Yes, I am assuming a "retarded Green's function" solution. But the "zero at infinity" boundary is only relevant after an infinite amount of time, i.e. is not physically relevant.
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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Take a charge. Not a theoretical one, take a real one. Now accelerate it. It will emit radiation that can be measured. And what happens at a distance L (e.g. the walls of my lab) only influence it after a time of order L/c. The walls of my lab matter, but not immediately.
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As an empirical matter, a uniformly accelerated charge does not radiate. Dirac calculates the radiation field in the vicinity of an accelerating point electron, and finds that it vanishes when the acceleration is constant. The question is: why? That's where this discussion began.
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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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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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