Like all my questions, I fear this will be very naive, because my physics background is very limited. Please bear with me.
I think of the electromagnetic field as a section of a vector bundle over spacetime, but I think nothing will be lost if we just treat it as a function $f:{\mathbb R}^4\rightarrow{\mathbb R}^6$.
Now suppose I want to send you a wireless message (by radio, cellphone, TV, whatever). I encode my message by jigglinig some electrons around so as to change the electromagnetic field from $f$ to a new function $f+g$. You are able to observe some values of this function $f+g$, or at least to infer something about some values of that function from the way the electrons in your receiver are jiggling around.
So at the very most, you are able to observe the function $f+g$. How on earth, then, can you infer anything about the function $g$?
I understand that you can decompose the field $f+g$ into Fourier components, but given that the ambient field $f$ can be anything at all, I don't see how that helps. The ambient field $f$ is affected by all sorts of things, from other people's communications to cosmic rays, to the fact that I happen to have just carried a 9-volt battery across my living room, much of which you have no knowledge of. Now I add a function $g$ to this completely unknown function $f$, and you're supposed to recover $g$ by observing the sum.
Clearly, this can't work. Clearly it goes ahead and works anyway. What am I missing?
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