March 11, 2000

This Week's Finds in Mathematical Physics (Week 146)

John Baez

Paper in white the floor of the room, and rule it off in one- foot squares. Down on one's hands and knees, write in the first square a set of equations conceived as able to govern the physics of the universe. Think more overnight. Next day put a better set of equations into square two. Invite one's most respected colleagues to contribue to other squares. At the end of these labors, one has worked oneself out into the doorway. Stand up, look back on all those equations, some perhaps more hopeful than others, raise one's finger commandingly, and give the order "Fly!" Not one of those equations will put on wings, take off, or fly. Yet the universe "flies".

Some principle uniquely right and compelling must, when one knows it, be also so compelling that it is clear the universe is built, and must be built, in such and such a way, and that it could not be otherwise. But how can one disover that principle?

John Wheeler was undoubtedly the author of these words, which appear near the end of Misner, Thorne and Wheeler's textbook "Gravitation", published in 1972. Since then, more people than ever before in the history of the world have tried their best to find this uniquely compelling principle. A lot of interesting ideas, but no success yet.

But what if Wheeler was wrong? What if there is not a uniquely compelling principle or set of equations that governs our universe? For example, what if all equations govern universes? In other words, what if all mathematical structures have just as much "physical existence" (whatever that means!) as those describing our universe do? Many of them will not contain patterns we could call awareness or intelligence, but some will, and these would be "seen from within" as "universes" by their inhabitants. In this scenario, there's nothing special about our universe except that we happen to be in this one.

In other words: what if there is ultimately no difference between mathematical possibility and physical existence? This may seem crazy, but personally I believe that most alternatives, when carefully pondered, turn out to be even more crazy.

Of course, it's fun to think about such ideas and difficult to get anywhere with them. But tonight when I was nosing around the web, I found out that someone has already developed and published this idea:

1) Max Tegmark, Is the "theory of everything" merely the ultimate ensemble theory?, Ann. Phys. 270 (1998), 1-51, preprint available as gr-qc/9704009.

Max Tegmark, Which mathematical structure is isomorphic to the universe?,

3) Marcus Chown, Anything goes, New Scientist 158 (1998) 26-30, also available at

As far as I can tell, most of the time Max Tegmark is a perfectly respectable physicist at the University of Pennsylvania; he works on the cosmic microwave background radiation, the large-scale structure of the universe (superclusters and the like), and type 1A supernovae. But he has written a fascinating paper on the above hypothesis, which I urge you to read. It's less far-out than you may think.

Okay, now on to quantum gravity. Jan Ambjorn and Renate Loll have teamed up to work on discrete models of spacetime geometry, with an emphasis on the Lorentzian geometry of triangulated manifolds. Much more has been done over on the Riemannian side of things, so it's high time to focus more energy on the physically realistic Lorentzian case. Of course, if the metric is fixed you can often use a trick called "Wick rotation" to turn results about quantum field theory on Riemannian spacetime into results for Lorentzian spacetime. But it's never been clear that this works when the geometry of spacetime is a variable - and quantized, for that matter. So we need both more work on Wick rotation in this context and also work that goes straight for the jugular: the Lorentzian context.

Here are some of their papers:

4) J. Ambjorn, J. Correia, C. Kristjansen, and R. Loll, On the relation between Euclidean and Lorentzian 2d quantum gravity, preprint avilable as hep-th/9912267.

J. Ambjorn, J. Jurkiewicz and R. Loll, Lorentzian and Euclidean quantum gravity - analytical and numerical results, preprint available as hep-th/0001124.

J. Ambjorn, J. Jurkiewicz and R. Loll, A non-perturbative Lorentzian path integral for gravity, preprint avilable as hep-th/0002050.

The last paper is especially interesting to me, since it tackles the problem of defining a path integral for 3+1-dimensional Lorentzian quantum gravity. They describe a path integral where you first slice spacetime like a salami using surfaces of constant time, and then pack each slice with simplices having edges with specified lengths - the edges being spacelike within each surface, and timelike when they go from one surface to the next. They allow the number of simplices in each slice to be variable. Actually they focus on the 2+1-dimensional case, but they say the 3+1-dimensional case works similarly, and I actually trust them enough to believe them about this - especially since nothing they do relies on the fact that 2+1-dimensional gravity lacks local degrees of freedom. They can Wick-rotate this picture and get a time evolution operator that's self-adjoint and positive, just like you'd expect of an operator of the form exp(-tH).

Speaking of Wick rotations in quantum gravity, here's another paper to think about:

5) Abhay Ashtekar, Donald Marolf, Jose Mourao and Thomas Thiemann, Osterwalder-Schrader reconstruction and diffeomorphism invariance, preprint available as quant-ph/9904094.

The Osterwalder-Shrader theorem is the result people use when they want to rigorously justify Wick rotations. Here these authors generalize it so that it applies to a large class of background-free field theories - perhaps even quantum gravity! It turns out not to be hard, once you go about it properly. Quite a surprise.

I've been working with Ashtekar and Krasnov for a couple of years now on computing the entropy of black holes using loop quantum gravity. I talked about this in "week112", right after we came out with a short paper sketching the calculation. Now we're almost done with the detailed paper. In the meantime, Ashtekar has written a couple of pedagogical accounts explaining the basic idea. I mentioned one he wrote with Krasnov in "week120", and here's another:

6) Abhay Ashtekar, Interface of general relativity, quantum physics and statistical mechanics: some recent developments, to appear in Annalen der Physik, preprint available as gr-qc/9910101.

Let me just quote the abstract - I can't bear to talk about this any more until the actual paper is finished:

The arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of `isolated horizons.' Unlike event horizons, isolated horizons can be located in a space-time quasi-locally. Furthermore, they need not be Killing horizons. In particular, a space-time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the zeroth and first laws of black hole mechanics extend to isolated horizons. Furthermore, by carrying out a systematic, non-perturbative quantization, one can explore the quantum geometry of isolated horizons and account for their entropy from statistical mechanical considerations. After a general introduction to black hole thermodynamics as a whole, these recent developments are briefly summarized.

There have also been a number of papers working out the details of the classical notion of "isolated horizon" - I've mentioned some already, but let me just list them all here:

7) Abhay Ashtekar, Alejandro Corichi, and Kirill Krasnov, Isolated horizons: the classical phase space, preprint available as gr-qc/9905089.

Abhay Ashtekar, Christopher Beetle, and Stephen Fairhurst, Mechanics of isolated horizons, Class. Quant. Grav. 17 (2000) 253-298, preprint available as gr-qc/9907068.

Abhay Ashtekar and Alejandro Corichi, Laws governing isolated horizons: inclusion of dilaton couplings, preprint available as gr-qc/9910068.

Jerzy Lewandowski, Space-times admitting isolated horizons, preprint available as gr-qc/9907058.

Lewandowski's paper is important because it gets serious about studying rotating isolated horizons - this makes me feel a lot more optimistic that we'll eventually be able to extend the entropy calculation to rotating black holes (so far it's just done for the nonrotating case).

Okay, now let me turn my attention to spin foams. Last month, Reisenberger and Rovelli came out with a couple of papers that push forward the general picture of spin foams as Feynman diagrams, generalizing the old work of Boulatov and Ooguri, and the newer work of De Pietri et al. Again, I'll just quote the abstracts....

8) Michael Reisenberger and Carlo Rovelli, Spin foams as Feynman diagrams, preprint available as gr-qc/0002083.

It has been recently shown that a certain non-topological spin foam model can be obtained from the Feynman expansion of a field theory over a group. The field theory defines a natural "sum over triangulations", which removes the cutoff on the number of degrees of freedom and restores full covariance. The resulting formulation is completely background independent: spacetime emerges as a Feynman diagram, as it did in the old two-dimensional matrix models. We show here that any spin foam model can be obtained from a field theory in this manner. We give the explicit form of the field theory action for an arbitrary spin foam model. In this way, any model can be naturally extended to a sum over triangulations. More precisely, it is extended to a sum over 2-complexes.

9) Michael Reisenberger and Carlo Rovelli, Spacetime as a Feynman diagram: the connection formulation, preprint available as gr-qc/0002095.

Spin foam models are the path integral counterparts to loop quantized canonical theories. In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper [gr-qc/0002083]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models.

I'm completely biased, but I think this is the way to go in quantum gravity... we need to think more about the Lorentzian side of things, like Barrett and Crane have been doing, but these spin foam models are so darn simple and elegant I can't help but think there's something right about them - especially when you see the sum over triangulations pop out automatically from the Feynman diagram expansion of the relevant path integral.

There's also been some good work on the relation between canonical quantum gravity and Vassiliev invariants. The idea is to use this class of knot invariants as a basis for a Hilbert space of diffeomorphism- invariant states - a tempting alternative to the Hilbert space having spin networks as a basis. Maybe everything will start making sense when we see how these two choices fit together. But anyway, these papers tackle the crucial issue of the Hamiltonian constraint using this Vassiliev approach, and get results startlingly similar to those obtained by Thiemann using the spin network approach:

10) Cayetano Di Bartolo, Rodolfo Gambini, Jorge Griego, and Jorge Pullin, Consistent canonical quantization of general relativity in the space of Vassiliev invariants, preprint available as gr-qc/9909063.

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure, preprint available as gr-qc/9911009.

Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra, preprint available as gr-qc/9911010.

Finally, Martin Bojowald has written a couple of papers applying the loop approach to quantum cosmology. The idea is to apply loop quantization to a "minisuperspace" - a phase space describing only those solutions of general relativity that have a certain large symmetry group.

11) Martin Bojowald, Loop Quantum Cosmology I: Kinematics, preprint available as gr-qc/9910103.

Martin Bojowald, Loop Quantum Cosmology II: Volume Operators, gr-qc/9910104.

© 2000 John Baez