Could physics still be local? Here's what I mean:
The Schrodinger/Dirac equations allow for quantum entanglement, right? So in that sense they are non-local physically. But they are mathematically local in the sense of being point-wise partial differential equations. So if you imagine that the QM wavefunction corresponds to a true field, as yet to be identified, or at least a close approximation to it, and particles are due to a true collapse of that field, whatever "collapse" means in reality, then you could imagine that physics might still be local.
Whereas, if the QFT wavefunction has as its domain, instead of spacetime, the Hilbert space of generally delocalized field states, then what does that say about locality? Does that mean that, even in principle, there is no way to write the laws of physics in a point-wise mathematical form with respect to spacetime? And in that sense, is it the case that the non-locality of QFT is much more unavoidable, more essential than that of regular QM?
I saw there's also a topic called non-local Lagrangians, but I'm not trying to go there...I just want to understand the inherent local/non-local character of QFT in general.