The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional

Understand

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.