= Hegerfeldt theorem
{c}
Let $H$ be a self-adjoint Hamiltonian bounded below, $A$ a bounded positive operator and $p_A(t)=\langle e^{-iHt}\psi,Ae^{-iHt}\psi\rangle$. 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.
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