separating vector

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**field theory**: classical, pre-quantum, quantum, perturbative quantum

Let $\mathcal{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$.

A vector $x \in \mathcal{H}$ is a **separating vector** if $M(x) = 0$ implies $M = 0$ for all $M \in \mathcal{M}$.

The notions of separating vector is dual to that of cyclic vector with respect to the commutant $\mathcal{M}'$, that is a vector is cyclic for $\mathcal{M}$ iff it is separating for $\mathcal{M}'$.

In the context of AQFT separating vectors appear as *vacuum states* . See Reeh-Schlieder theorem.

Revised on November 30, 2010 11:43:54
by Urs Schreiber
(131.211.232.96)