August 26, 1996

This Week's Finds in Mathematical Physics (Week 88)

John Baez

This issue concludes my report of what happened at the Mathematical Problems of Quantum Gravity workshop in Vienna. I left the workshop at the end of July, so my reportage ends there, but the workshop went on for a few more weeks after that. I'll be really bummed out if I find out that they solved all the problems with quantum gravity after I left.

Before I launch into my day-by-day account of what happened, let me note that I've written a little introduction to Thiemann's work on the Hamiltonian constraint, which he presented at the workshop (see "week85"):

1) John Baez, The Hamiltonian constraint in the loop representation of quantum gravity, available at

A less technical version of this appears in Jorge Pullin's newsletter Matters of Gravity, issue 8, at

Okay... I'll start out simple today since there is something nice and simple to ponder:

Tuesday, July 23rd - Ted Jacobson spoke on the "Geometry and Evolution of Degenerate Metrics". One of the interesting things about Ashtekar's reformulation of general relativity is that it extends general relativity to the case of degenerate metrics, that is, metrics where there are vectors whose dot product with all other vectors is zero. However, one needs to be very careful because different versions of Ashtekar's formulation give different ways of handling degenerate metrics.

To see why in a simple example, remember that the usual metric on Minkowski spacetime is nondegenerate and in nice coordinates looks like

-dt^2 + dx^2 + dy^2 + dz^2

Here we are setting the speed of light equal to 1. In general relativity, one way people describe the metric is using a tensor g_{ab}, where the indices a and b go from 0 to 3. In Minkowski space this tensor equals

-1  0  0  0 
 0  1  0  0
 0  0  1  0
 0  0  0  1

What this tensor means is that if you have two vectors v and w, their dot product is g_{ab} v^a w^b, where as usual we multiply the entries of the metric tensor and the vectors v and w as indicated, and then sum over repeated indices. So, for example, the dot product of the vector

v = (1, 1, 0, 0)

with itself is 0, though its dot product with other vectors needn't be zero. There is a bunch of vectors whose dot products with themselves are zero, and these are called lightlike vectors, because light travels in these directions, moving one unit in space for each unit in time. There is actually a cone of lightlike vectors, called the lightcone.

One can imagine a world where the metric g_{ab} is

-1  0  0  0
 0  1  0  0
 0  0  k  0
 0  0  0  k

for some k > 0. This world isn't really so different from Minkowski space, because you can also think of it as Minkowski space described in screwy coordinates where you are measuring distances in the y and z directions in different units than the x direction. When k gets small, you can check that the lightcone gets stretched out in the y and z directions. Alternatively, when k gets big, the lightcone gets squashed in the y and z directions.

Another way to formulate general relativity uses the inverse metric g^{ab}. This is just the inverse of the matrix g_{ab}, which is indeed invertible when the metric is nondegenerate. So for example in the above case the inverse metric g^{ab} is

-1  0  0  0
 0  1  0  0
 0  0  K  0
 0  0  0  K

where K = 1/k. You can think of K as the speed of light in the y and z directions, which is different from the speed of light in the x direction.

Now there are two different limiting cases we can consider, depending on whether we work with the metric or the inverse metric. If we work with the metric, we can let k = 0. This corresponds to making the speed of light in the y and z directions infinite, so that information can go as fast as it likes in those directions and the lightcone gets completely stretched out in those directions. Note that now the metric g_{ab} is

-1  0  0  0
 0  1  0  0
 0  0  0  0
 0  0  0  0

so the inverse metric doesn't even make sense --- you can't invert this matrix. If we extend general relativity to degenerate metrics, we are allowing ourselves to study weird worlds like this. Why we'd want to --- well, that's another matter.

If we work with the inverse metric, we can't let k = 0, but we can let K = 0. This corresponds to making the speed of light in the y and z directions zero, so that information can't go at all in those directions: the lightcone is squashed down onto the t-x plane. Now it's the inverse metric that equals

-1  0  0  0
 0  1  0  0
 0  0  0  0
 0  0  0  0

and the metric doesn't even make sense.

Ted Jacobson's talk was about doing general relativity in weird worlds like this, where the inverse metric is degenerate. Here information flows only along surfaces, like the x-t plane in the example above, and these different surfaces don't really talk to each other very much. It's as if the world was split up (or in math jargon, foliated) into lots of different 2-dimensional worlds, which didn't know about each other. Jacobson showed that in this case, the equations of general relativity (extended in a certain way to degenerate inverse metrics) boil down to saying that there are two kinds of massless spin-1/2 particle living on all these 2-dimensional worlds.

This got me quite excited because it reminded me of string theory, which is all about massless particles (or in physics jargon, conformal fields) living on the 2-dimensional string worldsheet. I am always hunting around for relationships between string theory and the loop representation of quantum gravity, and I think this is an important clue. The reason is that I think the loop representation can be thought of as a quantum version of the theory of degenerate solutions of general relativity where the metric is zero most places and less degenerate (but still degenerate) on certain surfaces. When you slice one of these surfaces with the hyperplane t = 0 you get a bunch of loops (or more generally a graph), and these are the loops of the loop representation. Jacobson's talk may give a way to understand the conformal field theory living on these surfaces, which one needs if one wants to think of these surfaces as the "string worldsheets" of string theory fame. Anyway, I am busily thrashing this stuff out and trying to write a paper on it, but it may or may not hang together.

Jacobson's talk is based on a short paper he'd just been editing the galley proofs for; so it should come out soon:

2) Ted Jacobson, 1+1 sector of 3+1 gravity, Class. Quant. Grav. 13 (1996), L1-L6.

Now around this time the Erwin Schroedinger Institute, where the workshop was being held, moved from its comfortable old spot on Pasteurgasse to a more spacious location on Boltzmanngasse, near the physics department. (In Germany the word "Gasse" means "alley", and one might find it disrespectful that Pasteur and Boltzmann have mere alleys named after them, but in Vienna even lots of large streets are called "Gasse", when in Germany they'd be called "Strasse". But then even the word for potato is different in Austria; it's all part of the charm of the place.) The move disrupted the schedule of the talks a bit, and it also seems to have disrupted my note-taking, which gets more sketchy from here on out. Some of the dates below might be a bit off.

Thursday, July 25th - I spoke on "Topological Quantum Field Theory". I am always talking about this on This Week's Finds so I won't bore you with the details. Basically I summarized what is known about BF theory (a particular topological quantum field theory) in dimensions 2, 3, and 4, and the discrete formulation of BF theory where you chop spacetime into simplices and label the edges and so on with spins and the like --- so-called "state sum models". You can read more about this in "week38".

Later that day, Jerzy Lewandowski spoke on "Degenerate Metrics". Being somewhat less degenerate than Ted Jacobson, he spoke about extending general relativity to cases where the inverse metric looks like

-1  0  0  0
 0  1  0  0
 0  0  1  0 
 0  0  0  0

In other words, where the speed of light is zero only in the z direction. Basically what happens is that spacetime gets foliated with a lot of 3-dimensional slices, and on each one you get the equations of 3-dimensional general relativity.

Friday, July 26th - Thomas Strobl spoke on 2-dimensional gravity. I don't understand his work well enough yet to have anything much to say, but the most interesting thing about it to me is that it allows one to see how quantum groups emerge from the G/G gauged Wess-Zumino-Witten model (a certain 2-dimensional topological quantum field theory), by describing this theory as the quantization of a Poisson σ-model --- a field theory where the fields take values in a Poisson manifold. For more, try:

3) Peter Schaller and Thomas Strobl, A brief introduction to Poisson σ-models, preprint available as hep-th/9507020.

Peter Schaller and Thomas Strobl, Poisson σ-models: a generalization of 2d gravity-Yang-Mills systems, preprint available as hep-th/9411163.

Later, I had a great conversation with Mike Reisenberger and Carlo Rovelli on reformulating the loop representation of quantum gravity in terms of surfaces embedded in spacetime. This again touched upon my interest in relating string theory and the loop representation. They are writing a paper on this which should be on the preprint servers pretty soon, so I'll wait until then to talk about it.

Saturday, July 27th - Carlo Rovelli explained some things about the problem of time to me.

Monday, July 30th - I spoke about relative states and entanglement entropy in two-part quantum systems (see "week27" and the applications of these ideas to topological quantum field theory and quantum gravity. A lot of this came from my attempts to understand the relation between quantum gravity and Chern-Simons theory, and Lee Smolin's work where he tries to use this relation to derive the Bekenstein bound on the entropy of a system in terms of its surface area (see "week56").

An interesting little fact that I needed to use is that if you have a two-part quantum system in a pure state --- a state of zero entropy --- the two parts, regarded individually, can themselves have entropy, but the entropies of the two parts are equal. I worked this out using the symmetry of the situation but Walter Thirring, who attended the talk, pointed out that it can also be derived from a wonderful general fact: the triangle inequality! Namely, if your two-part system has entropy S, and the two parts individually have entropies S1 and S2, then S can never be less than |S1 - S2| or greater than S1 + S2. (In classical mechanics it's also true that S can never be less than either S1 or S2, but this fails in quantum mechanics, where for example you can have S be zero but S1 = S2 > 0.)

Wednesday, August 1st - Full of excitement and new ideas, I somewhat regretfully left the workshop and flew to London. Then I spent most of August working at Imperial College, thanks to a kind offer of office space from Chris Isham. I had some nice talks with Isham and his students on quantum gravity and the decoherent histories approach to quantum mechanics. I'll say a bit about this in a while, but next Week I am going to talk about triality and the secret inner meaning of E8.

© 1996 John Baez