August 6, 1996

This Week's Finds in Mathematical Physics (Week 86)

John Baez

Let me continue my reportage of what happened at the Mathematical Problems of Quantum Gravity workshop in Vienna. I will only write about quantum gravity aspects here. I had an interesting conversation with Bertram Kostant in which he explained to me the deep inner secrets of the exceptional Lie group E8. However, my writeup of that has grown to the point where I will save it for some other week.

By the way, my course on n-category theory is not over! I'm merely taking a break from it, and will return to it after this workshop.


Wednesday, July 10th - Jerzy Lewandowski gave a talk on the "Spectrum of the Area Operator". As I've mentioned a few times before, one of the exciting things about the loop representation of quantum gravity is that the spectrum of the area operator associated to any surface is discrete. In other words, area is quantized!

Let me describe the area operator a bit more precisely. Before I tell you what the area operator is, I have to tell you what it operates on. Remember from "week43" that there are various Hilbert spaces floating around in the canonical quantization of gravity. First there is the "kinematical state space". In the old-fashioned metric approach to quantum gravity, known as geometrodynamics, this was supposed to be L^2(Met), where Met is the space of Riemannian metrics on space. (We take as space some 3-manifold S, which for simplicity we assume is compact). The problem was that nobody knew how to rigorously define this Hilbert space L^2(Met). In the "new variables" approach to quantum gravity, the kinematical state space is taken instead to be L^2(A), where A is the space of connections on space on some trivial SU(2) bundle over S. This can be defined rigorously.

Here's the idea, roughly. Fix any graph g, with finitely many edges and vertices, embedded in S. Let A_g, the space of connections on that graph, be SU(2)^n where n is the number of edges in e. Thus a connection on a graph tells us how to parallel transport things along each edge of that graph --- an idea familiar from lattice gauge theory. L^2(A_g) is well-defined because SU(2) has a nice measure on it, namely Haar measure, so A_g gets a nice measure on it as well.

Now if one graph g is contained in another graph h, the space L^2(A_g) is contained in the space L^2(A_h) in an obvious way. So we can form the union of all the Hilbert spaces L^2(A_g) and get a big Hilbert space L^2(A). Mathematicians would say that L^2(A) is the "projective limit" of the Hilbert spaces L^2(A_g), but let's not worry about that.

So that's how we get the space of "kinematical states" in the loop representation of quantum gravity. The space of physical states is then obtained by imposing constraints: the Gauss law constraint (i.e., gauge-invariance), the diffeomorphism constraint (i.e., invariance under diffeomorphisms of S) and the Hamiltonian constraint (i.e., invariance under time evolution). States in the physical state space are supposed to only contain information that's invariant under all coordinate transformations and gauge transformations --- the really physical information.

I explained these constraints to some extent in "week43", and I don't really want to worry about them here. But let me just mention that the Gauss law constraint is easy to impose in a mathematically rigorous way. The diffeomorphism constraint is harder but still possible, and the Hamiltonian constraint is the big thorny question plaguing quantum gravity --- see "week85" for some recent progress on this. The area operators I'll be talking about are self-adjoint operators on the space of kinematical states, L^2(A), and are a preliminary version of some related operators one hopes eventually to get on the physical state space, after much struggle and sweat.

To define an operator on L^2(A) it's enough to define it on L^2(A_g) for every graph g and then check that these definitions fit together consistently to give an operator on the big space L^2(A). So let's take a graph g and a surface s in space. The area operator we're after is supposed to be the quantum analog of the usual classical formula for the area of s. The usual classical area is a function of the metric on space; similarly, the quantum area is an operator on the space L^2(A).

The area operator only cares about the points where the graph intersects the surface. We assume that there are only finitely many points where it does so, apart from points where the edges are tangent to the surface. (To make this assumption reasonable, we need to assume, e.g., that the space S has a real-analytic structure and the surface and graph are analytic --- an annoying technicality that I have been seeking to eliminate.)

The area operator is built using three operators on L^2(SU(2)) called J_1, J_2, and J_3, the self-adjoint operators corresponding to the 3 generators of SU(2) --- which often show up in physics as the three components of angular momentum! Alternatively, we can think of all three together as one vector-valued operator J, the "angular momentum operator". Note that L^2(A_g) is just the tensor product of one copy of the Hilbert space L^2(SU(2)) for each edge of our graph g. Thus for any edge e we get an angular momentum operator J(e) that acts on the copy of L^2(SU(2)) corresponding to the edge e in question, leaving the other copies alone.

This, then, is how we define the operator on L^2(A_g) corresponding to the area of the surface s. Pick an orientation for the surface s. For any point where the graph g intersects s, let J(in) denote the sum of the angular momentum operators of all edges intersecting s at the point in question and pointing "inwards" relative to our chosen orientation. Similarly, let J(out) be the sum of the angular momentum operators of edges intersecting s at the point in question and pointing "outwards". (Note: edges tangent to the surface contribute neither to J(in) nor J(out).) Now sum up, over all points where the graph intersects the surface, the following quantity:

sqrt[(J(in) - J(out)) . ((J(in) - J(out))]

where the dot denotes the obvious sort of dot product of vector-valued operators. Multiply by half the Planck length squared and you've got the area operator!

This very beautiful and simple formula was derived by Ashtekar and Lewandowski, but the first people to try to quantize the area operator were Rovelli and Smolin; see

1) Discreteness of area and volume in quantum gravity, by Carlo Rovelli and Lee Smolin, 36 pages in LaTeX format, 13 figures uuencoded, available as gr-qc/9411005.

Abhay Ashtekar and Jerzy Lewandowski, Quantum theory of geometry I: area operators, 31 pages in LaTeX format, to appear in the Classical and Quantum Gravity special issue dedicated to Andrzej Trautman, preprint available as gr-qc/9602046.

In his talk Jerzy showed how to work the spectrum of the area operator (which is discrete) and showed how it could depend on whether the surface s cuts space into 2 parts or not.

Later that day, Mike Reisenberger, Matthias Blau, Carlo Rovelli and I talked about the relation between string theory and the loop representation of quantum gravity.

Mike has been working on a very interesting "state sum model" for quantum gravity; that is, a discretized model in which spacetime is made of 4-simplices (the 4d version of tetrahedra), fields are thought of ways of labelling the faces, edges and so on by spins, elements of SU(2) and the like, and the path integral is replaced by a sum over these labellings. This works out quite nicely for quantum gravity in 3 dimensions --- see "week16" --- but it's much more challenging in 4 dimensions.

One nice feature of these state sum models for quantum gravity is that they may be reinterpreted as sums over "worldsheets" --- surfaces mapped into spacetime. Since the spacetime is discrete, so are these surfaces --- they're made of lots of triangles --- but apart from that, having a path integral that's a sum over worldsheets is pleasantly reminscent of string theory. Indeed, once upon a time I proposed that the loop representation of quantum gravity and string theory were two aspects of some theory still waiting to be fully understood:

2) John Baez, Strings, loops, knots, and gauge fields, in "Knots and Quantum Gravity", ed. J. Baez, Oxford U. Press, Oxford, 1994, preprint available in LaTeX form as hep-th/9309067, 34 pages.

The problem was getting a concrete way to relate the Lagrangian for the string theory to the Lagrangian for gravity (or whatever gauge theory one started with). Iwasaki tackled this problem was tackled in 3d quantum gravity using state sum models:

3) Junichi Iwasaki, A reformulation of the Ponzano-Regge quantum gravity model in terms of surfaces, University of Pittsburgh, 11 pages in LaTeX format available as gr-qc/9410010.

Later, Reisenberger extended this approach to deal with certain 4d theories which are simpler than quantum gravity, like BF theory:

4) Michael Reisenberger, Worldsheet formulations of gauge theories and Gravity, University of Utrecht preprint, 1994, available as gr-qc/9412035.

In all of these theories, one computes the action for the worldsheets by summing something over places where they intersect. In other words, they "interact" at intersections.

But the really exciting thing would be to do something like this for Mike's new state sum model for 4d quantum gravity. And the real challenge would be to relate this --- if possible! --- to conventional string theory. In a coffeeshop I suggested that one might do this by using the usual formula for the action in (bosonic) string theory. This is simply the area of the string worldsheet with respect to some background metric. The loop representation of quantum gravity doesn't make reference to any background metric; the closest approximation to a classical metric is a "weave" state in which space is tightly packed with lots of loops or spin networks. From the 4d point of view, we'd expect this to correspond to a spacetime packed with lots of worldsheets. Now, given the relation between area and intersection number in the loop representation (see above!), one might expect the area of a given worldsheet to be roughly proportional to the number of its intersections with the other worldsheets in this "weave". But this is what one would expect in any theory where the worldsheets interact at intersections. So, one could hope that Mike's state sum model would be approximately equivalent to a string theory of the sort string theorists study.

There are lots of obvious problems with this idea, but it led to an interesting conversation, and I am still not convinced that it is crazy.

Thursday, July 11th - Jorge Pullin spoke on skein relations and the Hamiltonian constraint in lattice quantum gravity. His idea was that the Hamiltonian constraint contains a "topological factor" which serves as a skein relation on loop states.

Friday, July 12th - Abhay Ashtekar gave a talk on "Noncommutativity of Area Operators". This explained how the rather shocking fact that the area operators for two intersecting surfaces needn't commute actually has a perfect analog in classical general relativity.

Mike Reisenberger spoke on "Euclidean Simplicial GR". This presented the details of his state sum model. Since he hasn't published this yet, and since I am getting a bit tired out, I won't describe it here.

Monday, July 15th - Renate Loll gave a talk on the volume and area operators in lattice gravity. I wrote a bit about her work on the volume operator in "week55", and more can be found in:

5) Renate Loll, The volume operator in discretized quantum gravity, preprint available as gr-qc/9506014, 15 pages.

Renate Loll, Spectrum of the volume operator in quantum gravity, preprint available as gr-qc/9511030, 14 pages.

Also, Jerzy Lewandowski spoke on his work with Ashtekar on the volume operator in the continuum theory:

6) Jerzy Lewandowski, Volume and quantizations, preprint available as gr-qc/9602035, 8 pages.

Abhay Ashtekar and Jerzy Lewandowski, Quantum theory of geometry II: volume operators, manuscript in preparation.

The volume operator is more tricky than the area operator, and various proposed formulas for it do not agree. This is summarized quite clearly in Jerzy's paper.

In fact, I have already left Vienna by now. I was too busy there to keep up with This Week's Finds, but my life is a bit calmer now and I will try to finish these reports soon.

© 1996 John Baez