deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 0c3bb9c..444c743 100644  --- a/untitled.tex  +++ b/untitled.tex 
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P^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi}\sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) (\Sigma_{0,\alpha\delta}^{>}+ \Sigma_{0,\alpha\delta}^{>,h}) G^a_{\delta \gamma}(\epsilon)\Sigma_{\gamma\beta}^{<,h}(\omega+\epsilon)\delta_{kq}  \end{eqnarray}  \begin{eqnarray}  P^{>,2}(\omega) = \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\alpha\delta\tau\theta} G^r_{\beta \alpha}(\epsilon) (\Sigma_{0,\alpha\delta}^{>}+ \Sigma_{0,\alpha\delta}^{>,h}) G^a_{\delta \gamma}(\epsilon) \gamma}(\epsilon)\Sigma_{\gamma\tau}^{r,h}(\omega+\epsilon) G_{\tau\theta}^r(\omega+\epsilon)\Sigma_{\theta\beta}^{<,h}(\omega+\epsilon)  \end{eqnarray}  \begin{eqnarray}  P^{>,3}(\omega) = =\frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\alpha\delta\tau\theta} G^r_{\beta \alpha}(\epsilon) (\Sigma_{0,\alpha\delta}^{>}+ \Sigma_{0,\alpha\delta}^{>,h}) G^a_{\delta \gamma}(\epsilon)  \Sigma_{\gamma\tau}^{r,h}(\omega+\epsilon) G_{\tau\theta}^r(\omega+\epsilon)\Sigma_{\theta\beta}^{r,<}(\omega+\epsilon) + \Sigma_{\gamma\tau}^{r,h}(\omega+\epsilon) G_{\tau\theta}^<(\omega+\epsilon)\Sigma_{\theta\beta}^{a,<}(\omega+\epsilon) G_{\tau\theta}^<(\omega+\epsilon)\Sigma_{\theta\beta}^{a,h}(\omega+\epsilon)  \end{eqnarray}  \begin{eqnarray}  P^{>,4}(\omega) =\Sigma_{\gamma\tau}^{r,h}(\omega+\epsilon) G_{\tau\theta}^r(\omega+\epsilon)\Sigma_{\theta\beta}^{r,<}(\omega+\epsilon) + \Sigma_{\gamma\tau}^{<,h}(\omega+\epsilon) G_{\tau\theta}^a(\omega+\epsilon)\Sigma_{\theta\beta}^{a,<}(\omega+\epsilon)  \end{eqnarray}  They can be reformulated in terms of self-energies as  \begin{eqnarray}  P^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon)\Sigma^{h,<}_{0,\alpha\delta}(\epsilon+\omega)  \end{eqnarray}  \begin{eqnarray}  P^{>,2}(\omega) = \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon) \Sigma^{h,r }_{0,\gamma\gamma}(\epsilon+\omega) G^{r}_{\gamma\beta}(\omega+\epsilon) \Sigma^{h,<}_{0,\beta\beta}(\epsilon+\omega) ]  \end{eqnarray}  \begin{eqnarray}  P^{>,3}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon) \Sigma^{h,r }_{0,\gamma\gamma}(\epsilon+\omega) G^{<}_{\gamma\beta}(\omega+\epsilon) \Sigma^{h,a}_{0,\beta\beta}(\epsilon+\omega) ]  \end{eqnarray}  \begin{eqnarray}  P^{>,4}(\omega)= \frac{e^2}{\hbar^2} =\frac{e^2}{\hbar^2}  \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} \sum_{k,q,p\beta\gamma\alpha\delta\tau\theta}  G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) (\Sigma_{0,\alpha\delta}^{>}+ \Sigma_{0,\alpha\delta}^{>,h})  G^a_{\delta \gamma}(\epsilon) \Sigma^{h,< }_{0,\gamma\gamma}(\epsilon+\omega) G^{a}_{\gamma\beta}(\omega+\epsilon) \Sigma^{h,a}_{0,\beta\beta}(\epsilon+\omega) ] \gamma}(\epsilon)\Sigma_{\gamma\tau}^{<,h}(\omega+\epsilon) G_{\tau\theta}^a(\omega+\epsilon)\Sigma_{\theta\beta}^{a,h}(\omega+\epsilon)  \end{eqnarray}  Now we explicitely write down the expressions for the self-energies  \begin{equation}