What is the difference "between action at a distance" and "field theory"?

Answer 1

Take Newtonian and Einsteinian gravitation theories. Newton proposed that bodies exterted their pulls through "action at a distance", although he never explains how this occurs, and doesn't suggest that the space between them plays any role in this action. Einstein, on the other hand, rejected this view, and instead proposed that it's the local curvature of spacetime that is responsible for the effects of gravitation.

Likewise, the Schrodinger wave equation is the Hamiltonian form of quantum mechanics, and once again only potentials and forces "at a distance" are presumed to be actors in any quantum interaction. Quantum field theory, which is the Lagrangian form developed by physicists such as Feynman decades after Schrodinger, proposes that space is never empty, and is in fact roiling with virtual particles and virtual interactions. It is through such a virtual world that things really do not "act at a distance", but are carried by virtual intermediaries.

Field theories offer far richer milieus than simpler "action at a distance" theories, spawning a diverse array of theories such as string theory and loop quantum gravity. They also provide a more satisfying alternative than the overly simplistic belief that a vacuum is a case of pure nothingness, by suggesting that perhaps it's meaningless to speak of a perfect nothingness. No matter how much something appears to be nothing, it may be far from "nothing".

Addendum: Actually, jim m's answer was not a bad one, pointing out that earlier "action at a distance" theories relied on instantaneous transmission of influences, a problem even Newton acknowledged. Yet, we do know now that quantum attributes can be transmitted "non-locally".

Answer 2

Action at a distance is a holdover from Newtonian physics, where massive bodies instantaneously exert 1/r^2 forces on each other. However Einstein''s relativity says that fields are "local" in the sense that quanta of momentum are exchanged between bodies, travelling at the speed of light, i.e., not instantaneously and hence not action at a distance.
The Einstein-Rosen-Podolsky paradox and J.S Bell's theorem have revived the action at a distance because measurement of the quantum state of either partner of certain correlated quanta automatically determines the state of the partner quanta instantaneously on the opposite side of the disturbance.