Reeh–Schlieder Theorem

Definition

The Reeh–Schlieder (RS) theorem states that in a QFT satisfying the Wightman / Haag–Kastler axioms, the vacuum $|\Omega ⟩$ is cyclic and separating for the local algebra $\mathcal{A}(\mathcal{O})$ of any open spacetime region $\mathcal{O}$, no matter how small.

• Cyclic: $\{a|\Omega ⟩:a\in \mathcal{A}(\mathcal{O})\}$ is dense in the full Hilbert space $\mathcal{H}$. Acting with operators localized in a tiny region can approximate any global state — including states describing particles arbitrarily far away.
• Separating: if $a|\Omega ⟩=0$ for $a\in \mathcal{A}(\mathcal{O})$ then $a=0$. No non-zero local operator annihilates the vacuum.

The two properties are dual: $|\Omega ⟩$ is cyclic for $\mathcal{A}(\mathcal{O})$ iff it is separating for the commutant $\mathcal{A}(\mathcal{O}{)}^{\prime }$, which by Haag duality is $\mathcal{A}({\mathcal{O}}^{\prime })$ (the causal complement). The proof rests on the analyticity of vacuum correlators in the energy (the spectrum condition): the vacuum has no lowest-energy localized excitation to annihilate.

The physically decisive corollary: a cyclic-separating vector defines a faithful normal state, which is exactly the hypothesis of Tomita–Takesaki modular theory. Moreover RS holds not only for the vacuum but for any state of bounded energy (finite energy states are also cyclic-separating), and any state that resembles the vacuum at short distances inherits the property — so it covers essentially all physical states: $k$-particle states, conformal primaries and descendants, etc.

Role in Research

Reeh–Schlieder is the theorem that guarantees the group’s modular-theoretic machinery applies at all, and it recurs as a hypothesis across the program:

• Modular theory is available everywhere. Because RS makes the vacuum cyclic-separating for every local algebra, Tomita–Takesaki theory — and hence the Bisognano–Wichmann modular operator, the Araki–Uhlmann relative entropy, and the Bell-CHSH construction — is defined on the algebra of any region.
• Vacuum entanglement / Bell violation. RS is the original source of the statement that the vacuum is entangled across any bipartition, which is what allows vacuum Bell-CHSH violation on complementary wedges.
• Magic. In Benedetti–Dabholkar–Dalmonte, RS is the step that upgrades “the vacuum is magical” to “every physical state is magical”: all cyclic-separating (faithful normal) states have a continuous modular spectrum and hence a non-flat entanglement spectrum.
• Gribov–Zwanziger frontier. RS assumes the standard axioms; whether it holds for the RGZ/confining vacuum, where the functional integral is restricted by the non-local horizon condition, is itself an open problem — see entanglement-as-confinement-probe and magic-and-the-gribov-horizon.
• Impossible measurements. RS-driven non-locality (a local operation can affect distant expectation values) is part of the tension behind impossible measurements in relativistic QFT.

Relations

• tomita-takesaki-modular-theory — RS supplies the cyclic-separating vector that modular theory requires
• bisognano-wichmann-theorem — RS provides the cyclic-separating vacuum whose wedge modular flow BW identifies with a boost
• type-iii-von-neumann-algebras — that every faithful normal state is cyclic-separating (no pure states) is a hallmark of type III₁; RS is how this shows up for the vacuum
• araki-uhlmann-relative-entropy — well-defined precisely because RS guarantees faithful normal states
• magic-nonstabilizerness — RS extends “magical” from the vacuum to all low-energy states
• rindler-wedges, causal-diamonds — the local regions for which the vacuum is cyclic-separating

Papers

• H. Reeh, S. Schlieder, Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern, Nuovo Cimento 22 (1961) 1051 — the original theorem.
• E. Witten, Notes on some entanglement properties of QFT, Rev. Mod. Phys. 90 (2018) 045003 [1803.04993] — modern pedagogical account (cyclic-separating, modular theory, type III).
• Derived in the AQFT course: week-09-reeh-schlieder-and-local-algebras.
• Used as a hypothesis in 2026-benedetti-magic-in-qft and the group’s Bell / relative-entropy work; reviewed in 2025-liu-lectures-entanglement-vna.

Notes

• RS is often stated as “the vacuum is highly entangled” but the precise content is cyclicity/separability, which is stronger and more useful: it is the exact hypothesis of modular theory, not merely a statement about entanglement magnitude.
• The theorem is sometimes felt to be paradoxical (local operators “create” distant particles), but the operators required are highly non-local in effect and enormous in norm; RS constrains what is possible in principle, not what is achievable with bounded resources — again the backdrop to impossible-measurements-qft.
• RS fails for theories without a mass gap in the naive form only through domain/technical subtleties; the cyclic-separating conclusion is robust for standard QFTs and is what the algebraic program uses.