deletions | additions
diff --git a/untitled.tex b/untitled.tex
index c575528..dae24c4 100644
--- a/untitled.tex
+++ b/untitled.tex
...
[G^t_{\beta\gamma}(t,t') G^{h,t}_{qk}(t',t) - G^{t}_{\beta q}(t,t')G^{h,t}_{\gamma k}(t',t)]\\ \nonumber
&& + V_{\beta k}^{*}V_{\gamma q} [G^{h,t}_{kq}(t,t') G{t}_{\gamma\beta}(t',t) - G^{h,t}_{k\gamma}(t,t')G^{t}_{q \beta}(t',t)]
\end{eqnarray}
Finally, we employ the following relation to obtain
$S^{<(>)(t,t')}$: $S^{<(>)}(t,t')$:
\begin{equation}
S(t,t') = A(t,t') B(t',t) \rightarrow S^{>(<)}(t,t') = A^{>(<)}(t,t') B^{<(>)}(t',t)
\end{equation}
...
\end{equation}
Then, inserting the previous expression into the equation for $G_{qk}^{h,t}(t,t')$, we obtain
\begin{equation}
G_{qk}^{h,t}(t,t') = G_{qk}^{h,t}(t,t')\delta_{kq} +
\sum_{\beta\gamma}\int \sum_{\tau\theta}\int \frac{-dt_1}{\hbar}\frac{-dt_2}{\hbar} g_{q}^{h,t}(t,t_1)
V_{\gamma V_{\tau q}
G^{t}_{\gamma\beta}(t_1,t_2)V_{\beta G^{t}_{\tau\theta}(t_1,t_2)V_{\theta k}^* g_{k}^{h,t}(t_2,t')
\end{equation}
The rest of equations for the Green functions that appear in the noise expression are already in J. S note.
Now we employ the following definition for the Fourier transform
...
Now we enter the expression for $G_{kq}^{h,t}(t,t')$ in the frequency domain
\begin{equation}
G_{qk}^{h,t}(\omega) = g_{q}^{h,t}(\omega)\delta_{kq} +
\sum_{\beta\gamma} \sum_{\tau\theta} g_{q}^{h,t}(\omega)
V_{\gamma V_{\tau q}
G^{t}_{\gamma\beta}(\omega)V_{\beta G^{t}_{\tau\theta}(\omega)V_{\theta k}^* g_{k}^{h,t}(\omega)\,,
\end{equation}
\begin{multline}
...
V_{\beta k} V_{\gamma q}^{*} [G^>_{\beta\gamma}(\epsilon) G^{h,<}_{qk}(\omega+\epsilon) - G^{>}_{\beta q}(\epsilon)G^{h,<}_{\gamma k}(\epsilon+\omega)]
+ V_{\beta k}^{*}V_{\gamma q} [G^{>,h}_{kq}(\epsilon_1) G^{<}_{\gamma\beta}(\epsilon+\omega) - G^{h,>}_{k\gamma}(\epsilon)G^{<}_{q \beta}(\epsilon+\omega)]\Biggr\}\,,
\end{multline}
Let us treat first the following term:
$\sum_{k\beta,q\gamma} $e^2/\hbar^2\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*} [G^>_{\beta\gamma}(\epsilon)
G^{h,<}_{qk}(\omega+\epsilon) - G^{>}_{\beta q}(\epsilon)G^{h,<}_{\gamma k}(\epsilon+\omega)]$ G^{h,<}_{qk}(\omega+\epsilon)]$
Then,
\begin{eqnarray}
&&G_{qk}^{h,<}(\omega+\epsilon) = g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} +
\sum_{\beta\gamma} \sum_{\tau\theta} [g_{q}^{h,r}(\omega+\epsilon)
V_{\gamma V_{\tau q}
G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{r}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,<}(\omega)\nonumber
\\
&&+g_{q}^{h,r}(\omega+\epsilon)
V_{\gamma V_{\tau q}
G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{<}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,a}(\omega+\epsilon)+g_{q}^{h,<}(\omega+\epsilon)
V_{\gamma V_{\tau q}
G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{a}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,a}(\omega+\epsilon)\,,
\end{eqnarray}
On the other hand we have for $G^>_{\beta\gamma}(\epsilon)$ (accordingly with J.S note)
\begin{eqnarray}
G^>_{\beta\gamma}(\epsilon) =
\sum_{k\alpha\delta} \sum_{p\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha
k} g^<_{k}(\epsilon)V_{\delta k} p} g^>_{p}(\epsilon)V_{\delta p} + V_{\alpha
k} g^{h,<}_{k}(\epsilon) p} g^{h,>}_{p}(\epsilon) V^*_{\delta
k}]G^a_{\delta p}]G^a_{\delta \gamma}(\epsilon)
\end{eqnarray}
We need to compute the following product of Green functions:
$P^>(t,t')=G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$ $P^>(t,t')=e^2/\hbar^2\sum_{k\beta,q\gamma} G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\begin{eqnarray}
&&G^>_{\beta\gamma}(\epsilon) &&\sum_{k\alpha\delta} V_{\gamma q}^* V_{\beta k} G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta}
V_{\gamma q}^* V_{\beta k}\Biggr\{ G^r_{\beta \alpha}(\epsilon) [V^*_{k\alpha} g^>_{k}(\omega)V_{\delta k} + V_{\alpha k} g^{h,>}_{k}(\omega) V^*_{\delta k}]G^a_{\delta \gamma}(\epsilon) g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} \\ \nonumber
&+&
\sum_{p\beta\gamma\alpha\gamma} \sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon)
[g_{q}^{h,r}(\omega+\epsilon) [ V_{\gamma
q}^* g_{q}^{h,r}(\omega+\epsilon) V_{\tau q}
G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{r}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,<}(\omega)
V_{\beta k} \nonumber
\\
&&+G^r_{\beta &&\sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon)
[g_{q}^{h,r}(\omega+\epsilon) [ V_{\gamma
q}^* g_{q}^{h,r}(\omega+\epsilon) V_{\tau q}
G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{<}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^*
g_{k}^{h,a}(\omega+\epsilon)]\\ g_{k}^{h,a}(\omega) V_{\beta k}
\nonumber
&&+G^r_{\beta \\
&&+ \sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta
\gamma}(\epsilon)[g_{q}^{h,<}(\omega+\epsilon) \gamma}(\epsilon) [ V_{\gamma
q}^* g_{q}^{h,<}(\omega+\epsilon) V_{\tau q}
G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{a}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^*
g_{k}^{h,a}(\omega+\epsilon)]\,, g_{k}^{h,a}(\omega) V_{\beta k} \Biggr\}
\,,
\end{eqnarray}
We now compute separately the different parts of the previous expression for the ac noise
\begin{eqnarray}
