January 4, 1999

This Week's Finds in Mathematical Physics (Week 128)

John Baez

This week I'd like to catch you up on the latest developments in quantum gravity. First, a book that everyone can enjoy:

1) John Archibald Wheeler and Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics, Norton, New York, 1998.

This is John Wheeler's autobiography. If Wheeler's only contribution to physics was being Bohr's student and Feynman's thesis advisor, that in itself would have been enough. But he did much more. He played a crucial role in the Manhattan project and the subsequent development of the hydrogen bomb. He worked on nuclear physics, cosmic rays, muons and other elementary particles. And he was also one of the earlier people to get really excited about the more outlandish implications of general relativity. For example, he found solutions of Einstein's equation that correspond to regions of gravitational field held together only by their own gravity, which he called "geons". He was not the first to study black holes, but he was one of the first people to take them seriously, and he invented the term "black hole". And the reason he is my hero is that he took seriously the challenge of reconciling general relativity and quantum theory. Moreover, he recognized how radical the ideas needed to accomplish this would be - for example, the idea that spacetime might not be truly be a continuum at short distance scales, but instead some sort of "quantum foam".

Anyone interested in the amazing developments in physics during the 20th century should read this book! Here is the story of how he first met Feynman:

Dick Feynman, who had earned his bachelor's degree at MIT, showed up at my office door as a brash and appealing twenty-one- year-old in the fall of 1939 because, as a new student with a teaching assistantship, he had been assigned to grade papers for me in my mechanics course. As we sat down to talk about the course and his duties, I pulled out and placed on the table between us a pocket watch. Inspired by my father's keenness for time-and- motion studies, I was keeping track of how much time I spent on teaching and teaching-related activities, how much on research, and how much on departmental or university chores. This meeting was in the category of teaching-related. Feynman may have been a little taken aback by the watch but he was not one to be intimidated. He went out and bought a dollar watch (as I learned later), so he would be ready for our next meeting. When we got together again, I pulled out my watch and put it on the table between us. Without cracking a smile, Feynman pulled out his watch and put it on the table next to mine. His theatrical sense was perfect. I broke down laughing, and soon he was laughing as hard as I, until both of us had tears in our eyes. It took quite a while for us to sober up and get on with our discussion. This set the tone for a wonderful friendship that endured for the rest of his life.
Next for something a wee bit more technical:

2) Steven Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge University Press, 1998.

If you want to learn about quantum gravity in 2+1 dimensions this is the place to start, because Carlip is the world's expert on this subject, and he's pretty good at explaining things.

(By the way, physicists write "2+1 dimensions", not because they can't add, but to emphasize that they are talking about 2 dimensions of space and 1 dimension of time.)

Quantum gravity in 2+1 dimensions is just a warmup for what physicists are really interested in - quantum gravity in 3+1 dimensions. Going down a dimension really simplifies things, because Einstein's equations in 2+1 dimensions say that the energy and momentum flowing through a given point of spacetime completely determine the curvature there, unlike in higher dimensions. In particular, spacetime is flat in the vacuum in 2+1 dimensions, so there's no gravitational radiation. Nonetheless, quantum gravity in 2+1 dimensions is very interesting, for a number of reasons. Most importantly, we can solve the equations exactly, so we can use it as a nice testing-ground for all sorts of ideas people have about quantum gravity in 3+1 dimensions.

Quantum gravity is hard for various reasons, but most of all it's hard because, unlike traditional quantum field theory, it's a "background-free" theory. What I mean by this is that there's no fixed way of measuring times and distances. Instead, times and distances must be measured with the help of the geometry of spacetime, and this geometry undergoes quantum fluctuations. That throws most of our usual methods for doing physics right out the window! Quantum gravity in 2+1 dimensions gives us, for the first time, an example of a background-free theory where we can work out everything in detail.

Here's the table of contents of Carlip's book:

1. Why (2+1)-dimensional gravity?

2. Classical general relativity in 2+1 dimensions

3. A field guide to the (2+1)-dimensional spacetimes

4. Geometric structures and Chern-Simons theory

5. Canonical quantization in reduced phase space

6. The connection representation

7. Operator algebras and loops

8. The Wheeler-DeWitt equation

9. Lorentzian path integrals

10. Euclidean path integrals and quantum cosmology

11. Lattice methods

12. The (2+1)-dimensional black hole

13. Next steps

A. Appendix: The topology of manifolds

B. Appendix: Lorentzian metrics and causal structure

C. Appendix: Differential geometry and fiber bundles

And now for some stuff that's available online. First of all, anyone who wants to keep up with research on gravity should remember to read "Matters of Gravity". I've talked about it before, but here's the latest edition:

3) Jorge Pullin, editor, Matters of Gravity, vol. 12, available at gr-qc/9809031 and at http://vishnu.nirvana.phys.psu.edu/mog.html

There's a lot of good stuff in here. Quantum gravity buffs will especially be interested in Gary Horowitz's article "A nonperturbative formulation of string theory?" and Lee Smolin's "Neohistorical approaches to quantum gravity". The curious title of Smolin's article refers to new work on quantum gravity involving a sum over histories - or in other words, spin foam models.

Even if you can't go to a physics talk, these days you can sometimes find it on the world-wide web. Here's one by John Barrett:

4) John W. Barrett, State sum models for quantum gravity, Penn State relativity seminar, August 27, 1998, audio and text of transparencies available at http://vishnu.nirvana.phys.psu.edu/online/Html/Seminars/Fall1998/Barrett/

Barrett and Crane have a theory of quantum gravity, which I've also worked on; I discussed it last in "week113" and "week120". Before I describe it I should warn the experts that this theory deals with Riemannian rather than Lorentzian quantum gravity (though Barrett and Crane are working on a Lorentzian version, and I hear Friedel and Krasnov are also working on this). Also, it only deals with vacuum quantum gravity - empty spacetime, no matter.

In this theory, spacetime is chopped up into 4-simplices. A 4-simplex is the 4-dimensional analog of a tetrahedron. To understand what I'm going to say next, you really need to understand 4-simplices, so let's start with them.

It's easy to draw a 4-simplex. Just draw 5 dots in a kind of circle and connect them all to each other! You get a pentagon with a pentagram inscribed in it. This is a perspective picture of a 4-simplex projected down onto your 2-dimensional paper. If you stare at this picture you will see the 4-simplex has 5 tetrahedra, 10 triangles, 10 edges and 5 vertices in it.

The shape of a 4-simplex is determined by 10 numbers. You can take these numbers to be the lengths of its edges, but if you want to be sneaky you can also use the areas of its triangles. Of course, there are some constraints on what areas you can choose for there to exist a 4-simplex having triangles with those areas. Also, there are some choices of areas that fail to make the shape unique: for one of these bad choices, the 4-simplex can flop around while keeping the areas of all its triangles fixed. But generically, this non-uniqueness doesn't happen.

In Barrett and Crane's theory, we chop spacetime into 4-simplices and describe the geometry of spacetime by specifying the area of each triangle. But the geometry is "quantized", meaning that the area takes a discrete spectrum of possible values, given by


where the "spin" j is a number of the form 0, 1/2, 1, 3/2, etc. This formula will be familiar to you if you've studied the quantum mechanics of angular momentum. And that's no coincidence! The cool thing about this theory of quantum gravity is that you can discover it just by thinking a long time about general relativity and the quantum mechanics of angular momentum, as long as you also make the assumption that spacetime is chopped into 4-simplices.

So: in Barrett and Crane's theory the geometry of spacetime is described by chopping spacetime into 4-simplices and labelling each triangle with a spin. Let's call such a labelling a "quantum 4-geometry". Similarly, the geometry of space is described by chopping space up into tetrahedra and labelling each triangle with a spin. Let's call this a "quantum 3-geometry".

The meat of the theory is a formula for computing a complex number called an "amplitude" for any quantum 4-geometry. This number plays the usual role that amplitudes do in quantum theory. In quantum theory, if you want to compute the probability that the world starts in some state ψ and ends up in some state ψ', you just look at all the ways the world can get from ψ to ψ', compute an amplitude for each way, add them all up, and take the square of the absolute value of the result. In the special case of quantum gravity, the states are quantum 3-geometries, and the ways to get from one state to another are quantum 4-geometries.

So, what's the formula for the amplitude of a quantum 4-geometry? It takes a bit of work to explain this, so I'll just vaguely sketch how it goes. First we compute amplitudes for each 4-simplex and multiply all these together. Then we compute amplitudes for each triangle and multiply all these together. Then we multiply these two numbers.

(This is analogous to how we compute amplitudes for Feynman diagrams in ordinary quantum field theory. A Feynman diagram is a graph whose edges have certain labellings. To compute its amplitude, first we compute amplitudes for each edge and multiply them all together. Then we compute amplitudes for each vertex and multiply them all together. Then we multiply these two numbers. One goal of work on "spin foam models" is to more deeply understand this analogy with Feynman diagrams.)

Anyway, to convince oneself that this formula is "good", one would like to relate it to other approaches to quantum gravity that also involve 4-simplices. For example, there is the Regge calculus, which is a discretized version of classical general relativity. In this approach you chop spacetime into 4-simplices and describe the shape of each 4-simplex by specifying the lengths of its edges. Regge invented a formula for the "action" of such a geometry which approaches the usual action for classical general relativity in the continuum limit. I explained the formula for this "Regge action" in "week120".

Now if everything were working perfectly, the amplitude for a 4-simplex in the Barrett-Crane model would be close to exp(iS), where S is the Regge action of that 4-simplex. This would mean that the Barrett-Crane model was really a lot like a path integral in quantum gravity. Of course, in the Barrett-Crane model all we know is the areas of the triangles in each 4-simplex, while in the Regge calculus we know the lengths of its edges. But we can translate between the two, at least generically, so this is no big deal.

Recently, Barrett and Williams came up with a nice argument saying that in the limit where the triangles have large areas, the amplitude for a 4-simplex in the Barrett-Crane theory is proportional, not to exp(iS), but to cos(S):

5) John W. Barrett and Ruth M. Williams, The asymptotics of an amplitude for the 4-simplex, preprint available as gr-qc/9809032.

This argument is not rigorous - it uses a stationary phase approximation that requires further justification. But Regge and Ponzano used a similar argument to show the same sort of thing for quantum gravity in 3 dimensions, and their argument was recently made rigorous by Justin Roberts, with a lot of help from Barrett:

6) Justin Roberts, Classical 6j-symbols and the tetrahedron, preprint available as math-ph/9812013.

So one expects that with work, one can make Barrett and Williams' argument rigorous.

But what does it mean? Why does he get cos(S) instead of exp(iS)? Well, as I said, the same thing happens one dimension down in the so-called Ponzano-Regge model of 3-dimensional Riemannian quantum gravity, and people have been scratching their heads for decades trying to figure out why. And by now they know the answer, and the same answer applies to the Barrett-Crane model.

The problem is that if you describe 4-simplex using the areas of its triangles, you don't completely know its shape. (See, I lied to you before - that's why you gotta read the whole thing.) You only know it up to reflection. You can't tell the difference between a 4-simplex and its mirror-image twin using only the areas of its triangles! When one of these has Regge action S, the other has action -S. The Barrett- Crane model, not knowing any better, simply averages over both of them, getting

(1/2)(exp(iS) + exp(-iS)) = cos(S)

So it's not really all that bad; it's doing the best it can under the circumstances. Whether this is good enough remains to be seen.

(Actually I didn't really lie to you before; I just didn't tell you my definition of "shape", so you couldn't tell whether mirror-image 4-simplices should count as having the same shape. Expository prose darts between the Scylla of overwhelming detail and the Charybdis of vagueness.)

Okay, on to a related issue. In the Barrett-Crane model one describes a quantum 4-geometry by labelling all the triangles with spins. This sounds reasonable if you think about how the shape of a 4-simplex is almost determined by the areas of its triangles. But if you actually examine the derivation of the model, it starts looking more odd. What you really do is take the space of geometries of a tetrahedron embedded in R4, and use a trick called geometric quantization to get something called the "Hilbert space of a quantum tetrahedron in 4 dimensions". You then build your 4-simplices out of these quantum tetrahedra.

Now the Hilbert space of a quantum tetrahedron has a basis labelled by the eigenvalues of operators corresponding to the areas of its 4 triangular faces. In physics lingo, it takes 4 "quantum numbers" to describe the shape of a quantum tetrahedron in 4 dimensions.

But classically, the shape of a tetrahedron is not determined by the areas of its triangles: it takes 6 numbers to specify its shape, not just 4. So there is something funny going on.

At first some people thought there might be more states of the quantum tetrahedron than the ones Barrett and Crane found. But Barbieri came up with a nice argument suggesting that Barrett and Crane had really found all of them:

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

While convincing, this argument was not definitive, since it assumed something plausible but not yet proven - namely, that the "6j symbols don't have too many exceptional zeros". Later, Mike Reisenberger came up with a completely rigorous argument:

8) Michael P. Reisenberger, On relativistic spin network vertices, preprint available as gr-qc/9809067.

But while this settled the facts of the matter, it left open the question of "why" --- why does it take 6 numbers to describe the shape of classical tetrahedron in 4 dimensions but only 4 numbers to describe the shape of a quantum one? John Barrett and I have almost finished a paper on this, so I'll give away the answer.

Not surprisingly, the key is that in quantum mechanics, not all observables commute. You only use the eigenvalues of commuting observables to label a basis of states. The areas of the quantum tetrahedron's faces commute, and there aren't any other independent commuting observables. It's a bit like how in classical mechanics you can specify both the position and momentum of a particle, but in quantum mechanics you can only specify one.

This isn't news, of course. And indeed, people knew perfectly well that for this reason, it takes only 5 numbers to describe the shape of a quantum tetrahedron in 3 dimensions. The real puzzle was why it takes even fewer numbers when your quantum tetrahedron lives in 4 dimensions! It seemed bizarre that adding an extra dimension would reduce the number of degrees of freedom! But it's true, and it's just a spinoff of the uncertainty principle. Crudely speaking, in 4 dimensions the fact that you know your tetrahedron lies in some hyperplane makes you unable to know as much about its shape.

Here are some other talks available on the web:

9) Abhay Ashtekar, Chris Beetle and Steve Fairhurst, Mazatlan lectures on black holes, slides available at http://vishnu.nirvana.phys.psu.edu/online/Html/Conferences/Mazatlan/

These explain a new concept of "nonrotating isolated horizon" which allow one to formulate and prove the zeroth and first laws of black hole mechanics in a way that only refers to the geometry of spacetime near the horizon. For more details try:

10) Abhay Ashtekar, Chris Beetle and S. Fairhurst, Isolated horizons: a generalization of black hole mechanics, preprint available as gr-qc/9812065.

This concept also serves as the basis for a forthcoming 2-part paper where Ashtekar, Corichi, Krasnov and I compute the entropy of a quantum black hole (see "week112" for more on this).

Finally, here are a couple more papers. I don't have time to say much about them, but they're both pretty neat:

11) Matthias Arnsdorf and R. S. Garcia, Existence of spinorial states in pure loop quantum gravity, preprint available as gr-qc/9812006.

I'll just quote the abstract:

We demonstrate the existence of spinorial states in a theory of canonical quantum gravity without matter. This should be regarded as evidence towards the conjecture that bound states with particle properties appear in association with spatial regions of non-trivial topology. In asymptotically trivial general relativity the momentum constraint generates only a subgroup of the spatial diffeomorphisms. The remaining diffeomorphisms give rise to the mapping class group, which acts as a symmetry group on the phase space. This action induces a unitary representation on the loop state space of the Ashtekar formalism. Certain elements of the diffeomorphism group can be regarded as asymptotic rotations of space relative to its surroundings. We construct states that transform non-trivially under a 2 π rotation: gravitational quantum states with fractional spin.

14) Steve Carlip, Black hole entropy from conformal field theory in any dimension, preprint available as hep-th/9812013.

Again, here's the abstract:

When restricted to the horizon of a black hole, the 'gauge' algebra of surface deformations in general relativity contains a physically important Virasoro subalgebra with a calculable central charge. The fields in any quantum theory of gravity must transform under this algebra; that is, they must admit a conformal field theory description. With the aid of Cardy's formula for the asymptotic density of states in a conformal field theory, I use this description to derive the Bekenstein-Hawking entropy. This method is universal - it holds for any black hole, in any dimension, and requires no details of quantum gravity - but it is also explicitly statistical mechanical, based on the counting of microscopic states.

On Thursday I'm flying to Schladming, Austria to attend a workshop on geometry and physics organized by Harald Grosse and Helmut Gausterer. Some cool physicists will be there, like Daniel Kastler and Julius Wess. If I understand what they're talking about I'll try to explain it here. Happy new year!

Addendum: Above I wrote:
Recently, Barrett and Williams came up with a nice argument saying that in the limit where the triangles have large areas, the amplitude for a 4-simplex in the Barrett-Crane theory is proportional, not to exp(iS), but to cos(S)....

This argument is not rigorous - it uses a stationary phase approximation that requires further justification. But similar argument to show the same sort of thing for quantum gravity in 3 dimensions, and their argument was recently made rigorous by Justin Roberts, with a lot of help from Barrett....

So one expects that with work, one can make Barrett and Williams' argument rigorous.

In fact one can't make it rigorous: it's wrong! In the limit of large areas the amplitude for a 4-simplex in the Barrett-Crane model is wildly different from cos(S), or exp(iS), or anything like that. Dan Christensen, Greg Egan and I showed this in a couple of papers that I discuss in "week170" and "week198". Our results were confirmed by John Barrett, Chris Steel, Laurent Friedel and David Louapre.

By now — I'm writing this in 2009 — it's generally agreed that the Barrett-Crane model is wrong and another model is better. To read about this new model, see "week280".

© 1999 John Baez