xii Preface
and few of the important questions can yet be answered with certainty.
I believe, however, that we now know enough to warrant a book on
quantum gravity in 2+1 dimensions.
In fact, we know enough for two books. Over the past few years, the
field has split into two weakly interacting sectors, one dealing with the
scattering of point particles and the other with empty space 'quantum
cosmology'. This book is about the latter. To try to keep some focus, I
have omitted almost all discussion of the very interesting work on point
sources and matter interactions. Even with this restriction, the treatment
here is incomplete and idiosyncratic: I discuss certain aspects of quantum
gravity - those I understand best - in some detail, and treat others rather
sketchily or not at all. Perhaps a reader will be inspired to write 'Volume
IF.
I have assumed that the reader understands basic general relativity,
at the level of the first chapters of Wald [274] or Track 1 of Misner,
Thorne, and Wheeler
[198].
Some familiarity with a physicist's version of
differential topology will be helpful, although the book does not require
knowledge of the intricacies of two- and three-manifold topology. I have
also assumed that the reader is reasonably comfortable with quantum
mechanics (canonical quantization, the Heisenberg and Schrodinger pic-
tures,
constraints, gauge invariance and gauge-fixing), and has had some
exposure to quantum field theory. For the most part, however, I have
not used very deep or difficult results, and when such complications were
necessary, I have tried to explain them reasonably well. I end the book
with three appendices, on the topology of manifolds, causal structures,
and differential geometry and fiber bundles. These are not substitutes for
texts,
but they may help the reader through some of the more obscure
sections of this work.
This book is the product of countless conversations, collaborations,
arguments, and patient explanations by those who understood more than
I did. Much of what I know about quantum gravity was taught to me by
Bryce and Cecile DeWitt. My original interest in the (2+l)-dimensional
model was inspired by Ed Witten. The idea for a book was first suggested
by Gianluca Grignani and Pasquale Sodano, with whom I worked on
an early version. I have learned much from my collaborators, Russell
Cosgrove, Shanta de Alwis, Jack Gegenberg, Ian Kogan, Robert Mann,
Jeanette Nelson, and Claudio Teitelboim. A number of mathematicians
have helped me steer through the shoals of three-dimensional geometry
and topology, among them William
Abikoff,
Scott Axelrod, Walter Carlip,
William Goldman, Joel Hass, Wolfgang Luck, Geoff Mess, Alan Reid,
and Scott Wolpert. I can only mention a few of the many physicists who
have assisted me along the way: Arley Anderson, Abhay Ashtekar, Max
Banados, David Brown, Marc Henneaux, Akio Hosoya, Ted Jacobson,
Cambridge Books Online © Cambridge University Press, 2009