simplest compact 4-dimensional manifold and triangulate it in the simplest possible way: the 4-sphere, triangulated as the boundary of a 5-simplex. Then we compared various recipes for computing the amplitudes of spin foams that fit neatly into this triangulation. In Riemannian quantum gravity, a spin foam living in this triangulation simply amounts to a way of labelling each triangle with a spin j = 0, 1/2, 1, 3/2,... which describes its area. More precisely, the area is proportional to sqrt(j(j+1)) - a formula familiar from other contexts in quantum mechanics.

The job of a spin foam model is to compute an amplitude from all these spins. If the amplitudes we get are biggest when lots of the spins are large, we'll know the model favors quantum histories where the discrete geometry of spacetime is visible at large scales. If the amplitudes are biggest when most of the spins are small, it means the discreteness is visible only at very small scales. This is just step down the long road towards seeing whether a model reduces to general relativity at large distance scales.

However, when you do these calculations, at first you get amplitudes that aren't normalized. To normalize them, you must divide by the sum of these unnormalized amplitudes over all spin foams, which is called the "partition function". So, one of the very first questions to ask about a specific model is whether this sum converges!

The models we compared were all variants of the Riemannian Barrett-Crane model (see "week128" and the previous issues referred to there). Barrett and Crane left some aspects of their model unspecified, and different ways of filling in the details turn out to give drastically different results. The first people to tackle this were De Pietri, Freidel, Krasnov and Rovelli:

9) Roberto De Pietri, Laurent Freidel, Kirill Krasnov, and Carlo Rovelli, Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, preprint available as hep-th/9907154.

Our calculations show that the partition function diverges very rapidly in their model. In fact, when we summed the amplitudes of spin foams with all triangles labelled by spins less than or equal to J, we got the following results:

spin cutoff               cutoff partition function 

J = 0                         1.000 
J = 1/2                       3.722 x 105
J = 1                         7.812 x 109
J = 3/2                       2.128 x 1013
J = 2                         1.345 x 1016

In fact, Perez and Rovelli had already suspected problems with a divergent partition function in this model. They came up with an elegant variant in which they could show the partition function converges:

10) Alejandro Perez and Carlo Rovelli, A spin foam model without bubble divergences, Nucl. Phys. B599 (2001), 255-282. Also available as gr-qc/0006107.

Alejandro Perez, Finiteness of a spin foam model for Euclidean quantum general relativity, Nucl. Phys. B599 (2001), 427-434. Also available as gr-qc/0011058.

Alejandro Perez, Group quantum field theories and spin foam models for quantum gravity, to appear.

The funny thing is, our calculations show the partition function converges really fast in this model. Taking the same triangulation of the 4-sphere, we got these results:

spin cutoff               cutoff partition function 

J = 0                        1.000000000000 
J = 1/2                      1.000014319178                 
J = 1                        1.000014323656                 
J = 3/2                      1.000014323670                 
J = 2                        1.000014323670                 

It's hard to be sure that this "spin-zero dominance" is a bad thing, but at the very least, it makes computer simulations a bit dull. Eventually we started trying to come up with a model that darts between the Scylla of a divergent partition function and the Charybdis of spin-zero dominance. This turns out to be quite tricky, at least if one wants a theory that's neatly expressed in the language of category theory - which is what underlies spin foam models. But eventually we found one! And amusingly, it was the very simplest, prettiest model of all... for some silly reason we'd overlooked it until we were really desperate.

Here is what our calculations gave for this model:

spin cutoff               cutoff partition function 

J = 0                         1.000000000000  
J = 1/2                       2.342658607645   
J = 1                         3.378038633798   
J = 3/2                       3.966290480574   
J = 2                         4.293589340364   
J = 5/2                       4.480621474940   

To get more information on whether the partition function converges, one has to use a sneakier method: the Metropolis algorithm. This is a technique widely used in statistical mechanics and quantum field theory. In our case, it amounts to designing a random walk that samples spin foams with a frequency equal to their amplitude. This only works thanks to a special property of the Barrett-Crane model which we proved in an earlier paper: the amplitudes are nonnegative! See "week172" for details.) One can't use the Metropolis algorithm to compute the partition function, but one can use it to compute expectation values of observables. We proved that a very simple observable - the average area of a triangle - can only have a finite expectation value if the partition function converges. Then we used the Metropolis algorithm to compute the expectation value of this observable... and it came out nice and finite: in fact, close to 0.401507.

So, we feel quite sure the partition function converges in our new model - at least for this particular triangulated 4-manifold. We also came up with an argument that should apply more generally, but it has some holes in it, which we'll try to plug in our next paper, with Greg Egan.

Anyway, it's fun seeing actual numbers coming out of spin foam models, and seeing computer calculations lead to new questions and even new theories. I think this sort of interaction between theory and "experiment" is a good thing, especially since we can't do real quantum gravity experiments yet. I hope more people working on quantum gravity do some numerical calculations, instead of focusing solely on analytically solvable problems, which is very limiting, especially for nonperturbative work.

Indeed, over in string theory there have already been some interesting calculations done in the IKKT matrix model - see "week172" for references on that. These are actually quite similar to our spin foam calculations: you take a manifold, triangulate it, label it in various ways and compute an amplitude for each labelling....