The Reeh–Schlieder Theorem

The Reeh–Schlieder theorem says that the vacuum orbit of every nonempty local algebra is dense, and—when the causal complement has interior—that the vacuum is also separating. Positive-energy analyticity propagates a local vanishing condition to all translations; locality then converts cyclicity of the complementary region into separation. The theorem is qualitative: it provides no uniform norm, energy, probability, or complexity bound for a local approximation.

Required background. Wightman Functions and Spectral Support supplies the positive-energy boundary values. Tube Domains, Complex Lorentz Covariance, and Analyticity supplies the analytic continuation mechanism. Haag–Kastler Nets and Locality supplies the regional algebras, and States, GNS Representations, and Folia supplies the vacuum representation.

Helpful background. Reeh–Schlieder Property and Limits of Localization develops the operational interpretation.

Cyclicity and separation

Consider a Poincaré-covariant net in its vacuum representation $(\mathcal H,\pi,\Omega)$. Assume:

• translations are strongly continuous and have joint spectrum in $\overline V_+$;
• $\Omega$ is translation invariant and cyclic for the global quasilocal algebra;
• the net is weakly additive, so translates of any nonempty local algebra generate the global algebra;
• the region $O$ contains a smaller nonempty open region whose sufficiently small translates remain in $O$.

Then

$$\overline{\mathfrak A(O)\Omega}=\mathcal H.$$

If Einstein causality also holds and $O'$ contains a nonempty open region, then

$$A\Omega=0,\quad A\in\mathfrak A(O) \quad\Longrightarrow\quad A=0;$$

that is, $\Omega$ is separating for $\mathfrak A(O)$. Variants replace weak additivity by field-theoretic irreducibility and use Wightman fields directly. The region, spectrum, and generation hypotheses must therefore be stated with the conclusion. The original field theorem is Reeh and Schlieder 1961, pp. 1051–1068; the net proof with explicit assumptions is Fewster and Rejzner 2020, § 5.2, Theorem 26, pp. 25–27.

Why positive energy propagates local information

Suppose $\Psi$ is orthogonal to $\mathfrak A(O)\Omega$. Choose $O_0$ with $\overline O_0\subset O$ and $Q_1,\ldots,Q_n\in\mathfrak A(O_0)$. For small real translations $x_j$, all translated factors remain local enough that

$$\langle\Psi, U(x_1)Q_1U(x_2-x_1)Q_2\cdots U(x_n-x_{n-1})Q_n\Omega\rangle=0.$$

The spectrum condition extends this matrix element to a holomorphic function when the imaginary translation increments lie in the forward cone. Since its boundary value vanishes on a real open set, edge-of-the-wedge uniqueness makes it vanish for all real translations. Weak additivity and global cyclicity then imply $\Psi=0$.

For separation, take $A\in\mathfrak A(O)$ with $A\Omega=0$. If $B\in\mathfrak A(O')$, locality gives

$$AB\Omega=BA\Omega=0.$$

Cyclicity of $\Omega$ for a nonempty subregion of $O'$ makes $\mathfrak A(O')\Omega$ dense, so the bounded operator $A$ vanishes. Notice the direction: spectrum plus generation yields cyclicity; cyclicity of the complement plus locality yields separation. Cyclicity alone does not imply either positive energy or locality.

The theorem also has no converse in the form often desired. A cyclic and separating vector for one algebra is standard-form data, but it need not be a translation-invariant vacuum and does not supply a Poincaré action. Even cyclicity for every member of a net does not reconstruct the spectrum condition: positive energy entered through the tube analyticity used in the proof. Conversely, locality without weak additivity can leave a proper translation-invariant subspace invisible to the chosen region, so the generation hypothesis cannot simply be dropped.

First application: the free scalar vacuum

Reeh–Schlieder Property and Limits of Localization owns the localization consequences of this free-field application.

For the massive free scalar in its vacuum representation, the two-point function has Fourier support on the positive-energy mass shell. If a one-particle wavefunction is orthogonal to every $\phi(f)\Omega$ with $\operatorname{supp}f\subset O$, its associated positive-frequency Klein–Gordon solution has vanishing distributional boundary values on $O$. Tube analyticity forces that solution to vanish identically. Repeating the argument for polynomial field vectors, or applying the net theorem to the Weyl algebra, gives

$$\overline{\mathfrak A(O)\Omega}=\mathcal H_F$$

for every nonempty double cone $O$. If $O'$ has interior, spacelike commutativity and cyclicity for a double cone inside $O'$ make $\Omega$ separating for $\mathfrak A(O)$.

Failure test: demanding cheap remote preparation

Let $\Psi$ be a normalized wavepacket concentrated far from $O$. The theorem gives $A_n\in\mathfrak A(O)$ such that $A_n\Omega\to\Psi$, but it gives no bound on

$$\lVert A_n\rVert, \qquad \frac{\langle A_n\Omega,H A_n\Omega\rangle} {\lVert A_n\Omega\rVert^2}, \qquad \text{or the success probability of a selective realization.}$$

Requiring all three to remain uniformly controlled while the error tends to zero is an added quantitative claim. Cyclicity supplies density only. Moreover, applying $A_n$ and renormalizing describes a postselected branch unless a trace-preserving localized operation and its outcome probability are supplied. No superluminal signal follows.

Independent check

Given an approximation sequence, plot the error $\epsilon_n=\lVert A_n\Omega-\Psi\rVert$ together with $\lVert A_n\rVert$ and the energy expectation. Decreasing $\epsilon_n$ alone confirms only the qualitative orbit-density statement. Separately verify that deterministic operations localized in $O$ leave expectation values in spacelike algebras unchanged. These are independent mathematical and operational checks.

Exercises

Assuming cyclicity for $\mathfrak A(O')$ and locality between $O$ and $O'$, prove that $\Omega$ is separating for $\mathfrak A(O)$.

Solution

If $A\Omega=0$ with $A\in\mathfrak A(O)$, then for every $B\in\mathfrak A(O')$ locality gives $AB\Omega=BA\Omega=0$. Thus $A$ vanishes on the dense set $\mathfrak A(O')\Omega$. Since $A$ is bounded, $A=0$ on all of $\mathcal H$.

References

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI; Open PDF.
• Reeh, Helmut, and Siegfried Schlieder. “Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern.” Il Nuovo Cimento 22 (1961): 1051–1068. DOI.