Finite entanglement entropy and spectral dimension in quantum gravity

Understand

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.