 Hegerfeldt theorem

Let $H$ be a self-adjoint Hamiltonian bounded below, $A$ a bounded positive operator and $p_A(t)=⟨e^{-iHt}\psi ,A{e}^{-iHt}\psi ⟩$. The Hegerfeldt theorem states that either $p_A(t)$ vanishes identically or it is nonzero for almost every real time. In particular, vanishing throughout an open interval forces identically vanishing. The proof uses analytic continuation of $A^{1/2}{e}^{-iHt}\psi$ into the lower half-plane. Applied to an assumed sharp localization effect, this obstructs strict finite-speed localization for positive-energy one-particle dynamics. It does not identify a controllable superluminal signal: the localization observables and admissible local preparation procedures need separate analysis.