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Let's talk about two kinds of nonlocal theories. The first one frequently derives from integrating out part of the degrees of freedom to obtain a kind of effective theory. Probably, we get an integral potential or something more complex. The second one is the case where we have operators like $\sqrt{-\hbar^2c^2\nabla^2+m^2c^2}$ acting on fields.

In some sense, people say they are problematic, at least not fundamental. I wonder if one issue is violation of causality per se. And as far as I know, some researchers also distinguish causality and locality, e.g. E.P.R. paradox.

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