About locality in classical physics

[karlzr]

In wikipedia it says that "Coulomb's law of electric forces was initially also formulated as instantaneous action at a distance, but was later superseded by Maxwell's Equations of electromagnetism which obey locality."

I don't really understand in which sense Maxwell's equations solve the problem of locality? Is it the invention of vector potential A, which mediates the action of charges at a distance?
I am interpreting locality in this way:
what locality requires is that when we write an equation of motion, all physical quantities present should be evaluated at the same position, such as Maxwell's equations, Schrödinger's equation, Einstein's equations, etc. But locality doesn't say anything about the solution. Take the Maxwell's equation in the presence of external currents as an example, the solution might include integration over all space.
Thanks,

 

[mfb]

Is it the invention of vector potential A, which mediates the action of charges at a distance?

No, the key point is the final propagation speed of changes in this field. With Coulomb's law, changing a charge distribution directly changes the potential everywhere. With the Maxwell equation (and the electromagnetic vector potential), this does not happen any more.

Take the Maxwell's equation in the presence of external currents as an example, the solution might include integration over all space.

Only in the static case, and then signal distribution speed does not matter.