Do tachyons travel more slowly than light?

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Abstract
The propagation of solutions of the Klein-Gordon equation with an arbitrary complex mass is investigated. Owing to the strict hyperbolicity of the Klein-Gordon operator, the global Cauchy problem with initial data on a spacelike hyperplane is well posed. In the process of constructing the solution of this Cauchy problem, a Lorentz-invariant retarded Green's function is calculated. Thus the Klein-Gordon causal order relation between pairs of events is invariant under any orthochronous Lorentz transformation and the field propagates no faster than light. In particular, this limitation on the propagation speed is valid for the imaginary-mass Klein-Gordon field. The relation between the causal order and Green's functions is examined. It is shown that only those Green's functions associated with the global Cauchy problem are relevant to the causal order. Finally, it is shown that the group velocity is not physically significant when the dispersion is anomalous.

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