= Fourier Green function in a spatially varying background
A static background permits a <Fourier transform> of time difference, but a spatially varying coefficient generally prevents dependence on position difference alone. A momentum-space <Green function> then carries an incoming and an outgoing <momentum>, or equivalently one <momentum> together with the source position. Multiplication by a background field becomes a convolution, with a source-position phase when the Fourier basis is written as $e^{i\mathbf k\cdot(\mathbf r-\mathbf r')}$. A diagonal single-momentum representation is justified only with additional spatial <translation> symmetry or an explicitly fixed source origin.
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