Suppose I have a system that is invariant under both space and time translations. The Lehmann representation of the propagator says that the poles of the propagator are the exact eigenenergies of the N+1 particles system and they all have infinitesimal imaginary part. Therefore the poles are:
$\omega= \omega_a^{N+1} (k) - i \eta$.
On the other hand when defining quasiparticles one uses the poles of the propagator:
$G(k, \omega) = \frac{1}{\omega - \omega_k^{unpert} - \Sigma(k, \omega)} $, which in general has poles $\omega^{N+1} \left( k \right) = \omega_{real} ^{N+1} (k) + i \omega_{imm} ^{N+1} (k) $.
But aren't the poles in the two representations the same? They must be because it's the same object. This means that these poles must have infinitesimal imaginary part: $\omega_{imm} ^{N+1} (k) = - \eta $ and therefore quasiparticles have infinite lifetime. What am I missing?
