3.7.2 LANGRETH Theorem

The next step is replacing contour by real time integrals in the DYSON equation. In that equation one encounters the following contour integrals

 \displaystyle D(t,t') \ = \ \int_{\mathrm{C}} d\tau \ A(t,\tau)
 B(\tau,t') \ ,
 \end{array}\end{displaymath} (3.53)

and their generalizations involving products of three or more terms. To evaluate (3.53) one can assume that $ t$ is on the first half of the contour and $ t'$ is on the latter half. In view of the discussion of (3.51), we are thus analyzing a lesser function. The next step is to deform the contour as indicated in Fig. 3.10. Thus (3.53) becomes

 \displaystyle D^\mathrm{<}(t,t') \ = \ \int...
... \
 A^\mathrm{<}(t,\tau) B^\mathrm{<}(\tau,t') \ .
 \end{array}\end{displaymath} (3.54)

Here, in appending the label $ <$ to the function B in the first term we made use of the fact that as long as the integration variable $ \tau $ is confined on the contour $ \mathrm{C_1}$ it is less than $ t'$ (in the contour sense) . A similar argument applies to the second term. Considering the first term in (3.54) the integration can be split into two parts

 \displaystyle \int_{\mathrm{C_1}} d\tau \ ...
...1 \ \ A^\mathrm{r}(t,t_1) B^\mathrm{<}(t_1,t') \ ,
 \end{array}\end{displaymath} (3.55)

where the definition of the retarded function (3.52) has been used. A similar analysis can be applied to the second term involving contour $ \mathrm{C_2}$, where the advanced function is generated. Putting the two terms together, one gets the first of LANGRETH's results [185]

\begin{displaymath}\begin{array}{l} \displaystyle
 D^\mathrm{<}(t,t') \ = \ \int...
... \ A^\mathrm{<}(t,t_1)
 B^\mathrm{a}(t_1,t') ] \ .
 \end{array}\end{displaymath} (3.56)

The same result applies for the greater function just by replacing the $ <$ labels by the $ >$ labels. It is easy to generalize the result (3.56) to a product of three functions. The retarded and analogously the advanced component of a product of functions defined on the contour can be derived by repeated use of the definitions (3.51) and (3.52) and the result (3.56).

 \displaystyle D^\mathrm{r}(t,t') &\display...
... dt_1\ A^\mathrm{r}(t,t_1) B^\mathrm{r}(t_1,t') \ 
 \end{array}\end{displaymath} (3.57)

Figure 3.10: Deformation of contour C into contours C$ _1$ and C$ _2$.
In the self-energies another structure occurs

 D(\tau,\tau') \ &\displaystyle = \ A(\tau,\tau')B(\tau,\tau') \ ,<tex2html_comment_mark>567 \end{array}\end{displaymath} (3.58)

where $ \tau $ and $ \tau'$ are contour variables. The derivation of the required formula is similar to the analysis presented above [185]

 D^\mathrm{\gtrless}(t,t') \ &\displaystyle...
...t')B^\mathrm{r}(t,t') \ .<tex2html_comment_mark>571 \end{array}\end{displaymath} (3.59)

The rules provided by the LANGRETH theorem are summarized in Table 3.1.

Table 3.1: Rules for analytic continuation derived from the LANGRETH theorem.
Contour Real axis
$ D=\displaystyle\int_{\mathrm{C}}

$ \begin{array}{ll}\displaystyle
D^\mathrm{\gtrless} &\displaystyle=\ \int_{\ma...
&\displaystyle =\ \int_{\mathrm{t}} \ A^\mathrm{r}B^\mathrm{r}

$ D=\displaystyle\int_{\mathrm{C}} ABC$

$ \begin{array}{ll}
D^\mathrm{\gtrless} &\displaystyle=\ \int_{\mathrm{t}}[
...tyle=\ \int_{\mathrm{t}} \

$ \displaystyle

$ \begin{array}{ll}
D^\mathrm{\gtrless}(t,t')\hspace*{-15pt} &\displaystyle=
...}(t,t')B^\mathrm{<}(t,t') + A^\mathrm{r}(t,t')B^\mathrm{r}(t,t')

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