deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 8c8ae44..3280181 100644  --- a/untitled.tex  +++ b/untitled.tex 
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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  \begin{eqnarray}  M^>(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k}^{*}V_{\gamma q} [G^{h,>}_{kq}(t,t') G{<}_{\gamma\beta}(t',t) G^{<}_{\gamma\beta}(t',t)  \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^{<}_{\gamma\beta}(\epsilon)  \end{eqnarray}  We replace now  \begin{eqnarray}  G_{kq}^{h,>}(\omega+\epsilon) &&G_{kq}^{h,>}(\omega+\epsilon)  = \sum_{\beta\gamma} \sum_{\alpha\delta}  [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma V_{\alpha  k} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{r}_{\alpha\delta}(\omega+\epsilon)V_{\delta  q}^* g_{q}^{h,>}(\omega) +g_{k}^{h,r}(\omega+\epsilon) V_{\gamma V_{\alpha  k} G^{>}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{>}_{\alpha\delta}(\omega+\epsilon)V_{\delta  q}^* g_{q}^{h,a}(\omega) g_{q}^{h,a}(\omega+\epsilon)]  \\  \nonumber  &&g_{k}^{h,>}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega) ]\,,  \end{eqnarray}  Then we get  \begin{eqnarray}  &&M^>(\omega)=\frac{2 i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta,\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{  V_{\beta k}^{*} g_{k}^{h,r}(\omega+\epsilon) V_{\alpha k} G^{r}_{\alpha\delta}(\omega+\epsilon) V_{\delta q}^* g_{q}^{h,>}(\omega+\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\epsilon)(f_e(\epsilon)+f_h(\epsilon))+ \\ \nonumber  &&  V_{\beta k}^{*} g_{k}^{h,r}(\omega+\epsilon) V_{\alpha k} G^{>}_{\alpha\delta}(\omega+\epsilon) V_{\delta q}^* g_{q}^{h,<}(\omega+\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\epsilon)(f_e(\epsilon)+f_h(\epsilon))\\ \nonumber  &&  V_{\beta k}^{*} g_{k}^{h,>}(\omega+\epsilon) V_{\alpha k} G^{a}_{\alpha\delta}(\omega+\epsilon) V_{\delta q}^* g_{q}^{h,a}(\omega+\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\epsilon) (f_e(\epsilon)+f_h(\epsilon))\Biggr\}  \end{eqnarray}  We now use the explicit expressions for the self-energies  \begin{eqnarray}  &&M^>(\omega)=\frac{2i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{ [\Sigma^{r,h}_{0,\beta\alpha}(\omega+\epsilon) G^{r}_{\alpha\delta}(\omega+\epsilon) \Sigma^{h,>}_{\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon)) +   \\   \nonumber  &&[\Sigma^{r,h}_{0,\beta\alpha}(\omega+\epsilon) G^{>}_{\alpha\delta}(\omega+\epsilon) \Sigma^{h,a}_{\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon))+   \\   \nonumber  && [\Sigma^{>,h}_{0,\beta\alpha}(\omega+\epsilon) G^{a}_{\alpha\delta}(\omega+\epsilon) \Sigma^{h,a}_{\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon)) \Biggr\}  \end{eqnarray}  Then we obtain,