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5 What Can We Still Learn?

We know immensely more about (2+1)-dimensional quantum gravity than we did twenty years ago. But we still have an enormous amount to learn. In particular, it is only quite recently that the general tools developed over the past few years have been brought to bear on particular physical problems - the resolution of singularities, for example, and the question of whether space is discrete at the Planck scale. A sketchy and rather personal list of open questions would include the following:
A key question in quantum gravity is whether quantized spacetime “resolves” the singularities of classical general relativity. This is a difficult question - already classically, it is highly nontrivial to even define a singularity  [92], and the quantum extensions of the classical definitions are far from obvious. This is an area in which (2+1)-dimensional gravity provides a natural arena, but results so far are highly preliminary  [227202] .
Sums over topologies
Another long-standing question in quantum gravity is whether spacetime topology can (or must) undergo quantum fluctuations. As we saw in Section  3.11, some real progress has been made in 2+1 dimensions. Often, though, the results require saddle point approximations, and pick out particular classes of saddle points. The nonperturbative summation techniques discussed at the end of Section  3.11 promise much deeper results, and may point toward a measure on the space of topologies analogous to the measure on the space of geometries induced by the DeWitt metric.
Quantized geometry
We saw above that there is some evidence for quantization of timelike intervals in (2+ 1)-dimensional gravity. A systematic exploration of this issue might teach us a good deal about differences among approaches to quantization. In particular, it would be very interesting to see whether any corresponding result appears in reduced phase space quantization, Wheeler-DeWitt quantization, or path integral approaches. To address this problem properly, one must introduce genuine observables for quantities such as length and area, either by adding point particles  [215] or by looking at shortest geodesics around noncontractible cycles. Note that for the torus universe, the moduli can be considered as ratios of lengths, and there is no sign that these need be discrete. This does not contradict the claims of  [130], since the lengths in question are spacelike, but it does suggest an interesting dilemma in Euclidean quantum gravity, where spacelike as well as timelike intervals might naturally be quantized  [238] .
Euclidean vs.  Lorentzian gravity
In the Chern-Simons formalism of Section  2.3, “Euclidean” and “Lorentzian” quantum gravity seem to be dramatically inequivalent: They have different gauge groups, different holonomies, and very different behaviors under the actions of large diffeomorphisms. In the ADM approach of Section  2.4, on the other hand, the differences are almost invisible. This suggests that further study might finally tell us whether Euclideanization is merely a technical trick, analogous to Wick rotation in ordinary quantum field theory, or whether it gives a genuinely different theory; and, if the latter, just how different the Euclidean and Lorentzian theories are. In canonical quantization, a key step would be to relate Chern-Simons and ADM amplitudes in the Euclidean theory, perhaps using the methods of Section  3.4 . In spin foam and path integral approaches, it might be possible to explicitly compare amplitudes.
Which approaches are equivalent?
A more general problem is to understand which of the approaches described here are equivalent. In particular, it is not obvious how much of the difference among various methods of quantization can be attributed to operator ordering ambiguities, and how much reflects a deeper inequivalence, as reflected (for instance) in different length spectra or different possibilities for topology change. An answer might help us understand just how nonunique quantum gravity in higher dimensions will be.
Higher genus
Most of the detailed, explicit results in (2+1)-dimensional quantum gravity hold only for the torus universe R × T2 . As noted in Section  2.7, this topology has some exceptional features, and might not be completely representative. In particular, the relationship between the ADM and Chern-Simons quantizations in Section  3.4 relied on a particularly simple operator ordering; it is not obvious that such an ordering can be found for the higher genus case  [207] . An extension to arbitrary genus might be too difficult, but a full treatment of the genus two topology, using the relation to hyperelliptic curves or the sigma model description of  [264], may be possible. It could also be worthwhile to further explore the case of spatially nonorientable manifolds  [181] to see whether any important new features arise.
Coupling matter
This review has dealt almost exclusively with vacuum quantum gravity. We know remarkably little about how to couple matter to this theory. Some limited progress has been made: For example, there is some evidence that (2+1)-dimensional gravity is renormalizable in the 1/N expansion when coupled to scalar fields  [174205] . This is apparently no longer the case when gravity is coupled to fermions and a U(1) Chern-Simons gauge theory  [22], although Anselmi has argued that if coupling constants are tuned to exact values, renormalizability can be restored, and in fact the theory can be made finite  [21] . Certain matter couplings in supergravity have been studied  [104196], and work on circularly symmetric “midi-superspace models” has led to some surprising results, including unexpected bounds on the Hamiltonian  [27Jump To The Next Citation Point32Jump To The Next Citation Point13753265224] . But the general problem of coupling matter remains very difficult, not least because - except in the special case of “topological matter”  [14085] - we lose the ability to represent diffeomorphisms as ISO(2, 1) gauge transformations.

Difficult as it is, however, an understanding of matter couplings may be the key to many of the conceptual issues of quantum gravity. One can explore the properties of a singularity, for example, by investigating the reaction of nearby matter, and one can look for quantization of time by examining the behavior of physical clocks. Moreover, some of the deep questions of quantum gravity can be answered only in the presence of matter. For example, does gravity cut off ultraviolet divergences in quantum field theory? This idea is an old one  [109165166], and it gets some support from the boundedness of the Hamiltonian in midi-superspace models  [32], but it is only in the context of a full quantum field theory that a final answer can be given.

The cosmological constant
Undoubtedly, the biggest embarrassment in quantum gravity today is the apparent prediction, at least in effective field theory, that the cosmological constant should be some 120 orders of magnitude larger than the observed limit. Several attempts have been made to address this problem in the context of (2+1)-dimensional quantum gravity. First, Witten has suggested a novel mechanism by which supersymmetry in 2+1 dimensions might cancel radiative corrections to /\ without requiring the equality of superpartner masses, essentially because even if the vacuum is supersymmetric, the asymptotics forbid the existence of unbroken supercharges for massive states  [28152] . This argument requires special features of 2+1 dimensions, though, and it is not at all clear that it can be generalized to 3+1 dimensions (although some attempts have been made in the context of “deconstruction”  [168]).

Second, the discovery that the sum over topologies can lead to a divergent partition function has been extended to 3+1 dimensions, at least for /\ < 0, and it has been argued that this behavior might signal a phase transition that could prohibit a conventional cosmology with a negative cosmological constant  [7980] . The crucial case of a positive cosmological constant is not yet understood, however, and if a phase change does indeed occur, its nature is still highly obscure. It may be that the nonperturbative summation over topologies discussed at the end of Section  3.11 could cast light on this question.

One might also hope that a careful analysis of the coupling of matter in 2+1 dimensions could reveal useful details concerning the vacuum energy contribution to /\, perhaps in a setting that goes beyond the usual effective field theory approach. For example, there is evidence that the matter Hamiltonian is bounded above in (2+1)-dimensional gravity  [27] ; perhaps this could cut off radiative contributions to the cosmological constant at an interesting scale.

Again, (2+1)-dimensional gravity as a test bed
As new approaches to quantum gravity are developed, the (2+1)-dimensional model will undoubtedly remain important as a simplified test bed. For example, a bit of work has been done on the null surface formulation of classical gravity in 2+1 dimensions  [123] ; a quantum treatment might be possible, and could tell us more about the utility of this approach in 3+ 1 dimensions. Similarly, (2+1)-dimensional gravity has recently been examined as an arena in which to test for a new partially discrete, constraint-free formulation of quantum gravity  [138] .

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