I have some naive questions about the Reeh-Schlieder theorem

[antaris]

Hello,

do local operator algebras only matter when you’re constructing a QFT or making approximations? In Witten’s paper (arXiv:1803.04993), he argues quite the opposite: the full Hilbert space H cannot be factorized into H_V ⊗ H_V′, since that would imply the vacuum splits into unentangled regions, contradicting the universal UV divergence of entanglement entropy. From his argument I understand that every short‑distance (“UV”) state looks like the vacuum at small scales—an important point when thinking about UV fixed points.

Edward Witten arXiv:1803.04993:

The Reeh-Schlieder theorem says that, in quantum field theory, if AV and AV′ are the algebras
of operators supported in complementary regions of spacetime, then similarly the vacuum is a cyclic separating vector for this pair of algebras.8 This might make one suspect that the Hilbert space H should be factored as H = HV ⊗ HV′ , with the vacuum being a fully entangled vector in the sense that the coefficients analogous to ck are all nonzero. This is technically not correct. If it were correct, then picking ψ ∈ HV , χ ∈ HV′ , we would get a vector ψ ⊗ χ ∈ H with no entanglement between observables in V and those in V′. This is not what happens in quantum field theory.

In quantum field theory, the entanglement entropy between adjacent regions has a universal ultraviolet divergence, independent of the states considered. The leading ultraviolet divergence is the same in any state as it is in the vacuum, because every state looks like the vacuum at short distances. The universality of this ultraviolet divergence means that it reflects not a property of any particular state but rather the fact that H cannot be factored as HV ⊗ HV′ .

Have I misunderstood something?

More precisely: if one wants to describe IR and UV effects on all scales within a single model, must one turn to algebraic QFT (AQFT), because in standard QFT there is no single, global Fock space? The usual Fock‑space decomposition is an excellent approximation only for the free (non‑interacting) vacuum—once interactions in a realistic model cannot be neglected, that decomposition fails? Does this mean we cannot build a fully realistic QFT by “gluing together” simpler pieces of theories?

Thanks in advance.

 

[SergejMaterov]

Local operator algebras are not just a technicality you only need when building a QFT or doing approximations. They capture a structural, state-independent feature of continuum QFTs (why the vacuum is highly entangled across every boundary, why reduced density matrices are ill-defined in the continuum, why modular flow/Bisognano–Wichmann phenomena exist, etc.). Witten’s point is exactly this: the failure of a global factorization H=HV⊗HV′ is a physical statement about the continuum theory, not merely a bookkeeping annoyance of some constructions.

The universality of the UV divergence of entanglement entropy (area law) means the following:
Explicit computations in free fields (Bombelli , Srednicki, etc) show that the leading divergence of entanglement entropy for a region scales with the area of the entangling surface and is independent of the global state — i.e. every short-distance state looks vacuum-like near the boundary. That universality is precisely what you would expect if there is no factorization at arbitrarily short distances: the divergence is not a property of a special state but of the kinematic structure of the continuum theory.