Self-energy for two correlators

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SUMMARY

The discussion centers on the calculation of the interacting single particle Green function, represented as \( G(t-t') = \langle T d(t) d^{\dagger}(t') \rangle \), and the correlator \( g(t-t') = \langle T d^{\dagger}(t) d(t') \rangle \). The key inquiry is whether the self-energy \( \Sigma \) remains unchanged when calculating the correlator \( g(t-t') \) and if the Dyson equation applies to this correlator. The consensus indicates that while the self-energy may not change, the application of the Dyson equation to \( g(t-t') \) is valid within the context of perturbation theory.

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gonadas91
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Imagine I want to calculate the INTERACTING single particle Green function in a model:

\begin{eqnarray}
G(t-t')=\langle Td(t)d^{\dagger}(t')\rangle
\end{eqnarray}

We know from Dyson equation that this is:\\
\begin{eqnarray}
G=G_{0} + G_{0}\Sigma G
\end{eqnarray}
Where $\Sigma$ is the self-energy. My question is, if we want to calculate the correlator :\\
\begin{eqnarray}
g(t-t')=\langle Td(t)^{\dagger}d(t')\rangle
\end{eqnarray}
Does the self-energy $\Sigma$ change for this correlator? In my intuition it shouldnt, but maybe I am wrong... Thanks!
 
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Oh and by the way I forgot, does Dyson equation apply to this correlator as well ($g(t-t')$)? (Even not being the definition of a Green function, can this infinite series be hold in a correlator like that when we do perturbation theory? )
 

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