deletions | additions
diff --git a/untitled.tex b/untitled.tex
index 0e6f464..2254a62 100644
--- a/untitled.tex
+++ b/untitled.tex
...
Then, we have
\begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} V_{\beta k} V^*_{\gamma q}
[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon)+ G_{\beta\nu}^>(+\epsilon) V_{\nu q}
g^{a}_{q}(\omega+\epsilon)] g^{a}_{q}(\epsilon)]
\\ \nonumber
&[G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon)+ G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon)]
\end{align*}
...
&N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\omega+\epsilon) \Gamma_{\mu\beta} (1-f_e(\epsilon) f_{h}(\omega+\epsilon)] \\
&+\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^r(\omega+\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_e(\epsilon)(f_{h}(\epsilon+\omega)+f_e(\omega+\epsilon))] \\
&+\sum_{\lambda\delta} G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}]
G_{\gamma\mu}^r(\omega+\epsilon)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\ G_{\gamma\mu}^r(\omega+\epsilon)(1-f_{e}(\epsilon)+1-f_{h}(\epsilon))f_h(\omega+\epsilon)\\
&+\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega+\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\epsilon)+1-f_{h}(\epsilon))(f_{e}(\omega+\epsilon+)+f_{h}(\epsilon+\omega))] \Biggr\}
\end{align*}
Finally, to obtain $N<(t,t)$ we just change $(1-f)\rightarrow f$, and $f\rightarrow (1-f)$.
\begin{align*}
&N(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\epsilon+\omega) \Gamma_{\mu\beta}
F_{ee} F_{eh} \\
&+\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^r(\omega+\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon+\omega)[i\Gamma_{\mu\beta}](F_{eh}+F_{ee}) \\
&+\sum_{\lambda\delta} G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega+\epsilon)(F_{eh}+F_{hh})\\
&+\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega+\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](F_{ee}+F_{hh}+(F_{eh}+F_{he})) \Biggr\}
\end{align*}
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^{>,h}_{kq}(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}(\epsilon) [G^{>,h}_{kq}(\epsilon) G^{<}_{\gamma\beta}(\omega+\epsilon)
\end{eqnarray}
We replace now
\begin{align*}
&G_{kq}^{h,>}(\epsilon) &G_{kq}^{>,h}(\epsilon) =
g_{q}^{h,>}(\epsilon)+ g_{q}^{h,>}(\epsilon)\delta_{kq}+ \sum_{\alpha\delta}
[g_{k}^{h,r}(\epsilon) [g_{k}^{r,h}(\epsilon) V_{\alpha k} G^{r}_{\alpha\delta}(\epsilon)V_{\delta q}^* g_{q}^{h,>}(\epsilon)
+g_{k}^{h,r}(\epsilon) +g_{k}^{r,h}(\epsilon) V_{\alpha k} G^{>}_{\alpha\delta}(\epsilon)V_{\delta q}^*
g_{q}^{h,a}(\epsilon)] g_{q}^{a,h}(\epsilon)]
\\
\nonumber
&+g_{k}^{h,>}(\epsilon) V_{\gamma &+g_{k}^{>,h}(\epsilon) V_{\alpha k}
G^{a}_{\gamma\beta}(\epsilon)V_{\beta G^{a}_{\alpha\delta}(\epsilon)V_{\delta q}^*
g_{q}^{h,a}(\epsilon) g_{q}^{a,h}(\epsilon) ]\,,
\end{align*}
Then we get
\begin{align*}
&M^>(\omega)=\frac{2 i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\nu\mu} \int \frac{d\epsilon}{2\pi}\
V_{\beta k}^{*}
g_{q}^{h,>}(\epsilon) V_{\alpha k} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon)) \\
&+\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_{q}^{h,>}(\epsilon) V_{\alpha k} g_{q}^{>,h}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
&
+ \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}(\epsilon) g_{k}^{r,h}(\epsilon) V_{\alpha k} G^{r}_{\alpha\delta}(\epsilon) V_{\delta q}^*
g_{q}^{h,>}(\epsilon) g_{q}^{>,h}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
&+V_{\beta k}^{*}
g_{k}^{h,r}(\epsilon) g_{k}^{r,h}(\epsilon) V_{\alpha k} G^{>}_{\alpha\delta}(\epsilon) V_{\delta q}^*
g_{q}^{h,<}(\epsilon) g_{q}^{<,h}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
&+V_{\beta k}^{*}
g_{k}^{h,>}(\epsilon) g_{k}^{>,h}(\epsilon) V_{\alpha k} G^{a}_{\alpha\delta}(\epsilon) V_{\delta q}^*
g_{q}^{h,a}(\epsilon) g_{q}^{a,h}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon) (f_e(\omega+\epsilon)+f_h(\omega+\epsilon))\Biggr\}
\end{align*}
We now use the explicit expressions for the self-energies
\begin{align*}
&M^>(\omega)=\frac{2 i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\nu\mu} \int \frac{d\epsilon}{2\pi}\
V_{\beta k}^{*} g_{q}^{h,>}(\epsilon) V_{\gamma
k} q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon)) \\
&\frac{2i &+ \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}(\epsilon) G^{r}_{\alpha\delta}(\epsilon)
\Sigma^{h,>}_{0,\delta\gamma}(\omega+\epsilon) \Sigma^{h,>}_{0,\delta\gamma}(\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) +
\\
\nonumber
&[\Sigma^{r,h}_{0,\beta\alpha}(\epsilon) G^{>}_{\alpha\delta}(\epsilon) \Sigma^{h,a}_{0,\delta\gamma}
(\omega+\epsilon)G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))+ (\epsilon)G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))+
\\
\nonumber
& [\Sigma^{>,h}_{0,\beta\alpha}(\epsilon) G^{a}_{\alpha\delta}(\epsilon)
\Sigma^{h,a}_{0,\delta\gamma}(\omega+\epsilon) \Sigma^{h,a}_{0,\delta\gamma}(\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) \Biggr\}
\end{align*}
Then we obtain,
\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_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))
(1-f_{e}(\epsilon)] (1-f_{h}(\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}(\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)]
...
\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})] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](F_{hh}+F_{he})] \\
& [-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}) G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\mu\beta}(\omega+\epsilon)](F_{hh}+F_{he}) +
\\
&[\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}) (F_{he}+F_{hh}) \Biggr\}
\end{align*}
The last term that we need to compute is
$Q>(t,t')+Q<(t,t')= $Q^>(t,t')+Q^<(t,t')= G^{h,>}_{k\gamma}(t,t')G^{<}_{q \beta}(t',t)+ G^{h,<}_{k\gamma}(t,t')G^{>}_{q \beta}(t',t)$. 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)]
\end{eqnarray}
...
\end{eqnarray}
Then
\begin{align*}
&Q^>(\omega)\frac{
e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha} e^2}{\hbar^2}\sum_{k\beta,q\gamma,\delta} \int \frac{d\epsilon}{2\pi}\\
&V_{\beta k}^* &V_{k\beta}^*\ V_{\gamma q}[g^{r,h}_{k}(\epsilon) V_{\alpha k}
G_{\alpha\gamma}^>(\epsilon)+g^{>,h}_{k}(\epsilon) V_{\alpha G_{\alpha\gamma}^>(\epsilon)+V_{k\beta}^*g^{>,h}_{k}(\epsilon) V_{\delta k}
G_{\alpha\gamma}^a(\epsilon)] G_{\delta\gamma}^a(\epsilon)] [ V_{\gamma q} g^{r}_{q}(\omega+\epsilon)
V^*_{\alpha V^*_{\delta q}
G_{\alpha\beta}^<(\omega+\epsilon)+ G_{\delta\beta}^<(\omega+\epsilon)+ V_{\gamma q} g^{<}_{q}(\epsilon)
V^*_{\alpha V^*_{\delta q}
G_{\alpha\beta}^a(\omega+\epsilon) G_{\delta\beta}^a(\omega+\epsilon) ]
\end{align*}
Then we have
\begin{equation}
Q^>(\omega) = \frac{
e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha} e^2}{\hbar^2}\sum_{k\beta,q\gamma,\delta} \int \frac{d\epsilon}{2\pi}[\Sigma_{0,\beta\alpha}^r (\epsilon)G_{\alpha\gamma}^>(\epsilon)+ \Sigma_{0,\beta\alpha}^> (\epsilon) G_{\alpha\gamma}^a(\epsilon)] [\Sigma^r_{0,\gamma
\alpha}(\omega+\epsilon) G_{\alpha\beta}^<(\omega+\epsilon)+ \delta}(\omega+\epsilon) G_{\delta\beta}^<(\omega+\epsilon)+ \Sigma^<_{0,\gamma
\alpha}(\omega+\epsilon)G_{\alpha\beta}^a(\omega+\epsilon)] \delta}(\omega+\epsilon)G_{\delta\beta}^a(\omega+\epsilon)]
\end{equation}
We can split $Q^>(\omega)$ in four contributions
\begin{align*}
...
\end{align*}
\begin{eqnarray}
Q^{>,2}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau} \int \frac{d\epsilon}{2\pi}[\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\epsilon)
[i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\omega+\epsilon)\Gamma_{\mu\nu} [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\omega+\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\omega+\epsilon)
(1-f_e(\epsilon))(f_e(\omega+\epsilon)+f_h(\omega+\epsilon)) (1-f_h(\epsilon))(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\end{eqnarray}
\begin{eqnarray}
...
\end{eqnarray}
\begin{eqnarray}
Q^{>,4}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau } \int \frac{d\epsilon}{2\pi}[\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\epsilon)
[-i\Gamma_{\gamma\tau}]G_{\tau\beta}^a(\omega+\epsilon)f_e(\omega+\epsilon)(1-f_e(\epsilon)) [\Gamma_{\gamma\tau}]G_{\tau\beta}^a(\omega+\epsilon)f_e(\omega+\epsilon)(1-f_h(\epsilon))
\end{eqnarray}
...
\end{align*}
\begin{eqnarray}
Q^{>,2}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau} \int \frac{d\epsilon}{2\pi}[\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\epsilon) [i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\omega+\epsilon)\Gamma_{\mu\nu}
G_{\nu\beta}^a(\omega+\epsilon)(F_{eh}+F_{ee}) G_{\nu\beta}^a(\omega+\epsilon)(F_{he}+F_{hh})
\end{eqnarray}
\begin{eqnarray}
...
\end{eqnarray}
\begin{eqnarray}
Q^{>,4}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau } \int \frac{d\epsilon}{2\pi}[\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\epsilon)
[-i\Gamma_{\gamma\tau}]G_{\tau\beta}^a(\omega+\epsilon)F_{ee} [-i\Gamma_{\gamma\tau}]G_{\tau\beta}^a(\omega+\epsilon)F_{he}
\end{eqnarray}
We can now express the total contribution for $Q^>$ and $Q^<$ as
