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Formalismo de Keldysh

v. 3.141. junio 2015
F. P. Marín
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La Solución ${\rm G}\pars{t,t'}$

La ecuación de movimiento para ${\rm G}\pars{t,t'}$ puede ser expresada como una ecuación integral, que acopla ${\rm G}$ con ${\rm F}_{\eta\vec{k}}$, de acuerdo a:

\begin{align} {\rm G}\pars{t,t'} & = {\rm g}\pars{t,t'} + \oint_{\ck}\dd t''\,{\rm g}\pars{t,t''} \sum_{\eta\vec{k}}V_{\eta\vec{k}}\,{\rm F}_{\eta\vec{k}}\pars{t'',t'} \nonumber\\ & = {\rm g}\pars{t,t'} + \oint_{\ck}\dd t''\,{\rm g}\pars{t,t''} \sum_{\eta\vec{k}}V_{\eta\vec{k}}\ \ub{\oint_{\ck}\dd t'''\,{\rm g}_{\eta\vec{k}}\pars{t'',t'''} V_{\eta\vec{k}}^{*}{\rm G}\pars{t''',t'}} {\ds{{\rm F}_{\eta\vec{k}}\pars{t'',t'}}} \label{GFuncOfF} \end{align} y $\LARGE\mbox{(}$ ver expresión previa para ${\rm F}_{\eta\vec{k}}\pars{t,t'}$ $\LARGE\mbox{)}$ $$ {\rm g}\pars{t,t'} = -\ic\angles{\tk\,b\pars{t}b^{\dagger}\pars{t'}}_{0} $$ Note que ${\rm g}\pars{t,t'}$ satisface la ecuación de movimiento $$ \pars{\ic\,{\partial \over \partial t} - \xi}{\rm g}\pars{t,t'} = \delta\pars{t,t'} $$ Ec. \eqref{GFuncOfF} muestra que ${\rm G}\pars{t,t'}$ satisface la Ecuación de Dyson \begin{align} {\rm G}\pars{t,t'} & = {\rm g}\pars{t,t'} + \oint_{\ck}\oint_{\ck}\dd t''\,\dd t'''\, {\rm g}\pars{t,t''}\Sigma\pars{t'',t'''}{\rm G}\pars{t''',t'} \label{GAsFuncG} \\ &\mbox{donde}\quad \Sigma\pars{t,t'} \equiv \sum_{\eta}\Sigma_{\eta}\pars{t,t'} = \sum_{\eta\vec{k}} \verts{V_{\eta\vec{k}}}^{2}{\rm g}_{\eta\vec{k}}\pars{t,t'} \label{defSigmaConGEtaK78190} \end{align} Usemos reiteradamente las reglas de Langreth con el fin de obtener una ecuación integral para ${\rm G}^{<}\pars{t,t'}$ a partir de la ecuación integral \eqref{GAsFuncG}. Puesto que ${\rm G}^{<}\pars{t,t'}$ se acopla a ${\rm G}^{\rm\pars{a}}\pars{t,t'}$, es necesario a su vez, usar las reglas de Langreth para obtener una ecuación integral para ${\rm G}^{\rm\pars{a}}\pars{t,t'}$:
\begin{align} {\rm G}^{<}\pars{t,t'} &= {\rm g}^{<}\pars{t - t'} + \int_{t_{0}}^{\infty}\dd t''\int_{t_{0}}^{\infty}\dd t'''\nonumber \\[3mm]& \left\lbrace {\rm g}^{\rm\pars{r}}\pars{t - t''}\bracks{% \Sigma^{\rm\pars{r}}\pars{t'' - t'''}{\rm G}^{<}\pars{t''',t'} + \Sigma^{<}\pars{t'' - t'''} {\rm G}^{\rm\pars{a}}\pars{t''' - t'}\vphantom{{\LARGE A}}}\right. \nonumber \\& \left.\mbox{} + {\rm g}^{<}\pars{t - t''}\,\bracks{\Sigma^{\rm\pars{a}}\pars{t'' - t'''} {\rm G}^{\rm\pars{a}}\pars{t''',t'}\vphantom{{\LARGE A}}} \vphantom{{\huge A}}\right\rbrace \label{GMenost0} \\[5mm] {\rm G^{\pars{a}}}\pars{t,t'} &= {\rm g^{\pars{a}}}\pars{t - t'}\nonumber \\[3mm]&+ \int_{t_{0}}^{\infty}\dd t''\,\int_{t_{0}}^{\infty}\dd t'''\, {\rm g^{\pars{a}}}\pars{t - t''} \Sigma^{\pars{a}}\pars{t'' - t'''} {\rm G^{\pars{a}}}\pars{t''',t'} \label{Gat0} \end{align} Note que: \begin{equation} \hspace{-2cm}\left\lbrace\begin{array}{rclrcl} {\rm g}^{<}\pars{t} & = & \ic\angles{b\+b\pars{t}}_{0} = \ic\angles{b\+b}_{0}\expo{-\ic\xi t}\,,& \pars{\ic\,\partiald{}{t} - \xi}{\rm g}^{<}\pars{t} & = & 0 \\[1mm] {\rm g}^{\rm\pars{a}}\pars{t} & = & \ic\Theta\pars{-t} \angles{\braces{b\pars{t},b\+}}_{0} = \ic\Theta\pars{-t}\expo{-\ic\xi t}\,,& \pars{\ic\,\partiald{}{t} - \xi}{\rm g}^{\rm\pars{a}}\pars{t} & = & \delta\pars{t} \\[1mm] {\rm g}^{\rm\pars{r}}\pars{t} & = & -\ic\Theta\pars{t} \angles{\braces{b\pars{t},b\+}}_{0} = -\ic\Theta\pars{t}\expo{-\ic\xi t}\,,& \pars{\ic\,\partiald{}{t} - \xi}{\rm g}^{\rm\pars{r}}\pars{t} & = & \delta\pars{t} \end{array}\right. \label{freegf9876} \end{equation}
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