# The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional

## TL;DR

This paper argues that the Hilbert space in quantum gravity is inherently locally finite-dimensional, implying that quantum field theory cannot be the fundamental framework of nature due to the finite nature of local degrees of freedom.

## Contribution

It provides a model-independent argument that the Hilbert space of quantum gravity is locally finite-dimensional, emphasizing the importance of decoherence in defining spatial regions.

## Key findings

- Hilbert space in quantum gravity is locally finite-dimensional.
- Quantum field theory cannot be the fundamental description of nature.
- Implications for the structure of quantum gravity and spacetime.

## Abstract

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.