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Is there a physical interpretation of the existence of poles for a Green function? In particular how can we interpret the fact that a pole is purely real or purely imaginary? It's a general question but I would be interested in the interpretation in quantum systems.

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Can you be more specific and maybe give an example from Quantum Mechanics? –  Turion Jun 5 '12 at 8:42
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The pole of Green's function is related to the spectrum of the particle which is propagating. One dimension for example $$\tilde{G}(\omega)= \frac{i}{\omega-(\epsilon+i\Gamma)}$$ If pure real, G(t) is some oscillation function which shows that the particle is stable. If pure imaginary, G(t) has some exponential decay behavior which shows that the particle is unstable.

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For classical linear response the GF can be neither purely real nor purely imaginary on any finite interval because of Kramers-Kronig. I.e., a real GF violates causality except in vacuum. Is this not true in QM? –  user27777 Aug 12 '13 at 15:22
    
The picture is similar. For free case without interaction, GF is real. For intacting case, there is a imaginary part which is related to self energy. Linear response is some kinds of interacting. –  Craig Thone Aug 13 '13 at 3:07
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