
This Week's Finds in Mathematical Physics (Week 24) John Baez
I will now revert to topics more directly connected to physics and start catching on the papers that have been accumulating. First, two very nice review papers:
1) Prima facie questions in quantum gravity, by Chris Isham, lecture at Bad Honeff, September 1993, preprint available in LaTeX form as grqc/9310031.
If one wants to know why people make such a fuss about quantum gravity, one could not do better than to start here. There are many approaches to the project of reconciling quantum mechanics with gravity, all of them rather technical, but here Isham focuses on the "prima facie" questions that present themselves no matter what approach one uses. He even explains why we should study quantum gravity  a nontrivial question, given how difficult it has been and how little practical payoff there has been so far! Let me quote his answers and urge you to read the rest of this paper:
Gravitational singularities. The classical theory of general relativity is notorious for the existence of unavoidable spacetime singularities. It has long been suggested that a quantum theory of gravity might cure this disease by some sort of `quantum smearing'.
Quantum cosmology. A particularly interesting singularity is that at the beginning of a cosmological model described by, say, a RobertsonWalker metric. Classical physics breaks down here, but one of the aims of quantum gravity has always been to describe the `origin' of the universe as some type of quantum event.
The end state of the Hawking radiation process. One of the most striking results involving general relativity and quantum theory is undoubtedly Hawking's famous discovery of the quantum thermal radiation produced by a black hole. Very little is known of the final fate of such a system, and this is often taken to be another task for a quantum theory of gravity.
The unification of fundamental forces. The weak and electromagnetic forces are neatly unified in the SalamWeinberg model, and there has also been a partial unification with the strong force. It is an attractive idea that a consistent quantum theory of gravity must include a unification of all the fundamental forces.
The possibility of a radical change in basic physics. The deep incompatibilities between the basic structures of general relativity and of quantum theory have lead many people to feel that the construction of a consistent theory of quantum gravity requires a profound revision of the most fundamental ideas of modern physics. The hope of securing such a paradigm shift has always been a major reason for studying the subject.
2) Lectures on 2d gauge theories: topological aspects and path integral techniques, by Matthias Blau and George Thompson, 70 pages, preprint available in LaTeX form as hepth/9310144.
Most of the basic laws of physics appear to be gauge theories. Gauge theories are tricky to deal with because they are inherently nonlinear. (At least the "nonabelian" ones are  the main example of an abelian gauge theory is Maxwell's equations.) People have been working hard for quite some time trying to develop tools to study gauge theories on their own terms, and one reason for the interest in gauge theories in 2dimensional spacetime is that life is simple enough in this case to exactly solve the theories and see precisely what's going on. Another reason is that in string theory one becomes interested in gauge fields living on the 2dimesional "string worldsheet."
This paper is a thorough review of two kinds of gauge theories in 2 dimensions: topological YangMills theory (also called BF theory) and the G/G gauged WessZuminoWitten model. Both of these are of great mathematical interest in addition to their physical relevance. Studying the BF theory gives a way to do integrals on the moduli space of flat connections on a bundle over a Riemann surface, while studying the G/G model amounts to a geometric construction of the categories of representations of quantum groups at roots of unity. (Take my word for it, mathematicians find these important!)
I have found this review a bit rough going so far because the authors like to use supersymmetry to study these models. But I will continue digging in, since the authors consider the following topics (and I quote): solution of YangMills theory on arbitrary surfaces; calculation of intersection numbers of moduli spaces of flat connections; coupling of YangMills theory to coadjoint orbits and intersection numbers of moduli spaces of parabolic bundles; derivation of the Verlinde formula from the G/G model; derivation of the shift k to k+h in the G/G model via the index of the twisted Dolbeault complex.
3) Semiclassical limits of simplicial quantum gravity, by J. W. Barrett and T. J. Foxon, preprint available as grqc/9310016.
This paper looks at quantum gravity in 3 spacetime dimesions formulated along the lines of Ponzano and Regge, that is, with the spacetime manifold replaced by a bunch of tetrahedra (a "simplicial complex"). I describe some work along these lines in "week16". Here the Feynman path integral is replaced by a discrete sum over states, in which the edges of the tetrahedra are assigned integer or halfinteger lengths, which really correspond to "spins," and the formula for the action is given in terms of 6jsymbols. The authors look for stationary points of this action and find that some correspond to Riemannian metrics and some correspond to Lorentzian metrics. This is strongly reminiscent of Hartle and Hawking's work on quantum cosmology,
4) Wave function of the universe, by J. B. Hartle and S. W. Hawking, Phys. Rev. D28 (1983), 2960.
in which there is both a Euclidean and a Lorentzian regime (providing a most fascinating answer to the old question, "what came before the big bang!). Here, however, the path integral is oscillatory in the Euclidean regime and exponential in the Lorentzian one  the opposite of what Hartle and Hawking had. This puzzles me.
5) Generalized measures in gauge theory, by John Baez, available in LaTeX as hepth/9310201.
Path integrals in gauge theory typically invoke the concept of Lebesgue measure on the space of connections. This is roughly an infinitedimensional vector space, and there is no ``Lebesgue measure'' on an infinitedimensional vector space. So what is going on? Physicists are able to do calculations using this concept and get useful answers  mixed in with infinities that have to be carefully ``renormalized.'' Some of the infinities here are supposedly due to the fact that one should really be working no on the space of connections, but on a quotient space, the connections modulo gauge transformations. But not all the infinities are removed this way, and mathematically the whole situation is enormously mysterious.
Recently Ashtekar, Isham, Lewandowski and myself have been looking at a way to generalize the concept of measure, suggested by earlier work on the ``loop representation'' of gauge theories. Ashtekar and Lewandowski managed to rigorously construct a kind of ``generalized measure'' on the space of connections modulo gauge transformations that acts formally quite a bit like what might hope for. In this paper I show how can define generalized measures directly on the space of connections. All of these project down to generalized measures on the space of connections modulo gauge transformations, but even when one is interested in gaugeinvariant quantities, it is sometimes easier to work ``upstairs.'' In particular, when the gauge group is compact, there is a ``uniform'' generalized measure on the space of connections that projects down to the measure constructed by Ashtekar and Lewandowski. This generalized measure is in some respects a rigorous substitute for the illdefined ``Lebesgue measure,'' but it is actually built using Haar measure on G. I also define generalized measures on the group of gauge transformations (which is an infinitedimensional group), and when G is compact I construct a natural example that is a rigorous substitute for Haar measure on the group of gauge transformations . As an application of this ``generalized Haar measure'' I show that any generalized measure on the space of connections can be averaged against generalized Haar measure to give a gaugeinvariant generalized measure on the space of connections.
This doesn't, by the way, mean the problems I mentioned at the beginning are solved!
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
