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HalFinney comments on Which Basis Is More Fundamental? - Less Wrong
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The problem with locality and the position basis is that the Schrodinger equation doesn't fully enforce locality. With a single particle, it does, but with a multi-particle configuration, conditions near particle 1 can affect the evolution of a configuration that involves particle 2. Somehow these kinds of correlations and influences happen while still not enabling FTL communication, but I don't know of anything in the formalism that clearly enforces this limitation.
FTL communication is not ruled out by the Schrodinger equation, but this is irrelevant because the Schrodinger equation is not valid for systems which include fast-moving particles. Instead, you have to use quantum field theory, of which the Schrodinger equation is the limit as the speed of light approaches infinity. In QFT, FTL communication is indeed ruled out by the formalism, as you suggest. Specifically, it's the commutativity or anticommutativity of field operators based at points which are spacelike separated that does it. For further details I would suggest reading the short paper of Eberhard and Ross. (Unfortunately you need an institutional affiliation to view the link, but I can send a PDF to anyone who wants it.)