June 24, 1998

This Week's Finds in Mathematical Physics (Week 122)

John Baez

In summertime, academics leave the roost and fly hither and thither, seeking conferences and conversations in far-flung corners of the world. At the end of May, everyone started leaving the Center for Gravitational Physics and Geometry: Lee Smolin for the Santa Fe Institute, Abhay Ashtekar for Uruguay and Argentina, Kirill Krasnov for his native Ukraine, and so on. It got so quiet that I could actually get some work done, were it not for the fact that I, too, flew the coop: first for Chicago, then Portugal, and then to one of the most isolated, technologically backwards areas on earth: my parents' house. Connected to cyberspace by only the thinnest of threads, writing new issues of This Week's Finds became almost impossible....

I did, however, read some newsgroups, and by this means Jim Carr informed me that an article on spin foam models of quantum gravity had appeared in Science News. I can't resist mentioning it, since it quotes me:

1) Ivars Peterson, Loops of gravity: calculating a foamy quantum space-time, Science News, June 13, 1998, Vol. 153, No. 24, 376-377.

It gives a little history of loop quantum gravity, spin networks, and the new burst of interest in spin foams. Nothing very technical - but good if you're just getting started. If you want something more detailed, but still user-friendly, try Rovelli's new paper:

2) Carlo Rovelli and Peush Upadhya, Loop quantum gravity and quanta of space: a primer, available as gr-qc/9806079.

I haven't read it yet, since I'm still in a rather low-tech portion of the globe, but it gives simplified derivations of some of the basic results of loop quantum gravity, like the formula for the eigenvalues of the area operator. As explained in "week110", one of the main predictions of loop quantum gravity is that geometrical observables such as the area of any surface take on a discrete spectrum of values, much like the energy levels of a hydrogen atom. At first the calculation of the eigenvalues of the area operator seemed rather complicated, but by now it's well-understood, so Rovelli and Upadhya are able to give a simpler treatment.

While I'm talking about the area operator, I should mention another paper by Rovelli, in which he shows that its spectrum is not affected by the presence of matter (or more precisely, fermions):

3) Carlo Rovelli and Merced Montesinos, The fermionic contribution to the spectrum of the area operator in nonperturbative quantum gravity, available as gr-qc/9806120.

This is especially interesting because it fits in with other pieces of evidence that fermions could simply be the ends of wormholes - an old idea of John Wheeler (see "week109").

I should also mention some other good review articles that have turned up recently. Rovelli has written a survey comparing string theory, the loop representation, and other approaches to quantum gravity, which is very good because it points out the flaws in all these approaches, which their proponents are usually all too willing to keep quiet about:

4) Carlo Rovelli, Strings, loops and others: a critical survey of the present approaches to quantum gravity. Plenary lecture on quantum gravity at the GR15 conference, Pune, India, available as gr-qc/9803024.

Also, Loll has written a review of approaches to quantum gravity that assume spacetime is discrete. It does not discuss the spin foam approach, which is too new; instead it mainly talks about lattice quantum gravity, the Regge calculus, and the dynamical triangulations approach. In lattice quantum gravity you treat spacetime as a fixed lattice, usually a hypercubical one, and work with discrete versions of the usual fields appearing in general relativity. In the Regge calculus you triangulate your 4-dimensional spacetime - i.e., chop it into a bunch of 4-dimensional simplices - and use the lengths of the edges of these simplices as your basic variables. (For more details see "week120".) In the dynamical triangulations approach you also triangulate spacetime, but not in a fixed way - you consider all possible triangulations. However, you assume all the edges of all the simplices have the same length - the Planck length, say. Thus all the information about the geometry of spacetime is in the triangulation itself - hence the name "dynamical triangulations". Everything becomes purely combinatorial - there are no real numbers in our description of spacetime geometry anymore. This makes the dynamical triangulations approach great for computer simulations. Computer simulations of quantum gravity! Loll reports on the results of a lot of these:

5) Renate Loll, Discrete approaches to quantum gravity in four dimensions, available as gr-qc/9805049, also available as a webpage on Living Reviews in Relativity at http://www.livingreviews.org/Articles/Volume1/1998-13loll/

By the way, "Living Reviews in Relativity" is a cool website run by the AEI, the Albert Einstein Institute for gravitational physics, located in Potsdam, Germany. The idea is that experts will write review articles on various subjects and keep them up to date as new developments occur. You can find this as follows:

6) Living Reviews in Relativity, http://www.livingreviews.org

Here are some other good places to learn about the dynamical triangulations approach to quantum gravity:

7) J. Ambjorn, Quantum gravity represented as dynamical triangulations, Class. Quant. Grav. 12 (1995) 2079-2134.

8) J. Ambjorn, M. Carfora, and A. Marzuoli, The Geometry of Dynamical Triangulations, Springer-Verlag, Berlin, 1998. Also available electronically as hep-th/9612069 - watch out, this is 166 pages long!

I can't resist pointing out an amusing relationship between dynamical triangulations and mathematical logic, which Ambjorn mentions in his review article. In computer simulations using the dynamical triangulations approach, one wants to compute the average of certain quantities over all triangulations of a fixed compact manifold - e.g., the 4-dimensional sphere, S⁴. The typical way to do this is to start with a particular triangulation and then keep changing it using various operations - "Pachner moves" - that are guaranteed to eventually take you from any triangulation of a compact 4-dimensional manifold to any other.

Now here's where the mathematical logic comes in. Markov's theorem says there is no algorithm that can decide whether or not two triangulations are triangulations of the same compact 4-dimensional manifold. (Technically, by "the same" I mean "piecewise linearly homeomorphic", but don't worry about that!) If they are triangulations of the same manifold, blundering about using the Pachner moves will eventually get you from one to the other, but if they are not, you may never know for sure.

On the other hand, S⁴ may be special. It's an open question whether or not S⁴ is "algorithmically detectable". In other words, it's an open question whether or not there's an algorithm that can decide whether or not a triangulation is a triangulation of the 4-dimensional sphere.

Now, suppose S⁴ is not algorithmically detectable. Then the maximum number of Pachner moves it takes to get between two triangulations of the 4-sphere must grow really fast: faster than any computable function! After all, if it didn't, we could use this upper bound to know when to give up when using Pachner moves to try to reduce our triangulation to a known triangulation of S⁴. So there must be "bottlenecks" that make it hard to efficiently explore the set of all triangulations of S⁴ using Pachner moves. For example, there must be pairs of triangulations such that getting from one to other via Pachner moves requires going through triangulations with a lot more 4-simplices.

However, computer simulations using triangulations with up to 65,536 4-simplices have not yet detected such "bottlenecks". What's going on? Well, maybe S⁴ actually is algorithmically detectable. Or perhaps it's not, but the bottlenecks only occur for triangulations that have more than 65,536 4-simplices to begin with. Interestingly, one dimension up, it's known that the 5-dimensional sphere is not algorithmically detectable, so in this case bottlenecks must exist - but computer simulations still haven't seen them.

I should emphasize that in addition to this funny computability stuff, there is also a whole lot of interesting physics coming out of the dynamical triangulations approach to quantum gravity. Unfortunately I don't have the energy to explain this now - so read those review articles, and check out that nice book by Ambjorn, Carfora and Marzuoli!

On another front... Ambjorn and Loll, who are both hanging out at the AEI these days, have recently teamed up to study causality in a lattice model of 2-dimensional Lorentzian quantum gravity:

9) J. Ambjorn and R. Loll, Non-perturbative Lorentzian quantum gravity, causality and topology change, available as hep-th/9805108.

I'll just quote the abstract:

We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory.

I've been hearing rumbles off in the distance about some interesting work by Kreimer relating renormalization, Feynman diagrams, and Hopf algebras. A friendly student of Kreimer named Mathias Mertens handed me a couple of the basic papers when I was in Portugal:

10) Dirk Kreimer, Renormalization and knot theory, Journal of Knot Theory and its Ramifications, 6 (1997), 479-581. Preprint available as q-alg/9607022 - beware, this is 103 pages long!

Dirk Kreimer, On the Hopf algebra structure of perturbative quantum field theories, available as q-alg/9707029.

I'm looking through them but I don't really understand them yet. The basic idea seems to be something like this. In quantum field theory you compute the probability for some reaction among particles by doing integrals which correspond in a certain way to pictures called Feynman diagrams. Often these integrals give infinite answers, which forces you to do a trick called renormalization to cancel the infinities and get finite answers. Part of why this trick works is that while your integrals diverge, they usually diverge at a well-defined rate. For example, you might get something asymptotic to a constant times 1/d^k, where d is the spatial cutoff you put in to get a finite answer. And the constant you get here can be explicitly computed. For example, it often involves numbers like ζ(n), where ζ is the Riemann zeta function, much beloved by number theorists:

ζ(n) = 1/1ⁿ + 1/2ⁿ + 1/3ⁿ + ....

Kreimer noticed that if you take the Feynman diagram and do some tricks to turn it into a drawing of a knot or link, the constant you get is related in interesting ways to the topology of this knot or link! More complicated knots or links give fancier constants, and there are all sorts of suggestive patterns. He worked out a bunch of examples in the first paper cited above, and since then people have worked out lots more, which you can find in the references.

Apparently the secret underlying reason for these patterns comes from the combinatorics of renormalization, which Kreimer was able to summarize in a certain algebraic structure called a Hopf algebra. Hopf algebras are important in both combinatorics and physics, so perhaps this shouldn't be surprising. But there is still a lot of mysterious stuff going on, at least as far as I can tell.

What's really intriguing about all this is which quantum field theories Kreimer was studying when he discovered this stuff: not topological quantum field theories like Chern-Simons theory, which already have well-understood relationship to knot theory, but instead, field theories that ordinary particle physicists have been thinking about for decades, like quantum electrodynamics, φ⁴ theory in 4 dimensions, and φ³ theory in 6 dimensions - field theories where renormalization is a deadly serious business, thanks to nasty problems like "overlapping divergences".

The idea that knot theory is relevant to these field theories is exciting but also somewhat puzzling, since they don't live in 3-dimensional spacetime the way Chern-Simons theory does. People familiar with Chern-Simons theory have already been seeing fascinating patterns relating knot theory, quantum field theory and number theory. Is this new stuff related? Or is it something completely different? Kreimer seems to think it's related.

According to Kirill Krasnov, the famous mathematician Alain Connes is going around telling people to learn about this stuff. Apparently Connes is now writing a paper on it with Kreimer, and it was Connes who got the authors of this paper interested in the subject:

11) Thomas Krajewski and Raimar Wulkenhaar, On Kreimer's Hopf algebra structure of Feynman graphs, available as hep-th/9805098.

Since I haven't plunged in yet, I'll just quote the abstract:

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories. In Kreimer's original work, overlapping divergences were first disentangled into a linear combination of disjoint and nested ones using the Schwinger-Dyson equation. The linear combination then was tackled by the Hopf algebra operations. We present a formulation where the coproduct itself produces the linear combination, without reference to external input.