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
[149]
- 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
[246]
. 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
[76]
and an earlier review
[74], and much of the material can be found in more detail in a
book
[81
]
. 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
[82]
.
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
the author(s)
Problems/comments to |