In quantum field theory, the propagator $D(x-y)$ doesn't vanish for space-like separation. In Zee's book, he claims that this means a particle can leak out of the light-cone. Feynman also gives this interpretation.
What is wrong, then, with this thought experiment:
Bob and Alice synchronize their watches and space-like separate themselves by some distance. Bob tells Alice that at exactly 3 o'clock, he intends to test his new particle oven, which claims to make a gazillion particles. Since Bob's oven makes so many particles, there is a pretty good chance that Alice will detect a particle right at 3 o'clock. If she detects such a particle, Bob will have transmitted a piece of information (whether or not the oven works) instantaneously - way faster than the speed of light.
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