Three Roads To Quantum Gravity

KNOTS, LINKS AND KINKS
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satisfying that there is a complete correspondence between
the classical and quantum pictures of geometry. In classical
geometry the volumes of regions and the areas of the surfaces
depend on the values of gravitational ®elds. They are coded in
certain complicated collections of mathematical functions,
known collectively as the metric tensor. On the other hand, in
the quantum picture the geometry is coded in the choice of a
spin network. These spin networks correspond to the classical
description in that, given any classical geometry, one can
®nd a spin network which describes, to some level of
approximation, the same geometry (Figure 27).
In classical general relativity the geometry of space evolves
in time. For example, when a gravitational wave passes a
surface, the area of that surface will oscillate in time. There is
an equivalent quantum picture in which the structures of the
spin networks may evolve in time in response to the passage
of a gravitational wave. Figure 28 shows some of the simple
steps by which a spin network evolves in time. If we let a spin
network evolve, we get a discrete spacetime structure. The
events of this discrete spacetime are the processes by which
changes of the form shown in Figure 28 occur. We can draw
pictures of evolving spin networks; they look like Figures 29
to 31. An evolving spin network is very like a spacetime, but it
is discrete rather than continuous. We can say what the causal
relations are among the events, so it has light cones. But it also
has more, for we can draw slices through it that correspond to
moments of time. As in relativity theory, there are many
different ways of slicing an evolving spin network, so as to see
it as a succession of states evolving in time. Thus, the picture
of spacetime given by loop quantum gravity agrees with the
fundamental principle that in the theory of relativity there are
no things, only processes.
John Wheeler used to say that on the Planck scale spacetime
would no longer be smooth, but would resemble a foam, which
he called spacetime foam. In tribute to Wheeler, the mathematician
John Baez has suggested that evolving spin networks be
called spin foam. The study of spin foam has sprung up since
the mid-1990s. There are several different versions presently
under study, invented by Mike Reisenberger, by Louis Crane