= Causality versus stability of a Green function
With inverse <Fourier transform> $G(t)=\int G(\omega)e^{-i\omega t}d\omega/(2\pi)$, a tempered retarded response vanishes for $t<0$ and is analytic in the upper frequency half-plane with suitable large-frequency growth bounds. Poles strictly below the real axis describe decaying modes; upper-half-plane poles obstruct a bounded retarded real-axis Fourier response. However causality alone does not imply stability: a retarded exponentially growing Green function can exist without an ordinary Fourier transform. Its <Laplace transform> inversion contour must pass above the growth poles in the frequency plane. Moving the contour to the real axis changes the boundary condition and can introduce a pre-response.
Back to article page
