Lorentz invariance of positive energy solutions to the Klein-Gordon equation I am reading Arthur Jaffe's Introduction to Quantum Field Theory. (You can find it here.) There is an interesting question posed in Exercise 2.5.1:

Solutions to the Klein-Gordon equation propagate with finite speed. But $f^t$ instantly spreads from its localization at the origin [...] Does this fact not contradict the laws of special relativity?

$f^t$ is defined in equation (2.30) and it is explicitly a solution to the KG equation, see (2.32).  
Is the correct answer that $f^t$ is not a propagating solution? I.e., the group velocity isn't well-defined so nothing can be said to propagate/violate finite speed information flow? Or is there something more subtle here?
 A: Please, if this is a homework from Arthur to you and if you will use any information in the answer below, indicate it clearly in your answer (and send my best regards to Arthur).
Your comments about the group velocity have nothing to do with the right resolution to Arthur's "would-be violation of relativity". The right solution is that ${\mathfrak f}(\vec x,t)$ solves the first-order (in time) Schrödinger equation (2.5). As sketched in (2.34) etc., the solution immediately becomes nonzero at time $t+dt$ everywhere in space although ${\mathfrak f}=0$ were true everywhere except for a small region at time $t$.
However, because (2.5) is a first-order equation that is nonlocal in space, it follows that even $\partial {\mathfrak f}/\partial t$ is nonzero (almost) everywhere in space at time $t+dt$.  This first time derivative is proportional to the action of the nonlocal Hamiltonian $H$ on the Gothic $f$.
This fact prevents us from saying that the original "impulse" was confined to the small region. To claim so in the context of solutions to the actual second-order (in time) Klein-Gordon equation, we need all the initial conditions i.e. both ${\mathfrak f}$ and $\partial {\mathfrak f}/\partial t$ (canonical coordinates and canonical momenta) to vanish everywhere outside the small region. The latter doesn't vanish in the initial time $t$ so it's not true that the initial "impulse" was confined to the small region, and it's therefore allowed for the responses to appear everywhere in space at $t+dt$, too.
If one could construct any solutions that vanish and whose first time derivative vanish everywhere outside a small region at time $t$, but that are nonzero arbitrarily far at the time $t+dt$, it would prove that the theory violates relativity, regardless of any additional excuses about ill-defined group velocities or anything else.
