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JessRiedel comments on The Quantum Arena - Less Wrong
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Psy-Kosh: In Quantum Field Theory, the fields (the analog of wavefunctions in non-relativistic Quantum Mechanics) evolve locally on the spacetime. This is given a precise, observer-independant (i.e. covariant) meaning. This property reduces to the spatially-local evolution of the wavefunction in QM which Eliezer is describing. Further, this indeed identifies position-space as "special", compared to momentum-space or any other decomposition of the Hilbert space.
Eliezer: The wavefunctions in QM (and the fields in QFT) evolve locally under normal (Hermitian) evolution. However, Bell-type experiments show that wavefunction collapse is a non-local process (be it the preposterous Copenhagen-style collapse, or some flavor of decoherence). As far as I have read, the source of this non-locality is not understood.
I'm pretty sure Many Worlds doesn't have waveform collapse. Also, I don't think they're talking about configuration space. They're saying that particle a being in point A and particle b being in point B interacting is non-local. That configuration is one point, so it's completely local.