Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student.

I have an integral/summation as a product of a retarded and advanced Green's functions, looks simply like $$\sum_{p^{\prime}}\frac{1}{p^{2}-{p^{\prime}}^{2}-i\epsilon}\frac{1}{p^{2}-{p^{\prime}}^{2}+i\epsilon}$$ where I have omitted the mass $m$ to make the notation simple; $\epsilon$ is a positive infinitesimal.

I can convert the summation to an integral over momentum $p$ or energy, this is standard. Then I follow the complex analysis and identify 4 poles (if the integration is over momentum) and further apply the residue theorem. But, what I get is an expression proportional to $\frac{1}{\epsilon}$. This means it is divergent since $\epsilon\rightarrow 0$.

Could any one please point out to me what have I missed in this calculation? What mistakes did I make? Many many thanks.

  • 1
    $\begingroup$ It would help if you could present the 'calculations' you refer to. $\endgroup$
    – JamalS
    Aug 29, 2014 at 18:35
  • $\begingroup$ This is actually the calculation I have to do. A product of a retarded and an advanced Green's functions (or propagator), for which I have to sum over the $p^{\prime}$. $\endgroup$
    – Haag Neder
    Aug 29, 2014 at 19:25
  • $\begingroup$ Related post. $\endgroup$ Jul 6, 2017 at 1:43

1 Answer 1


In general one can write

\begin{equation} G^R_{k_1}(z)\; G^A_{k_2}(z)=\frac{G^R_{k_1}(z)-G^A_{k_2}(z)}{\varepsilon_{k_1}-\varepsilon_{k_2}}, \end{equation}

This way you can perform the integral easily. Take a look at Economou's book on Green's functions...


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