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I will now start catching up on some of the papers that have accumulated over the summer. This time I'll say a bit about recent developments in quantum field theory and 4-dimensional topology.
The quantum field theories that describe three of the forces of nature (electromagnetic, strong and weak) depend for their formulation on a fixed metric on spacetime - that is, a way of measuring distance and time. Indeed, it seems pretty close to being true that spacetime is R^4, and that the "interval" between any two points in 4-dimensional space is given by the Minkowski metric
dt^2 - dx^2 - dy^2 - dz^2
where dt is the change in the time, or t, coordinate, dx is the change in the spatial x coordinate, and so on. However, it's apparently not quite true. In fact, the presence of matter or energy distorts this metric a little, and the effect of the resulting "curvature of spacetime" is perceived as gravity. This is the basic idea of general relativity, which is nicely illustrated by the way in which the presence of the sun bends starlight that passes nearby.
Gravity is thus quite different from the other forces, at least to our limited understanding. The other forces we have quantum theories of, and these theories depend on a fixed (that is, pre-given) metric. We have no quantum theory of gravity yet, only a classical theory, and this theory is precisely a set of equations describing a variable metric, that is, one dependent upon the state of the universe. These are, of course, Einstein's equations.
In fact it is no coincidence that we have no quantum theory of gravity. For most of the last 50 years or so physicists have been working very hard at inventing and understanding quantum field theories that rely for their formulation on a fixed metric. Indeed, physicists spent huge amounts of effort trying to make a theory of quantum gravity along essentially these lines! This is what one calls "perturbative" quantum gravity. Here one says, "Well, we know the metric isn't quite the Minkowski metric, but it's awfully close, so we'll write it as the Minksowski metric plus a small perturbation, derive equations for this perturbation from Einstein's equations, and make a quantum field theory based on those equations." That way we could use the good old Minkowski metric as a "background metric" and thus use all the methods that work for other quantum field theories. This was awfully fishy from the standpoint of elegance, but if it had worked it might have been a very good thing, and indeed we learned a lot from its failure to work. Mainly, though, we learned that we need to bite the bullet and figure out how to do quantum field theory without any background metric.
A recent big step was made when people (in particular Witten and Atiyah) formulated the notion of a "topological quantum field theory." This is a precise list of properties one would like a quantum field theory independent of any background metric to satisfy. A wish list, as it were. One of the best-understood examples of such a "TQFT" is Chern-Simons theory. This is a quantum field theory that makes sense in 3-dimensional spacetime, not 4d spacetime, so in a sense it has no shot at being "true." However, it connects up to honest 4d physics in some very interesting ways, it serves as warmup for more serious physics yet to come, AND it has done wonders for the study of topology.
It is also worth noting that one particular case of Chern-Simons theory is equivalent to quantum gravity in 3d spacetime. Here I am being a bit sloppy; there are various ways of doing quantum gravity in 3 dimensions and they are not all equivalent, but the approach that relates to Chern-Simons theory is, in my opinion, the nicest. This approach to 3d quantum gravity was the advantage that it can also be described using a "triangulation" of spacetime. In other words, if we prefer the discrete to the continuum, we can "triangulate" it, or cut it up into tetrahedra, and formulate the theory solely in terms of this triangulation. Of course, it's pretty common in numerical simulations to approximate spacetime by a lattice or grid like this. What's amazing here is that one gets exact answers that are independent of the triangulation one picks. The idea for doing this goes back to Ponzano and Regge, but it was all done quite rigorously for 3d quantum gravity by Turaev and Viro just a few years ago. In particular, they were able to show the 3d quantum gravity is a TQFT using only triangulations, no "continuum" stuff.
It is tempting to try to do something like this for 4 dimensions. But it is unlikely to be so simple. A number of people have recently tried to construct 4d TQFTs copying tricks that worked in 3d. Some papers along these lines that I have mentioned before are:
A categorical construction of 4d topological quantum field theories, by Louis Crane and David Yetter, preprint available as hep-th/9301062 in latex. (Week 2)
Surgical invariants of four-manifolds, by Boguslaw Broda, preprint available as hep-th/9302092. (Weeks 9 and 10)
(I have listed which "Week" I discussed these in case anyone wants to go back and check out some of the details. See below for how to find old "Weeks".)
These papers ran into stiff opposition as soon as they came out! First Ocneanu claimed that the Crane-Yetter construction was trivial, in the sense that the number it associated to any compact 4-dimensional spacetime manifold was 1. (This number is called the partition function of the quantum field theory, and having it be 1 for all spacetimes means the theory is deadly dull.)
A note on simplicial dimension shifting, Adrian Ocneanu, preprint, available in AMSLaTeX as hep-th/9302028. (Week 5)
Crane and Yetter wrote a rebuttal noting that Ocneanu was not dealing with quite the same theory:
We are not stuck with gluing, by David Yetter and Louis Crane, preprint available as hep-th/9302118 in latex form, 2 pages. (Week 7)
They also presented, at their conference this spring, calculations showing that their partition function was not equal to 1 for certain examples.
In my discussions of Broda's work I extensively quoted some correspondence with Dan Ruberman, who showed that in Broda's original construction, the partition function of a 4-dimensional manifold was just a function of its signature and possibly some Betti numbers - these being well-known invariants, it's not especially exciting from the point of view of topology. This was also shown by Justin Roberts:
Skein theory and Turaev-Viro invariants, by Justin Roberts, Pembroke College preprint, April 14, 1993. (Roberts is at J.D.Roberts@pmms.cam.ac.uk) (Week 14).
He suggested that the Crane-Yetter partition function was also a function of the signature and Betti numbers, but did not check their precise normalization conventions, and so did not quite prove this. However, more recently Crane and Yetter, together with Kauffman, have shown this themselves:
1) Evaluating the Crane-Yetter Invariant, Louis Crane, Louis H. Kauffman, David N. Yetter, 4 pages, AMSTeX, preprint available as hep-th/9309063.
Abstract: We provide an explicit formula for the invariant of 4-manifolds introduced by Crane and Yetter (in hep-th 9301062). A consequence of our result is the existence of a combinatorial formula for the signature of a 4-manifold in terms of local data from a triangulation. Potential physical applications of our result exist in light of the fact that the Crane-Yetter invariant is a rigorous version of ideas of Ooguri on B wedge F theory.
They also have shown that Broda's original construction, and also a souped-up construction of his, give a partition function that depends only on the signature:
2) On the Classicality of Broda's SU(2) Invariants of 4-Manifolds, Louis Crane, Louis H. Kauffman, David N. Yetter, 4 pages LaTeX version 2.09, preprint available as hep-th/9309102.
Abstract: Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane-Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1- and 2- handles are treated differently. We use a refinement of Roberts' techniques developed by the authors in hep-th/9309063 to show that the "improved" surgical invariant of Broda also depends only on the signature and Euler character.
Now let me say just a little bit about what this episode might mean for physics as well as mathematics. The key is the "B wedge F" theory alluded to above. This is a quantum field theory that makes sense in 4 dimensions. I have found that the nicest place to read about it is:
3) Exactly soluble diffeomorphism-invariant theories, Gary Horowitz, Comm. Math. Phys., 125 (1989) 417-437.
This theory is a kind of simplified version of 4d quantum gravity that is a lot closer in character to Chern-Simons theory. Like Chern-Simons theory, there are no "local degrees of freedom" - every solution looks pretty much like every other one as long as we don't take a big tour of space and notice that funny things happen when we go around a noncontractible loop, which is the sort of thing that can only exist if space has a nontrivial topology. 4d quantum gravity, on the other hand, should have loads of local degrees of freedom - the local curving of spacetime!
What Crane and Yetter were dreaming of doing was constructing 4d quantum gravity as a TQFT using triangulations of spacetime. What they really did, it turns out, was to construct B wedge F theory as a TQFT using triangulations. (Broda constructed it another way.) On the one hand, the simplicity of B wedge F theory compared to honest-to-goodness 4d quantum gravity makes it possible to understand it a lot better, and calculate it out explicitly. On the other hand, B wedge F theory is so simple that it doesn't tell us much new about topology, at least not the topology of 4-dimensional manifolds per se. Via Donaldson theory and the work of Kronheimer and Mrowka it's probably telling us a lot about the topology of 2-dimensional surfaces embedded in 4-dimensional manifolds - but alas, I don't understand this stuff very well yet!
Getting our hands on 4d quantum gravity as a TQFT along these lines is still, therefore, an unfinished business. But we are, at last, able to study some examples of 4d TQFTs and ponder more deeply what it means to do quantum field theory without any background metric. The real thing missing is local degrees of freedom. Without them, any model is really just a "toy model" not much like physics as we know it. The loop representation of quantum gravity has these local degrees of freedom (to the extent that we understand the loop representation!), and so the challenge (well, one challenge!) is to better relate it to what we know about TQFTs.
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
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