The obstacles to quantizing gravity are in part technical. General relativity is a complicated nonlinear theory, and one should expect it to be more difficult than, say, electrodynamics. Moreover, viewed as an ordinary field theory, general relativity has a coupling constant with dimensions of an inverse mass, and standard power-counting arguments - finally confirmed in 1986 by explicit computations  - indicate that the theory is nonrenormalizable. But the problem of finding a consistent quantum theory of gravity goes deeper. General relativity is a geometric theory of spacetime, and quantizing gravity means quantizing spacetime itself. In a very basic sense, we do not know what this means. For example:
Faced with such problems, it is natural to look for simpler models that share the important conceptual features of general relativity while avoiding some of the computational difficulties. General relativity in 2+1 dimensions - two dimensions of space plus one of time - is one such model. As a generally covariant theory of spacetime geometry, (2+1)-dimensional gravity has the same conceptual foundation as realistic (3+ 1)-dimensional general relativity, and many of the fundamental issues of quantum gravity carry over to the lower dimensional setting. At the same time, however, the (2+1)-dimensional model is vastly simpler, mathematically and physically, and one can actually write down viable candidates for a quantum theory. With a few exceptions, (2+1)-dimensional solutions are physically quite different from those in 3+1 dimensions, and the (2+1)-dimensional model is not very helpful for understanding the dynamics of realistic quantum gravity. In particular, the theory does not have a good Newtonian limit [107, 49, 94] . But for understanding conceptual problems - the nature of time, the construction of states and observables, the role of topology and topology change, the relationships among different approaches to quantization - the model has proven highly instructive.
Work on (2+1)-dimensional gravity dates back to 1963, when Staruszkiewicz first described the behavior of static solutions with point sources  . Progress continued sporadically over the next twenty years, but the modern rebirth of the subject can be traced to the seminal work of Deser, Jackiw, ’t Hooft, and Witten in the mid-1980s [107, 105, 106, 249, 103, 277, 279] . Over the past twenty years, (2+ 1)-dimensional gravity has become an active field of research, drawing insights from general relativity, differential geometry and topology, high energy particle theory, topological field theory, and string theory.
As I will explain below, general relativity in 2+1 dimensions has no local dynamical degrees of freedom. Classical solutions to the vacuum field equations are all locally diffeomorphic to spacetimes of constant curvature, that is, Minkowski, de Sitter, or anti-de Sitter space. Broadly speaking, three ways to introduce dynamics have been considered:
In this paper, I will limit myself to the third case, (2+1)-dimensional vacuum “quantum cosmology.” This review is based in part on a series of lectures in  and an earlier review , and much of the material can be found in more detail in a book  . There is not yet a comprehensive review of gravitating point particles in 2+1 dimensions, although [65, 197, 195, 37, 36, 199, 63, 183] will give an overview of some results. Several good general reviews of the (2+1)-dimensional black hole exist [75, 39], although a great deal of the quantum mechanics is not yet understood  .
Although string theory is perhaps the most popular current approach to quantum gravity, I will have little to say about it here: While some interesting results exist in 2+1 dimensions, almost all of them are in the context of black holes (see, for example, [157, 170, 187, 188, 189]). I will also have little to say about (2+1)-dimensional supergravity, although many of the results described below can be generalized fairly easily, and I will not address the coupling of matter except for a brief discussion in Section 5 .
Throughout, I will use units and unless otherwise noted.
|http://www.livingreviews.org/lrr-2005-1||© Max Planck Society and