May 6, 1998

This Week's Finds in Mathematical Physics (Week 120)

John Baez

Now that I'm hanging out with the gravity crowd at Penn State, I might as well describe what's been going on here lately.

First of all, Ashtekar and Krasnov have written an expository account of their work on the entropy of quantum black holes:

1) Abhay Ashtekar and Kirill Krasnov, Quantum geometry and black holes, preprint available as gr-qc/9804039.

But if you prefer to see a picture of a quantum black hole without any equations, try:

2) Kirill Krasnov's research webpage.

You'll see a bunch of spin networks poking the horizon, giving it area and curvature. Of course, this is just a theory.

Second, there's been a burst of new work studying quantum gravity in terms of spin foams. A spin foam looks a bit like a bunch of soap suds - with the faces of the bubbles and the edges where the bubbles meet labelled by spins j = 0, 1/2, 1, 3/2, etc.. Spin foams are an attempt at a quantum description of the geometry of spacetime. If you slice a spin foam with a hyperplane representing "t = 0" you get a spin network: a graph with its edges and vertices labelled by spins. Spin networks have been used in quantum gravity for a while now to describe the geometry of space at a given time, so it's natural to hope that they're a slice of something that describes the geometry of spacetime.

As usual in quantum gravity, it's too early to tell if this approach will work. As usual, it has lots of serious problems. But before going into the problems, let me remind you how spin foams are supposed to work.

To relate spin foams to more traditional ideas about spacetime, one can consider spin foams living in a triangulated 4-manifold: one spin foam vertex sitting in each 4-simplex, one spin foam edge poking through each tetrahedron, and one spin foam face intersecting each triangle. Labelling the spin foam edges and faces with spins is supposed to endow the triangulated 4-manifold with a "quantum 4-geometry". In other words, it should let us compute things like the areas of the triangles, the volumes of the tetrahedra, and the 4-volumes of the 4-simplices. There are some arguments going on now about the right way to do this, but it's far from an arbitrary business: the interplay between group representation theory and geometry says a lot about how it should go. In the simplified case of 3-dimensional spacetime, it's fairly well understood - the hard part, and the fun part, is getting it to work in 4 dimensions.

Assuming we can do this, the next trick is to compute an amplitude for each spin foam vertex in a nice way, much as one computes amplitudes for vertices of Feynman diagrams. A spin foam vertex is supposed to represent an "event" - if we slice the spin foam by a hyperplane we get a spin network, and as we slide this slice "forwards in time", the spin network changes its topology whenever we pass a spin foam vertex. The amplitude for a vertex tells us how likely it is for this event to happen. As usual in quantum theory, we need to take the absolute value of an amplitude and square it to get a probability.

We also need to compute amplitudes for spin foam edges and faces, called "propagators", in analogy to the amplitudes one computes for the edges of Feynman diagrams. Multiplying all the vertex amplitudes and propagators for a given spin foam, one gets the amplitude for the whole spin foam. This tells us how likely it is for the whole spin foam to happen.

Barrett and Crane came up with a specific way to do all this stuff, Reisenberger came up with a different way, I came up with a general formalism for understanding this stuff, and now people are busy arguing about the merits of different approaches. Here are some papers on the subject - I'll pick up where I left off in "week113".

3) Louis Crane, David N. Yetter, On the classical limit of the balanced state sum, preprint available as gr-qc/9712087.

The goal here is to show that in the limit of large spins, the amplitude given by Barrett and Crane's formula approaches

exp(iS)

where S is the action for classical general relativity - suitably discretized, and in signature ++++. The key trick is to use an idea invented by Regge in 1961.

Regge came up with a discrete analog of the usual formula for the action in classical general relativity. His formula applies to a triangulated 4-manifold whose edges have specified lengths. In this situation, each triangle has an "angle deficit" associated to it. It's easier to visualize this two dimensions down, where each vertex in a triangulated 2-manifold has an angle deficit given by adding up angles for all the triangles having it as a corner, and then subtracting 2 π. No angle deficit means no curvature: the triangles sit flat in a plane. The idea works similarly in 4 dimensions. Here's Regge's formula for the action: take each triangle in your triangulated 4-manifold, take its area, multiply it by its angle deficit, and then sum over all the triangles.

Simple, huh? In the continuum limit, Regge's action approaches the integral of the Ricci scalar curvature - the usual action in general relativity. For more see:

4) T. Regge, General relativity without coordinates, Nuovo Cimento 19 (1961), 558-571.

So, Crane and Yetter try to show that in the limit of large spins, the Barrett-Crane spin foam amplitude approaches exp(iS) where S is the Regge action. There argument is interesting but rather sketchy. Someone should try to fill in the details!

However, it's not clear to me that the large spin limit is physically revelant. If spacetime is really made of lots of 4-simplices labelled by spins, the 4-simplices have got to be quite small, so the spins labelling them should be fairly small. It seems to me that the right limit to study is the limit where you triangulate your 4-manifold with a huge number of 4-simplices labelled by fairly small spins. After all, in the spin network picture of the quantum black hole, it seems that spin network edges labelled by spin 1/2 contribute most of the states (see "week112").

When you take a spin foam living in a triangulated 4-manifold and slice it in a way that's compatible with the triangulation, the spin network you get is a 4-valent graph. Thus it's not surprising that Barrett and Crane's formula for vertex amplitudes is related to an invariant of 4-valent graphs with edges labelled by spins. There's already a branch of math relating such invariants to representations of groups and quantum groups, and their formula fits right in. Yetter has figured out how to generalize this graph invariant to n-valent graphs with edges labelled by spins, and he's also studied more carefully what happens when one "q-deforms" the whole business - replacing the group by the corresponding quantum group. This should be related to quantum gravity with nonzero cosmological constant, if all the mathematical clues aren't lying to us. See:

5) David N. Yetter, Generalized Barrett-Crane vertices and invariants of embedded graphs, preprint available as math.QA/9801131.

Barrett has also given a nice formula in terms of integrals for the invariant of 4-valent graphs labelled by spins. This is motivated by the physics and illuminates it nicely:

6) John W. Barrett, The classical evaluation of relativistic spin networks, preprint available as math.QA/9803063.

Let me quote the abstract:

The evaluation of a relativistic spin network for the classical case of the Lie group SU(2) is given by an integral formula over copies of SU(2). For the graph determined by a 4-simplex this gives the evaluation as an integral over a space of geometries for a 4-simplex.

Their approach is based on a certain way to describe the geometry of a 4-simplex. Instead of specifying lengths of edges as in the old Regge approach, we specify bivectors for all its faces. Geometrically, a bivector is just an "oriented area element"; technically, the space of bivectors is the dual of the space of 2-forms. If we have a 4-simplex in R⁴ and we choose orientations for its triangular faces, there's an obvious way to associate a bivector to each face. We get 10 bivectors this way.

What constraints do these 10 bivectors satisfy? They can't be arbitrary! First, for any four triangles that are all the faces of the same tetrahedron, the corresponding bivectors must sum to zero. Second, every bivector must be "simple" - it must be the wedge product of two vectors. Third, whenever two triangles are the faces of the same tetrahedron, the sum of the corresponding bivectors must be simple.

It turns out that these constraints are almost but not quite enough to imply that 10 bivectors come from a 4-simplex. Generically, it there are four possibilities: our bivectors come from a 4-simplex, the negatives of our bivectors come from a 4-simplex, their Hodge duals come from a 4-simplex, or the negatives of their Hodge duals come from a 4-simplex.

If we ignore this and describe the 4-simplex using bivectors satisfying the three constraints above, and then quantize this description, we get the picture of a "quantum 4-simplex" that is the starting-point for the Barrett-Crane model. But clearly it's dangerous to ignore this problem.

Actually, I learned about this problem from Robert Bryant over on sci.math.research, and I discussed it in my paper on spin foam models, citing Bryant of course. Barrett and Crane overlooked this problem in the first version of their paper, but now they recognize its importance. Two papers have recently appeared which investigate it further:

7) Michael P. Reisenberger, Classical Euclidean general relativity from ``left-handed area = right-handed area'', preprint available as gr-qc/9804061.

8) Roberto De Pietri and Laurent Freidel, so(4) Plebanski Action and relativistic spin foam model, preprint available as gr-qc/9804071.

These papers study classical general relativity formulated as a constrained SO(4) BF theory. The constraints needed here are mathematically just the same as the constraints needed to ensure that 10 bivectors come from the faces of an actual 4-simplex! This is part of the magic of this approach. But again, if one only imposes the three constraints I listed above, it's not quite enough: one gets fields that are either solutions of general relativity or solutions of three other theories! This raises the worry that the Barrett-Crane model is a quantization, not exactly of general relativity, but of general relativity mixed in with these extra theories.

Here's another recent product of the Center for Classical and Quantum Gravity here at Penn State:

9) Laurent Freidel and Kirill Krasnov, Discrete space-time volume for 3-dimensional BF theory and quantum gravity, preprint available as hep-th/9804185.

Freidel and Krasnov study the volume of a single 3-simplex as an observable in the context of the Turaev-Viro model - a topological quantum field theory which is closely related to quantum gravity in spacetime dimension 3.

And here are some other recent papers on quantum gravity written by folks who either work here at the CGPG or at least occasionally drift through. I'll just quote the abstracts of these:

10) Ted Jacobson, Black hole thermodynamics today, to appear in Proceedings of the Eighth Marcel Grossmann Meeting, World Scientific, 1998, preprint available as gr-qc/9801015.

A brief survey of the major themes and developments of black hole thermodynamics in the 1990's is given, followed by summaries of the talks on this subject at MG8 together with a bit of commentary, and closing with a look towards the future.

We suggest that in a recently proposed framework for quantum gravity, where Vassiliev invariants span the the space of states, the latter is dramatically reduced if one has a non-vanishing cosmological constant. This naturally suggests that the initial state of the universe should have been one with Λ=0.

We draw parallels between the recently introduced ``Immirzi ambiguity'' of the Ashtekar-like formulation of canonical quantum gravity and other ambiguities that appear in Yang-Mills theories, like the θ ambiguity. We also discuss ambiguities in the Maxwell case, and implication for the loop quantization of these theories.

We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P → M is a smooth principal G-bundle. A `cylinder function' on the space of smooth connections on P is a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M. We construct diffeomorphism-invariant functionals on the space of cylinder functions from `spin networks': graphs in M with edges labeled by representations of G and vertices labeled by intertwining operators. Using the `group averaging' technique of Ashtekar, Marolf, Mourao and Thiemann, we equip the space spanned by these `diffeomorphism-invariant spin network states' with a natural inner product.

13) John H. Schwarz and Nathan Seiberg, String theory, supersymmetry, unification, and all that, to appear in the American Physical Society Centenary issue of Reviews of Modern Physics, March 1999, preprint available as hep-th/9803179.

14) Keith R. Dienes and Christopher Kolda, Twenty open questions in supersymmetric particle physics, 64 pages, preprint available as hep-ph/9712322.

I'm afraid I'll slack off on my "tour of homotopy theory" this week. I want to get to fun stuff like model categories and E_&infin spaces, but it's turning out to be a fair amount of work to reach that goal! That's what always happens with This Week's Finds: I start learning about something and think "oh boy, this stuff is great; I'll write it up really carefully so that everyone can understand it," but then this turns out to be so much work that by the time I'm halfway through I'm off on some other kick.