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 $e^{i\mathbf{k}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })}$. A diagonal single-momentum representation is justified only with additional spatial translation symmetry or an explicitly fixed source origin.