Why should I need to assume anything about the field at a distance greater than c t? (You seem more concerned about the mathematical structure of the theory than about the Physics the theory strives to describe. I am not)
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Replying to @j_bertolotti
I recommend reading the Master: http://bit.ly/2yHQgmk
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Replying to @MathPrinceps
Interesting, but irrelevant to our discussion. In that paper Dirac tries to solve the problem of the field singularity for a point electron within the Maxwell eqs framework, but it never claims (like you do) that classical electrodynamics is nonlocal.
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Replying to @j_bertolotti
Sigh. Did you read the paper? All of its conclusions depend upon choices of Green functions, which in turn depend upon boundary conditions. Physical laws do not determine boundary conditions. Imposing them in general, which you are doing, is adding a non-local law to physics.
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Replying to @MathPrinceps @j_bertolotti
Consider the simpler case of Newtonian gravity. This may be formulated in two ways: via action at a distance (which is non-local), and via a gravitational potential function subject to a Poisson equation (which is local) AND a boundary condition at infinity (which is non-local.)
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Replying to @MathPrinceps @j_bertolotti
If you drop the boundary condition on the Newtonian potential function, then an arbitrary homogeneous solution may be added to it. And this arbitrariness makes the gradient of the Newtonian potential -- i.e., the gravitational "force" -- arbitrary. It can be annulled at a point.
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Replying to @MathPrinceps @j_bertolotti
A purely local formulation of Newtonian gravity theory, then, leads naturally and inevitably to the Equivalence Principle. In such a theory, the notion of gravitational "force" is meaningless. This notion can only be recovered by adding a non-local "law" to the theory.
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Replying to @MathPrinceps @j_bertolotti
The same situation obtains in Maxwell theory. It is a purely local theory, whose laws take the form of partial differential equations. In this theory, no meaningful statements can be made about the "radiation" emitted by a source, until and unless boundary conditions are imposed.
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Replying to @MathPrinceps @j_bertolotti
So no theory of electromagnetic radiation can ever be purely local. In a purely local theory, no constraints are possible on the radiative content of the theory; the radiation field can be whatever you want it to be -- like the gradient of the Newtonian gravitational potential.
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Replying to @MathPrinceps
Ok, I start wondering if we are talking different languages. What do you mean EXACTLY by non-local? Because to me it means essentially "spooky action at distance", i.e. the ability to influence the system from far away without having to wait for the wave to propagate 'till there.
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I certainly do NOT mean non-local in the quantum-mechanical sense, nor even in the relativistic sense, but rather in the classical mathematical sense -- "local" means "completely determined by information discernible by inspecting the immediate vicinity of the point in question."
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Replying to @MathPrinceps @j_bertolotti
It is in this precise sense that the Equivalence Principle is local, and the theory of electromagnetic radiation (and of Unruh radiation) is not.
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