June 8, 1999

This Week's Finds in Mathematical Physics (Week 134)

John Baez

My production of "This Week's Finds" has slowed to a trickle as I've been struggling to write up a bunch of papers. Deadlines, deadlines! I hate deadlines, but when you write things for other people, or with other people, that's what you get into. I'll do my best to avoid them in the future. Now I'm done with my chores and I want to have some fun.

I spent last weekend with a bunch of people talking about quantum gravity in a hunting lodge by a lake in Minnowbrook, New York:

1) Minnowbrook Symposium on Space-Time Structure, program and transparencies of talks available at http://www.phy.syr.edu/research/he_theory/minnowbrook/#PROGRAM

The idea of this get-together, organized by Kameshwar Wali and some other physicists at Syracuse University, was to bring together people working on string theory, loop quantum gravity, noncommutative geometry, and various discrete approaches to spacetime. People from these different schools of thought don't talk to each other as much as they should, so this was a good idea. People gave lots of talks, asked lots of tough questions, argued, and learned what each other were doing. But I came away with a sense that we're quite far from understanding quantum gravity: every approach has obvious flaws.

One big problem with string theory is that people only know how to study it on a spacetime with a fixed background metric. Even worse, things are poorly understood except when the metric is static - that is, roughly speaking, when geometry of space does not change with the passage of time.

For example, people understand a lot about string theory on spacetimes that are the product of Minkowski spacetime and a fixed Calabi-Yau manifold. There are lots of Calabi-Yau manifolds, organized in continuous multi-parameter families called moduli spaces. This suggests the idea that the geometry of the Calabi-Yau manifold could change with time. This idea is lurking behind a lot of interesting work. For example, Brian Greene gave a nice talk on "mirror symmetry". Different Calabi-Yau manifolds sometimes give the same physics; these are called "mirror manifolds". Because of this, a curve in one moduli space of Calabi-Yau manifolds can be physically equivalent to a curve in some other moduli space, which sometimes lets you continue the curve beyond a singularity in the first moduli space. Physicists like to think of these curves as representing spacetime geometries where the Calabi-Yau manifold changes with time. The problem is, there's no fully worked out version of string theory that allows for a time-dependent Calabi-Yau manifold!

There's a good reason for this: one shouldn't expect anything so simple to make sense, except in the "adiabatic approximation" where things change very slowly with time. The product of Minkowski spacetime with a fixed Calabi-Yau manifold is a solution of the 10-dimensional Einstein equations, and this is part of why this kind of spacetime serves as a good background for string theory. But we do not get a solution if the geometry of the Calabi-Yau manifold varies from point to point in Minkowski spacetime - except in the adiabatic approximation.

There are also problems with "unitarity" in string theory when the geometry of space changes with time. This is already familiar from ordinary quantum field theory on curved spacetime. In quantum field theory, people usually like to describe time evolution using unitary operators on a Hilbert space of states. But this approach breaks down when the geometry of space changes with time. People have studied this problem in detail, and there seems to be no completely satisfactory way to get around it. No way, that is, except the radical step of ceasing to treat the geometry of spacetime as a fixed "background". In other words: stop doing quantum field theory on spacetime with a pre-established metric, and invent a background-free theory of quantum gravity! But this is not so easy - see "week132" for more on what it would entail.

Apparently this issue is coming to the attention of string theorists now that they are trying to study their theory on non-static background metrics, such as anti-de Sitter spacetime. Indeed, someone at the conference said that a bunch of top string theorists recently got together to hammer out a strategy for where string theory should go next, but they got completely stuck due to this problem. I think this is good: it means string theorists are starting to take the foundational issues of quantum gravity more seriously. These issues are deep and difficult.

However, lest I seem to be picking on string theory unduly, I should immediately add that all the other approaches have equally serious flaws. For example, loop quantum gravity is wonderfully background-free, but so far it is almost solely a theory of kinematics, rather than dynamics. In other words, it provides a description of the geometry of space at the quantum level, but says little about spacetime. Recently people have begun to study dynamics with the help of "spin foams", but we still can't compute anything well enough to be sure we're on the right track. So, pessimistically speaking, it's possible that the background-free quality of loop quantum gravity has only been achieved by simplifying assumptions that will later prevent us from understanding dynamics.

Alain Connes expressed this worry during Abhay Ashtekar's talk, as did Arthur Jaffe afterwards. Technically speaking, the main issue is that loop quantum gravity assumes that unsmeared Wilson loops are sensible observables at the kinematical level, while in other theories, like Yang-Mills theory, one always needs to smear the Wilson loops. Of course these other theories aren't background-free, so loop quantum gravity probably should be different. But until we know that loop quantum gravity really gives gravity (or some fancier theory like supergravity) in the large-scale limit, we can't be sure it should be different in this particular way. It's a legitimate worry... but only time will tell!

I could continue listing approaches and their flaws, including Connes' own approach using noncommutative geometry, but let me stop here. The only really good news is that different approaches have different flaws. Thus, by comparing them, one might learn something!

Some more papers have come out recently which delve into the philosophical aspects of this muddle:

2) Carlo Rovelli, Quantum spacetime: what do we know?, to appear in Physics Meets Philosophy at the Planck Scale, eds. Craig Callender and Nick Huggett, Cambridge U. Press. Preprint available as gr-qc/9903045.

3) J. Butterfield and C. J. Isham, Spacetime and the philosophical challenge of quantum gravity, to appear in Physics Meets Philosophy at the Planck Scale, eds. Craig Callender and Nick Huggett, Cambridge U. Press. Preprint available as gr-qc/9903072.

Rovelli's paper is a bit sketchy, but it outlines ideas which I find very appealing - I always find him to be very clear-headed about the conceptual issues of quantum gravity. I found the latter paper a bit frustrating, because it lays out a wide variety of possible positions regarding quantum gravity, but doesn't make a commitment to any one of them. However, this is probably good when one is writing to an audience of philosophers: one should explain the problems instead of trying to sell them on a particular claimed solution, because the proposed solutions come and go rather rapidly, while the problems remain. Let me quote the abstract:

We survey some philosophical aspects of the search for a quantum theory of gravity, emphasising how quantum gravity throws into doubt the treatment of spacetime common to the two `ingredient theories' (quantum theory and general relativity), as a 4-dimensional manifold equipped with a Lorentzian metric. After an introduction, we briefly review the conceptual problems of the ingredient theories and introduce the enterprise of quantum gravity. We then describe how three main research programmes in quantum gravity treat four topics of particular importance: the scope of standard quantum theory; the nature of spacetime; spacetime diffeomorphisms, and the so-called problem of time. By and large, these programmes accept most of the ingredient theories' treatment of spacetime, albeit with a metric with some type of quantum nature; but they also suggest that the treatment has fundamental limitations. This prompts the idea of going further: either by quantizing structures other than the metric, such as the topology; or by regarding such structures as phenomenological. We discuss this in Section 5.

Now let me mention a few more technical papers that have come out in the last few months:

4) John Baez and John Barrett, The quantum tetrahedron in 3 and 4 dimensions, preprint available as gr-qc/9903060.

The idea here is to form a classical phase whose points represent geometries of a tetrahedron in 3 or 4 dimensions, and then apply geometric quantization to obtain a Hilbert space of states. These Hilbert spaces play an important role in spin foam models of quantum gravity. The main goal of the paper is to explain why the quantum tetrahedron has fewer degrees of freedom in 4 dimensions than in 3 dimensions. Let me quote from the introduction:

State sum models for quantum field theories are constructed by giving amplitudes for the simplexes in a triangulated manifold. The simplexes are labelled with data from some discrete set, and the amplitudes depend on this labelling. The amplitudes are then summed over this set of labellings, to give a discrete version of a path integral. When the discrete set is a finite set, then the sum always exists, so this procedure provides a bona fide definition of the path integral.

State sum models for quantum gravity have been proposed based on the Lie algebra so(3) and its q-deformation. Part of the labelling scheme is then to assign irreducible representations of this Lie algebra to simplexes of the appropriate dimension. Using the q-deformation, the set of irreducible representations becomes finite. However, we will consider the undeformed case here as the geometry is more elementary.

Irreducible representations of so(3) are indexed by a non-negative half-integers j called spins. The spins have different interpretations in different models. In the Ponzano-Regge model of 3-dimensional quantum gravity, spins label the edges of a triangulated 3-manifold, and are interpreted as the quantized lengths of these edges. In the Ooguri-Crane-Yetter state sum model, spins label triangles of a triangulated 4-manifold, and the spin is interpreted as the norm of a component of the B-field in a BF Lagrangian. There is also a state sum model of 4-dimensional quantum gravity in which spins label triangles. Here the spins are interpreted as areas.

Many of these constructions have a topologically dual formulation. The dual 1-skeleton of a triangulated surface is a trivalent graph, each of whose edges intersect exactly one edge in the original triangulation. The spin labels can be thought of as labelling the edges of this graph, thus defining a spin network. In the Ponzano-Regge model, transition amplitudes between spin networks can be computed as a sum over labellings of faces of the dual 2-skeleton of a triangulated 3-manifold. Formulated this way, we call the theory a `spin foam model'.

A similar dual picture exists for 4-dimensional quantum gravity. The dual 1-skeleton of a triangulated 3-manifold is a 4-valent graph each of whose edges intersect one triangle in the original triangulation. The labels on the triangles in the 3-manifold can thus be thought of as labelling the edges of this graph. The graph is then called a `relativistic spin network'. Transition amplitudes between relativistic spin networks can be computed using a spin foam model. The path integral is then a sum over labellings of faces of a 2-complex interpolating between two relativistic spin networks.

In this paper we consider the nature of the quantized geometry of a tetrahedron which occurs in some of these models, and its relation to the phase space of geometries of a classical tetrahedron in 3 or 4 dimensions. Our main goal is to solve the following puzzle: why does the quantum tetrahedron have fewer degrees of freedom in 4 dimensions than in 3 dimensions? This seeming paradox turns out to have a simple explanation in terms of geometric quantization. The picture we develop is that the four face areas of a quantum tetrahedron in four dimensions can be freely specified, but that the remaining parameters cannot, due to the uncertainty principle.

Naively one would expect the quantum tetrahedron to have the same number of degrees of freedom in 3 and 4 dimensions (since one is considering tetrahedra mod rotations). However, quantum mechanics is funny about these things! For example, the Hilbert space of two spin-1/2 particles whose angular momenta point in opposite directions is smaller than the Hilbert space of a single spin-1/2 particle, even though classically you might think both systems have the same number of degrees of freedom. In fact a very similar thing happens for the quantum tetrahedron in 3 and 4 dimensions.

5) Abhay Ashtekar, Alejandro Corichi and Kirill Krasnov, Isolated horizons: the classical phase space, preprint available as gr-qc/9905089.

This paper explains in more detail the classical aspects of the calculation of the entropy of a black hole in loop quantum gravity (see "week112" for a description of this calculation). Let me quote the abstract:

A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are "isolated" near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration. A precise notion of non-rotating isolated horizons is formulated to capture these ideas. With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon. The required term turns out to be precisely the Chern-Simons action for the self-dual connection. The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure. We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics. In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons. It may also have applications in classical general relativity, particularly in the investigation of analytic issues that arise in the numerical studies of black hole collisions.

The following are some review articles on spin networks, spin foams and the like:

6) Roberto De Pietri, Canonical "loop" quantum gravity and spin foam models, to appear in the proceedings of the XXIIIth Congress of the Italian Society for General Relativity and Gravitational Physics (SIGRAV), 1998, preprint available as gr-qc/9903076.

7) Seth Major, A spin network primer, to appear in Amer. Jour. Phys., preprint available as gr-qc/9905020.

8) Seth Major, Operators for quantized directions, preprint available as gr-qc/9905019.

9) John Baez, An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics, eds. Helmut Gausterer and Harald Grosse, Lecture Notes in Physics, Springer-Verlag, Berlin, 2000, pp. 25-93. Preprint available as gr-qc/9905087.

By the way, Barrett and Crane have come out with a paper sketching a spin foam model for Lorentzian (as opposed to Riemannian) quantum gravity:

10) John Barrett and Louis Crane, A Lorentzian signature model for quantum general relativity, preprint available as gr-qc/9904025.

However, this model is so far purely formal, because it involves infinite sums that probably diverge. We need to keep working on this! Now that I'm getting a bit of free time, I want to tackle this issue. Meanwhile, Iwasaki has come out with an alternative spin foam model of Riemannian quantum gravity:

11) Junichi Iwasaki, A surface theoretic model of quantum gravity, preprint available as gr-qc/9903112.

Alas, I don't really understand this model yet. Finally, to wrap things up, something completely different:

12) Richard E. Borcherds, Quantum vertex algebras, preprint available as math.QA/9903038.

I like how the abstract of this paper starts: "The purpose of this paper is to make the theory of vertex algebras trivial". Good! Trivial is not bad, it's good. Anything one understands is automatically trivial.

© 1999 John Baez