Do lightwaves exist? and what exactly is light?

[Dale]

Consider a point source that starts to radiate at the time t=0. Clearly you cannot decompose this wave into plane waves, because plane waves have the full amplitude at t=0 as far from the point source as we wish.

This does not follow. The superposition of the waves can sum to 0 even though the individual basis functions are generally non-zero.

 

[htg]

This does not follow. The superposition of the waves can sum to 0 even though the individual basis functions are generally non-zero.

We are talking about an L^2 (R^n) space here, and either about an orthonormal basis or about Fourier transform.
When you take a linear combination of basis elements, the L^2 norm of this combination is equal to SQRT(sum of squares of moduli of coefficients), so if at least one coefficient is non-zero, you get a non-zero linear combination.
Probably what you meant is that a non-trivial linear combination of basis functions can be zero on some proper subset of R^n.
And you still confuse waves and basis functions - IT IS NOT THE SAME.

 

[htg]

Why? I don't think so...

Consider the Fourier transform of restriction of sin(w*t) to an interval [a,b].
Ther Fourier transform decomposes this time-limited signal into a continuum of complex exponential functions defined on the whole real line. So if we want to say that they are signals, we must be able to detect them before our time-restricted signal began to be non-zero. Obviously it would violate causality.

 

[Dale]

Probably what you meant is that a non-trivial linear combination of basis functions can be zero on some proper subset of R^n.

Yes, I understood that to be your objection. That the basis functions were non-zero in a region far away from the origin at t=0 despite the fact that the function was 0 in that region.

 

[htg]

Fourier transform (as distinguished from Fourier series) does not decompose into basis functions. L^2 (R^n) is a countably dimensional space and has a countable basis.

 

[Dale]

So according to you we should not call them waves and we should not call them basis functions. What terminology would you prefer? I am out of good words.

If you don't want to decompose your EM functions then by all means don't do it. Nobody is forcing you to do so. But nothing you have said diminishes the utility of doing it. There is no requirement that basis functions be compact, nor that they be constrained to a specific frequency or wavelength, nor even that they be solutions to the same problem. And in this case they are mathematically plane waves.

 

[htg]

"components of the Fourier transform decomposition" is the proper term.