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4 What Have We Learned?

The world is not (2+1)-dimensional, and (2+1)-dimensional quantum gravity is certainly not a realistic model of our Universe. Nonetheless, the (2+1)-dimensional model reflects many of the fundamental conceptual issues of real world quantum gravity, and work in this field has provided some valuable insights.
Existence and nonuniqueness
Perhaps the most important lesson of (2+1)-dimensional quantum gravity is that general relativity can, in fact, be quantized. While additional ingredients - strings, for instance - may have their own attractions, they are evidently not necessary for the existence of quantum gravity. More than an “existence theorem”, though, the (2+1)-dimensional models also provide a “nonuniqueness theorem”: Many approaches to the quantum theory are possible, and they are not all equivalent. This is perhaps a bit of a disappointment, since many in this field had hoped that once we found a self-consistent quantum theory of gravity, the consistency conditions might be stringent enough to make that theory unique. In retrospect, though, we should not be so surprised: Quantum gravity is presumably more fundamental than classical general relativity, and it is not so strange to learn that more than one quantum theory can have the same classical limit.
(2+1)-dimensional gravity as a test bed
General relativity in 2+1 dimensions has provided a valuable test bed for a number of specific proposals for quantum gravity. Some of these are “classics” - the Wheeler-DeWitt equation, for instance, and reduced phase space quantization - while others, like spin foams, Lorentzian dynamical triangulations, and covariant canonical quantization, are less well established.

We have discovered some rather unexpected features, such as the difficulties caused by spatial diffeomorphism invariance and the consequent nonlocality in Wheeler-DeWitt quantization, and the necessity of understanding the representations of the group of large diffeomorphisms in almost all approaches. For particular quantization programs, (2+1)-dimensional models have also offered more specific guidance: Special properties of the loop operators (55View Equation), methods for treating noncompact groups in spin foam models, and properties of the sums over topologies described in Section  3.11 have all been generalized to 3+1 dimensions.

Lorentzian dynamical triangulations
A particular application of (2+1)-dimensional gravity as a test bed is important enough to deserve special mention. The program of “Lorentzian dynamical triangulations” described in Section  3.7 is a genuinely new approach to quantum gravity. Given the failures of ordinary “Euclidean dynamical triangulations”, one might normally be quite skeptical of such a method. But the success in reproducing semiclassical states in 2+1 dimensions, although still fairly limited, provides a strong argument that the approach should be taken seriously.
Observables and the “problem of time”
One of the deepest conceptual difficulties in quantum gravity has been the problem of reconstructing local, dynamical spacetime from the nonlocal diffeomorphism-invariant observables required by quantum gravity. The notorious “problem of time” is a special case of this more general problem of observables. As we saw in Section  3.4, (2+1)-dimensional quantum gravity points toward a solution, allowing the construction of families of “local” and “time-dependent” observables that nevertheless commute with all constraints.

The idea that “frozen time” quantum gravity is a Heisenberg picture corresponding to a fixed-time-slicing Schrödinger picture is a central insight of (2+1)-dimensional gravity. In practice, though, we have also seen that the transformation between these pictures relies on our having a detailed description of the space of classical solutions of the field equations. We cannot expect such a fortunate circumstance to carry over to full (3+1)-dimensional quantum gravity; it is an open question, currently under investigation, whether one can use a perturbative analysis of classical solutions to find suitable approximate observables  [50] .

It has long been hoped that quantum gravity might smooth out the singularities of classical general relativity. Although the (2+1)-dimensional model has not yet provided a definitive test of this idea, some progress has been made. Puzio, for example, has shown that a wave packet initially concentrated away from the singular points in moduli space will remain nonsingular  [227Jump To The Next Citation Point] . On the other hand, Minassian has recently demonstrated  [202Jump To The Next Citation Point] that quantum fluctuations do not “push singularities off to infinity” (as suggested in  [158]), and that several classically singular (2+1)-dimensional quantum spacetimes also have singular “quantum b -boundaries”.
Is length quantized?
Another long-standing expectation has been that quantum gravity will lead to discrete, quantized lengths, with a minimum length on the order of the Planck length. Partial results in quantum geometry and spin foam approaches to (2+1)-dimensional quantum gravity suggest that this may be true, but also that the problem is a bit subtle  [197Jump To The Next Citation Point251241] . The most recent result in this area  [130Jump To The Next Citation Point] relates the spectrum of lengths to representations of the (2+ 1)-dimensional Lorentz group, which can be discrete or continuous. Freidel et al. argue that spacelike intervals are continuous, while timelike intervals are discrete, with a spectrum of the form V~ --------- n(n - 1)lP . The analysis is a bit tricky, since the length “observables” do not, in general, commute with the Hamiltonian constraint. A first step towards defining truly invariant operators describing distances between point particles supports this picture  [215Jump To The Next Citation Point], but the results are not yet conclusive.
“Doubly special relativity”
Quantum gravity contains two fundamental dimensionful constants, the Planck length lP and the speed of light c . This has suggested to some that special relativity might itself be altered so that both lP and c are constants. This requires a nonlinear deformation of the Poincaré algebra, and leads to a set of theories collectively called “doubly special relativity”  [17185171] . It has recently been pointed out that (2+1)-dimensional gravity automatically displays such a deformation  [197Jump To The Next Citation Point18126] . A few attempts have been made to connect this picture to noncommutative spacetime, mainly in the context of point particles  [19727538], but it seems too early to evaluate them.
Topology change
Does consistent quantum gravity require spatial topology change? The answer in 2+1 dimensions is unequivocally no: Canonical quantization gives a perfectly consistent description of a universe with a fixed spatial topology. On the other hand, the path integrals of Section  3.10 seem to allow the computation of amplitudes for tunneling from one topology to another. Problems with these topology-changing amplitudes remain, particularly in the regulation of divergent integrals over zero-modes. If these can be resolved, however, we will have to conclude that we have found genuinely and deeply inequivalent quantum theories of gravity.
Sums over topologies
In conventional descriptions of the Hartle-Hawking wave function, and in other Euclidean path integral descriptions of quantum cosmology, it is usually assumed that a few simple contributions dominate the sum over topologies. The results of (2+1)-dimensional quantum gravity indicate that such claims should be treated with skepticism; as discussed in Section  3.11, the sum over topologies is generally dominated by an infinite number of complicated topologies, each individually exponentially suppressed. This is a new and unexpected result, whose implications for realistic (3+1)-dimensional gravity are just starting to be explored.

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