
Things are moving very fast in the quantum gravity/4d topology game, so I feel I should break my vow not to continue this series until after next weekend's conference on Knots and Quantum Gravity.
Maybe I should recall where things were when I left off. The physics problem motivating a lot of work in theoretical physics today is reconciling general relativity and quantum theory. The key feature of general relativity is that time and space do not appear as a "background structure," but rather are dynamical variables. In mathematical terms, this just means that there is not a fixed metric; instead gravity is the metric, and the metric evolves with time like any other physical field, satisfying some field equations called the Einstein equations.
But it is worth stepping back from the mathematics and trying to put into simple words why this makes general relativity so special. Of course, it's very hard to put this sort of thing into words. But roughly, we can say this: in Newtonian mechanics, there is a universal notion of time, the "t" coordinate that appears in all the equations of physics, and we assume that anyone with a decent watch will be able to keep in synch with everyone else, so there is no confusion about what this "t" is (apart from choosing when to call t = 0, which is a small sort of arbitrariness one has to live with). In special relativity this is no longer true; watches moving relative to each other will no longer stay in synch, so we need to pick an "inertial frame," a notion of rest, in order to have a "t" coordinate to play with. Once we pick this inertial frame, we can write the laws of physics as equations involving "t". This is not too bad, because there is only a finiteparameter family of inertial frames, and simple recipes to translate between them, and also because nothing going on will screw up the functioning of our (idealized) clocks: that is, the "t" coordinate doesn't give a damn about the state of the universe. That's what is meant by saying a "background structure"  it's some aspect of the universe that is unaffected by everything else that's going on.
In general relativity, things get much more interesting: there is no such thing as an inertial frame that defines coordinates on spacetime, because there is no way you can get a lot of things at different places to remain at rest with each other  this is what is meant by saying that spacetime is curved. You can measure time with your watch, socalled "proper time," but this applies only near you. More interestingly still, to compare what your watch is doing to what someone else's is doing, you actually need to know a lot about the state of the universe, e.g., whether there are any heavy masses around that are curving spacetime. The "metric," whereby one measures distances and proper time, depends on the state of the universe  or more properly, it is part of the state of the universe.
Trying to do quantum theory in this context has always been too hard for people. Part of the reason why is that built into the heart of traditional quantum theory is the "Hamiltonian," which describes the evolution of the state of the system relative to a Godgiven "background" notion of "t". Anyone who has taken quantum mechanics will know that the star of the show is the Schrodinger equation:
i dPsi/dt = H Ψ
saying how the wavefunction Ψ changes with time in a way depending on the Hamiltonian H. No "t," no "H"  this is one basic problem with trying to reconcile quantum theory with general relativity.
Actually, it turns out that the analog to Schrodinger's equation for quantum gravity is the WheelerDeWitt equation. The Hamiltonian is replaced by an operator called the "Hamiltonian constraint" and we have
H Ψ = 0.
Note how this cleverly avoids mentioning "t"! The problem is, people still aren't quite sure what to do with the solutions to this equation  we're so used to working with Schrodinger's equation.
Now in 1988 Witten wrote a paper in which he coined the term "topological quantum field theory," or TQFT, for short. This was meant to capture in a rigorous way what field theories like quantum gravity should be like. Actually, Witten was working on a different theory called Donaldson theory, which also has the property of having no background structures. Shortly thereafter the mathematician Atiyah came up with a formal definition of a TQFT. To get an idea of this definition, try my notes on symmetries and (if you don't know what categories are) categories. For a serious tour of TQFTs and the like, try his book:
The Geometry and Physics of Knots, by Michael Atiyah, Cambridge U. Press, 1990.
One can think of a TQFT as a framework in which a WheelerDeWittlike equation governs the dynamics of a quantum field theory. Experts may snicker here, but it is true, if not as enlightening as other things one can say.
I won't bother to define TQFTs here, but I think Smolin put it very well when he said the idea of TQFTs really helped us break out of our traditional idea of fields as being something defined at every point of spacetime, wiggling around, and allowed us to see field theory from many new angles. For example, TQFTs let us wiggle out of the old conundrum of whether spacetime is continuous or discrete, because many TQFTs can be equivalently described in either of two ways: via a continuum model of spacetime, or via a discrete one in which spacetime is given a "simplicial structure," like a big tetrahedral tinkertoy lattice kind of thing. The latter idea appears to be due to Turaev and Viro, although certainly physicists have had similar ideas for years, going back to Ponzano and Regge, who worked on simplicial quantum gravity.
Now the odd thing is that while interesting 3d TQFTs have been found, the most notable being ChernSimons theory, nobody has quite been able to make 4d TQFTs rigorous. Witten's original work on Donaldson theory has led to many interesting things, but not yet a fullfledged TQFT in the rigorous sense of Atiyah. And quantum gravity still resists being formulated as a TQFT.
A while back I noted that Crane and Yetter had invented a 4d TQFT using the simplicial approach. There has been a lot of argument over whether this TQFT is interesting or "trivial." Of course, trivial is not a precise concept. For a while Ocneanu claimed that the partition function of every compact 4manifold equalled 1 in this TQFT, which counts as very trivial. But this appears not to be the case. Broda invented another 4d TQFT and here on "This Week's Finds" Ruberman showed it was trivial in the sense that the partition function of any compact 4manifold was a function of the "signature" of the 4manifold. This is trivial because the signature is a wellunderstood invariant and if we are trying to do something new and interesting that just isn't good enough.
In the following paper:
1) Skein theory and TuraevViro invariants, by Justin Roberts, Pembroke College preprint, April 14, 1993 (Roberts is at J.D.Roberts@pmms.cam.ac.uk)
Roberts almost claims to show that the CraneYetter invariant is trivial in the same sense, namely that the partition function of any compact 4manifold is an exponential of the signature. Now if Crane and Yetter's own computations are correct, this cannot be the case, but it could be an exponential of a linear combination of the signature and the Euler characteristic, as far as I know. The catch is that Roberts does not normalize his version of the CraneYetter invariant in the same way that Crane and Yetter do, so it is hard to compare results. But Roberts says: "The normalisations here do not agree with those in Crane and Yetter, and I have not checked the relationship. However, when dealing with the [3d TQFT] invariants, different normalisations of the initial data change the invariants by factors depending on standard topological invariants (for example Betti numbers), so there is every reason to belive that these [4d TQFT] invariants are trivial (that is, they differ from 1 only by standard invariant factors) in all normalisations."
This is a bit of a disappointment, because Crane at least had hoped that their TQFT might actually turn out to be quantum gravity. This was not idle dreaming; it was because the CraneYetter construction was a rigorous analog of some work by Ooguri on simplicial quantum gravity.
Then, about a week ago, Rovelli put a paper onto the net:
2) The basis of the PonzanoReggeTuraevViroOoguri model is the loop representation basis, 16 pages in LaTeX, Friday April 30, available as hepth/9304164.
This is a remarkable paper that I have not been able to absorb yet. First it goes over 3d quantum gravity  which has been made into a rigorous TQFT. It works with the simplicial formulation of the theory. That is, we consider our (3dimensional) spacetime as being chopped up into tetrahedra, and assign to each edge a length, which is required to be 0,1/2,1,3/2,.... This idea of quantized edgelengths goes back to 4d work of Ponzano and Regge, but recently Ooguri showed that in 3d this assumption gives the same answers as Witten's continuum approach to 3d quantum gravity. The "halfintegers" 0,1/2,1,3/2, etc. should remind physicists of spin, which is quantized in the same way, and mathematically this is exactly what is going on: we are really labelling edges with representations of the group SU(2), that is, spins. What Rovelli shows is that if one starts with the loop representation of 3d quantum gravity (yet another approach), one can prove it equivalent to Ooguri's approach, and what's more, using the loop representation one can calculate the lengths of edges of triangles in a given state of space (space here is a 2dimensional triangulated surface) and show that lengths are quantized in units of the Planck length over 2. (Here the Planck length L is the fundamental length scale in quantum gravity, about 1.6 times 10^{33} meters.)
And, most tantalizing of all, he sketches a generalization of the above to 4d. In 4d it is known that in the loop representation of quantum gravity it is areas of surfaces that are quantized in units of L^2/2, rather than lengths. Rovelli considers an approach where one chops 4dimensional spacetime up into simplices and assigns to each 2dimensional face a halfinteger area. He uses this to write down a formula for the inner product in the Hilbert space of quantum gravity  thus, at least formally, answering the longstanding "inner product problem" in quantum gravity. The problem is that, unlike in 3d quantum gravity, here one must sum over ways of dividing spacetime into simplices, so the formula for the inner product involves a sum that does not obviously converge. This is however sort of what one might expect, since in 4d quantum gravity, unlike 3d, there are "local excitations"  local wigglings of the metric, if you will  and this makes the Hilbert space be infinitedimensional, whereas the 3d TQFTs have finitedimensional Hilbert spaces.
I think I'll quote him here. It's a bit technical in patches, but worth it...
We conclude with a consideration on the formal structure of 4d quantum gravity, which is important to understand the above construction. Standard quantum field theories, as QED and QCD, as well as their generalizations like quantum field theories on curved spaces and perturbative string theory, are defined on metric spaces. Witten's introduction of the topological quantum field theories has shown that one can construct quantum field theories defined on a manifold which has only its differential structure, and no fixed metric structure. The theories introduced by Witten and axiomatized by Atiyah have the following peculiar feature: they have a finite number of degrees of freedom, or, equivalently, their quantum mechanical Hilbert spaces are finite dimensional; classically this follows from the fact that the number of fields is equal to the number of gauge transformations. However, not any diffinvariant field theory on a manifold has a finite number of degrees of freedom. Witten's gravity in 3d is given by the actionS[A,E] = integral(F ^ E)
which has finite number of degrees of freedom. Consider the action
S[A,E] = integral(F ^ e ^ e)
in 3+1 dimensions, for a (self dual) SO(3,1) connection A and a (real) one form e with values in the vector representation of SO(3,1). This theory has a strong resemblance with its 2+1 dimensional analog: it is still defined on a differential manifold without any fixed metric structure, and is diffeomorphism invariant. We expect that a consistent quantization of such a theory should be found along lines which are more similar to the quantization of the integral(F ^ E), theory than to the quantization of theories on flat space, based on the Wightman axioms namely on npoints functions and related objects. Still, the theory integral(F ^ e ^ e) has genuine field degrees of freedom: its physical phase space is infinite dimensional, and we expect that its Hilbert state space will also be infinite dimensional. There is a popular belief that a theory defined on a differential manifold without metric and diffeomorphism invariant has necessarily a finite number of degrees of freedom ("because thanks to general covariance we can gauge away any local excitation"). This belief is of course wrong. A theory as the one defined by the action integral(F ^ e ^ e) is a theory that shares many features with the topological theories, in particular, no quantity defined "in a specific point" is gauge invariant; but at the same time it has genuinely infinite degrees of freedom. Indeed, this theory is of course nothing but (Ashtekar's form of) standard general relativity.
The fact that "local" quantities like the npoint functions are not appropriate to describe quantum gravity nonperturbatively has been repeatedly noted in the literature. As a consequence, the issue of what are the quantities in terms of which a quantum theory of gravity can be constructed is a much debated issue. The above discussion indicates a way to face the problem: The topological quantum field theories studied by Witten and Atiyah provide a framework in terms of which quantum gravity itself may be framed, in spite of the infinite degrees of freedom. In particular, Atiyah's axiomatization of the topological field theories provides us with a clean way of formulating the problem. Of course, we have to relax the requirement that the theory has a finite number of degrees of freedom. These considerations leads us to propose that the correct general axiomatic scheme for a physical quantum theory of gravity is simply Atiyah's set of axioms up to finite dimensionality of the Hilbert state space. We denote a structure that satisfies all Atiyah's axioms, except the finite dimensionality of the state space, as a generalized topological theory.
The theory we have sketched is an example of such a generalized topological theory. We associate to the connected components of the boundary of M the infinite dimensional state space of the Loop Representation of quantum gravity. Eq. 5 [the magic formula I alluded to  jb], then, provides a map, in Atiyah's sense, between the state spaces constructed on two of these boundary components. Equivalently, it provides the definition of the Hilbert product in the state space.
One could argue that the framework we have described cannot be consistent, because it cannot allow us to recover the "broken phase of gravity" in which we have a nondegenerate background metric: in the proposed framework one has only nonlocal observables on the boundaries, while in the broken phase a local field in M has nonvanishing vacuum expectation value. We think that this argument is weak because it disregards the diffeomorphism invariance of the theory: in classical general relativity no experiment can distinguish a Minkowskian spacetime metric from a nonMinkowkian flat metric. The two are physically equivalent, as two gaugerelated Maxwell potentials. For the same reason, no experiment could detect the absolute position of, say, a gravitational wave, (while of course the position of an e.m. wave is observable in Maxwell theory). Physical locality in general relativity is only defined as coincidence of some physical variable with some other physical variable, while in non general relativistic physics locality is defined with respect to a fixed metric structure. In classical general relativity, there is no gaugeinvariant obervable which is local in the coordinates. Thus, any observation can be described by means of the value of the fields on arbitrary boundaries of spacetime. This is the correct consequence of the fact that "thanks to general covariance we can gauge away any local excitation", and this is the reason for which one can have the ADM "frozen time" formalism. The spacetime manifold of general relativity is, in a sense, a much weaker physical object than the spacetime metric manifold of ordinary theories. All the general relativistic physics can be read from the boundaries of this manifold. At the same time, however, these boundaries still carry an infinite dimensional number of degrees of freedom.
Next, let me take the liberty of describing some work of my own:
3) Diffeomorphisminvariant generalized measures on the space of connections modulo gauge transformations, by John Baez, to appear in the proceedings of the Conference on Quantum Topology, Manhattan, Kansas, May 8, 1993, available as state.tex.
This is an extremely interesting paper by a very good mathematician. Whoops! Let's be objective here. In the loop representation of quantum gravity, states of quantum gravity are given naively by certain "measures" on a space A/G of connections modulo gauge transformations. The ChernSimons path integral is also such a "measure". In both cases, the "measure" in question is invariant under diffeomorphisms of space. And in both cases, the loop transform allows one to think of these measures as instead being functions of multiloops (collections of loops in space). This is the origin of the relationship to knot theory.
The problem, as always in quantum field theory, is that the notion of "measure" must be taken with a big grain of salt  it's not the sort of measure they taught you about in real analysis. Instead, these measures are a kind of "generalized measure" that allows you to integrate not all continuous functions on A/G but only those lying in an algebra called the "holonomy algebra," defined by Ashtekar, Isham and Lewandowski. To be precise and technical, this is the closure in the L^infty norm of the algebra of functions on A/G generated by "Wilson loops," or traced holonomies around loops. So what we are really interested in is not diffeomorphisminvariant measures on A/G, but diffeomorphism invariant elements of the dual of the holonomy algebra. I begin with a review of generalized measures, introduce the holonomy algebra, and then do a bunch of new work in which I show how to rigorously construct lots of diffeomorphisminvariant elements of the dual of the holonomy algebra by doing lattice gauge theory on graphs embedded in space. Again, as with the work discussed above, we see that the discrete and continuum approaches to space go handinhand! And we see that there are some interesting connections between singularity theory and group representation theory showing up when we try to understand "measures" on the space A/G.
The following is a part of a paper discussed in "week5", now available from grqc:
4) Completeness of Wilson loop functionals on the moduli space of SL(2,C) and SU(1,1)connections, Abhay Ashtekar and Jerzy Lewandowski, Plain TeX, 7 pages, available as grqc/9304044.
I didn't discuss this aspect of the paper, so let me quote the abstract:
The structure of the moduli spaces M := A/G of (all, not just flat) SL(2,C) and SU(1,1) connections on a nmanifold is analysed. For any topology on the corresponding spaces A of all connections which satisfies the weak requirement of compatibility with the affine structure of A, the moduli space M is shown to be nonHausdorff. It is then shown that the Wilson loop functionals  i.e., the traces of holonomies of connections around closed loops  are complete in the sense that they suffice to separate all separable points of M. The methods are general enough to allow the underlying nmanifold to be topologically nontrivial and for connections to be defined on nontrivial bundles. The results have implications for canonical quantum general relativity in 4 and 3 dimensions.
By the way, someone should extend this result to more general noncompact semisimple Lie groups, and also show that for all compact semisimple Lie groups the Wilson loop functionals in any faithful representation do separate points (this is known for the fundamental representation of SU(n)). If I had a bunch of grad students I would get one to do so.
The following was discussed in an earlier edition of this series, "week11," but is now available from grqc:
5) An algebraic approach to the quantization of constrained systems: finite dimensional examples, by Ranjeet S. Tate, (Ph.D. Dissertation, Syracuse University), 124 pages, LaTeX (run thrice before printing), available as grqc/9304043.
I haven't read the following one but it seems like an interesting application of loop variables to more downtoearth physics; Gambini was one of the originators of the loop representation, and intended it for use in QCD:
6) SU(2) QCD in the path representation, by Rodolfo Gambini and Leonardo Setaro, LaTeX 37 pages (7 figures included), available as heplat/9305001. ("heplat" is the computational and lattice physics preprint list, at heplat@ftp.scri.fsu.edu.)
Let me quote the abstract:
We introduce a pathdependent hamiltonian representation (the path representation) for SU(2) with fermions in 3 + 1 dimensions. The gaugeinvariant operators and hamiltonian are realized in a Hilbert space of open path and loop functionals. We obtain a new type of relation, analogous to the Mandelstam identity of second kind, that connects open path operators with loop operators. Also, we describe the cluster approximation that permits to accomplish explicit calculations of the vacuum energy density and the mass gap.
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
