deletions | additions       diff --git a/untitled.tex b/untitled.tex  index f212764..5c74137 100644  --- a/untitled.tex  +++ b/untitled.tex 
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&M^>(\omega)=\frac{4 e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{ \Gamma_{\beta\gamma} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) (1-f_{e}(\epsilon)] \\  & [-i\Gamma_{\beta\alpha}] G^{r}_{\alpha\delta}(\epsilon) \Gamma_{\delta\gamma} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))(1-f_h(\epsilon)) +   \\   &[\sum_{\theta\tau} [-i\Gamma_{\beta\alpha}] G^{r}_{\alpha\theta}(\omega+\epsilon)\Gamma_{\theta\tau}G^{a}_{\tau\delta}(\epsilon) G^{r}_{\alpha\theta}(\epsilon)\Gamma_{\theta\tau}G^{a}_{\tau\delta}(\epsilon)  [i\Gamma_{\delta\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G ^{a}_{\mu\beta}(\omega+\epsilon)] \\  &(f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) (1-f_{e}(\epsilon)+1-f_h(\epsilon)) +   [\Gamma_{\beta\alpha}(\epsilon) G^{a}_{\alpha\delta}(\epsilon) [i\Gamma_{\delta\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G ^{a}_{\mu\beta}(\omega+\epsilon)]  \\  & (f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)(1-f_h(\epsilon))) \Biggr\}  \end{align*}  Again, the "lesser" term for $M(t,t')$ is obtained by exchanging $1-f$ by $f$ and viceversa. The whole contribution for $M(t,t')$ becomes  \begin{align*}  &M(\omega)=\frac{4 e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{ \Gamma_{\beta\gamma} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](F_{ee}+F_{eh})] \\  & [-i\Gamma_{\beta\alpha}] G^{r}_{\alpha\delta}(\epsilon) \Gamma_{\delta\gamma} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](F_{eh}+F_{hh}) +   \\   &[\sum_{\theta\tau} [-i\Gamma_{\beta\alpha}] G^{r}_{\alpha\theta}(\epsilon)\Gamma_{\theta\tau}G^{a}_{\tau\delta}(\epsilon) [i\Gamma_{\delta\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G ^{a}_{\mu\beta}(\omega+\epsilon)(F_{ee}+F_{hh})+(F_{eh}+F_{he})]  \\  & + [\Gamma_{\beta\alpha}(\epsilon) G^{a}_{\alpha\delta}(\epsilon) [i\Gamma_{\delta\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G ^{a}_{\mu\beta}(\omega+\epsilon)] (F_{eh}+F_{ee}) \Biggr\}  \end{align*}  The last term that we need to compute is $Q>(t,t')+Q<(t,t')= G^{h,>}_{k\gamma}(t,t')G^{<}_{q \beta}(t',t)+ G^{h,<}_{k\gamma}(t,t')G^{>}_{q \beta}(t',t)$. We only calculate $Q^>(t,t')$. For such calculation we employ  \begin{eqnarray}  G^{h,>}_{k\gamma}(\epsilon) = \sum_\alpha [g^{r,h}_{k}(\epsilon) V_{\alpha k} G_{\alpha\gamma}^>(\epsilon)+g^{>,h}_{k}(\epsilon) V_{\alpha k} G_{\alpha\gamma}^a(\epsilon)]