= Causal support and contour conditions
A <retarded Green function> obeys $G(\tau)=0$ for $\tau<0$ and
$$
(\partial_\tau^2+1-\varepsilon\partial_\tau^3)G(\tau)=\delta(\tau).
$$
For a bounded or suitably tempered response represented by the real-axis Fourier integral, analyticity in the upper <frequency> half-plane is required, with suitable behaviour at infinity. For negative $\tau$, $e^{-i\omega\tau}$ decays on a large upper semicircle. If that half-plane has no singularities, contour closure gives zero response before the source. Lower-half-plane poles give factors decaying for positive time. Real poles are approached from below for a retarded prescription; at $\varepsilon=0$, for example, $G_R(\tau)=\Theta(\tau)\sin\tau$.
The <causality versus stability of a Green function> distinction matters: an unstable initial-value problem can still have a strictly causal response that grows exponentially. Such a response has no ordinary real-axis <Fourier transform>. Its inverse <Laplace transform> must run on a <frequency> contour above its growth poles. Thus “upper poles violate causality” is correct only with the stable/tempered real-axis Fourier assumptions, not as an unconditional theorem about differential equations.
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