January 12, 1998

This Week's Finds in Mathematical Physics (Week 114)

John Baez

Classes have started! But I just flew back yesterday from the Joint Mathematics Meetings in Baltimore - the big annual conference organized by the AMS, the MAA, SIAM, and other societies. Over 4000 mathematicians could be seen wandering in clumps about the glitzy harbor area and surrounding crime-ridden slums, arguing about abstractions, largely oblivious to the world around them. Everyone ate the obligatory crab cakes for which Baltimore is justly famous. Some of us drank a bit too much beer, too.

Witten gave a plenary talk on "M-theory", which was great fun even though he didn't actually say what M-theory is. Steve Sawin and I ran a session on quantum gravity and low-dimensional topology, so I'll say a bit about what went on there. There was also a nice session on homotopy theory in honor of J. Michael Boardman. I'll talk about that and various other things next week.

A lot of the buzz in our session concerned the new "spin foam" approach to quantum gravity which I discussed in "week113". The big questions are: how do you test this approach without impractical computer simulations? Lee Smolin's paper below suggests one way. Should you only sum over spin foams that are dual to a particular triangulation of spacetime, or should you sum over all spin foams that fit in a particular 4-dimensional spacetime manifold, or should you sum over all spin foams? There was a lot of argument about this. In addition to the question of what is physically appropriate, there's the mathematical problem of avoiding divergent infinite sums. Perhaps the sum required to answer any truly physical question only involves finitely many spin foams - that's what I hope. Finally, should the time evolution operators constructed using spin foams be thought of as describing true time evolution, or merely the projection onto the kernel of the Hamiltonian constraint? While it sounds a bit technical, this question is crucial for the interpretation of the theory; it's part of what they call "the problem of time".

Carlo Rovelli spoke about how spin foams arise in canonical quantum gravity, while John Barrett and Louis Crane discussed them in the context of discretized path integrals for quantum gravity, also known as state sum models. As in the more traditional "Regge calculus" approach, these models start by chopping spacetime into simplices. The biggest difference is that now areas of triangles play a more important role than lengths of edges. But Barrett, Crane and others are starting to explore the relationships:

1) John W. Barrett, Martin Rocek, Ruth M. Williams, A note on area variables in Regge calculus, preprint available as gr-qc/9710056.

2) Jarmo Makela, Variation of area variables in Regge calculus preprint available as gr-qc/9801022.

Also, there's been some progress on extracting Einstein's equation for general relativity as a classical limit of the Barrett-Crane state sum model. Let me quote the abstract of this paper:

3) Louis Crane and David N. Yetter, On the classical limit of the balanced state sum, preprint available as gr-qc/9712087.

"The purpose of this note is to make several advances in the interpretation of the balanced state sum model by Barrett and Crane in gr-qc/9709028 as a quantum theory of gravity. First, we outline a shortcoming of the definition of the model pointed out to us by Barrett and Baez in private communication, and explain how to correct it. Second, we show that the classical limit of our state sum reproduces the Einstein-Hilbert lagrangian whenever the term in the state sum to which it is applied has a geometrical interpretation. Next we outline a program to demonstrate that the classical limit of the state sum is in fact dominated by terms with geometrical meaning. This uses in an essential way the alteration we have made to the model in order to fix the shortcoming discussed in the first section. Finally, we make a brief discussion of the Minkowski signature version of the model."

Lee Smolin talked about his ideas for relating spin foam models to string theory. He has a new paper on this, so I'll just quote the abstract:

4) Lee Smolin, Strings as perturbations of evolving spin-networks, preprint available as hep-th/9801022.

"A connection between non-perturbative formulations of quantum gravity and perturbative string theory is exhibited, based on a formulation of the non-perturbative dynamics due to Markopoulou. In this formulation the dynamics of spin network states and their generalizations is described in terms of histories which have discrete analogues of the causal structure and many fingered time of Lorentzian spacetimes. Perturbations of these histories turn out to be described in terms of spin systems defined on 2-dimensional timelike surfaces embedded in the discrete spacetime. When the history has a classical limit which is Minkowski spacetime, the action of the perturbation theory is given to leading order by the spacetime area of the surface, as in bosonic string theory. This map between a non-perturbative formulation of quantum gravity and a 1+1 dimensional theory generalizes to a large class of theories in which the group SU(2) is extended to any quantum group or supergroup. It is argued that a necessary condition for the non-perturbative theory to have a good classical limit is that the resulting 1+1 dimensional theory defines a consistent and stable perturbative string theory."

Fotini Markopolou spoke about her recent work with Smolin on formulating spin foam models in a manifestly local, causal way.

5) Fotini Markopoulou and Lee Smolin, Quantum geometry with intrinsic local causality, preprint available as gr-qc/9712067.

"The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group G_q over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism."

The main technical idea in this paper is to work with "thickened" or "framed" spin networks, which amounts to replacing graphs by solid handlebodies. One expects this "framing" business to be important for quantum gravity with nonzero cosmological constant. This framing business also appears in the q-deformed version of Barrett and Crane's model and in my "abstract" version of their model, which assumes no background spacetime manifold. Markopoulou and Smolin don't specify a choice of dynamics; instead, they describe a class of theories which has my model as a special case, though their approach to causality is better suited to Lorentzian theories, while mine is Euclidean.

As I've often noted, spin foams are about spacetime geometry, or dynamics, while spin networks are a way of describing the geometry of space, or kinematics. Kinematics is always easier than dynamics, so the spin network approach to the quantum geometry of space has been much better worked out than the new spin foam stuff. Abhay Ashtekar gave an overview of these kinematical issues in his talk on "quantum Riemannian geometry", and Kirill Krasnov described how our understanding of these already allows us to compute the entropy of black holes (see "week112"). Here it's worth mentioning that the second part of Ashtekar's paper with Jerzy Lewandowski is finally out:

6) Abhay Ashtekar and Jerzy Lewandowski, Quantum theory of geometry II: volume operators, preprint available as gr-qc/9711031.

"A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics."

Before spin foam ideas came along, the basic strategy in the loop representation of quantum gravity was to start with general relativity on a smooth manifold and try to quantize it using the "canonical quantization" approach. Here the most important and difficult thing is to implement the "Hamiltonian constraint" as an operator on the Hilbert space of kinematical states, so you can write down the Wheeler-deWitt equation, which is, quite roughly speaking, the quantum gravity analog of Schrodinger's equation. (For a summary of this approach, try "week43".)

The most careful attempt to do this so far is the work of Thiemann:

7) Thomas Thiemann, Quantum spin dynamics (QSD), preprint available as gr-qc/9606089.

Quantum spin dynamics (QSD) II, preprint available as gr-qc/9606090.

QSD III: Quantum constraint algebra and physical scalar product in quantum general relativity, preprint available as gr-qc/9705017.

QSD IV: 2+1 Euclidean quantum gravity as a model to test 3+1 Lorentzian quantum gravity, preprint available as gr-qc/9705018.

QSD V: Quantum gravity as the natural regulator of matter quantum field theories, preprint available as gr-qc/9705019.

QSD VI: Quantum Poincare algebra and a quantum positivity of energy theorem for canonical quantum gravity, preprint available as gr-qc/9705020

Kinematical Hilbert spaces for fermionic and Higgs quantum field theories, gr-qc/9705021

If everything worked as smoothly as possible, the Hamiltonian constraint would satisfy nice commutation relations with the other constraints of the theory, giving a representation of something called the "Dirac algebra". However, as Don Marolf explained in his talk, this doesn't really happen, at least in a large class of approaches including Thiemann's:

8) Jerzy Lewandowski and Donald Marolf, Loop constraints: A habitat and their algebra, preprint available as gr-qc/9710016.

9) Rodolfo Gambini, Jerzy Lewandowski, Donald Marolf, and Jorge Pullin, On the consistency of the constraint algebra in spin network quantum gravity, preprint available as gr-qc/9710018.

This is very worrisome... as everything concerning quantum gravity always is. Personally these results make me want to spend less time on the Hamiltonian constraint, especially to the extent that it assumes a the old picture of spacetime as a smooth manifold, and more time on approaches that start with a discrete picture of spacetime. However, the only way to make serious progress is for different people to push on different fronts simultaneously.

There were a lot of other interesting talks, but since I'm concentrating on quantum gravity here I won't describe the ones that were mainly about topology. I'll wrap up by mentioning Steve Carlip's talk on spacetime foam. He gave a nice illustration to how hard it is to "sum over topologies" by arguing that this sum diverges for negative values of the cosmological constant. He has a paper out on this:

10) Steven Carlip, Spacetime foam and the cosmological constant, Phys. Rev. Lett. 79 (1997) 4071-4074, preprint available as gr-qc/9708026.

Again, I'll quote the abstract:

"In the saddle point approximation, the Euclidean path integral for quantum gravity closely resembles a thermodynamic partition function, with the cosmological constant Λ playing the role of temperature and the ``density of topologies'' acting as an effective density of states. For Λ < 0, the density of topologies grows superexponentially, and the sum over topologies diverges. In thermodynamics, such a divergence can signal the existence of a maximum temperature. The same may be true in quantum gravity: the effective cosmological constant may be driven to zero by a rapid rise in the density of topologies."

© 1998 John Baez