6
$\begingroup$

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.

$\endgroup$
3
  • 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

1
$\begingroup$

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...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.