November 5, 1994

This Week's Finds in Mathematical Physics (Week 43)

John Baez

It is very exciting, yet somewhat scary, as work continues on the loop representation of quantum gravity. On the one hand, researches are busy making it mathematically rigorous; on the other hand, they are beginning to understand its physical significance. The reasons for excitement are obvious, but the scary part is that until the final touches are put on the mathematical rigor, we don't know if the theory really exists!

Of course there is the whole separate issue of whether the theory will find experimental confirmation. If the theory were experimentally confirmed, questions of mathematical rigor wouldn't be quite such a big deal. But experimental verification will probably take a long time! Also, we don't really expect a theory of "pure gravity" to be experimentally confirmed. One will need to figure out how all the other particles and fields fit in --- except perhaps for very general, qualitative issues. (See the paper mentioned at the very end of this article for some of those.) So here the suspense is of a long-term sort. Luckily, the question of whether the theory makes mathematical sense is already very interesting, since so many theories of quantum gravity have already been shot down on that basis, and the loop representation approach seems so pretty. Either it will make sense, or we will run into some obstacle, which is bound to be enlightening.

Let me briefly review the loop representation, without too many technical details. For more details try the original paper by Rovelli and Smolin (see "week42" for a reference), the book by Ashtekar (see "week7"), or, especially if you're a mathematician, my review article "Knots and quantum gravity: progress and prospects".

There are 3 basic steps in the "canonical quantization" of general relativity. At each step there is a vector space of quantum states, but only in the last do we really need a Hilbert space of states, since only when we're done do we want to be able to compute expectation values of observables, which takes an inner product.

In what follows I'll talk about the simplest situation, where we have the vacuum Einstein equations

                          G = 0

where G is the "Einstein tensor" cooked up from the curvature of spacetime. Say spacetime is of the form R x S, where R is the real numbers (time) and S is a 3-dimensional manifold (space). We will think of S as the "t = 0 slice" of R x S.

I) The first stage is to get the space of "kinematical states". In the quantum mechanics of a point particle on the line, the space of wavefunctions is a space of functions on the real line. Similarly, in quantum gravity we naively expect kinematical states to be functions on the space of Riemannian metrics on the 3-dimensional manifold S we're taking to be "space". In the loop representation one does something a bit more clever, but let's move on and then come back to that.

II) The second stage is getting the space of "diffeomorphism-invariant states". In fact, Einstein's equations in coordinates look like

                          G_{μ ν} = 0

where the indices μ, ν range from 0 to 3. It's customary to work in coordinates x_{μ} where x_0 is "time" and the other three coordinates are the "space" coordinates on S. Then classically, the equations G_{0 μ} = 0 serve as constraints on the initial data for Einstein's equations, while the remaining equations describe time evolution. I.e., only for certain choices of a metric and its first time derivative at t = 0 can we get a solution of Einstein's equations. In fact, G_{0 μ} can be calculated knowing only the metric and its first time derivative at t = 0, and the equations saying they are zero are the constraints that this data must satisfy to get a solution of Einstein's equations.

Following the usual recipes of quantum theory, we want to turn these constaints into operators on the kinematical Hilbert space of stage I, and then demand that the states relevant for physics be annihilated by these operators. The "diffeomorphism-invariant subspace" is the subspace of the kinematical state space that is annihilated by the constraints corresponding to G_{0i} where i = 1, 2, 3. Let us put off for a moment why it's called what it is!

III) The third and final stage is getting the space of "physical states". Here we look at the subspace of diffeomorphism-invariant states that are also annihilated by the constaint corresponding to G_{00}. The equation saying that a diffeomorphism-invariant state is annihilated by this constraint is called the "Wheeler-DeWitt equation", and this is generally regarded as the fundamental equation of quantum gravity.

Now, it should make some sense why we call the "physical states" what we do. These are quantum states satisfying the quantum analogues of the constraints that the classical initial data must satisfy to be initial data for a solution of Einstein's equations. But why do we impose the constraints G_{μ ν} = 0 in two separate stages, and call the states in part II "diffeomorphism-invariant states"?

This is a very important question which gives quantum gravity much of its curious character. In classical general relativity, G_{0i} not only gives one of Einstein's equations, namely G_{0i} = 0, it also "generates diffeomorphisms" of the 3-dimensional manifold S representing space. If you don't quite know what this means, let me simply say that in classical mechanics, observables give rise to one-parameter families of symmetries. For example, momentum gives rise to spatial translations, while energy (aka the Hamiltonian) gives rise to time translations. We say that the observable "generates" the one-parameter family of symmetries. This is (roughly) what I mean by saying that G_{0i} generates diffeomorphisms of S. Similarly, G_{00} generates diffeomorphisms of the spacetime R x S corresponding to time evolution.

A similar thing happens in quantum theory. BUT: in quantum theory, if a state is annihilated by some observable, it implies that the state is invariant under the one-parameter family of symmetries generated by that observable. This is not true in classical mechanics. Indeed, it's rather odd. But what it implies is that in step II we are really restricting ourselves to kinematical states that are invariant under diffeomorphisms of the spatial manifold S. This is why we call them "diffeomorphism-invariant" states. Similarly, in step III we're further restricting ourselves to states that are invariant under time evolution. The final "physical states" are, at least heuristically, invariant under ALL DIFFEOMORPHISMS OF SPACETIME. (So maybe the physical states are the ones that really should be called "diffeomorphism-invariant" --- but it's too late now.) While this may seem odd, all it really means is that in the quantum theory of gravity --- at least when one does it this way --- the physical states describe only those aspects of the world that are independent of any choice of coordinate system. That has a certain charm, philosophically speaking. It is, however, not something physicists are used to.

Now, the general scheme outlined above has been around ever since the work of DeWitt:

1) Quantum theory of gravity, I-III by Bryce S. DeWitt, Phys. Rev. 160 (1967), 1113-1148, 162 (1967) 1195-1239, 1239-1256.

However, the problem has always been making the scheme mathematically rigorous, or else to do some kind of calculations that shed some light on the meaning of it all! There are lots of problems. Let me not delve into them now, but simply cut directly to the "new variables" idea for handling these problems. The key idea of Ashtekar was to use as basic variables, not the metric on S and its first time derivative, but the "chiral spin connection" on S and a "complex frame field". To describe these would require a digression into differential geometry that I'm not in the mood for right now, especially since I already explained this stuff a bit in "week7". (There I call the chiral spin connection the "right-handed" connection.)

I do, however, want to emphasize that the new variables rely heavily upon some of the basic group-theoretic facts about 3 and 4 dimensions. The group of rotations in 3d space is called SO(3), because mathematically these are 3x3 orthogonal matrices with determinant 1. Now, a key fact in math and physics is that this group has the group SU(2) of 2x2 complex unitary matrices with determinant 1 as a "double cover". This means roughly that there are two elements of this other group corresponding to each element of SO(3). It's this fact that allows the existence of spin-1/2 particles!

Now, SU(2) is sitting inside a bigger group, SL(2,C), the group of all 2x2 complex matrices with determinant 1, not necessarily unitary. Just as SU(2) is used to describe the symmetries of spin-1/2 particles in space, SL(2,C) describes the symmetries of spin-1/2 particles in spacetime. The reason is that SL(2,C) is the double cover of the group SO(3,1) of Lorentz transformations.

Given a Riemannian metric on the space S, there is always an "SO(3) connection" describing how objects rotate when you move them around a loop, due to the curvature of space. This is called the Levi-Civita connection. With a little work we can also think of this as an SU(2) connection. However, Ashtekar works instead with the chiral spin connection, which is an SL(2,C) connection cooked up from the Levi-Civita connection and the first time derivative of the metric (which turns out to be closely related to the "extrinsic curvature" of S as it sits in the spacetime R x S.)

The great advantage of Ashtekar's "new variables" is that the Hamiltonian and diffeomorphism constraints are simpler in these variables. Unfortunately, they lead to a curious new issue which at first seemed very nasty --- the problem of "reality conditions". This has a lot to do with going from SU(2), which is a "real" group in a technical sense, to SL(2,C), which is a "complex" group that's roughly twice as big. Essentially, Ashtekar's formalism seems at first to be better suited to general relativity with a complex-valued metric than to good old "real" general relativity. For quite a while people didn't know quite what to do about this, so a lot of work on the new variables more or less ignores this issue. Luckily, there is now a very elegant approach to handling it, worked out by Ashtekar and collaborators. They are coming out with a couple of papers on this, hopefully by mid-November:

2) Coherent State Transforms for Spaces of Connections, by Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao and Thomas Thiemann, draft, to appear on gr/qc.

Quantum geometrodynamics, by A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, in progress, to appear on gr/qc.

The first paper constructs a kind of transform that takes functions on the space of SU(2) connections on S into functions on the space of SL(2,C) connections on S. The "kinematical states" in Ashtekar's approach are, roughly speaking, functions of the latter kind. (Really they are more like "measures".) This is some really pretty mathematics --- it's a kind of generalization of the Bargmann-Segal transform to the case of functions on spaces of connections.

Physically, the transform allows us to relate Ashtekar's approach to the traditional "metric" approach much more clearly, since, as I described, SU(2) connections are closely related to metrics on S. The second paper should treat the physics behind this in more detail, and also describe a rigorous construction of "loop states" --- a large class of diffeomorphism-invariant states which Rovelli and Smolin have claimed are actually physical states. (For more on these, see below.) This means that to check Rovelli and Smolin's claim, the main thing we need is a rigorous treatment of the Hamiltonian constraint in quantum gravity.

Unfortunately, this is where it gets scary, since the Hamiltonian constraint is a very tricky thing. For more on it, try:

3) The Hamiltonian constraint in quantum gravity, M. Blencowe, Nucl. Phys. B341 (1990), 213-251.

On the constraints of quantum gravity in the loop representation, Bernd Bruegmann and Jorge Pullin, Nucl. Phys. B390 (1993), 399-438.

On the constraints of quantum general relativity in the loop representation, Bernd Bruegmann, Ph.D. Thesis, Syracuse University (May 1993)

The loop states of Rovelli and Smolin are in one-to-one correspondence with knots in space, or more precisely, isotopy classes of knots. (Roughly, two knots are isotopic if you can get one from the other by applying a diffeomorphism of space that can be continuously deformed to the identity.)

What is the physical meaning of these loop states? Roughly it's this. Say you take a spin-1/2 particle and move it around in a path that traces out a knot. When you do this using the Levi-Civita connection, it comes back "rotated" by some SU(2) matrix. If you take the trace of this matrix (sum of diagonal entries) and divide by two, you get a number between -1 and 1. This number is called a "Wilson loop".

This should remind you of the Bohm-Aharonov effect where a split electron beam takes two paths from A to B. Depending on the magnetic flux through the loop, one can have constructive or destructive interference in the split beam experiment. Mathematically, one can imagine moving the electron around a loop that starts at A, goes to B by one path, and goes back to A by the other path. If this phase corresponding to going is 1 we get total constructive interference in the split beam experiment, while if it's -1 we get total destructive interference. So, just as the Bohm-Aharonov effect measures interference effects due to the magnetic field, the the above Wilson loop sort of measures the interference effects due to GRAVITY!

Now, in the Rovelli-Smolin loop state corresponding to a particular knot K, the expectation value of a Wilson loop around any knot K' will be 1 if K and K' are isotopic, and 0 otherwise! That's the physical meaning of the loop states: they describe quantum states of geometry in terms of the resulting interference effects on spin-1/2 particles.

Now, there is a more general kind of diffeomorphism-state than than the loop states. These are the spin network states! Here one fancies up the Wilson loop idea and imagines a graph embedded in space --- i.e. a bunch of edges and vertices --- where each edge is labelled by a spin that can be 0,1/2,1,3/2, etc. In the simplest flavor of spin network, one only allows 3 edges to meet at each vertex, and requires j3 to be of the form

j3 = |j1-j2|, |j1-j2| + 1, ...., j1+j2-1, j1+j2.

where j1, j2, j3 are the spins labelling the edges adjacent to the given vertex. For example, we can have the the three spins be 1/2,3, and 5/2, because it's possible for a spin-1/2 particle and a spin-3 particle to interact and form a spin-5/2 particle. Here by "possible" I simply mean that it doesn't violate conservation of angular momentum. Mathematicians would say the spins should be thought of as irreducible representations of SU(2), and the condition above is just the condition that the representation j3 appears as a summand in the tensor product of the representations j1 and j2.

Just as we can compute a kind of "Wilson loop" number from a knot that a spin-1/2 particle goes around, we can compute a number from a spin network. I've thought about spin networks for quite a while, since they are very important in topological quantum field theories. A great introduction to how they show up in TQFTs, by the way, is:

4) State-sum invariants of manifolds, I, by Louis Crane, Louis H. Kauffman, and David N. Yetter, 46 pages, LaTeX (Sun release 4.1) source code produces many error messages, but a correct dvi-file, available as hep-th/9409167.

This explains how to cook up 3d quantum gravity (or more precisely, the Turaev-Viro model) and a 4d TQFT field theory called the Crane-Yetter model using spin networks.

However, Rovelli's talk on spin network states in quantum gravity (see "week41"), followed by some good conversations, got me motivated to write up something on spin network states:

5) Spin network states in gauge theory, by John Baez, 19 pages in LaTeX format available as gr-qc/9411007 (or as "spin.tex" by ftp as described at the end of this article).

Basically, I show that in the loop representation of any gauge theory, states at the kinematical level can be described by spin networks, slightly generalized. Heck, I'll quote my abstract:

Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P → M, there is a canonical `generalized measure' on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L^2(A/G). Here we construct a set of vectors spanning L^2(A/G). These vectors are described in terms of `spin networks': graphs φ embedded in M, with oriented edges labelled by irreducible unitary representations of G, and with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We also describe an orthonormal basis of spin networks associated to any fixed graph φ. We conclude with a discussion of spin networks in the loop representation of quantum gravity, and give a category-theoretic interpretation of the spin network states.

I'm now hard at work trying to show that spin networks also give a complete description of states at the diffeomorphism-invariant level. Well, actually right NOW I'm goofing off by writing this darn thing, but you know what I mean.

Rovelli and Smolin have come out with one of their papers on spin networks and they should be coming out with another soon. These are not about the rigorous mathematics of spin network states, but how to use them to really understand the physics of quantum gravity. The first one out is:

6) Discreteness of area and volume in quantum gravity, by Carlo Rovelli and Lee Smolin, 36 pages in LaTeX format, 13 figures uuencoded, available as gr-qc/9411005.

This is perhaps the most careful computation so far that derives discreteness of geometrical quantities directly from Einstein's equations and the principles of quantum theory! Let me quote the abstract:

We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.

© 1994 John Baez