  We argue in a model-independent way that the Hilbert space of quantum gravity
is locally finite-dimensional. In other words, the density operator describing
the state corresponding to a small region of space, when such a notion makes
sense, is defined on a finite-dimensional factor of a larger Hilbert space.
Because quantum gravity potentially describes superpo- sitions of different
geometries, it is crucial that we associate Hilbert-space factors with spatial
regions only on individual decohered branches of the universal wave function.
We discuss some implications of this claim, including the fact that quantum
field theory cannot be a fundamental description of Nature.
