Cherenkov radiation of superluminal particles

Any charged particle moving faster than light through a medium emits Cherenkov radiation. We show that charged particles moving faster than light through the vacuum emit Cherenkov radiation. How can a particle move faster than light? The weak speed of a charged particle can exceed the speed of light. By definition, the weak velocity $〈\mathbf{v}{〉}_w$ is $〈{\Psi }_{\mathrm{fin}}|\mathbf{v}|{\Psi }_{\mathrm{in}}〉/〈{\Psi }_{\mathrm{fin}}|{\Psi }_{\mathrm{in}}〉,$ where $\mathbf{v}$ is the velocity operator and $|{\Psi }_{\mathrm{in}}〉$ and $|{\Psi }_{\mathrm{fin}}〉$ are, respectively, the states of a particle before and after a velocity measurement. We discuss the consistency of weak values and show that superluminal weak speed is consistent with relativistic causality.