January 14, 1994

This Week's Finds in Mathematical Physics (Week 30)

John Baez

For the most part, this is a terse description of some papers dealing with quantum gravity. Some look to be quite important, but as I have not had time to read them as thoroughly as I would like, I won't say much.

First, however, let me note some books:

1) QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga, by Silvan S. Schweber, Princeton Series in Physics, Princeton U. Press, 784 pages, available May 1994.

Back in the 1930s there was a crisis in physics: nobody knew how to reconcile quantum theory with special relativity. This book describes the history of how people struggled with this problem and achieved a marvelous, but flawed, solution: quantum electrodynamics (QED). Marvelous, because it made verified predictions of unparalleled accuracy, involves striking new concepts, and gave birth to beautiful new mathematics. Flawed, only because nobody yet knows for sure whether the theory is mathematically well-defined -- for reasons profoundly related to physics at ultra-short distance scales. This story should give some inspiration to those currently attempting to reconcile quantum theory with general relativity! Feynman, Schwinger, and Tomonaga won Nobel prizes for QED, but Dyson was also instrumental in inventing the theory, and the book is mainly a story of these 4 men.

2) The Music of the Heavens: Kepler's Harmonic Astronomy, by Bruce Stephenson, Princeton U. Press, 296 pages, available July 1994.

Kepler's Physical Astronomy, by Bruce Stephenson, Princeton U. Press, 218 pages, paperback available June 1994.

Kepler's work on astronomy was in part based on the notion of the "music of the spheres," and in his Harmonice Mundi (1619) he sought to relate planetary velocities to the notes of a chord. He was also fascinated with geometry, and sought to relate the radii of the planetary orbits to the Platonic solids. While this may seem a bit silly nowadays, it's clear that this faith that mathematical patterns pervade the heavens was a crucial part of how Kepler found his famous laws of planetary motion. Also important, of course, was his use of what we would now call "physical" reasoning to understand the heavens -- that is, the use of analogies between the motions of heavenly bodies and that of ordinary terrestial matter. But even this is not as straightforward as one might hope, since (Stephenson argues in the second book) this physical reasoning was what we would now consider incorrect, even though it led to valid laws. More inspiration for those now struggling amid error to understand what the universe is really like!

3) Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, by Louis Kauffman and Sostenes Lins, Annals of Mathematics Studies No. 133, Princeton U. Press, 304 pages, available July 1994.

I described this briefly in "week17," before I had spent much time on it. Let me recall the main point: in the late 80's Jones invented a new invariant of knots and links in ordinary 3d space, but then Witten recognized that this invariant came from a quantum field theory, and thus could be extended to obtain an invariant of links in arbitrary 3d manifolds. (In particular, taking the link to be empty, one obtains a 3-manifold invariant.) In fact, there is a whole family of such invariants, essentially one for each semisimple Lie algebra, and Jones original example corresponded to the case su(2). In this case the combinatorics of the invariants are so simple that one can write a nice exposition in which one forgets the underlying, fairly sophisticated, mathematical physics (quantum groups, conformal field theory and the like) and simply presents the "how-to" using a kind of diagrammatic calculus known as "skein relations," or what Kauffman calls "Temperley-Lieb recoupling theory." That is the approach the authors take here. The curious reader will naturally want to know more! For example, anyone familiar with quantum theory and "6j symbols" will sense that this kind of thing is lurking in the background, and indeed, it is.

Now for the papers:

4) The physical hamiltonian in quantum gravity, by C. Rovelli and L. Smolin, 11 pages, preprint available in LaTeX form as gr-qc/9308002.

Fermions in quantum gravity, by H. A. Marales-Tecotl and C. Rovelli, 37 pages, preprint available in LaTeX form as gr-qc/9401011.

The Rovelli-Smolin loop variables program proceeds apace! In the former paper, Rovelli and Smolin consider quantum gravity coupled to a scalar matter field which plays the role of a clock. (Using part of the system described to play the role of a clock is a standard idea for dealing with the "problem of time," which arises in quantum theories on spacetimes having no preferred coordinates, like quantum gravity. However, getting this idea to actually work is not at all easy. For a bit on this issue see "week27".) Only after choosing this "clock field" can one work out a Hamiltonian for the theory, write down the analog of Schrodinger's equation, and examine the dynamics. Before, there is only a "Hamiltonian constraint" equation, also known as the Wheeler-DeWitt equation.

In the latter paper, Rovelli and Marales-Tecotl discuss how to include fermions. The beautiful thing here is that fermions are described in the loop representation by the ends of arcs, while pure gravity is described by loops. This is completely analogous to the old string theory of mesons, in which mesons were represented as arcs of "string" -- the gluon field -- connecting two fermionic "ends" -- the quarks.

5) Extended loops: a new arena for nonperturbative quantum gravity, by C. Di Bartolo, R. Gambini, J. Griego and J. Pullin, 12 pages, preprint available in Revtex form as gr-qc/9312029.

For a while now, Gambini and collaborators have been developing a modified version of the loop representation that appears to be especially handy for doing perturbative calculations (perturbing in the coupling constant, that is, Newton's gravitational constant -- not perturbing about a fixed flat "background" spacetime, which is regarded as a "no-no" in this philosophy). The mathematical basis for this "extended" loop representation is something quite charming in itself: it amounts to embedding the loop group into an (infinite-dimensional) Lie group. The "perturbative" calculations described above are thus analogous to how one uses Lie algebras to study Lie groups. In fact, this analogy is a deep one, since the extended loop representation also permits perturbative calculations in Chern-Simons theory, allowing one to calculate "Vassiliev invariants" starting just from Lie-algebraic data. In fact this was done by Bar-Natan (cf "week3"), who was using the extended loop representation without particularly knowing about that fact!

This paper puts the extended loop representation to practical use by finding some new solutions to the quantum version of Einstein's equations. These solutions are essentially Vassiliev invariants! (See also the paper by Gambini and Pullin listed in "week23").

6) Ashtekar variables in classical general relativity, by Domenico Giulini, 43 pages, preprint available in TeX form as gr-qc/9312032.

This was a lecture given at the 117th WE-Heraeus Seminar: ``The Canonical Formalism in Classical and Quantum General Relativity'', 13-17 September 1993, Bad-Honnef, Germany, the goal of which was to give an introduction to Ashtekar's "new variables" for general relativity.

© 1994 John Baez