
A nice crop of papers has built up while I've been taking a break... In "week15" I talked a bit about constructing topological quantum field theories starting with a triangulation of spacetime, and how this seems to sneak around the old "is spacetime continuous or discrete" argument. Let me describe a bit about one of the more mathematically elegant physics papers I've run across in a while, which treats exactly this issue. Then I'll describe two review articles, one on gravity in 2+1 dimensions (which is closely related to the lattice business), and one on Lagrangians for quantum gravity.
1) Structure of Topological Lattice Field Theories in Three Dimensions, by Stephenwei Chung, Masafumi Fukuma and Alfred Shapere, preprint, available as hepth/9305080 (make sure to get the pictures if possible)!
What's a 2dimensional "topological lattice field theory"? According to the definition used in this paper, it goes like this. First take a compact oriented 2manifold without boundary M, that is, an nholed torus. (One could also discuss the case when there is a boundary, but to keep life simple we won't here.) We want to calculate a number Z(M), the partition function of M, since the partition function is a basic ingredient in Feynman's approach to quantum field theory. We first triangulate M... so a patch might look like:
____ /\ / / \ / /____\/ \ / \ / \/
Then "disassemble" M into separate triangles, like this:
____ /\ \ / / \ \ / /____\ \/ ____ \ / \ / \/
Now assign to each edge of the disassembled version of M a "color" taken from a fixed finite set S. Note that there are twice as many edges in the disassembled version of M as in the original triangulation of M. Any way of assigning a color to each edge of the disassembled M will be called a "coloring". We think of a coloring as a "history of the world" and we will compute Z(M) by summing a certain quantity over all colorings.
To compute this quantity, we need two pieces of data that determine our theory. First, for each i,j in S, we fix a complex number g^{ij}. We require that the matrix g^{ij} be invertible. We define g_{ij} to be the matrix inverse of g^{ij}. We can raise and lower indices with g as if it were a metric. The matrix g will be used when we glue two edges of the disassembled M together in the process of rebuilding M. Second, for each i,j,k in S, we fix a number c_{ijk}. This number comes in because each triangle has three edges.
Here's how we calculate Z(M). Write down one index next to each edge of the dissasembled M  by "index" I mean something like i,j,k running over S. Then write down the obvious factor of g for each pair of edges that get glued together when we form M, and write down the obvious factor of c for each triangle in M. Finally, sum over all colorings to get Z(M).
For example, if M were a torus that we triangulated with two triangles like this
/\ / \ /____\ \ / \ / \/
 with opposite edges of the parallelogram identified  we would dissasemble M and label the edges like this, say:
/\ l m /__j_\ _i__ \ / n k \/
To form M we glue i to j, k to l, and m to n. So we write down
g^{ij}g^{kl}g^{mn}c_{jml}c_{ink}
and then sum over i,j,k,l,m,n to get Z(M). Notice that for this procedure to be well defined it had better not matter whether we write g^{ij} or g^{ji}, since we have no way of knowing which to use. So g had better be symmetric. Similarly, we had better have c_{ijk} = c_{jki}  invariance under cyclic permutations. Note that since M is oriented we can (and will) require that we go around each triangle counterclockwise when writing down things like c_{ink}, as we have done above.
Okay, this is a pretty scheme, but the real point is that it should be independent of the triangulation of M we chose, for us to have something that deserves to be called "topological." This imposes extra conditions on g and c. Here it is handy to know that we can get between any two triangulations of M using a sequence of two moves and their inverses. The first move is called the "(2,2) move." It looks like this, where I will draw the vertices as O's:
O O /\ / \ /  \ / \ /  \ / \ O  O <> OO \  / \ / \  / \ / \/ \ / O O
It is called the (2,2) move since it really amounts to taking 2 faces of a tetrahedron and replacing them with the other 2 faces! There is a similar (3,1) move that takes 3 faces of a tetrahedron and replaces them with the other 1, as follows:
O O /\ / \ /  \ / \ /  \ / \ /  \ / \ / _O_ \ <> / \ / _/ \_ \ / \ / _/ \_ \ / \ /_/ \_\ / \ OO OO
(This drawing done by my friend Bruce Smith in a fit of insomnia!) These are examples of the "Pachner moves," and the same idea works in any dimension. But in 2 dimensions we can use a move called the "bubble move" instead of the (3,1) move. Here is where drawing vertices as O's is crucial:
O O /\  /  \  /  \   O  <>  \  /  \  /  \/  O O
In the left, we have two hideously deformed triangles (remember, this is topology!) that are attached along TWO edges, leaving two edges exposed, and in the right we have collapsed them down to a single edge. We leave it as a fun exercise to show that you can do anything with the (2,2) move and the bubble move that you can do with the (2,2) move and the (3,1) move.
Requiring that Z(M) be invariant under the (2,2) moves amounts to the following equation  if you check it, you will make sure you understand what's going on:
c_{xy}^u c_{uz}^w = c_{xu}^w c_{yz}^u.
Here I have raised indices using the "metric" g. This equation looks sort of hairy, but it's actually something very nice in disguise. We need to tease out its inner essence! Suppose we take a vector space A having the colors in S as a basis, and use the tensor c_{ij}^k to define a bilinear map from A x A to A. Then the equation above says this map is an associative product! If you ponder the picture of the (2,2) move for a while, this should become obvious to you. Think of each triangle as being a gadget that you can feed vectors into from two sides and have the "product" pop out on the third side. Then the equation
O O /\ / \ /  \ / \ /  \ / \ O  O = OO \  / \ / \  / \ / \/ \ / O O
really is just associativity! To understand this in a deeper way, read Kapranov and Voevodsky's paper (reviewed in "week4"), especially the section on the "associahedron".
Requiring that Z(M) be invariant under the bubble move amounts to the following:
c_{xu}^v c_{yv}^u = g_{xy}
Here g_{xy} is the matrix inverse of g^{xy}. Again, I leave it as an exercise to show this is the right equation. It is a formula expressing the "metric" g on A in terms of the product on A! In fact, it has a beautiful algebraic interpretation: it says that the algebra A is "semisimple." A semisimple algebra is just a direct sum of matrix algebras, and in such algebras the inner product g(a,b) of any two elements is just equal to tr(a^T b), where a^T is the transpose of a.
So we discover a charming fact: there is a onetoone correspondence between topological lattice field theories in 2 dimensions and finitedimensional semsimple algebras over the complex numbers!
Actually this was apparently already shown by
2) C. Bachas and P. M. S. Petropoulos, Comm. Math. Phys. 152 (1993) 191.
and
3) Lattice topological field theory in twodimensions, by M. Fukuma, S. Hosono and H. Kawai, preprint available as hepth/921254.
The big result of the present paper is to generalize this to 3 dimensions. The authors consider a specific definition of 3d topological lattice field theories in which one chops a 3d manifold up into tetrahedra and assigns colors to edges. They claim to get a onetoone correspondence between these and finitedimensional Hopf algebras for which the antipode squared is the identity! If you don't know what a Hopf algebra is, let me simply say it is a very beautiful sort of thing that has both a product and a "coproduct," and they come up all the time in group theory, knot theory, and the study of quantum groups. So we are seeing that there is a profound correspondence between topology and algebra, with higherdimensional topology giving more subtle algebra.
(In fact, I am a little worried that the authors haven't stated the theorem quite precisely enough to have it be quite true, but it's basically right  I am afraid only only gets a particular class of Hopf algebras, those which are semisimple and cosemisimple. I may be missing something.)
Let me conclude with a few very exciting open problems.
A. One could instead consider theories in which colors are only assigned to faces. This turns out not to broaden the class of examples: any Hopf algebra has a dual Hopf algebra, and one just gets the theory associated to the dual Hopf algebra this way! But if one considers theories in which colors are assigned BOTH to edges AND faces one apparently gets a larger class of 3d examples. What algebraic structure do these correspond to?
B. The TuraevViro theory of quantum gravity  described below  is a 3d topological lattice field theory of some sort. Where does it fit into this picture? The authors ask this question but don't answer it. Also, a more difficult problem  where does ChernSimons gauge theory fit into this picture?
C. The 64,000 dollar question: how does all this generalize to 4 dimensions? What sort of algebraic structure corresponds to a 4d topological lattice field theory? It is becoming increasingly clear that 4d field theories will involve some kind of "higher algebra" that we are only beginning to understand.
4) Six ways to quantize (2+1)dimensional gravity, by Steven Carlip (carlip@nsfitp.itp.ucsb.edu), 21 pages, preprint available in LaTeX form as grqc/9305020.
While we have no real way to quantize gravity in 3+1 dimensions  although lots of good ideas  we have six, count 'em, six, ways to do it in 2+1 dimensions! Sometimes this sort of thing makes one yearn to be a physicist in some other, lowerdimensional universe. However, lest one make such wish prematurely to a genie passing by, one should note that life in 2+1 dimensions is boring compared to our 3+1dimensional world. The reason can be seen from the following count of the number of independent components of the Riemann tensor R_{ijkl} which vanishes when spacetime is flat, and the Einstein tensor, which vanishes when the vacuum Einstein equations hold:
dimension Riemann Einstein 1 0 0 2 1 1 3 6 6 4 20 10
What this means is that, until one gets up to dimension 4, the vacuum Einstein equations imply that spacetime is flat. That means that there are no gravitational waves in empty space; there are only global, topological effects. Typically this means that if space is compact there are only finitely many degrees of freedom. This means that 2+1 quantum gravity is really quantum mechanics, not fullfledged quantum field theory (which deals with local excitations  wiggles in the metric and such  and infinitely many degrees of freedom). The good news is, this means that 2+1 gravity is somewhat understandable  no nasty infinities or illdefined integrals needed, etc.. The bad news is, it means 2+1 gravity is not too much like 3+1 gravity. But still, many of the most puzzling qualitative features of quantum gravity are present in the 2+1 case. In particular, one has a testing ground in which to look at the interlocking triad of problems that stump us in the 3+1 case: the problem of time, the problem of observables, and the inner product problem. In brief these are: what is time evolution in quantum gravity, what are the observables in quantum gravity, and what is the inner product on the space of states of quantum gravity? As you can see, we are overwhelmingly ignorant about quantum gravity! I think that work on 2+1 gravity has given us some interesting clues about these problems.
Carlip describes 6 approaches to 2+1 gravity. I'll list them and comment on them briefly below. But one point to make is that these approaches have not all been shown to be equivalent; on the contrary, they seem to give different answers. Part of the problem in my opinion is that we do not have enough criteria for a "good" theory of 2+1 quantum gravity. Certainly one would like to see that in the h → 0 limit the theory reduces to classical gravity in some sense or other (but this is a bit vague). Perhaps another thing one could hope for is that the theory be a 2+1dimensional TQFT. I am not sure which of the approaches below give a TQFT (although #6 definitely does and probably so does #2):
The "ADM" or ArnowittDeserMisner formalism amounts to what people would typically call canonical quantization: one writes down a description of the phase space of quantum gravity in terms of initial data, figures out the Poisson brackets of functions on this phase space, and then tries to quantize by turning them into commutators. In gravity the roles of "position" and "momentum" variables are played by the metric on space at a given time, and the extrinsic curvature (or more precisely, something cooked up from it).
This is essentially the 2+1 analog of Ashtekar's approach in 3+1 dimensions, in that a connection and triad field play the main role, rather than the metric. However, in 2+1 dimensions we can lump the triad field and the connection together to get an "ISO(2,1) connection"  where ISO(2,1) is mildly terrifying notation for the Poincare group in 2+1 dimensions (or "inhomogeneous Lorentz group," hence the "I"). The action for the theory then becomes the ChernSimons action, as noted by Witten.
This might sound oxymoronic to some, but what it means is that the phase space of solutions is describe in a manifestly covariant way, rather than in terms of initial data, and then one tries to turn Poisson brackets into commutators.
The loop representation of quantum gravity starts with the connection representation and then takes traces of holonomies around loops  socalled Wilson loops  as the basic variables to quantize. This suffers irritating technical problems in 2+1 dimensions, as noted in the following recent paper:
5) An illustration of 2+1 gravity loop transform troubles, by Donald Marolf (MAROLF@SUHEP.PHY.SYR.EDU), 6 pages, preprint available in LaTeX form as grqc/9305015.
I know that Ashtekar and Loll are attacking these problems this right now; Loll discussed this a bit in a lecture she gave in my seminar.
Here we proceed as in approach #1 but attempt to impose the Hamiltonian and diffeomorphism constraints after quantizing. That is, we start with an overly large phase space of initial data for general relativity  overly large because a given solution of Einstein's equations will have many different initial data on different spacelike slices  quantize by turning Poisson brackets into commutators, and THEN try to take care of the mistake we made by defining the "physical" states to be those annihilated by certain operators, the Hamiltonian and diffeomorphism constraints. I gave a brief intro to this in "week11". This is the most traditional approach in 3+1 gravity.
These are closely related to the topological lattice field theories described above. Here we treat spacetime as discrete, that is, as a kind of lattice. One approach here is due to Regge and Ponzano, and recently worked out rigorously by Turaev and Viro. To get going in this theory, you "triangulate" your 3dimensional spacetime, that is, chop it into tetrahedra. All we need to work with is this "simplicial complex" consisting of tetrahedra, their triangular faces, their linesegment edges, and the vertex points. We assume for simplicity that spacetime is compact, so we can use finitely many tetrahedra. Thus everything in sight is finite and discrete. A "history of the world" in this theory amounts to labelling each edge with a length, or "spin", that must be 0,1/2,1,3/2,... or j/2. There are thus finitely many possible histories. To do calculations in this theory, we follow Feynman's procedure and "sum over histories"  write down a formula for the quantity we are interested in, and add up its value for all histories, weighted by a quantity depending on the history, the exponential of the action of that history, to obtain the vacuum expectation value of the quantity. The formula for the action is very familiar to folks knowledgeable about quantum theory. Each tetrahedron has 6 edges labelled by spins, and we calculate a quantity called the "6j symbol" from these spins and then add it up for all tetrahedra. In the TuraevViro version, we have replaced the gauge group SU(2) by the corresponding quantum group, with the quantum parameter q a root of unity, so there are only finitely many irreducible representations, or spins, to sum over. (See "week5" for the vaguest of introductions to quantum groups and their representations!) The beauty of this theory is that the answer one gets is independent of the triangulation one has chosen.
While I'm at it, let me list some key references to the subject of lattice 2+1 gravity, a subject I'm fascinated by these days.
The grandaddy of the all, the PonzanoRegge paper, is:
6) G. Ponzano and T. Regge: in Bloch, F. (ed.), Spectroscopic and Group Theoretical Methods in Physics, Amsterdam: NorthHolland 1968.
Then there are:
7) E. Witten, Nucl. Phys. B311 (1988), 46.
8) State sum invariants of 3manifolds and quantum 6jsymbols, by V. G. Turaev and O. Y. Viro, Topology 31 (1992), 865.
Also Ooguri wrote a paper on 3+1 lattice gravity that has been quite influential:
9) H. Ooguri, Mod. Phys. Lett. A7 (1992), 2799.
And there is also the recent paper by Rovelli, which I discussed in "week14". This is very readable (once you know what's going on!) and conceptual.
10) Actions for Gravity, with Generalizations: A Review, by Peter Peldan (tfepp@fy.chalmers.se), 61 pages, preprint available as grqc/9305011
The classic action principle for general relativity is the EinsteinHilbert action: the Ricci scalar times the volume form associated to the metric. An important modification, often called the Palatini action, takes a connection and tetrad (aka vierbein or frame field) as basic. More recently, Plebanski invented an action using the selfdual part of the connection and a tetrad field; this turns out to be closely related to an action naturally associated with the Ashtekar "new variables" (a selfdual connection and tetrad field), although this was realized only subsequently by Capovilla, Dell, and Jacobson. More recently still, there is the CapovillaDellJacobson action. These new action principles shed a very interesting new light on gravity, particularly when it comes to quantizing it. Of course it must be remembered that actions that give the same classical dynamics can (and typically DO) give different quantum theories. So a traditionalist might question whether these new actions give the "right" quantum theory of gravity. Of course, the correct response to such a traditionalist is "well, you come up with the `right' quantum theory of gravity and then we can compare!" The point is that the good old EinsteinHilbert action is extremely intractable when it comes to quantization  so perhaps it is not the "right" one, and any quantization is more enlightening than none at this stage.
Peldan presents a grand tour of the various Lagrangian formulations of gravity, and on page 3 of this large manuscript there is a large diagram of the main Lagrangian and Hamiltonian approaches to gravity in 3+1 dimensions, while on page 35 there is a somewhat smaller chart for 2+1 gravity. (A very brief preliminary warmup on some of these formulations appears in my earlier article, "week7".) I plan on going through this carefully in order to be able to make up for years of neglect on my part of this sort of thing.
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
