Is self-energy $\mathrm{Im}\Sigma^r<0$ always true?

Question

Consider a one-particle retarded Green's function $$G^r(\alpha)=[\omega+i\eta-\varepsilon(\alpha)-\Sigma^r(\alpha)]^{-1}$$ with self-energy $\Sigma^r(\alpha)$ for some quantum number $\alpha$. It is argued that $-\mathrm{Im}\Sigma^r>0$ always holds as it signifies the quasiparticle lifetime. Is this true always?

Answer 1

Let us consider the following:

• $\Sigma^r$ is a retarded self-energy, i.e., it is given by expressions like $$\Sigma^r(\omega)=\sum_k |V_k|^2G^r(\omega).$$ Here $V_k$ might be replaced by vertex parts with more complex structure, but the presence of the retarded Green's function, $\Im \left[G^r\right]<0$ suggests that $\Im [\Sigma^r]<0$.
• $G^r$ is a retarded Green's function - it would not be the case, if we had $\Im \left[\Sigma^r\right]>0$.

I cannot say, if there is a rigorous mathematical proof that $\Im[\Sigma^r]\leq 0$, for any type of dispersion laws and interactions. However, if this relation breaks, we are facing the breakdown of the causality in our calculations, which means that some of the premises of our derivations are wrong - the initial Hamiltonian requires revision (keeping in mind that sometimes the breakdowns of theory are meaningful - e.g., many quantities diverge near phase transitions).