Does Lorentz-Invariance Imply Causality?

We investigate whether faster-than-light propagation of particles and wave motions is consistent with Lorentz invariance of the $S$ matrix in the general context of relativistic, off-shell scattering theories. We do three things. First, we show that Lorentz invariance of the Green's functions implies that of certain weight functions; in a field theory, these weight functions are the Wightman functions. Secondly, we show that any off-shell scattering theory with Lorentz-invariant Green's functions arises from a field theory with causal equal-time commutators. Thirdly, we show that in particle theories with normal connectedness structure, the equal-time commutators are necessarily canonical. These results show that there are great difficulties facing theories of faster-than-light motion which aspire to relativistic invariance.