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**Can someone explain to me "Locality" in field theory?**

I know that two fields have to commute at space-like separations so that observations don't affect each other at these separation.

what about the thing where the Lagrangian can't couple fields at different locations and time, is it so that we get a normal (local) differential equation and not one that involves the field at different coordinates.

any other information would be great,, Thanks.