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 〈v〉w is 〈Ψfin|v|Ψᵢₙ〉/〈Ψfin|Ψᵢₙ〉, where v is the velocity operator and |Ψᵢₙ〉 and |Ψfin〉 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.