Far in the future, the mathematical theory of "quantum graph theory" is the theory of
physics. Unlike the current theories of relativity and quantum physics,
which are obviously approximations that fall apart when pushed too far, QGT
seems to be an entirely correct description of reality from the beginning
of the universe onwards: a true theory of everything. And so everyone is
surprised when an experiment  one that QGT predicts to be entirely safe
 goes wrong, producing an expanding "novovacuum" that will apparently
destroy our universe. The main focus of the book is the moral and
scientific questions surrounding the attempts to stop this disaster.
It doesn't take long for this book to get into the mathematics; We see it
right on the first page! As in
several of Egan's other books, a form of mathematical physics that builds
on current theories of quantum gravity is the focal point. The socalled
"quantum graph theory" is described in interesting and believable detail.
Unlike some other mathematical fiction where the mathematics comes off
sounding obviously fake, Egan is able to capture the essence and cadence of
mathematical dialogue while saying things that, presumably, are equally
fake. Here, for instance, the scientists brainstorm to try to figure out
where they went wrong and eventually come up with an explanation that
parallels the current attempts to explain the quantum measurement problem
in terms of decoherence.
It is always difficult for me, personally, to read SF stories that take
place so far in the future that the characters no longer really seem to be
human. In this book, for example, bodies have become optional. The fact
that I find it more difficult to identify with these characters however is
counterbalanced by two wonderful consequences of this plot twist:
 Egan makes use of this to bring up some interesting questions
surrounding the question of "identity" in philosophy. Suppose we
could just leave our bodies like a person stepping out of a car, and
have a new one built...to what extent do we then identify with our bodies?
More mathematically, the title of the book is brought to bear on
this old philosophical question. I must admit, I'd never heard of
"Schild's Ladder" before, but it is apparently a constructive explanation
of the notion of parallel transport of vectors on a manifold. The modern
notion of "tangent vectors" in mathematics involves a "tangent bundle",
which is to say a separate "tangent space" at each point on the manifold.
Although this works well in general, it becomes difficult to say when the
tangent vector at one point is the same vector that appears at another.
So, the notion of "carrying" a vector around on a manifold becomes a subtle
one of
practical importance. Here, not only is this notion used to explain (by
analogy) how we can tell from moment to moment whether we are the same
person, but the mathematical fact that sometimes parallel transport of the
same starting vector around
different paths with the same endpoints will produce different resulting
vectors!
 One character who has only temporarily taken on the form of a physical
human body, is asked by some colleagues about what it was like where he
grew up. His response is my favorite line in the book. He explains
that he was raised in CP^{4} (which looks locally like a 4
dimensional complex vector space but has a very different global topology)
and that he only sometimes visited 3 dimensional spaces to work on certain
bizarre physics problems...but even then, he prefers working in symplectic
manifolds where you can separate the positions and momenta!
Contributed by
John C. Konrath
I found this work intellectually satisfying. Mr. Egan challenges the reader with plentiful mathematical, physical and biological theories, as well as raising several philosophical questions. This story is an entertaining blend of sciencefiction, adventure and mystery. While the characters are unremarkable and at times the plot stretches beyond the believable, overall this book is well worth reading.

