Green's functions: a "smooth" source cannot create high-frequency waves

We have been claiming that destructive interference wipes out high-frequency waves in natural phenomena. We can now present a more sophisticated argument for our claim.

Let us have a field. The field can be a drum skin, the Dirac field, or whatever. Let the wave equation for the undisturbed field be a homogeneous linear wave equation

       W(ψ(t, x)) = 0.

If the field is disturbed, we can calculate the approximate response of the field with the same linear wave equation, except that now we have a source I(t, x):

       W(ψ(t, x)) = I(t, x).

The disturbance to the field is typically an interaction with another field in which we have a wave packet. The letter "I" stands for interaction.

Let us assume that the Fourier transform of the source I(t, x) contains no high frequencies. The source I(t, x) is a smooth wave packet.

Since the wave equation is linear, we can build the response of the field by summing the responses to individual components

        S(t, x) = exp(-i (-E t + p x))

of the Fourier transform of the source I(t, x).

We build the response from a Green's function of the wave equation for our field. A Green's function is the response of the field to a Dirac delta source. For example, we can hit a drum skin with a sharp hammer. That constitutes a Dirac delta source in the drum skin wave equation.

The Fourier transform of the Green's function typically contains arbitrarily high frequencies, because it is the response to a very "sharp" impulse.

Let us then look at the response to the source S(t, x). What happens to high frequency components of the Green's function? They are totally wiped out by destructive interference. The source S(t, x) spans the whole spacetime. The only surviving component of the Green's function is the one which "syncs" with the source S(t, x).

Note that when an electron "absorbs" a photon, it is just this syncing mechanism behind the process.

We have found the reason why in typical natural phenomena, low-frequency waves cannot give birth to high-frequency waves.

Are there processes where low frequencies can produce high frequencies? Yes. A gearbox can turn slow oscillation into fast oscillation. Such gearboxes do not exist in typical natural phenomena.