deletions | additions       diff --git a/untitled.tex b/untitled.tex  index df4a5ad..3942568 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\end{align*}  Now we collect $P^{>,2}(\omega)+P^{>,4}(\omega)$  \begin{eqnarray}  &&P^{>,2}(\omega)+ P^{>,2}(\omega)+  S^{>,4}(\omega) = -i\frac{4e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon)\Gamma_{\gamma\tau} \\ \nonumber  && [G^{r}_{\tau\theta}(\omega+\epsilon)-G^{a}_{\tau\theta}(\omega+\epsilon)] \gamma}(\epsilon)\Gamma_{\gamma\tau}[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{eqnarray}  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} 
...
\end{eqnarray}  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}_{\gtau\nu}(\omega+\epsilon)\Gamma_{\nu\mu} 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}] \end{eqnarray}  and  \begin{eqnarray}