## This Week's Finds in Mathematical Physics (Week 113)

#### John Baez

This week I'd like to talk about "spin foams". People have already thought a lot about using spin networks to describe the quantum geometry of 3-dimensional space at a given time. This is great for kinematical aspects of quantum gravity, but not so good for dynamics. For dynamics, it would be nice to have a description of the quantum geometry of 4-dimensional spacetime. That's where spin foams come in.

If we use spin networks to describe the geometry of space at the Planck scale, how might we describe spacetime? Well, space is supposed to be a kind of slice of spacetime, so let's recall what a spin network is, and see what it could be a slice of.

First of all, spin network is a graph: a bunch of vertices connected by edges. What gives a graph when you slice it? Foam! Consider the soap suds you get while washing the dishes. If we idealize it as a bunch of 2-dimensional surfaces meeting along edges, and imagine intersecting it with a plane, we see that the result is typically a graph.

Topologists call this sort of space a "2-dimensional complex". It's a generalization of a graph because we can form it by starting with a bunch of "vertices", then connecting these with a bunch of "edges" to obtain a graph, and then taking a bunch of 2-dimensional discs or "faces" and attaching them along their boundaries to this graph. Mathematically, there's no reason to stop in dimension 2. Topologists are interested in complexes of all dimensions. However, 2-dimensional complexes have been given special attention, thanks to a number of famous unsolved problems involving them. A great place to learn about them is:

1) C. Hog-Angeloni, W. Metzler, and A. Sieradski, Two-dimensional Homotopy and Combinatorial Group Theory, London Mathematical Society Lecture Note Series 197, Cambridge U. Press, Cambridge, 1993.

However, a spin network is not merely a graph: it's a graph with edges labelled by irreducible representations of some symmetry group and vertices labelled by intertwiners. If you don't know what this means, don't panic! If we take our symmetry group to be SU(2), things simplify tremendously. If we take our graph to have 4 edges meeting at every vertex, things simplify even more. In this case, all we need to do is label each vertex and each edge with a number j = 0, 1/2, 1, 3/2,... called a "spin".

In this special case, we can get our spin network as a slice of a 2-dimensional complex with faces and edges labelled by spins. Such a thing looks a bit like a foam of soap bubbles with edges and faces labelled by spins - hence the term "spin foam"! More generally, a spin foam is a 2-dimensional complex with faces labelled by irreducible representations of some group and edges labelled by intertwining operators. When we slice a spin foam, each of its faces gives a spin network edge, and each of its edges gives a spin network vertex.

Actually, if you've ever looked carefully at soap suds, you'll know that generically 3 faces meet along each edge. Spin foams like this are important for quantum gravity in 3 spacetime dimensions. In 4 spacetime dimensions it seems especially interesting to use spin foams of a different sort, with 4 faces meeting along each edge. When we slice one of these, we get a spin network with 4 edges meeting at each vertex.

What's so interesting about spin foams with 4 faces meeting along each edge? Well, suppose we take a 4-dimensional manifold representing spacetime and triangulate it - that is, chop it up into 4-simplices, the 4-dimensional analogs of tetrahedra. We get a mess of 4-simplices, which have tetrahedra as faces, which in turn have triangles as faces.

Now we can form a spin foam with one vertex in the middle of each 4-simplex, one edge intersecting each tetrahedron, and one face intersecting each triangle. This trick is called "Poincare duality": each d-dimensional piece of our spin foam intersects a (4-d)-dimensional piece of our triangulation. Since each tetrahedron in our triangulated manifold has 4 triangular faces, our spin foam will dually have 4 faces meeting at each edge. Since each 4-simplex has 5 tetrahedra and 10 triangles, each spin foam vertex will have 5 edges and 10 faces meeting at it.

This seems to be a particularly interesting sort of spin foam for quantum gravity in 4 dimensions: a spin foam dual to a triangulation of spacetime. If we slice such a spin foam, we generically get a spin network dual to a triangulation of space!

I discussed Barbieri's work on such spin networks in "week110". A spin network like this has a nice interpretation as a "3-dimensional quantum geometry", that is, a quantum state of the geometry of space. Each spin network edge labelled by spin j gives an area proportional to sqrt(j(j+1)) to the triangle it intersects. There's also a formula for the volume of each tetrahedron, involving the spin on the corresponding spin network vertex, together with the spins on the 4 spin network edges that meet there.

It would be nice to have a similar geometrical interpretation of spin foams dual to triangulations of spacetime. Some recent steps towards this can be found in the following papers:

2) John Barrett and Louis Crane, Relativistic spin networks and quantum gravity, 9 pages, preprint available as gr-qc/9709028.

3) John Baez, Spin foam models, 39 pages, preprint available as gr-qc/9709052 or in Postscript form as http://math.ucr.edu/home/baez/foam.ps

Perhaps I can be forgiven some personal history here. Michael Reisenberger has been pushing the idea of spin foams (though not the terminology) for quite a while... see for example his paper:

4) Michael Reisenberger, Worldsheet formulations of gauge theories and gravity, preprint available as gr-qc/9412035.

More recently, Carlo Rovelli and he gave a heuristic derivation of a spin foam approach to quantum gravity starting with the so-called canonical quantization approach:

5) Michael Reisenberger and Carlo Rovelli, ``Sum over surfaces'' form of loop quantum gravity, Phys. Rev. D56 (1997), 3490-3508, preprint available as gr-qc/9612035.

I started giving talks about spin foams in the spring of this year. Following the ideas of Reisenberger and Rovelli, I was trying to persuade everyone to think of spin foams as higher-dimensional analogs of Feynman diagrams.

Mathematically, a Feynman diagram is just a graph with edges labelled by representations of some group. But physically, a Feynman diagram describes a process in which a bunch of particles interact. Its edges correspond to the worldlines traced out by some particles as time passes, while its vertices represent interactions. Different quantum field theories use Feynman diagrams with different kinds of vertices. For any Feynman diagram in our theory, we want to compute a number called an "amplitude". The absolute value squared of this amplitude gives the probability that the process in question will occur.

We calculate this amplitude by computing a number for each for each edge and each vertex and multiplying all these numbers together. The numbers for edges are called "propagators" - they describe the amplitude for a particle to go from here to there. The numbers for vertices are called "vertex amplitudes" - they describe the amplitude for various interactions to occur.

Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwiners. Each spin foam face corresponds to the "worldsheet" traced out by a spin network edge as time passes. So, in addition to thinking of a spin foam as a "4-dimensional quantum geometry", we can think of it as a kind of process. The goal of the spin foam approach to quantum gravity is to compute an amplitude for each spin foam. Following what we know about Feynman diagrams, it seems reasonable to do it by computing a number for each spin foam face, edge, and vertex, and then multiplying them all together.

Quantum gravity is related to a simpler theory called BF theory, which has a spin foam formulation known as the Crane-Yetter model - see "week36", "week58", and "week98". There are various clues suggesting that that the numbers for faces and edges - the "propagators" - should be computed in quantum gravity just as in the Crane-Yetter model. The problem is the vertex amplitudes! The vertices are crucial because these represent the interactions: the places where something really "happens". The number we compute for a vertex represents the amplitude for the corresponding interaction to occur. Until we know this, we don't know the dynamics of our theory!

The "spin foam vertex amplitudes for quantum gravity" became my holy grail. Whenever I gave a talk on this stuff I would go around afterwards asking everyone if they could help me figure them out. I would lay out all the clues I had and beg for assistance... or at least a spark of inspiration. In March I gave a talk a talk at Penn State proposing a candidate for these vertex amplitudes - a candidate I no longer believe in. Afterwards Carlo Rovelli told me about his attempts to work out something similar with Louis Crane and Lee Smolin... attempts that never quite got anywhere. We had a crack at it together but it didn't quite gel. In May I asked John Barrett about the vertex amplitudes at a conference in Warsaw. He said he couldn't guess them. I couldn't get anyone to guess an answer. In June, at a quantum gravity workshop in Vienna, I asked Roger Penrose a bunch of questions about spinors, hoping that this might be the key - see "week109". I learned a lot of interesting stuff, but I didn't find the holy grail.

I kept on thinking. I started getting some promising ideas, and by the summer I was hard at work on the problem, calculating furiously. I was also writing a big fat paper about spin foams: the general formalism, the relation to triangulations, the relationships to category theory, and so on. I was very happy with it - but I didn't want to finish it until I knew the spin foam vertex amplitudes. That would be the icing on the cake, I thought.

Then one weekend Louis Crane sent me email saying he and John Barrett had written a paper proposing a model of quantum gravity. Aaargh! Had they beat me to the holy grail? I frantically wrote up everything I had while waiting for Monday, when their paper would appear on the preprint server gr-qc. On Monday I downloaded it and yes, they had beaten me. It was a skinny little paper and I absorbed it more or less instantly. They didn't say a word about spin foams - they were working dually with triangulations - but from my viewpoint, what they had done was to give a formula for the spin foam vertex amplitudes.

Oh well. When you can't beat 'em, join 'em! I finished up my paper, explaining how their formula fit in with what I'd written already, and put it on the the preprint server the following weekend.

What did they do to get their formula? Well, the key trick was not to use SU(2) as the symmetry group, but instead use SU(2) x SU(2). This is the double cover of SO(4), the rotation group in 4 dimensions. Following the idea behind Ashtekar's new variables for general relativity, I was only using the "left-handed half" of this group, that is, one of the SU(2) factors. But the geometry of the 4-simplex, and its relation to quantum theory, is in some ways more easily understood using the full SU(2) x SU(2) symmetry group.

Not surprisingly, an irreducible representation of SU(2) x SU(2) is described by a pair of spins (j,k). The reason is that we can take the spin-j representation of the "left-handed" SU(2) and the spin-k representation of the "right-handed" SU(2) and tensor them to get an irreducible representation of SU(2) x SU(2). If we use SU(2) x SU(2) as our group, our spin foams dual to triangulations will thus have every face and every edge labelled by a pair of spins. However, Barrett and Crane's work suggests that the only spin foams with nonzero amplitudes are those for which both spins labelling a face or edge are equal! Thus in a way we are back down to SU(2) - but we think of it all a bit differently.

I'm tempted to go into detail and explain exactly how the model works, because it involves a lot of beautiful geometry. But it takes a while, so I won't. First you need to really grok the phase space of all possible geometries of the 4-simplex. Then you need to quantize this phase space, obtaining the "Hilbert space of a quantum 4-simplex". Then you need to note that there's a special linear functional on this Hilbert space, called the "Penrose evaluation" - see "week110". Putting all this together gives the vertex amplitudes for quantum gravity... we hope.

Anyway, back to my little personal story....

Though I'd been working on my paper before Barrett and Crane started, and they finished before me, Michael Reisenberger started one even earlier and finished even later! Indeed, he has been working on a spin foam model of quantum gravity for several years now - see "week86". He took a purely left-handed SU(2) approach, a bit different what I'd been trying, but closely related. He told lots of people about it, but unfortunately he's very slow to publish.

When I heard Barrett and Crane were about to come out with their paper, I emailed Reisenberger warning him of this. He doesn't like being scooped any more than I do. Unfortunately I only had his old email address in Canada, and now he's down in Uruguay, so he never got that email. Thus Barrett and Crane's paper, followed by mine, came as a a big shock to him! Luckily, this motivated him to hurry and come out with a version of his paper:

6) Michael Reisenberger, A lattice worldsheet sum for 4-d Euclidean general relativity, 50 pages, preprint available as gr-qc/9711052.

Let me quote the abstract:

`A lattice model for four dimensional Euclidean quantum general relativity is proposed for a simplicial spacetime. It is shown how this model can be expressed in terms of a sum over worldsheets of spin networks, and an interpretation of these worldsheets as spacetime geometries is given, based on the geometry defined by spin networks in canonical loop quantized GR. The spacetime geometry has a Planck scale discreteness which arises "naturally" from the discrete spectrum of spins of SU(2) representations (and not from the use of a spacetime lattice). The lattice model of the dynamics is a formal quantization of the classical lattice model of [Reisenberger's paper "A left-handed simplicial action for euclidean general relativity"], which reproduces, in a continuum limit, Euclidean general relativity.'

To wrap up my little history, let me say what's been happening lately. There is still a lot of puzzlement and mystery concerning these spin foam models... it's far from clear that they really work as hoped for. We may be doing things a little bit wrong, or we may be on a completely wrong track. The phase space of the 4-simplex involves some tricky constraint equations which could kill us if we're not dealing with them right. Barbieri has suggested a modified version of Barrett and Crane's approach which may overcome some problems with the constraints:

7) Andrea Barbieri, Space of the vertices of relativistic spin networks, 2 pages, preprint available as gr-qc/9709076.

John Barrett visited me last week and we made some progress on this issue, but it's still very touchy.

Also, all the work cited above deals with so-called "Euclidean" quantum gravity - that's why it uses the double cover of the rotation group SO(4). For "Lorentzian" quantum gravity we'd need instead to use the double cover of the Lorentz group SO(3,1). This group is isomorphic to SL(2,C). As explained in "week109", the finite-dimensional irreducible representations of SL(2,C) are also described by pairs of spins, so the Lorentzian theory should be similar to the Euclidean theory. However, most work so far has dealt with the Euclidean case; this needs to be addressed.

Finally, Crane has written a bit more about the geometrical significance of his work with Barrett:

8) Louis Crane, On the interpretation of relativistic spin networks and the balanced state sum, 4 pages, preprint available as gr-qc/9710108.

Next week I'll talk about other developments in the loop representation of quantum gravity, some arising from Thiemann's work on the Hamiltonian constraint. After that, I think I want to talk about something completely different, like homotopy theory. Lately I've been trying to make "This Week's Finds" very elementary and readable - relatively speaking, I mean - but I'm getting in the mood for writing in a more technical and incomprehensible manner, and homotopy theory is an ideal subject for that sort of writing.