13 Next steps 215
has changed this. Black holes in 2+1 dimensions differ in impor-
tant respects from their (3+l)-dimensional counterparts, but the
similarities are significant enough to make the (2+l)-dimensional
model quite valuable. In particular, while the statistical mechan-
ical explanation of black hole entropy of chapter 12 has not yet
been generalized to 3+1 dimensions, the (2+l)-dimensional model
has certainly suggested a new and interesting place to look for an
answer to one of the perennial questions of quantum gravity.
Where do we go from here? There are at least three broad areas in
which research in (2+l)-dimensional quantum gravity should advance.
We need to extend existing methods of quantization; we need to apply
the results to new conceptual issues in quantum gravity; and we need to
figure out how to incorporate nontrivial couplings with matter.
Much of this book has focused on 'quantum cosmologies' in which
spatial slices have the topology of a torus. Perhaps the biggest open
problem within the framework developed here is the extension of these
results to more complicated topologies. The torus topology is atypical - it
has an abelian fundamental group, for example, and an unusual mapping
class group action. For conclusions based on spacetimes with the topology
[0,1] x T
2
to be fully trusted, they really ought to be checked in at least
one other topology. This is a difficult problem, but the case of genus 2
seems within reach.
It may be that exact solutions are no longer possible in the case of
more complicated topologies. If this is the case - and even if it is not
- we need to develop suitable perturbation theories. Some preliminary
steps in this direction have been taken in the classical context [214, 215],
but quantum perturbation theory is almost completely unexplored. In
particular, virtually nothing is known about perturbative approaches to
the covariant canonical quantization of chapter 6.
I have tried to point out other open issues throughout the course of
this book, ranging from the fundamental - 'solve the (2+l)-dimensional
Wheeler-DeWitt equation' - to the more technical - 'work out the action
of the mapping class group on the space of wave functions vK
r
2~>
r
2~)'-
There is clearly no shortage of interesting questions.
Even within the limits of our present understanding, however, important
conceptual problems have yet to be addressed. For example, what is the
fate of
a
singularity in quantum gravity? An operator like the Hamiltonian
H
re
d
of equation (6.22) is clearly badly behaved at T = 0. On the other
hand, Puzio has argued that in the ADM quantization of chapter 5, a
wave packet initially concentrated away from the singular points in moduli
space will remain nonsingular
[222].
We do not really know how to ask
the right quantum mechanical questions about singularities; Hosoya has
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