# QFT Hilbert space and analysis of quantum black holes

QFT Hilbert space is infinite dimensional and it is known that given a region $$A$$ and its complement $$A^c$$ of the spacetime, the QFT Hilbert space $$\textbf{does not}$$ decompose into $$\mathcal{H}_A \otimes \mathcal{H}_{A^c}$$ as such a decomposition contradicts the universal ultraviolet divergence in the entanglement entropy.

But when is it safe to do a calculation under the assumption of the above decomposition even though, strictly speaking, such a decomposition is not allowed in QFT? For example, I have read several articles studying quantum black holes in which they assume that the total Hilbert space factorizes into $$\mathcal{H}_{\text{inside the horizon}} \otimes \mathcal{H}_{\text{outside the horizon}}$$. There are several papers approaching the information paradox under the assumption that the laws of QM applies to black holes and uses quantum information theory to try to address this question. But, in a realistic situation with some quantum fields in a black hole geometry, the Hilbert space of the system is infinite dimensional.

So, is it 'safe' to apply quantum information theory to black holes without carefully analyzing the subtleties involving infinite dimensional Hilbert spaces?

PS: I am aware of a development involving Tomita-Takesaki modular theory to carefully define a relative entropy for QFT Hilbert spaces. But, I am not sure whether we have enough machinery yet to apply this approach to interesting problems as the modular Hamiltonian is known only for very few cases.