I guess the following obvious question is answered by any flavor of relativistic Quantum Mechanics, but I just wanted to check whether I understand correctly:
Is it correct that nonrelativistic QM violates locality (allows "statistical superluminal communication") in the following way:
Let Alice and Bob be far away from each other (and at relative rest). Assume we have a particle determined at $t=0$ to be in a "small" region around Alice (and therefore with quite undetermined momentum, but not so undetermined that it is possible to reach Bob in a "very small time"). Alice and Bob agreed that Alice would at $t=0$ measure the momentum with "extremely high precision" iff she wants to send Bob a signal. (This would make the position very undetermined, and thus make it possible for the particle to be at Bob's position). At $t=0$ (or a "very small time afterwards") Bob tries to find the particle at his position. In the unlikely event that he succeeds, he knows that Alice must have tried to send the signal. (If he does not find it, he doesn't know anything.)
A weak point of this example might be that it is probably (?) not possible to have wave function with compact support in position space ("close to Alice") as well as in momentum space (not able to reach Bob "instantaneously") if you look at the Fourier transform. However, if you look at the Schrödinger Equation, it seems to be that case that a free particle cannot "instantaneously" enter a region separated from the support of the wave function (position space) at a given time? I have to admit that this confuses me and I cannot come up with reasonable examples (the Gauss curve being the only normalized example for a free particle I have seen thus far, which obviously does not have a compact support). But I would be surprised if the non-locality effect above would depend on such technical issues?