November 11, 1995

This Week's Finds in Mathematical Physics (Week 69)

John Baez

One of the great things about starting to work on quantum gravity was getting to know some of the people in the field. Ever since the development of string theory and the loop representation of quantum gravity, there has been a fair amount of interest in understanding how quantum theory and gravity fit together. Indeed, now that the Standard Model seems to be giving a spectacularly accurate description of all the forces except gravity, quantum gravity is one of the few really big mysteries left when it comes to working out the basic laws of physics --- or at least, one of the few obvious big mysteries. (As soon as one mystery starts becoming less mysterious, new mysteries tend to become more visible.) But back when particle physics was big business, only a few rather special sorts of people were seriously devoted to quantum gravity. These people seem to be often more than averagely interested in philosophy, often more interested in mathematics (which is one of the few solid handholds in this slippery subject), and always more resigned to the fact that Nature does not reveal all her secrets very readily.

One of these folks is Chris Isham, whom I first saw at a conference in Seattle in 1991. The conference was on classical field theory but somehow he, Abhay Ashtekar, and Renate Loll sneaked in and gave some talks on the loop representation of quantum gravity. This is when I first became really interested in this subject, which I was later to work on quite a bit. I remember Isham saying how he had been working on quantum gravity for many years, and that he'd gotten used to the fact that nothing ever worked, but that this approach seemed to be working so far. He went on to talk about work he'd done with Ashtekar on making the loop representation rigorous, which was based on Gelfand-Naimark spectral theory. He said that as a student, when he'd learned about this theory, he was really excited, because it completely depends on the fact that if we have a space X, we can think of any point x in X as a functional on the space of functions on X, basically defining by defining


to be


He said this with a laugh, but I knew what he meant, because I too had found this idea tremendously exciting when I first learned the Gelfand-Naimark theory. I guess it's something about how what seems at first like some sort of bizarre joke can turn out to be very useful....

Anyway, later, when I decided to work on this sort of thing and was trying to learn more about quantum gravity, I found his review article on the problem of time (see "week9") tremendously helpful, and I constantly recommend it to everyone who is trying to get their teeth into this somewhat elusive issue. So it's not surprising that Isham figures prominently in the following nice popular article on the problem of time:

1) Marcia Bartusiak, When the universe began, what time was it?, Technology Review (edited at the Massachusetts Institute of Technology), November/December 1995, pp. 54-63.

If you can find this, read it: it also features Karel Kuchar and Carlo Rovelli.

This spring, I visited Isham at Imperial College in London and found him to be just as interesting in person as in print, and not at all scary... a bit of an cynic about all existing approaches to quantum gravity (probably because he sees so clearly how flawed they all are), but thoroughly good-humored about it and perfectly open to all sorts of ideas, even my own nutty ideas about n-categories and physics.

Anyway, Isham has recently written a review article on quantum gravity that gives a nice overview of the basic issues of the field:

2) C. J. Isham, Structural issues in quantum gravity, plenary session lecture given at the GR14 conference, Florence, August 1995, preprint available as gr-qc/9510063.

One interesting thing about it is the emphasis on the question of whether spacetime is really a manifold the way we all usually think, or perhaps something that just looks like a manifold at sufficiently large distance scales. This is one of those fundamental issues that is rather hard to make direct progress on; one has to sort of sneak up on it, but it's nice to see someone boldly holding the problem up for examination. Often the most important issues are the ones everyone is scared to talk about, because they are so intractable.

Much of Abhay Ashtekar's early work dealt with asymptotically flat solutions of Einstein's equation, but in about 1986 he somehow invented a new formulation of general relativity, which everyone now calls the "new variables" or "Ashtekar variables". In terms of these new variables general relativity looks a whole lot more like Yang-Mills theory (the theory of all the forces except gravity), and this let Rovelli and Smolin formulate a radical new approach to quantum gravity, the "loop representation". (For a fun, nontechnical introduction to this, try the article by Bartusiak reviewed in "week10".)

Nowadays, Ashtekar is the main person behind the drive to make the loop representation of quantum gravity into a mathematically rigorous theory. Thus it's natural that after that first time in Seattle I would wind up seeing him pretty often... first at Syracuse University and then at the Center for Gravitational Physics and Geometry which he started at Penn State. It's really impressive how he has organized people into an effective team there... and how he is systematically converting people's hopes and dreams concerning the loop representation into a beautiful set of rigorous theorems. For a good mathematical introduction to his program, see his paper reviewed in "week7". A less mathematical introduction is:

3) Abhay Ashtekar, Polymer geometry at Planck scale and quantum Einstein equations.

This will probably appear on gr-qc in a while.

I have also seen Renate Loll fairly often in the years since that Seattle conference. She is younger than Ashtekar and Isham (in fact, she was a postdoc with Isham at one point), hence less intimidating to me, which meant that I really enjoyed pestering her with stupid questions when I was just starting to learn about this loop representation stuff. One of her specialties is lattice gauge theory, and recently she has developed a lattice version of quantum gravity that is eminently suitable for computer calculations. The last time I saw her was at a conference in Warsaw this spring (as reported in "week55" and "week56"). In the process of working on her lattice approach, she gave Rovelli and Smolin a big shock by turning up an error in their computation of the volume operator in quantum gravity. A state of quantum gravity can be visualized roughly as a graph embedded in space, with edges labelled by spins. Rovelli and Smolin had thought there were states of nonzero volume corresponding to graphs with only trivalent vertices (3 edges meeting a vertex, that is). As it turns out, they'd made a sign error, and these states have zero volume; you need a quadrivalent vertex to get some volume. She has just written a paper on this topic:

4) Renate Loll, Spectrum of the volume operator in quantum gravity, 14 pages in plain tex, with 4 figures (postscript, compressed and uu-encoded), available as gr-qc/9511030.

The abstract reads as follows:

The volume operator is an important kinematical quantity in the non-perturbative approach to four-dimensional quantum gravity in the connection formulation. We give a general algorithm for computing its spectrum when acting on four-valent spin network states, evaluate some of the eigenvalue formulae explicitly, and discuss the role played by the Mandelstam constraints.

Quote of the week: "Nothing is too wonderful to be true, if it be consistent with the laws of nature, and in such things as these, experiment is the best test of such consistency." Faraday, laboratory diaries, entry 10,040, March 19, 1849.

© 1995 John Baez