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What was wrong with action a distance?
It is usually said that the idea of fields was introduced (electric and magnetic fields) in electricity and magnetism after Coulomb's law to cure the conceptual problems of action at a distance.
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How can one know if a theory allow action at a distance effects or not?
1-In general, if a theory has action at a distance effects, where can that appear exactly in the theory?
2-Does it appear in the dynamical law of the theory? (does it appear in Newton's 2nd law? ...