Reeh–Schlieder theorem

Remarks

The idea of the proof is analyticity. Take $\Psi \in H$ orthogonal to $\mathcal{P}(O)\Omega$. By the spectrum condition, the function $(x_1, \dots, x_n) \mapsto \ip{\Psi}{\phi(x_1) \cdots \phi(x_n)\Omega}$ is the boundary value of a function analytic in a complex tube. It vanishes when all $x_i$ lie in $O$, so by analytic continuation it vanishes for all $x_i$. So $\Psi$ is orthogonal to the dense set given by the cyclicity axiom; hence $\Psi = 0$.

A consequence: suppose the set of points spacelike to all of $O$ has a nonempty interior $O'$. If $A \in \mathcal{P}(O)$ and $A\Omega = 0$, then $A$ vanishes on the dense subspace $\mathcal{P}(O')\Omega$. Indeed $A B \Omega = B A \Omega = 0$ for $B \in \mathcal{P}(O')$, by locality. In this sense $\Omega$ is separating for the local fields. The same results hold for the local von Neumann algebras of a net satisfying the Haag–Kastler axioms with a vacuum state, under a weak additivity assumption.

Going deeper

For a student

The theorem is a central structural result of axiomatic and algebraic quantum field theory. The separating property means no local operator can annihilate the vacuum, so there are no strictly local projections onto "a particle here". In the algebraic setting the cyclic and separating vector makes Tomita–Takesaki modular theory available, and the Bisognano–Wichmann theorem identifies the modular group of a wedge algebra with Lorentz boosts. The theorem is also used in the proofs of the spin–statistics theorem and of results on entanglement entropy in field theory.