# Finite entanglement entropy and spectral dimension in quantum gravity

## TL;DR

This paper investigates the conditions under which quantum gravity models can have finite entanglement entropy density, revealing that spectral dimension positivity prevents finiteness and exploring potential solutions like alternative definitions or analytic continuation.

## Contribution

It proves that ultraviolet finiteness alone does not ensure finite entanglement entropy and that positive spectral dimension at all scales prevents entropy finiteness, with implications for quantum gravity models.

## Key findings

- Ultraviolet finiteness does not guarantee finite entropy density.
- Positive spectral dimension at all scales prevents finite entanglement entropy.
- Alternative definitions or analytic continuation can yield finite entropy in some models.

## Abstract

What are the conditions on a field theoretic model leading to a finite entanglement entropy density? We prove two very general results: 1) Ultraviolet finiteness of a theory does not guarantee finiteness of the entropy density; 2) If the spectral dimension of the spatial boundary across which the entropy is calculated is non-negative at all scales, then the entanglement entropy cannot be finite. These conclusions, which we verify in several examples, negatively affect all quantum-gravity models, since their spectral dimension is always positive. Possible ways out are considered, including abandoning the definition of the entanglement entropy in terms of the boundary return probability or admitting an analytic continuation (not a regularization) of the usual definition. In the second case, one can get a finite entanglement entropy density in multi-fractional theories and causal dynamical triangulations.