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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.

Could someone explain what are the conceptual and physical difficulties or contradictions that one might have with action at a distance?

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Could you be more specific, this seems to be a fairly basic conceptual problem so it isn't clear what you are asking. – user11547 Feb 2 '13 at 2:02
@HalSwyers What I am asking is sharply stated – Revo Feb 2 '13 at 2:11
This is very similar to – joshphysics Feb 2 '13 at 2:35
You're right replace "very similar to" with "somewhat tangentially related to but may nonetheless help clarify aspects of action at a distance" – joshphysics Feb 2 '13 at 3:10
It is precisely the model of the mechanism of mediation that illuminates precisely what is happening with action at a distance, so I think your critcism is misplaced. – joshphysics Feb 2 '13 at 3:18

In Newtonian physics, there was no problem with action at a distance, and indeed Newton explicitly formulated his theory of gravitation in such terms. It may be that this was criticised from a philosophical standpoint (I don't know whether it was or not), but there were no fundamental mathematical difficulties with the idea.

However, in relativity the picture changes quite dramatically. The problem is this: action at a distance means that one object is able to influence another instantaneously, but in relativity this idea doesn't really make sense, as I will now explain.

In the Newtonian picture of gravity, if we have two massive objects separated by some distance, and one of them starts moving (perhaps because some distant object has exerted a force on it via a long cable) then the forces on the other one will change at exactly the same moment in time. But in relativity, what happens at "exactly the same time" is different depending on the velocity of the observer. If you and I are travelling at different speeds, and I observe the two objects' motion to change simultaneously, you might see the second object moving before the first one, which would be very strange indeed.

Because of this, you can't have action at a distance in relativistic space-time. (Well, I guess you can if there's a special preferred reference frame, but this causes conceptual difficulties relating to causality as hinted at above, and it can easily be refuted experimentally.) From this you can conclude that all influences must be local. In general relativity, gravitational influences propagate no faster than the speed of light.

But having said all that, Maxwells equations were developed before relativity rather than the other way around, so historically it isn't true that the idea of a field was developed to resolve this particular difficulty.

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Nothing was wrong. It was just inconvenient and now the preference is to the local theories.

You are not quite correct saying that the fields were introduced to save the locality. At school I was taught that if you consider the system of the particles which interact electromagnetically then the energy of the particles is not conserved. People do not like to drop the idea of energy and momentum conservation, so it was decided to count the field energy and the energy and momentum conservation were saved. The resulting picture was local. Now people often start from the postulate of locality, but it is not a fundamental rule or something. In some sense, the local theories are simpler and more convenient and it is tricky to fit non-local theories into relativistic physics as Nathaniel mentioned. That's all.

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Contrary to a common misconception in physics, there is nothing wrong with action-at-a-distance.

In fact, it is field theory which has well-known difficulties: divergences, violation of causality, inability to deal with general two-body motion...

As shown by Feynman and others, the well-known difficulties of the field theory of electromagnetism are solved by action-at-a-distance:

Classical Electrodynamics in Terms of Direct Interparticle Action

Cosmology and action-at-a-distance electrodynamics

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