deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 00bcb42..2875ed8 100644  --- a/untitled.tex  +++ b/untitled.tex 
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\\ [G^{r}_{\tau\theta}(\omega+\epsilon)-G^{a}_{\tau\theta}(\omega+\epsilon)] \Gamma_{\theta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]   \end{multline}  Now we replace $[G^{r}_{\gamma\tau}(\omega+\epsilon)-G^{a}_{\gamma\beta}(\omega+\epsilon)]= -4iG^r_{\gamma\nu}\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\mu\beta}(\omega+\epsilon)$, then  \begin{eqnarray}  &&P^{>,2}(\omega)+ \begin{align*}  &P^{>,2}(\omega)+  P^{>,4}(\omega) = \frac{-16e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\mu\nu} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta}(\epsilon) \\  &  G^a_{\delta \gamma}(\epsilon)\Gamma_{\gamma\tau}[G^r_{\tau\nu}(\omega+\epsilon)\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\mu\theta}(\omega+\epsilon)] \Gamma_{\theta\beta}  \\ \nonumber  &&[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) \Gamma_{\theta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon))  f_{h}(\epsilon+\omega)] \end{eqnarray} \end{align*}  We now define $F_{\tau\tau'} =f_\tau(\epsilon)(1-f_\tau'(\epsilon+\omega))+f_\tau(\epsilon+\omega)(1-f_\tau'(\epsilon))$ with $\tau=e,h$. Then collecting all the terms for $P$ (including the two pieces $P^>$ and $P^<$ we have  \begin{eqnarray}  P^{A}(\omega)= \frac{4 e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\nu\mu} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [\Gamma_{\gamma\tau}] G^{r}_{\tau\nu}(\omega+\epsilon)\Gamma_{\nu\mu} G^{a}_{\mu\theta}(\omega+\epsilon) [\Gamma_{\theta\beta}][F_{ee}+F_{hh}+F_{eh}+F_{he}] 
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\\   &[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]  \\   &&[G_{\beta\nu}^>(\epsilon) &[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}] \\   &&[G_{\beta\nu}^>(\epsilon) &[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}(\omega+\epsilon) V_{\beta k}] \end{align*}  Inserting the expressions for the self-energies we get  \begin{align*}  &N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}   [G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^r(\epsilon+\omega) \Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)]   \\   &[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)][G_{\beta\nu}^r(\epsilon)  \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]\\   &[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]  \\   &[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)][G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon)  G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)] \end{align*}  \begin{align*}