Reeh-Schlieder theorem

The Reeh–Schlieder theorem is a result in relativistic local quantum field theory published by Helmut Reeh and Siegfried Schlieder (1918-2003) in 1961.

The theorem states that the vacuum state ${\displaystyle \vert \Omega \rangle }$ is a cyclic vector for the field algebra ${\displaystyle {\mathcal {A}}({\mathcal {O}})}$ corresponding any open set ${\displaystyle {\mathcal {O}}}$ in Minkowski space. That is, any state ${\displaystyle \vert \psi \rangle }$ can be approximated to arbitrary precision by acting on the vacuum with an operator selected from the local algebra, even for ${\displaystyle \vert \psi \rangle }$ that contain excitations arbitrarily far away in space. In this sense, states created by applying elements of the local algebra to the vacuum state are not localized to the region ${\displaystyle {\mathcal {O}}}$.

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