deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 7358647..f606251 100644  --- a/untitled.tex  +++ b/untitled.tex 
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\\ \nonumber  &&[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta k}]  \\ \nonumber  &&[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon) g^{a,h}_{k}(\omega+\epsilon)  V_{\beta k}] \end{eqnarray}  Inserting the expressions for the self-energies we get  \begin{eqnarray} 
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&&N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{  [G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\epsilon+\omega) \Gamma_{\mu\beta} (1-f_e(\epsilon) f_{h}(\epsilon+\omega)] \\ \nonumber  &&\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^r(\omega+\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon+\omega)[i\Gamma_{\mu\beta}](1-f_e(\epsilon)(f_{h}(\epsilon+\omega)+f_e(\epsilon+\omega))] \\ \nonumber  &&\sum_{\lambda\delta} G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega+\omega)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\ G_{\gamma\mu}^r(\omega+\epsilon)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\  \nonumber &&\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega+\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\epsilon)+1-f_{h}(\omega+\epsilon))(f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega))] G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\epsilon)+1-f_{h}(\epsilon))(f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega))]  \Biggr\} \end{eqnarray}  Finally, to obtain $N<(t,t)$ we just change $(1-f)\rightarrow f$, and $f\rightarrow (1-f)$.   The next term is $M(t,t') = M^>(t,t´) + M^<(t,t´)$ with 
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\end{eqnarray}  Then,  \begin{eqnarray}  M^>(\omega)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k}^{*}V_{\gamma q}\int \frac{d\epsilon}{2\pi} [G^{h,>}_{kq}(\omega+\epsilon) G^{<}_{\gamma\beta}(\epsilon) [G^{h,>}_{kq}(\epsilon) G^{<}_{\gamma\beta}(\omega+\epsilon)  \end{eqnarray}  We replace now  \begin{eqnarray}