I think an important thing to consider with this change of basis is that fourier modes are the eigenvectors of translation. As such any linear operation which commutes with translation will also have fourier modes as eigenvectors. As long as the laws of physics are expressed in such a way that they do not work differently in different places, they will treat fourier modes independently.
I think an important thing to consider with this change of basis is that fourier modes are the eigenvectors of translation. As such any linear operation which commutes with translation will also have fourier modes as eigenvectors. As long as the laws of physics are expressed in such a way that they do not work differently in different places, they will treat fourier modes independently.